One hundred years of general relativity. Vol.2 9789814678490

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Foreword

General Relativity (GR) is founded on the observation of Mercury perihelion precession anomaly discovered by Le Verrier and improved by Newcomb, the Michelson– Morley experiment, and the precision E¨ otv¨ os experiment. Its theoretical basis is based on Special Relativity (previously called the restricted theory of relativity), Einstein Equivalence Principle (EEP) and the realization that the metric is the dynamical quantity of gravity together with the Principle of General Covariance and the absence of other dynamical quantities. The establishment of GR in 1915 is a community effort with Albert Einstein clearly playing the dominant role. For one hundred years, its applicability through solar system to cosmology is prevailing. If one includes the cosmological constant (proposed in 1917 by Einstein) in GR, there have not been any fully established non-applicable places. The only possible potential exception is the missing mass (dark matter)-deficient acceleration issue. Dark energy and quantum gravity are needed in the present theoretical foundation of physics; however, more experimental clues are needed. The framework applicability of GR is already demonstrated in theoretical inflation models with quantum fluctuations leading to structure formation with experimentally observed spectrum. To celebrate the GR centennial, we solicit the writing of 23 chapters in these two volumes consisting of five parts: Part I. Part II. Part III. Part IV. Part V.

Genesis, Solutions and Energy. Empirical Foundations. Gravitational Waves. Cosmology. Quantum Gravity.

Volume 1 consists of Part I, Part II and Part III; Volume 2 consists of Part IV and Part V. In Part I, Valerie Messager and Christophe Letellier start in Chapter 1 with a genesis of special relativity to set the stage. They rely on the original literature to make the development clear and connected. Thanks to many thorough researches in the last 50 years, the path to general relativity is clear. A concise exposition of the path is presented in Chapter 2. In Chapter 3, Christian Heinicke and Friedrich Hehl present the historical development and detailed properties of the basic and fundamental spherical Schwarzschild and axisymmetric Kerr solutions. In Chapter 4, Chiang-Mei Chen, James Nester and Roh-Sung Tung expound the important and useful concept of energy with its many facets and various applications. v

vi

Foreword

In Part II, the empirical foundations of GR are examined. First, the cornerstone Einstein Equivalence Principle (EEP) is explored. Ever since about 100 ps (the time of electroweak phase transition or the equivalent/substitute) at the quasiequilibrium Higgs/intermediate boson energy scale from the Big Bang (or equivalent/substitute), photons and charged particles are abundant. With the premetric formulation of electrodynamics, we examine the tests of EEP via the metric-induced spacetime constitutive tensor density. The non-birefringence of the cosmic electromagnetic wave propagation in spacetime is observed to ultrahigh precision. This constrains the spacetime constitutive tensor density to Maxwell–Lorentz (metric) form plus a scalar (dilaton) degree of freedom and a pseudoscalar (axion) degree of freedom to high precision. The accurate agreement of cosmic microwave background spectrum with the Planck spectrum constrains the fractional change of the cosmic dilaton to be less than 8 × 10−4 . The Galileo weak equivalence principle (WEP) experiments (E¨ otv¨ os-type experiments) constrain the fractional dilatonic change in the solar system to be less than 10−10 . Accompanying the axion degree of freedom is the rotation of linear polarization in the cosmic propagation of electromagnetic waves called cosmic polarization rotation (CPR). Sperello di Serego Alighieri reviews the constraints from radio galaxy observations and CMB polarization observations to give a general constraint of 0.02 rad for the mean (uniform) CPR and also a constraint of 0.02 rad for the CPR fluctuations. In many inflation models dilatons and axions play important roles; these investigations are crucial to give clues or constraints on the models. Frequency and time are the most precise metrological quantities. Their uses in gravity experiments are unavoidable. The use of GR in time synchronization and in GPS, GLONNESS, Galileo and Beidou becomes a folk talk. There are two good ways to compare precision clocks: (i) fiber ´ links; (ii) space optical links using laser ranging. Etienne Samain expounds the space optical link approach and addresses the laser ranging missions T2L2 (Time transfer by Laser link), LRO (Lunar Reconnaissance Orbiter) and LTT (Laser Time Transfer) together with future space mission proposals for fundamental physics, solar system science/navigation in which laser links are of prime importance. Solarsystem observation provides the original impetus and the first confirmation of GR. Chapter 8 summarizes the progress of classical solar system tests and explores its potential in the future. Improvement of three or more orders of magnitude is still possible. Perhaps the most dramatic development in testing relativistic gravity and in improving the dynamical foundations of general relativity is the discovery and observation of pulsars, binary pulsars, millisecond pulsars and double pulsars since 1967, 1974, 1982 and 2003 respectively. Richard Manchester reviews the pulsar observation in its relation with gravity in Chapter 9 with a brief introduction to basic pulsar properties and pulsar timing. He presents a rather thorough account of dynamical tests of GR and the strong equivalence principle together with a lucid but in-depth account of GW detection using pulsar timing arrays (PTAs). See front cover for an illustrative schematic of a PTA.

Foreword

vii

In 1916 Einstein predicted gravitational waves (GWs) in GR almost immediately after his founding of it. The existence of gravitational waves is the direct consequence of general relativity and unavoidable consequences of all relativistic gravity theories with finite velocity of propagation. Their importance in GR is like that electromagnetic waves in Maxwell–Lorentz theory of electromagnetism. Einstein’s general relativity and relativistic gravity theories predict the existence of gravitational waves. Gravitational waves propagate in spacetime forming ripples of spacetime geometry. In the introductory chapter of Part III, Kazuaki Kuroda, Wei-Ping Pan and I review and summarize the complete GW spectrum, the methods of detection, and the detection sensitivities in various frequency bands with a brief introduction to GW sources. At the time Einstein predicted GWs in GR, he estimated that GWs were experimentally not detectable due to feeble strengths. However, thanks to one hundred years of development of experimental methods and technology together with the discovery of various astrophysical compact objects and cosmological sources, GWs are now on the verge of detection in three frequency bands. The very low frequency band (10 fHz–300 pHz) GWs are on the verge of detection by the PTAs; Richard Manchester covers this part in his chapter on pulsars and gravity in Part II. As mentioned in Chapter 10, the observation of PTAs has already constrained the isotropic GW background to a level excluding most current models of supermassive black hole formation. This is a strong signal that PTA observation is on the verge of detecting GWs. The high frequency band (10 Hz–100 kHz) GWs are on the verge of detection by ground-based interferometers; Kazuaki Kuroda addresses the detection methods and the sources in the second chapter of Part III. The extremely low (Hubble) frequency band (1 aHz–10 fHz) GWs may also be on the verge of detection by CMB polarization observations; the present status is briefly reviewed in the introduction chapter of Part III. The low frequency band (100 nHz–100 mHz) and the middle frequency band detections will have the greatest S/N ratios according to the present expectation. We review the sources, goal sensitivities, various mission proposals together with the current supporting activities in the third chapter of Part III. The GW quadrupole radiation formula has already been verified by the binary pulsar observations. In the next hundred years we will see great discoveries and immense focused activities toward the establishment and flourish of GW astronomy and GW cosmology. GW physics and GW astronomy will become a precision discipline in the coming century. The development of cosmology is most dramatic during the last hundred years. From Kapteyn universe in 1915 of observed disk star system of 10 kpc diameter and 2 kpc thickness with the Sun near its center to full-fledged precision cosmology now is monumental in the human history. It is fortunate that the development of observational cosmology has GR theory as a theoretical basis and goes hand-in-hand with the development of general relativity. This is fortunate both for observational cosmology and for GR. Using the Cosmological Principle Einstein looked into cosmological solutions in GR in 1917. The fast development of observational distance ladder around that time soon extends the reach of astronomy to modern cosmos.

viii

Foreword

Studies in the fundamental issues on the origins of cosmos lead to anthropological principle, cosmic inflation, and cosmic landscape scenarios. The cosmos is believed to be open in (extended beyond) the Hubble distance scale. Part III consists of seven chapters: Martin Bucher and I present some introductory remarks with a discussion of missing mass-deficient acceleration issue in the first chapter; Marc Davis reviews the observation and evolution of cosmic structure; Martin Bucher give a rather comprehensive exposition of the physics (almost on every aspect of cosmology) of CMB; Xiangcun Meng, Yan Gao and Zhanwen Han review the SNIa as a standardizable distance candle, its nature, its progenitors and its role in the cosmology together with related current issues; Toshifumi Futamase on the gravitational lensing in cosmology; K. Sato and Juni’ichi on cosmic inflation with a brief historical exposition on the development in Japan and Russia; David Chernoff and Henry Tye on inflation and cosmic strings from the point of view of string theory. The quest for a satisfactory quantum description of gravity began very early. Einstein thought that quantum effects must modify general relativity in his first paper on GWs in 1916. Klein argued that the quantum theory must ultimately modify the role of spatiotemporal concepts in fundamental physics in 1927. Part V on Quantum Gravity consists of 4 chapters. Chapter 20 gives a bird’s-eye survey on the development of fundamental ideas of quantum gravity together with possible observations of quantum gravitational effects in the foreseeable future. The classical age (1958–1969; according to the chronological classification of Rovelli) started with ADM canonical formalism and concluded with DeWitt–Wheeler equation and DeWitt’s derivation of Feynman rules for perturbative GR. In the middle ages (1970–1983), the discovery of black hole thermodynamics and Hawking’s derivation of black hole radiation radically affected our understanding of general relativity. In the renaissance period (1984–1994), there are two influential developments. From the covariant approach, attempts to get rid of infinities merge into string theory. The use of strings and branes extends the theoretical framework of quantum field theory. From the canonical approach, background-independent loop quantum gravity emerged 20 years after DeWitt–Wheeler equation. In Chapter 21, Richard Woodard starts with experiences of two personal academic careers through the classical and middle ages, advocates that the cosmological data from the epoch of primordial inflation is catalyzing the maturation of quantum gravity from speculation into a hard science, explains why quantum gravitational effects from primordial inflation are observable, reviews what has been done in perturbative quantum gravity, tells us what the future holds both theoretically and observationally, and discusses what this tells us about quantum gravity. In Chapter 22, Steven Carlip reviews the discovery of black hole thermodynamics and summarizes the many independent ways of obtaining the thermodynamic and statistical mechanical properties of black holes. This has offered us some early hints about the nature of quantum gravity. Steven then describes some of the remaining puzzles, including the nature of the quantum microstates, the problem of universality, and the information loss paradox. In the last chapter, Dah-Wei Chiou gives us a rather self-contained introductory review

Foreword

ix

on loop quantum gravity — a background-independent nonperturbative approach to a consistent quantum theory of gravity placing emphasis on the fundamental ideas and their significance. The review presents the canonical formulation of loop quantum gravity as the central topic and covers briefly the spin foam theory, the relation to black hole thermodynamics and the loop quantum cosmology with current directions and open issues summarized. Although we do not yet have a consistent calculable quantum gravity theory which has a good degree of completeness like quantum electrodynamics or quantum chromodynamics, the efforts to find one already led to the consistent renormalization of the gauge theory in 1960’s. The new development since 1980’s together with more understanding and further development of perturbation theory may give clues to a consistent theory. During these endeavors, the quest for a well-developed quantum gravity phenomenology including the quest to find a correct inflationary (or non-inflationary) scenario may play a significant role. The hope is that we will have one within a generation. This book is written and assembled for graduate students and general scientific-oriented readers alike. Each chapter is basically a review article. The five Parts are interconnected. Different combinations can be designed for special topics for graduate students and advanced undergraduates. For example, following combinations are suitable for each topic named: (i) Basics (Selected Topics in GR): Part I, Chapters 8, 9, 10, 13, 20; (ii) Empirical Foundations (Empirical Foundations of Relativistic Gravity): Chapter 2, Part II, Chapters 10, 11, 13, 14, 16, 20; (iii) Gravitational Waves: Chapters 2, 9, Part III, Chapters 15, 18, 19; (iv) Cosmology: Chapters 5, 6, 10, 12, Part IV, Chapter 20; (v) Quantum Gravity: Chapters 3, 4, 10, 18, 19, Part V. There can be various other combinations too. We are grateful to all contributors for agreeing to write comprehensive reviews to make this publication possible. We would also like to thank all the referees for their valuable comments and suggestions: Martin Bucher, Stephen Carlip, DahWei Chiou, Sperello di Serego Alighieri, Angela Di Virgilio, John Eldridge, Jeremy Gray, Friedrich Hehl, Jim Hough, Ekaterina Koptelova, Ettore Majorana, James Nester, Ulrich Schreiber, Alexei Starobinsky, David Tanner, Richard Woodard, AnMing Wu, Masahide Yamaguchi. We thank the World Scientific staff, especially Dr. K. K. Phua and Kah Fee Ng for their generous support in completing the book. We dedicate this two-volume GR centennial book to the founders of GR and various communities who have contributed to this dramatic century of development and applications of GR. Wei-Tou Ni November, 2015

x

Foreword

Note Added in Proof After the foreword was written, LIGO Scientific and Virgo Collaborations announced in February 2016 and in June 2016 the first direct detections of gravitational waves (GWs) by LIGO Hanford and LIGO Livingston detectors in September 2015 and in December 2015. With the LIGO discovery announcements, two important things are verified: (i) GWs are directly detected in the solar-system; (ii) Black holes (BHs), binary BHs and BH coalescences are discovered and measured experimentally and directly with the distances reached more than 1 billion light years. These discoveries constitute the best celebration of the centennial of the genesis of general relativity. We refer the readers to Refs. 1 and 2 for the discovery and Refs. 3 and 4 for a brief history of gravitational wave research. A web page will be set up for updates of the reviews of these two volumes. Please see http://astrod.wikispaces.com/ for announcement. References 1. B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), Observation of gravitational waves from a binary black hole merger, Phys. Rev. Lett. 116 (2016) 061102. 2. B. P. Abbott et al. (LIGO Scientific and Virgo Collaborations), GW151226: Observation of gravitational waves from a 22-solar-mass binary black hole coalescence, Phys. Rev. Lett. 116 (2016) 241103. 3. J. L. Cervantes-Cota, S. Galindo-Uribarri and G.F. Smoot, A brief history of gravitational waves, Universe 2(22) (2016) 09400. 4. C.-M. Chen, J. M. Nester, W.-T. Ni, A brief history of gravitational wave research, Chinese Journal of Physics 55 (2017) 142–169.

Contents

Volume 1 Foreword

v

Color plates

I-CP1

Part I. Genesis, Solutions and Energy 1. A genesis of special relativity Val´erie Messager and Christophe Letellier IJMPD 24 (2015) 1530024 1. Introduction 2. The Ether: From Celestial Body Motion to Light Propagation 2.1. Its origin 2.2. The luminiferous ether 3. Galileo’s Composition Law for Velocities 4. Questioning the Nature of Light: Waves or Corpuscles? 5. From Electrodynamics to Light 5.1. Amp`ere’s law 5.2. Maxwell’s electromagnetic waves as light 5.3. Helmholtz’s theory 5.4. Hertzs experiments for validating Maxwell’s theory 6. Invariance of the Field Equations from a Frame to Another One 6.1. Hertz’s electrodynamic theory 6.2. Voigt’s wave equation 6.3. Lorentz’s electrodynamical theory 6.4. Larmor’s theory 7. Poincar´e’s Contribution 8. Einstein’s 1905 Contribution 9. Conclusion Appendices A. 1. Fizeau’s experiments xi

I-1 I-3

I-3 I-5 I-5 I-8 I-11 I-15 I-24 I-24 I-28 I-32 I-33 I-37 I-37 I-41 I-42 I-50 I-51 I-72 I-76 I-77 I-77

xii

Contents

A. 2. Michelson and Morley’s experiments 2. Genesis of general relativity — A concise exposition Wei-Tou Ni IJMPD 25 (2016) 1630004

I-77 I-85

1. Prelude — Before 1905

I-86

2. The Period of Searching for Directions and New Ingredients: 1905–1910

I-91

3. The Period of Various Trial Theories: 1911–1914 4. The Synthesis and Consolidation: 1915–1916

I-96 I-100

5. Epilogue

I-103

3. Schwarzschild and Kerr solutions of Einstein’s field equation: An Introduction Christian Heinicke and Friederich W. Hehl IJMPD 24 (2015) 1530006 1. Prelude 1.1. Newtonian gravity 1.2. Minkowski space

I-109

I-109 I-109 I-114

1.2.1. Null coordinates

I-115

1.2.2. Penrose diagram 1.3. Einstein’s field equation

I-115 I-118

2. The Schwarzschild Metric (1916)

I-120

2.1. Historical remarks 2.2. Approaching the Schwarzschild metric

I-120 I-122

2.3. Six classical representations of the Schwarzschild metric

I-126

2.4. The concept of a Schwarzschild black hole 2.4.1. Event horizon

I-126 I-128

2.4.2. Killing horizon

I-130

2.4.3. Surface gravity 2.4.4. Infinite redshift

I-131 I-131

2.5. Using light rays as coordinate lines 2.5.1. Eddington–Finkelstein coordinates 2.5.2. Kruskal–Szekeres coordinates 2.6. Penrose–Kruskal diagram 2.7. Adding electric charge and the cosmological constant: Reissner–Nordstr¨ om 2.8. The interior Schwarzschild solution and the TOV equation

I-131 I-132 I-133 I-135 I-136 I-137

Contents

xiii

3. The Kerr Metric (1963)

I-141

3.1. Historical remarks

I-141

3.2. Approaching the Kerr metric

I-144

3.2.1. Papapetrou line element and vacuum field equation

I-144

3.2.2. Ernst equation (1968)

I-147

3.2.3. From Ernst back to Kerr

I-148

3.3. Three classical representations of the Kerr metric

I-149

3.4. The concept of a Kerr black hole

I-151

3.4.1. Depicting Kerr geometry 3.5. The ergoregion

I-152 I-155

3.5.1. Constrained rotation

I-155

3.5.2. Rotation of the event horizon

I-156

3.5.2. Penrose process and black hole thermodynamics

I-156

3.6. Beyond the horizons 3.6.1. Using light rays as coordinate lines

I-157 I-158

3.7. Penrose–Carter diagram and Cauchy horizon

I-160

3.8. Gravitoelectromagnetism, multipole moments

I-161

3.8.1. Gravitoelectromagnetic field strength

I-163

3.8.2. Quadratic invariants

I-165

3.8.3. Gravitomagnetic clock effect of Mashhoon, Cohen et al.

I-166

3.8.4. Multipole moments: Gravitoelectric and gravitomagnetic ones

I-167

3.9. Adding electric charge and the cosmological constant: Kerr–Newman metric

I-168

3.10. On the uniqueness of the Kerr black hole

I-170

3.11. On interior solutions with material sources

I-171

4. Kerr Beyond Einstein

I-172

4.1. Kerr metric accompanied by a propagating linear connection

I-172

4.2. Kerr metric in higher dimensions and in string theory

I-174

Appendix

I-175

A.1. Exterior calculus and computer algebra

I-175

xiv

Contents

4. Gravitational energy for GR and Poincar´e gauge theories: A covariant Hamiltonian approach Chiang-Mei Chen, James Nester and Roh-Suan Tung IJMPD 24 (2015) 1530026

I-187

1. Introduction

I-188

2. Background

I-189

2.1. Some brief early history

I-189

2.2. From Einstein’s correspondence

I-190

2.3. Noether’s contribution

I-192

2.4. Noether’s result

I-193

3. The Noether Energy–Momentum Current Ambiguity 4. Pseudotensors

I-194 I-196

4.1. Einstein, Klein and superpotentials

I-197

4.2. Other GR pseudotensors

I-198

4.3. Pseudotensors and the Hamiltonian

I-200

5. The Quasi-Local View

I-201

6. Currents as Generators

I-201

7. Gauge and Geometry

I-202

8. Dynamical Spacetime Geometry and the Hamiltonian 8.1. Pseudotensors and the Hamiltonian

I-203 I-204

8.2. Some comments 9. Differential Forms 10. Variational Principle for Form Fields

I-204 I-204 I-206

10.1. Hamiltons principle

I-207

10.2. Compact representation

I-207

11. Some Simple Examples of the Noether Theorems

I-208

11.1. Noether’s first theorem: Energy–momentum 11.2. Noether’s second theorem: Gauge fields

I-208 I-209

11.3. Field equations with local gauge theory

I-211

12. First-Order Formulation 13. The Hamiltonian and the 3 + 1 Spacetime Split

I-213 I-214

13.1. Canonical Hamiltonian formalism

I-215

13.2. The differential form of the spacetime decomposition

I-215

13.3. Spacetime decomposition of the variational formalism

I-217

Contents

14. The Hamiltonian and Its Boundary Term

xv

I-218

14.1. The translational Noether current

I-219

14.2. The Hamiltonian formulation

I-220

14.3. Boundary terms: The boundary condition and reference

I-221

14.4. Covariant-symplectic Hamiltonian boundary terms

I-222

15. Standard Asymptotics

I-223

15.1. Spatial infinity

I-224

15.2. Null infinity

I-224

15.3. Energy flux

I-225

16. Application to Electromagnetism

I-225

17. Geometry: Covariant Differential Formulation

I-227

17.1. Metric and connection

I-228

17.2. Riemann–Cartan geometry

I-229

17.3. Regarding geometry and gauge

I-230

17.4. On the affine connection and gauge theory

I-230

18. Variational Principles for Dynamic Spacetime Geometry

I-232

18.1. The Lagrangian and its variation

I-232

18.2. Local gauge symmetries, Noether currents and differential identities

I-233

18.3. Interpretation of the differential identities

I-238

19. First-Order Form and the Hamiltonian

I-240

19.1. First-order Lagrangian and local gauge symmetries

I-240

19.2. Generalized Hamiltonian and differential identities

I-241

19.3. General geometric Hamiltonian boundary terms

I-244

19.4. Quasi-local boundary terms

I-245

19.5. A preferred choice

I-245

19.6. Einstein’s GR

I-246

19.7. Preferred boundary term for GR

I-247

20. A “Best Matched” Reference

I-248

20.1. The choice of reference

I-249

20.2. Isometric matching of the 2-surface

I-250

20.3. Complete 4D isometric matching

I-251

xvi

Contents

20.4. Complete 4D isometric matching 21. Concluding Discussion Part II. Empirical Foundations 5. Equivalence principles, spacetime structure and the cosmic connection Wei-Tou Ni IJMPD 25 (2016) 1630002

I-251 I-252 I-263 I-265

1. Introduction

I-265

2. Meaning of Various Equivalence Principles

I-270

2.1. Ancient concepts of inequivalence

I-271

2.2. Macroscopic equivalence principles

I-271

2.3. Equivalence principles for photons (wave packets of light)

I-273

2.4. Microscopic equivalence principles

I-273

2.5. Equivalence principles including gravity (Strong equivalence principles)

I-276

2.6. Inequivalence and interrelations of various equivalence principles

I-277

3. Gravitational Coupling to Electromagnetism and the Structure of Spacetime

I-278

3.1. Premetric electrodynamics as a framework to study gravitational coupling to electromagnetism

I-278

3.2. Wave propagation and the dispersion relation

I-279

3.2.1. The condition of vanishing of B(1) and B(2) for all directions of wave propagation

I-282

3.2.2. The condition of (Sk) B(1) = (P)B(1) = 0 and A(1) = A(2) for all directions of wave propagation

I-284

3.3. Nonbirefringence condition for the skewonless case

I-284

3.4. Wave propagation and the dispersion relation in dilaton field and axion field

I-288

3.5. No amplification/no attenuation and no polarization rotation constraints on cosmic dilaton field and cosmic axion field

I-292

3.6. Spacetime constitutive relation including skewons

I-293

Contents

xvii

3.7. Constitutive tensor from asymmetric metric and Fresnel equation

I-297

3.8. Empirical foundation of the closure relation for skewonless case

I-300

4. From Galileo Equivalence Principle to Einstein Equivalence Principle

I-303

5. EEP and Universal Metrology

I-305

6. Gyrogravitational Ratio

I-307

7. An Update of Search for Long Range/Intermediate Range Spin–Spin, Spin–Monopole and Spin–Cosmos Interactions

I-308

8. Prospects

I-309

6. Cosmic polarization rotation: An astrophysical test of fundamental physics Sperello di Serego Alighieri IJMPD 24 (2015) 1530016

I-317

1. Introduction

I-317

2. Impact of CPR on Fundamental Physics

I-318

3. Constraints from the Radio Polarization of RGs

I-319

4. Constraints from the UV Polarization of RGs

I-320

5. Constraints from the Polarization of the CMB Radiation

I-321

6. Other Constraints

I-325

7. Discussion

I-326

8. Outlook

I-327

7. Clock comparison based on laser ranging technologies ´ Etienne Samain IJMPD 24 (2015) 1530021

I-331

1. Introduction

I-331

2. Scientific Objectives

I-335

2.1. Time and frequency metrology

I-335

2.2. Fundamental physics

I-338

2.3. Solar System science

I-340

2.4. Solar System navigation based on clock comparison

I-341

3. Time Transfer by Laser Link: T2L2 on Jason-2

I-341

3.1. Principle

I-341

3.2. Laser station ground segment

I-342

xviii

Contents

3.3. Space instrument

I-344

3.4. Time equation

I-347

3.5. Error budget

I-349

3.6. Link budget

I-351

3.7. Exploitation

I-352

4. One-Way Lunar Laser Link on LRO Spacecraft

I-357

5. Prospective

I-361

6. Conclusion and Outlook

I-364

8. Solar-system tests of relativistic gravity Wei-Tou Ni

I-371

IJMPD 25 (2016) 1630003 1. Introduction and Summary

I-371

2. Post-Newtonian Approximation, PPN Framework, Shapiro Time Delay and Light Deflection

I-374

2.1. Post-Newtonian approximation

I-375

2.2. PPN framework

I-377

2.3. Shapiro time delay

I-380

2.4. Light deflection

I-381

3. Solar System Ephemerides

I-382

4. Solar System Tests

I-385

5. Outlook — On Going and Next-Generation Tests 9. Pulsars and gravity R. N. Manchester

I-393 I-407

IJMPD 24 (2015) 1530018 1. Introduction 1.1. Pulsar timing 2. Tests of Relativistic Gravity 2.1. Tests of general relativity with double-neutron-star systems 2.1.1. The Hulse–Taylor binary, PSR B1913 + 16 2.1.2. PSR B1534 + 12 2.1.3. The double pulsar, PSR J0737 − 3039A/B 2.1.4. Measured post-Keplerian parameters 2.2. Tests of equivalence principles and alternative theories of gravitation 2.2.1. Limits on PPN parameters

I-407 I-410 I-412 I-412 I-412 I-415 I-417 I-421 I-421 I-423

Contents

2.2.2. Dipolar gravitational waves and the constancy of G 2.2.3. General scalar–tensor and scalar–vector–tensor theories 2.3. Future prospects 3. The Quest for Gravitational-Wave Detection 3.1. Pulsar timing arrays 3.2. Nanohertz gravitational-wave sources 3.2.1. Massive black-hole binary systems

xix

I-427 I-429 I-431 I-432 I-432 I-435 I-435

3.2.2. Cosmic strings and the early universe

I-439

3.2.3. Transient or burst GW sources

I-440

3.3. Pulsar timing arrays and current results 3.3.1. Existing PTAs 3.3.2. Limits on the nanohertz GW background 3.3.3. Limits on GW emission from individual black-hole binary systems 3.4. Future prospects 4. Summary and Conclusion Part III. Gravitational Waves 10. Gravitational waves: Classification, methods of detection, sensitivities, and sources Kazuaki Kuroda, Wei-Tou Ni and Wei-Ping Pan IJMPD 24 (2015) 1530031

I-443 I-444 I-445 I-446 I-450 I-452 I-459 I-461

1. Introduction and Classification

I-461

2. GWs in GR

I-464

3. Methods of GW Detection, and Their Sensitivities 3.1. Sensitivities

I-470 I-471

3.2. Very high frequency band (100 kHz–1 THz) and ultrahigh frequency band (above 1 THz)

I-477

3.3. High frequency band (10 Hz–100 kHz)

I-478

3.4. Doppler tracking of spacecraft (1 μHz–1 mHz in the low-frequency band)

I-480

3.5. Space interferometers (low-frequency band, 100 nHz–100 mHz; middle-frequency band, 100 mHz–10 Hz)

I-481

3.6. Very-low-frequency band (300 pHz–100 nHz)

I-486

3.7. Ultra-low-frequency band (10 fHz–300 pHz)

I-488

xx

Contents

3.8. Extremely-low (Hubble)-frequency band (1 aHz–10 fHz) 4. Sources of GWs

I-489 I-491

4.1. GWs from compact binaries

I-491

4.2. GWs from supernovae

I-492

4.3. GWs from massive black holes and their coevolution with galaxies

I-493

4.4. GWs from extreme mass ratio inspirals (EMRIs)

I-495

4.5. Primordial/inflationary/relic GWs

I-495

4.6. Very-high-frequency and ultra-high-frequency GW sources

I-496

4.7. Other possible sources

I-496

5. Discussion and Outlook 11. Ground-based gravitational-wave detectors Kazuaki Kuroda IJMPD 24 (2015) 1530032 1. Introduction to Ground-Based Gravitational-Wave Detectors 1.1. Gravitational-wave sources

I-497 I-505

I-505 I-506

1.1.1. Achieved sensitivities of large projects

I-506

1.1.2. Coalescences of binary neutron stars

I-508

1.1.3. Coalescences of binary black holes

I-508

1.1.4. Supernova explosion

I-509

1.1.5. Quasi-normal mode oscillation at the birth of black hole

I-509

1.1.6. Unstable fast rotating neutron star

I-510

1.2. Acceleration due to a gravitational wave

I-510

1.3. Response of a resonant antenna

I-512

1.4. Response of a resonant antenna

I-515

1.4.1. Directivity

I-516

1.4.1. Positioning

I-518

1.5. Comparison of a resonant antenna and an interferometer 2. Resonant Antennae

I-519 I-519

2.1. Development of resonant antennae

I-520

2.2. Dynamical model of a resonant antenna with two modes

I-523

2.3. Signal-to-noise ratio and noise temperature

I-525

Contents

2.4. Comparison of five resonant antennae 3. Interferometers 3.1. First stage against technical noises in prototype interferometers

xxi

I-526 I-527 I-528

3.1.1. 3 m-Garching interferometer 3.1.2. 30 m-Garching interferometer

I-528 I-530

3.1.3. Glasgow 10 m-Fabry–Perot Michelson interferometer

I-533

3.1.4. Caltech 40 m-Fabry–Perot Michelson interferometer 3.1.5. ISAS 10 m and 100 m delay-line interferometer

I-535 I-536

3.2. Further R&D efforts in the first-generation detectors 3.2.1. Power recycling

I-536 I-537

3.2.2. Signal recycling and resonant side-band extraction 3.3. Fighting with thermal noise of the second stage

I-538 I-539

3.3.1. Mirror and suspension thermal noise 3.3.2. Thermal noise of optical coating 3.4. Fighting against quantum noises and squeezing 3.4.1. Radiation pressure noise 3.4.2. Squeezing 4. Large Scale Projects

I-540 I-542 I-543 I-543 I-544 I-546

4.1. LIGO project 4.2. Virgo project

I-546 I-548

4.3. GEO project 4.4. TAMA/CLIO/LCGT(KAGRA) project

I-552 I-555

4.4.1. TAMA

I-555

4.4.2. CLIO 4.4.3. LCGT (KAGRA)

I-558 I-561

4.4.4. Einstein telescope 5. Summary

I-565 I-566

Appendix A. Thermal Noise

I-567

A.1. Nyquist theorem A.2. Thermal noise of a harmonic oscillator

I-567 I-568

Appendix B. Modulation Appendix C. Fabry–Perot Interferometer

I-569 I-571

xxii

Contents

C.1. Fabry–Perot cavity

I-571

C.2. Frequency response of a Fabry–Perot Michelson interferometer Appendix D. Newtonian Noise

I-572 I-573

12. Gravitational wave detection in space Wei-Tou Ni IJMPD 25 (2016) 1630001

I-579

1. Introduction

I-579

2. Gravity and Orbit Observations/Experiments in the Solar System 3. Doppler Tracking of Spacecraft

I-586 I-589

4. Interferometric Space Missions

I-591

5. Frequency Sensitivity Spectrum 6. Scientific Goals

I-596 I-601

6.1. Massive black holes and their co-evolution with galaxies

I-601

6.2. Extreme mass ratio inspirals 6.3. Testing relativistic gravity

I-603 I-603

6.4. Dark energy and cosmology

I-603

6.5. Compact binaries 6.6. Relic GWs

I-604 I-604

7. Basic Orbit Configuration, Angular Resolution and Multi-Formation Configurations 7.1. Basic LISA-like orbit configuration

I-605 I-605

7.2. Basic ASTROD orbit configuration

I-607

7.3. Angular resolution 7.4. Six/twelve spacecraft formation

I-611 I-612

8. Orbit Design and Orbit Optimization Using Ephemerides

I-612

8.1. CGC ephemeris 8.2. Numerical orbit design and orbit optimization for eLISA/NGO

I-613

8.3. Orbit optimization for ASTROD-GW 8.3.1. CGC 2.7.1 ephemeris

I-616 I-616

8.3.2. Initial choice of spacecraft initial conditions 8.3.3. Method of optimization 9. Deployment of Formation in Earthlike Solar Orbit 10. Time Delay Interferometry

I-614

I-616 I-617 I-619 I-619

Contents

xxiii

11. Payload Concept

I-622

12. Outlook

I-624

Subject Index

I

Author Index

XIII

Volume 2 Foreword

v

Color plates

II-CP1

Part IV. Cosmology 13. General Relativity and Cosmology Martin Bucher and Wei-Tou Ni IJMPD 24 (2015) 1530030 14. Cosmic Structure Marc Davis

II-1 II-3

II-19

IJMPD 23 (2014) 1430021 1. History of Cosmic Discovery

II-19

2. Measurement of the Galaxy Correlation Function

II-22

2.1. Before 1980

II-22

2.2. After 1980

II-23

2.3. Remarkable large-scale structure in simulations

II-25

2.4. Measurement of the BAO effect

II-26

2.5. Further measurements of the power spectrum

II-28

2.6. Lyman-α clouds

II-29

3. Large Scale Flows

II-31

4. Dwarf Galaxies as a Probe of Dark Matter

II-34

5. Gravitational Lensing

II-38

5.1. Double images

II-38

5.2. Bullet cluster

II-38

5.3. Substructure of gravitational lenses

II-38

6. Conclusion 15. Physics of the cosmic microwave background anisotropy Martin Bucher IJMPD 24 (2015) 1530004

II-40 II-43

1. Observing the Microwave Sky: A Short History and Observational Overview

II-43

2. Brief Thermal History of the Universe

II-54

xxiv

Contents

3. Cosmological Perturbation Theory: Describing a Nearly Perfect Universe Using General Relativity

II-58

4. Characterizing the Primordial Power Spectrum

II-61

5. Recombination, Blackbody Spectrum, and Spectral Distortions

II-62

6. Sachs–Wolfe Formula and More Exact Anisotropy Calculations

II-63

7. What Can We Learn From the CMB Temperature and Polarization Anisotropies?

II-69

7.1. Character of primordial perturbations: Adiabatic growing mode versus field ordering

II-69

7.2. Boltzmann hierarchy evolution

II-71

7.3. Angular diameter distance

II-76

7.4. Integrated Sachs–Wolfe effect

II-77

7.5. Reionization

II-78

7.6. What we have not mentioned

II-83

8. Gravitational Lensing of the CMB

II-84

9. CMB Statistics

II-86

9.1. Gaussianity, non-Gaussianity, and all that

II-86

9.2. Non-Gaussian alternatives

II-92

10. Bispectral Non-Gaussianity

II-92

11. B Modes: A New Probe of Inflation

II-94

11.1. Suborbital searches for primordial B modes

II-95

11.2. Space based searches for primordial B modes

II-96

12. CMB Anomalies 13. Sunyaev–Zeldovich Effects 14. Experimental Aspects of CMB Observations

II-96 II-98 II-100

14.1. Intrinsic photon counting noise: Ideal detector behavior

II-102

14.2. CMB detector technology

II-104

14.3. Special techniques for polarization

II-106

15. CMB Statistics Revisited: Dealing with Realistic Observations

II-110

16. Galactic Synchrotron Emission

II-112

17. Free–Free Emission

II-113

18. Thermal Dust Emission

II-114

19. Dust Polarization and Grain Alignment

II-116

19.1. Why do dust grains spin?

II-117

Contents

19.2. 19.3. 19.4. 19.5.

About which axis do dust grains spin? A stochastic differential equation for L(t) Suprathermal rotation Dust grain dynamics and the galactic magnetic field 19.5.1. Origin of a magnetic moment along L 19.6. Magnetic precession 19.6.1. Barnett dissipation 19.7. Davis–Greenstein magnetic dissipation 19.8. Alignment along B without Davis–Greenstein dissipation 19.9. Radiative torques 19.10. Small dust grains and anomalous microwave emission (AME) 20. Compact Sources 20.1. Radio galaxies 20.2. Infrared galaxies 21. Other Effects 21.1. Patchy reionization 21.2. Molecular lines 21.3. Zodiacal emission 22. Extracting the Primordial CMB Anisotropies 23. Concluding Remarks 16. SNe Ia as a cosmological probe Xiangcun Meng, Yan Gao and Zhanwen Han IJMPD 24 (2015) 1530029 1. Introduction 2. SNe Ia as a Standardizable Distance Candle 3. Progenitors of SNe Ia 4. Effect of SN Ia Populations on Their Brightness 5. SN Ia’s Role in Cosmology 6. Issues and Prospects 17. Gravitational Lensing in Cosmology Toshifumi Futamase IJMPD 24 (2015) 1530011 1. Introduction and History 2. Basic Properties for Lens Equation 2.1. Derivation of the cosmological lens equation

xxv

II-118 II-118 II-119 II-120 II-121 II-122 II-122 II-124 II-125 II-126 II-128 II-130 II-131 II-132 II-132 II-132 II-132 II-133 II-133 II-134 II-151

II-151 II-152 II-157 II-160 II-163 II-167 II-173

II-173 II-176 II-176

xxvi

Contents

2.2. Properties of lens mapping 2.3. Caustic and critical curves 2.3.1. Circular lenses 2.3.2. The Einstein radius and radial arcs 2.3.3. Non-circular lenses 3. Strong Lensing 3.1. Methods of solving the lens equation: LTM and non-LTM 3.2. Image magnification 3.3. Time delays 3.4. Comparison of lens model software 3.4.1. Non-light traces mass software 3.4.2. Light traces mass software 3.5. Lens statistics 4. Weak Lensing 4.1. Basic method 4.1.1. Shape measurements 4.2. E/B decomposition 4.3. Magnification bias 4.3.1. Simulation test 4.3.2. Higher-order weak lensing-flexion and HOLICs 4.4. Cluster mass reconstruction 4.4.1. Density profile 4.4.2. Dark matter subhalos in the coma cluster 4.5. Cosmic shear 4.5.1. How to measure the cosmic density field 5. Conclusion and Future 18. Inflationary cosmology: First 30+ years Katsuhiko Sato and Jun’ichi Yokoyama IJMPD 24 (2015) 1530025 1. Introduction 1.1. Developments in Japan 1.2. Developments in Russia 1.3. Inflation paradigm 2. Resolution of Fundamental Problems 3. Realization of Inflation

II-179 II-183 II-184 II-187 II-189 II-190 II-190 II-191 II-191 II-194 II-194 II-194 II-195 II-196 II-197 II-199 II-203 II-206 II-206 II-207 II-208 II-211 II-212 II-214 II-217 II-219 II-225

II-225 II-227 II-228 II-230 II-231 II-233

Contents

xxvii

3.1. Three mechanisms 3.2. Inflation scenario 4. Slow-Roll Inflation Models 4.1. Large-field models 4.2. Small-field model 4.3. Hybrid inflation 5. Reheating 6. Generation of Quantum Fluctuations that Eventually Behave Classically 7. Cosmological Perturbation 8. Generation of Curvature Fluctuations in Inflationary Cosmology 9. Tensor Perturbation 10. The Most General Single-Field Inflation 10.1. Homogeneous background equations 10.2. Kinetically driven G-inflation 10.3. Potential-driven slow-roll G-inflation 11. Power Spectrum of Perturbations in Generalized G-inflation 11.1. Tensor perturbations 11.2. Scalar perturbations 12. Inflationary Cosmology and Observations 12.1. Large-field models 12.2. Small-field model 12.3. Hybrid inflation model 12.4. Noncanonical models and multi-field models 13. Conclusion 19. Inflation, string theory and cosmic strings David F. Chernoff and S.-H. Henry Tye IJMPD 24 (2015) 1530010

II-233 II-234 II-236 II-236 II-237 II-238 II-239

1. Introduction 2. The Inflationary Universe 3. String Theory and Inflation 3.1. String theory and flux compactification 3.2. Inflation in string theory 4. Small r Scenarios 4.1. Brane inflation ¯ 4.1.1. D3-D3-brane inflation

II-273 II-277 II-280 II-281 II-282 II-283 II-284 II-285

II-242 II-244 II-246 II-249 II-250 II-251 II-253 II-254 II-255 II-255 II-258 II-261 II-264 II-265 II-266 II-266 II-267 II-273

xxviii

Contents

4.1.2. Inflection point inflation 4.1.3. DBI model 4.1.4. D3-D7-brane inflation 4.2. K¨ ahler moduli inflation 5. Large r Scenarios 5.1. The Kim–Nilles–Peloso mechanism 5.1.1. Natural inflation 5.1.2. N-flation 5.1.3. Helical inflation 5.2. Axion monodromy 5.3. Discussions 6. Relics: Low Tension Cosmic Strings 6.1. Strings in brane world cosmology 6.2. Current bounds on string tension Gμ and probability of intercommutation p 7. Scaling, Slowing, Clustering and Evaporating 7.1. Large-scale string distribution 7.2. Local string distribution 8. Detection 8.1. Detection via Microlensing 8.2. WFIRST microlensing rates 8.3. Gravitational waves 9. Summary Part V. Quantum Gravity 20. Quantum gravity: A brief history of ideas and some outlooks Steven Carlip, Dah-Wei Chiou, Wei-Tou Ni and Richard Woodard IJMPD 24 (2015) 1530028 1. Prelude 2. Perturbative Quantum Gravity 3. String Theory 4. Loop Quantum Gravity 5. Black Hole Thermodynamics 6. Quantum Gravity Phenomenology 21. Perturbative quantum gravity comes of age R. P. Woodard IJMPD 23 (2014) 1430020

II-286 II-286 II-287 II-287 II-288 II-288 II-288 II-288 II-290 II-291 II-292 II-293 II-296 II-297 II-299 II-302 II-305 II-307 II-307 II-307 II-311 II-314 II-323 II-325

II-325 II-327 II-328 II-332 II-334 II-337 II-349

Contents

1. Introduction 2. Why Quantum Gravitational Effects from Primordial Inflation are Observable 2.1. The background geometry 2.2. Inflationary particle production 3. Tree Order Power Spectra 3.1. The background for single-scalar inflation 3.2. Gauge-fixed, constrained action 3.3. Tree order power spectra 3.4. The controversy over adiabatic regularization 3.5. Why these are quantum gravitational effects 4. Loop Corrections to the Power Spectra 4.1. How to make computations 4.2. -Suppression and late-time growth 4.3. Nonlinear extensions 4.4. The promise of 21 cm radiation 5. Other Quantum Gravitational Effects 5.1. Linearized effective field equations 5.2. Propagators and tensor 1PI functions 5.3. Results and open problems 5.4. Back-Reaction 6. Conclusions 22. Black hole thermodynamics S. Carlip IJMPD 23 (2014) 1430023 1. Introduction 2. Prehistory: Black Hole Mechanics and Wheeler’s Cup of Tea 3. Hawking Radiation 3.1. Quantum field theory in curved spacetime 3.2. Hawking’s calculation 4. Back-of-the-Envelope Estimates 4.1. Entropy 4.2. Temperature 5. The Many Derivations of Black Hole Thermodynamics 5.1. Other settings 5.2. Unruh radiation

xxix

II-349 II-351 II-351 II-355 II-358 II-359 II-360 II-363 II-369 II-369 II-371 II-372 II-376 II-380 II-382 II-384 II-384 II-386 II-395 II-399 II-402 II-415

II-415 II-416 II-418 II-419 II-420 II-422 II-422 II-423 II-424 II-425 II-425

xxx

Contents

5.3. 5.4. 5.5. 5.6. 5.7. 5.8. 5.9.

6.

7.

8. 9.

10.

Particle detectors Tunneling Hawking radiation from anomalies Periodic Greens functions Periodic Gravitational partition function Periodic Pair production of black holes Periodic Quantum field theory and the eternal black hole 5.10. Periodic Quantized gravity and classical matter 5.11. Periodic Other approaches Thermodynamic Properties of Black Holes 6.1. Periodic Black hole evaporation 6.2. Periodic Heat capacity 6.3. Periodic Phase transitions 6.4. Periodic Thermodynamic volume 6.5. Periodic Lorentz violation and perpetual motion machines Approaches to Black Hole Statistical Mechanics 7.1. Periodic “Phenomenology” 7.2. Periodic Entanglement entropy 7.3. Periodic String theory 7.3.1. Weakly coupled strings and branes 7.3.2. Fuzzballs 7.3.3. The AdS/CFT correspondence 7.4. Loop quantum gravity 7.4.1. Microcanonical approach 7.4.2. Microcanonical approach 7.5. Other ensembles 7.6. Induced gravity 7.7. Logarithmic corrections The Holographic Conjecture The Problem of Universality 9.1. State-counting in conformal field theory 9.2. Application to black holes 9.3. Effective descriptions The Information Loss Problem 10.1. Nonunitary evolution

II-426 II-426 II-427 II-428 II-429 II-431 II-431 II-432 II-433 II-433 II-434 II-434 II-435 II-435 II-436 II-437 II-437 II-438 II-440 II-440 II-441 II-441 II-442 II-442 II-444 II-445 II-445 II-446 II-446 II-448 II-449 II-450 II-451 II-451 II-452

Contents

xxxi

10.2. No black holes

II-452

10.3. Remnants and baby universes

II-453

10.4. Hawking radiation as a pure state

II-454

11. Conclusion

II-455

Appendix A. Classical Black Holes

II-456

23. Loop quantum gravity Dah-Wei Chiou IJMPD 24 (2015) 1530005

II-467

1. Introduction

II-467

2. Motivations

II-469

2.1. Why quantum gravity?

II-469

2.2. Difficulties of quantum gravity

II-470

2.3. Background-independent approach

II-470

3. Connection Theories of General Relativity 3.1. Connection dynamics

II-471 II-471

3.2. Canonical (Hamiltonian) formulation

II-473

3.3. Remarks on connection theories

II-476

4. Quantum Kinematics 4.1. Quantization scheme

II-478 II-478

4.2. Cylindrical functions

II-479

4.3. Spin networks

II-481

4.4. S-knots

II-483

5. Operators and Quantum Geometry 5.1. Holonomy operator

II-486 II-486

5.2. Area operator

II-487

5.3. Volume operator

II-489

5.4. Quantum geometry

II-490

6. Scalar Constraint and Quantum Dynamics

II-492

6.1. Regulated classical scalar constraint

II-492

6.2. Quantum scalar constraint

II-495

6.3. Solutions to the scalar constraint

II-498

6.4. Quantum dynamics

II-500

7. Inclusion of Matter Fields

II-503

7.1. Yang–Mills fields

II-503

7.2. Fermions

II-504

7.3. Scalar fields

II-505

7.4. S-knots of geometry and matter

II-506

xxxii

Contents

8. Low-Energy Physics 8.1. Weave states 8.2. Loop states versus Fock states 8.3. Holomorphic coherent states 9. Spin Foam Theory 9.1. From s-knots to spin foams 9.2. Spin foam formalism 10. Black Hole Thermodynamics 10.1. Statistical ensemble 10.2. Bekenstein–Hawking entropy 10.3. More on black hole entropy 11. Loop Quantum Cosmology 11.1. Symmetry reduction 11.2. Quantum kinematics 11.3. Quantum constraint operator 11.4. Physical Hilbert space 11.5. Quantum dynamics 11.6. Other models 12. Current Directions and Open Issues 12.1. The master constraint program 12.2. Algebraic quantum gravity 12.3. Reduced phase space quantization 12.4. Off-shell closure of quantum constraints 12.5. Loop quantum gravity versus spin foam theory 12.6. Covariant loop quantum gravity 12.7. Spin foam cosmology 12.8. Quantum reduced loop gravity 12.9. Cosmological perturbations in the Planck era 12.10. Spherically symmetric loop gravity 12.11. Planck stars and black hole fireworks 12.12. Information loss problem 12.13. Quantum gravity phenomenology 12.14. Supersymmetry and other dimensions Subject Index Author Index

II-507 II-507 II-508 II-508 II-511 II-511 II-514 II-515 II-516 II-517 II-519 II-520 II-520 II-522 II-524 II-526 II-527 II-528 II-529 II-529 II-530 II-530 II-532 II-533 II-533 II-534 II-534 II-534 II-535 II-535 II-536 II-537 II-537 I XIII

Chapter 13, Fig. 2; Chapter 14, Fig. 13.

Chapter 13, Fig. 3.

II-CP1

II-CP2

Chapter 13, Fig. 4.

Chapter 14, Fig. 1.

II-CP3

Chapter 14, Fig. 5.

Chapter 14, Fig. 6.

II-CP4

Chapter 14, Fig. 8.

Chapter 14, Fig. 9.

II-CP5

Chapter 15, Fig. 2.

Chapter 15, Fig. 3.

Chapter 15, Fig. 4.

II-CP6

Chapter 15, Fig. 6.

II-CP7

Chapter 15, Fig. 7.

II-CP8

Chapter 15, Fig. 8.

II-CP9

Chapter 15, Fig. 9.

Chapter 15, Fig. 17.

II-CP10

Chapter 16, Fig. 1.

Chapter 16, Fig. 5.

II-CP11

Chapter 16, Fig. 6.

Chapter 16, Fig. 7.

II-CP12

Chapter 17, Fig. 8.

Chapter 17, Fig. 9.

Chapter 17, Fig. 13.

II-CP13

Chapter 17, Fig. 14.

Chapter 17, Fig. 15.

II-CP14

Chapter 19, Fig. 1.

Chapter 19, Fig. 2.

Chapter 19, Fig. 3.

II-CP15

Chapter 19, Fig. 4.

G c2 10 Log10 n kpc

15

10

8

3

5

R kpc 5

10

15

5

10

Chapter 19, Fig. 6.

Chapter 19, Fig. 7.

20

25

II-CP16

Flux

Time Chapter 19, Fig. 8.

Log10 4

2

0

14

12

2

4

Chapter 19, Fig. 9.

10

8

Log10

May 4, 2017 9:55 b2167-ch13

Chapter 13 General relativity and cosmology

Martin Bucher†,‡ and Wei-Tou Ni§ †Laboratoire

APC, Universit´ e Paris 7/CNRS, Bˆ atiment Condorcet, Case 7020, 75205 Paris Cedex 13, France [email protected]

‡Astrophysics and Cosmology Research Unit and School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4041, South Africa §Center for Gravitation and Cosmology, Department of Physics, National Tsing Hua University, Hsinchu, Taiwan 30013, Republic of China

[email protected]

This year marks the 100th anniversary of Einstein’s 1915 landmark paper “Die Feldgleichungen der Gravitation” in which the field equations of general relativity were correctly formulated for the first time, thus rendering general relativity a complete theory. Over the subsequent hundred years, physicists and astronomers have struggled with uncovering the consequences and applications of these equations. This paper, which was written as an introduction to six chapters dealing with the connection between general relativity and cosmology that will appear in the two-volume book One Hundred Years of General Relativity: From Genesis and Empirical Foundations to Gravitational Waves, Cosmology and Quantum Gravity, endeavors to provide a historical overview of the connection between general relativity and cosmology, two areas whose development has been closely intertwined. Keywords: General relativity; cosmology; centennial connection. PACS Number(s): 04, 98.80.−k

One hundred years ago, the best model of the universe summarizing the state of the observations at that time was the Kapteyn universe, which consists of a system of stars distributed more or less uniformly within a disk about 10 kpc in diameter and 2 kpc thick. In this model, the Sun is situated near the center of the disk. In 1917, Einstein1 proposed a static cosmological model based on the “cosmological principle,” a generalization of the Copernican principle postulating that the homogeneity and isotropy of space in the large should be extended to include the time dimension as well. Using distance determinations to about 100 globular clusters, Shapley in 1918 pushed back the boundaries of the measured universe and concluded that the Sun lies near the edge of this distribution. As a II-3

May 4, 2017 9:55 b2167-ch13

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result of rapid improvements in astronomical observations, the size of our observed universe soon grew to extend almost to our causal horizon. This next group of chapters comprising Part IV of GR100 book deals with Cosmology, applying Einstein’s theory of general relativity to the universe as a whole. Cosmology considers the universe on very large scales and evolving over very long times, comparable to the age of the universe. Cosmology today is often described as being a “precision” science and the insistence on this term reflects that cosmology had not always been seen as such. When Einstein formulated his general theory of relativity, little was known about the universe beyond our own galaxy, and although distant galaxies had been observed as “nebulous” unresolved spiral blotches, it was not at all clear that these “nebulae” consisted of numerous stars much like our own galaxy. A glimpse of the state of affairs slightly after the formulation of general relativity may be gained by reading the written summaries2 of the “Great Debate” held at the Smithsonian institution in 1920, where Harlow Shapley and Heber Curtis sparred over the question of the “scale of the universe.” The former argued that the “nebulae” were simply clouds lying on the periphery of our own galaxy, whereas the latter maintained that the observations at the time suggested that the “nebulae” were distant “island universes” much like our own galaxy. A key observation that helped settle this question in favor of the latter point of view was the discovery by Edwin Hubble of Cepheid variable stars in the Andromeda galaxy, which established that Andromeda lies far beyond the confines of our own galaxy. On the observational side, as telescopes and other observational techniques greatly improved, our view of the universe progressively expanded to greater and greater distances. Once it had been established that the universe was populated by galaxies of different sizes, a recurrent question became whether the universe was homogeneous and isotropic on the largest observable scales, and it is only recently that galaxy surveys sufficiently deep and with sufficient statistics became available to settle this question definitively. An excellent historical account of these early debates and their role of the development of modern cosmology can be found in Peebles’ 1980 book3 (see also Ref. 4). The theory underlying the hot big bang model as we know it today was developed before the observational issues mentioned above had been settled. The geometry and time evolution of the universe as predicted by Einstein’s theory are given by what is now known as the Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) model, which describes the solutions to Einstein’s field equations for a spatially homogeneous and isotropic universe whose scale factor varies with time. This solution to the Einstein field equations was first put forward by Alexander Friedmann in 19225,6 and later independently by Georges Lamaˆıtre7 (see also Ref. 8). Robertson9 and Walker10 subsequently showed that this was the only solution to the field equations consistent with spatial homogeneity and isotropy. As discussed above, in 1917 Einstein1 had put forth a theory of a static universe — a solution of the general relativity field equations that is not only homogeneous and isotropic

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in the three spatial dimensions but also homogeneous in time. Given the lack of compelling observational evidence to the contrary at the time, Einstein believed that an eternal universe, in which the Copernican principle held not only in three spatial dimensions but also in time, was more elegant and hence more plausible. In order to satisfy the gravitational field equations, Einstein had to introduce a cosmological constant term, denoted by Λ, a proposal that according to George Gamow, Einstein had later once described as his “greatest blunder,” although the authenticity of this quote is doubted by some. Of the following six chapters of this volume dealing with the connection between cosmology and general relativity, the first three chapters deal primarily with the observations underpinning our modern conception of the cosmos. The first chapter of this group “Cosmic structure” provides a comprehensive overview of galaxy surveys and the information provided by them regarding the large-scale distribution of mass in the universe.11 The second chapter discusses the physics of the 2.725 K cosmic microwave background (CMB) radiation.12 The CMB confirms the big bang story in two ways. First, the excellent agreement between the observed frequency spectrum of the microwave sky and a perfect blackbody spectrum at T = 2.725 K provides strong confirmation of the expanding universe scenario, in which the universe at early times was much smaller and hotter, and thus once in a state extremely close to thermal equilibrium. Second, the measurement of the small departures from homogeneity and isotropy confirms the gravitational instability hypothesis for the origin of structure. This chapter describes in detail how the precise mapping of the anisotropies of the blackbody temperature in both intensity and polarization can be used to test theories of the very early universe based on new physics far beyond the standard model and how these fluctuations provide the initial conditions for the subsequent evolution leading to the formation of structure. The third chapter deals with the SN Ia distance scale and the discovery of the accelerated expansion of the universe.13 The fourth chapter “Gravitational lensing in cosmology” details the present state of gravitational lensing probes.14 The observation of the deflection of light rays as predicted by general relativity of course dates back to the famous expedition led by Arthur Eddington in 1919 to the island Princip´e off the west coast of Africa in order to measure the bending of light by the Sun during a total eclipse. Since that celebrated and crucial experiment confirming the theory of general relativity, gravitational lensing has evolved into a powerful tool for mapping the inhomogeneity in the distribution of mass. As pointed out by Robert Dicke in the late 1960s,15,16 a universe that is governed by general relativity (through the FLRW solution) and filled with ordinary matter (i.e., a combination of nonrelativistic and relativistic particles with w = p/ρ ranging from 0 to 1/3) provides a model that is unsatisfactory in several respects including (i) the so-called “horizon problem,” whereby distant regions in opposite directions in the sky appear similar but could never have been in causal contact subsequent to the apparent initial singularity17,18 and (ii) the “flatness

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problem,” under which the relation between the expansion rate and mean density at very early times usually expressed in terms of the dimensionless density parameter Ω(t) = 8πGρmean (t)/3H 2 (t) initially had to be tuned near one with incredible precision to avoid a universe that today is either nearly empty or already re-collapsed.15,16 Although Dicke did not propose a concrete alternative, he suggested that the big bang model as formulated at the time might not be the whole story. “Cosmic inflation,” reviewed in the fifth chapter,19 is a theory postulating that the very early universe underwent a period of quasi-exponential inflation during which whatever inhomogeneities may have existed prior to inflation are effectively erased and replaced with quantum fluctuations of the quantum fields relevant during inflation. The final chapter “Inflation, string theory and cosmic strings”20 discusses possible connections between cosmic inflation and ideas from superstring theory concerning how inflation might be realized as part of a theory unifying all the fundamental interactions. Inflationary cosmology as developed in the early 1980s offered an attractive proposal, solving many of the problems that one would face if one naively extrapolated a conventional matter-radiation equation of state backward in time all the way to the putative big bang singularity. Moreover, inflation offers a predictive mechanism for how the primordial perturbations from a perfectly homogeneous and isotropic FLRW background solution are generated. But inflation is an incomplete theory whose predictions are not completely defined without specifying a broader particle physics model within which inflation is realized. This final chapter reviews work on how inflation might be realized within the framework of superstring theory. These chapters serve to summarize the role of general relativity in modern cosmology. Two key issues in modern cosmology are the dark matter problem and the dark energy problem. Since we have not included separate chapters for these issues, we give a brief review here. Both problems lead to postulating a new contribution to the stress–energy tensor because the known components combined with gravity as we understand it cannot account for the observations. In general, matter and objects in the universe have been discovered through nongravitational means, for example, through the electromagnetic radiation that they emit or scatter, but the history of objects first discovered through their gravitational pull on other objects dates quite far back, to the discovery of new outer planets. Since Herschel’s discovery of the planet Uranus in 1781, its observed orbit persistently deviated from its predicted Newtonian trajectory according to the ephemeris calculations at the time. Hussey suggested in 1834 that this disagreement could be explained by perturbations arising from an undiscovered planet. In 1846, Le Verrier predicted the position of this new planet. On September 25, 1846, Galle and d’Arrest discovered this new planet, now known as Neptune, to within a degree of Le Verrier’s prediction. This discovery was a great triumph for Newton’s gravitational theory and was the first example where deviations of an observed orbit from the predicted orbit led to the discovery of missing mass.21

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When sufficient data from meridian and transit observations of Mercury had been accumulated, in 1859 Le Verrier discovered a discrepancy between the observations and Newtonian gravity. This discrepancy may be described as the anomalous perihelion advance of Mercury,a at the rate of 38" per century. Using improved calculations and data sets, Newcomb in 1882 measured this discrepancy more precisely, obtaining 42".95 per century. A more recent value is 42".98 ± 0.04 per century.22 In the last half of the 19th century, efforts to account for the anomalous perihelion advance of Mercury explored two general directions: (i) Searching for a putative planet “Vulcan” or other matter inside Mercury’s orbit and (ii) postulating an ad hoc modified gravitational force law. Both these directions proved unsuccessful. Proposed modifications of the gravitational law included Clairaut’s force law (of the form A/r2 + B/r4 ), Hall’s hypothesis (that the gravitational attraction is proportional to the inverse of distance to the (2 + δ) power instead of the square) and velocity-dependent force laws. (The reader is referred Ref. 23 for an in depth history of the measurement and understanding of Mercury’s perihelion advance.) A compelling solution to this problem had to await the development of general relativity. When general relativity is taken as the correct theory for predicting corrections to Newton’s theory, we understand why when the observations reached an accuracy of the order of 1" per century (transit observations), a discrepancy would be seen. Over a century, Mercury orbits around the Sun 400 times, amounting to a total angle of 5 × 108 arcsec. The fractional relativistic correction (perihelion advance anomaly) of Mercury’s orbit is of order ξGM Sun /dc2 , (i.e. 8 × 10−8 ) with ξ = 3 and d being the distance of Mercury to the Sun. Therefore, the relativistic correction for perihelion advance is about 40 arcsec per century. As the orbit determination of Mercury reached an accuracy of order 10−8 , the relativistic corrections to Newtonian gravity became manifest. We thus see how gravitational anomalies can lead either to the discovery of missing matter or to a modification of the fundamental theory for gravity. But there is also a third more mundane possibility. The secular contribution to the change of Moon’s orbit owing to the back reaction of lunar tides of Earth is such an example.b When such orbital anomalies were found, their solution involved neither missing mass nor modified Newtonian dynamics. Moreover, the discovery of Pluto might be an example of an accidental discovery resulting from the interplay between theory and experiment. Observations of Neptune in the late 19th century made astronomers suspect that there could be another planet besides Neptune perturbing Uranus’ orbit. In the early 20 century, Lowell searched in vain for such a planet at the Lowell observatory, which he founded in Flagstaff. In 1931, Tombaugh a Here,

“anomalous” means after the much larger corrections from perturbations of other planets have been subtracted. Only this residual, or anomalous part, constitutes a sign of something new. b The action on the Earth is to slow down its rotation so that after some time leap seconds need to be inserted.

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at Lowell observatory discovered Pluto 6◦ off its predicted position. However, its mass was much smaller than the predicted mass, so its influence on the orbits of Uranus and Neptune is negligible given the precision at that time. Pluto thus became the ninth major planet. However subsequently, as a result of a redefinition by the IAU of what constitutes a major planet, Pluto was downgraded to become reclassified as a “trans-Neptunian” object. In 1992, the next trans-Neptunian object was discovered and since then more than 1200 such objects have been discovered with Eris (discovered in 2005) more massive than Pluto. We now turn to dark matter, which was first introduced to account for the dynamics of clusters and the rotation curves of spiral galaxies, because the ordinary visible baryonic matter in the form of stars assuming plausible mass-to-light ratios, and also of the gas in the case of galaxy clusters, was unable to account for the observations if the correctness of ordinary Newtonian gravity is assumed. (In this context, corrections from special and general relativity are negligible.) The first hints of the need to postulate a dark matter component date back to the 1930s, when Zwicky found that the virial mass of galaxy clusters (i.e., the mass deduced by applying the virial theorem to the observed internal velocities of their member galaxies) greatly exceeds the total mass inferred from the luminous matter based on plausible mass-to-light ratios.24 X-ray observations in the 1970s and 1980s alleviated this discrepancy without completely resolving it. The need to postulate an additional dark matter component of some sort also arose from studies of the dynamics of spiral galaxies. In a series of papers in the 1970s, Rubin, Ford, and Thonnard measured the rotation curves of a number of disk galaxies and found that rotation speeds were larger than would be expected from the gravitational attraction arising from the visible mass distribution.25−27 Moreover, the shape of the rotation curves inferred from the visible mass did not agree with the observations, which found a rotation velocity almost constant with varying radius. The authors interpreted their findings as evidence for a new dark matter component. Logically this conflict, known as the missing mass problem, could arise from a mass discrepancy, an acceleration discrepancy, or possibly even both. Many people believe that the missing mass is the dark matter, whose presence is made manifest only through its gravitational interaction with the visible baryonic matter. Others however explored the possibility that Newtonian gravitational dynamics should be modified. In 1982, Milgrom proposed the phenomenological modified Newtonian dynamics (MOND) law for small accelerations.28 Under this hypothesis, the gravitational dynamics become modified when the acceleration is smaller than a0 ∼ 10−10 ms−2 (Fig. 1). It was later shown possible to explain such a phenomenological Ansatz in the framework of a relativistic theory of gravity with additional degrees of freedom; however, none of these theories are yet satisfactory from a theoretical standpoint.31

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Fig. 1. Critical acceleration from galactic rotation curves. The ratio of dynamical to baryonic mass is shown at each point along rotation curves as a function of the centripetal acceleration at that point. The panel shows data for 74 galaxies (see Ref. 29). The presence of missing or “dark” matter is the conventional explanation for why in spiral galaxies the measured rotation curves do not agree, neither in magnitude nor in shape, with the rotation curves predicted assuming a reasonable mass-to-light ratio and taking into account only the contribution from the disk. However, if the MOND hypothesis is adopted under which the gravitational dynamics becomes modified when the predicted acceleration is smaller than a0 (∼ 10−10 ms−2 ), the rotation curves can be fit assuming a constant mass-to-light ratio and no halo dark matter component. (Note: Figure reprinted with permission from Ref. 30.)

Perhaps today one of the strongest arguments for the need for a dark matter component arises from CMB anisotropy measurements as explained in the chapter on the CMB.12 The standard six-parameter model, which includes a dark matter component comprising ∼ 24% of the critical mass, provides an adequate fit to the observations and it is not possible to account for the observations with a model having only baryonic matter. Compared to other probes, the CMB is a particularly clean probe of cosmological models and parameters because its interpretation is based on linear theory except for small and calculable nonlinear corrections. Except for a few degeneracies (which can be lifted by combining with a few other reliable ancillary data sets, such as BAO), the errors on the cosmological parameters as determined using the CMB are typically of order 1% and characterized by a well understood error budget. See Refs. 32 and 33 for a detailed discussion of the current status of CMB constraints. Another dramatic though less quantitative recent illustration of the need for dark matter arises from analyzing the Bullet Cluster as observed by gravitational lensing (which provides a clean probe of the mass) and as observed in X-rays. The Bullet Cluster, shown in Fig. 2,34 is in fact the merger of two galaxy clusters. The X-ray image highlights the intracluster gas, which has been shocked and thus heated up as the result of the collision. This intracluster gas provides the dominant contribution to the baryonic mass, the mass from stars being subdominant. If there were no dark matter, the mass of the merged system would be concentrated in the center of the collision and would roughly coincide with the X-ray hot spot. But this is not what is observed in the gravitational lensing reconstruction of the projected mass density, which includes all types of mass whether visible or not. Instead, the

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Fig. 2. Chandra X-ray image of the “Bullet Cluster” (red) superposed with lensing image (blue). In this merger seen in the X-ray, we see the hot baryonic intracluster gas that was shocked at the center of the collision. However in the gravitational lensing reconstruction we see two presumably dark matter halos comprising the majority of the mass which have been able to pass through each other without interaction. (Credit: NASA, see Ref. 34 for a detailed explanation.) (For color version, see page II-CP1.)

lensing reconstruction shows two halos which have hardly been disrupted by the collision apparently simply having gone through each other, as one might expect from a weakly interacting dark matter component. While striking, this special system should not be overinterpreted, because it is difficult to render quantitative this qualitative interpretation. As the case for a dark matter component strengthened, more and more experiments to search for and ultimately characterize the dark matter have come on line. These experiments fall into two classes: direct and indirect. The direct dark matter search experiments look for the recoil of nuclei in the detector due to scattering of nuclei by dark matter particles. Backgrounds, for example from cosmic ray muons or from radioactive decays within or near the detector, obviously constitute a formidable obstacle. Indirect dark matter experiments search for an overproduction (or anomaly) of particles of various kinds from dark matter decays or annihilations. The current status of direct dark matter searches is summarized in Fig. 3.35 Indirect dark matter search experiments or observations may be subject to conflicting interpretations. The data from the ATIC and PAMELA experiments, for example, of the electron and positron fluxes have been interpreted as evidence for dark matter annihilation, although other explanations have been put forth. See the Ref. 36 and references therein for a recent discussion of the methodology and

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Fig. 3. Particle dark matter searches: Current status of constraints on WIMP dark matter from direct detection. Here spin-independent couplings have been assumed. (Note: Figure reprinted with permission from Ref. 35). (For color version, see page II-CP1.)

current status of indirect searches. To date, no confirmed dark matter candidates have been found. However, these experiments have made considerable progress in constraining models and in demonstrating and perfecting experimental techniques. Search experiments for example have already excluded a large part of the cMSSMpreLHC model parameter space. The discovery of flat spiral galaxy rotation curves and the subsequent dark matter search takes us to the early 1990s when the favored cosmological model includes radiation (contributing negligibly to the mean density today), ordinary matter, and a weakly interacting dark matter component with a contribution such that the sum of the components yields a spatially flat universe. Despite the apparent beauty of a spatially flat universe with a vanishing cosmological constant, there were a few wrinkles to this story given the evidence at that time including (i) the inconsistency between the high measured value of the Hubble constant and the ages of the oldest known objects in the universe, (ii) the inconsistency between cluster baryon fraction and nucleosynthesis predictions, and (iii) the inconsistency with the value of Ωmatter inferred from large-scale flows. (See, for example, Refs. 37 and 38 and references therein.) This model was simpler than the six-parameter minimal model presently in vogue and by the end of the 1990s, it became clear that another component was needed to explain the current acceleration of the universe. This could be either a cosmological constant or some other form of stress–energy with a similar equation of state (i.e., a component with a large negative pressure) that subsequently became known as “dark energy.”

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In 1970s, when astrophysical and cosmological observational data accumulated, the improvement in the determination of the age of globular clusters,39,40 the Hubble constant, and the abundance of elements led to tensions in fitting FLRW cosmological models using only visible and dark matter. There were arguments that these measurements might already indicate a nonzero cosmological constant; however, the evidence was not yet compelling (see, e.g. Refs. 41–43). In 1998, when the light curve correction to the intrinsic variability in absolute luminosities of the type Ia supernova had been perfected and a sufficiently large sample of such supernovae at intermediate redshift had been accumulated, it was shown that the expansion of the universe was accelerating. Acceleration of the scale factor was inconsistent with the minimal Ωmatter = 1 cosmological model in vogue at the time and could be explained by a nonzero cosmological constant or other equally radical extension of the then accepted cosmological model. See the chapter SNe Ia as a cosmological probe for a detailed account of this development.31 A key challenge of contemporary observational cosmology is to characterize the nature of the dark energy. If we ignore how the dark energy responds to cosmological perturbations, the problem can be expressed as measuring the run of the dimensionless parameter w = p/ρ through cosmic history, where p is the pressure and ρ is the density of the dark energy. If the dark energy were a cosmological constant, it follows that w = −1 at all times, but a host of other scenarios have been proposed where w is not exactly −1. Observations of the cosmic microwave background are ill-suited to characterizing the nature of the dark energy because the effect of the dark energy on the CMB anisotropies can for the most part be encapsulated into a single number — the angular diameter distance to the surface of last scatter of the CMB photons at z ≈ 1100. What is needed is a survey of the geometry of the universe out to large redshifts. From cosmological observations, it has been inferred that our universe is very close to being spatially flat. In a FLRW universe, the luminosity distance as a function of redshift is given by the integral
z  − 1 −1 dz  Ωm (1 + z  )3 + ΩDE (1 + z  )3(1+w) 2 , dL (z) = (1 + z)(H0 ) 0

where ΩDE is the present dark energy density parameter and the equation of state of the dark energy w is assumed to be constant. For nonconstant w or a nonflat FLRW universe, the expression is similar but slightly more complicated. To determine w(z), luminosity distances and angular diameter distances need to be measured at many redshifts where dark energy plays a role. These can be measured for example by using (i) type Ia supernovae (as standard candles), (ii) baryon acoustic oscillations (BAO) in the matter power spectrum (as standard rulers), (iii) gamma ray bursts (as standards candle, although at present this method remains speculative) and (iv) gravitational waves (GW) from compact binaries and SMBHs (as standard “sirens” or candles). All these methods suffer from dispersion and bias due to gravitational lensing of distant objects, which can substantially alter their apparent luminosities and sizes. GW methods have the potential for high precision,

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but one will likely have to wait about 20 years for a space-based GW observatory (e.g., eLISA44 ) to be put in place (see Chap. 12 for a review on GW mission proposal studies [45]). GW methods are also limited by gravitational lensing. However, since the intrinsic measurement uncertainty can be made very small (better than 0.1% with enough events), lensing uncertainty can be reduced by accumulating enough events for a detailed statistical analysis.46 Assuming ΛCDM, we can investigate whether the parameter values found using the CMB observations are consistent with the parameter choices indicated by completely independent probes such as observations of type Ia supernovae and observations of the scale of the baryonic acoustic oscillations at various redshifts, as shown in Fig. 4. The fact that the three ellipsoids overlap indicates that ΛCDM is telling a consistent story.

Fig. 4. Concordance of CMB, BAO and SNIa observations. The respective error ellipsoids are shown on the Ωm −ΩΛ plane. The supernova data are from the updated Union2.1 compilation of 580 Supernovae. (Note: Figure reprinted with permission from Ref. 47.) (For color version, see page II-CP2.)

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On smaller scales and at later times when nonlinear effects difficult to model are dominant, the success of the ΛCDM is less compelling. This is not because there is a clear contradiction between the predictions and what is observed, but rather because the predictions of the theory are notoriously difficult to calculate because of the presence of nonlinear physics and poorly understood processes such as star formation and galaxy formation for which much of the modeling is at present still largely ad hoc. It has been argued by some that alternative models of gravity are able to account for the observations without positing an additional dark matter component. For example, in Fig. 1, it is argued that MOND theory offers a better explanation of galactic rotation curves. In MOND proposed by Milgrom28 in 1983, Newtonian theory remains valid for large accelerations, but becomes modified for centripetal accelerations smaller than the critical acceleration a0 ∼ 10−10 ms−2 . Numerically, the critical acceleration is related to the cosmological constant Λ : a0 ∼ Λ1/2 in natural units. Gravitational acceleration, of course, is not an invariant in Einstein’s theory, so in order to incorporate MOND into general relativity, some additional field would be required to define a preferred time direction or equivalently a foliation of spacetime into three-dimensional hypersurfaces. Stratified theories of gravity and preferred-frame theories of gravity introduce a preferred foliation. With a preferred foliation and extra degrees of freedom other than the physical metric, the theories still need to satisfy the strong equivalence principle to experimental precision to be viable. This is one reason why stratified theories or preferred-frame theories are not easy to construct in an empirically viable manner, so that they are consistent with experiments and observations on various noncosmological scales and with local experiments. Such a theory was constructed in 197348,49 and also constructed for MOND phenomenology in 2004.31 Nevertheless, both these theories have encountered restrictions as empirical evidence is accumulated. For example, pulse timing observations on the relativistic pulsar–white dwarf binary PSR J1738+ 0333 provide stringent tests of these theories.50 However, GR is covariant, but cosmology is stratified in the large. In studies of the microscopic origin of gravity and quantum gravity, the question arises whether Lorentz invariance is fundamental or derived, especially in the canonical formulation.51 The histories of cosmology and relativity theory over the last 100 years have been closely intertwined. On the one hand, models of the universe rely on general relativity theory for part of their dynamics. On the other hand, the universe provides a testing ground for general relativity where Einstein’s theory can be confronted with observation under the most extreme conditions: on the largest length scales, over the longest time intervals, and at the highest energy scales, close to the putative big bang singularity. We close by identifying four areas where despite the existence of plausible hypotheses, it is likely that the last word has not been said and surprises may be forthcoming: (i) The dark matter acceleration discrepancy problem, (ii) the dark energy problem, (iii) the inflationary epoch, and (iv) a possible quantum gravity phase. It is likely that over the coming 100 years, part of the story told here may be substantially rewritten.

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References 1. A. Einstein, “Kosmologische Betrachtungen zur allgemeinen Relativit¨ atstheorie,” Sitzungsberichte der Preussischen Akademie der Wissenschaften (Berlin) (8 February 1917), p. 142, reprinted in English in H. Lorentz, A. Einstein, H. Minkowski and H. Weyl, The Principle of Relativity: A Collection of Original Memoirs, 1923, Methuen, London. Paperback reprint, Dover, New York. 2. H. Shapley and H. Curtis, “The scale of the Universe,” Bull. Natl. Res. Counc. 2 (1921) 171, https://archive.org/ details/scaleofuniverse00shap. 3. P. J. E. Peebles, The Large-Scale Structure of the Universe (Princeton University Press, Princeton, 1980). 4. P. J. E. Peebles, Principles of Physical Cosmology (Princeton University Press, Princeton, 1993). ¨ 5. A. Friedmann, “Uber die Kr¨ ummung des Raumes,” Z. Phys. A 10 (1922) 377. ¨ 6. A. Friedmann, “Uber die M¨ oglichkeit einer Welt mit konstanter negativer Kr¨ ummung des Raumes,” Z. Phys. A 21 (1924) 326. 7. G. Lemaˆıtre, “Un Univers homog`ene de masse constante et de rayon croissant rendant compte de la vitesse radiale des n´ebuleuses extragalactiques,” Ann. Soc. Sci. Brux. 47 (1927) 49. 8. A. S. Eddington, “On the instability of Einstein’s spherical world,” Mon. Not. R. Astron. Soc. 90 (1930) 668. 9. H. P. Robertson, “Kinematics and world structure,” Astrophys. J. 82 (1935) 284. 10. A. G. Walker, “On Milne’s theory of world structure,” Proc. Lond. Math. Soc. s2–42 (1937) 90. 11. M. Davis, “Cosmic structure”, in One Hundred Years of General Relativity: From Genesis and Empirical Foundations to Gravitational Waves, Cosmology and Quantum Gravity, ed. W.-T. Ni (World Scientific, Singapore, 2015); Int. J. Mod. Phys. D 23 (2014) 1430011. 12. M. Bucher, “Physics of the cosmic microwave background anisotropy”, in One Hundred Years of General Relativity: From Genesis and Empirical Foundations to Gravitational Waves, Cosmology and Quantum Gravity, ed. W.-T. Ni (World Scientific, Singapore, 2015); Int. J. Mod. Phys. D 24 (2015) 1530004. 13. X. Meng, Y. Gao and Z. Han, “SNe Ia as a cosmological probe”, in One Hundred Years of General Relativity: From Genesis and Empirical Foundations to Gravitational Waves, Cosmology and Quantum Gravity, ed. W.-T. Ni (World Scientific, Singapore, 2015); Int. J. Mod. Phys. D 24 (2015) 1530029. 14. T. Futamase, “Gravitational lensing in cosmology”, in One Hundred Years of General Relativity: From Genesis and Empirical Foundations to Gravitational Waves, Cosmology and Quantum Gravity, ed. W.-T. Ni (World Scientific, Singapore, 2015); Int. J. Mod. Phys. D 24 (2015) 1530011. 15. R. H. Dicke, Gravitation and the Universe, Jayne Lectures for 1969 (American Philosophical Society, Philadelphia, 1970). 16. R. H. Dicke and P. J. E. Peebles, “The Big Bang cosmology: Enigmas and nostrums”, in General Relativity: An Einstein Centenary Survey, eds. S. Hawking and W. Israel (Cambridge University Press, Cambridge, 1979). 17. W. Rindler, “Visual horizons in world models,” Mon. Not. R. Astron. Soc. 116 (1956) 663. 18. C. W. Misner, “Mixmaster Universe,” Phys. Rev. Lett. 22 (1969) 1071. 19. K. Sato and J. Yokoyama, “Inflationary cosmology: First 30+ Years,” in One Hundred Years of General Relativity: From Genesis and Empirical Foundations to Gravitational Waves, Cosmology and Quantum Gravity, ed. W.-T. Ni (World Scientific, Singapore, 2015); Int. J. Mod. Phys. D 24 (2015) 1530025.

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20. D. Chernoff and H. Tye, “Inflation, string theory and cosmic strings”, in One Hundred Years of General Relativity: From Genesis and Empirical Foundations to Gravitational Waves, Cosmology and Quantum Gravity, ed. W.-T. Ni (World Scientific, Singapore, 2015); Int. J. Mod. Phys. D 24 (2015) 1530010. 21. P. Moore, The Story of Astronomy, 5th edn. (Grosset & Dunlap Publishing, New York, 1977). 22. I. I. Shapiro, “Solar system tests of GR: Recent results and present plans”, in General Relativity and Gravitation, eds. N. Ashby, D. F. Bartlett and W. Wyss (Cambridge University Press, Cambridge, 1990), p. 313. 23. N. T. Roseveare, Mercury’s Perihelion from Le Verrier to Einstein (Clarendon Press, Oxford, 1982). 24. F. Zwicky, “On the masses of nebulae and clusters of nebulae,” Astrophys. J. 86 (1937) 217. 25. V. Rubin and W. K. Ford Jr., “Rotation of the Andromeda Nebula from a spectroscopic survey of emission regions,” Astrophys. J. 159 (1970) 379. 26. V. C. Rubin, N. Thonnard and W. K. Ford Jr., “Extended rotation curves of highluminosity spiral galaxies. IV — Systematic dynamical properties, Sa through Sc,” Astrophys. J. 225 (1978) L107. 27. V. Rubin, N. Thonnard and W. K. Ford Jr., “Rotational Properties of 21 Sc Galaxies with a Large Range of Luminosities and Radii from NGC 4605 (R = 4 kpc) to UGC 2885 (R = 122 kpc),” Astrophys. J. 238 (1980) 471. 28. M. Milgrom, “A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis,” Astrophys. J. 270 (1983) 365. 29. S. S. McGaugh, “The mass discrepancy–acceleration relation: Disk mass and the dark matter distribution,” Astrophys. J. 609 (2004) 652, arXiv:astro-ph/0403610. 30. S. S. McGaugh, “A tale of two paradigms: The mutual incommensurability of ΛCDM and MOND,” Can. J. Phys. 93 (2015) 250, arXiv:1404.7525. 31. J. D. Bekenstein, “Relativistic gravitation theory for the modified Newtonian dynamics paradigm,” Phys. Rev. D 70 (2004) 083509. 32. Planck Collab. (P. A. R. Ade et al.), “Planck 2013 results. XVI. Cosmological parameters,” Astron. Astrophys. 571 (2014) A16, arXiv:1303.5076. 33. Planck Collab. (P. A. R. Ade et al.), “Planck 2015 results. XIII. Cosmological parameters,” arXiv:1502.01589. 34. M. Markevitch et al., “Direct constraints on the dark matter self-interaction crosssection from the merging galaxy cluster 1E0657-56,” Astrophys. J. 606 (2004) 819, arXiv:astro-ph/0309303. 35. K. A. Olive et al., “Review of Particle Physics,” Chin. Phys. C 38 (2014) 090001, http://pdg.lbl.gov/2014/reviews/rpp2014-rev-dark-matter.pdf. 36. M. Cirelli, “Indirect searches for dark matter: A status review,” Pramana 79 (2012) 1021, astro-ph/1202.1454. 37. M. Turner, “The case for ΛCDM”, in Critical Dialogues in Cosmology, ed. N. Turok (World Scientific, Singapore, 1997). 38. J. P. Ostriker and P. J. Steinhardt, “The observational case for a low-density Universe with a non-zero cosmological constant,” Nature 377 (1995) 600. 39. P. Demarque and R. D. McClure, in The Evolution of Galaxies and Stellar Populations, eds. B. M. Tinsley and R. B. Larson (Yale University Observatory, New Haven, 1977). 40. H. Saio, “Ages of globular clusters. II,” Astrophys. Space Sci. 50 (1977) 93. 41. V. Petrosian, “Confrontation of Lemaˆıtre models and the cosmological constant with observations,” in Confrontation of Cosmological Theories with Observational Data, ed. M. S. Longair (Reidel, Dordrecht, 1974).

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42. J. E. Gunn and B. M. Tinsley, “An accelerating Universe,” Nature 257 (1975) 454. 43. B. M. Tinsley, “Accelerating Universe revisited,” Nature 273 (1978) 208. 44. http://www.esa.int/Our Activities/Space Science/ESA s new vision to study the invisible Universe. 45. W.-T. Ni, “Gravitational wave detection in space,” in One Hundred Years of General Relativity: From Genesis and Empirical Foundations to Gravitational Waves, Cosmology and Quantum Gravity, ed. W.-T. Ni (World Scientific, Singapore, 2015); Int. J. Mod. Phys. D 24 (2016) 1630001. 46. T. D. Saini, S. K. Sethi and V. Sahni, “Possible use of self-calibration to reduce systematic uncertainties in determining distance-redshift relation via gravitational radiation from merging binaries,” Phys. Rev. D 81 (2010) 103009, arXiv:1005.4489. 47. The Supernova Cosmology Project (N. Suzuki et al.), “The Hubble Space Telescope Cluster Supernova Survey: V. Improving the Dark Energy Constraints Above z > 1 and Building an Early-Type-Hosted Supernova Sample,” Astrophys. J. 746 (2012) 85, arXiv:1105.3470. 48. W.-T. Ni, “A new theory of gravity,” Phys. Rev. D 7 (1973) 2880. 49. News and Views, “Preponderance of theories of gravity,” Nature 244 (1973) 537. 50. P. C. C. Freire et al., “The relativistic pulsar–white dwarf binary PSR J1738+0333 – II. The most stringent test of scalar-tensor gravity,” Mon. Not. R. Astron. Soc. 423 (2012) 3328, arXiv:1205.1450. 51. C. Soo and H.-L. Yu, “General Relativity without paradigm of space-time covariance, and resolution of the problem of time,” Prog. Theor. Exp. Phys. 2014 (2014) 013E01, arXiv:1201.3164.

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Chapter 14 Cosmic structure

Marc Davis Astronomy Department, University of California at Berkeley, Berkeley, CA 94720-3411, United States [email protected]

The history of cosmic structure goes back to the time of Einstein’s youth, although few scientists actually thought of the problem of galaxy and cluster formation. The data and ideas were collected slowly as astronomers slowly realized the nature of the problem of large-scale structure. This paper will review several of the key episodes in the history of the field. Starting with the discovery of dark matter in the 30s, the CMBR discovery in the 1960s to the idea of an early episode of inflation in the 1980s, the field has had an acceleration of discovery. In the 80s it was realized that the initial conditions of the universe were specified by the cold dark matter (CDM). Now initial conditions for the formation of structure could be specified for any type of dark matter. With the advent of computing resources, highly nonlinear phases of galaxy formation could be simulated and scientists could ask whether cold dark matter was the correct theory, even on the scale of dwarf spheroidal galaxies, or do the properties of the dwarfs require a different type of dark matter? In an idiosyncratic list, we review several of the key events of the history of cosmic structure, including the first measurements of ξ(r), then the remarkable success of Λ CDM explanations of the large-scale universe. We next turn to velocity fields, the large-scale flow problem, a field which was so promising 20 years ago, and to the baryon acoustic oscillations, a field of remarkable promise today. We review the problem of dwarf galaxies and Lyman-α absorption systems, asking whether the evidence is pointing toward a major switch in our understanding of the nature of dark matter. Finally, we discuss flux anomalies in multiply-lensed systems, which set constraints on the number of dwarf galaxies associated with the lensing galaxy, a topic that is now very interesting since simulations have indicated there should be hundreds of dwarfs orbiting the Milky Way, rather than the 10 that are known. It is quite remarkable that many of the today’s results are dependent on techniques first used by Einstein. Keywords: Cosmology; large scale structure. PACS Number(s): 95.35.+d, 98.52.−b, 98.65.−r, 98.80.−k

1. History of Cosmic Discovery On the birthdate of general relativity, the discovery of the Hubble law,1 that the large luminous nebulae were nearby galaxies receding from the Milky Way with velocity proportional to their distance, was more than 10 years in the future. That the galaxies were clustered was realized in the 1920’s but was first discussed by Shapley and Ames2 upon the completion of their uniform plate survey of the 1000 II-19

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brightest galaxies over the entire sky. The thought that the stars seen in the heavens were but a small fraction of the mass of the universe was not realized until 1933. The idea that the nature of the clustering would tell us anything about the nature of the dark matter was far into the future.3 The existence of dark matter had been known for the last 70 years, although at first few acknowledged it. Zwicky,4 after 10 years of observations and comments on the strange results he was finding for such nearby clusters as Coma, concluded that the velocity dispersion within clusters of galaxies was so high, ∼ 1500 km/s, that they could not exist for longer than 1 billion years unless there was unseen mass holding them together. It was not until the 1980s that reconstruction of the mass profile by data from the Uhuru X-ray satellite observed the Coma cluster of galaxies and reported that its mass profile was consistent with Zwicky’s earlier estimates.5 Since then X-ray observations of clusters show this thermal Bremstrallung emission to always be present in galaxy clusters. Now, strong lensing of background galaxies by rich foreground clusters unambiguously shows the presence of dark matter in the foreground. Gravitational lensing is another tool which shows that the clusters contain enormous amounts of dark matter, just like Zwicky argued 70 years ago. An example is shown in Fig. 1.

Fig. 1. Gravitational lensing of Abell A2218 taken by the HST (Ref. 6). This beautiful image would certainly impress Einstein! The remarkable circular images of faint background galaxies can only be produced if there is 50 times the visible mass within the cluster, exactly as Zwicky had argued. (For color version, see page II-CP2.)

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Evidence for dark matter in individual galaxies began to accumulate with Rubin et al.’s observation of emission line regions in the Andromeda galaxy,7 which they observed to have a flat rotation curve, showing that something exterior to the starlight dominates its mass distribution. The evidence that the rotation curves of many galaxies was flat could not be gathered until image tubes were available for spectrographs,8 and with such a device they found that every galaxy in their study had a flat rotation curve (Fig. 2) rather than the expected ρ(r) ∝ r−1/2 , which immediately tells us the dark matter in galaxies has a mass profile ρ(r) ∝ r−2 , indicating a large amount of dark matter in the outer regions of galaxies. This came as a shock to the cosmological community, but they already suspected the galaxies and clusters to be hiding vast amounts of dark matter.

Fig. 2. Rotation curves of 21 spiral galaxies, spectra taken by Rubin et al. (Ref. 8). Many of the galaxies show remarkably flat rotation, indicative of dark matter in the outer portions of the galaxy.

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The outrageous realization that our universe is dominated by an unseen dark matter has led to alternative theories, chief among them the so-called MOND theory (Modified Newtonian Dynamics).9 MOND modifies Newton’s second law by the addition of a constant, a0 , a rather draconian solution to the dark matter problem. But it is healthy for the field to see alternatives and to see how much of the cosmological phenomenology can be reproduced by a “simple” but radical modification. Of course, this must ultimately be explained by a modified field theory.10 However, later work showed that MOND does not even match the linear theory result seen on larger scales,11 which is a fatal error, and therefore it shall not be covered in this paper. The notion that our universe underwent an inflationary phase12,13 was an incredible motivator of an enormous body of work to flesh out details. Finally, we had an answer to the question of why the universe should be approximately homogeneous and isotropic, and perhaps there would eventually emerge an explanation of the amplitude of the fluctuations. Inflation finally gave us explanations for these and why the density of the universe was close to critical. It furthermore motivated the experimental search for the dark matter. Now, cosmologists realized that the large-scale structure of the universe was telling them the nature of the fluctuation growth at a very early time, and publications exploded. This paper, will be an idiosyncratic review of key aspects of the historical development of cosmic structure. The field is large and life is short. I apologize if I do not not discuss your particular field. 2. Measurement of the Galaxy Correlation Function 2.1. Before 1980 Peebles14 wrote a series of 12 papers in which he applied power spectra analysis to all extragalactic catalogs then in existence. The utility of its transform, the covariance function ξ(r), provided a powerful and discriminating cosmological test, a tool to determine the veracity of a given theory. First, a few definitions given by Peebles which are now the standard. In a catalog listing only the right ascension and declination of galaxies, define solid angle dΩ and catalog density N . From any galaxy in the catalog, the probability that a neighboring cell is occupied is defined as dP = (N dΩ)[1 + ω(θ)], where ω(θ) is the angular correlation function, with θ the angle between the two galaxies. If the galaxies are unclustered then ω(θ) = 0. This of course is sensitive to the distances of galaxies in the catalog. More interesting is the three-dimensional definition of correlations, dP = ndV [1 + ξ(r)], where n is the density and dV is the volume element and ξ(r) is the deviation from uniformity as a function of the distance r between the two galaxies. Of course, the galaxy correlations were presumed to be closely related to the underlying mass correlations, an assumption that has held up well. It was shown later that the galaxy fluctuations δg can be generally represented as δg = bδdm ,

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where δdm are the fluctuations in dark matter and b is the galaxy bias, b ∼ 1.3, although if it is examined using a sufficiently large catalog of galaxies,15 one can see it to be dependent on the galaxy’s mass and color, effects which were known but ignored in Peebles earlier work. Peebles showed that the spatial correlation function, ξ(r), of different catalogs obeyed simple scaling laws, indicating a reproducibility that must be telling us something quite important about the nature of structure formation. An example of the angular correlation scaling Peebles emphasized is shown in Fig. 3, where the angular correlations of the Zwicky catalog is plotted against the Shane–Wirtanen correlations and the Jagellonian field correlations. The nonlinear correlation function has a power law slope of γ = 1.8, that is ξ(r) = (r/r0 )−γ with r0 = 5.4 h−1 Mpc. Once the catalogs are adjusted for their differing depths, the correlation functions are seen to match. What was the shape of the correlation function telling us? 2.2. After 1980 The first redshift survey of galaxies was published in 1982,17 and it opened our eyes, giving cosmologists a new, definitive look at what the universe contained, with enormous filaments of galaxies, huge voids, and rich clusters at the intersections of the filaments. It was clear that the universe was not the result of power-law initial conditions throughout the scales of galaxies and clusters, as had been advocated prior to 1980. Power-law initial conditions were simply a holding pattern in the absence of any better theory, but the birth of the inflationary models12,13 changed everything.

Fig. 3. Left: Angular covariance function for the Jagellonian catalog (squares), Shane–Wirtanen catalog (circles), and Zwicky catalog (triangles). Right: Test of the scaling relationship, where the w values have been scaled to the Zwicky catalog and the abscissas adjusted for different depths (Ref. 16).

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Until the advent of the inflationary model, it was not clear how to interpret the covariance function and the large-scale structure that cosmologists were beginning to see.17 At this time the data was clearly ahead of theory. Detailed calculations of the temperature fluctuations in the cosmic background radiation for universes dominated by massive collisionless relics of the big bang were just becoming available19,20 as well as analytic approximations for the transfer function in the early universe, T (k). A first analytic approximation of the density perturbation spectrum was |δk |2  =

Ak , (1 + αk + βk 3/2 + γk 2 )2

with α = 1.7 Γ, β = 9.0 Γ3/2 , γ = 1.0 Γ2 and Γ = (Ωm h2 ), with Ωm the matter density of the universe and h = H0 /100. The natural expectation was that Γ = 0.5, for Ωm = 1 and h = 0.7. With this power spectrum one could begin to run n-body codes to see if any of this made sense and could match the data.21 After careful calibration of the APM catalog of the full southern sky, Efstathiou and collaborators18 were able to measure the angular correlation function, w(θ), over large enough scales to definitively show that Γ = 0.2, and not Γ = 0.5 as most of the cosmological community believed. Figure 4 is what convinced the community of the problem. Conventionally it had been assumed that the cosmological density parameter Ωm = 1 and that the cosmological constant Λ is zero. But the data

Fig. 4. Data points from the APM catalog (Ref. 18). The heavy black line is the fit for Γ = 0.2 while the light dotted line is for Γ = 0.5 (Ref. 18). Note that it falls well below the data for scales greater than 2 degrees, thus ruling out the standard model at the time. The thin solid and dashed lines show the results of the linear theory for the scale invariant CDM models with h = 1 and 0.75, respectively. This dotted line indicates that something is wrong with the Ωm = 1 CDM model.

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favored a low-density model, either an open cosmology or one dominated by the cosmological constant, with more large-scale power than in the standard Ωm = 1 CDM model. A Λ cold matter model (ΛCDM) was immediately accepted, since an open cosmology, Ωm < 1, ΩΛ = 0, was even more abhorrent.

2.3. Remarkable large-scale structure in simulations Measurement of the detailed large-scale galaxy correlations would have to await the completion of extremely large redshift surveys such as the SDSS and 2dFGRS. Beautiful redshift space maps from both of then, is shown in the top and left portions of Fig. 5. These projects were only possible with teams of astronomers using dedicated telescope time and specialized instrumentation, principally multiple fibers allowing the spectroscopic observation of hundreds of galaxies per exposure, as opposed to one object at a time in the first redshift surveys. Now, the total

Fig. 5. Redshift space slices of the CfA2 stickman and the deeper walls seen with the SDSS and the 2dFGRS, as well as equivalent slices taken from a simulation of a ΛCDM cosmogony (Ref. 22). The galaxies are colored blue, while the simulation points are colored red. (For color version, see page II-CP3.)

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surveys were several hundred thousand galaxies each. The advance of technology made this possible at a reasonable price and telescope time. Computer simulations were essential to understanding the models predicted by inflation. The most powerful computers available for cosmological simulations in the 80’s were made by Digital Equipment Corporation and featured all of 4 MB of memory. The earliest models21 did show that CDM models could produce the remarkable filaments and voids that had recently been discovered. The models were very small, 323 particles, but they were sufficient to make the point. As computing power increased, the models grew with them and cosmological simulations used a substantial fraction of available computing resources. The Millennium Run22 is one of the largest simulations of the formation of structure within the ΛCDM cosmogony so far carried out. It uses 1010 particles to follow the dark matter distribution in a cubic region 500 h−1 Mpc on a side, and has a spatial resolution of 5 h−1 kpc. Equivalent slices from the n-body simulations are shown as red points in Fig. 5. Note how well the Millennium simulation matches the statistics of the 2dFRGS (left and right). While the simulation does not have a rich cluster like Coma in the foreground (top), the statistics are still extremely similar in the bottom images. Note the very long stretch of connected filaments in the simulation, matching those of the observations. Pictures such as this demonstrate that ΛCDM fits the data extremely well and contains a large grain of truth. The observations show the presence of extremely large filaments, and the clusters reside at the filament intersections. Note also the elongation of galaxies pointing toward the observer, but these are redshif t space maps. For example, in the center of the CfA2 “stick man,” the cluster is elongated, pointing to us. This is the Coma cluster, showing the high virialized velocity dispersion of the galaxies in the cluster, just as Zwicky observed 70 years ago. Now with these huge redshift surveys it is possible to measure galaxy correlations to extremely large scales. Furthermore, they have shown that the clustering strength of the galaxies is stronger for those objects of higher luminosity and red color, as expected in detailed models that were becoming available. 2.4. Measurement of the BAO effect Since 2000, the acoustic peaks in the cosmic microwave background anisotropy power spectrum have clearly become one of the strongest cosmological probes.23 They measure the contents and curvature of the universe, demonstrate that the cosmic perturbations are generated early (z ∼ 1100) and are dominantly adiabatic, and by their mere existence largely validate the simple theory used to support their interpretation. The acoustic peaks occur because the cosmological perturbations are sound waves in the fully ionized relativistic plasma of the early universe. The recombination to a neutral gas at redshift z ∼ 1100 abruptly decreases the sound speed and effectively ends the wave propagation of the CMBR and entrained baryons.

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These fluctuations give rise to Doppler peaks of the CMBR, as well as to fluctuations known as the baryon acoustic oscillations (BAO), which have size of the first doppler peak. Because the universe has a significant ratio of baryons to DM, they will be imprinted onto the late-time power spectrum of the nonrelativistic matter. The baryonic perturbations continue to move after recombination, reaching a scale of 150 Mpc, where they begin to attract the dark matter. The resulting peak is perfect for performing the classical angular diameter test to redshift z  1 with galaxy samples from the BOSS survey, or to redshifts z ∼ 3 with absorption lines from the quasar sample of the BOSS survey.24 For example, Fig. 6 is a first measurement of the BAO effect, from an early analysis of 47,000 galaxies from the SDSS. The bump seen in the correlations is weak but it tells a remarkable story. First, recall the definition of angular distances. The comoving sizes, in a universe with Λ and matter (Ωm ), of an object or a feature at redshift z in line-of-sight (r
) and transverse (r⊥ ) directions are related to the observed sizes ∆z and ∆θ by H(z) and DA (z): r
=

c∆z , H(z)

r⊥ = (1 + z)DA (z)∆θ.

The angular distance is given by c DA (z) = 1+z


0

z

dz . H(z)

The Hubble parameter is given by 


H(z) = h Ωm (1 + z) + (1 − Ωm )exp 3 3

0

z

1 + w(z) dz 1+z

1/2

Fig. 6. A first detection of the BAO. The plot of s2 ξ(s) from the SDSS survey (Ref. 23). The magenta line is a pure CDM model without a BAO, and the other lines are different values of Ωm h2 . (For color version, see page II-CP3.)

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Fig. 7. Plot of distance via the BAO technique versus redshift for four separate teams. A ΛCDM model, normalized by data from the Planck satellite, is a perfect fit to the data. This indicates that the equation of state parameter w must be close to constancy, as a cosmological constant model (Ref. 25).

and w, the equation of state of the dark energy is p  w(z) =  . ρ z Since the measurement involves an integral of H(z), it is most informative to measure the BAO as a function of redshift by combining the results of different surveys. An example is given in Fig. 7 showing that the data is exactly consistent with an interpretation of w = −1 as a constant, as with the cosmological constant Λ. The BAO measurements are proving to be an excellent means of constraining w(z). The growth of studies of the BAO is stunning to this old observer. In the CfA1 redshift survey17 of bright (mB < 14.5) galaxies, we were at the cutting edge of technology and galaxies were measured one at a time. The shot noise was too large and the volume too small to allow ξ(r) to be measured for any scale r > 20 Mpc. Now the realistic goal of the observing teams is to measure galaxies ∼ 1000 at a time with a target of 106 galaxies, or more, giving them the ability to measure ξ(r) reliably to scales r ∼ 150 Mpc. Their motivation is to measure the scale of the BAO peak to better than 1%, and then to use this as a “standard ruler” for a precise cosmological test, setting strong constraints on possible models of dark energy, in particular the equation state parameter w. The measurements are going very well and the constraint on w is becoming tighter and tighter. This is a worthwhile goal as deviations from the standard DE model might be very modest. 2.5. Further measurements of the power spectrum Figure 8 is a plot of the observed power spectrum of structure with wavelength spanning four orders of magnitude, where the solid line is ΛCDM. Asymptotically the curve varies as k 1 on large-scale and as k −3 on small-scale. Data ranges from measurements of the cosmic microwave background on large-scale to Lyman-α forest

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Fig. 8. A power spectra summary covering four orders of magnitude from a variety of cosmological studies, ranging from CMB measurements to constraints from Ly α clouds (Refs. 26 and 27). The agreement with the curve, a ΛCDM model, is remarkable. (For color version, see page II-CP4.)

clouds on small scales. A cold dark matter (CDM) particle is nonrelativistic as different scales come within the horizon. Scales on the left-hand side, where the curve is rising, came within their horizon after the universe was dominated by nonrelativistic matter, that is matter dominated, while scales on the right-hand side of the figure came within their horizon while the universe was radiation dominated. If the dark matter is relativistic it will free-stream until the temperature of the CMB drops below its mass, and thus perturbations of that scale are erased. This is why the perturbations of a light particle, e.g. a particle of rest mass less than 1 keV, produce a cut-off in the observable range. If the dark matter is, for example, a warm dark matter candidate (WDM), the power spectrum on small scales, high values of k, will show a cut-off when the temperature of the universe was equal was equal to the rest mass of the particle. The precise slope of the small-scale measurements is sensitive to the masses of the neutrino background particle, which will cause the slope to bend down from the small scale asymptotic slope of −3. It is quite remarkable that the full spectrum of cosmic structure fits the ΛCDM model so well. A tremendous amount of work went into these tests in the last 30 years, and they all set constraints on cosmic structure as defined by ΛCDM. 2.6. Lyman-α clouds Observations of redshifted Lyman α forest absorption at z ∼ 2−4 in the spectra of QSOs provide a sensitive probe of the distribution of gaseous matter in the universe,

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Fig. 9. A simulation of the Lyman-α forest showing the typical spectrum of a high redshift quasar (Ref. 28). The optically thin Ly-α lines are the crossings of the line of sight through the large scale filaments. A representation of large scale structure of the filaments is shown as well. (For color version, see page II-CP4.)

indicating it to be highly ionized. The understanding of the origin of the Lyman-α clouds became clear only after n-body simulations in the 1980s demonstrated the ubiquity of the filamentery clustering throughout the intergalactic medium in a CDM universe. A model of the Lyman-α forest is given in Fig. 9 where it is seen that the quasar line of sight pierces a filament is the site of a cloud. Prior to this realization the community did not have any realistic models for the forest, but this model appears to satisfy all the constraints. The precise measurements of the clustering of Lyman-α forest absorption lines provides further support for cosmological models based on inflation, CDM and vacuum energy, as seen in Fig. 8. The Lyman-α forest, due to its spectral nature, probes the matter power spectrum in velocity space. But the measurement is also sensitive to the mass of the dark matter particle which would generate a small-scale cut-off in the linear power spectrum, with a cut-off that scales approximately inversely with the particle mass and is a result of free streaming of the dark matter. If this mass is of order 1 keV, the cut-off occurs on the scale of dwarf galaxies, and the dark matter particle would be in the class of warm dark matter (WDM).29 If the WDM is a thermal relic of mass 1 keV, the cut-off occurs for k(h/Mpc) > 10, which is slightly beyond the reach of the power spectrum plotted in Fig. 8. However, a further probe is to measure the individual line shapes of Lyman-α clouds, which will be broadened by the thermal velocity, and this can be observed with ground-based spectrographs on large telescopes. By requiring that there be enough small-scale power in the linear power spectrum to reproduce the observed properties of the Lyman-α forest in quasar spectra, Narayanan et al.30 arrive at a lower limit to the mass of the WDM particle of 0.75 keV. They argue that any model that suppresses the CDM linear power spectra

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more severely than a 0.75 keV particle cannot reproduce the Lyman-α forest. Recent progress in this field is reviewed by Viel29 who cites upper limits of 4.5 keV. Improving these observations is important because a cut-off in small-scale power of the dark matter would corroborate the possible cut-off as indicated by dwarf galaxies, discussed below. 3. Large Scale Flows Soon after the discovery of the CMBR, the search began for large-scale inhomogeneities in the CMB, and the dipole anisotropy was detected in 1976 in competition between groups at Princeton and Berkeley.31,32 With the advent of the COBE satellite in 1989 we finally measured the first ∼ 20 spherical harmonics, including a beautiful picture of the dipole anisotropy. The Milky Way and local group of galaxies are moving at 640 km/s, 40◦ away from the Virgo local supercluster, which had been thought to be the attractive center. The question immediately raised by this observation was what was responsible for our motion? Was the flow generated gravitationally, or does it result from some nongravitational effect? Expectation of the CDM models was that the dipole anisotropy should be generated by local perturbations, with essentially nothing generated from great distances, z > 0.15. This would be the same type of motion which is apparent in n-body models, where galaxies and large-scale structure are moving together, simply expressing the local gravitational field. Because the gravity field is the vector sum of all the perturbations on every scale, it makes no sense to break down our motion into components generated by named clusters structures, since they do not dominate our gravitational acceleration. Rather, the gravity is controlled by the total mass distribution which is dominated by the voids and filaments. About all that can be done is to plot the net acceleration of the local group as a function of distance. It was recognized that the measurement of large-scale flow could serve as a check on Ωm , as it is a measure of the gravity on scales even larger than clusters, ∼ 50 Mpc. One presumes that the galaxy distribution, multiplied by a bias factor “b,” is a substitute for the large-scale mass field. It need not be precise, merely tracing the rough outline of the mass distribution. Because of the long decay time from the time, z ≈ 1100, when the gravity and velocity fields were released from the radiation, any random velocity field has decayed, eliminating rotational and other decaying modes. The velocity field present now has an especially simple relationship with the responsible gravitational field. Those components of the velocity field which are not coherent with the density fluctuations have decayed as the universe expands, and so at late times one expects the velocity field to be aligned with the gravity field, at least in the limit of small amplitude fluctuations. In the linear regime, this relation implies a simple proportionality between the gravity field g and the velocity field vg , namely vg ∝ gt where the only possible time t is the Hubble time. The exact expression depends on the

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mean cosmological mass density Ωm and is given by34 vg (r) =

2f (Ωm ) g(r), 3H0 Ωm

where f (Ω) is the linear growth rate. Given complete knowledge of the mass fluctuation field, δρ (r), over all space, the gravity field g(r) is
r − r g(r) = G¯ ρ d3 r δρ (r )  , |r − r|3 where ρ¯ is the mean mass density of the universe. Replacing the integral over space with a sum over the galaxies in a catalog, we have H0 β  1 ri − r H0 β vg (r) = + r, 4π¯ n i φ(ri ) |ri − r|3 3 where n ¯ is the true mean galaxy density in the sample, β ≡ f (Ω)/b with f ≈ Ω0.55 the linear growth factor with Λ included, and with radial selection function φ(r).a The work is greatly aided by supplementing it with secondary distance indicators of as many galaxies as possible. These secondary distance indicators can be for example, galaxies with measured magnitudes and the amplitude of its asymptotic rotation curve, which is sufficient to predict a galaxy’s distance to approximately 20%.35 Of course with the relatively poor distance indicator, it will take many galaxies to produce a meaningful signal, but the samples now include approximately 5000 galaxies drawn from the full sky. The techniques for finding the best flow-field have undergone considerable evolution, a recent flow-field is show in Fig. 10.33 The velocity field is determined from the galaxies having secondary distance indicators by a technique known as the inverse Tully–Fisher method, to which approximately 40 orthogonal modes are fit. The gravity field is determined by a full-scale sample of 45,000 galaxies in the 2MRS catalog, which is useful to a distance of z < 0.04. The figures all show the flow as seen from the local group frame, but in order to see the deviations beyond the dipole, 400 km/s of dipole anisotropy is subtracted from the LG motion of all the figures. The top figure is the distribution of galaxies less than cz < 2000 km/s where the red is outflowing from our perspective and the blue is inflowing, with the size of the symbol being proportional to the velocity. The second row is for galaxies with 2000 < cz < 4000 km/s, the third row is for galaxies with 4000 < cz < 6000 km/s, and the fourth row is for galaxies with 6000 < cz < 10000 km/s. The figures on the right are the gravity field computed from the full sky distribution of galaxies, with the same change of coordinates as for the figures on the left. This gravity field was generated with Ωm = 0.3. Note that the velocity field is not a simple dipole, as the local velocity is responding to all the multipoles of the local gravity. Note also the quite amazing agreement between the curves on the a φ(r)

is defined as the fraction of the luminosity distribution function observable at distance r for a given flux limit.

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Fig. 10. The velocity field of local galaxies (left) and the gravity field (right) derived from the 2MRS sample (Ref. 33). The stars are outflowing from the local group frame of reference and the circles are inflowing. The size of the symbol is proportional to the velocity, with a caption of sizes drawn at the bottom of the figure. To better see the fields, a 400 km/s dipole has been subtracted from each plot. The four rows divide the samples by distance. It is important to note that the two fields were independently derived and show essentially perfect agreement, just as linear theory predicts.

left with those on the right. This tells us that for the most part linear perturbation theory does describe the motion of galaxies in the universe, a reassuring check. It is apparent that the source of the bulk of the 640 km/s dipole pattern seen in the local CMBR has 500 km/s due to the galaxy distribution up to a distance of 100 h−1 Mpc, with the rest generated by larger scale structure, as expected in ΛCDM models. There is no need for exotic explanations of our dipole velocity field.

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The agreement between the velocity field and the gravitational field is particularly reassuring for our understanding of cosmological theory. Such a conclusion is a comfort for linear perturbation theory in an expanding universe and was certainly expected, because the effective smoothing of the velocity and gravity fields has left only linear perturbations, which appears to be adequate for the large scales tested by this method. We see no evidence that the dark matter does not follow the galaxy distribution, and it is consistent with constant bias on large scales. There is no evidence for a nonlinear bias in the local flows. A smooth, unclustered component to the dark matter in the universe can only be tested by the agreement between β measured in the large-scale flows and Ωm measured by the CMBR. To the accuracy of the measurement, the two do agree. Furthermore, the velocity–gravity comparison measures the acceleration on scales up to 30–50 Mpc and since we derived a similar value of β as for clusters of galaxies, we conclude that dark matter appears to fully participate in the clustering on scales of a few megaparsec and larger. It is interesting that the counts of galaxies give the best possible gravity field, reinforcing the old idea that the mass of the halo around a galaxy is not very well correlated with the luminosity of that galaxy. We find no evidence for large-scale flows beyond those indicated by the gravity field. Note that our analysis has not used the CMBR dipole, but at the end we see a velocity field that is consistent with it. We see no evidence that the dipole in the CMBR is produced by anything other than our motion in the universe. 4. Dwarf Galaxies as a Probe of Dark Matter Astronomers did not realize that there was a missing satellite problem for the Milky Way until large simulations showed an enormous number of substructures to every large galaxy. Simulated galaxies retain a large amount of substructure formed by earlier collapses on smaller scales, predicting hundreds or thousands of subhalos in contrast to the ∼ 10 dwarf spheroidal satellites of the Milky Way. N -body simulations of the formation of dark matter halos robustly predict that approximately 10% of the mass of any galactic dark halo should be in the form of sub-halos with numbers that rise steeply toward lower masses. For the past 10 years, astronomers have been aware of this disagreement between observed dwarf spheroidal galaxies and their numerical simulations of normal galaxies.36 Boylin-Kolchin et al.37 used the aquarius simulations to show that the most massive subhaloes in detailed ΛCDM galaxy-mass simulations of dark matter haloes are grossly inconsistent with the dynamics of the brightest Milky Way dwarf spheroidal galaxies. The best-fitting hosts of the MW dwarf spheroidals in all CDM simulations predict at least 10 haloes with rotation curves double that amount observed in MW dwarfs. Another question is what has happened to the missing 90% of condensed halo’s with the mass of MW dwarfs? The dwarf spheroidals have lower dispersion and therefore lower central density or lower mass than predicted for the most massive satellites37 as shown in Fig. 11.

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Fig. 11. Observed Vcirc values of the nine bright, LV > 105 L
, dwarf spheroidals in the MW along with rotation curves corresponding to dwarfs with Vmax = (12, 18, 24, 40) km/s (Ref. 37). The absence of any dwarfs in the Milky Way with potential of ∼ 40 km/s is a serious problem for the CDM cosmogeny as there are far more than 10 dwarfs in the simulations with potential this deep.

These dwarfs, and the absence of the abundant sub-halos, do not seem to match the expectations of ΛCDM, and this is the first prediction of the theory that fails to match the observations. However, the dark matter predictions hold only if the velocity dispersion of the dark matter particle is very low, which would be the case if the dark matter mass is sufficiently high,  100 keV. We know so little about the nature of the dark matter. Is this a clue to its mass or its history? All we know is that the DM is collisionless and has a rest mass of at least 1 keV, with guesses that the DM is the supersymmetric partner to an ordinary particle with a mass in the range 10–100 GeV. After 30 years of effort, we still have no idea if dark matter is a supersymmetric particle with mass greater than a baryon. The missing satellite problem seems like it could be solved fairly easily by baryonic physics, which is not included in the simulations above. For example, the velocity threshold at which subhalo satellites and dwarf satellite counts diverge is close to 30 km/s, a value that is reached by the heating of the ultraviolet photoionizing background which suppresses gas accretion in small halos with low velocity dispersion. Perhaps these halos are dark matter only, and the baryons are simply too hot to condense?

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Even if the baryons condense, the difficulty of understanding galaxies is complex because the gas will form stars and supernovae, which can reheat the gas, causing it to outflow, possibly changing the orbits of the dark matter particles so as to reduce the DM central density.38 Whether this is possible for the very low mass, ∼ 107 M dwarfs, is not clear. Analytical models39 suggest that with so few stars there is not enough energy in supernovae alone to create sufficiently large dark matter cores. Alternatively, supernova and stellar winds from the first generation of stars could drive remaining gas out of the shallow potential wells of these low mass halos making them impossible to see. But it has been known for some time that physics including more than gravitation was essential to the evolution of the baryons if the simulations are to make galaxies with observed properties. The addition of supernova feedback and hot winds from early type stars, in addition to heating by AGNs, reduces the density of baryons and DM in the galactic center and are now realized as critical in galaxy formation.40 Thus, it is necessary to understand these additional heating mechanisms very well before concluding that there is a problem with standard CDM. The situation is at present controversial, with different simulators producing different results.41 Dark matter halos offer a unique method of determining the mass of DM. The transfer function of fluctuations on galaxy sizes varies as k −3 (k is the wave number), provided the DM particle is collisionless and effectively at rest, with small free streaming length compared to the size of the fluctuation. WDM is a simple modification of CDM in which the dark matter particles have initial velocities due either to their having decoupled as thermal relics or to their having been formed via nonequilibrium decay. The free streaming length of the DM, the distance a dark matter particle will travel in the early universe, erases structure of that mass and smaller. An important check on the rest mass of the particle is the smallest astronomical structure that is dominated by dark matter. If the rest mass of the dark matter is sufficiently high then the dark matter has ever more negligible peculiar velocities, but for rest masses of order 5 keV the particles today have a primordial velocity dispersion of approximately 20 km/s, and this is sufficient to play an important role in the formation of dwarf galaxies. The structure of galaxies in CDM is self-similar because the free streaming length is extremely small so there is no scale defined by the collapsing structure. In WDM, the smallest galaxies would not be self-similar, but their DM would present a scale, the free streaming scale, of order the drift distance a particle would travel in the age of the universe. The WDM cosmogony thus erases sufficiently small perturbations. CDM cannot pick a scale at the present epoch and any DM particle of rest mass greater than ∼ 100 keV therefore is a CDM candidate. Could it be that WDM erased the small-scale perturbations? No CDM-based model of the satellite population of the Milky Way explains this result. The problem lies in the densities: The number density of galaxies in the model is typically set by matching halos to the galaxy counts in the sky and in so

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doing requires the dwarf spheroidals to have DM haloes that are a factor of 5 more massive than is observed. Thus, the problem of the shallow potential wells remains. Rather than erasing small-scale perturbations, as in WDM models, another method of smoothing the core of dwarf galaxies is to have the dark matter be selfinteracting (SIDM).42 This model has been discussed several times in the literature starting with Spergel and Steinhardt.43 If the dark matter scatters it will form a core in the center of galaxies, making the potential well shallower, reducing the velocities of the simulation. The central density is reduced as shown in Fig. 12, bringing the simulations into closer agreement with the data. But the dark matter would need a strongly interacting cross-section which seems unlikely. Rocha et al.42 argue that SIDM with σ/m  0.1 cm2 /g appears capable of reproducing reported core sizes and central densities of dwarfs. (1 cm2 /g = 1 barn/GeV, a nuclear-scale cross-section.) They have also shown that self-interaction of that strength does not force the galaxies to become spherical,44 which previously was an argument against SIDM models. SIDM models usually have assumed the simplest scattering cross-section. The particles are like billiard balls with no velocity dependence in their cross-section. In the absence of a theory of the dark matter, it is best to keep the simulations simplest. However, a few theorists are braver,45,46 who use a nonpower-law velocity dependence of the cross-section which is motivated by a Yukawa-like term. As in the other SIDM models, they find the density profile of dwarfs is unchanged except for the constant density core. These models offer an interesting alternative to the standard cold dark matter models that can reduce the discrepancy between the brightest Milky Way satellites and the dense subhaloes found in CDM simulations. The understanding of the nature of the dwarf spheroidal galaxies is clearly a point of tension with the CDM models, a problem which must be solved, either

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through astrophysics or particle physics. Perhaps some clarification will emerge from detailed high-resolution study of Lyman-α clouds, or from a realistic assessment of the effects of the baryonic component? It is quite remarkable that the dwarf spheroidals might actually set an extremely important particle physics parameter. 5. Gravitational Lensing Einstein knew all about gravitational lensing, and it is appropriate to celebrate the 100th year of GR with a section detailing how gravitational lensing is used today. This section shall discuss three separate projects where gravitational lensing was essential. This is not a complete list but represents the first projects I would desire to discuss with Einstein. Reviews of lensing are given by Blandford and Narayan.47 5.1. Double images In 1979, the astronomical community was greatly surprised when Walsh et al. announced the discovery of a gravitational lens.48 Two images of a quasar at z = 1.4, separated by 5.7 arc seconds, were the first documented case of a gravitational lens. The lens was a cluster of galaxies plus an elliptical galaxy in a cluster at an intermediate redshift. The separation of the lens is twice that expected for most galaxies and the quasars are off-center from the galaxy, suggesting the galaxy cluster is partly responsible for the lensing. Only a few scientists realized the power of gravitational lensing but soon the field exploded. An example of the spectacular use of lensing is that of Markevitch,49 see Fig. 13. Now gravitational lensing has become a wonderful, unexpected tool. 5.2. Bullet cluster The bullet cluster is an example of two clusters that have slammed through each other and in another ∼ 109 years will merge into one, a process that is completely normal in the growth of structure, as shown in the beautiful n-body movies. The baryons emitting the X-rays are colored red in Fig. 13, while the weak gravitationallensing results showing the existence of dark matter is colored blue. Galaxies are mixed with the dark matter, as expected since their cross-section is small, but the collisional hot gas shows a bow shock created when the two clouds collided. The discovery of this remarkable “bullet cluster”49 has shown quite clearly that the dark matter is collisionless while the hot gas behaves like normal baryons. This has emerged as a very illustrative example that is a favorite to discuss in class with students. The problems with the speed of the merging clusters strikes me as a minor issue. A MOND explanation for the bullet cluster is a bit far-fetched. 5.3. Substructure of gravitational lenses Another way to search for the missing dark matter substructure is to monitor the flux of multiply-imaged quasars behind foreground galaxies. If the galaxy is

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Fig. 13. Chandra X-ray image of the “bullet cluster” (red) superposed with lensing image of two clusters (blue) (Ref. 49). (For color version, see page II-CP1.)

a smooth potential, then the relative brightness and positions of the quasars can be predicted, but if there are multiple dwarf galaxies within the foreground, then the microscopic details of the quasars will be anomalous. Kochanek and Dalal50 detail results of seven cases and conclude that lowmass subhalos remain the best explanation of the galaxy’s potential. They conclude that a large fraction (∼ 1%– 2%) of the projected mass at the Einstein radius, ∼ 5 kpc, must be in the form of local substructure. The number of anomalies is far higher than the 10 dwarf ellipticals in the Milky Way. When this result was first announced, the astronomical community was skeptical, but Dalal and Kochanek have brought together data from the radio and the optical which eliminates foreground dust as the explanation. This is strong evidence in favor of the ΛCDM model but the uncertainties of the measurement are quite large because of the small sample size. Analysis of the SDSS sample will find more multiply-lensed objects, but they will have small angular size, making them difficult for detailed followup. The multiply-lensed quasars are giving results that appear consistent with the standard model but the number of suitable candidates are insufficient to give details. A new technique for detecting dwarfs in other galaxies involves ALMA imaging of galaxies that are strongly lensing background galaxies containing dust.51 The lensed dusty galaxies can be found in the data from the CMB experiments at the South Pole. Most galaxies have dust in their nuclear regions of size ∼ 1 kpc, corresponding to an angular scale of 0.2, a size large enough to prevent lensing by a single star in the foreground galaxy. But it will be lensed by dwarf galaxies of order 108 M◦ , which have an Einstein radius of the same angular extent. The dusty nuclear regions

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typically show strong velocity gradients, some indicating fast-rotating disks and others indicating major mergers. The spatially resolved spectroscopy of ALMA thus will permit the enhancement of the spatial resolution. Simulations of the procedure appear promising, and they estimate as many as one dwarf object per target. This is a brilliant technique for utilizing the magnification of gravitational lenses. 6. Conclusion It is remarkable how developed the ΛCDM story has become in a relatively short time, a beautiful tale that we would love to share with Einstein. It is a new spin on his “greatest blunder.” And Einstein would certainly love the uses to which gravitational lensing, his tool of 100 years ago, has been applied. The field has developed in the last century in ways that were unimaginable while Einstein was alive. The search for the dark matter is still outstanding after 30 years of effort, but hopefully it will soon be discovered either in an underground experiment or at CERN’s LHC. The inability to detect the dark matter as regions of phase space are steadily excluded will become a problem for the WIMP explanation of dark matter unless the next generation of experiments has a positive detection. The possible hints of problems with the ΛCDM model on the scale of dwarfs are extremely puzzling and will hopefully be resolved in a few years. Detecting fainter dwarfs is certainly worthwhile. The prospect of using ALMA to observe strong galactic lenses that are magnifying background dusty galaxies is an excellent idea and we hope the experiment works. Or perhaps the dwarf data is telling us that the notion of self-interacting dark matter should be taken seriously? The next few years will, with the new telescopes under development and further study of the remarkable recent advances in the inflationary early universe, lead to new, tighter cosmological constraints that will have implications for particle physics that I cannot imagine today. References 1. E. Hubble, Contributions from the Mount Wilson Observatory, Vol. 3 (1927), pp. 23– 28. 2. H. Shapley and A. Ames, Harvard College Obs. Bull. 887 (1932) 1–6. 3. A. G. Doroshkevich, Y. Zeldovich and I. D. Novikov, Pis’ma Zh. Ehksp. Teor. Fiz. 8 (1968) 90–95. 4. F. Zwicky, Helvetica Phys. Acta 6 (1933) 110–127. 5. K. I. Tanaka, Y. Fujishima and M. Fujimoto, Publ. Astron. Soc. Jpn. 34 (1982) 147. 6. J. P. Kneib, R. S. Ellis, I. Smail, W. J. Couch and R. M. Sharples, Astrophys. J. 471 (1996) 643. 7. V. C. Rubin and W. K. Ford, Astrophys. J. 159 (1970) 379. 8. V. C. Rubin, W. K. Ford and N. Thonnard, Astrophys. J. 238 (1980) 471–487. 9. M. Milgrom, Astrophys. J. 270 (1983) 365–370. 10. J. D. Bekenstein, Phys. Rev. D 70 (2004) 3509. 11. A. Nusser, Mon. Not. R. Astron. Soc. 331 (2002) 909–916.

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12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46.

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A. H. Guth, Phys. Rev. D 23 (1981) 347–356. A. D. Linde, Phys. Lett. B 100 (1981) 37–40. P. J. E. Peebles, Astrophys. J. 185 (1973) 413–440. I. Zehavi, Z. Zheng, D. H. Weinberg, J. A. Frieman and the SDSS team, Astrophys. J. 630 (2005) 1–27. P. J. E. Peebles, Astrophys. J. 196 (1975) 647–652. M. Davis, J. Huchra, D. W. Latham and J. Tonry, Astrophys. J. 253 (1983) 423. G. Efstathiou, W. J. Sutherland and S. J. Maddox, Nature 348 (1990) 705–707. G. R. Blumenthal and J. R. Primack, in Fourth Workshop on Grand Unification, eds. H. A. Weldon, P. Langacker and P. J. Steinhardt (Birkhauser, Boston, 1983), p. 256. J. R. Bond and G. Efstathiou, Astrophys. J. 285 (1984) L45–L48. M. Davis, G. Efstathiou, C. Frenk and S. D. M. White, Astrophys. J. 292 (1983) 371. G. Lemson and the Virgo Consortium, arXiv:astro-ph/0608019. D. Eisenstein, I. Zehavi, D. W. Hogg, R. Scoccimarro, M. R. Blanton and SDSS team, arXiv:astro-ph:/0501171. D. Eisenstein, F. Beutler, A. S. Bolton, A. Burden and BOSS team, Am. Astron. Soc. 223 (2014) 245–246. L. Anderson, E. Aubourg, S. Bailey and BOSS team, arXiv:astro-ph:/1312.4877v2. M. Tegmark and M. Zaldarriaga, arXiv:astro-ph/0207047. R. Croft, D. Weinberg, M. Bolte, S. Burles, L. Hernquist, N. Katz, D. Kirkman and D. Tytler (2002), arXiv:astro-ph/0012324. C. Stark, Private communication (2014). M. Viel, G. Becker, J. Bolton and M. Haehnelt, arXiv:1306.2314 [astro-ph.CO]. V. K. Narayanan, D. N. Spergel, R. Dave and C. P. Ma, Astrophys. J. 543 (2000) L103–L106. B. E. Corey and D. T. Wilkinson, Bull. AAS 8 (1978) 351. G. F. Smoot, M. V. Gorenstein and R. A. Muller, Phys. Rev. Lett. 39 (1978) 898. M. Davis, A. Nusser, K. L. Masters, C. Springob, J. P. Huchra and G. Lemson, Mon. Not. R. Astron. Soc. 413 (2011) 2906. P. J. E. Peebles, The Large-Scale Structure of the Universe (Princeton University Press, Princeton, 1980). C. M. Springob, K. L. Masters, M. P. Haynes, R. Giovanelli and C. Marinoni, Astrophys. J. Suppl. 172 (2007) 599. P. Bode, J. P. Ostriker and N. Turok, Astrophys. J. 556 (2001) 93–107. M. Boylan-Kolchin, J. S. Bullock and M. Kaplinghat, Mon. Not. R. Astron. Soc. 422 (2012) 1203–1218. D. H. Weinberg, J. S. Bullock, F. Governato, R. K. de Naray and A. H. G. Peter, arXiv:1306.0913 [astro-ph.CO]. J. Penarrubia, A. Pontzen, M. G. Walker and S. E. Koposov, Astrophys. J. Lett. 759 (2012) L42. A. Pontzen and F. Governato, Nature 506 (2014) 171–178. A. Brooks, Annalen der Physik (2014) arXiv: 1407.7544. M. Rocha, A. Peter, J. Bullock, M. Kaplinghat, S. Garrison-Kimmel, J. Onorbe and L. Moustakas, Mon. Not. R. Astron. Soc. 430 (2013) 81. D. N. Spergel and P. J. Steinhardt, Phys. Rev. Lett. 84 (2000) 3760. A. H. G. Peter, M. Rocha, J. S. Bullock and M. Kaplinghat, Mon. Not. R. Astron. Soc. 430 (2013) 105–120. A. Loeb and N. Weiner, arXiv:1011.6374 [astro-ph.CO]. M. Vogelsberger, J. Zavala and A. Loeb, Mon. Not. R. Astron. Soc. 423 (2012) 3740– 3752.

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R. D. Blandford and R. Narayan, Ann. Rev. Astron. Astrophys. 30 (1992) 311–358. D. Walsh, R. F. Carswell and R. J. Weymann, Nature 279 (1979) 381–384. M. Markevitch et al., Astrophys. J. 606 (2004) 819. C. S. Kochanek and N. Dalal, Astrophys. J. 610 (2004) 69–79. Y. Hezaveh, N. Dalal, G. Holder, M. Kuhlen, D. Marrone, N. Murray and J. Vieira, Astrophys. J. 767 (2013) 9–18.

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Chapter 15 Physics of the cosmic microwave background anisotropy

Martin Bucher Laboratoire APC, Universit´ e Paris 7/CNRS, Bˆ atiment Condorcet, Case 7020, 75205 Paris Cedex 13, France Astrophysics and Cosmology Research Unit and School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4041, South Africa [email protected]

Observations of the cosmic microwave background (CMB), especially of its frequency spectrum and its anisotropies, both in temperature and in polarization, have played a key role in the development of modern cosmology and of our understanding of the very early universe. We review the underlying physics of the CMB and how the primordial temperature and polarization anisotropies were imprinted. Possibilities for distinguishing competing cosmological models are emphasized. The current status of CMB experiments and experimental techniques with an emphasis toward future observations, particularly in polarization, is reviewed. The physics of foreground emissions, especially of polarized dust, is discussed in detail, since this area is likely to become crucial for measurements of the B modes of the CMB polarization at ever greater sensitivity.

1. Observing the Microwave Sky: A Short History and Observational Overview In their 1965 landmark paper A. Penzias and R. Wilson,153 who were investigating the origin of radio interference at what at the time were considered high frequencies, reported a 3.5 K signal from the sky at 4 GHz that was “isotropic, unpolarized, and free from seasonal variations” within the limits of their observations. Their apparatus was a 20 foot horn directed at the zenith coupled to a maser amplifier and a radiometer (see Fig. 1). The maser amplifier and radiometer were switched between the sky and a liquid helium cooled reference source used for comparison. Alternative explanations such as ground pick-up from the sidelobes of their antenna were ruled out in their analysis, and they noted that known radio sources would contribute negligibly at this frequency because their apparent temperature falls rapidly with frequency.

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Fig. 1. Horn antenna used in 1964 by Penzias and Wilson to discover the CMB. (Credit: NASA image)

In a companion paper published in the same issue of the Astrophysical Journal, Dicke et al.46 proposed the explanation that the isotropic sky signal seen by Penzias and Wilson was in fact emanating from a hot big bang, as had been suggested in the 1948 paper of Alpher et al.6 suggesting the presence of the photon blackbody component having a temperature of approximately a few K. Their prediction was based on considering the conditions required for successful nucleosynthesis in an expanding universe — that is, to create an appreciable fraction of primordial helium from the neutrons that are decaying as the universe is expanding. The importance of this discovery became almost immediately apparent, and others set out to better characterize this excess emission, which later would become known as the Cosmic Microwave Background (CMB), or sometimes in the older literature the Cosmic Microwave Background Radiation (CMBR). The two principal questions were: (i) To what extent is this background isotropic? (ii) How close is the spectrum to a perfect Planckian blackbody spectrum? The main obstacle to answering these questions was the lack of adequate instrumentation, and this was the main reason why the first detection of the CMB anisotropy had to wait until 1992, when the COBE team announced their observation of a statistically significant anisotropy of primordial origin after the dipole due to our motion with respect to the CMB had been subtracted205 (see Figs. 2 and 3). The COBE satellite, in a low-Earth orbit, carried three instruments: the differential microwave radiometer (DMR),204 the far infrared absolute spectrophotometer (FIRAS),133 and the diffuse infrared background experiment (DIRBE). By today’s standards, the measurement of the cosmic microwave background anisotropy was crude. The angular resolution was low — the width of the beam was 7◦ (FWHM),

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Fig. 2. The microwave sky as seen by the COBE DMR (differential microwave radiometer) instrument. The top panel shows the microwave sky as seen on a linear temperature scale including zero. No anisotropies are visible in this image, because the CMB monopole at 2.725 K dominates. In the middle panel, the monopole component has been subtracted. Apart from some slight contamination from the galaxy near the equator (corresponding to the plane of our Galaxy), one sees only a nearly perfect dipole pattern, owing to our peculiar motion with respect to the rest frame defined by the CMB. In the bottom panel, both the monopole and dipole components have been removed. Except for the galactic contamination around the equator, one sees the cosmic microwave background anisotropy along with some noise. (Credit: NASA/COBE Science Team) (For color version, see page II-CP5.)

and the sky map used for the analysis was smoothed to 10◦ to suppress beam artefacts. The contribution from instrument noise was large by contemporary standards. The COBE noise was 43, 16, and 22 mK Hz−1/2 for the 31, 53, and 90 GHz channels, respectively.21,a Nevertheless, COBE did provide a convincing first detection of the CMB anisotropy, and most importantly established the overall level of the primordial cosmological density perturbations, which played a crucial role in determining the viability of the cosmological models in vogue at that time, which were much more numerous and varied than today. comparison, for WMAP the detector sensitivities were 0.7, 0.7, 0.9, 1.1, and 1.5 mK · Hz−1/2 for the K, Ka, Q, V, and W bands, respectively.98 a By

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Fig. 3. COBE DMR individual frequency maps (with monopole and dipole components removed). Taking data at several frequencies is key to proving the primordial origin of the signal and removing galactic contaminants. Here are the three frequency maps from the COBE DMR observations. (Credit: NASA/COBE Science Team) (For color version, see page II-CP5.)

The COBE detection was followed by an intense effort to characterize the CMB anisotropy at greater sensitivity and on smaller angular scales. There were numerous experiments from the ground at locations where the column density of water in the atmosphere is particularly low, such as Saskatoon, the Atacama Desert in Chile, the South Pole, and the Canary Islands, as well as from stratospheric balloons. Figure 4 shows the state of play about four years after COBE, with two competing theoretical models plotted together with the data points available at that time. In Sec. 6 we shall present the physics of the CMB systematically, but jumping ahead a bit, we give here a few words of explanation for understanding this plot. In most theoretical models, the CMB is generated by an isotropic Gaussian stochastic process, in which case all the available information concerning the underlying theoretical model can be extracted by measuring the angular power spectrum of the CMB anisotropies. Because of isotropy, one may expand the map in spherical harmonics to extract its angular power spectrum, defined as C
T T,obs =

+

1 |aT
m |2 . (2
+ 1) m=−


(1)

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Fig. 4. State of CMB observations in 1998. After the COBE DMR detection at very large angular scales at the so-called Sachs–Wolfe plateau, numerous groups sought to discover the acoustic peaks, or to find their absence. This compilation from 1998 shows the state of play at that time. While an unmistakable rise toward the first acoustic peak is apparent, it is not so apparent what happens toward higher
. Two theories broadly compatible with the data at that time are plotted, one with Ωk = 0 and another with Ωk ≈ 0.7, which predicts a higher amplitude of the primordial perturbations and an acoustic peak shifted to smaller angular scales, as explained in Sec. 7.3. (Credit: Max Tegmark) (For color version, see page II-CP5.)

It is customary to plot the quantity
(
+ 1)C
/(2π), which would be constant for a scale invariant pattern on the sky.b This is the quantity plotted in Fig. 4, where the question posed at the time was whether there is a rise to a first acoustic peak followed by several decaying secondary acoustic peaks, as indicated by the solid theoretical curves. This structure is a prediction of simple inflationary models — or more precisely, of models where only the adiabatic growing mode is excited with an approximately scale invariant primordial spectrum. In this plot, one sees fairly convincing evidence for a rise in the angular power, but the continuation of the curve is unclear. Several experiments contributed to providing the first clear detection of the acoustic oscillations, namely TOCO,134 MAXIMA,77 and Boomerang.15,140 speaking, scale invariance on the sky can be defined only asymptotically in the
→ ∞ limit, because the curvature of the celestial sphere breaks scale invariance. The use of
(
+ 1) rather than
2 here is an historical convention. b Strictly

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Fig. 5. Boomerang observation of the acoustic oscillations. The C
T T power spectrum as measured by Boomerang is shown on the left without a fit to a theoretical model and on the right with the theoretical predictions for a spatially flat cosmological model with an exactly scale invariant primordial power spectrum for the adiabatic growing mode. (Credit: Boomerang Collaboration)

Figure 5 shows the power spectrum as measured by one of these experiments, namely Boomerang, where a series of well defined acoustic oscillations is clearly visible. In the meantime, parallel efforts were underway in Europe and in the United States to prepare for another CMB space mission to follow on COBE at higher sensitivity and angular resolution. The COBE beam, which was 7◦ FWHM, did not use a telescope but rather microwave feed horns pointed directly at the sky. While not allowing for a high angular resolution, the feed horns had the advantage of producing well defined beams with rapidly falling sidelobes. The US NASA WMAP satellite, launched in 2001, delivered its one-year data release in 2003 (including TT and TE),216–234 and its first polarization data (including also EE) in 2006235–238 (see Figs. 6 and 7). WMAP continued taking data for nine years, and released installments of papers based on the five-, seven-, and nine-year data releases, in which the results were further refined benefitting from a longer integration time √ (which nominally would shrink error bars in proportion to 1/ tobs ) as well as improved instrument modeling.239–251 WMAP used horns pointed at a 1.4 m×1.6 m off-axis Gregorian mirror to obtain diffraction limited beams (although the mirror was under-illuminated somewhat to reduce far sidelobes). The 20 detectors were based on coherent amplification using HEMTs (high electron mobility transistors) — the state of the art in coherent low-noise amplification at the time. The coherent detection technology used by WMAP had the advantage that the electronics could be passively cooled. The competing incoherent bolometric detection technology permits superior sensitivity, which operates almost at the quantum noise

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Fig. 6. WMAP single temperature frequency maps. WMAP observed in five frequency bands. (Credit: NASA/WMAP Science Team) (For color version, see page II-CP6.)

limit of the incident photons, but requires cooling to approximately 100 mK. The cryogenic system required was judged to present substantial risk for a space mission at the time, and this was likely one of the reasons why WMAP was selected over one of its competitor missions, which was similar to the European Planck HFI proposal. The successor to WMAP was the European Space Agency (ESA) Planck satellite, which was launched in May 2009.155–181 Planck consisted of two instruments: (i) a low-frequency instrument with three channels (at 30, 44, and 70 GHz) using a coherent detection technology, and (ii) a high-frequency instrument using cryogenically-cooled bolometric detectors observing in six bands (100, 143, 217, 353, 545, and 857 GHz. All bands except the highest two bands were polarization sensitive and had a diffraction limited angular resolution, starting at 33 arcmin for the 30 GHz channel and going down to 5.5 arcmin for the 217 GHz channel. (The top three channels have an angular resolution of approximately 5’.) Planck HFI took data from August 2009 to January 2012, when the coolant for its high-frequency instrument ran out. The Planck Collaboration reported its first results for cosmology in March 2013 based on its temperature anisotropy data. The first results for cosmology using the polarization data collected by Planck were delivered in February 2015.

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Fig. 7. WMAP single frequency polarization maps. The lines (which may be thought of as doubleheaded vectors) show the amplitude and orientation of the linear polarization of the measured p CMB anisotropy in the indicated bands. The amplitude P = Q2 + U 2 is also shown using the indicated color scale. (Credit: NASA/WMAP Science Team) (For color version, see page II-CP7.)

Figure 8 shows the Planck full-sky maps of the intensity of the microwave emission in nine frequency bands ranging from 30 GHz to 857 GHz. Figure 9 shows a cleaned full-sky map where a linear combination of the single band maps has been taken in order to isolate the primordial cosmic microwave background signal. While the fluctuations in the cleaned map in Fig. 9 do not appear to single out any particular direction in the sky and appear consistent with an isotropic Gaussian random process, the maps in Fig. 8 show a clear excess in the galactic plane. These full-sky maps use a Molleweide projection in galactic coordinates, with the galactic center at the center of the projection. Even though the galactic contamination depends largely on the angle to the galactic plane, with a lesser tendency to increase toward the galactic center, considerable structure can be seen in the galactic emission over a broad range of angular scales. The central bands include the least amount of galactic contamination, which visibly is much larger for the lowest and the highest frequencies shown. The CMB dipole amplitude is ∆T = 3.365 ± 0.027 mK and directed toward (l, b) = (264.4◦ ± 0.3◦ , 48.4 ± 0.5◦ ) in galactic coordinates.104 The maps have been processed to remove the 2.725 K CMB monopole component as well as the smaller

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Fig. 8. Planck single frequency temperature maps. The ESA Planck satellite surveyed the sky in nine broad [i.e., (∆ν)/ν ≈ 0.3] frequency bands, centered at 30, 44, 70, 100, 143, 217, 353, 545, and 857 GHz, shown in galactic coordinates. The units are CMB thermodynamic temperature [see Eq. (3)]. The nonlinear scale avoids saturation in regions of high galactic emission. (Credit: ESA/Planck Collaboration) (For color version, see page II-CP8.)

CMB dipole having a peak-to-peak amplitude of approximately 6.73 mK, so that only the spherical harmonic multipoles with
≥ 2 are included. CMB angular power is typically expressed in terms of D
=
(
+ 1)C
/2π, which in the flat sky approximation corresponds to rms power (in µK2 ) per unit logarithmic interval in the spatial frequency. A scale invariant temperature spectrum on the celestial sphere would correspond to
2 C
being constant. For comparison we give here a

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Fig. 9. Planck CMB map obtained from combining the single-frequency maps. A linear combination of the Planck single frequency maps (shown in Fig. 8) is taken. This linear combination is optimized to filter out any unwanted contaminants based on their differing frequency dependence. The success of this procedure to remove the galactic contaminants is evident from the disappearance of excess power along the equator corresponding to the galactic plane. (For more details, see Ref. 161.) (Credit: ESA/Planck Collaboration) (For color version, see page II-CP9.)

few ballpark numbers characterizing the strength of the primordial CMB temperature anisotropy. Figure 10 plots the CMB power spectrum as observed by Planck together with the predictions of the fit to a six-parameter theoretical model, which we will discuss further below. The magnitude at low
, before the rise to the first acoustic, or Doppler, peak at
≈ 220, is D
≈ 103 µK2 , which would correspond to an rms temperature in the neighborhood of 30 µK. The rise to the first acoustic peak increases the power by about a factor of six, and then after approximately five oscillations, damping effects take over, making the oscillations less apparent and causing the spectrum to suffer a quasi-exponential decay. Although the detectors (to the extent that their response is ideal or linear) measure the intensity expressed as the “spectral radiance” or “specific intensity” Iν (having units of erg s−1 cm−2 Hz−1 str−1 ) averaged over a frequency band defined by the detector, it is convenient to re-express these intensities in terms of an effective temperature. There are two types of effective temperature: the Rayleigh– Jeans (R–J) temperature and the thermodynamic temperature. It is important to keep in mind the distinction, especially at high frequencies [in this context ν > ∼ νCMB = h−1 (kB TCMB ) = 57 GHz] where the factor relating the two becomes large. In the R–J limit, where (hν/kB T )  1, the blackbody spectral radiance may be approximated as Iν = Bν (T ) ≈ 2(ν/c)2 (kB T ), as one would obtain classically from the density of states of the radiation field and assigning an energy (kB T ) to each

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Fig. 10. CMB angular power spectrum as measured by Planck. The binned C
T T power spectrum as given in the Planck 2013 release is plotted with error bars that combine uncertainties from cosmic variance (dominant at low
) and instrumental noise (dominant at high
). The solid curve indicates the theoretical predictions of the six-parameter concordance model with the 1σ cosmic variance for the adopted binning scheme. The lower panel shows the residuals. For more details, see Ref. 166. The exquisite fit, seen here up to about
≈ 2200, into the damping tail, has been shown to extend to smaller angular scales by ACT58 and SPT.102 (Credit: ESA/Planck Collaboration)

harmonic oscillator degree of freedom. If we invert, assuming the R–J limit above (regardless of whether this limit is valid), we obtain the following definition for the R–J temperature 1 1  c 2 TR−J (ν) = Iν . (2) 2 kB ν The “thermodynamic” temperature corresponding to a specific intensity Iν at a certain frequency, on the other hand, is obtained by inverting the unapproximated blackbody expression Iν = Bν (T ) = (2hν 3 /c2 )[exp(hν/kB T ) − 1]−1 . For the CMB, where the variations about the average CMB temperature are small (i.e., ∆T /T ≈ 10−5 ), linearized perturbation theory is valid and one can invert this equation, obtaining δTCMB =

(ex − 1)2 δTR−J , x2 ex

(3)

where x = hν/kB T. This factor is approximately one for x < ∼ 1, but rises exponen> tially for x ∼ 1. It should be remembered that for adding optically thin emissions, it is the intensities, and not the thermodynamic temperatures, that add. Moreover, foreground

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emission power laws know nothing about TCMB and thus are naturally expressed using either the specific intensity, or equivalently the R–J temperature. Other important milestones of CMB observation include the DASI discovery of the polarization of the CMB in 2002111 and the observations at small angular scales carried out by the ACT and SPT teams, which at the time of writing constitute the best measurements of the microwave anisotropies on large patches of the sky at high angular resolution, measuring the power spectrum up to
≈ 10,000. The DASI experiment was carried out using several feed horns pointed directly at the sky. Correlations were taken of the signals from pairs of horns, thus measuring the sky signal interferometrically. The 6 m diameter Atacama Cosmology Telescope (ACT) situated in the Atacama desert in Chile58,109 and the 10 m SPT (South Pole Telescope) located at the Amundsen–Scott South Pole Station in Antarctica102,190,194 probe the microwave sky on very small angular scales, beyond the angular resolution of Planck. CMB observations are almost always diffraction limited, so that the angular resolution is inversely proportional to the telescope diameter. Ground based instruments, despite all their handicaps (i.e., atmospheric interference, lack of stability over long timescales, ground pickup from far sidelobes) will always outperform space based experiments in angular resolution. There are dozens of other suborbital experiments not mentioned here but which paved the road for contemporary CMB observation. These experiments were important not only because of their observations, but also because of their role in technology development and in the development of new data analysis techniques. More information on these experiments can be found in other earlier reviews.c 2. Brief Thermal History of the Universe The big bang model of the universe is an unfinished story. Certain aspects of the big bang model are well established observationally, while other aspects are more provisional and represent our present best bet speculation. In the account below, we endeavor to distinguish what is relatively certain and what is more speculative. If we assume: (i) the correctness of general relativity, (ii) that the universe is homogeneous and isotropic on large scales (at least up to the size of that part of the universe presently observable to us), and (iii) that for calculating the behavior of the universe, small-scale anisotropies can be averaged over, we obtain the Friedmann–Lamaˆıtre–Robertson–Walker (FLRW) solutions to the Einstein field equations, which we now describe. The metric for this family of solutions takes the following form: ds2 = −dt2 + a2 (t)γij dxi dxj ,

(4)

c See http://lambda.gsfc.nasa.gov/links/experimental sites.cfm for an extensive compilation of CMB experiments with links to their websites. The book147 provides an insightful account emphasizing the early history of CMB observations with contributions from many of the major participants.

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where (i, j = 1, 2, 3) and the line element d
2 = γij dxi dxj describes a maximally symmetric three-dimensional space that may be Euclidean (flat), hyperbolic, or spherical.d For the flat case, which is the most important and most often discussed, Eq. (4) takes the form ds2 = −dt2 + a2 (t)[dx21 + dx2 2 + dx3 2 ].

(5)

Here a(t) represents the scale factor of the universe. Under the assumption of homogeneity and isotropy, the stress–energy tensor (expressed as a mixed tensor with one contravariant and one covariant index) must take the form   ρ(t) 0 0 0  0 p(t) 0 0  . Tµ ν =  (6)  0 0 p(t) 0  0 0 0 p(t) The Einstein field equations Gµν = Rµν − (1/2)gµν R = (8πG)Tµν may be reduced to the two equations H 2 (t) ≡

a˙ 2 (t) 8πGN k = ρ(t) − 2 2 a (t) 3 a (t)

(7)

and a ¨(t) 4πG =− (ρ + 3p). a(t) 3

(8)

(For more details, see for example the discussion in the books by Weinberg221 or Wald.218 ) The above discussion, which included only general relativity, is incomplete because no details concerning the dynamics of ρ(t) and p(t) have been given. The only constraint imposed by general relativity is stress–energy conservation, which in the most general case is expressed as Tµν ;ν = 0, but in the special case above takes the form dρ(t) = −3H(t)[ρ(t) + p(t)]. dt d More

(9)

generally, the three-dimensional line element ds2 = γij dxi dxj = „

dx1 2 + dx2 2 + dx3 2 „ « «2 , k 1+ (x1 2 + x2 2 + x3 2 ) 4

where k = +1, 0, or −1 corresponds to a spherical, flat (Euclidean), or hyperbolic threedimensional geometry, respectively. Apart from an overall change of scale, these are the only geometries satisfying the hypotheses of spatial homogeneity and isotropy. By substituting tan(χ/2) = r/2 or tanh(χ/2) = r/2 for the cases k = +1 or k = −1, we may obtain the more familiar representations ds2 = dχ2 + sin χ2 dΩ(2) 2 or ds2 = dχ2 + sinh χ2 dΩ(2) 2 , respectively, where dΩ(2) 2 = dθ 2 + sin θ 2 dφ2 .

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The relationship between ρ and p is known as the equation of state, and for the special case of a perfect fluid p depends only on ρ and on no other variables. Two special cases are of particular interest: a nonrelativistic fluid, for which p/ρ = 0, and an ultrarelativistic fluid, for which p/ρ = 1/3. Until the mid-90s it was commonly believed that except at the very beginning of the universe, a two-component fluid consisting of a radiation component and matter component suffices to account for the stress–energy filling the universe. In such a model

ρ(t) = ρm,0

a0 a(t)



3 + ρr,0

a0 a(t)

4 .

(10)

Here the subscript 0 refers to the present time. For the radiation component, there is obviously the contribution from the CMB photons, whose discovery was recounted above in Sec. 1. If we assume that the universe started as a plasma at some very large redshift with a certain baryon number density, most conveniently parameterized as a baryon-to-photon or baryon-to-entropy ratio, we would conclude that there was statistical equilibrium early on between the various species, and from this assumption we can calculate the neutrino density today. It turns out that the neutrinos are colder than the photons because the neutrinos fell out of statistical equilibrium before the electrons and positrons annihilated. The e+ e− annihilation had the effect of heating the photons relative to the neutrinos, because all the entropy that was in the electrons and positrons was dumped into the photons. Consequently, assuming a minimal neutrino sector, we can calculate the number of effective bosonic degrees of freedom due to the cosmic neutrino background, which according to big bang theory must be present but has never been detected. Measuring the number of baryons in the universe today is more difficult, and is the subject of an extensive literature. The realization of a need for some additional nonbaryonic “dark” matter dates back to Fritz Zwicky’s study of the Coma cluster in 1933 and is a long story that we do not have time to go into. Today “cold dark matter” is part of the concordance model, where the name simply means that whatever the details of this extra component may be (the lightest supersymmetric partner of the ordinary particles in the standard electroweak model, the axion, or yet something else), for analyzing structure formation in the universe, we can treat this component as nonrelativistic (cold) particles that can be idealized as interacting only through gravitation. In the mid-1990s it was realized that a model where the stress–energy included only matter and radiation components could not account for the observations. A crucial observation for many researchers was the measurement of the apparent luminosities of Type Ia supernovae as a function of redshift,154,192 although at that time there were already several other discordant observations indicating that the universe with only matter and radiation could not account for the observations.e To e See

for example some of the contributions in Ref. 213 and references therein.

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reconcile the observations with theory, another component having a large negative pressure, such as would arise from a cosmological constant, was required. A cosmological constant makes a contribution to the stress–energy Tµν proportional to the metric tensor gµν meaning that p = −ρ exactly. This relation inserted into the right-hand side of Eq. (9) implies that any expansion (or contraction) does not change the contribution to the density from the cosmological constant. With a cosmological constant, the density of the universe scales in the following manner:



0

3 4 a0 a0 a0 + ρm,0 ρ(t) = ρΛ,0 + ρr,0 . (11) a(t) a(t) a(t) The cosmological constant is most important at late times, when its density rapidly comes to dominate over the contribution from nonrelativistic matter (consisting according to our best current understanding of baryons and a so-called cold dark matter component). It is customary to express the contributions of each component as a fraction of the critical density ρcrit = 3H 2 /8πG, which would equal the total density if the universe were exactly spatially flat. For the ith component, we define Ωi = (ρi /ρcrit). From the present data, we do not know whether the new component (or components) with a large negative pressure is a cosmological constant, although current observations indicate that w = p/ρ for this component must lie somewhere near w = −1. A key question of many of the major initiatives of contemporary observational cosmology is to measure w (and its evolution in time) in order to detect a deviation of the behavior of the dark energy from that predicted by a cosmological constant. The earliest probe of the hot big bang model arises from comparing the primordial light element abundances as inferred from observations to the theory of primordial nucleosynthesis. In a hot big bang scenario, there is only one free parameter the baryon-to-photon ratio ηB , or equivalently, the baryon-to-entropy ratio ηS , which is related to the former by a constant factor. As discussed above, in the hot big bang theory at early times, in the limit as a(t) → 0, we have ρ → ∞, and T → ∞. For calculating the primordial light element abundances, we do not need to extrapolate all the way back to the initial singularity, where these quantities diverge. Rather it suffices to begin the calculation at a temperature sufficiently low so that all the baryon number of the universe is concentrated in nucleons (rather than in free quarks), but high enough so that there are only protons and neutrons rather than bound nuclei consisting of several nucleons, and so that the reaction p + e−
n + νe ,

(12)

(as well as related reactions) proceeds at a rate faster than the expansion rate of the universe H. Under this assumption, the ratio of protons to neutrons is determined

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by an equilibrium condition expressed in terms of chemical potentials µ(p) + µ(e− ) = µ(n) + µ(νe ) + ∆M,

(13)

where ∆M = MN − MP − Me = 939.56 MeV − 938.27 MeV − 511 KeV = 0.78 MeV. (14) At first, when T (∆M ), the protons and neutrons have almost precisely equal abundances, but then, as the universe cools down and expands, the neutrons become less abundant than the protons owing to the neutron’s slightly greater mass. Then the reactions (12) freeze out — that is, their rate becomes small compared to H — and the neutron–proton ratio becomes frozen in. Subsequently protons and neutrons fuse to form deuterium, almost all of which combines to form 4 He, either directly from a pair of deuterons or through first forming 3 He, which subsequently fuses with a free neutron. Almost all the neutrons that do not decay end up in the form 4 He, because of its large binding energy relative to other low A nuclei, but trace amounts of other light elements are also produced. Even though other heavier nuclei have a larger binding energy per nucleon and their production would be favored by equilibrium considerations, the rates for their production are too small, principally because of the large Coulomb barrier that must be overcome. In primordial big bang nucleosynthesis (BBN) the only free parameter is ηB . Since nucleosynthesis takes place far into the epoch of radiation domination, the density as a function of temperature is determined by the equation ρ(T ) = (π 2 /30)Nrel T 4 where Nrel is the effective number of bosonic degrees of freedom. The expansion rate then is important because it determines for example how many neutrons are able to be integrated into nuclei heavier than hydrogen before decaying. Based on the particles known to us that are ultrarelativistic during nucleosynthesis, the photon, electron, positron, neutrinos, and antineutrinos, we think that we know what value Nrel should have. But nucleosynthesis can be used to test for the presence of extra relativistic degrees of freedom. We shall see later that the CMB can also be used to constrain Nrel . For a review with more details about primordial nucleosynthesis, see Ref. 219. 3. Cosmological Perturbation Theory: Describing a Nearly Perfect Universe Using General Relativity The broad brush account of the history of the universe presented in the previous section tells an average story, where spatial homogeneity and isotropy have been assumed. As we shall see, early on, at large z, this story is not far from the truth, especially on large scales. But a universe that is exactly homogeneous and isotropic would be quite unlike our own, in that there would be no clustering of matter, observed very concretely in the form of galaxies, clusters of galaxies, and so forth.

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In this section, we present a brief sketch of the theory of cosmological perturbations. For excellent reviews with a thorough discussion of the theory of cosmological perturbations in the framework of general relativity, see Refs. 103 and 139. For the “scalar” perturbations,f we may write the line element for the metric with its linearized perturbations Φ(x, η) and Ψ(x, η) in the following form: ds2 = a2 (η)[−(1 + 2Φ)dη 2 + (1 − 2Ψ)dx2 ].

(15)

The functions Ψ(x, η) and Φ(x, η) are the Newtonian gravitational potentials. At low velocities Φ(x, η) is most relevant, but Ψ(x, η) can be probed observationally using light or some other type of ultrarelativistic particles. When there are no anisotropic stresses (i.e., all the partial pressures of the stress–energy tensor are equal), Φ = Ψ. For a universe filled with a perfect fluid with an equation of state p = p(ρ), the evolution of the Newtonian gravitational potential Φ(x, η) is governed by the equation (see Ref. 139 for a derivation and more details) Φ + 3H(1 + cs 2 )Φ − cs 2 ∇2 Φ + [2H + (1 + 3cs 2 )H2 ]Φ = 0,

(16)

where cs 2 = dp/dρ, H = a /a, and  = d/dη. Here η is the conformal time, related to the more physical proper time by the relation dt = a(η)dη. If the spatial derivative term (i.e., ∇2 Φ) is neglected, which is an approximation valid on superhorizon scales, then the derived quantity10,83 ζ =Φ+

2 (Φ + H−1 Φ ) 3 (1 + w)

(17)

(where w = p/ρ) is conserved on superhorizon scales. This approximate invariant, which can be related to the spatial curvature of the surfaces of constant density, is invaluable for relating perturbations at the end of inflation (or some other very early noninflationary epoch) to the later times of interest in this chapter. This property is particularly useful given our ignorance of the intervening epoch, the period of “reheating” or “entropy generation,” the details of which are unknown and highly model dependent. To understand the qualitative behavior of the above equation, it is important to focus on the role of the Hubble parameter H(t) = a(t)/a(t), ˙ which at time t defines the dividing line between superhorizon modes [for which |kphys | < ∼ H(t)] and subg H(t)]. This distinction relates to the relevance horizon modes [for which |kphys | > ∼ f In

the theory of cosmological perturbations, the terms “scalar,” “vector,” and “tensor” are given a special meaning. A tensor of any rank that can be constructed using derivatives acting on a scalar function and the Kronecker delta multiplying a scalar function is regarded as a “scalar”. A quantity that is not a “scalar” but can be constructed in the same way from a 3-vector field is regarded as “vector,” and a “tensor” is a second-rank tensor with its “scalar” and “vector” components removed. Under these definitions, the linearized perturbation equations reduce to a block diagonal form in which the “scalar,” “vector,” and “tensor” sectors do not talk to each other. g Here “horizon” means “apparent horizon,” which depends only on the instantaneous expansion rate, unlike the “causal horizon,” which depends on the entire previous expansion history.

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of the spatial gradient terms in the evolution equations for the cosmological perturbations. This same distinction may be re-expressed in terms of the conformal time ˙ η and the co-moving wavenumber k, where H = a /a takes the place of H = a/a, > H(t). Here with superhorizon meaning |k| < H(t) and subhorizon meaning |k| ∼ ∼ dots denote derivatives with respect to proper time whereas primes denote derivatives with respect to conformal time. It turns out that for calculations, working with conformal time and co-moving wavenumbers is much more convenient than working with physical variables. What happens to the perturbations as the universe expands depends crucially on whether the co-moving horizon size is expanding or contracting. During inflation, or more generally when w < −1/3, the co-moving horizon size is contracting. This means that the dynamics of Fourier modes, which initially hardly felt the expansion of the universe, become more and more affected by the expansion of the universe. Terms proportional to H and H 2 in the evolution equations are becoming increasingly relevant whereas the spatial gradient terms are becoming increasingly irrelevant as the mode “exits” the horizon, finally to become completely “frozen in” on superhorizon scales. During inflation a mode starts far within the horizon and crosses the horizon at some moment during inflation. At the end of inflation, all the modes of interest lie frozen in, far outside the horizon. During inflation, w = p/ρ was slightly more positive than −1. Formally inflation ends when w crosses the milestone w = −1/3, which is when the co-moving horizon stops shrinking and starts to expand. This moment roughly corresponds to the onset of “reheating” or “entropy generation,” when the vacuum energy of the inflaton field gets converted into radiation, or equivalently ultrarelativistic particles, which afterward presumably constitute the dominant contribution to the stress–energy Tµ ν , so that w ≈ 1/3. During the radiation epoch following “entropy generation,” modes successively re-enter the horizon again becoming dynamical. The degrees of freedom describing the modes are not the same as previously during inflation. This is at present our best-bet story of what likely happened in the very early universe. In this chapter, we shall be interested in the later history of the universe, when modes are re-entering the horizon. We shall not enter into the details of how the primordial perturbations are generated from quantum fluctuations of the vacuum during inflation, instead referring the reader to the chapter by K. Sato and J. Yokoyama on Inflationary Cosmology for this early part of the story. Nor shall we discuss alternatives to inflation. The original papers in which the scalar perturbations generated from inflation were first calculated include Mukhanov,136–138 Hawking,86 Guth,75 Starobinsky,209 and Bardeen et al.11 For early discussion of the generation of gravity waves during inflation and their subsequent imprint on the CMB, see Refs. 258–260, 3, and 261. A more pedagogical account may be found for example in the books by Liddle and Lyth84 and by Peacock,146 as well as in several review articles, including for example Ref. 126.

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We emphasize the simplest type of cosmological perturbations, known as primordial “adiabatic” perturbations. The fact that the perturbations are “primordial” means that at the late times of interest to us, only the growing adiabatic mode is present, because whatever the amplitude of the decaying mode may initially have been, sufficient time has passed for its amplitude to decay away and thus become irrelevant. The term “adiabatic” deserves some explanation because its customary meaning in thermodynamics does not precisely correspond to how it is used in the context of cosmological perturbations. In a universe with only adiabatic perturbations, at early times all the components initially shared a common equation of state and common velocity field, and moreover on surfaces of constant total density, the partial densities of the components are also spatially constant. This is the state of affairs initially, on superhorizon scales, but subsequently after the modes enter the horizon, the different components can and do separate. Adiabatic modes are the simplest possibility for the primordial cosmological perturbations, but they are not the only possibility. Isocurvature perturbations where the equation of state varies spatially are also possible, as discussed in detail for example in Refs. 22, 28 and 257 references therein. For isocurvature perturbations, the ratios of components vary on a constant density surface, on which the spatial curvature is constant as well. However the ratios between the components are allowed to vary. As the universe expands, these variations in the equation of state eventually also generate curvature perturbations. It might be argued that the isocurvature modes should generically be expected to have been excited with some nonvanishing amplitude in multi-field models of inflation. For recent constraints on isocurvature perturbations from the CMB, see Planck Inflation 2013171 and the extensive list of references therein. 4. Characterizing the Primordial Power Spectrum In the previous section, we defined the dimensionless invariant ζ(x), conserved on superhorizon scales, which can be used to characterize the primordial cosmological perturbations at some convenient moment in the early universe when all the relevant scales of interest lie far outside the horizon. Because this quantity is conserved on superhorizon scales, the precise moment chosen is arbitrary within a certain range. The existence of a quantity conserved on superhorizon scales allows us to make precise predictions despite our nearly complete ignorance of the details of reheating. Assuming a spatially flat universe, we may expand ζ(x) into Fourier modes, so that d3 k ζ(k) exp[ik · x]. (18) ζ(x) = (2π)3 If spatial homogeneity and isotropy are assumed, the correlation function of the Fourier coefficients must take the form ζ(k)ζ(k ) = (2π)3 k −3 P(k)δ 3 (k − k ),

(19)

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which also serves as the definition of the function P(k) known as the primordial power spectrum. The k −3 factor is present so that P(k) is likewise dimensionless. An exactly scale invariant power spectrum corresponds to P(k) ∼ k 0 . The meaning of scale invariance can be understood by writing the mean square fluctuation at a point as the following integral over wavenumber: +∞ d[ln(k)]P(k). (20) ζ 2 (x = 0) = −∞

Changes of scale correspond to rigid translations in ln(k), and the only choice for P(k) that does not single out a particular scale is P(k) = (constant). 5. Recombination, Blackbody Spectrum, and Spectral Distortions It is sometimes said that the CMB is the best blackbody in the universe. This is not true. Observationally the only way to test how close the CMB is from a perfect blackbody is to construct a possibly better artificial blackbody and perform a differential measurement between the emission from the sky and this artificial blackbody as a function of frequency. In our discussion of the CMB, we stressed that theory predicted a frequency spectrum having a blackbody form to a very high accuracy. This indeed was one of the striking predictions of the big bang theory, and the lack of any measurable deviation from the perfect blackbody spectral shape over a broad range of frequencies was probably one of the most important observational facts that led to the demise of the alternative steady state cosmological model. To date, the best measurements of the frequency spectrum of the CMB are still those made by the COBE FIRAS instrument.65,131,132 FIRAS133 made differential measurements comparing the CMB frequency spectrum on the sky to an artificial blackbody. Despite the many years that have gone by since COBE and the importance of measuring the absolute spectrum of the CMB, no better measurement has been carried out because improving on FIRAS would require going to space in order to avoid the spectral imprint of the Earth’s atmosphere. However a space mission concept called PIXIE has been proposed that would essentially redo FIRAS but with over two orders of magnitude better sensitivity and with polarization sensitivity included.105 While differences in temperature between different directions in the sky, particularly on small angular scales, can be measured from the ground, albeit with great difficulty, the same is not possible for measurements of the absolute spectrum. From a theoretical point of view, even in the simplest minimal cosmological model with no new physics, deviations from a perfect blackbody spectrum are predicted that could be measured with improved observations of the absolute spectrum having a sensitivity significantly beyond that of FIRAS. It is often believed that when we look at the CMB, we are probing conditions in the universe at around last scattering — that is, around z ≈ 1100 — and that the CMB photons are

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unaffected by what happened during significantly earlier epochs. This is not true. Although when z > ∼ 1100, photons are frequently scattered by electrons randomizing their direction and causing them to move diffusively, this frequent scattering is inefficient at equilibrating the kinetic temperature of the electrons to the energy spectrum of the photons and vice versa owing to the smallness of the dimensionless parameter Eγ /me , where Eγ is a typical photon energy. This ratio gives the order of magnitude of the fractional energy exchange due to electron recoil for a typical collision. Given that in this limit the approach to a blackbody spectrum is a diffusive process, we would estimate that (Eγ /me )2 collisions correspond to one decay time of the deviation of the photon energy spectrum from the equilibrium spectrum. Plugging in Eγ ≈ kB Tz=1100 ≈ 0.3 eV, we get (Eγ /me )2 ≈ 4 × 1010 ! Thomson scattering, however, cannot equilibrate the photon number density with the available energy density, so with Thomson scattering alone, an initially out of equilibrium photon energy distribution, for example as the result of some sort of energy injection, would generically settle down to a photon phase space distribution having a positive chemical potential. Processes such as Bremsstrahlung and its inverse or double Compton scattering are required to make the chemical potential decay to zero, and these processes freeze out at z ≈ 106 .97 Therefore any energy injection between z ≈ 106 and z ≈ 103 will leave its mark on the absolute spectrum of the CMB photons, either in the form of a so-called µ distortion, or for energy injected at later times, in the form of a more complicated energy dependence. A broad range of interesting early universe science can be explored by searching for deviations from a perfect blackbody spectrum. Some of the spectral distortions are nearly certain to be present, while others are of a more speculative nature. On the one hand, y-distortions in the field (that is, away from galaxy clusters where the y-distortion is localized and particularly large) constitute a nearly certain signal, as do the spectral lines from cosmological recombination. But sources like decaying dark matter31 may or may not be present. Another interesting source of energy injection arise from Silk damping91,201 of the primordial power spectrum on small scales, beyond the range of scales where the primordial power spectrum can be probed by other means. If a space-based instrument is deployed with the requisite sensitivity, a formidable challenge will be to distinguish these signals from each other, and also from the more mundane galactic and extragalactic backgrounds. From this discussion one might conclude, borrowing terminology from stellar astrophysics, that the CMB “photosphere” extends back to around z ≈ 106 . 6. Sachs–Wolfe Formula and More Exact Anisotropy Calculations At early times (z < ∼ 1100) the universe is almost completely ionized and the photons scatter frequently with free charged particles, primarily with electrons because the proton–photon scattering cross-section is suppressed by a factor of (me /mp ). During this period the plasma consisting of baryons, leptons, and photons may be

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regarded as a single tightly-coupled fluid component almost inviscid but with a viscosity relevant on the shortest scales of interest for the calculation of the CMB temperature and polarization anisotropies. Later, as the universe cools the electrons and protons (and other nuclei) combine to form neutral atoms whose size greatly exceeds the thermal wavelength of the photons, rendering the universe almost transparent. Subsequent to this process, known as “recombination” (despite the fact that the universe had never previously been neutral), the photons cease to rescatter and free stream toward us today.149 This last statement is only 90% accurate, because the late-time reionization starting at about z ≈ 6–7 causes about 10% of the CMB photons to be rescattered. The effect of this rescattering on the CMB anisotropies will be discussed in detail in Sec. 7.5. It is during this transitional epoch, situated around z ≈ 1100, that the CMB temperature and polarization anisotropies were imprinted. Thus when we observe the microwave anisotropies on the sky today, we are probing the physical conditions on a sphere approximately 47 billion light years in radius today,h or more precisely the intersection of our past light cone at the surface of constant cosmic time at z ≈ 1100. This sphere is known as the surface of last scatter or the last scattering surface. The transition from a universe that is completely opaque to the blackbody photons, in which the tight-coupling approximation holds for the baryon–lepton plasma because the photon mean free path is negligible, to a universe that is virtually transparent is not instantaneous. This fact implies that the last scattering surface (LSS) is not infinitely thin, rather having a finite width that must be taken into account for calculating the small angle CMB anisotropies, because this profile of finite thickness smears out the small-scale three-dimensional inhomogeneities as they are projected onto the two-dimensional celestial sphere. Figure 11 shows a plot of the ionization history of the universe (not taking into account the late-time reionization at z > ∼ 6 mentioned above). (For a more detailed discussion of the early ionization history of the universe and the physics by which it is determined, see for example Refs. 87, 196, 197, 211 and 30). The first calculation of the CMB temperature anisotropies predicted in a universe with linearized cosmological perturbations was given by Sachs and Wolfe in their classic 1967 paper195 (see also Ref. 150). In their treatment, the LSS surface is idealized as the surface of a three-dimensional sphere — in other words, the transition from tight-coupling to transparency is idealized to be instantaneous. Locally, on this surface the photon–baryon fluid is subject to two kinds of perturbations: (i) perturbations in density δγ−b and (ii) velocity perturbations vb . The former translate into fluctuations in the photon blackbody temperature Tγ , into “intrinsic” temperature fluctuations at the last scattering surface with δTγ /T¯γ and the second translate into a Doppler shift of the CMB temperature. If there were no

h The radius of this sphere in terms of today’s comoving units is larger than the age of the universe (approximately 13.8 Gyr) converted into a distance owing to the expansion of the universe.

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Fig. 11. Ionization fraction as function of redshift with effect of late-time ionizing radiation ignored. Calculated using the code CosmoRec by Jens Chluba (for details see, Ref. 262).

metric perturbations, we would simply have ˆ ˆ δTf (Ω) δTi (Ω) 1 ˆ = − δvγ · Ω, ¯ ¯ c Tf Ti

(21)

ˆ is the unit outward normal on the last scattering surface. But there are also where Ω additional terms due to the metric perturbations, and careful calculations along the perturbed geodesics yield the following modification to the above equation: ηf ˆ ˆ δTf (Ω) δTi (Ω) 1 ˆ = − δv · Ω + Φ + dη(Φ + Ψ ), (22) γ c T¯f T¯i ηi where the metric perturbations are parametrized as in Eq. (15) using conformal Newtonian gauge.i This equation is known as the Sachs–Wolfe formula. Because of the approximations involved, the Sachs–Wolfe formula is not used for accurate calculations (especially on small angular scales). However, it offers a good approximation to the CMB anisotropies on large angular scales and provides invaluable intuition that is lacking in more precise treatments. A few words about gauge dependence,j an issue that renders linearized perturbation theory within the framework of general relativity somewhat messy. While the ˆ T¯f is the same in all coordinate systems (except final observed anisotropy δTf (Ω)/ i The Sachs–Wolfe formula as given here includes only the “scalar” perturbations. The formula can be generalized to include “vector” and “tensor” contributions, for which the only contributions are from the Doppler and ISW (integrated Sachs-Wolfe) terms. j “Gauge dependence” here means invariance under general coordinate transformations, which (consistent with the linear approximation above) may be truncated at linear order.

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for the monopole and dipole terms), the attribution of the total anisotropy among the various terms (i.e., intrinsic temperature fluctuation, Doppler, gravitational redshift, integrated Sachs–Wolfe) depends on the choice of coordinates. We may for example choose coordinates so that the photon temperature serves as the time coordinate, making the first term disappear, or alternatively, we may make the Doppler term disappear by making our co-moving observers move with the local photon rest frame. Students of cosmology are sometimes misled into believing the solution to these ambiguities is to use conformal Newtonian gauge, or equivalently the “gauge invariant” formalism, which is equivalent to transforming to conformal Newtonian gauge. It is believed that this gauge choice is somehow more “physical” because at least to linear order, the gauge conditions lead to a unique choice of gauge. It is certainly nice to have a gauge condition leading to a unique choice of coordinates, but this uniqueness comes at a price. The Newtonian gauge condition is “nonlocal” because of its reliance on the decomposition of hµν into “scalar”, “vector”, and “tensor” components, which requires information extending all the way out to spatial infinity. But no one has ever seen all the way out to spatial infinity. For this reason, the Newtonian potentials Ψ and Φ are unphysical because they cannot be measured. Their determination would require information extending beyond our horizon. We may further simplify Eq. (22) assuming that only the adiabatic growing mode has been excited. Using the fact that δTi and v are not independent of Φ, we obtain the often cited result δT Φ ˆ = − − v · Ω + dη(Φ + Ψ ), (23) T 3 although the −1/3 factor is not quite right because a matter-dominated evolution for a(t) is assumed rather than a more careful treatment taking into account matterradiation equality occurring at around zeq ≈ 3280. (For a discussion see Ref. 224.) Using this result and ignoring the integrated Sachs–Wolfe term, we obtain Φ δT ˆ ˆ (Ω) = − − v · Ω. T 3

(24)

On large angular scales — that is, large compared to the angle subtended by the horizon on the last scattering surface — the first term dominates. The modes labeled by k obey an oscillatory system of coupled ODEs, and at the putative big bang each mode starts with a definite sharp phase corresponding to the part of the cycle where v = 0. It is only at horizon crossing that the phase has evolved sufficiently for the velocity term to contribute appreciably to the CMB anisotropy. Figure 12 shows the calculated form of the CMB temperature power spectrum. For
< ∼ 100, ∆T /T ≈ −Φ/3 provides an adequate explanation for this leftmost plateau of the curve, but at higher
a rise to a first acoustic peak, situated at about
≈ 220, is observed followed by a series of decaying secondary oscillations.

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Fig. 12. Theoretical CMB spectrum. CMB temperature power spectrum predicted for a model with only the adiabatic growing mode excited, as in the standard concordance cosmological model, is shown. The theoretical model here assumes the best-fit cosmological parameters taken from the Planck 2013 Results for Cosmology.166

A more accurate integration of the evolution of the adiabatic mode up to last scattering can be obtained in the fluid approximation. This approximation assumes that the stress–energy content of the universe can be described by a fluid description consisting of two components coupled to each other only by gravity.199 However more accurate calculations must go beyond the fluid approximation using a Boltzmann formalism that includes higher order moments. However, before describing the Boltzmann formalism, capable of describing this intermediate regime for the photons, we note one final shortcoming of the approximation with tight-coupling and instantaneous recombination — namely, the treatment of polarization. We present here an approximate, heuristic treatment in order to provide intuition. The scattering of photons by electrons is polarization dependent, and this effect leads to a polarization of the CMB anisotropy when the fact that recombination does not occur instantaneously is taken into account. The Thomson scattering cross-section of a photon off an electron is given by dσ = dΩ



e2 mc2

2 (ˆ
i ·
ˆf )2 ,

(25)

where
ˆi and
ˆf are the initial and final polarization vectors, respectively. To see qualitatively how the polarization of the CMB comes about, let us for the moment assume that the photons coming from next-to-last scattering are

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Fig. 13. Anisotropy of polarized Thomson scattering and the origin of the CMB polarization. We show two examples of photon scattering from (NLS) (next-to-last scattering) to (LS) (last scattering) and finally to the observer (at O). The three segments in the diagram have been chosen mutually at right angles in order to maximize the effect, and for simplicity we assume that the radiation emanating from (NLS)1 and (NLS)2 is unpolarized. All the radiation that scatters coming from (NLS)1 to (LS) and then towards O is completely linearly polarized in the
ˆ2 direction, because the other polarization is parallel to (LS)O and thus cannot be scattered in that direction. Likewise, the scattered photons emanating from (NLS)2 are completely linearly polarized in the orthogonal direction for the same reason. Consequently if there were only two sources as above, measuring the Stokes parameter Q at O amounts to measuring the difference in intensity between these two sources.

unpolarized and calculate the polarization introduced at last scattering, as indicated in Fig. 13. Measuring photons with one linear polarization selects photons that propagated from next-to-last to last scattering at a small angle to the axis of linear polarization, while photons with the other linear polarization tend to come from a direction from next-to-last to last scattering with a small angle with the other axis. Consequently, measuring the polarization amounts to measuring the temperature quadrupole as seen by the electron of last scattering. If we make the simplifying assumption that the radiation emanating from next-to-last scattering is unpolarized, we obtain the following expression for the Stokes parameters from the linear polarization,



 ρ=ρmax Q ˆ sin2 θ cos φ = d(−exp[−τ (rl.s. + ρ)]) dΩ sin φ ρ=0 U l.s. × T (ρ sin θ cos φ, ρ sin θ sin φ, rl.s. + ρ cos θ).

(26)

Here we assume that the line of sight is along the z direction. The expression gives two of the five components of quadrupole moment of the temperature as seen by the electron at last scattering, and after integration along the line of sight would give the total linear polarization with the polarization at next-to-last scattering neglected.

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7. What Can We Learn From the CMB Temperature and Polarization Anisotropies? The previous section showed how starting from Gaussian isotropic and homogeneous initial conditions on superhorizon scales, the predicted CMB temperature and polarization anisotropies are calculated, describing in detail all the relevant physical processes at play. In this section, we turn to the question of what we can learn about the universe from these observations. We focus on how to exploit the temperature and polarization two-point functions, which under the assumption of Gaussianity would summarize all the available information characterizing the underlying stochastic process. Gaussianity is a hypothesis to be tested using the data, as discussed separately in Sec. 9. 7.1. Character of primordial perturbations: Adiabatic growing mode versus field ordering In the aftermath of the COBE/DMR announcement of the detection of the CMB temperature anisotropy, two paradigms offered competing explanations for the origin of structure in the early universe. On the one hand, there was cosmic inflation, which in its simplest incarnations predicts homogeneous and isotropic Gaussian initial perturbations where only the adiabatic growing mode is excited. On the other hand, there was also another class of models in which the universe is postulated initially to have been perfectly homogeneous and isotropic. However, subsequently a symmetry breaking phase transition takes place, with the order parameter field taking uncorrelated values in causally disconnected regions of spacetime. Then, as the universe expands and the co-moving size of the horizon grows, the field aligns itself over domains of increasingly large co-moving size, generally in a self-similar way. In many of these models, topological defects of varying co-dimension — such as monopoles, cosmic strings, and domain walls — arise after the phase transition, but textures where the spatial gradient energy is more diffusely distributed are also possible.k In these models the contribution of the field ordering sector to the total energy is always subdominant, and cosmological perturbations are generated in a continuous manner extending to the present time. The stress–energy Θµν from this sector sources perturbations in the metric perturbation hµν . These metric perturbations in turn generate perturbations in the dominant components contributing to the stress–energy: the baryons, photons, neutrinos, and cold dark matter. In these models, the cosmological perturbations are not primordial, but rather are continuously generated so that the perturbations on the scale k are generated primarily as the mode k enters the apparent horizon.152 This property implies that the decaying mode as well as the growing mode of the adiabatic perturbations is excited, and this fact has a spectacular effect on the shape of the predicted CMB power k For

the physics of topological defects, see Ref. 184, and for a comprehensive account emphasizing the connection to cosmology, see Ref. 217.

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spectrum. While inflationary models, which excite only the growing mode, predict a series of sharp, well-defined acoustic oscillations, field ordering models predict broad, washed out oscillations or no oscillations at all in the angular CMB power spectrum.4,36,151 Heuristically one can understand this behavior by arguing that the positions of the peaks reflect the phase of the oscillations. Therefore, if both the growing and decaying modes are present in exactly equal proportion, there should be no oscillations in the angular spectrum. However, precise predictions require difficult numerical simulations and depend on the precise model for the topological defect or field ordering sector. Figure 14 shows the shape of the predicted CMB TT power spectra for some field ordering models, to be compared with the predictions for minimal inflation shown in Fig. 12. After COBE, a big question was whether improved degree-scale CMB observations would unveil the first Doppler peak followed by a succession of decaying secondary peaks at smaller angular scales, as predicted by an approximately scale invariant primordial power spectrum for the growing adiabatic mode, or whether some other shape would be observed, perhaps the one predicted by a field ordering model.

Fig. 14. Structure of acoustic oscillations: field ordering predictions. These acoustic oscillation predictions from four field-ordering models are to be contrasted with the sharp, well-defined peaks in Fig. 12 as predicted by inflation. (Reprinted with permission from Ref. 151.) (Credit: Pen, Seljak and Turok)

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We already presented (in Sec. 1) the experimental results for the first observations of the acoustic oscillations and their subsequent precise mapping (see in particular Figs. 4, 5 and 10.) The data clearly favor simple models of inflation producing adiabatic growing mode perturbations with an approximately scale invariant power spectrum and exclude scenarios where topological defects serve as the primary source for the initial density perturbations. However, models including a small admixture of defect induced perturbations cannot be ruled out.174 7.2. Boltzmann hierarchy evolution In the previous section, we derived the Sachs–Wolfe formula, which provides an intuitive understanding of how the CMB anisotropies are imprinted. The Sachs– Wolfe treatment provides an approximate calculation of the CMB temperature anisotropies, correctly capturing their qualitative features. However to calculate predictions accurate at the sub-percent level, as is required for confronting current observations to theoretical models, the transition from the tight-coupling regime to transparency, or the free-streaming regime, must be modeled more realistically. This requires going beyond the fluid approximation, where the photon gas can be described by its first few moments. Modeling this transition faithfully requires a formalism with an infinite number of moments — that is, a formalism in which the fundamental dynamical variable is the photon phase space distribution function ˆ , ˆ
) where ν is the photon frequency, and the unit vectors n ˆ and ˆ
denote f (x, t, ν, n the photon propagation direction and electric field polarization, respectively. The perturbations about a perfect blackbody spectrum are small, so truncating at linear order provides an adequate approximation. Moreover, Thomson scattering is independent of frequency, and thus (to linear order) does not alter the blackbody character of the spectrum. Consequently, a description in terms of a perturbation of the blackbody temperature that depends on spacetime position, propagation direction, and polarization direction can be used in place of the full six-dimensional phase space density, thus reducing the number of dimensions of the phase space by one.l For a calculation correct to second order, a description in terms of a perturbed temperature would not suffice. As before we consider a single “scalar” Fourier mode k of flat three-dimensional space, so that an exp[ik · x] spatial dependence is implied. A complication in describing polarization arises in specifying a basis, which preferably is accomplished ˆ using only scalar quantities. We describe the polarization using the unit vectors n ˆ to define the unit vectors and k ˆ ˆ (ˆ ˆ n · k) ˆ = k−n , θ ˆ−n ˆ ˆ (ˆ

k n · k)

ˆ ˆ×k ˆ= n φ , ˆ

ˆ n × k

(27)

l Although written as a continuum variable, ˆ
is in reality a discrete variable corresponding to the Stokes parameters (I, Q and U ).

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ˆ form an orthonormal basis. which together with photon propagation direction n ˆ as We thus describe the Stokes parameters of the radiation propagating along n follows: ˆ ab + Q(ˆ ˆ θ ˆ⊗θ ˆ−φ ˆ ⊗ φ] ˆ ab . n, ν)Eb (ˆ n, ν) = I(ˆ n, ν; k)δ n, ν; k)[ Ea (ˆ

(28)

For the “scalar” mode, the Stokes parameters U and V vanish in this basis. The Thomson scattering cross-section does not vary with frequency, a property that provides substantial simplification. Suppose that at some initial time the photon distribution function has a blackbody dependence on frequency, with a blackbody temperature suffering only small linearized perturbations about an average temperature, and that these temperature perturbations depend on the spacetime position, propagation direction, and linear polarization direction. The frequency independence of Thomson scattering ensures that a photon distribution function initially having this special form will retain this special form during its subsequent time evolution. Initially, when tight-coupling is an excellent approximation, any deviation in the frequency dependence from a perfect blackbody spectrum decays rapidly. Moreover, any initial polarization is erased by the frequent scattering. Therefore a blackbody form described by a perturbation in the spectral radiance of the form ∂Bν (T ) n, ˆ
, x, t) = T¯γ ∆ab (ˆ n, x, t)ˆ a ˆb δIν (ˆ ∂T

(29)

is well motivated. Here

Bν (T ) =

2hν 3 c2



  −1 hν exp −1 kB T

(30)

is the spectral radiance for a blackbody at temperature T . The unperturbed spectral radiance is Iν,0 (ˆ n, ˆ
, x, t) = Bν (Tγ (t)), where Tγ (t) is the photon temperature at cosmic time t in the unperturbed background expanding FLRW spacetime. (In Sec. 5, however, we discuss some interesting possible caveats to this assumption.) For the “scalar” mode of wavenumber k, we may decompose (scl)

n, x, t) = ∆T ∆ab (ˆ

(scl)

(ˆ n, x, t)δab + ∆P

(ˆ n, x, t)(θˆa θˆb − φˆa φˆb ). (scl)

(31)

(scl)

We now turn to the evolution of the variables ∆T (ˆ n, t) and ∆P (ˆ n, t). It is ˆ to expand convenient, exploiting the symmetry under rotations about k, (scl)

∆T

(ˆ n, t; k) =

∞


(scl) ˆ ·n ˆ ), ∆T,
(t; k)(−i)
(2
+ 1)P
(k


=0 (scl) ∆P (ˆ n, t; k)

=

∞

=0

(32) (scl) ∆P,
(t; k)(−i)
(2


ˆ ·n ˆ ). + 1)P
(k

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(scl)

The evolution of ∆T,
and ∆P,
is governed by the following infinite system of equations127 : (scl)


∆T 0

(scl)


∆T 1

(scl)


∆T 2

(scl)


∆T


(scl)


∆P


= −k∆T 1 + φ , (scl)

v  k (scl) b (scl) (scl) (∆T 0 − 2∆T 2 + ψ) + a(t)ne (t)σT − ∆T 1 , 3 3

 k Π (scl) (scl) (scl) = (2∆T 1 − 3∆T 3 ) + a(t)ne (t)σT + ∆T 2 , 5 10

=

= =

k (scl) (scl) (scl) {
∆T (
−1) − (
+ 1)∆T (
+1) } − a(t)ne (t)σT ∆T
, (2
+ 1) k (scl) (scl) (scl) {
∆P (
−1) − (
+ 1)∆P (
+1) } − a(t)ne (t)σT ∆P
(2
+ 1)

 1 1 + a(t)ne (t)σT Π δ
,0 + δ
,2 , for
≥ 0, 2 5

for
≥ 3,

(33)

where (scl)

(scl)

(scl)

Π = ∆T 0 + ∆P 0 + ∆P 2 .

(34)

These equations are known as the Boltzmann hierarchy for the photon phase space distribution function. In the last line of Eq. (33) δab denotes the Kronecker delta function. For practical numerical calculations, this infinite set of coupled equations must be truncated at some
max . In principle, ignoring questions of computational efficiency, we could truncate the system of equations in (33) at some large
max , sufficiently above the maximum multipole number of interest today, and integrate the coupled system from some sufficiently early initial time to the present time t0 to find the “scalar” temperature and polarization anisotropies today, which would be given by the following integrals over plane wave modes: (scl) (scl) n, t0 ) = d3 k∆T (ˆ n, t0 ; k), ∆T (ˆ (scl) ∆P (ˆ n, t0 )

=

(35) d

3

(scl) k∆P (ˆ n, t0 ; k).

This was the approach used in the COSMICS code,16 one of the first codes providing accurate computations of the CMB angular power spectrum. However in this approach much computational effort is expended at late times when there is virtually no scattering. A computationally simpler but mathematically completely equivalent approach known as the line of sight formalism was introduced in Ref. 200. The crucial idea is to express the anisotropies today (at x = 0 and η = η0 ) in terms of a line of sight

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integral, so that



(S)

∆T (ˆ n) =

(S) ∆P (ˆ n)

dη exp[−τ (η)] exp[ik(η − η0 )]

0

× and

η0







  dτ 1 − ∆T 0 + iµvB + P2 (µ)Π + (φ − ikµψ) dη 2

(36)

 dτ dη exp[−τ (η)] exp[ik(η − η0 )] − (1 − P2 (µ))Π dη

(37)

η0

= 0

(S)

(S)

(S)

where Π = ∆T 2 +∆P 2 +∆P 0 . We may think of exp[−τ (η)](−dτ /dη)dη = V(η)dη as a measure along the line of sight weighted according to where last scattering takes place. We shall call V(η) the visibility function. In Eq. (36), the first term represents the nonpolarized contribution at last scatter and the second term represents the contribution from the integrated Sachs–Wolfe effect. In Eq. (37) for the polarized anisotropy, the only contribution is from last scatter. It is convenient to rewrite the above integrals using integration by parts so that all spatial derivatives along the line of sight and occurrences of µ are eliminated, and in the process we also omit the monopole surface term at the endpoint corresponding to the observer as this contribution is not measurable. After this integration by parts, the integrals above take the form η0 (S) (S) n) = dη exp[ikµ(η0 − η)]ST,P ((η0 − η)ˆ n, η), (38) ∆T,P (ˆ 0

where the scalar source functions are (S)

ST

 v 3Π = exp[−τ (η)](φ + ψ  ) + V ∆T 0 + ψ + b + 2 k 4k

 vb 3Π 3Π + V + 2 + V  2 , k 4k 4k

(39)

3 (V(k 2 Π + Π ) + 2V  Π + V  Π), 4k 2 and the primes denote derivatives with respect to conformal time η. Now that the spatial derivatives have been eliminated, it is no longer necessary to use the plane waves exp[ikµ(η0 − η)] as the eigenfunctions of the Laplacian operator with eigenvalue −k 2 . It is convenient instead to use a spherical wave expansion with j
(k(η0 − η)) η0     (S) (S) ∆
;A = dηj
k(η0 − η) SA (η0 − η)ˆ n, η , (40) (S)

SP

=

0

where A = T, P. The line of sight formulation results in a significant reduction of the computational effort required for computing the high
coefficients for two reasons. First, the Boltzmann hierarchy can be truncated at low
, because only moments up to

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= 2 appear in the integral. Second, when the universe has become transparent, there is no longer any contribution from the Thomson scattering. We express the initial state for the growing adiabatic mode at some early reference time whose exact value must be chosen after the epoch of entropy generation following inflation but before any of the relevant modes re-entered the horizon. We use the variable ζ(x), whose value is conserved on superhorizon scales, to characterize the primordial perturbations in the adiabatic growing mode. We may expand, either using plane wave Fourier modes or a spherical wave expansion, so that (41) ζ(x) = d3 kζ(k) exp[ik · x], or ζ(x) =

+
∞



∞

dkζ
,m (k)j
(kr)Y
m (θ, φ).

(42)

0


=0 m=−


We characterize the homogeneous and isotropic Gaussian statistical ensemble for ζ(x) by means of a power spectrum P(k), which suffices to completely define the statistical process generating ζ(x). We define P(k) according to the following expectation values: ζ(k)ζ(k ) = (2π)3 δ 3 (k − k )P(k),

(43)

ζ
m (k)ζ

m
(k  ) = δ
,

δm,m
δ(k − k  )P(k).

(44)

or equivalently

It follows that (S),T a
m

∞

=



η0

dk

0

0

  (S) dηj
k(η0 − η) ST (η, k)ζ
m (k),

(45)

so that (S),T T C


=

(S),T (S),T [a
m ]∗ a
m

= 0

∞

(S),T

dkP(k)∆


(S),T

(k)∆


(k),

(46)

where (S),T ∆
(k)

= 0

η0

    (S) dηj
k(η0 − η) ST η, k .

(47)

The above approach was first implemented in the publicly available code CMBFAST200 and later in the code CAMB.121 m More details about the computational issues may be found in these papers. m For

a download and information on CAMB, see www.camb.info.

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7.3. Angular diameter distance The radius of the last scattering surface expressed in terms of co-moving units today (for example in light-years or Mpc) is given by the integral xls = H0 −1



1

als

da 1 √ 2 −4 a Ωr a + Ωm a−3 + ΩΛ

(48)

for a universe whose stress–energy content includes radiation, matter, and a cosmological constant. Here the scale factor is given by als = (zls + 1)−1 . For other equations of state the argument of the square root above is modified accordingly. A physical scale d at last scattering is converted to a co-moving scale today x according to x = (zls + 1)d, and under the assumption of a flat spatial geometry, the angle in radians subtended by an arc of length d on the last scattering sphere is θ = (d/dls ). Except for the integrated Sachs–Wolfe effect (discussed below in Sec. 7.4), which is very hard to measure because the S/N is negligible but for the very first multipoles, the projection effect described above is the only way in which late-time physics enters into determining the angular power spectrum. Possible spatial curvature, parameterized through Ωk , the details of a possible quintessence field, or similar new physics that alter the expansion history at late time enters into the CMB power spectra only through a rescaling of the relation between angular scales, on the one hand, and physical scales, on the other, around last scattering, and this rescaling can be encapsulated into a single variable dls . If the family of FLRW solutions is extended to admit spatial manifolds of constant positive or negative curvature, the expression for the angular size must be modified to the following for a negatively curved (hyperbolic) universe 1/2

θ=

Ωk dls 1/2 sinh[Ωk dls ]

d , dls

(49)

or in the case of positive curvature (i.e., a spherical universe) θ=

(−Ωk )1/2 dls d . sin[(−Ωk )1/2 dls ] dls

(50)

In the former case, the additional prefactor giving the contribution for the nonEuclidean character of the geometry has a demagnifying effect and increases the angular diameter distance, whereas for the spherical case the effect is the opposite. In the early 1990s, many theorists working in early universe cosmology fervently believed on grounds of simplicity in a cosmological model with Ωk = 0 (supposedly an inexorable prediction of inflation) and no cosmological constant (regarded as extremely finely tuned and “unnatural”) containing only matter and

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radiation so that Ωr + Ωm = 1 exactly. This belief was maintained despite a number of discordant astronomical observations. Many in the astronomical community, however, rather adopted the attitude “I believe only what I see,” and thus favored some kind of a low-density universe, either a negatively curved universe where Ωk = 1 − Ωm − Ωr or a universe with a nonzero cosmological constant, where ΩΛ = 1 − Ωm − Ωr .n Although it was widely claimed that spatial flatness was a prediction of inflation, the basic idea of how to produce a negatively curved universe within the framework of inflation had already been proposed in 1982 by Gott and Statler.72,73 However, since they had not calculated the perturbations predicted in such a model, it was not clear that such a model would work. In 1995 Bucher et al.26,27 and Yamamoto et al.252 calculated the perturbations for single bubble open inflation, showing them to be consistent with all the observations available at the time. Present constraints from the CMB limit |Ωk − 1| to a few percent at most, the exact number depending on the precise parametric model assumed.166,171 7.4. Integrated Sachs–Wolfe effect The discussion above claimed that the impact of late-time physics on the CMB power spectrum could be completely encapsulated into a single parameter: the angular diameter distance to last scattering dls . This is only approximately true, because the geometric argument above assumed that all the CMB anisotropy can be localized on the last scattering surface and ignored the integrated Sachs–Wolfe term in Eq. (22), which gives the part of the linearized CMB anisotropy imprinted after last scattering. Intuitively, the integrated Sachs–Wolfe term can be understood as follows. The Newtonian gravitational potential blueshifts the CMB photons as they fall into potential wells and similarly redshifts them as they again climb out. If the depth of the potential well does not change with time, the two effects cancel. In this case, there is no integrated Sachs–Wolfe contribution because the integral can be reduced to the sum of two surface terms. But if the depth of the potential well changes with time, in particular if the overall scale of the potential is decaying, the two effects no longer cancel and an integrated Sachs–Wolfe contribution is imprinted. In the linear theory, the gravitational potential at late times may be factorized in the following way: Φ(x, t) = Φ(x)T (t),

(51)

and we may normalize T (t) to one when the universe has become matter dominated, but before the cosmological constant — or some other dark energy sector — has started to alter the expansion history. n For a snapshot of the debates in cosmology in the mid-1990s, see the proceedings of the conference Critical dialogues in cosmology held in 1995 at Princeton University.213

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From the CMB anisotropies alone, it is hard to determine the contribution of the integrated Sachs–Wolfe effect to the total CMB anisotropy on large angular scales. However one can try to isolate its contribution by measuring the cross-correlation of the CMB temperature anisotropy with the large-scale structure using a broad radial window function extending to large redshifts as was first proposed in Ref. 37, whose aim was to find evidence for a nonzero cosmological constant. The evolution equation for the linearized density contrast of a single pressureless matter component is148 3 δ¨m (t) + 2H(t)δ˙m (t) − H 2 (t)Ωm (t)δm (t) = 0, 2

(52)

where the dots denote derivatives with respect to proper time and H(t) = a(t)/a(t). ˙ Here we treat the baryons and the CDM (cold dark matter) as a single component ignoring any relative velocity between them. This is not a bad approximation at late times, particularly on very large scales. For an Einstein–de Sitter universe (for which Ωr is negligible and ΩΛ = 0), which is a good approximation for our universe at intermediate redshifts, because 4 for z < ∼ zeq ≈ 3.4 × 10 radiation is unimportant and after the decoupling of the photons and baryons, it is valid to set the velocity of sound equal to zero as above. Let zmΛ = 1/(1 + amΛ ) be the redshift where the nonrelativistic matter and the putative cosmological constant contribute equally to the mean density of the universe, so that Ωm = amΛ 3 /(1 + amΛ 3 ) and ΩΛ = 1/(1 + amΛ 3 ). For ΩΛ = 0.68, for example, we would have zmΛ = 0.3 and amΛ = 0.78. For the Einstein–de Sitter case, a(t) = t2/3 and the growing and decaying solution to Eq. (52) are t2/3 and t−1 , respectively. (1/a2 )∇2 φ = (3/2)H 2 Ωm δm , and for the growing mode Φ(x) is independent of time, implying that there is no integrated Sachs–Wolfe contribution at late times. However, as the cosmological constant begins to take over and ΩΛ starts to rise toward one, Φ starts to decay, as sketched in Fig. 15. As explained in detail in Ref. 37, the maximum signal-to-noise (S/N) that can be extracted from the ISW correlation, even with an ideal measurement of the large-scale structure anisotropy within our past light cone, is modest because of the “noise” from the primary CMB anisotropies emanating from the last scattering surface. This cosmic variance noise cuts off the available S/N because on small angular scales, the relative contribution from the ISW term plummets. The S/N is centered around
≈ 20 and integrates to about S/N ≈ 6.5 (assuming ΩΛ0 = 0.7). For an early detection of a correlation, see Ref. 25, and for the most recent work in this area see Ref. 168 and references therein.

7.5. Reionization According to our best current understanding of the ionization history of the universe, the early ionization fraction around recombination is accurately modeled assuming homogeneity and using atomic rate and radiative transfer equations. As

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Fig. 15. Decay of the gravitational potential for the late-time integrated Sachs–Wolfe effect. We sketch the time dependence of the gravitational potential for a universe with ΩΛ0 = 0.9, 0.8, 0.7, and 0.6 (from top to bottom on the right) with a = 1 today. The qualitative behavior will be similar for other kinds of dark energy.

a rough estimate, one may assume the reionization fraction predicted by equilibrium thermodynamics given by the Saha equation, in which there is an exponential sensitivity to changes in temperature. After x ≈ 12 , which occurs at about z ≈ 1270, the ionization fraction x(z) rapidly decays to almost zero, except for trace amounts of ionized hydrogen (denoted as HII in the astrophysical literature) and of free electrons, due to the inefficiency of recombination in the presence of trace concentrations. If it were not for inhomogeneities and structure formation, this nearly perfect neutral phase would persist to the present day. However, the initially small primordial inhomogeneities grow and subsequently evolve nonlinearly. The first highly nonlinear, gravitationally collapsed regions give rise to the first generation of stars and quasars, which serve as a source of UV radiation that reionizes the neutral hydrogen almost completely, except in a minority of regions of extremely high gas density, where recombination occurs sufficiently frequently to counteract the ionizing flux of UV photons. The first observational evidence for such nearly complete reionization came from observations of the spectra of quasars at large redshift in the interval 10.4 eV/(1 + zquasar ) < Eγ < 10.4 eV — that is, blueward of the redshifted Lyman α line.74 If the majority of the hydrogen along the line of sight to the quasar were neutral, resonant Lyα (2p → 1s) scattering would deflect the photons in this frequency range out of the line of sight, and this part of the spectrum would be devoid of photons. The fact that the spectrum is not completely blocked in this range indicates that the universe along the line of sight was reionized. Almost complete absorption by neutral hydrogen — that is, the so-called Gunn–Peterson trough — was not observed in

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quasar spectra until 2001, when spectroscopic follow-up of several high-redshift quasars discovered by the Sloan Digital Sky Survey (SDSS) detected such a trough from z ≈ 5.7 to z ≈ 6.3, where the continuum emission blueward of the Lyα appears to have been completely absorbed by neutral hydrogen.14 The big question here is exactly when did the universe first become reionized. Or said another way: When did the first generation of stars and quasars form and become capable of generating enough ionizing radiation to convert the neutral gas to its present ionized state? This is a vast subject of very current research, sometimes described as the exploration of the “Dark Ages” of the universe. While we believe that we know a lot about the conditions around the time when the CMB anisotropies were imprinted, and we also know a lot about the recent universe extending to moderate redshift, little is known with any degree of certainty concerning this intermediate epoch. Good reviews of this subject include Ref. 12. Future observations of the 21 cm hyperfine transition of atomic hydrogen (HI) by the next generation of radio telescopes, in particular the Square Kilometer Array (SKA),o promise to provide three-dimensional maps of the neutral hydrogen in our past light cone. Here we limit ourselves to discussing the impact of reionization on the CMB anisotropies. The most simplistic model — or caricature — of reionization postulates that the ionization fraction x(z) changes instantaneously from x = 0 for z > zreion to x = 1 for z < zreion. Under this assumption, the optical depth for rescattering by the reionized electrons is zreion d
dzσT ne (z) . τ= (53) dz 0 The Thomson scattering optical depth from redshift z to the present day is given by the integral τ (z) =

=

cσT H0



1

a(z)

cσT ne0 H0



da ne (a) −4 a (Ωr a + Ωm a−3 + ΩΛ )1/2 1

a(z)

da xe (a) , a4 (Ωr a−4 + Ωm a−3 + ΩΛ )1/2

(54)

where a(z) = 1/(z+1), σT is the Thomson scattering cross-section, H0 is the present value of the Hubble constant (in units of inverse length), and ne0 is the electron density today, assuming xe = 1. The functions ne (a) and xe (a) are the electron densityp and the ionization fraction at redshift z = 1/a − 1, respectively. o https://www.skatelescope.org/.

is customary to define xe as ne /nB , so that xe slightly exceeds one when the helium is completely ionized as well.

p It

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On small angular scales, the effect of reionization on the predicted CMB temperature and polarization anisotropy spectra is straightforward to calculate. Of the CMB photons emanating from the LSS, a fraction exp(−τ ) does not undergo rescattering by the reionized electrons and their fractional temperature perturbations are preserved. However, for the remaining fraction (1 − exp(−τ )) ≈ τ that is rescattered, the small-angle CMB anisotropies are completely erased. The reionized electrons nearly completely wash out small-scale detail much like a window pane of ground glass. The net effect is that the small-scale CMB anisotropies are attenuated by a factor of exp(−τ ) in amplitude and by a factor of exp(−2τ ) in the power spectrum. On larger angular scales, the attenuation of the anisotropy from the rescattered photon component is incomplete, and a more detailed calculation is needed to determine the detailed form of the
-dependent attenuation factor. The order of magnitude of the angular scale where the attenuation starts to become incomplete is given by the ratio of the co-moving distance travelled from next-tolast to last scattering by a rescattered photon dreion to the radius of the LSS dLS . We thus obtain an angular scale θreion ≈ dreion /dLS for this transition. For determining the cosmological model from the CTT power spectrum alone, the reionization optical depth is highly degenerate with the overall amplitude of the cosmological perturbations A. This is because except on very large scales, where cosmic variance introduces substantial uncertainty, it is the combination A exp(−2τ ) that determines the observed amplitude of the CMB angular power spectrum. The polarization power spectra, meaning C EE and also C TE , help to break this degeneracy by providing a rather model independent measurement of τreion . Recall our order of magnitude estimate for the polarization introduced by the finite thickness of the LSS ∂2T P ∼ d2 2 , (55) ∂x where d and x are expressed in co-moving units and d is the mean co-moving distance that the photon travels from next-to-last scattering to last scattering. The second derivative is that of the temperature anisotropy evaluated at the position of a typical electron of last scatter. Let us compare the polarization imprinted during recombination to that imprinted later for the fraction of photons (1 − exp(−τ )) ≈ τ rescattered by the free electrons from reionization, for the moment assuming that the second derivative factors are comparable for the two cases. However, drec  dreion ; therefore, we may expect that Preion /Prec ≈ τ (drec /dreion )2 1. This polarization from reionization, however, is concentrated almost exclusively in the first few multipoles, because directions separated by a small angle see the same quadrupole anisotropy from last scattering from the vantage point of a typical free electron after reionization. Figure 16 shows the scalar power spectra for four values of τ keeping the other cosmological parameters fixed. We shall see that the same amplification applies to the tensor modes, or primordial gravitational waves, presumably generated during inflation.

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M. Bucher

Fig. 16. CMB power spectra for four values of the reionization optical depth. We plot the indicated power spectra for τ = 0.00, 0.04, 0.08, and 0.12 assuming the same concordance values for the other cosmological parameters. We multiply the spectra by exp(2τ ) in order to remove the trivial dependence on τ at small angular scales and thus highlight what occurs at low
.

We now turn to observations allowing us to fix the value of τ using the low
C spectra and C TE spectra based on the qualitative theoretical arguments above but made more precise by calculating τ within the framework of the previously described six-parameter concordance model. While the first observations of the polarization of the CMB were reported by the DASI experiment,111 it was the WMAP large-angle polarization results, first the TE correlations reported in the one-year release, later followed by the EE correlation results from the third-year results, that provided the first determination of τreion using the CMB, and subsequently refined in later updates to the WMAP results with more integration time. As part of their first-year results,231 the WMAP team reported a value for the reionization optical depth of τ = 0.17 ± 0.04 at 68% confidence level, which would correspond to 11 < zr < 30 at 95% confidence where a step function profile for the ionization fraction is assumed. This determination was based on exploiting only the measurement of the TE cross-correlation power spectrum, because for this first release the analysis of the polarized sky maps was not sufficiently advanced to include a reliable EE auto-correlation power spectrum measurement. The reported value of τ was larger than expected. In their third-year release,237 when WMAP presented a full analysis of the polarization with EE included, the determination of τ shifted downward, namely to τ = 0.10 ± 0.03 (using only EE data) and τ = 0.09 ± 0.03 with (TT, TE and EE all included). The final nine-year WMAP release251 cites a value of τ ≈ 0.089 ± 0.014 using WMAP alone. Including other external data sets results in small shifts about this value. EE

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The Planck 2013 Results for Cosmology161,166,171 release did not include an analysis of the Planck polarization data. Therefore the likelihoods used to determine the cosmological parameters included the WMAP determination of τ as a prior, and in some cases the WMAP low-
polarization likelihood was used in order to break the degeneracy between τ and A described above that arises when only temperature data is used. It has been pointed out that reionization histories, which are completely characterized by function xe (z), cannot be reduced to a single number. Some work has been done to characterize what further information can be extracted from the low-
CMB concerning the reionization history, as described in Refs. 95 and 101, but the conclusion is that from the CMB data alone at most a few nonzero numbers can be extracted at S/N > ∼ 1 given the large cosmic variance at low
. In the above discussion we have assumed that reionization occurs homogeneously in space, an approximation adequate for calculating its effect on the anisotropies at large and intermediate angular scales. However, according to our best understanding, the universe becomes reionized by the formation of bubbles, or Str¨ omgren spheres, surrounding the first sources of ionizing radiation, which grow and become more numerous, and eventually percolate. While this basic picture is highly plausible, the details of modeling exactly how this happens remain speculative. The effect on the CMB is significant only at very small angular scales, far into the damping tail of the primordial anisotropies. Power from the low-
anisotropies is transformed into power at very high
. This can be understood by considering the power in Fourier space resulting from the rather sharp bubble walls, which in projection look almost like jump discontinuities. Some simulations of this effect, along with other effects at large
, can be found for example in Ref. 253. 7.6. What we have not mentioned The preceding subsections provide a sampling of what can be learned from the CMB TT and polarization power spectra with each point discussed in some detail. For lack of space, we cannot cover all the parameters and extensions of the simple six-parameter concordance model that can be explored, and here we provide a few words regarding what we have not covered giving some general references for more details. The structure of the acoustic oscillations and the damping tail (relative heights, positions, etc.) of the CMB power spectra depends on the cosmological parameters, as forecast and explained pedagogically in Refs. 92, 99, and most recently applied to the Planck 2013 data for determining the cosmological parameters.166 Some of the constraints are obtained in the framework of the standard six-parameter concordance model, but many extensions to this model can be constrained, such as theories with varying αQED , nonminimal numbers of neutrinos, varying neutrino masses, isocurvature modes, nonstandard recombination, to name just a few examples. Moreover the question can be addressed whether there exists statistically

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significant evidence in favor of extending the basic six-parameter model to a model having additional parameters. See Refs. 166 and 171 for an extensive set of references. 8. Gravitational Lensing of the CMB The previous sections presented a simplified view of the CMB where the CMB anisotropies are imprinted on the so-called last scattering surface, situated around z ≈ 1100. It is assumed that we look back to this surface along straight line geodesics in the unperturbed coordinate system. This is not a bad approximation, but it is not the whole story. Inhomogeneities in the distribution of matter, particularly at late times when the clustering of matter has become nonlinear, act to curve the photon, or geometric optics, trajectories. These nonuniform deflections distort the appearance of the surface of last scatter, much like a fun house mirror. Gravitational lensing of the CMB is a small effect but large enough so that it has to be taken into account in order to compare models of the primordial universe with the observations correctly.264 In one sense, gravitational lensing may be viewed as a nuisance effect to be removed. But gravitational lensing of the CMB may also be regarded as an invaluable tool for probing the inhomogeneities of the matter distribution between us and the last scattering surface. Gravitational lensing is presently a very active area of astronomy, and the CMB is only one of the many types of objects whose gravitational lensing may be measured in order to probe the underlying mass inhomogeneities in the universe. Gravitational lensing of the CMB may be contrasted with competing weak lensing probes in that: (i) for the CMB, the sources (i.e., the objects being lensed) are the most distant possible, situated at z ≈ 1100, (ii) because of the above, CMB lensing is sensitive to clustering at larger redshifts, and thus less sensitive to nonlinear corrections, and (iii) unlike observations of the correlations of the observed ellipticities of galaxies, where there are systematic errors due to intrinsic alignments, for CMB lensing there is no such problem. In the linear theory (which is a good approximation, especially for a heuristic discussion), the deflection due to lensing may be described by means of a lensing ˆ on the celestial sphere. Let Ω ˆ unlens potential, defined as a function of position Ω coordinatize the last scattering surface as it would look in the absence of lensing, ˆ lens to be the position on the last scattering surface with lensing taken and define Ω into account. We may express ˆ unlens + ∇Φlens (Ω). ˆ ˆ lens = Ω Ω

(56)

Tlens = Tunlens − (∇Φlens ) · (∇Tunlens ),

(57)

It follows that

and similar expressions may be derived for the polarization. As in weak lensing, we may derive a projected Poisson equation for the lensing potential xsource ˆ = ˆ x), dx W (x; xsource ) δρ(Ω, (58) ∇2 Φlens (Ω) 0

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where ∇2 is the Laplacian operator on the sphere. More details about the lensing of the CMB and what can be learned from it may be found in the review123 and references therein. The first discovery of gravitational lensing of the CMB involved detecting a statistically significant nonzero cross-correlation between a lensing map reconstructed using the WMAP data and infrared galaxies.203 The use of a cross-correlation allows one to detect a small signal within a noisy map. Figure 17 shows the gravitational lensing spectrum as measured by Planck.167 Measurements of gravitational lensing have also been made by ACT39 and SPT,214 and the Polarbear experiment has recently reported the discovery of B modes due to lensing.182 The lensing field described above (which can be expressed either as a potential, as a deflection field, or as a dilatation and shear field) can in principle be recovered from distortions in the small-scale power spectrum due to the lensing field on larger scales. As a general rule, the higher the resolution of the survey, the better the reconstruction. Hu and Okamoto143,93 proposed a reconstruction based on Fourier modes, and later many other estimators have been proposed, which are all quite similar because they are all exploiting the same signal. In the future, as observations at higher sensitivity and in particular due to B modes at small scales become available, lensing promises to become a powerful probe of the clustering of matter and halo structure. It is hoped that lensing will eventually be able to determine the absolute neutrino masses, even if they are as low as allowed by present neutrino oscillation data.120 Lensing

Fig. 17. Power spectrum of CMB gravitational lensing potential as measured by Planck. The angular power spectrum for the lensing potential as reported in the Planck 2013 results is shown above. The overall statistical significance for a lensing detection is greater than 25σ. The solid curve indicates the expectation based on the concordance model whose six parameters have been fixed by the Planck data combined with ancillary data. For more details, see Ref. 167. (Credit: ESA/Planck Collaboration) (For color version, see page II-CP9.)

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detected through B modes is particularly advantageous because the fact that the B modes should be zero in the absence of lensing means that cosmic variance does not intervene and that only the instrument noise is an issue.q

9. CMB Statistics 9.1. Gaussianity, non-Gaussianity, and all that For the simplest models analyzed at linear order, the primordial CMB signal is the outcome of an isotropic Gaussian random process on the celestial sphere. While any particular realization of this random process produces a sky temperature map that is not isotropic, the hypothesis is that the underlying stochastic process is isotropic. Concretely the hypotheses of isotropy and Gaussianity imply that the ˆ whose multipole expansion in terms of probability of obtaining a sky map T (Ω), spherical harmonic coefficients is given by ˆ = T (Ω)

∞
+



ˆ a
m Y
m (Ω),

(59)


=0 m=−


is as follows: P ({a
m }) =

  1 1 |a
m |2  exp − . 2 C
th 2πC
th
=0 m=−
+
∞  

(60)

ˆ = (θ, φ) denotes a position on the celestial sphere and the set of posiHere Ω tive coefficients {C
th } represents the angular power spectrum of the underlying stochastic process, or the theoretical power spectrum. If we had postulated Gaussianity alone without the additional hypothesis of isotropy, we would instead have obtained a more general probability distribution function having the form  

1 −1/2 (2πCth ) exp− (61) a
m ∗ (Cth )−1
m,

m
a

m
. P ({a
m }) = det 2


,m
,m

Here Cth is the covariance matrix for the spherical harmonic multipole coefficients. This Ansatz is too general to be useful because there are many more parameters than observables, and in cosmology we can observe only a single sky — or said another way, only a single realization of the stochastic process defined in Eq. (61). The assumption of isotropy is very strong, greatly restricting the number of degrees conclusion hardly depends on r whatever its final value may turn out to be, because the angular scales best for probing lensing lie beyond the recombination bump of the B modes.

q This

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of freedom by setting all the off-diagonal elements to zero and requiring that the diagonal elements depend only on
. Happily, the present data, subject to some caveats (such as gravitational lensing, discussed above in Sec. 8), seem consistent with the hypotheses of isotropy and Gaussianity, notwithstanding some “anomalies” at moderate statistical significance (discussed in detail below in Sec. 12). These anomalies could be a sign of something new, but given their limited statistical significance, the argument that they are simply statistical “flukes” cannot be rejected. For completeness, we now indicate how the above formalism is extended to include the polarization of the primordial CMB signal. There are several formalisms for describing the polarization of the CMB, for example, using spin-weighted spherical harmonics, or using the Stokes parameters I, Q, U, V .23 (The polarization of the CMB and its description are discussed in Refs. 35, 96, and 108.) We may define ˆ = Ea (Ω) ˆ ∗ Eb (Ω) ˆ time where Ea (Ω) ˆ is the electromagnetic field component Tab (Ω) ˆ propagating from the direction Ω and units are chosen so that for a unit electric ˆ a εb represents the CMB thermodynamic temperature polarization vector ε, Tab (Ω)ε when by means of a linear polarizer only the component along ε is measured. We may decompose ˆ = T (Ω)δ ˆ ab + Pab (Ω), ˆ Tab (Ω)

(62)

where the polarization tensor (containing the Stokes parameters Q, U , and V ) is trace-free and Hermitian (and also symmetric in the absence of circular polarization). We must now choose a convenient basis for the polarization. If we considered a single point on the celestial sphere, we could using Stokes parameters set Q = T11 − T22 and U = 2T12 , but no natural choice for the directions of the orthonormal ˆ can be singled out as preferred. unit vectors eˆ1 and eˆ2 in the tangent space at Ω Had we instead chosen  

 cos θ sin θ eˆ1 eˆ1 = , (63)  −sin θ cos θ eˆ2 eˆ2 we would instead have the Stokes parameters

 Q U

=

cos 2θ

sin 2θ

−sin 2θ

cos 2θ

  Q U

.

(64)

The transformation law in Eq. (64) indicates that the traceless part of the polarization is of spin-2. If the sky were flat with the geometry of R2 instead of S 2 and we had to include homogeneous modes as well, there would be no way out of this quandary. But there is a celebrated result of differential geometry that a fur on a sphere of even dimension cannot be combed without somewhere leaving a bald spot

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or singularity.r A traceless tensor field on the sphere cannot have an everywhere vanishing covariant derivative. We can generate a complete basis for the polarization on the celestial sphere by ˆ as means of derivative operators acting on the spherical harmonics. Using Y
m (Ω) a starting point, we may define an E mode basis vector as follows:

 ˆ = N E,B ∇a ∇b − 1 δab ∇2 Y
m (Ω), ˆ Eab;
m (Ω) (65)
2 where N
E,B is a normalization factor. Here a, b = 1, 2 indicate components with respect to an arbitrary orthonormal basis on S 2 . Without introducing more structure, this is the only way to produce a second-rank traceless tensor on the sphere ˆ in a way that does not break isotropy, with derivative operators acting on Y
m (Ω) ˆ a feature that ensures that Eab;
m (Ω) transforms under rotations according to the quantum numbers
m. However, there is another polarization, rotated with respect to the E mode by 45◦ , and in order to write down a basis for this other polarization, known as the B modes, it is necessary to introduce an orientation on the sphere through the unit antisymmetric tensor with 12 = 1. This choice of orientation, or volume element in the language of differential geometry, defines a handedness, or direction in which the 45◦ rotation is made. We may define Bab =

1 (δaa
+ aa
)(δbb
+ bb
)Ea
b
, 2

(66)

or in terms of derivative operators Bab =

1 E,B ˆ N (ac δbd + bc δad )∇c ∇d Y
m (Ω). 2


(67)

The difference between E modes and B modes is illustrated in Fig. 18. We want to normalize so that ˆ ∗

(Ω) ˆ = dΩB ˆ ab,
m (Ω)B ˆ ∗

(Ω) ˆ = δ
,

δm,m
(68) ˆ ab,
m (Ω)E dΩE ab,
m ab,
m and



ˆ ab,
m (Ω)B ˆ ∗

(Ω) ˆ = 0, dΩE ab,
m

(69)

so we require that100  N
E,B

r This

=

2 . (
+ 2)(
+ 1)
(
− 1)

is a consequence of the “hairy ball” theorem.60

(70)

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Fig. 18. E and B modes of the CMB polarization. The polarization pattern in the left panel can be expressed as the double derivative acting on a potential with its trace removed (in this case a Gaussian potential) and thus is an E mode. The B mode pattern on the right, however, cannot be represented in this way, but can be represented by a pseudo-scalar potential whose double derivative with the trace removed followed by a 45◦ rotation gives the indicated pattern.

Although the above discussion emphasized the role of spherical harmonics, the distinction between E and B modes does not rely on spherical harmonics, and much confusion has resulted from not fully appreciating this point. The absence of an E mode component or a B mode component may be formulated in a completely local manner using covariant derivative operators acting on the polarization tensor ˆ If ∇a Pab = 0, then the E mode is absent, and similarly if ab ∇a Pbc = 0, Pab (Ω). then the B mode is absent. Consequently, if we are willing to differentiate the data, we could detect B modes locally with absolutely no leakage. However, the above does not imply that a unique decomposition of Pab into E and B on an incomplete sky is possible because nonsingular vector fields (akin to harmonic functions) can be found that satisfy both conditions, although no nontrivial such vector fields exist defined on the full sky. Where global topology does enter is in eliminating configurations that satisfy both the above differential constraints. Having defined the E and B modes, we now indicate how the statistical distribution defined in Eq. (60) is modified to include polarization. We modify our notation somewhat to avoid unnecessary clutter. When polarization is included, T E B our vector {aT
m } is generalized to become {a
m , a
m , a
m }, which we shall denote as the vector (t, e, b). For each
we have a positive definite covariance matrix of the form 

C
TT

  C
= C
ET 

C
BT

C
TE

C
TB



C
EE

  C
EB . 

C
BE

C
BB

(71)

If we assume that the underlying physics imprinting the cosmological perturbations is invariant under spatial inversion, then the above covariance matrix

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simplifies to  TT C
  C
= C
ET  0

C
TE C
EE 0

0



  0 , 

(72)

C
BB

because under spatial inversion C
TB → −C
TB ,

C
EB → −C
EB ,

(73)

whereas the other power spectra coefficients preserve their sign. If parity is a symmetry of the physical processes generating the primordial perturbations and imprinting the CMB anisotropies, the expectation values C
TB,th and C
EB,th must vanish, although for any particular sky realization C
TB,obs and C
EB,obs will not vanish because of cosmic variance. (Recall that C
TB,th and C
EB,th is the average over a fictitious ensemble including an infinite number of sky realizations.) Studying C
TB,obs and C
EB,obs to search for statistically significant deviations from zero is a way to search for parity violation in the very early universe. However, in practice adjusting experimental parameters to minimize C
TB,obs and C
EB,obs is often used to calibrate detector angles in B mode polarization experiments. The propagation of light, or electromagnetic radiation, at least historically and conceptually, occupies a privileged place in the theories of special and general relativity, because of the constancy of the speed of light in vacuum and the fact that at least in the eikonal approximation, light travels along null geodesics, with its electric polarization vector propagated along by parallel transport. It is by thinking of bundles of light rays that we construct for ourselves a physical picture of the causal structure of spacetime. A modification of the propagation of light (for example, from dilaton and axion couplings,141 a possible birefringence of spacetime,32,62,82,206 or alternative theories of electromagnetism70,71,142 ) would not necessarily undermine the foundations of relativity theory, but would probably merely lead to a more complicated theory, possibly only in the electromagnetic sector. The CMB may be considered an extreme environment because of the exceptionally long travel time of the photons, and one can search for new effects beyond the known interactions with matter (e.g., Thomson and other scattering, collective plasma effects such as dispersion and Faraday rotation). Some such effects would lead to spectral distortions, for example through missing photons. The rotation of the polarization vector during this long journey would lead to mixing of E and B modes, and since the primordial E modes are so much larger, leads to interesting constraints on birefringence, which will greatly improve as more data comes in from searches for primordial B modes. The statistical description in Eqs. (71) and (72) suffices for confronting the predictions of theory with idealized measurements of the microwave sky with complete sky coverage and no instrument noise. We must further assume that there are no

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secondary anisotropies, nor galactic foregrounds, both of which have been perfectly removed. Even under these idealized assumptions, it is not possible to pin down the theory completely because of a phenomenon known as cosmic variance. The problem is that while we would like to characterize the covariance of the underlying theory C
AB,th (where A, B = T, E, B), we have only a single realization of the microwave sky, and C
AB,obs is only an estimate of C
AB,th with fluctuations about its expectation value, which is equal to C
AB,th . If we consider just the temperature fluctuations, we may define the sample variance for the
th multipole C
TT,obs =

+

1 |a
m |2 . (2
+ 1)

(74)

m=−


The random variable C
TT,obs obeys a χ2 -distribution with (2
+ 1) degrees of free dom and thus has fractional variance of 2/(2
+ 1). If we consider a microwave sky map bandlimited up to
max , there are approximately
max 2 degrees of freedom, so for example the overall amplitude of the cosmological perturbations may be determined with a fractional accuracy of approximately 1/
max . Most analyses of the CMB data (possibly combined with other data sets) assume a model where the theoretical power spectrum depends on a number of cosmological parameters, which we may abstractly denote as α1 , α2 , . . . , αD , also written more compactly as the vector α. For example, in the six-parameter minimal cosmological model claimed sufficient in the 2013 Planck Analysis166 to explain the present data, the parameters comprising this vector were AS , ns , H0 , ωb , ωCDM , and τ. Whether one adopts a frequentist or Bayesian analysis to analyze the data, the input is always the relative likelihood of the competing models given the observations, or a ratio of the form: P (observations | α1 ) . P (observations | α2 )

(75)

In this ratio, the measure for the probability density of the outcome, represented by a continuous variable, cancels. In this abridged discussion, we shall follow a Bayesian analysis where a prior probability Pprior (α) is assumed on the space of models. In Bayesian statistics, Bayes’ theorem is used to tell us how we should rationally update our prior beliefs (or prejudices) in light of the data or observations to yield a posterior probability to characterize our updated beliefs given by the formula Pposterior (α) = 

P (observations | α)Pprior (α) . D d α P (observations | α )Pprior (α )

(76)

There is no reason why the posterior distribution should lend itself to a simple analytic form, and a common procedure is to explore the form of the posterior

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distribution using Markov Chain Monte Carlo (MCMC) methods,57,122 which are particularly well suited to explore distributions of large dimensionality. However, it often occurs that the likelihood and the posterior distribution can be adequately represented as a Gaussian by expanding about its maximum likelihood value. In this case, the maximum of the log of the probability can be found using a numerical optimization routine, and then by computing second derivatives using numerical finite differences, the Gaussian approximation to the posterior can be found. This infinitesimal analysis is often referred to as a Fisher analysis.

9.2. Non-Gaussian alternatives It is incredibly difficult to test the Gaussianity of the primordial microwave sky without some guidance from theory as to what non-Gaussian alternatives are well motivated. The space of non-Gaussian stochastic models is dauntingly vast, with the space of Gaussian models occupying by any measure an almost negligible fraction of the model space. If one tests enough models, one is guaranteed to produce a result beyond any given threshold of significance. Consequently, one of the questions lurking behind any claim of a detection of non-Gaussianity is how many similar models did one test.

10. Bispectral Non-Gaussianity Although it was recognized that inflation, in its simplest form described by Einstein gravity and a scalar field minimally coupled to gravity, would have some nonlinear corrections, early analyses of the cosmological perturbations generated by inflation linearized about a homogeneous solution and calculated the perturbations in the framework of a linearized (free) field theory. According to the lore at the time, the nonlinear corrections to this approximation would be too small to be observed. The first calculations of the leading nonlinear corrections were given by Maldacena128 and Aquaviva et al.,7 who calculated the bispectrum or three-point correlations of the primordial cosmological perturbations within the framework of single-field inflation. Subsequent work indicated that in models of multi-field inflation, bispectral nonGaussianity of an amplitude much larger than for the minimal single-field inflationary models can be obtained. The predictions of many of these multi-field models were within the range that would be detectable by the Planck satellite. For many of these models, the non-Gaussianity, owing to the fact that it is generated by dynamics on superhorizon scales where derivative terms are unimportant, is well described by the “local” Ansatz for bispectral non-Gaussianity, under which the non-Gaussian field ζNL is generated from an underlying Gaussian field ζL according to the rule  2 ζNL (x) = ζL (x) + ζL (x) ,

(77)

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where fNL is a dimensionless parameter that quantifies the degree of nonGaussianity. Here fNL is independent of wavenumber, but a dependence on wavenumber can also be contemplated, so that the above relation is generalized to the following (now working in wavenumber space): ζNL (k) =

d k1

d3 k2 fNL (k; k1 ; k2 )δ 3 (k − k1 − k2 )ζL (k1 )ζL (k2 ).

3

(78)

Here as a consequence of the hypothesis of isotropy, we have expressed fNL as a function depending only on the lengths of the vectors. Any two triangles (where the three vectors close) that may be mapped into each other by an isometry are assigned the same fNL . For bispectral non-Gaussianity of the local form, as defined in Eq. (77), we briefly sketch how the order of magnitude of (S/N )2 for a detection of fNL from a CMB map extending up to
max may be estimated. We simplify the estimate by employing the flat sky approximation, allowing us to replace the unintuitive discrete sums and Wigner 3j symbols with the more transparent continuum integrals and a two-dimensional δ-function. Approximating C
≈ C0
−2 (in other words, we ignore acoustic oscillations, damping tails, and all that, and we assume that the CMB is scale invariant on the sky, as would occur if one could extrapolate the simplified Sachs–Wolfe formula (∆T )/T = −Φ/3 to arbitrarily small scales), we may write

S N

2

= O(1)

d2  (2π)2



d2 2 (2π)2



d2 3 (2π)2 δ 2 (1 + 2 + 3 ) (2π)2

fNL 2 C0 4 (
1 −2
2 −2 +
2 −2
3 −2 +
3 −2
1 −2 )2 C0 3
1 −2
2 −2
3 −2


max 2 = O(1)(fNL ) C0
max 2 ln .
min ×

(79)

The quantity in the first line simply approximates the sum over distinguishable triangles, the combinatorial factors being absorbed in the O(1) factor, and the second line gives the signal-to-noise squared of an individual triangle, as calculated diagrammatically. More exact calculations taking into account more details confirm this order of magnitude estimate. Numerical calculations are needed to determine the O(1) factor and deal with
max more carefully. Nevertheless this rough estimate reveals several characteristics of local non-Gaussianity, namely: (i) the presence of the ln(
max /
min) factor indicates that the bulk of the signal arises from the coupling of large angle modes and small angle modes (i.e., the modulation of the smallest scale power near the resolution limit of the survey by the very large angle modes (i.e.,
= 2, 3, . . .)), and (ii) the cosmic variance of the estimator derives from the inability to measure the small-scale power accurately, and thus decreases

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in proportion to the effective number of resolution elements of the survey. Cosmic variance on large scales is not an issue. As long as we know the particular realization on large scales accurately (which is not subject to cosmic variance), we know what kind of modulation to look for in the small-scale anisotropies. Theoretical models for other shapes for the bispectral anisotropy can be motivated by models for fundamental physics (see for example Refs. 9 and 107 and references therein) and have been searched for as well. The Planck 2013 Results gave equil local ortho = 2.7 ± 5.8, fNL = −42 ± 75, and fNL = −25 ± 39. the following limits173 : fNL This result rules out many of the models developed to explain a result at low statistical significance from some analyses of the WMAP data and was a profound disappointment to those who hoped that Planck would turn up striking evidence against the simplest inflationary models. One may anticipate that the results from Planck 2014 will improve modestly on the 2013 results by including more modes because of the use of polarization. If there is a next-generation all-sky CMB polarization satellite having good angular resolution so that S/N≈ 1 maps may be obtained up to
max ≈ 3000, an improvement on the limits on fNL by a factor of a few may be envisaged. However, beyond
≈ 3000, the primary CMB anisotropies become a sideshow. As one pushes upward in
, other nonprimordial sources of anisotropy take over, such as gravitational lensing, the Sunyaev–Zeldovich effect, and point sources of various sorts. These contaminants have angular power spectra that rise steeply with
while the CMB damping tail is falling rapidly, in fact almost exponentially. For including more modes, polarization may be helpful. On the one hand, polarization measurements present formidable instrumental challenges because of their lower amplitude compared to the temperature anisotropies. However, for the polarization, the ratio of foregrounds to the primordial CMB signal is favorable up to higher
than for temperature. For searching for non-Gaussianity, the primordial CMB is highly linear, unlike other tracers of the primordial cosmological perturbations. However given that the basic observables are two-dimensional maps, the small number of modes compared to three-dimensional tracers of the primordial perturbations is a handicap.

11. B Modes: A New Probe of Inflation As discussed above, some limits on the possible contribution from tensor modes, or primordial gravitational waves, can be obtained from analyzing the C T T power spectrum. However, given that r is not large, these limits become highly model dependent, and because of cosmic variance as well as uncertainty as to the correct parametric model, cannot be improved to any substantial extent. This difficulty is illustrated by the results from the Planck 2013 analysis,171 which did not include the data on polarized anisotropies collected by Planck but only the TT power spectrum. If one assumes the minimal six-parameter model plus r added as an extra parameter, a limit of r < 0.11 (95% C.L.) is obtained. However, if the primordial

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power spectrum in this model (where a form P (k) ∼ k ns −1 is assumed) is generalized to P (k) ∼ (k/k ∗ )ns −1 exp(α(ln(k/k ∗ ))2 ), the limits on r loosen by about a factor of two to become r < 0.2 (95% confidence level). This lack of robustness to the assumptions of the parametric model highlights the fragility of this approach and demonstrates that the statistical error bars are not to be trusted unless one has strong reason to trust the underlying parametric model. However, given our current understanding of inflationary potentials and dynamics, we have no reason to trust any of the parametric models, which serve more as fitting functions for summarizing the current state of the observations. Searching for tensor modes using B modes, on the other hand, allows us to probe much lower values of r in a manner that is substantially model independent because in the linear theory “scalar” perturbations cannot generate any B modes, and this conclusion is independent of the parametric model assumed. 11.1. Suborbital searches for primordial B modes In March 2014, the BICEP2 Collaboration17,18 claimed a detection of primordial gravitational waves based primarily on observations at a single frequency (150 GHz) over a small patch of the sky (1% of the sky) that was believed to be of particularly low polarized dust emission. Using a telescope based at the South Pole, the BICEP2 team created a map at 150 GHz of the polarized sky emission from which the B mode contribution was extracted. The BICEP2 analysis included many null tests to estimate and exclude systematic errors as well as to estimate the statistical noise in the measurement. However, in order to claim a detection of primordial gravitational waves, the BICEP2 team had to exclude a nonprimordial origin for the observed B mode signal, for which the main suspect would be polarized thermal dust emission. An independent analysis66 called into question the BICEP2 estimate of the likely dust contribution in their field, claiming that all the observed signal could be attributed to dust. A subsequent Planck analysis estimating the dust contribution in the BICEP2 field by extrapolating polarized dust maps at higher frequencies where dust is dominant down to 150 GHz confirmed this finding.159 While this analysis does not necessarily exclude a nonzero primordial contribution to the B modes observed by BICEP2 at 150 GHz, the argument crucial to the BICEP2 claim that a dominant contribution from dust can be excluded collapses in light of this new finding from Planck concerning the expected contribution from thermal dust emission. A cross-correlation study is now underway as a joint project of the two teams that will combine the BICEP2 map at 150 GHz with the Planck polarization maps at higher frequencies (which are dominated by dust). It remains to be seen whether this effort can establish a limit on r better than what is currently available or whether this combined analysis might even result in a detection of primordial B modes albeit at a lower level of r. In any case, a number of experiments are now underway involving observations at multiple frequencies that will push down the limits on r using B modes if not

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result in a first detection if r is not too small. The BICEP2 team has a series of upgrades to their experiment expanding their frequency coverage as well as massively increasing the number of detectors and hence their sensitivity. Other competing efforts include POLARBEAR,8 SPIDER34 and QUBIC,13 as well as ACTPol and SPTPol. Moreover, farther into the future, a more ambitious upgrade to ground based efforts is contemplated through the US Stage 4 (S4)1,2 CMB experiment currently under consideration by the US DOE. S4 contemplates deploying approximately a total of 2×105 detectors from the ground in order to achieve a massive improvement in raw sensitivity. Although it is claimed that an improvement in the control of systematic errors to a comparable level can be achieved, it remains to be seen whether S4 can realize its projected performance. 11.2. Space based searches for primordial B modes Several groups around the world have proposed space missions specifically dedicated to mapping the microwave polarization over the full sky with a sensitivity that would permit the near ultimate measurement of the CMB B modes. In Europe, three ESA missions had been proposed: BPol in 2007,43 COrE33 in 2010, both of which were medium-class, and the large-class mission PRISM185 in 2013. None of these missions was selected. A mission proposal called COrE+ is currently in the process of being submitted. In Japan a mission called LiteBird81 has been developed, but final approval is still pending. In the US, CMBPol/EPIC61 has proposed a number of options for a dedicated CMB polarization space mission,s but none of these has been successful in securing funding. All the above concepts involve a focal plane including several thousand single-mode detectors in order to achieve the required sensitivity. Another proposal endeavors to achieve the needed sensitivity by another means: multi-moded detectors. In the Pixie105 proposal, a Martin–Pupplet Fourier spectrometer comparing the sky signal to that of an artificial blackbody is proposed. The setup is similar to the COBE FIRAS instrument, except that polarization sensitive bolometers and a more modern technology are used, which will allow an improvement in the measurement of the absolute spectrum about two orders of magnitude better than FIRAS. 12. CMB Anomalies The story recounted so far has emphasized the agreement of the observations accumulated over the years with the six-parameter so-called “concordance” model. But this would not be a fair and accurate account without reporting a few wrinkles to this remarkable success story. These wrinkles may either be taken as hints of new physics, or discounted as statistical flukes, perhaps using the pejorative term s See

http://cmbpol.uchicago.edu/papers.php for a list of papers on this effort.

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a posteriori statistics. On the one hand, one can argue that if one looks at enough models that are in some sense independent, one is bound to turn up something at high statistical significance, and by most standards the significance of these anomalies is not high. Which interpretation is preferred is presently under debate. Ultimately, despite all the fancy statistical terminology used in this discourse, what one concludes inevitably relies on a subjective judgment of theoretical plausibility. One anomaly explored over the years, known under several names including “hemispherical asymmetry,” “dipolar modulation,” and “bipolar disorder,” asks the question whether the local angular power spectrum is identical when compared between opposite directions on the celestial sphere. There is no unique way to pose this question precisely. The statistical significance, amplitude, and direction of the asymmetry depend somewhat on the precise formulation chosen. However, the WMAP finding of a dipolar modulation with an amplitude of ≈ 7% and a significance ranging within about 2−4σ, and pointing approximately toward (l, b) = (237◦ , 20◦ ) in galactic coordinates is broadly confirmed by the Planck 2013 Results. This confirmation by Planck, owing to the additional high frequency coverage, renders less plausible a nonprimordial explanation based on foreground residuals. The sensitivity to dipolar modulation on these angular scales is primarily limited by cosmic variance for both WMAP and Planck, so future experiments cannot hope to improve substantially on the significance of these results. Planck, however, because of its superior angular resolution, was able to probe for dipolar modulation pushing to smaller scales, where cosmic variance is less of a problem and a possible modulation can be constrained more tightly. [The measurement uncertainty in the scale-dependent dipole modulation of the amplitude of the perturbations ∆A(
) (assuming broad binning) scales as 1/
as long as cosmic variance is the limiting factor.] The Planck 2013 data find no evidence of dipolar modulation at the same amplitude extending to small angular scales, suggesting that if the dipolar modulation observed by WMAP is not a statistical fluke, a scale-dependent theoretical mechanism for dipolar modulation is needed. Moreover, Planck does not see evidence of modulation associated with higher multipoles (e.g., quadrupolar and higher order disorders). This last point is important for constraining theoretical models producing statistical anisotropy by means of extra fields disordered during inflation. Another anomaly is the so-called “cold spot.” Rather than limiting ourselves to two-point statistics, or to three-point statistics, searching for a bispectral signal using theoretically motivated templates, as described in Sec. 10, we may ask whether the most extreme values of an appropriately filtered pure CMB map lie within the range that may be expected assuming an isotropic Gaussian stochastic process, or whether their p-values (i.e., probability to exceed) render the Gaussian explanation implausible. There are many ways to formulate questions of this sort, a fact that renders the assessment of statistical significance difficult. One approach that has been applied to the data is to filter the sky maps with a spherical Mexican hat wavelet (SMHW) filter, which on the sphere would have the profile of the Laplacian

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operator applied to a Gaussian kernel of a given width σ, giving a broad [i.e., (∆
)/
= O(1)] two-dimensional spatial bandpass filter. (See Ref. 215 for more details concerning the methodology and Ref. 216 for the original claim of a cold spot detection. See also Ref. 248 for an assessment by the WMAP team as well as Ref. 172 for the analysis from Planck 2013.) The final anomaly concerns possible alignments of the low-
multipoles, imaginatively named the “axis of evil” by Land and Magueijo.114 A Gaussian isotropic theory for the temperature anisotropyt predicts that the multipole vectors a
= {A
m }+
m=−
are each Gaussian and independently distributed. If we add three or more such vectors together according to the tensor product of representations of SO(3) L1 ⊗ L2 ⊗ · · · ⊗ LN ,

(80)

which may be decomposed into a direct sum of irreducible representations, provided that the triangle inequality is satisfied, we may extract one or more scalars (transforming according to L = 0) from the above tensor product. The mathematics is simply that of the usual addition of angular momentum. For each of these scalars, we may ask whether the observed value lies within the range expected from cosmic variance in the framework of a Gaussian theory taking into account the uncertainty in the determination of the underlying power spectrum. The approach just described n)| was not the approach of Land and Magueijo, who maximized maxm∈[−
,+
] |a
m (ˆ ˆ where a
m (ˆ n) is the expansion coefficient in the coordinate sysas a function of n ˆ pointing to the north pole. In this way, for each generic value of the tem with n multipole number
a unique double headed vector, or axis, ±ˆ n may be extracted, and the alignments between these axes may be assessed. Land and Magueijo report an alignment of the
= 3,
= 4, and
= 5 axes with a p-value less than 10−3 in the Gaussian theory. Accurate p values may be obtained for each of these invariant quantities by resorting to MC simulations to account for the practicalities of dealing with a cut sky, etc. However, there still remains a subjective element to assessing statistical significance. One may ask how many invariants were tried before arriving at a reportable statistically significant anomaly. 13. Sunyaev–Zeldovich Effects In the simplified discussion of reionization presented in Sec. 7.5, it was assumed that the electrons responsible for the rescattering are at rest with respect to the cosmic rest frame. This approximation treats the reionized gas as a cold plasma having a vanishing peculiar velocity field. As pointed out by Sunyaev and Zeldovich,208 corrections to this approximation arise in two ways: (i) In the so-called thermal Sunyaev–Zeldovich effect (tSZ), the random thermal motions of the free electrons t The discussion can readily be generalized to polarization, but we shall stick to the temperature anisotropy alone in our discussion.

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Fig. 19. Frequency dependence of thermal Sunyaev–Zeldovich spectral distortions. The solid and dashed curves show the fraction spectral distortion in terms of the CMB thermodynamic and R–J temperatures, respectively. For low frequencies (ν < 217 GHz), there is a decrement in temperature, corresponding to the fact that low frequency photons are on the average Doppler shifted to higher frequencies by the hot electrons. On the other hand, for ν > 217 GHz the net effect is to increase the photon phase space density.

alter the spectrum of the rescattered CMB photons through the Doppler effect and Compton recoil. Because v = 0, this effect is second order in the velocity, or linear in the electron temperature, and results in a spectral distortion not respecting the frequency dependence of a blackbody spectrum with a perturbed temperature. (See Fig. 19.) (ii) In the kinetic Sunyaev–Zeldovich (kSZ) effect the peculiar velocity of the gas results in a shift in temperature proportional to the component of the peculiar velocity along the line of sight. The kSZ effect has the same frequency dependence as the underlying primary CMB perturbations, thus making it hard to detect given its small magnitude. There is also a much smaller effect where the transverse peculiar velocity imparts a linear polarization to the scattered photons for which the polarization tensor Pij is proportional to (v⊥ ⊗ v⊥ − 12 v⊥ 2 δ ⊥ ), where v⊥ is the peculiar velocity perpendicular to the line of sight and δ ⊥ is the Kronecker delta function in the plane perpendicular to the line of sight. Galaxy clusters are filled with hot gas [Tgas = O(10 KeV)] that emits primarily at X-ray frequencies. The hot electrons of this fully ionized gas, or plasma, scatter CMB photons by Thomson, or Compton, scattering, shifting their frequency by a factor of approximately (1 + β cos θ) where β = kB T /me c2 and we have ignored higher order corrections in β. The y-distortion parameter along a line of sight is given by the integral y=

dτ

kB Te = mc2

d
σT ne

kB Te , mc2

(81)

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and the fractional perturbation in the CMB thermodynamic temperature in the nonrelativistic approximation is given by

 ˆ δTCMB (ν, Ω) x(ex + 1) ˆ = − 4 y(Ω), (82) ex − 1 T¯CMB ˆ is the y-distortion map as defined where x = hν/kB TCMB = ν/(57 GHz) and y(Ω) by the line of sight integral in Eq. (81). On the one hand, the thermal Sunyaev–Zeldovich effect provides a powerful probe of the dynamics of galaxy clusters, which can be used to discover new clusters and to probe the structure of known clusters away from their central core. On the other hand, for observing the primordial CMB, the thermal Sunyaev–Zeldovich effect constitutes a contaminant that must either be removed or modelled. The review by Birkinshaw19 recounts the early history of tSZ measurements. See also the reviews by Carlstrom et al.29 and by Rephaeli.191 For the observation of the kSZ, see Ref. 79. For more recent measurements, see the SZ survey papers from the ACT,130 SPT,223 and Planck169,170 collaborations. See Ref. 185 for a discussion of what might ultimately be possible from space in the future. 14. Experimental Aspects of CMB Observations Like astronomical observations at other wavelengths, most modern microwave experiments consist of a telescope and a number of detectors situated at its focal plane. The COBE experiment was an exception, as it used single-moded feed horns pointed directly at the sky to define its beam, which was relatively broad (7◦ FWHM). However for measurements at higher angular resolution, it is difficult to construct a feed horn forming a sufficiently narrow beam on the sky without the help of intermediate optics, unless one resorts to interferometry as for example in the DASI experiment. Observations of the sky at microwave frequencies present a number of challenges unique to this frequency range. Unlike at other frequencies, the microwave sky is remarkably isotropic. The CMB monopole moment, and to a much lesser extent the dipole moment, dominates. In the bands where other contaminants contribute least—that is, roughly in the range 70–150 GHz—virtually all the photons collected result from the isotropic 2.725 K background. The brightest feature superimposed on this uniform background is the CMB dipole, resulting from our proper motion with respect to the cosmic rest frame, at the level of 0.1%. When this dipole component is removed, the dominant residual is the primordial CMB anisotropy, whose amplitude on large angular scales is about 35 µK, or roughly 10−5 . This situation is to be contrasted for example with X-ray astronomy where only some 50 photons need to be collected to make an adequate galaxy cluster detection. For the CMB about 1010 photons must be collected in each pixel just to obtain the temperature anisotropy at a signal-to-noise ratio of one, and many more photons are needed for measuring polarization at the same marginal signal-to-noise. The challenge is to

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measure minute differences in temperature between different points on the celestial sphere without introducing spurious effects. These can arise from a variety of sources: drifts in the zero point of the detector, detector noise, hot objects in the far sidelobes of the beam, and diffraction around the edges of the mirrors, to name some of the most common problems. For this reason, almost all measurements of the microwave sky are in some sense differential. Drifts in the zero point of the detector (known as 1/f noise) can be mitigated by rapidly switching either between the sky and an artificial cold load with a long time constant or between different parts of the sky. This can be accomplished by moving the beam rapidly across the sky, so that in one way or the other the basic observable becomes differences in intensity of pixels whose measurements are closely separated in time. The point to take away is that one absolutely avoids making direct rather than differential measurements of the sky temperature. Another specificity of astronomical observations at microwave frequencies is that extremely stringent requirements must be imposed limiting the magnitude of the far sidelobes of the beam. Given the smallness of the differences in temperature of interest, contamination from the ground, the earth, the sun, and moon, and also from the galaxy near the galactic plane, can easily introduce spurious signals if the beam does not fall sufficiently rapidly away from its central peak. Ground pickup (from a spherical angle of 2π) is particularly challenging to shield, and this is why the WMAP and Planck satellites were situated at L2 (the second Lagrange point, where the earth and the moon have a tiny angular diameter) rather than in a lowearth orbit, which would equally well avoid atmospheric emission, which is the other major motivation for going to space. The second earth-sun Lagrange point (L2) is 1.5×106 km from the earth — that is, about 4 times the earth–moon distance. From L2 the earth subtends an angle of only 16 arcminutes. More importantly from L2, the sun, earth, and moon all appear at approximately the same position in the sky making it easier to keep away from the far sidelobes of the beam. A formidable challenge for suborbital observations is emission from the atmosphere, primarily but not exclusively from water vapor. The emission from the atmosphere has a complicated frequency dependence, as shown in Fig. 20, where the optical depth as a function of frequency is shown. This optical depth must be multiplied by the atmospheric temperature (i.e., ≈ 270 K) to obtain the sky brightness. The structure in frequency of the regions of high opacity implies that bands must be carefully chosen to avoid emission lines and bands, giving less flexibility for wide frequency coverage than from space. The problems arising from atmospheric emission can be mitigated to some extent by choosing a site on the ground with a particularly small water column density, such as the South Pole or the Atacama Desert in Chile, or by observing from a stratospheric balloon. Atmospheric emission is problematic in two respects. First, if the atmospheric emission were stable in time, it would merely introduce additional thermal loading on the detectors, which could be mitigated simply by deploying more detectors or by observing over

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Fig. 20. Atmospheric optical depth as a function of frequency for 1 mm of precipitable water vapor. Water vapor is an important but not the only source of atmospheric contamination. See Ref. 145 for more details. A column density of 0.5–1 mm of precipitable water vapor corresponds to good conditions in the Atacama Desert. Conditions at the South Pole are often better by a factor of a few. The black curve shows the opacity, or optical depth, and the red curve shows the opacity multiplied by 1/20 in order to highlight the structure around the resonances. The limited transmission 0 < t < 1 [where t = exp(−τ )], on the one hand, attenuates the desired sky signal. But more serious is the superimposed additive noise, having a brightness temperature (1 − t)Tatm . Here Tatm is the effective temperature of the atmosphere. (Credit: J. Cernicharo)

a longer time. If the contribution from the sky is Tsky , then the fluctuations in intensity at the detector (assumed perfect for the moment) would increase by a factor of (Tsky + TCMB )/TCMB , meaning that we could measure the sky temperature with the same error by increasing either the number of detectors or the total observation time by a factor of ((Tsky + TCMB )/TCMB )2 . At 150 GHz from the South Pole, for example, to choose a band where the sky brightness temperature is not very high, one has Tsky ≈ 16 K. This means that under the idealized assumptions made here, almost 40 times more detectors would be required than for a space-based experiment. But the situation is worse than this, because the atmosphere contamination is not stable in time but rather fluctuates in a complicated way characterized by a wide range of time scales (owing to the underlying turbulence). Moreover, the atmospheric load varies with zenith angle, roughly according to a secant law. Of course, observations from the ground are attractive because of their low cost, the fact that telescopes can be improved from season to season to address problems as they arise, and larger telescopes with many more detectors can be deployed. As experiments increase in sensitivity, requirements for mitigating these problems will become more stringent. At present it is not known what is the limit on the quality of measurements that can be achieved from the ground or from balloons.

14.1. Intrinsic photon counting noise: Ideal detector behavior In radio astronomy (where the frequency of observations is typically lower than for CMB observations) the following expression, known as the radio astronomers’ equation,45,110 gives the fractional error of an observation under slightly idealized

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assumptions:  2 δI Tsky + Tsys 1  = . I Tsky (∆B)tobs

(83)

Here tobs is the time of observation of a pixel and (∆B) is the bandwidth. Tsky is the sky brightness temperature averaged over the pixel and Tsys represents the additional noise introduced within the receiver. This expression can be understood by considering the source in the sky as having thermal statistics — that is, fluctuating in intensity as a Gaussian random field. In radio astronomy, detection is coherent and each detector picks up only a single transverse mode of the electromagnetic field incident from the sky, which is converted by an antenna or a feed horn into an electrical signal, which depends only on time. This signal is coherently amplified and nowadays digitized. Mathematically, in the absence of additional detector noise, the sky signal can be thought of as a signal whose complex amplitude fluctuates according to a Gaussian distribution with a variance proportional to Tsky . The coherence time is 1/(∆B), so in a time interval tobs , Nsamp = 2(∆B)tobs independent realizations of this Gaussian random process whose variance we are trying to measure are collected,  leading to a fractional error in the determination of the variance, or Tsky , of 1/ Nsamp . For a more realistic measurement noise is also introduced, and in the above formula the noise is idealized as an independent additive Gaussian random signal, characterized in terms of a system temperature Tsys , to be added to Tsky to obtain the variance actually measured. The additive noise lumped together and known as the “system” temperature includes thermal emission from the atmosphere, Johnson noise from dielectric losses in the feedlines, noise in the detector itself — in other words, everything other than the sky temperature that one would measure from space with an ideal measuring device. The above formula was derived treating the incoming electromagnetic field as entirely classical, and for a thermal source this treatment is valid in the R– J part of the spectrum — that is, when Tsource hν/kB . For the CMB, with TCMB = 2.725 K, this requires that ν < ∼ νCMB = kB TCMB /h = 57 GHz, meaning that quantum effects introduce significant corrections to the above formula, but one is still far from the regime where photons arrive in a nearly uncorrelated way, obeying Poissonian counting statistics, which is what happens for observations in the extreme Wien tail of the blackbody distribution. Before introducing the quantum corrections to Eq. (83), let us finish our discussion of this result based on classical theory extracting all the lessons to be learnt from this result. In the R–J regime, where N 1, N being the photon occupation number, roughly speaking, photons arrive in bunches of N photons, so the discreteness of the photons is not an issue. As long as a fraction greater than about 1/N of the photons is captured, little additional noise is introduced. The intrinsic noise of the incoming electromagnetic field is almost entirely classical in origin and can be modeled faithfully using stochastic

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classical electromagnetic formalism. (For a more detailed discussion of these issues, see Ref. 129.) From the above formula, we learn that noise can be reduced by the following measures: (i) choosing as wide a bandwidth as possible, (ii) increasing the observation time, and (iii) increasing the number of detectors. We also learn that although lowering the internal noise of the detection system increases the accuracy of the measurements, once the system temperature is approximately equal to the brightness temperature of the source, lowering the system temperature further leads only to marginal improvement in sensitivity. We now derive the quantum corrected version of the radio astronomer’s equation, using a Planck distribution instead of the classical description based on a deterministic stochastic Gaussian field. For the Planck distribution, where pN = xN /(1 − x) with x = exp(−hν/kB T ), ¯2 + N ¯. (δN )2 = N 2 + N = N

(84)

¯ 1, the first term dominates, reproducing the result In the R–J regime, where N obtained using a classical random field as described above. But as the frequency is increased and one starts to pass toward the Wien tail of the distribution, the second term increases in importance. In the extreme Wien tail, where the second term dominates, we observe the fluctuations characterized by Poisson statistics, where the (rare) arrivals of photons are completely uncorrelated. This means that the fractional error for the number of photons counted must be increased by a factor of 



hν 1 , (85) 1 + ¯ = exp 2kB (Tsky + Tsys ) N and the quantum-corrected version of Eq. (83) becomes113 δI = I



Tsky + Tsys Tsky

2

 1 hν  exp . 2kB (Tsky + Tsys ) (∆B)tobs

(86)

14.2. CMB detector technology Having analyzed the performance of an ideal detector limited only by the intrinsic fluctuations of the incoming electromagnetic field, we now review the state of the art of existing detector technologies, which may be divided into two broad classes: (i) coherent detectors, and (ii) incoherent detectors. Coherent detectors transfer the signal from a feed horn or antenna onto a transmission line and then coherently amplify the signal using a low-noise amplifier, generally using high electron mobility transistors (HEMTs) and switching with a cold load or between different points on the sky in order to mitigate 1/f noise.183 Coherent detection is an older technology. Coherent detectors have the advantage

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that they can operate at much higher temperatures than the competing incoherent bolometric detectors. Avoiding active cooling through a cryogenic system is a definite plus, especially in space, where reducing risk of failure is important. However, unlike their incoherent competitors, at the high frequencies of interest for CMB observations, coherent detectors have noise levels far above the quantum noise limit, and until now coherent detectors have not been able to reach frequencies above circa 100 GHz, although some believe that this situation may improve. It is not presently known how coherent amplification would function in the Wien part of the blackbody spectrum. Coherent detectors were used for the COBE DMR experiment, for WMAP (including the frequencies 20, 30, 40, 60, and 90 GHz), and the Planck LFI (low frequency instrument), which included the frequencies 30, 45, and 70 GHz. One advantage of coherent detectors is their insensitivity to cosmic rays. Another property of coherent detectors is that for a single mode measurement, all four Stokes parameters (i.e., I, Q, U, V) can be measured simultaneously. For incoherent detectors one can measure only two Stokes parameters simultaneously. However for measurements outside the R–J part of the spectrum, this advantage of coherent detectors disappears, because in the Wien part of the spectrum the photons are not bunched to any appreciable degree. On the other hand, for coherent detectors, it is not possible to construct multi-moded detectors, where the number of detectors can be greatly reduced by sacrificing angular resolution, but not at the cost of less sensitivity on large angular scales. Coherent detectors are always single-moded, sampling at the diffraction limit. Incoherent detectors, unlike their coherent counterparts, do not attempt to amplify the incident electromagnetic wave. Rather the incident electromagnetic wave, or equivalently the stream of photons, is directly converted into heat, changing the temperature of a small part of the detector of small heat capacity, whose temperature is monitored using a thermistor of some sort. Typically the detector is cooled to an average temperature much lower than that of the incident radiation. The bolometers of the Planck high frequency instrument (HFI) were cooled to 0.1 K. Consequently each CMB photon produces many phonons in the detector, and therefore the counting statistics of the phonons do not to any appreciable extent limit the measurement. Heat flows from the sky onto the detector and is then conducted to a base plate of high heat capacity and thermal stability. A thermal circuitry (which can be modelled as a sort of RC circuit with as many loops as time constants) is chosen with a carefully optimized time response. Several technologies are used for the thermistor. One of the most popular nowadays is transition edge sensors (TES),117 where a superconducting thin film is kept at the edge of the superconducting transition, so that minute changes in temperature give rise to large changes in resistivity. A feedback circuitry with heating is used to maintain the assembly near a fixed location on the normal-superconducting transition edge. One disadvantage of TES detectors is their low electrical impedance,

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which requires that SQUIDs (superconducting quantum interference devices), which need careful magnetic shielding, be used for the readout. Several new promising technologies are being developed and perfected for the next generation of bolometric detectors. One such technology is kinetic inductance detectors (also known as KIDs).41,47 For these detectors, a resonant electric circuit is constructed by depositing a pattern of superconducting film on a thick dielectric. Photons incident on the superconducting film break Cooper pairs causing its surface impedance to vary with the incident photon flux, which in turn alters the resonant frequency. The shifts in frequency of the resonator allow the KID to be read out in a simple way without using SQUIDs. The surface impedance of the superconducting film is largely determined by the inertia of the superconducting electrons, hence the name “kinetic inductance.” Another possibly promising technology is the cold electron bolometer,112 involving a small metal filament connected to a superconductor on both sides by means of a junction with a thin insulator in between. A bias current that passes through the assembly acts to cool the electrons in the thin wire through a Peltier-like effect to a temperature far below that of the substrate. CEBs are hoped to be more resistant to cosmic ray spikes than other detector technologies. Cosmic rays have proved to be more problematic than anticipated in the Planck experiment, and substantial effort was needed to remove the contamination from cosmic ray events in the raw time stream.181 Some of the data was vetoed and corrections were applied to the long-time tails of the larger magnitude events. While experiments from the ground benefit from substantial shielding of cosmic rays by the Earth’s atmosphere, how best to minimize interference from cosmic rays will be a major challenge for the next generation of experiments from space, which target sensitivities more than two orders of magnitude beyond the Planck HFI. Another challenge will be multiplexing, which is necessary to reduce the cooling requirements when the number of detectors is greatly expanded.

14.3. Special techniques for polarization If it were not for the fact that the polarized CMB anisotropy (i.e., the Q and U Stokes parameters) is much smaller than the anisotropy in the I Stokes parameter, measuring polarization would not pose any particular problems, because most detectors are polarization sensitive or can easily be made so, and there would be no need to discuss any techniques particular to polarization measurements. An inspection of the relative amplitudes of the spectra in Fig. 21 highlights the problems encountered. In particular if one wants to search for B modes at the level of r ≈ 10−3 or better, the requirements for preventing leakages of various types become exceedingly stringent, as we now describe. Let us characterize the problem for the most difficult polarization observation: measuring the primordial B modes. We may classify the leakages to be avoided

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Fig. 21. Summary of T,E, and B anisotropy power spectra from scalar and tensor modes. The green curves (from top to bottom) represent the TT, TE and EE CMB power spectra for the “scalar” mode, while the blue curves represent the CMB isotropies for the tensor mode. The broken blue curves (from top to bottom) and the top solid blue curve represent the TT, TE, EE and BB anisotropies assuming a value of the tensor-to-scalar ratio of r = 0.1. The lower two solid curves represent the predicted BB anisotropy for r = 0.01 and r = 0.001. The red curve shows the BB power spectrum arising nonlinearly from the scalar mode as a result of gravitational lensing. (Credit: M. Bucher)

by ordering them from most to least severe: (i) leakage of the CMB temperature monopole to the B mode, (ii) leakage of the T anisotropies to the B mode, and (iii) leakage of the E mode to the B mode. One of the solutions to many of these problems is polarization modulation. The idea is to place an element such as a rotating half-wave plate, whose orientation we shall denote by the angle θ, between the sky and the detector, preferably as the first element of the optical chain, as sketched in Fig. 22. Half-wave plates can be constructed using anisotropic crystals such as sapphire or using a mirror with a layer of wires a certain distance above it so that one linear polarization reflects off the wires while the other reflects off the mirror, resulting in a difference in path length. Mathematically, we have

(out)  Ex (out)

Ey

=

=

+ cos(θ)

−sin(θ)

+ sin(θ)

+ cos(θ)



+ cos(2θ)

+ sin(2θ)

−sin(2θ)

−cos(2θ)



1

0

0

−1



(in)

Ex

(in)

Ey

+ cos(θ)

+ sin(θ)

−sin(θ)

+ cos(θ)



(in)

Ex



(in)

Ey

 ,

(87)

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Fig. 22. Polarization modulation by means of a rotating half-wave plate. We show a transmissive rotating half-wave plate (HWP), here placed in front of a microwave feed horn pointing directly at the sky. However for modern CMB experiments, the horn would generally be replaced with a telescope having intermediate optical elements and horns on the focal plane. The rotating HWP is birefringent, with its fast polarization axis represented by the line OO  . As explained in the main text, by measuring only that part of the signal varying at an angular frequency 4Ω (where Ω is the angular velocity of the rotating HWP), one can prevent leakage from the much larger Stokes parameter I into the linearly polarized components Q and U. The presence of the HWP, while complicating the instrument design somewhat, allows many hardware requirements to be relaxed substantially compared to what would be needed without polarization modulation.

so that in terms of the Stokes parameters, expressed below as the matrix-valued expectation value Ei Ej∗ , so that

 I + Q U + iV U − iV I − Q det 

I +Q cos 2Ωt sin 2Ωt = U − iV sin 2Ωt −cos 2Ωt

=

U + iV I −Q



sky



cos 2Ωt

sin 2Ωt

sin 2Ωt

−cos 2Ωt

I + cos(4Ωt)Q + sin(4Ωt)U

−cos(4Ωt)U + sin(4Ωt)Q − iV

−cos(4Ωt)U + sin(4Ωt)Q + iV

I − cos(4Ωt)Q − sin(4Ωt)U

 . (88)

By measuring only the component that varies with an angular frequency 4Ω, we can measure the polarization without making two independent measurements of large numbers that are then subtracted from each other. The much larger Stokes intensity I does not mix because it remains constant in time. To see how polarization modulation helps, let us suppose that the vector (ideal) (ideal) (ideal) (ideal) (ideal) (actual) Pdet = (Idet , Qdet , Udet , Vdet ) is replaced by the vector Pdet , with (actual) (actual) (actual) (actual) (actual) (ideal) Pdet = (Idet , Qdet , Udet , Vdet ) = (I +
)Pdet , where I is the

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Fig. 23. Planck polarization sensitive bolometers. Shown above is a photograph of one of the Planck polarization sensitive bolometers consisting of two grids of parallel wires oriented orthogonally and placed on top of each other but separated by a small distance. Each grid has its own thermistor, allowing the two selected components of the linear polarizations to be read out separately. (Credit: ESA/Planck Collaboration)

identity matrix and
represents a hopefully small but unknown uncorrected residual error in the actual linear response of the detector pair. We discuss the setup for measuring polarization used in Planck as described above (see Fig. 23). We describe some of the possible systematic effects and how polarization modulation by means of a rotating half-wave plate would remove these systematic effects. Ideally the two orthogonal grids of wires would correspond but for the direction of the linear polarization to exactly the same circular beam on the sky and be perfectly calibrated, with no offset between the beam centers, nor any time-dependent drift of the differential gain. But in practice the beams are not identical and the gain and offset of the electronics drift in time. The centers of the beams do not coincide, and without corrections this would cause a local gradient of the T to be mistaken for E or B polarization. Differential ellipticity would cause the second derivative of T to leak into E and B. Moreover, the actual beams are ˆ Q(Ω), ˆ and U (Ω)] ˆ more complicated and require three functions [i.e., maps of I(Ω), for a complete description. All the above effects disappear, at least ideally, with the polarization modulation scheme described above. Polarization modulation can resolve or mitigate a lot of problems but is not a panacea. For example, polarization modulation cannot prevent E → B leakages. If there are any errors in the calibration of the angles of the direction of the linear polarization in the sky, these errors act

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to rotate E into B and vice versa. Moreover polarization modulation has been difficult to realize in practice for a number of reasons such as spurious signals due to microphonic coupling. Both continuously rotating and discretely stepped halfwave plates are options for polarization modulation. 15. CMB Statistics Revisited: Dealing with Realistic Observations The analysis of a CMB experiment going from the raw time stream of data taken, to sky maps at a given frequency, then to a clean sky map containing only the CMB, and finally to a likelihood function, whose input is a complete theoretical power spectrum, involves many steps and details. Space does not permit a complete discussion. The following Planck papers and references therein describe this analysis for the Planck experiment from the Planck 2013 Results series. See in particular the following papers: Overview of products and results,161 Low frequency instrument data processing,179 High frequency instrument data processing,180 Component separation,162 and HFI energetic particle effects.181 Similar descriptions can be found for other experiments. In this section, we limit ourselves to sketching a few issues and to describing an idealized statistical analysis. One of the main steps in CMB analysis is constructing a likelihood for the data given a particular theoretical model. Relative likelihood lies at the heart of almost all statistical inference, whether one uses frequentist methods, Bayesian methods, or an ecumenical approach borrowing from both doctrines. Here we shall discuss only how to formulate the likelihood, as how to exploit the likelihood is less specific to the CMB and details can be found in standard treatises on modern statistics. (See for example Ref. 125 for a brief overview and Ref. 210 for an authoritative treatment emphasizing the frequentist approach. For Bayesian sampling as applied to the CMB, see Refs. 122 and 57 and references therein.) Constructing a likelihood is straightforward for an idealized survey for which the noise in the sky maps is Gaussian and isotropic. For the simplest likelihood function, the input argument is a model for the CMB sky signal, or more precisely the parameters defining this model. This theoretical model defines an isotropic Gaussian stochastic process whose parameters one is trying to infer. In order to introduce some of the complications that arise, we start by writing down the likelihood L for the data given the model, or rather the variable −2 ln[L], which in some ways resembles χ2 in the Gaussian approximation. We have   TT,(sky)   ∞ TT,(th)
+ N
B
C
+ N
C
(2
+ 1) ln −2 ln(L) = + −1 . TT,(sky) TT,(th) C
+ N
B
C
+ N

=2 (89) Here the parameters N
represent the noise of the measurement. In the approximation of white instrument noise — that is, with no unequal time correlations — we have N
= N0 . The factor B
represents the smearing of the sky signal due to the

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finite beam width. For the beam profile approximated as having a Gaussian shape, where the sky intensity map is convolved with the Gaussian funcB
= exp[−σ 2
2 ]√ tion K(θ) = (1/ 2πσ 2 ) exp[−θ2 /(2σ 2 )].u Note that we ignore constant offsets in the log likelihood because for most statistical analysis only differences in the log likelihood are relevant. The presence of the normalization of the Gaussian implies that a quadratic representation of the log likelihood is not exact and higher order corrections are needed to avoid bias, especially at low
. To generalize the above expression to include polarization, we define the matrix

 C
TE C
TT + B
−1 N
TT , (90) C
= C
TE C
EE + B
−1 N
EE so that −2 ln(L) =

∞


  (th) (obs) (obs) (th) (2
+ 1) ln[C
(C
)−1 ] + tr[C
(C
)−1 ] − 2 .

(91)


=2

In order to generalize to less idealized (i.e., more realistic) situations, it is useful to rewrite the above results in a more abstract matrix notation, in which the expressions can be interpreted simultaneously as expressions in real (i.e., angular) space and as expressions in harmonic space, so that −2 ln[L] = det(Cth + N) + tT (Cth + N)−1 t.

(92)

Here the vector t represents the observed sky map, Cth is the covariance matrix of the underlying sky signal (in general depending on a number of parameters whose values one is trying to infer), and N represents the covariance matrix of the instrument (and other) noise. In the idealized case where everything is isotropic, this is simply a rewriting of Eq. (89), but in less idealized cases the likelihood in Eq. (92) is correct whereas Eqs. (89) and (91) are not applicable to the more general case. Let us now consider the effect of partial sky coverage, which may result from a survey that does not cover the whole sky or from masking portions of the sky such as the galaxy and bright point sources where contamination of the primordial signal cannot be corrected for in a reliable way. If the vectors and matrices in Eq. (92) are understood as representations in pixel space, at least formally there is no difficulty in evaluating this expression in the subspace of pixel space representing the truncated sky. If the dimensionality of the pixelized maps were small, there would be no difficulty in simply evaluating this expression by brute force. Another complication arises from inhomogeneous noise. Most surveys do not cover the sky uniformly because requirements such as avoiding the sun, the earth, and the planets as well as instrumental considerations lead to nonuniform sky coverage. This leads to a u Note that in the literature, beam widths are generally specified in terms of FWHM rather than using the Gaussian definition above.

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noise matrix that has a simple representation in pixel space (under the assumption of white noise), but not in harmonic space. Unfortunately evaluating likelihoods such as in Eq. (92) exactly is not possible for full-sky surveys at high angular resolution such as the Planck survey for which pixelized maps include up to about Npix = 5 × 107 pixels. While Cth is simple (i.e., sparse) in harmonic space, it is dense in pixel space, and the opposite holds for N (ignoring correlated noise), so no representation can be found to simplify the calculation in which all the matrices are sparse. Since even enumerating the matrices involves O(Npix 2 ) operations, and inverting these matrices or taking their determinant requires O(Npix 3 ) operations, it is immediately apparent that brute force will not work, and more clever, approximate techniques are required. Pixel based likelihoods are however used at low
, where other approximations have difficulty representing the likelihood, in large part because the cosmic variance is non-Gaussian. While the likelihoods above assumed an isotropic Gaussian stochastic process for the underlying theoretical model generating the anisotropy pattern in the sky, these likelihoods can be generalized to include nonlinear effects, such as nonGaussianity through fNL , gravitational lensing, and certain models of weak statistical anisotropy, to name just a few examples. Using the likelihood as a starting point, possibly subject to some approximations, has the virtue that one is guaranteed that an optimal statistical analysis will result without any special ingenuity. In the above simplified discussion, we have not touched on a number of very important issues, of which we list just a few examples: asymmetric beams, far sidelobe corrections, bandpass mismatch, errors from component separation, estimating noise from the data, and null tests (also known as “jackknives”). The reader is referred to the references at the beginning of this Section for a more complete discussion and references to relevant papers. 16. Galactic Synchrotron Emission An inspection of the single frequency temperature maps from the Planck space mission (see Fig. 8), shown here in galactic coordinates so that the equator corresponds to the galactic plane, suggests that the 70 GHz frequency map is the cleanest, as the excess emission around the galactic plane is the narrowest at this frequency. But at lower frequencies, the region where the galactic contamination dominates over the primordial CMB temperature anisotropies widens. This contamination at low frequencies is primarily due to galactic synchrotron emission, resulting from ultrarelativistic electrons spiralling in the galactic magnetic field, whose strength is of order a few microgauss. (See for example Ref. 78.) This emission is generally described as nonthermal because the energy spectrum of charged particles is non-Maxwellian, especially at high energies. The electrons in question reside in the high energy tail of the electron energy spectrum, described empirically as a falling power law spectrum. Models of cosmic ray acceleration, for example arising about

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a shock from an expanding supernova remnant, are able to explain spectra having such a power law form.20 If the radiating charged particles were nonrelativistic, their radiation would be emitted at the cyclotron frequency ωc = eB/mc, with very little emission in higher harmonics, and given the measured values of the galactic magnetic field, one could not thus explain the observed high frequency emission. However, if the charged particles are highly relativistic, due to beaming effects the bulk of the radiation is emitted in the very high order harmonics, allowing the observed emission to be explained for reasonable values of the galactic magnetic field.69 Although the data concerning the three-dimensional structure of the galactic magnetic field is at present in a very rudimentary state, and little is known observationally about the cosmic ray spectrum outside our immediate solar neighborhood, the physical mechanism underlying galactic synchrotron emission is well understood physics. One important point is that no matter what the details of the cosmic ray energy spectrum may be, the kernel of the integral transform relating this distribution to the frequency spectrum of the observed synchrotron emission applies considerable smoothing. Thus we can assume with a high degree of confidence that the synchrotron spectrum is smooth in frequency. Theory also predicts that this synchrotron emission will be highly polarized, at least to the extent that the component of the galactic magnetic field coherent over large scales is not negligible. This expectation is borne out by observations. It is customary to describe the synchrotron spectrum using a power law, so that its R–J brightness temperature may be fit by an Ansatz of the form

α ˆ ν) = TR−J (Ω, ˆ ν0 ) ν , (93) TR−J (Ω, ν0 where empirically it has been established that α ≈ 2.7–3.1. The same holds for the polarization — that is, the Stokes Q and U (and also V ) parameters. In the ultrarelativistic approximation this emission is primarily linearly polarized, but there is also a smaller circularly polarized component suppressed by a factor of 1/γ. Variations in the spectral index depending on position in the sky have been observed.67 A description of the WMAP full-sky observations of synchrotron emission appears in the first-year WMAP foregrounds paper229 and the polarization observations are described in the three-year WMAP paper on foreground polarization.235 A greater lever arm, in particular for studying the synchrotron spectral index, can be obtained by including the 408 MHz Haslam map covering almost the entire sky.80 17. Free–Free Emission Another source of low-frequency galactic contamination is free–free emission arising from Bremsstrahlung photons emitted from electron–electron collisions in the interstellar medium, and to a lesser extent from electron-ion collisions. Like galactic synchrotron emission, the free–free emission brightness temperature falls with

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increasing frequency, but the fall-off is slower than that of the galactic synchrotron emission. Unlike synchrotron emission, free–free emission is not polarized. Hα (n = 3 → n = 2) emission resulting from recombination of ionized atomic hydrogen can be used as a template for removing the free–free component, because the Hα emission likewise is proportional to the square of the electron density. However, because the ΓHα (T ) and Γfree–free (T, ν) do not have an identical temperature dependence, free– free removal using an Hα template has some intrinsic error. ΓHα (T ) is defined as the constant of proportionality in the emissivity relation: Hα = ΓHα (T )ne 2 where Hα is a bolometric emissivity (power per unit volume) and ne is the density of free electrons. Likewise free–free(ν) = Γfree–free (T, ν)ne 2 where the emissivity has units of power per unit frequency per unit volume per unit solid angle. 18. Thermal Dust Emission In Sec. 16 we discussed galactic synchrotron emission, whose R–J brightness temperature rises with decreasing frequency, making it exceedingly difficult to measure CMB anisotropies at frequencies below ≈ 20 GHz, especially close to the galactic plane where the galactic synchrotron emission is most intense. We also saw how the fact that the brightness temperature increases with decreasing frequency can be understood as a consequence of the synchrotron optical depth being much smaller than unity and increasing with decreasing frequency. Eventually when the frequency is low enough so that the optical depth is near one, the brightness temperature stops rising and approaches a temperature related to the cosmic ray electron kinetic energy. This section considers another component contributing to the microwave sky: the thermal emission from interstellar dust, whose brightness temperature increases in the opposite way — that is, the brightness temperature increases with increasing frequency. This behavior results because at microwave frequencies, the wavelength greatly exceeds the typical size of a dust grain. At long wavelengths, the crosssection for absorbing or elastically scattering electromagnetic radiation is much smaller than the geometric cross-section σgeom ≈ a2 where a is the effective grain radius. The dust grains can be approximated as dipole radiators in this frequency range because higher order multipoles are irrelevant. The dependence of the crosssection on frequency can be qualitatively understood by modeling the electric dipole moment of the grain as a damped harmonic oscillator where the resonant frequency is much higher than the frequencies of interest. In this low frequency approximation  a 2 a2 (94) σabs ∼ λ for absorption, and for the elastic, or Rayleigh, scattering component, which becomes subdominant at very low frequencies,  a 4 σelastic ∼ a2 . (95) λ

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This is a simplistic approximation that we later shall see is not accurate, but it does provide intuition about the qualitative behavior expected. Our knowledge of interstellar dust derives from combining different observations spanning a broad range of frequencies, from the radio to the UV and even X-ray bands, in order to put together a consistent theoretical model able to account simultaneously for all the observations. (For a nice recent overview, see for example Refs. 48 and 49.) In this section, we restrict ourselves to discussing the thermal microwave emission properties of the dust. An empirical law commonly used to model thermal dust emission at low frequencies is a Planck blackbody spectrum modulated by a power law emissivity, or graybody factor,

β ν B(ν, Tdust ) , (96) Idust (ν) = Idust (ν0 ) ν0 B(ν0 , Tdust ) where the Planck spectrum is given by B(ν; T ) =

2hν 3 1



 . hν c2 exp −1 kB T

(97)

The physical basis for this modification to the Planck spectrum can be partially motivated by the following argument based on the linear electric susceptibility of the dust grain, following a line of reasoning first suggested by EM Purcell in 1969.188 The simple argument, subject to a few caveats, suggests that β = 2. However the observational data do not bear out this prediction and instead are better fit by a power law with β ≈ 1.4 − 1.6.144,158 Let χ(ω) represent the linear susceptibility of a dust grain, which as a consequence of the Kramers–Kronig dispersion relations must include both nonvanishing real and imaginary parts. Here d(ω) = χ(ω)E(ω), where d(ω) and E(ω) are the grain electric dipole moment and the surrounding electric field, respectively. As a consequence of causality, χ(ω) is analytic on the lower half-plane. Its poles in the upper half-plane represent those decaying mode excitations of the grain that are coupled to the electromagnetic field. Moreover, on the real axis χ(−ω) = [χ(+ω)]∗ . If we assume analyticity in a neighborhood of the origin, we may expand as a power series about the origin, so that χ(ω) = ω0 + iχ1 ω + χ2 ω 2 + · · · ,

(98)

and χ0 , χ1 , χ2 , . . . are all real. It follows that the energy dissipated is characterized by the absorptive cross-section σabs (ω) = and at low frequencies

4πωIm[χ(ω)] , c

σabs (ω) ≈

4πχ1 c

(99)

 ω2,

(100)

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which corresponds to an emissivity index β = 2 as ω → 0. The perhaps questionable assumption here is that there exists a circle of nonzero radius about the origin of the ω-plane where χ(ω) has no singularities. If we postulate that there is an infinite number of poles of suitably decreasing strength whose accumulation point is ω = 0, we may evade the conclusion that asymptotically as ω → 0, β → 2. For an insightful discussion of how such low-frequency poles could arise, see C. Meny et al.135 19. Dust Polarization and Grain Alignment The fact that interstellar dust grains are aligned was first seen through the polarization of starlight at optical frequencies in 1949.76,89,90 Since this discovery an abundant literature has been amassed elucidating numerous aspects of dust grain dynamics. The best bet solution to the alignment problem has evolved over the years as additional relevant physical effects were pointed out. It is unclear whether the current understanding will remain the last word on this subject. The most naive solution to the alignment problem would be some sort of compass needle type mechanism with some dissipation. However, it was discovered early on that in view of the plausible magnitudes for the galactic magnetic field, such an explanation cannot work.207 The theoretical explanation that has evolved involves detailed modeling of the rotational dynamics of the grains, which as we shall see below almost certainly rotate suprathermally. For microwave observations of the primordial polarization of the CMB at ever greater sensitivity, the removal of contamination from polarized thermal dust emission is of the utmost importance. This fact is highlighted by the discussion of how to interpret the recent BICEP2 claim (discussed in Sec. 11.1) of a detection of a primordial B mode signal based on essentially a single frequency channel. This interpretation however seems unlikely given subsequent Planck estimates of the likely dust contribution.159 The problem is that little data is presently available on polarized dust emission at microwave frequencies. Current models of polarized dust involve a large degree of extrapolation and are in large part guided by simplicity. From a theoretical point of view, polarized dust could be much more complicated than the simplest empirical modeling suggests. For example there is no a priori reason to believe that polarized emission from dust must follow the same frequency spectrum as its emission in intensity. It could well be that the weighting over grain types gives a different frequency dependence, or that as more precise dust modeling is called for, the need to include several species of dust with spatially nonuniform proportions will become apparent. Moreover, based on theoretical considerations, it is not clear that there should be a simple proportionality between polarization and the strength of the magnetic field. If the BICEP2 B mode signal is confirmed as a dust artefact, we could see ourselves in the coming years trying to detect values of the tensor-to-scalar ratio around r ≈ 10−3 , which would require exceedingly accurate polarized dust removal from the individual frequency maps, and before this happens a wealth of new observational data on polarized dust emission will

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become available. If, on the other hand, a detection of B modes at a large value of r is made, as one tries to map out the precise primordial B mode spectrum, for example to measure nT and thus check the consistency between PS (k) and PT (k), the need for accurate dust removal will likewise be great, but for slightly different reasons. Since the observational inputs are likely to change, here we particularly emphasize the underlying physics of the alignment, which may be updated, but is likely to change less. The theoretical issues of dust alignment have so far not been widely discussed among those not specialized in the ISM (interstellar medium). Nevertheless, we believe that the questions raised will become increasingly relevant for CMB observations in the future, and this is why a thorough discussion is presented here. We shall find that the alignment of interstellar dust grains is closely linked to the rotational dynamics of a dust grain. We therefore start with a study of the rotation of a dust grain. Mathematically, the simplest case to analyze is a prolate (needle-like) or oblate (flattened spherical) dust grain with an axis of azimuthal symmetry, which constitutes a symmetric top. In this case, the motions can be solved in terms of elementary functions. For the more general case of an asymmetric grain (where none of the three principal values of the moment of inertia tensor are equal) integration of the equations of motion in the absence of torques is more difficult, but can be achieved using Jacobian elliptic functions.225 In the noninertial frame rotating with the dust grain, which is convenient because in this frame the moment of inertia tensor is constant in time, two types of inertial forces arise: (i) a centrifugal force, described by the potential Vctr (x)/m = −Ω2 (t)x2 + (Ω(t) · x)2 , and (ii) a Coriolis force Fcor /m = −2Ω(t) × v, which for our purposes is unimportant because the velocities are negligible in the co-rotating frame. In order to understand the alignment process, we must model the solid as being elastic with dissipation included. 19.1. Why do dust grains spin? A flippant answer to the above question might be: Why not? If we can associate a temperature with the three kinetic degrees of freedom of the dust grain, we would conclude that 3 1 2 Iω ∼ kB Trot . (101) 2 2 Early discussions of rotating dust grains supposed that the rotational degrees of freedom are subject to random torques, so that Γ = 0.

(102)

Later it was realized that suprathermal rotation is plausible, where the implied temperature of the rotation can greatly exceed the effective temperature of any of the other relevant degrees of freedom. The question now arises what temperature one should use for Trot . A natural candidate might be the kinetic temperature of the ambient gas Tkin . Indeed one

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expects molecules of the ambient gas to collide with the dust grains, and such collisions do provide random torques, causing the angular momentum to undergo a random walk accompanied by a dissipation mechanism, as must be present to prevent the angular velocity from diverging with time. 19.2. About which axis do dust grains spin? This question has two parts. First, we may ask what is the alignment of the corotating internal (body) coordinates with respect to the angular momentum vector. Second, we may ask how the angular momentum aligns itself with respect to the inertial coordinates of the ambient space. In this section, we consider only the first question, postponing the second question to a later section. But for exceptional situations where two or more eigenvalues of the moment of inertia tensor coincide, the principal axes of moment of inertia tensor provide a natural set of axes for the dust grain, and we may order the eigenvalues so that I1 < I2 < I3 . We argue that the dust grain tends to align itself so that its I3 axis is parallel to the angular momentum. This orientation of the grain minimizes the rotational energy subject to the constraint that the total angular momentum remains constant. If the grain starts in a state of random orientation, the grain wobbles. While L remains constant, Ω(t) = I−1 (t)L (expressed in inertial coordinates) does not, but rather fluctuates, exciting internal vibrational modes that dissipate. As the excess rotational energy is dissipated, an alignment as described above is achieved.v The relevant question is how this time scale compares to other time scales of the grain rotation dynamics. It turns out that this time scale, which we denote τI−L , is short. In the equations that follow, we exploit the shortness of τI−L compared to the other relevant time scales. This hierarchy of time scales allows us to simplify the equations by excluding the degrees of freedom that describe the lack of alignment of the principal axis of maximum moment of inertia from L, leading to a considerable simplification of the formalism. 19.3. A stochastic differential equation for L(t) Mathematically, we may describe the combined effects of the torque from random collisions and the associated damping through the coherent torque using a stochastic differential equation of the form 

˙ ¯ L(t) + Γran (t). (103) L(t) = −α L(t) − L |L(t)| v Much to the dismay of NASA engineers, this is exactly what happened to the Explorer 1 rocket, the first US satellite launched in 1958 following Sputnik, due primarily to dissipation from its whisker antennas. The elongated streamlined cylinder was initially set in rotation about its long axis, but subsequently, after an initial wobble phase, ended up spinning about an axis at right angles in the body coordinates. Since then rotational stability, also relevant in ballistics, has become a carefully considered design issue for spacecraft.

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In this equation, the torque described by the first term is coherent with respect to the body coordinates of the dust grain. Later we shall also consider torques that are coherent with respect to the ambient inertial coordinates, which must be modeled in a different way. Here the stochastic Gaussian forcing term has the expectation values Γran,i (t) = 0, Γran,i (t)Γran,j (t ) = µL δij δ(t − t ),

(104)

where the angular momentum diffusion coefficient has units [µL ] = (Torque)2 /(Time). Let us analyze the stationary ensemble defined by the above equation — in other words, the probability function p(L) toward which the biased random walk defined by the above equation evolves after a long period of time. In the low¯ with negligible fractional fluctuations, and the direction temperature limit, |L| ≈ L ˆ L = L(t)/|L(t)| follows a random walk on the of the angular momentum vector n ¯ 2 . In the opposite high surface of the sphere with diffusion coefficient µnˆ = µL /L ¯ temperature limit, the L term is a sideshow or small correction, and the main effect is the competition between the first and second terms leading to a thermal ensemble, which for an isotropic moment of inertial tensor would take the form

3/2 β p(L) = exp[−βL2 ], (105) 2π where β = α/µL . Without other sources of dissipation or random torques, we expect that β = 1/(2kB Tkin Imax ) where Imax is the largest eigenvalue of the moment of inertia tensor of the particular grain being modelled and Tkin is the kinetic temperature of the ambient gas. It is straightforward to generalize Eq. (105) to the intermediate case. It was however later noted that a number of mechanisms exist for which Γ = 0, which can give rise to what is known as “suprathermal rotation”189 — that is, rotation at an angular velocity larger than any temperature associated with the other degrees of freedom interacting with the dust grain. 19.4. Suprathermal rotation To see how such suprathermal rotation might come about, consider the caricature of a dust grain as sketched in Fig. 24. Let us assume that the dust grain emits radiation isotropically in the infrared, but that in the visible and UV bands radiation is absorbed or reflected by the surfaces indicated in the figure. Let us further assume an isotropic visible or UV illumination. The blackened (perfectly absorbing) faces suffer a radiation pressure equal to half that of the mirrored surfaces, giving a coherent torque in the direction indicated in the figure. The presence of two distinct temperatures, reminiscent of a thermal engine, is crucial to avoid thermal equilibrium for the rotational degree of freedom. As in the thermodynamics of an engine, there is a heat flow from a hotter temperature to a colder temperature with some

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Fig. 24.

Coherent torques from grain radiation pressure.

of the energy being siphoned off in the form of work exerted on the three rotational degrees of freedom of the dust grain. Other possibilities for coherent torques result from photoelectron emission at privileged sites and the catalysis of molecular hydrogen on favored sites on the grain surface. In the former case, the incident UV photon is almost pure energy due to its masslessness, but the emitted electron carries substantial momentum for which there is a recoil. Molecular hydrogen forms almost exclusively through catalysis on a grain surface. Unlike in ordinary chemistry where truly two-body collisions are an exception and activation barriers prevent reactions from establishing equilibrium, for molecular hydrogen formation in the cold interstellar medium, the bottleneck is the lack of three-body collisions and the lack of an efficient mechanism for carrying away the excess energy, so that the hydrogen atoms combine rather than simply bouncing off each other. In principle, the energy can be lost radiatively, but the rate for this is too slow. Given that 4.2 eV is liberated, the recoil at selected sites can produce a substantial coherent torque on the grain.

19.5. Dust grain dynamics and the galactic magnetic field Much of the dust grain rotational dynamics does not single out any preferred direction. To explain the polarization of starlight and of the thermal dust emission at microwave frequencies, some mechanism is needed that causes the dust grains to align themselves locally at least along a common direction such as that of the magnetic field.

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19.5.1. Origin of a magnetic moment along L Two effects generate a magnetic moment along the axis of L: (i) Inhomogeneous electric charge distribution in the dust grain. The second moment of the electric charge density about the principal axis of the largest eigenvalue of the moment of inertia tensor (denoted z) induces a magnetic dipole moment as the grain rotates about this axis, according to µ=ˆ zΩz d3 rρe (r)(x2 + y 2 ), (106) where ρe (r) is the electrostatic charge density of the dust grain. In general, and particularly when the grain structure is amorphous, the net charge density is nonvanishing, so the integral above does not vanish. This effect is known as the Rowland effect. (ii) The Barnett effect. Minimizing the free energy tends to align the free spins of a rotating body with the direction of rotation. In nonferromagnetic substances this effect is usually negligible, but rotating dust grains present an extreme environment because of their minute size and isolation from their environment. Interstellar dust grains are among the fastest rotating macroscopic objects known. To understand the origin of the Barnett effect, consider an isolated rotating body, whose conserved total angular momentum may be decomposed into two parts Ltot = Lbulk + Sfree .

(107)

We consider the energetics of the exchange of angular momentum between the bulk and spin degrees of freedom. We assume the spins to be free neglecting any interactions between them. In this case, the Hamiltonian for the system is H=

(Ltot − S)2 L2bulk = , 2I 2I

(108)

and retaining only the term linear in the spin, we obtain H = −Ω · S,

(109)

where Ω(t) = I−1 (t)L is the angular rotation velocity. The spins want to align themselves with the rotation to lower the rotational energy.115 So far we have not included any terms opposing complete alignment, so at T = 0 complete alignment would result with S = Savailable Ω/|Ω|. But at finite temperature entropic considerations oppose such alignment, and it is the free energy F = H − T S(S) or F (S; Ω) = −Ω · S + aT S2

(110)

that is minimized. Far from saturation, when the degree of alignment is small, this quadratic term provides a good approximation to the entropy of the spin system.

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19.6. Magnetic precession Having established that a spinning dust grain has a magnetic moment parallel to its angular momentum vector, which is aligned with the principal axis of the largest eigenvalue of the moment of inertia tensor, we now consider the effect of the torque resulting from the interaction with the ambient magnetic field. The interaction Hamiltonian is H = −µ · B,

(111)

L˙ = µ × B = γm L × B.

(112)

from which it follows that

In other words, the angular momentum vector precesses at a frequency ωprec = γm |B| around the magnetic field. Under this precession, the angle between L and µ remains constant, meaning that this effect neither aligns nor disaligns the grain with respect to the local magnetic field direction. We estimate the time scale of this precession. Let S denote the spin polarization density (angular momentum per unit volume), which under saturation (i.e., all free spins aligned) would have a magnitude Ssat , whose order of magnitude can be estimated by assuming approximately Avogadro’s number NA of spins having a magnitude
in a cubic centimeter volume, giving Ssat ≈ O(1)(4×10−4 )erg · s · cm−3 . We obtain the following free energy density (per unit volume):

2 S 1 F0 (S; T ) = (kB T )Nspin . (113) 2 Ssat Here for a paramagnetic substance the order of magnitude for Nspin might be around 1024 cm−3 but for a ferromagnetic substance could be much smaller, corresponding to the density of domains. Minimizing the free energy with respect to S gives the spin polarization density S=

2 Ssat = IS Ω. (kB T )Nspin

(114)

We express the constant of proportionality as if it were a moment of inertia because it has the same units. IS Nfree
2 ≈ IL kB T ρa2 ≈ (3 × 10

−5

)

Nfree 1024 cm−3



100 K T



3g cm−3 ρ



10 nm a

2 .

(115)

19.6.1. Barnett dissipation The coupling in the Barnett effect between the bulk and spin angular momenta does not act instantaneously. Rather as the instantaneous angular velocity Ω(t)

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changes, there is a lag in the response of S(t). In his landmark 1979 paper,189 Purcell pointed out that this lag would help align the body coordinates so that the axis of maximum moment of inertia is parallel to the angular momentum vector. Purcell coined the name Barnett dissipation for this effect, whose magnitude is greater than the mechanical dissipation mentioned above. Two types of free spins participate in the Barnett effect: the electronic spins and the nuclear spins. The electronic spins are much more relevant for establishing a magnetic moment along L as discussed above in Sec. 19.5.1 because electronic magnetic moments are larger than nuclear magnetic moments by a factor of about (mp /me ). But for dissipation, this strong coupling is a liability. The nuclear spins contribute predominantly to Barnett dissipation because of their longer lag time. Mathematically, the coupled equations ˙ S(t) = −γ(S(t) − αΩ(t)), ˙ L(t) = +γ(S(t) − αΩ(t)),

(116)

describe this effect. We may integrate out S(t) to obtain the following equation in inertial coordinates for the torque on L resulting from the exchange of angular momentum between the bulk and spin degrees of freedom ∞ ˙ L(t) = −αγΩ(t) + αγ 2 dτ exp[−γτ ]Ω(t − τ ). (117) 0

To calculate how fast this exchange of angular momentum aligns the axis of maximum moment of inertia with the angular momentum vector, we calculate the averaged rotational energy loss rate of the bulk degrees of freedom retaining only secular contributions. This calculation is simpler than keeping track of torques. The degree of alignment of Ω(t) with the axis of maximum moment of inertia is a function of the rotational energy in the bulk rotational degrees of freedom. The instantaneous power transferred from the bulk rotation to the spin degrees of freedom is given by ˙ ˙ = −Ω(t) · S(t) Ploss (t) = Ω(t) · L(t) = −γIs Ω(t) − γ Ω(t) − γ

∞

! dt exp[−γt ]Ω(t − t ) ,

(118)

0

and to obtain the averaged or secular contribution, we decompose Ω(t) into harmonic components of amplitude Ωa and frequency ωa , so that !# " ∞ 1
2 ωa 2 dt exp[−γt ]Ω(t − t ) = Ωa 2 Ω(t) · Ω(t) − γ . (119) 2 a γ + ωa 2 0 It follows that 1
2 ωa 2 Ploss = − IS Ωa 2 . 2 γ + ωa 2 a

(120)

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We see that the stationary component (with ωa = 0) does not contribute to the loss but the precessing components dissipate. We obtain an order of magnitude for the alignment rate by assuming that (δI)/I ≈ O(1) and that except for the stationary component ωa ≈ ωmax = L/Imax , so that the excess energy is approximately Eexcess ≈ O(1)Imax ωmax 2 sin2 θ

(121)

where θ is the disalignment angle, and the rate of dissipation is approximately 2 P¯loss ≈ O(1)γIS ωmax

γ2

ωmax 2 sin2 θ, + ωmax 2

(122)

so that γalign ≈ γ

IS ωmax 2 . I3 γ 2 + ωmax 2

(123)

The action of the imperfect coupling of the spins to the bulk angular momentum and bulk rotational velocity is somewhat analogous to that of a harmonic balancer on a rotating shaft (such as a crankshaft), a common device in mechanical engineering. Here however the weakly and dissipatively coupled flywheel is replaced by the internal spin degrees of freedom. 19.7. Davis–Greenstein magnetic dissipation If a grain rotates with an angular velocity along the direction of the magnetic field, in the frame co-rotating with the grain, the magnetic field is time independent. However the component of the angular velocity in the plane normal to the magnetic field Ω⊥ gives rise to an oscillating magnetic field of angular frequency Ω⊥ . In general, the magnetic susceptibility at nonzero frequency has a nonzero imaginary part χm,i , where χm (ω) = χm,r (ω) + iχm,i (ω).

(124)

The lag in the response of the magnetization to the applied driving field gives rise to dissipation accompanied by a torque that tends to align (or antialign) the spin with the magnetic field. This effect is known as the Davis–Greenstein mechanism.40 The alignment tendency of the Davis–Greenstein mechanism may be described by the pair of equations 1 L˙ ⊥ = − L⊥ , τDG

L˙  = 0,

(125)

where the decay time τDG indicates the time scale of the Davis–Greenstein relaxation process. In deriving this simple form, we have assumed that τprec  τDG , so that we may average over many precessions. We also assume that the alignment

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with the axis of maximum moment of inertia likewise is very fast. If these assumptions do not apply, an averaged equation of the simple form above does not hold, and these effects need to be analyzed simultaneously. A key issue in evaluating the viability of the Davis–Greenstein mechanism is the frequency of reversals of the direction of the coherent torque, termed “crossover” events in the literature. If we consider for example coherent torques from the formation of molecular hydrogen at a small number of catalysis sites on the grain surface, the orientation of this torque depends on the structure of the outermost surface atomic monolayer, and its structure is likely to change as new layers are deposited or as old layers are eroded away. Consequently, the long term coherence time of the “coherent” torque is uncertain. Under the assumption that the alignment of the body coordinates with the angular momentum is fast, it is the evolution of the coherent torque projected onto the axis of maximum moment of inertia that matters. While the grain rotates suprathermally, its direction is relatively stable against random perturbing torques, but when its rotation velocity is in the process of changing sign and of the same order as the would-be “thermal” rotation velocity, the grain is particularly susceptible to changing orientation as a result of random torques as discussed in detail in Refs. 262 and 263. In deciding whether τDG is too long for producing a significant degree of alignment, we must consider competing effects that tend to disalign the grains, which modify the above equation to become a stochastic differential equation of the form ¯ % 1 $ 1 ¯ eL + L n(t) (L − ˆ eB (ˆ eB · L)) − L − Lˆ L˙ = − 1/2 τDG τrel τrand

(126)

eL = L/|L|, and n(t) is a normalized white noise source whose where ˆ eB = B/|B|, ˆ expectation value is given by ni (t) nj (t ) = δ(t − t ).

(127)

If τrel < ∼ τrand , in the absence of a magnetic field, L undergoes a random walk on the ¯ If τDG  τrand the magnetic dissipation is strong surface of a sphere of radius L. enough to bring about almost complete alignment. However if, on the other hand, τDG τrand , the magnetic dissipation leads to a negligible amount of alignment. 19.8. Alignment along B without Davis–Greenstein dissipation Because of the magnetic precession discussed in Sec. 19.6, an average alignment with the magnetic field direction (or more precisely with a direction very close to the magnetic field direction) can be obtained if there is a tendency toward alignment with some other direction characterized by a time scale talign where talign tprec . The effect of precession about an axis Ω (presumably along ±B) in competition with alignment about another direction n ˆ is described by the ordinary differential

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equation ¯ L˙ = Ω × L − γ(L − n ˆ L).

(128)

This equation has two time scales: Ω−1 and γ −1 . We are primarily interested in fast precession (i.e., Ω γ). Equation (128) is a linear equation whose solution decomposes into a stationary part and transient spiralling in to approach the stationary attractor. We solve      −γ +Ω 0 L⊥1 n ˆ ⊥1     ¯ 0  L⊥2  = −γ L ˆ ⊥2 , (129) −Ω −γ n 0

0

−γ

L

n ˆ

so that L⊥1 =

γ2 ¯ γΩ ¯ Lˆ n⊥2 − 2 Lˆ n⊥1 , γ 2 + Ω2 γ + Ω2

L⊥2 =

γ2 ¯ γΩ ¯ Lˆ n⊥1 + 2 Lˆ n⊥2 , γ 2 + Ω2 γ + Ω2

(130)

¯ n . L = Lˆ ¯ In the fast precession regime of most interest to us, L = LΩ(Ω ·n ˆ )/Ω2 , with corrections suppressed by a factor (γ/Ω). The precession has no alignment tendency of its own, but if the precession period is shorter than the time scale of the alternative alignment mechanism, the precession acts to suppress alignment in the plane normal to Ω. If it turns out that radiative torques combined with the fast precession described above is the solution to the alignment problem, a number of interesting observational consequences follow. Blind data analysis might suggest that one should be able to predict polarization from the thermal dust emission intensity combined with a map of the galactic magnetic field, perhaps deduced from rotation measures and synchrotron emission. However, if the above mechanism is correct, this procedure will not work, even with a three-dimensional model of the unpolarized dust emission and galactic magnetic field. In the above equations, provided that the galactic magnetic field is strong enough, its direction serves to define the direction of the grain alignment but its precise magnitude has little relevance. The above mechanism can also accommodate a large amount of small-scale power in the polarized dust signal, even if both the galactic magnetic field and the dust density are smooth. One ¯ n(x) to might expect the field characterizing the radiative alignment tendency L(x)ˆ include an appreciable component varying on small scales. 19.9. Radiative torques Electromagnetic radiation scattered or absorbed by the dust grain in general exerts a torque on the dust grain, as already modeled in Sec. 19.4 for isotropic illumination,

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which is simpler because the torque direction is constant in the co-rotating body coordinates. Here we generalize to anisotropic illumination. Let Γ(ν, n ˆ ) expressed in body coordinates be the torque exerted on a dust grain by monochromatic electromagnetic radiation of frequency ν and unit spectral radiance propagating in the n ˆ direction. For an illumination described by the spectral radiance I(ν, n ˆ ), the torque exerted on the dust grain in inertial coordinates is given by ∞ dν dˆ nΓ(ν, R−1 (t)ˆ n)I(ν, n ˆ ). (131) Γ(t) = 0

S2

The complication here arises from the need to include both inertial and co-rotating body coordinates, related by the transformation xiner = R(t)xbody . Assuming that the time scale for radiative torques is long compared to the moment of inertia alignment time scale τI−L , we may take Ω to be aligned with L ¯ n ˆ L ), defined as and define an averaged kernel Γ(ν, ˆI ; n 1 ¯ n Γ(ν, ˆI ; n ˆ L) = 2π

0

2π

dχΓ(ν, R(χ,ˆ ˆ I ), nL ) n

(132)

where R(χ,ˆ ˆ L axis. Apart from a few nL ) represents a rotation by χ about the n general observations, little can be said about the form and properties of the radiative torque kernel, which differ from grain to grain. At very long wavelengths — that is, long compared to the size of the dust grain — it is a good approximation to treat the grain as a dipole (as one does for Rayleigh scattering). But in the dipole approximation there are no torques. Therefore, (a/λ) serves as an expansion parameter for which the torque approaches zero as this parameter tends toward zero. Likewise a spherical grain gives a vanishing torque at any frequency as the result of symmetry arguments. Convolving the incident radiation intensity I(ν, n ˆ I ) with this kernel, we obtain the angle-averaged torque Γtot,rad (ˆ nL ). If there were no magnetic field, we could incorporate the torque above into the previously derived stochastic differential equation for the evolution of L [Eq. (103)] to obtain

 ˙L(t) = −βL(t) + β L ¯ L(t) + Γtot,rad L(t) + nran (t), (133) |L(t)| |L(t)| where a coherent torque in body coordinates has been included as well. The above equation describes a random walk L(t) in three-dimensional angular momentum space. Owing to the stochastic random torque noise term, the rotational degree of freedom loses memory of its initial state after some characteristic decay time tran , after which its state can be described by a probability distribution p(L) calculable based on the above equation. The stochastic equation above has the general form L˙ = v(L) − n(t),

(134)

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for which the associated Fokker–Planck equation describing the evolution of the probability density is ∂p(L) = ∇2 p(L) + ∇ · (p(L)v(L)). (135) ∂t With the magnetic field included, we add the magnetic precession term to obtain

 ˙L(t) = −βL(t) + β L ¯ L(t) + Γtot,rad L(t) + γmag L × B + nran (t), (136) |L(t)| |L(t)| where the constant γmag relates the magnetic moment to the angular momentum according to µB = γmag L. When tran , trad tprec , it is possible to reduce the dimension of the stochastic differential equation to two dimensions, namely the angular velocity ω and the angle θBL between the vectors L and B, obtaining θ˙BL = F (θBL , ω) + nθ (t),

ω˙ = G(θBL , ω) + nω (t).

(137)

Unfortunately, very little can be said about the form of the functions F (θBL , ω) and G(θBL , ω), which depends on the detailed grain shapes and other properties. Draine and Weingartner53,54,222 studied three possible irregular grain shapes (shown in Fig. 25), calculating numerically the radiative torque as a function of illumination direction and frequency. Then assuming a certain anisotropic illumination, they calculated the functions F and G in Eq. (137) and studied the solutions to this equation, numerically establishing the orbits in the absence of noise. Their simulations conclude that radiative torques can provide a plausible alignment mechanism. The proposal that radiative torques resulting from anisotropic illumination align the interstellar dust grains with the magnetic field offers a promising alternative to the Davis–Greenstein magnetic dissipation mechanism whose characteristic time scale may be too long to provide the needed degree of alignment. However, its viability is hard to evaluate given our ignorance concerning the shapes of the dust grains. Without such shape information, the best case that can be put forward in favor of this scenario based on radiative torques and anisotropic illumination is to show that some candidate shapes exist that give the right orders of magnitude combined with the argument that the Davis–Greenstein scenario cannot account for the data. It is discouraging that it is not possible to formulate precise predictions for the radiative torque scenario. The origin of grain alignment seems to remain an open question. 19.10. Small dust grains and anomalous microwave emission (AME ) The thermal emission of interstellar dust grains discussed above was based on the assumption that its thermal state may be characterized by a mean temperature (around 20 K) and that fluctuations about this temperature subsequent to the absorption of an UV photon constitute a negligible correction. If a dust grain is

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Fig. 25. A selection of grain shapes for the study of radiative torques. In Ref. 53, Draine and Weingartner calculated the radiative torque on a dust grain as a function of frequency ν and ˆ for the three grain shape illustrated above. (Credit: B. Draine and J. incident photon direction n Weingartner)

sufficiently large, this is a good approximation, but for small molecules an FUV (far ultraviolet) photon can raise the grain temperature momentarily by an order of magnitude or more, allowing emission in bands where there would be almost no emission for a grain at its average temperature. The bulk of the energy absorbed from the UV photon is then rapidly re-emitted through IR fluorescence as the molecule rapidly cools back down to its mean temperature. Hints of such emission requiring a temperature significantly above the mean temperature were noted through the observation of aromatic infrared emission features consistent with C– H bonds on the edge of an aromatic molecule — that is, around 3.3 and 11.3 microns.55,118 Later, analysis of the IRAS dust maps showed that a significant fraction of the UV photons absorbed through extinction were re-emitted in bands of wavelength shorter than 60 µ, where under the assumption of a reasonable average temperature and a modified blackbody law there should be no emission.186 As a consequence, the dust models were revised to extend the size distribution to include small grains thought to be PAHs as a result of the infrared emission features described above. PAHs (polycyclic aromatic hydrocarbons) are essentially small monolayer planar sheets of graphite with hydrogen frills on their edges. A template for the small grain population can be constructed using maps of infrared emission at λ < ∼ 60µ where the thermal emission from large grains does not

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contribute. Since the short wavelength emission from the dust grains is proportional to the local FUV flux density, these templates can in principle be improved by dividing by the local FUV flux density to obtain a better map of the projected small dust grain column density, which in turn can be correlated with the AME (anomalous microwave emission) at low frequencies. It was also pointed out that such a population of very small dust grains would be expected to rotate extremely rapidly, giving rise to electric dipole emission at its rotation frequency and possibly at higher harmonics, since Pe ∼ µE ω 4 and ω 2 ≈ (kB Trot )/ma2 in the absence of suprathermal rotation.5,50,51 The possibility of magnetic dipole emission from small rotating grains has also been investigated.52 Observationally, emission at low frequencies, where diffuse thermal dust emission should be minuscule, was found correlated with high frequency dust maps.42,106,119 Such emission would be consistent with dipole emission from small rapidly rotating dust grains, as described above. As an alternative explanation for such a correlation, it was proposed that regions of free–free emission could be correlated with regions of thermal dust emission. However, this low frequency excess was found not to be correlated with Hα line emission, which serves as a tracer of free–free emission. One possibility to weaken such a correlation would be to postulate an extremely high temperature, so that Γfree–free (T )/ΓHα is large. However, it was found that such a high temperature would require an implausible energy injection rate to prevent this phase from cooling. The spectrum of anomalous microwave emission, presumably from spinning dust, has been measured by taking spectra in targeted regions having a large contribution from AME and subtracting the expected synchrotron, free–free, thermal dust, and primordial CMB signals.88,220 Figure 26 shows the spectrum thus obtained. The observed fall-off at low ν disfavors the hypothesis of free–free emission correlated with thermal dust emission as the origin of the anomalous emission, because in that case, the spectrum of the residual should rise as ν is lowered. Most theoretical studies have assumed that the small grains do not become aligned with the galactic magnetic field, but alignment mechanisms for this grain population have not yet been investigated in detail. Observational limits on the anomalous microwave emission constrain the degree of polarization to less than a few per cent.193 20. Compact Sources The above processes pertain primarily to diffuse galactic emission, which contaminates the primordial CMB signal most severely at low multipole number
. On smaller scales, however, compact sources become more of a concern and eventually become the dominant contribution at all frequencies. We use the term “compact” to denote both unresolved point sources and partially resolved, localized sources. The angular power spectrum of point sources distributed according to a Poissonian distribution would scale with multipole number as
0 . If ignoring the structure

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100

Du st

1000

(V ibr ati on al)

Haslam et al Reich & Reich Andersson et al COSMOSOMAS WMAP DIRBE

Dust (rotaional)

10

S␯ (Jy)

104

105

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1

Free–Free

1

10

100

1000

␯ (GHz) Fig. 26. Consistency of anomalous microwave emission with spinning dust. The multi-frequency observations shown here were taken for a molecular cloud believed to have strong anomalous dust emission. We observe that a linear combination of thermal dust and free–free emission cannot explain the observed shape. However, if an anomalous dust component with the spectral shape predicted by Draine and Lazarian for electric dipole emission from spinning dust is included (dotted curve), a reasonable fit may be obtained. For more details, see Ref. 220. (Credit: COSMOMAS Collaboration)

of the acoustic oscillations, one approximates the primordial CMB spectrum as having the shape
−2 , one would conclude that compact sources rapidly take over at high
. Because of Silk damping combined with the smoothing from the finite width of the last scattering surface, the fall-off of the CMB at large
is even more rapid. The fact that point sources are clustered rather than distributed in Poissonian manner causes the point source power spectrum to be clustered, leading to an excess of power at small
compared to an
0 power law.

20.1. Radio galaxies Section 16 described the physical process of synchrotron emission, emphasizing its contribution to the diffuse emission from our own galaxy. However the same process is also at work in galaxies other than our own, and in particular in the rare but highly luminous radio galaxies at high redshift, extreme processes fueled by a central black hole give rise to intense emission in many bands, including synchrotron emission in the radio and microwave bands. For our purposes, the most salient and relevant property of these sources is their inverted spectrum. For a detailed survey of radio sources, see Ref. 212. Since each source differs slightly in its spectral

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properties, the strategy for dealing with these sources is to mask out the brightest sources and then model the residual arising from the remaining unmasked sources. 20.2. Infrared galaxies In Sec. 18, we discussed at length the contamination from thermal emission from the interstellar dust in our galaxy. Thermal dust emission, however, is not restricted to our galaxy. All galaxies contribute some thermal dust emission. However, the galaxies at high redshift in which star formation is occurring at a rapid rate contribute predominantly to the extragalactic thermal dust emission in the microwave bands of interest to us. In fact, most of the energy of the radiation from these galaxies in which intense star formation is taking place is emitted in the infrared, because the light and UV emitted is absorbed by dust, which becomes heated and in turn re-emits in the infrared. Some of these infrared galaxies are resolved by current observation. Others remain unresolved by present day observations and these are attributed to the diffuse infrared background. These infrared galaxies do not all emit with the same frequency spectrum, in part because they are spread over a wide range of redshifts. For the early discovery papers on the CIB (cosmic infrared background) from COBE see Refs. 187, 85 and 63. For more recent observations from Planck, see Ref. 155. 21. Other Effects 21.1. Patchy reionization In Sec. 7.5, we idealized the reionized hot electron gas extending from intermediate redshifts to today as being isotropically distributed so that the free electron density depends only on redshift. Such a reionization scenario is implausible but provides a good approximation for the effect of reionization on the CMB power spectrum at low
. In a more realistic reionization scenario, the universe becomes reionized in an inhomogeneous manner. As the first stars and quasars form, their UV radiation creates around them spheres of almost completely ionized hydrogen (i.e., HII) that grow and eventually coalesce. These spheres have sharp edges that act to transform large-angle power from the primordial CMB into small-scale power. For some estimates of the impact due to patchy reionization of the angular CMB spectrum, see Ref. 253. 21.2. Molecular lines The sources of diffuse galactic emission discussed so far all have a broad continuum emission spectrum. But there also exist molecular lines, mostly from rotational transitions of molecules, that emit in the microwave bands of interest here. The brightest of these are the CO lines,38 of which the lowest is situated at 115.3 GHz.

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Since CO is a linear molecule, there are also transitions at integer multiples of this frequency. Removing CO emission from the Planck data has proven to be a challenge.163 For more sensitive future experiments, it is likely that other lines will also present concerns. 21.3. Zodiacal emission Besides the planets and asteroids, our solar system is filled with large dust grains, known as zodiacal dust, which on the average are much larger than the interstellar dust grains described above. These grains manifest themselves by a variety of means. In the visible they scatter light from the sun, and this background must be subtracted when analyzing diffuse emission in the optical and near infrared bands. The zodiacal dust is concentrated around the ecliptic plane. It is a good but not perfect approximation to model the distribution of zodiacal dust assuming azimuthal symmetry. The temperature of the zodiacal dust is much above that of the interstellar dust and roughly proportional to the inverse square of the distance from the sun. Subtracting zodiacal emission is challenging because its contribution to a given point on the celestial sphere depends on the time of year when the observation is made, because observations at different times of the year look through different columns of dust, even when azimuthal symmetry is assumed. The ideal observing program for characterizing the zodiacal dust would cover the whole sky many times over the year in order to accumulate as many lines of sight to the same point on the celestial sphere as possible and to carry out a sort of tomography. For a discussion of zodiacal emission at long wavelengths as measured by Planck, see Ref. 164. For previous work on zodiacal emission in the far infrared, see Ref. 64. 22. Extracting the Primordial CMB Anisotropies Most of the contaminants described above, with the exception of gravitational lensing and the kinetic SZ effect, have a frequency dependence that differs from that of the primary CMB anisotropies. This property can be exploited to remove these nonprimordial contaminants and lies at the heart of all component separation techniques. If there are N distinct components, each of which is characterized by a fixed frequency spectrum, in the absence of measurement noise, a perfect separation of these components is possible using sky maps in N frequency bands. With more bands, the separation is overdetermined allowing validation of the model. The component separation strategy sketched above, which may be described as a linear method, is based on a linear model, but this is not the only way to clean maps. More sophisticated nonlinear models may be formulated, often based on or inspired by Bayesian statistics that exploit other properties such as the positivity of certain components, spatial variation of spectral indices, and so on. Some methods are characterized as being “blind,” meaning that the number of components is indicated by data itself, while other methods are more physically based.

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Because of limitations of space, we do not go into the details or practicalities of component separation, instead referring the reader to a few excellent reviews for more details. For an early discussion of the component separation problem and various approaches, see Refs. 24 and 44. Reference 116 includes a comparison of methods used for the Planck analysis. Reference 56 describes foreground removal challenges for future CMB B mode polarization experiments and includes extensive references to the literature. 23. Concluding Remarks Prior to the discovery of the CMB anisotropy in 1992 by the COBE team, cosmology was a field with a questionable reputation. Many from other fields described cosmology as a field where theorists were free to speculate with almost no observational or experimental constraints to contradict them. Much was made of the supposed distinction between an “observation” and an “experiment.”w However, between 1992 and today cosmology has transformed into a precision science where it has become increasingly difficult to propose new models without contradicting present observation. Ironically, much science having only a tenuous connection to cosmology is now being explained to the public as serving to probe the big bang. In this review, we have tried to sketch the most important aspects of the physics of the CMB in a self-contained way. Other reviews with a different emphasis include Refs. 94, 255 and 256. We have seen how the CMB provides a snapshot of the state of the universe at around z ≈ 1100, when the primordial perturbations were still very nearly linear, and how the precise angular pattern on the sky is also sensitive to a number of other cosmological parameters, which can be determined in many cases to the percent level or better. We have also explored tests of non-Gaussianity. At present no non-Gaussianity of a primordial origin has been observed, ruling out many nonminimal models of inflation, in particular multi-field models developed to explain hints of non-Gaussianity in the WMAP data, which subsequently were excluded by Planck. We also saw how better measurements of the CMB absolute spectrum could reveal deviations from a perfect blackbody spectrum, some of which are to be expected and others of which would constitute signs of novel physics, such as energy injection at high-redshift. The absolute spectrum remains a promising frontier of CMB science. The other promising area of CMB research is the characterization of the polarized sky, in particular to search for primordial B modes from inflation, which constitute the most unique prediction of inflation. Maps of B modes on smaller angular scales would also allow a characterization of the gravitational lensing field at much w It

is not clear whether proton–antiproton colliders would be classified as constituting an “experiment” under this criterion, because the experimenter has virtually no control over parameters of the hard parton–parton collisions of interest due to the broad spread in x of the parton distribution functions.

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greater precision than is possible from observations of the temperature anisotropy. Such data would provide an important cross check of weak lensing surveys using the ellipticities of galaxies and provide a unique probe of absolute neutrino masses, perhaps allowing one to distinguish the normal from the inverted mass hierarchy. Since this volume is dedicated to Albert Einstein and the hundredth anniversary of his celebrated paper presenting the general theory of relativity, we close with a few remarks about Einstein, and in particular about his relation to the subject of this chapter. Einstein is remembered by today’s generation of physicists mostly for his role in the development of the special and general theories of relativity. This part of Einstein’s work sometimes overshadows Einstein’s numerous achievements in other areas and the extreme breadth of his scientific interests. Einstein is often viewed today as the paragon of the abstruse, highly mathematical theorist. Many physicists today would be surprised to learn that Einstein patented a design for an ammonia–butane refrigerator.59 Others might be surprised to learn that Einstein is the author of one experimental paper for the work after which the Einstein–de Haas effect is named. This effect may be thought of as the converse of the Barnett effect, which we saw in Sec. 19.5.1 plays a crucial role in the alignment of interstellar dust grains. Moreover, the stochastic differential equations in that section bear a close relation to Einstein’s work on Brownian motion. General relativity was once regarded as an abstruse, highly mathematical subfield of physics. Within astrophysics, relativity was first introduced through the emergence of a subfield known as “relativistic astrophysics.” But nowadays general relativity pervades almost all areas of astrophysics and has become one of the elements of the underlying fundamental physics that every astrophysicist must know. One can no longer neatly separate “relativistic astrophysics” from the rest of astrophysics. This chapter told the story of the big bang cosmology from a point of view emphasizing the observations of the relic blackbody radiation. Einstein’s theory of general relativity culminating in his celebrated 1915 formulation of the gravitational field equationsx,y enters as a key ingredient to this story in two ways. First there are the unperturbed spacetime solutions now known as the FLRW solutions. This family of spatially homogeneous and isotropic solutions to the Einstein equations was first found by Alexandre Friedmann in 1922 and independently by Georges Lemaˆıtre in 1927. At that time, the solutions were surprising because of their prediction of a non-eternal dynamical universe whose natural state was either expansion or contraction, an idea that was before its time. It was not until 1929 that Hubble discovered that the recessional velocities of distant galaxies except for some noise vary approximately linearly with distance (although in its first version x A.

Einstein, “Die Feldgleichungen der Gravitation,” Sitzungsberichte der K¨ onigliche Preussiche Akademie der Wissenschaften, p. 844 (25 November, 1915). y For an authoritative account from a scientific perspective of the path involving over a dozen papers and several reversals that finally led Einstein to the gravitational field equations as we know them today (i.e., Gµν = Rµν − 12 gµν R = (8πGN )Tµν ), see Abraham Pais, Subtle is the Lord (New York, Oxford University Press, 1982).

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the data was not statistically compelling). Einstein’s general theory was deduced by drawing mainly on considerations of mathematical simplicity and elegance and on the equivalence principle, but remarkably applied to the expanding universe through the FLRW solution and its linearized perturbations. General relativity remains the basis of cosmology despite the great improvement in the data that has taken place between the 1920s and now. Today there is much talk about “modified gravity,” largely motivated by the discovery of the so-called “dark energy,” which no one yet really understands. It can be argued whether the dark energy belongs to the left-hand side of the Einstein field equation, in which case it would constitute a revision to Einstein’s theory, or on the right-hand side, in which case it would simply constitute a new form of exotic stress–energy, leaving the gravitational sector unchanged. But so far Einstein’s theory of relativity as a classical field theory has survived without change. To be sure, we believe that gravitational theory as formulated by Einstein is an incomplete story. Where quantum effects become relevant, we know that there must be a more complete theory of which classical relativity is but a limiting case. But we do not yet know what that larger theory is. In the classical regime, Einstein’s theory has survived many stringent tests, and perhaps more importantly has become an indispensable foundation for our current understanding of the universe and its origins.

Acknowledgments MB would like to thank Ken Ganga, Kavilan Moodley, Heather Prince, Doris Rojas, and Andrea Tartari for many invaluable comments, corrections, and suggestions, and Jean-Luc Robert for making several of the figures. MB thanks the University of KwaZulu-Natal for its hospitality where a large part of this review was written. Note added in Proof: Since the writing of this chapter, the Planck 2015 Cosmological Results have appeared in the form of a series of papers of which a complete listing can be found in the overview article.265 Apart from an improved analysis of the temperature anisotropies, this release contains for the first time a detailed analysis of the Planck TE and EE polarization results (except at the lowest multipole numbers, which will be analyzed as part of the 2016 final release). The joint BICEP/Keck Array–Planck analysis of the B mode signal in the BICEP2 patch covering about a square degree of the sky resulted in a new upper limit on the tensorto-scalar ratio of (r < 0.12) at 95% confidence.266 This joint analysis exploited the Planck high-frequency maps dominated by polarized dust emission to clean the dust contribution from the BICEP2/Keck Array maps at 150 GHz. This new limit marks the first time that a limit on r from B modes alone is competitive with limits obtained from the temperature anisotropy. It should be stressed that the limits from B mode polarization alone are much more robust against model assumptions than limits obtained using the temperature anisotropy for the reasons explained in the main text.

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References 1. K. N. Abazajian et al., Neutrino physics from the cosmic microwave background and large scale structure, Astropart. Phys. 63 (2015) 66, arXiv:astro-ph/1309.5383. 2. K. N. Abazajian et al., Inflation physics from the cosmic microwave background and large scale structure, Astropart. Phys. 63 (2015) 55, arXiv:astro-ph/1309.5381. 3. L. F. Abbott and M. B. Wise, Constraints on generalized inflationary cosmologies, Nucl. Phys. B 244 (1984) 541. 4. A. Albrecht, D. Coulson, P. Ferreira and J. Magueijo, Causality, randomness, and the microwave background, Phys. Rev. Lett. 76 (1996) 1413. 5. Y. Ali-Ha¨ımoud, C. M. Hirata and C. Dickinson, A refined model for spinning dust radiation, Mon. Not. R. Astron. Soc. 395 (2009) 1055. 6. R. A. Alpher, H. Bethe and G. Gamov, The origin of chemical elements, Phys. Rev. 73 (1948) 803. 7. V. Acquaviva, N. Bartolo, S. Matarrese and A. Riotto, Gauge-invariant second-order perturbations and non-Gaussianity from inflation, Nucl. Phys. B 667 (2002) 119, arXiv:astro-ph/0209156. 8. K. Arnold et al., The POLARBEAR CMB polarization experiment, Proc. SPIE 7741 (2010) 77411E. 9. D. Babich, P. Creminelli and M. Zaldarriaga, The shape of non-Gaussianities, J. Cosmol. Astropart. Phys. 08 (2004) 009, arXiv:astro-ph/0405356. 10. J. M. Bardeen, Gauge-invariant cosmological perturbations, Phys. Rev. D 22 (1980) 1882. 11. J. M. Bardeen, P. J. Steinhardt and M. S. Turner, Spontaneous creation of almost scale-free density perturbations in an inflationary universe, Phys. Rev. D 28 (1983) 679. 12. R. Barkana and A. Loeb, In the beginning: The first sources of light and the reionization of the universe, Phys. Rep. 349 (2001) 125, arXiv:astro-ph/0010468. 13. E. Battistellie et al., QUBIC: The QU bolometric interferometer for cosmology, Astropart. Phys. 34 (2011) 705, arXiv:astro-ph/1010.0645. 14. R. H. Becker et al., Evidence for reionization at z ∼ 6: Detection of a Gunn–Peterson trough in a z = 6.28 quasar, Astron. J. 122 (2001) 2850, arXiv:astro-ph/0108097. 15. P. de Bernardis et al., A flat universe from high-resolution maps of the cosmic microwave background radiation, Nature 404 (2000) 955, arXiv:astro-ph/0004404. 16. E. Bertschinger, COSMICS: Cosmological initial conditions and microwave anisotropy codes, arXiv:astro-ph/9506070. 17. BICEP2 Collab. (P. A. R. Ade et al.), Detection of B-mode polarization at degree angular scales by BICEP2, Phys. Rev. Lett. 112 (2014) 241101. 18. BICEP2 Collab. (P. A. R. Ade et al.), BICEP2. II. Experiment and three-year data set, Astrophys. J. 792 (2014) 62, arXiv:astro-ph/1403.4302. 19. M. Birkinshaw, The Sunyaev–Zel’dovich effect, Phys. Rep. 310 (1999) 97, arXiv:astro-ph/9808050. 20. R. Blandford and D. Eichler, Particle acceleration at astrophysical shocks: A theory of cosmic ray origin, Phys. Rep. 154 (1987) 1. 21. N. W. Boggess et al., The COBE mission: Its design and performance two years after launch, Astrophys. J. 397 (1992) 420. 22. J. R. Bond and G. Efstathiou, Cosmic background radiation anisotropies in universes dominated by nonbaryonic dark matterm, Astrophys. J. 285 (1984) L45. 23. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1970). 24. F. R. Bouchet and R. Gispert, Foregrounds and CMB experiments: I. Semi-analytical estimates of contamination, New Astron. 4 (1999) 443, arXiv:astro-ph/9903176.

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II-138

M. Bucher

25. S. Boughn and R. Crittenden, A correlation between the cosmic microwave background and large-scale structure in the universe, Nature 427 (2004) 45, arXiv:astroph/0305001. 26. M. Bucher, A. S. Goldhaber and N. Turok, An open universe from inflation, Phys. Rev. D 52 (1995) 3314, arXiv:hep-ph/9411206. 27. M. Bucher and N. Turok, Open inflation with an arbitrary false vacuum mass, Phys. Rev. D 52 (1995) 5538. 28. M. Bucher, K. Moodley and N. Turok, General primordial cosmic perturbation, Phys. Rev. D 62 (2000) 083508, arXiv:astro-ph/9904231. 29. J. E. Carlstrom, G. P. Holder and E. D. Reese, Cosmology with the Sunyaev– Zel’dovich effect, Ann. Rev. Astron. Astrophys. 40 (2002) 643, arXiv:astroph/0208192. 30. J. Chluba and R. M. Thomas, Towards a complete treatment of the cosmological recombination problem, Mon. Not. R. Astron. Soc. 412 (2011) 748, arXiv:astroph/1010.3631. 31. J. Chluba and R. A. Sunyaev, Pre-recombinational energy release and narrow features in the CMB spectrum, Astron. Astrophys. 501 (2009) 29, arXiv:astroph/0803.3584. 32. C. R. Contaldi, J. Magueijo and L. Smolin, Anomalous CMB polarization and gravitational chirality, Phys. Rev. Lett. 101 (2008) 141101, arXiv: astro-ph/0806.3082. 33. COrE Collab. (F. Bouchet et al.), COrE (Cosmic Origins Explorer): A White Paper, arXiv:1102.2181. 34. B. P. Crill et al., SPIDER: A balloon-borne large-scale CMB polarimeter, Proc. SPIE 7010, Space Telescopes and Instrumentation 2008 : Optical, Infrared, and Millimeter, 70102P (July 12, 2008). 35. R. Crittenden, R. L. Davis and P. J. Steinhardt, Polarization of the microwave background due to primordial gravitational waves, Astrophys. J. 417 (1993) L13, arXiv:astro-ph/9306027. 36. R. G. Crittenden and N. Turok, Doppler peaks from cosmic texture, Phys. Rev. Lett. 75 (1995) 2642. 37. R. G. Crittenden and N. Turok, Looking for a cosmological constant with the ReesSciama effect, Phys. Rev. Lett. 76 (1996) 575, arXiv:astro-ph/9510072. 38. T. M. Dame et al., The milky way in molecular clouds: A new complete CO survey, Astrophys. J. 547 (2001) 792. 39. S. Das et al., Detection of the power spectrum of cosmic microwave background lensing by the Atacama Cosmology Telescope, Phys. Rev. Lett. 107 (2011) 021301, arXiv:astro-ph/1103.2124. 40. L. Davis and J. Greenstein, The polarization of starlight by aligned dust grains, Astrophys. J. 114 (1951) 206. 41. P. Day et al., A broadband superconducting detector suitable for use in large arrays, Nature 425 (2003) 817. 42. A. de Oliveira-Costa et al., Galactic microwave emission at degree angular scales, Astrophys. J. 482 (1997) L17, arXiv:astro-ph/9702172. 43. P. de Bernardis, M. Bucher, C. Burigana and L. Piccirillo, B-Pol: Detecting primordial gravitational waves generated during inflation, Exp. Astron. 23 (2009) 5, arXiv:0808.1881. 44. J. Delabrouille, J.-F. Cardoso and G. Patanchon, Multi-detector multi-component spectral matching and applications for CMB data analysis, Mon. Not. R. Astron. Soc. 346 (2003) 1089, arXiv:astro-ph/0211504.

May 4, 2017 9:55 b2167-ch15

Physics of the cosmic microwave background anisotropy

II-139

45. R. H. Dicke, The measurement of thermal radiation at microwave frequencies, Rev. Sci. Instrum. 17 (1946) 238. 46. R. H. Dicke, P. J. E. Peebles, P. G. Roll and D. T. Wilkinson, Cosmic black-body radiation, Astrophys. J. 142 (1965) 414. 47. S. Doyle, P. Mauskopf, J. Naylon, A. Porch and C. Duncombe, Lumped element kinetic inductance detectors, J. Low Temp. Phys. 151 (2008) 530. 48. B. T. Draine, Interstellar dust grains, Ann. Rev. Astron. Astrophys. 41 (2003) 241, arXiv:astro-ph/0304489. 49. B. T. Draine, Astrophysics of dust in cold clouds, in The Cold Universe, Saas-Fee Advanced Course 32, ed. D. Pfenniger (Springer-Verlag, Berlin, 2004), pp. 213–303, arXiv:astro-ph/0304488. 50. B. T. Draine and A. Lazarian, Electric dipole radiation from spinning dust grains, Astrophys. J. 508 (1998) 157, arXiv:astro-ph/9802239. 51. B. T. Draine and A. Lazarian, Diffuse galactic emission from spinning dust grains, Astrophys. J. 494 (1998) L19. 52. B. T. Draine and A. Lazarian, Magnetic dipole microwave emission from dust grains, Astrophys. J. 512 (1999) 740, arXiv:astro-ph/9807009. 53. B. T. Draine and J. C. Weingartner, Radiative torques on interstellar grains. I. Superthermal spin-up, Astrophys. J. 470 (1996) 551. 54. B. T. Draine, J. C. Weingartner and C. Joseph, Radiative torques on interstellar grains. II. Grain alignment, Astrophys. J. 480 (1997) 633. 55. W. W. Duley and D. A. Williams, The infrared spectrum of interstellar dust: Surface functional groups on carbon, Mon. Not. R. Astron. Soc. 196 (1981) 269. 56. J. Dunkley et al., CMBPol mission concept study: Prospects for polarized foreground removal, AIP Conf. Proc. 1141 (2009) 222, arXiv:astro-ph/0811.3915. 57. J. Dunkley, M. Bucher, P. G. Ferreira, K. Moodley and C. Skordis, Fast and reliable MCMC for cosmological parameter estimation, Mon. Not. R. Astron. Soc. 356 (2005) 925, arXiv:astro-ph/0405462. 58. J. Dunkley et al., The Atacama Cosmology Telescope: Cosmological parameters from the 2008 power spectrum, Astrophys. J. 739 (2011) 52, arXiv:astro-ph/1009.0866. 59. A. Einstein and L. Szil´ ard, Refrigeration (Appl: 16 December 1927; Priority: Germany, 16 December 1926) U. S. Patent 1,781,541, 11 November 1930. 60. M. Eisenberg and R. Guy, A proof of the hairy ball theorem, Am. Math. Mon. 86 (1979) 571. 61. EPIC Collab. (J. Bock et al.), Study of the experimental probe of inflationary cosmology EPIC-Intemediate mission for NASA’s einstein inflation probe, arXiv:0906.1188. 62. A. Fert´e and J. Grain, Detecting chiral gravity with the pure pseudospectrum reconstruction of the cosmic microwave background polarized anisotropies, Phys. Rev. D 89 (2014) 103516, arXiv:astro-ph/1404.6660. 63. D. J. Fixsen et al., The spectrum of the extragalactic far-infrared background from the COBE FIRAS observations, Astrophys. J. 508 (1998) 123. 64. D. J. Fixsen and E. Dwek, The zodiacal emission spectrum as determined by COBE and its implications, Astrophys. J. 578 (2002) 1009. 65. D. J. Fixsen et al., The cosmic microwave background spectrum from the full COBE FIRAS data set, Astrophys. J. 473 (1996) 576. 66. R. Flauger, J. C. Hill and D. N. Spergel, Toward an understanding of foreground emission in the BICEP2 region, arXiv:astro-ph/1405.7351. 67. U. Fuskeland, I. K. Wehus, H. K. Eriksen and S. K. Naess, Spatial variations in the spectral index of polarized synchrotron emission in the 9 yr WMAP sky maps, Astrophys. J. 790 (2014) 104, arXiv:astro-ph/1404.5323.

May 4, 2017 9:55 b2167-ch15

II-140

M. Bucher

68. M. Giard, G. Serra, E. Caux, F. Pajot and J. M. Lamarre, First detection of the aromatic 3.3-micron feature in the diffuse emission of the Galactic disk, Astron. Astrophys. 201 (1988) L1. 69. V. L. Ginzburg and S. I. Syrovatsk, Developments in the theory of synchrotron radiation and its reabsorption, Ann. Rev. Astron. Astrophys. 7 (1969) 375. 70. A. S. Goldhaber and M. Nieto, Terrestrial and extraterrestrial limits on the photon mass, Rev. Mod. Phys. 43 (1971) 277. 71. A. S. Goldhaber and M. Nieto, Photon and graviton mass limits, Rev. Mod. Phys. 82 (2010) 939, arXiv:hep-ph/0809.1003. 72. J. R. Gott III, Creation of open universes from de Sitter space, Nature 295 (1982) 304. 73. J. Richard Gott III and T. S. Statler, Constraints on the formation of bubble universes, Phys. Lett. B 136 (1984) 157. 74. J. E. Gunn and B. A. Peterson, On the density of neutral hydrogen in intergalactic space, Astrophys. J. 142 (1965) 1633. 75. A. H. Guth and S. Y. Pi, Fluctuations in the new inflationary universe, Phys. Rev. Lett. 49 (1982) 1110. 76. J. S. Hall, Observations of the polarized light from stars, Science 109 (1949) 166. 77. S. Hanany et al., MAXIMA-1: A measurement of the cosmic microwave background anisotropy on angular scales of 10’-5◦ , Astrophys. J. 545 (2000) L5, arXiv:astroph/0005123. 78. J. L. Han et al., Pulsar rotation measures and the large-scale structure of the galactic magnetic field, Astrophys. J. 642 (2006) 868, arXiv:astro-ph/0601357. 79. N. Hand et al., Evidence of galaxy cluster motions with the kinematic Sunyaev– Zel’dovich effect, Phys. Rev. Lett. 109 (2012) 041101, arXiv:astro-ph/1203.4219. 80. C. G. T. Haslam et al., A 408 MHz all-sky continuum survey. I — Observations at southern declinations and for the north polar region, Astron. Astrophys. 100 (1981) 209. 81. M. Hazumi et al., LiteBIRD: A small satellite for the study of B-mode polarization and inflation from cosmic background radiation detection, Proc. SPIE 8442, Space Telescopes and Instrumentation 2012 : Optical, Infrared, and Millimeter Wave, 844219 (August 22, 2012). 82. J. Kaufman, B. Keating and B. Johnson, Precision tests of parity violation over cosmological distances, arXiv:astro-ph/1409.8242. 83. D. Lyth, Large-scale energy-density perturbations and inflation, Phys. Rev. D 31 (1985) 1792. 84. A. R. Liddle and D. H. Lyth, Cosmological Inflation and Large-Scale Structure (Cambridge University Press, Cambridge, 2009). 85. M. G. Hauser et al., The COBE diffuse infrared background experiment search for the cosmic infrared background. I. Limits and detections, Astrophys. J. 508 (1998) 25. 86. S. W. Hawking, The development of irregularities in a single bubble inflationary universe, Phys. Lett. B 115 (1982) 295. 87. C. M. Hirata and E. R. Switzer, Primordial helium recombination. II. Two-photon processes, Phys. Rev. D 77 (2008) 083007, arXiv:astro-ph/0702144. 88. S. R. Hildebrandt, R. Rebolo, J. A. Rubi˜ no-Mart´ın, R. A. Watson, C. M. Guti´errez, R. J. Hoyland and E. S. Battistelli, COSMOSOMAS observations of the cosmic microwave background and Galactic foregrounds at 11 GHz: Evidence for anomalous microwave emission at high Galactic latitude, Mon. Not. R. Astron. Soc. 382 (2007) 594, arXiv:0706.1873.

May 4, 2017 9:55 b2167-ch15

Physics of the cosmic microwave background anisotropy

II-141

89. W. A. Hiltner, Polarization of light from distant stars by interstellar medium, Science 109 (1949) 165. 90. W. A. Hiltner, On the presence of polarization in the continuous radiation of stars. II., Astrophys. J. 109 (1949) 471. 91. W. Hu, D. Scott and J. Silk, Power spectrum constraints from spectral distortions in the cosmic microwave background, Astrophys. J. 430 (1994) L5, arXiv:astroph/9402045. 92. W. Hu and M. White, Acoustic signatures in the cosmic microwave background, Astrophys. J. 471 (1996) 30, arXiv:astro-ph/9602019. 93. W. Hu and T. Okamoto, Mass reconstruction with CMB polarization, Astrophys. J. 574 (2002) 566, arXiv:astro-ph/0111606. 94. W. Hu and S. Dodelson, Cosmic microwave background anisotropies, Ann. Rev. Astron. Astrophys. 40 (2002) 171–216, arXiv:astro-ph/0110414. 95. W. Hu and G. P. Holder, Model-independent reionization observables in the CMB, Phys. Rev. D 68 (2003) 023001, arXiv:astro-ph/0303400. 96. W. Hu and M. White, A CMB polarization primer, New Astron. 2 (1997) 323, arXiv:astro-ph/9706147. 97. A. F. Illarionov and R. A. Siuniaev, Comptonization, the background-radiation spectrum, and the thermal history of the universe, Sov. Astron. 18 (1975) 691. 98. N. Jarosik et al., First year Wilkinson microwave anisotropy probe (WMAP) observations: On-orbit radiometer characterization, Astrophys. J. 148 (2003) 29, arXiv:astro-ph/0302224. 99. G. Jungman, M. Kamionkowski, A. Kosowsky and D. N. Spergel, Cosmological parameter determination with microwave background maps, Phys. Rev. D 54 (1996) 1332, arXiv:astro-ph/9512139. 100. M. Kamionkowski, A. Kosowsky and A. Stebbins, Statistics of cosmic microwave background polarization, Phys. Rev. D 55 (1997) 7368, arXiv:astro-ph/9611125. 101. M. Kaplinghat, M. Chu, Z. Haiman, G. Holder, L. Knox and C. Skordis, Probing the reionization history of the universe using the cosmic microwave background polarization, Astrophys. J. 583 (2003) 24, arXiv:astro-ph/0207591. 102. R. Keisler et al., A measurement of the damping tail of the cosmic microwave background power spectrum with the South Pole Telescope, Astrophys. J. 743 (2011) 28, arXiv:astro-ph/1105.3182. 103. H. Kodama and M. Sasaki, Cosmological perturbation theory, Prog. Theor. Phys. Suppl. 78 (1984) 1. 104. A. Kogut et al., Dipole anisotropy in the COBE DMR first-year sky maps, Astrophys. J. 419 (1993) 1, arXiv:astro-ph/9312056. 105. A. Kogut et al., The primordial inflation explorer (PIXIE): A nulling polarimeter for cosmic microwave background observations, J. Cosmol. Astropart. Phys. 1107 (2011) 025, arXiv:astro-ph/1105.2044. 106. A. Kogut, A. J. Banday, C. L. Bennett, K. M. Gorski, G. Hinshaw and W. T. Reach, High-latitude galactic emission in the COBE differential microwave radiometer 2 year sky maps, Astrophys. J. 460 (1996) 1. 107. E. Komatsu and D. Spergel, Acoustic signatures of the primary microwave background bispectrum, Phys. Rev. D 63 (2001) 063002, arXiv:astro-ph/000503. 108. A. Kosowsky, Cosmic microwave background polarization, Ann. Phys. 246 (1996) 49, arXiv:astro-ph/9501045. 109. A. Kosowsky, The Atacama Cosmology Telescope, New Astron. Rev. 47 (2003) 939, arXiv:astro-ph/0402234. 110. J. Kraus, Radio Astronomy (McGraw-Hill, New York, 1966).

May 4, 2017 9:55 b2167-ch15

II-142

M. Bucher

111. J. Kovac et al., Detection of polarization in the cosmic microwave background using DASI, Nature 420 (2002) 772, arXiv:astro-ph/0209478. 112. L. Kuzmin, Ultimate cold-electron bolometer with strong electrothermal feedback, Proc. SPIE Conf., Millimeter and Submillimeter Detectors for Astronomy II, Vol. 5498 (2004), pp. 349–361. 113. J. M. Lamarre, Photon noise in photometric instruments at far infrared and submillimeter wavelengths, Appl. Opt. 25 (1986) 870. 114. K. Land and J. Magueijo, Examination of evidence for a preferred axis in the cosmic radiation anisotropy, Phys. Rev. Lett. 95 (2005) 071301, arXiv:astro-ph/0502237. 115. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media: Course of Theoretical Physics, Vol. 8 (First English edition) (Pergamon Press, New York, 1960). 116. S. M. Leach et al., Component separation methods for the Planck mission, Astron. Astrophys. 491 (2008) 597, arXiv:astro-ph/0805.0269. 117. A. T. Lee, P. L. Richards, S. W. Nam, B. Cabrera and K. D. Irwin, A superconducting bolometer with strong electrothermal feedback, Appl. Phys. Lett. 69 (1996) 1801. 118. A. Leger and J. L. Puget, Identification of the “unidentified” IR emission features of interstellar dust? Astron. Astrophys. 137 (1984) L5. 119. E. M. Leitch, A. C. S. Readhead, T. J. Pearson and S. T. Myers, An anomalous component of galactic emission, Astrophys. J. 486 (1997) L23, arXiv:astro-ph/9705241. 120. J. Lesgourgues and S. Pastor, Massive neutrinos and cosmology, Phys. Rep. 429 (2006) 307, arXiv:astro-ph/0603494. 121. A. Lewis, A. Challinor and A. Lasenby, Efficient computation of cosmic microwave background anisotropies in closed Friedmann–Robertson–Walker models, Astrophys. J. 538 (2000) 473. 122. A. Lewis and S. Bridle, Cosmological parameters from CMB and other data: A Monte–Carlo approach, Phys. Rev. D 66 (2002) 103511, arXiv:astro-ph/0205436. 123. A. Lewis and A. Challinor, Weak gravitational lensing of the CMB, Phys. Rep. 429 (2006) 1, arXiv:astro-ph/0601594. 124. E. M. Lifshitz and I. M. Khalatnikov, Investigations in relativistic cosmology, Adv. Phys. 12 (1963) 185. 125. R. Lupton, Statistics in Theory and Practice (Princeton University Press, Princeton, 1993). 126. D. H. Lyth and A. Riotto, Particle physics models of inflation and the cosmological density perturbation, Phys. Rep. 314 (1999) 1, arXiv:hep-ph/9807278. 127. C.-P. Ma and E. Bertschinger, Cosmological perturbation theory in the synchronous and conformal newtonian gauges, Astrophys. J. 455 (1995) 7, arXiv:astroph/9506072. 128. J. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models, J. High Energy Phys. 05 (2003) 013, arXiv:astro-ph/0210603. 129. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995). 130. T. A. Marriage et al., The Atacama Cosmology Telescope: Sunyaev–Zel’dovich — Selected galaxy clusters at 148 GHz in the 2008 survey, Astrophys. J. 737 (2011) 61, arXiv:astro-ph/1010.1065. 131. J. C. Mather et al., A preliminary measurement of the cosmic microwave background spectrum by the cosmic background explorer (COBE) satellite, Astrophys. J. 354 (1990) L37. 132. J. C. Mather et al., Measurement of the cosmic microwave background spectrum by the COBE FIRAS instrument, Astrophys. J. 420 (1994) 439.

May 4, 2017 9:55 b2167-ch15

Physics of the cosmic microwave background anisotropy

II-143

133. J. C. Mather, D. J. Fixsen and R. A. Shafer, Design for the COBE far-infrared absolute spectrophotometer, Proc. SPIE 2019, Infrared Spaceborne Remote Sensing (1993) 168. 134. A. D. Miller et al., A measurement of the angular power spectrum of the CMB from
= 100 to 400, Astrophys. J. 524 (1999) L1, arXiv:astro-ph/9906421. 135. C. Meny et al., Far-infrared to millimeter astrophysical dust emission I: A model based on physical properties of amorphous solids, Astron. Astrophys. 468 (2007) 171, arXiv:astro-ph/0701226. 136. V. F. Mukhanov, Gravitational instability of the universe filled with a scalar field, JETP Lett. 41 (1985) 493. 137. V. F. Mukhanov and G. V. Chibisov, Quantum fluctuation and nonsingular universe (in Russian), JETP Lett. 33 (1981) 532. 138. V. F. Mukhanov and G. V. Chibisov, The vacuum energy and large scale structure of the universe, Sov. Phys. JETP 56 (1982) 258. 139. V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger, Theory of cosmological perturbations, Phys. Rep. 215 (1992) 203. 140. C. B. Netterfield et al., A measurement by BOOMERANG of multiple peaks in the angular power spectrum of the cosmic microwave background, Astrophys. J. 571 (2002) 604, arXiv:astro-ph/0104460. 141. W.-T. Ni, Dilaton field and cosmic wave propagation, Phys. Lett. A 378 (2014) 3413, arXiv:1410.0126. 142. W.-T. Ni, Searches for the role of spin and polarization in gravity, Rep. Prog. Phys. 73 (2010) 056901, arXiv:0912.5057. 143. T. Okamoto and W. Hu, CMB lensing reconstruction on the full sky, Phys. Rev. D 67 (2003) 083002, arXiv:astro-ph/0301031. 144. D. Paradis et al., Variations of the spectral index of dust emissivity from HiGAL observations of the galactic plane, Astron. Astrophys. 520 (2010) L8, arXiv:1009.2779. 145. J. R. Pardo, J. Cernicharo and E. Serabyn, Atmospheric transmission at microwaves (ATM): An improved model for millimeter/submillimeter applications, IEEE Trans. Antennas Propagation 49 (2001) 1683. 146. J. A. Peacock, Cosmological Physics (Cambridge University Press, Cambridge, 1998). 147. P. J. E. Peebles, Lyman A. Page, Jr. and R. B. Partridge (eds.), Finding the Big Bang (Cambridge University Press, Cambridge, England, 2009). 148. P. J. E. Peebles, The Large-Scale Structure of the Universe (Princeton University Press, Princeton, 1980). 149. P. J. E. Peebles, Recombination of the primeval plasma, Astrophys. J. 153 (1968) 1. 150. P. J. E. Peebles and J. T. Yu, Primeval adiabatic perturbation in an expanding universe, Astrophys. J. 162 (1970) 815. 151. U.-L. Pen, U. Seljak and N. Turok, Power spectra in global defect theories of cosmic structure formation, Phys. Rev. Lett. 79 (1997) 1611. 152. U.-L. Pen, D. N. Spergel and N. Turok, Cosmic structure formation and microwave anisotropies from global field ordering, Phys. Rev. D 49 (1994) 692. 153. A. A. Penzias and R. W. Wilson, A measurement of excess antenna temperature at 4080 Mc/s, Astrophys. J. 142 (1965) 421. 154. S. Perlmutter et al., Measurements of Ω and Λ from 42 High-Redshift Supernovae, Astrophys. J. 517 (1999) 565, arXiv:astro-ph/9812133. 155. Planck Collab. (P. A. R. Ade et al.), Planck early results XVIII: The power spectrum of cosmic infrared background anisotropies, Astron. Astrophys. 536 (2011) A18, arXiv:astro-ph/1101.2028.

May 4, 2017 9:55 b2167-ch15

II-144

M. Bucher

156. Planck Collab. (P. A. R. Ade et al.), Planck early results. XX. New light on anomalous microwave emission from spinning dust grains, Astron. Astrophys. 536 (2011) A20, arXiv:1101.2031. 157. Planck Collab. (P. A. R. Ade et al.), Planck intermediate results. XV. A study of anomalous microwave emission in Galactic clouds, Astron. Astrophys. 565 (2014) A103, arXiv:1309.1357. 158. Planck Collab. (P. Ade et al.), Planck intermediate results. XIV. Dust emission at millimetre wavelengths in the Galactic plane, Astron. Astrophys. 564 (2014) A45, arXiv:1307.6815. 159. Planck Collab. (R. Adam et al.), Planck intermediate results. XXX. The angular power spectrum of polarized dust emission at intermediate and high galactic latitudes, arXiv:astro-ph/1409.5738. 160. Planck Collab. (P. Ade et al.), Planck 2013 results. XI. All-sky model of thermal dust emission, Astron. Astrophys. 571 (2014), arXiv:astro-ph/1312.1300. 161. Planck Collab. (P. Ade et al.), Planck 2013 results: I. Overview of products and results, Astron. Astrophys. 571 (2014) A1, arXiv:astro-ph/1303.5062. 162. Planck Collab. (P. Ade et al.), Planck 2013 results: XII. Diffuse component separation, Astron. Astrophys. 571 (2014) A12, arXiv:astro-ph/1303.5072. 163. Planck Collab. (P. Ade et al.), Planck 2013 results: XIII. Galactic CO emission, Astron. Astrophys. 571 (2014) A13, arXiv:astro-ph/1303.5073. 164. Planck Collab. (P. Ade et al.), Planck 2013 results: XIV. Zodiacal emission, Astron. Astrophys. 571 (2014) A14, arXiv:astro-ph/1303.5074. 165. Planck Collab. (P. Ade et al.), Planck 2013 results: XV. CMB power spectra and likelihood, Astron. Astrophys. 571 (2014) A15, arXiv:astro-ph/1303.5075. 166. Planck Collab. (P. Ade et al.), Planck 2013 results: XVI. Cosmological parameters, Astron. Astrophys. 571 (2014) A16, arXiv:astro-ph/1303.5076. 167. Planck Collab. (P. Ade et al.), Planck 2013 results: XVII. Gravitational lensing by large-scale structure, Astron. Astrophys. 571 (2014) A17, arXiv:astro-ph/1303.5077. 168. Planck Collab. (P. Ade et al.), Planck 2013 results: XIX. The integrated Sachs–Wolfe effect, Astron. Astrophys. 571 (2014) A19, arXiv:astro-ph/1303.5079. 169. Planck Collab. (P. Ade et al.), Planck 2013 results: XX. Cosmology from Sunyaev–Zeldovich cluster counts, Astron. Astrophys. 571 (2014) A20, arXiv:astroph/1303.5080. 170. Planck Collab. (P. Ade et al.), Planck 2013 results: XXI. All-sky Compton-parameter map and characterization, Astron. Astrophys. 571 (2014) A21, arXiv:astroph/1303.5081. 171. Planck Collab. (P. Ade et al.), Planck 2013 results: XXII. Constraints on inflation, Astron. Astrophys. 571 (2014) A22, arXiv:astro-ph/1303.5082. 172. Planck Collab. (P. Ade et al.), Planck 2013 results: XXIII. Isotropy and statistics of the CMB, Astron. Astrophys. 571 (2014) A23, arXiv:astro-ph/1303.5083. 173. Planck Collab. (P. Ade et al.), Planck 2013 results: XXIV. Constraints on primordial non-Gaussianity, Astron. Astrophys. 571 (2014) A24, arXiv:astro-ph/1303.5084. 174. Planck Collab. (P. Ade et al.), Planck 2013 results: XXV. Searches for cosmic strings and other topological defects, Astron. Astrophys. 571 (2014) A25, arXiv:astroph/1303.5085. 175. Planck Collab. (P. Ade et al.), Planck 2013 results: XXVI. Background geometry and topology of the Universe, Astron. Astrophys. 571 (2014) A26, arXiv:astroph/1303.5086.

May 4, 2017 9:55 b2167-ch15

Physics of the cosmic microwave background anisotropy

II-145

176. Planck Collab. (P. Ade et al.), Planck 2013 results: XXVII. Doppler boosting of the CMB: Eppur si muove, Astron. Astrophys. 571 (2014) A27, arXiv:astroph/1303.5087. 177. Planck Collab. (P. Ade et al.), Planck 2013 results: XXIX. The Planck catalogue of Sunyaev–Zeldovich sources, Astron. Astrophys. 571 (2014) A29, arXiv:astroph/1303.5089. 178. Planck Collab. (P. Ade et al.), Planck 2013 results. XXVIII. The Planck catalogue of compact sources, Astron. Astrophys. 571 (2014) A28, arXiv:astro-ph/1303.5088. 179. Planck Collab. (N. Aghanim et al.), Planck 2013 results. II. Low frequency instrument data processing, Astron. Astrophys. 571 (2014) A2, arXiv:1303.5063. 180. Planck Collab. (P. A. Ade et al.), Planck 2013 results. VI. High frequency instrument data processing, Astron. Astrophys. 571 (2014) A6, arXiv:1303.5067. 181. Planck Collab. (P. A. Ade et al.), Planck 2013 results. X. HFI energetic particle effects, Astron. Astrophys. 571 (2014) A10, arXiv:1303.5071. 182. Polarbear Collab. (P. A. R. Ade et al.), A measurement of the cosmic microwave background B-mode polarization power spectrum at sub-degree scales with POLARBEAR, Astrophys. J. 794 (2014) 171, arXiv:1403.2369. 183. M. Pospieszalski and E. J. Wollack, Ultralow noise InP field effect transistor radio astronomy receivers: State of the art, 13 Int. Conf. Microwaves, Radar and Wireless Communications, MIKON-2000. 184. J. Preskill, Vortices and monopoles, in Architecture of Fundamental Interactions at Short Distances, ed. P. Ramond (North-Holland, Amsterdam, 1987). 185. PRISM (Polarized Radiation Imaging and Spectroscopy Mission): An Extended White Paper PRISM Collab. (P. Andr´e et al., J. Cosmol. Astropart. Phys. 02 (2014) 006, arXiv:1310.1554. 186. J. L. Puget and A. Leger, A new component of the interstellar matter: Small grains and large aromatic molecules, Ann. Rev. Astron. Astrophys. 27 (1989) 161. 187. J.-L. Puget et al., Tentative detection of a cosmic far infrared background with COBE, Astron. Astrophys. 308 (1996) L5. 188. E. M. Purcell, On the absorption and emission of light by interstellar grains, Astrophys. J. 158 (1969) 433. 189. E. M. Purcell, Suprathermal rotation of interstellar grains, Astrophys. J. 231 (1979) 404. 190. C. L. Reichardt et al., A measurement of secondary cosmic microwave background anisotropies with two years of South Pole Telescope Observations, Astrophys. J. 755 (2012) 70, arXiv:astro-ph/1111.0932. 191. Y. Rephaeli, Comptonization of the cosmic microwave background: The Sunyaev– Zeldovich effect, Ann. Rev. Astron. Astrophys. 33 (1995) 541. 192. A. G. Riess et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant, Astron. J. 116 (1998) 1009, arXiv:astroph/9805201. 193. J. A. Rubi˜ no-Mart´ın, C. H. L´ opez-Caraballo, R. G´enova-Santos and R. Rebolo, Observations of the polarisation of the anomalous microwave emission: A review, Advances in Astronomy 2012 (2012) 351836. 194. J. Ruhl et al., The South Pole Telescope, Proc. SPIE 5498 (2004) 11, arXiv:astroph/0411122. 195. R. K. Sachs and A. M. Wolfe, Perturbations of a cosmological model and angular variations of the microwave background, Astrophys. J. 147 (1967) 73. 196. S. Seager, D, Sasselov and D. Scott, A new calculation of the recombination epoch, Astrophys. J. 523 (1999) L1, arXiv:astro-ph/9909275.

May 4, 2017 9:55 b2167-ch15

II-146

M. Bucher

197. S. Seager, D. Sasselov and D. Scott, How exactly did the universe become neutral? Astrophys. J. Suppl. 128 (2000) 407, arXiv:astro-ph/9912182. 198. K. Sellgren, The near-infrared continuum emission of visual reflection nebulae, Astrophys. J. 277 (1984) 623. 199. U. Seljak, A two-fluid approximation for calculating the cosmic microwave background anisotropies, Astrophys. J. 435 (1994) L87. 200. U. Seljak and M. Zaldarriaga, A line-of-sight integration approach to cosmic microwave background anisotropies, Astrophys. J. 469 (1996) 437, arXiv:astroph/9603033. 201. J. Silk, Cosmic black-body radiation and galaxy formation, Astrophys. J. 151 (1968) 459. 202. J. Silk and J. Chluba, Next steps for cosmology, Science 344 (2014) 586. 203. K. M. Smith, O. Zahn and O. Dor´e, Detection of gravitational lensing in the cosmic microwave background, Phys. Rev. D 76 (2007) 043510, arXiv:astro-ph/0705.3980. 204. G. Smoot et al., COBE differential microwave radiometers: Instrument design and implementation, Astrophys. J. 360 (1990) 685S. 205. G. F. Smoot et al., Structure in the COBE differential microwave radiometer firstyear maps, Astrophys. J. 396 (1992) L1. 206. L. Sorbo, Parity violation in the cosmic microwave background from a pseudoscalar inflaton, J. Cosmol. Astropart. Phys. 1106 (2011) 003, arXiv:astro-ph/1101. 1525. 207. L. Spitzer, Jr. and J. W. Tukey, A theory of interstellar polarization, Astrophys. J. 114 (1951) 187S. 208. R. A. Sunyaev and Y. B. Zeldovich, Small-scale fluctuations of relic radiation, Astrophys. Space Sci. 7 (1970) 3. 209. A. A. Starobinsky, Dynamics of phase transition in the new inflationary universe scenario and generation of perturbations, Phys. Lett. B 117 (1982) 175. 210. A. Stuart and K. Ord, Kendall’s Advanced Theory of Statistics, Classical Inference, Vol. 2 (Oxford University Press, New York, 1991). 211. E. R. Switzer and C. M. Hirata, Primordial helium recombination. I. Feedback, line transfer, and continuum opacity, Phys. Rev. D 77 (2008) 083006, arXiv:astroph/0702143. 212. A. Taylor, K. Grainge, M. Jones, G. Pooley, R. Saunders and E. Waldram, The radio source counts at 15 GHz and their implications for cm-wave CMB imaging, Mon. Not. R. Astron. Soc. 327 (2001) L1, arXiv:astro-ph/0102497. 213. N. Turok (ed.), Critical Dialogues in Cosmology (World Scientific Publishing, Singapore, 1997). 214. A. van Engelen et al., A measurement of gravitational lensing of the microwave background using South Pole Telescope data, Astrophys. J. 756 (2012) 142, arXiv:1202.0546. 215. P. Vielva, E. Mart´ınez-Gonz´ alez, M. Cruz, R. B. Barreiro and M. Tucci, Cosmic microwave background polarization as a probe of the anomalous nature of the cold spot, Mon. Not. R. Astron. Soc. 410 (2011) 33. 216. P. Vielva, E. Mart´ınez-Gonz´ alez, R. B. Barreiro, J. L. Sanz and L. Cay´ on, Detection of non-gaussianity in the Wilkinson Microwave Anisotropy Probe first-year data using spherical wavelets, Astrophys. J. 609 (2004) 22. 217. A. Vilenkin and E. P. S. Shellard, Cosmic Strings and Other Topological Defects (Cambridge University Press, Cambridge, 1994). 218. Wald, General Relativity (University of Chicago Press, Chicago, 1984).

May 4, 2017 9:55 b2167-ch15

Physics of the cosmic microwave background anisotropy

II-147

219. T. P. Walker, G. Steigman, H.-S. Kang, D. M. Schramm and K. A. Olive, Primordial nucleosynthesis redux, Astrophys. J. 376 (1991) 51. 220. R. A. Watson et al., Detection of anomalous microwave emission in the perseus molecular cloud with the COSMOSOMAS experiment, Astrophys. J. 624 (2005) L89, arXiv:astro-ph/0503714. 221. S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (J. Wiley & Sons, New York, 1972). 222. J. C. Weingartner and B. T. Draine, Radiative torques on interstellar grains. III. Dynamics with thermal relaxation, Astrophys. J. 589 (2003) 289. 223. R. Williamson et al., A Sunyaev–Zel’dovich-selected sample of the most massive galaxy clusters in the 2500 deg2 South Pole Telescope survey, Astrophys. J. 738 (2011) 139, arXiv:astro-ph/1101.1290. 224. M. White and W. Hu, The Sachs–Wolfe effects, Astron. Astrophys. 321 (1997) 8–9, arXiv:astro-ph/9609105. 225. E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (Cambridge University Press, Cambridge, 1917). 226. WMAP Collab. (C. L. Bennett et al.), First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Preliminary maps and basic results, Astrophys. J. Suppl. 148 (2003) 1. 227. WMAP Collab. (G. Hinshaw et al.), First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Data processing methods and systematic error limits, Astrophys. J. Suppl. 148 (2003) 63. 228. WMAP Collab. (D. N. Spergel et al.), First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Determination of cosmological parameters, Astrophys. J. Suppl. 148 (2003) 175. 229. WMAP Collab. (C. L. Bennett et al.), First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Foreground emission, Astrophys. J. Suppl. 148 (2003) 97. 230. WMAP Collab. (H. V. Peiris et al.), First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Implications for inflation, Astrophys. J. Suppl. 148 (2003) 213. 231. WMAP Collab. (L. Page et al.), First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Interpretation of the TT and TE angular power spectrum peaks, Astrophys. J. Suppl. 148 (2003) 233. 232. WMAP Collab. (A. Kogut et al.), First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Temperature-polarization correlation, Astrophys. J. Suppl. 148 (2003) 161. 233. WMAP Collab. (E. Komatsu et al.), First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Tests of gaussianity, Astrophys. J. Suppl. 148 (2003) 119. 234. WMAP Collab. (G. Hinshaw et al.), First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: The angular power spectrum, Astrophys. J. Suppl. 148 (2003) 135. 235. WMAP Collab. (A. Kogut et al.), Three-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Foreground polarization, Astrophys. J. Suppl. 665 (2007) 355. 236. WMAP Collab. (G. Hinshaw et al.), Three-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Temperature analysis, Astrophys. J. Suppl. 170 (2007) 288.

May 4, 2017 9:55 b2167-ch15

II-148

M. Bucher

237. WMAP Collab. (L. Page et al.), Three-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Polarization analysis, Astrophys. J. Suppl. 170 (2007) 335. 238. WMAP Collab. (D. N. Spergel et al.), Three-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Implications for cosmology, Astrophys. J. Suppl. 170 (2007) 377. 239. WMAP Collab. (J. Dunkley et al.), Five-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Bayesian estimation of CMB polarization maps, Astrophys. J. Suppl. 701 (2009) 1804–1813. 240. WMAP Collab. (G. Hinshaw et al.), Five-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Data processing, sky maps, and basic results, Astrophys. J. Suppl. 180 (2009) 225–245. 241. WMAP Collab. (B. Gold et al.), Five-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Galactic foreground emission, Astrophys. J. Suppl. 180 (2009) 265–282. 242. WMAP Collab. (M. Nolta et al.), Five-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Angular power spectra, Astrophys. J. Suppl. 180 (2009) 296– 305. 243. WMAP Collab. (J. Dunkley et al.), Five-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Likelihoods and parameters from WMAP Data, Astrophys. J. Suppl. 180 (2009) 306–329. 244. WMAP Collab. (E. Komatsu et al.), Five-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Cosmological interpretation, Astrophys. J. Suppl. 180 (2009) 330–376. 245. WMAP Collab. (N. Jarosik et al.), Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Sky maps, systematic errors, and basic results, Astrophys. J. Suppl. 192 (2011) 14J. 246. WMAP Collab. (B. Gold et al.), Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Galactic foreground emission, Astrophys. J. Suppl. 192 (2011) 15G. 247. WMAP Collab. (D. Larson et al.), Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Power spectra and WMAP-derived parameters, Astrophys. J. Suppl. 192 (2011) 16L. 248. WMAP Collab. (C. Bennett et al.), Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Are there cosmic microwave background anomalies? Astrophys. J. Suppl. 192 (2011) 17. 249. WMAP Collab. (E. Komatsu et al.), Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Cosmological interpretation, Astrophys. J. Suppl. 192 (20011) 18K. 250. WMAP Collab. (C. L. Bennett et al.), Nine-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Final maps and results, Astrophys. J. Suppl. 208 (2013) 20B. 251. WMAP Collab. (G. F. Hinshaw et al.), Nine-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Cosmology results, Astrophys. J. Suppl. 208 (2013) 19H. 252. K. Yamamoto, M. Sasaki and T. Tanaka, Large angle CMB anisotropy in an open universe in the one-bubble inflationary scenario, Astrophys. J. 455 (1995) 412, arXiv:astro-ph/9501109. 253. O. Zahn, M. Zaldarriaga, L. Hernquist and M. McQuinn, The influence of nonuniform reionization on the CMB, Astrophys. J. 630 (2005) 657, arXiv:astro-ph/0503166.

May 4, 2017 9:55 b2167-ch15

Physics of the cosmic microwave background anisotropy

II-149

254. M. Zaldarriaga and U. Seljak, All-sky analysis of polarization in the microwave background, Phys. Rev. D 55 (1997) 1830, arXiv:astro-ph/9609170. 255. P. J. E. Peebles, Cosmic background temperature anisotropy in a minimal isocurvature model for galaxy formation, Astrophys. J. 315 (1987) L73. 256. D. Samtleben, S. Staggs and B. Winstein, The cosmic microwave background for pedestrians: A review for particle and nuclear physicists, Ann. Rev. Nucl. Part. Sci. 57 (2007) 245, arXiv:0803.0834 [astro-ph]. 257. M. Kamionkowski and A. Kosowsky, The cosmic microwave background and particle physics, Ann. Rev. Nucl. Part. Sci. 49 (1999) 77, arXiv:astro-ph/9904108. 258. R. Fabbri and M. Pollock, The effect of primordially produced gravitons upon the anisotropy of the cosmological microwave background radiation, Phys. Lett. B 125 (1983) 445. 259. V. Rubakov, M. Sazhin and A. Veryaskin, Graviton creation in the inflationary universe and the grand unification scale, Phys. Lett. B 115 (1982) 189. 260. A. Starobinsky, Spectrum of relict gravitational radiation and the early state of the universe, JETP. Lett. 30 (1979) 682. 261. A. Starobinsky, Cosmic background anisotropy induced by isotropic flat-spectrum gravitatinal-wave perturbations, Sov. Astron. Lett. 11 (1985) 133. 262. L. Spitzer Jr. and T. A. McGlynn, Disorientation of interstellar grains in suprathermal rotation, Ap. J. 231 (1979) 417. 263. A. Lazarian and B. Draine, Thermal flipping and thermal trapping: New elements in grain dynamics, Ap. J. 516 (1999) 37. 264. A. Blanchard and J. Schneider, Gravitational lensing effect on the fluctuations of the cosmic background radiation, Astron. Astrophys. 184 (1987) 1. 265. Planck Collab. (R. Adam et al.), Planck 2015 results. I. Overview of products and scientific results, arXiv:1502.01582. 266. BICEP2/Keck and Planck Collab. (P. A. R. Ade et al.), A joint analysis of BICEP2/Keck Array and Planck data, Phys. Rev. Lett. 114 (2015) 101301, arXiv:1502.00612.

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Chapter 16 SNe Ia as a cosmological probe∗

Xiangcun Meng†, Yan Gao‡ and Zhanwen Han§ BPS Group, Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216, P. R. China †[email protected] ‡[email protected] §[email protected]

Type Ia supernovae (SNe Ia) luminosities can be corrected in order to render them useful as standard candles that are able to probe the expansion history of the universe. This technique was successfully applied to discover the present acceleration of the universe. As the number of SNe Ia observed at high redshift increases and analysis techniques are perfected, people aim to use this technique to probe the equation-of-state of the dark energy (EOSDE). Nevertheless, the nature of SNe Ia progenitors remains controversial and concerns persist about possible evolution effects that may be larger and harder to characterize than the more obvious statistical uncertainties. Keywords: SNe Ia; dark energy. PACS Number(s): 97.60.Bw, 98.80.−k, 95.36.+x

1. Introduction “Guest stars”, the name given to supernovae by ancient Chinese astronomers, have been recorded for several thousands of years, but the first modern study of supernovae began on 31 August 1885. On that day, Hartwig discovered a “nova” in M31, which disappeared 18 months later. In 1919, Lundmark1 noticed that the distance to M31 is about 0.7 million ly (later studies found it to be actually 2.5 million ly), which implies that the Hartwig “nova” is brighter than other novae by three orders of magnitude. The term “supernova” was coined shortly after. Later, it was realized that supernovae can be a powerful tool for measuring the distance to extragalactic sources. However, for want of detailed enough observed spectra, supernovae were not classified until 1940.2 With the advent of consistently accurate observed spectra, it was finally realized that there are two physically distinct classes of supernovae: thermonuclear supernovae, i.e. Type Ia supernovae (SNe Ia), and core collapse supernovae, including SN IIP, IIL, IIn, IIb, Ib and Ic.3 Type I supernovae are distinguished by the absence of hydrogen lines in their optical spectra. SNe Ia are defined by a deep absorption trough around 6150 ˚ A, which is now known to be due to silicon, additionally.3 II-151

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Due to the homogeneity of their properties, SNe Ia are often used to measure distance. Here, we will review the methods commonly used to standardize SNe Ia for this purpose in Sec. 2, and the progenitor problem in Sec. 3. In Sec. 4, we discuss the effect of different populations of SNe Ia on their brightness. In Sec. 5, we will present the application of SNe Ia in cosmology, i.e. the discovery of accelerating cosmic expansion, and the efforts to probe the equation-of-state of the dark energy (EOSDE). Finally, we summarize the future promise for SNe Ia as a probe to precision cosmology in Sec. 6. 2. SNe Ia as a Standardizable Distance Candle Generally, for two nearby objects (z
1) with astronomy apparent magnitudes m1 and m2 , an Euclidean spatial geometry can be assumed, the ratio of their apparent brightness E1 /E2 , is related by the following equation: m2 −m1 E1 = 100 5 , E2

(1)

which indicates that if the apparent magnitude of one object is lower than that of another one by 5 magnitudes, the object is brighter than the other by 100 times. For the same object, its apparent magnitude at 10 pc is defined as the absolute magnitude, which may reflect its intrinsic luminosity. Assuming the absence of extinction by dust, its apparent magnitude, m, and absolute magnitude, M , follow the equation: m−M d2 E10 = 100 5 = 2 , (2) E 10 where E10 is the apparent brightness at 10 pc, E the observed brightness, and d the distance to the object in units of parsecs. Equation (2) basically expresses the inverse-square law between apparent brightness and distance. Therefore, apparent magnitude, absolute magnitude and distance are related by the following equation:

M = m + 5 − 5 log d,

(3)

and m − M = 5 log d − 5 is called distance modulus. As we shall discuss later in Sec. 5, at larger redshifts, for which z
1 does not hold, the relation between apparent and absolute magnitudes is no longer so simple and depends on the details of the cosmological model assumed. If there is a method to independently obtain the absolute magnitude of an object, the distance may be obtained by measuring its apparent magnitude, like Eq. (3), and then this object may be taken as a standard candle to measure distance. SNe Ia are excellent candidates for standard candles after being corrected by some methods. Among all the subclasses of different supernovae,3 SNe Ia are the most homogeneous, with many practically identical properties, e.g. lightcurve shape, maximum luminosity, spectrum, and as a result, SNe Ia were taken to be perfect standard candles.4 However, 1991 was a fateful year for SNe Ia studies, two peculiar SNe Ia were

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Fig. 1. The B-band lightcurves for different SNe Ia observed. (For color version, see page IICP10.) Source: From Ref. 9.

found, SN 1991bg and 1991T. SN 1991bg was fainter than normal SNe Ia by about 2 magnitudes in the V-band,5,6 while SN 1991T was brighter than normal SNe Ia by 0.4 magnitudes.7,8 The discovery of these two peculiar SNe Ia implies a distance error of about 30% if SNe Ia are assumed to be perfect standard candles. As shown by the collection of different supernova lightcurves in Fig. 1, different SNe Ia have different peak brightnesses. It became a matter of critical importance to find a way to reduce the distance error originating from this heterogeneity of SNe Ia.106 In 1993, Phillips10 discovered that the absolute magnitude at maximum light of SNe Ia and the speed at which the luminosity fades in the B-band (blue light) over the first 15 days after the maximum light are related, as shown in Fig. 2. This relation implies that the brightness of SNe Ia is mainly dominated by one parameter, and it is widely agreed that this parameter is the amount of 56 Ni produced during the supernova explosion that determines its maximum luminosity. Actually, one may arrive at another conclusion from Fig. 2, that the intrinsic magnitude dispersion of SNe Ia in the I-band is smaller than those in the B and V bands, i.e. the infrared measurement of SNe Ia may yield a more precise distance. However, several years later, after increasingly dim SNe Ia were included, the linear relation was found to be a quadratic or an exponential relation as shown in Fig. 3.11,12 Although this relation is widely accredited to Phillips, it was originally discovered by Rust13 and Pskovskii,14,15 who noticed the correlation between the maximum light of SNe Ia and their light decline rate.

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Fig. 2. The correlation between the absolute magnitudes of SNe Ia in B, V and I bands and ∆m15 (B) which measures the B-band lightcurve drops during the first 15 days after the maximum light. Source: From Ref. 10.

Following the discovery by Phillips,10 several groups developed further improved methods to reduce the distance error, and SNe Ia were not a standard candle any more, but a distance indicator. Most notably, Riess et al.16,17 developed the multicolor lightcurve shapes (MLCS) method, which is known as the most mathematically sound method for inferring a distance. The method accounts for the nonuniform error on the distance from the effects of different bands and different supernova colors. In this method, a “training set” of several Type Ia supernova (SN Ia) lightcurves was constructed, and a complete model of the lightcurves is dependent on three parameters, i.e. a distance modulus, a brightness offset which is called “luminosity correction” and an extinction value. To calibrate the brightness of the training set, the distance modulus and the extinction value are obtained by independent methods. Actually, the brightness offset reflects that an SN Ia with a broader lightcurve has a higher peak luminosity, i.e. the Phillips relation. The reason that a correction was made for color is that redder SNe Ia are generally less luminous, both in intrinsic terms and for dust reddening considerations.16,18

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Fig. 3. Modified Phillips’ relation with the so-called peculiar SNe added (filled points). The solid line is an exponential fit for all the data. Source: From Ref. 12.

Compared to the Phillips relation, although the MLCS method does not significantly increase the precision of distance measurements, the method may be the most complete and rigorous one mathematically. Figure 4 shows the typical dispersions of light and color curves after correction by the MLCS method, and from the figure, we can see that the SNe Ia can act as very good distance indicators, because the dispersion at peak brightness is very small. Almost at the same time, Perlmutter et al.19,20 developed another tool, named the stretch factor method. This method has a similar distance precision compared with the MLCS method, as shown in Fig. 5, where the data are the same to those in Fig. 1. Again, SNe Ia are proven to be potent distance indicators. It is worth noting that the MLCS method and the stretch factor method essentially take advantage of the same underlying phenomena as that underlying the Phillips relation, i.e. the slower the lightcurve evolves, the brighter the SN Ia. To obtain the parameters crucial to the implementation of these methods, various algorithms have been developed to fit the lightcurves of SNe Ia data, such as BATM, MLCS2k2, SALT2 and SiFTO.21–26

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Fig. 4. Dispersions of noise-free light and color curves after correction by MLCS method, where the “gray snake” present 1σ-confidence region. Source: From Ref. 16.

Other research groups have also tried to find other methods to obtain more accurate SNe Ia distance measurements. For example, in 1995, Nugent et al.27 noticed a linear relation between the maximum luminosity of an SN Ia and the ratio of the fractional depths of the bluer to that of the redder absorption troughs of its silicon absorption line, or the flux ratio over the Ca II, H and K absorption features in a spectrum near maximum light (see also Refs. 28 and 29). SNe Ia were therefore calibrated to be practical standard candles, and have been applied successfully for cosmological purposes, ultimately leading to the discovery of the accelerating expansion of the universe.17,20 The above methods generally decrease the distance error to 5–10%. However, for a well-observed sample, the typical dispersion on a Hubble diagram may be as small as 0.12 magnitudes, indicating a typical distance accuracy of 5% for a single SN Ia. In 2005, Wang et al.30 found that the peak brightness of SNe Ia and

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Fig. 5. The lightcurves for different SNe Ia after correction by stretch factor method, where the data are the same to that in Fig. 1. (For color version, see page II-CP10.) Source: From Ref. 9.

their B–V colors at 12 days after peak B-band luminosity were strongly correlated. Calibrating SNe using this correlation, the typical error of the SNe Ia on a Hubble diagram can be reduced to less than 0.12 magnitudes. For a typical sample of SNe Ia with little reddening from host galaxies, the scatter may drop further to only 0.07 magnitudes, or 3–4% in distance.30,31 Here, it must be emphasized that for most cases, the observational error is not mainly due to imperfections in the observational apparatus, but rather from a lack of uninterrupted telescope time, i.e. it is difficult to obtain a complete lightcurve for each single object. Recently, in addition to the lightcurves and the colors of the SNe Ia, some spectra properties were found to be helpful for improving the accuracy of derived distances to SNe Ia.28,29,32,33 For precision cosmology, within most redshift intervals, systematic errors from the above correlations dominate over the statistical errors, the latter of which can be dealt with by increasing the size of an observational sample.34–36 The dominant systematic uncertainties for SNe Ia are survey-dependent. As far as what the uncertainties are and how they can be improved are concerned, one may refer to relevant detailed reviews.37–39 3. Progenitors of SNe Ia In 1960, Hoyle and Fowler40 suggested that a degenerate white dwarf (WD) may ignite a runaway thermonuclear fusion in its center, and the released energy may disrupt the whole WD, resulting in an SN Ia explosion. Owing to a series of breakthroughs on the subject since then, it has been indisputably proven that all SNe Ia arise from such explosions of carbon–oxygen white dwarfs (CO WDs) in binary

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systems, of which the most substantial evidence is derived from the observation of SN 2011fe.41 For a CO WD to undergo such an explosion, it must reach or exceed a mass of about 1.4 M
, henceforth referred to as the Chandrasekhar mass limit. The limit was named after Subrahmanyan Chandrasekhar, who was the first person to work out that there was a maximum mass for a WD to be supported by electron degeneracy pressure. He did this by combining quantum theory with relativity, and it was the first time to make a quantitative prediction about the maximum mass of WDs. In 1983, he was awarded the Nobel Prize in Physics mainly due to the discovery of the mass limit of WDs. However, the mass of a CO WD at birth cannot be more than 1.1 M
,42,43 according to canonical stellar evolutionary theory, and therefore the CO WD must accrete additional material somehow (i.e. from its companion in a binary system), and gradually increase its mass to the aforementioned maximum stable mass before it can achieve this state and undergo an SN explosion. The details of this accretion process will be discussed later. During an SN Ia explosion, nearly half of the mass involved is depleted and synthesized into radioactive 56 Ni, the radioactive energy of which is then injected into the supernova ejecta, heating it to the point where it becomes very luminous, ultimately resulting in the emissions which we observe.44 The amount of 56 Ni resulting from an SN explosion is the dominant determining factor behind the maximum luminosity of the SN Ia.45 However, despite all that we know, there still remain quite a few unresolved problems where SNe Ia are concerned. To name but a few: what are the progenitors of SNe Ia, or how does a CO WD increase its mass to the Chandrasekhar limit? How exactly does the WD explode? Do the basic properties of SNe Ia remain constant with redshift? Among all these basic problems, the most urgent is the one regarding the progenitors of SNe Ia, which can potentially affect the precision of distance measurement via SNe Ia, and hinder the development of precision cosmology. More specifically, the progenitor problem is important, for cosmology as well as for other areas of astrophysics, in the following ways. (i) When measuring the cosmological parameters with SNe Ia, one needs to know of any possible evolution of the luminosity and birth rate of SNe Ia as a function of redshift, which is mainly determined by the progenitor model. (ii) The progenitor model provides basic input parameters for explosion models of SNe Ia. (iii) Galactic chemical evolution models require some basic parameter input, such as the stellar feedback from SNe Ia, both chemical and radiative. These basic input parameters are closely related to the progenitor models of SNe Ia. (iv) The identification of the progenitor systems of SNe Ia may constrain binary evolution theory.46–49 Over the course of the last three decades, several progenitor models were discussed by many research groups. These models may be categorized based on the nature of the proposed companion of the CO WD, and we list the most notable ones below. (i) The single degenerate (SD) scenario50,51 : In this scenario, the companion may be a main sequence star, a subgiant, a red giant or a helium star.52–57 The accreted hydrogen-rich or helium-rich material undergoes stable burning on the

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surface of a CO WD, yields carbon and oxygen, which is then deposited onto the WD. When the WD mass reaches some value approaching the Chandrasekhar mass limit, an SN Ia is produced. The companion survives after the supernova explosion. This scenario is widely accepted and is the leading scenario. (ii) The double degenerate (DD) scenario: In this scenario, a binary system comprising two CO WDs loses its orbital angular momentum by means of gravitational radiation, and merges as a consequence. If the total mass of the merger exceeds the Chandrasekhar mass limit, an SN Ia may be expected.58–61 In this scenario, the merger is disrupted completely and the companion does not survive. Note, however, it is possible that the merger product may experience a core collapse rather than a thermonuclear explosion.62,63 The SD and DD scenarios are the two main competing scenarios in the SN Ia community. Generally, to form an SD system leading to an SN Ia, the primordial binary system needs to experience one common envelope phase, while to form a DD system, the situation is more complicated and one extra common envelope phase can be necessary. For example, in Fig. 6, three subchannels lead to a WD + main sequence system ending in an SN Ia explosion, and for every subchannel, the system experiences a single common envelope phase only (see details in Wang and Han48 ).

Fig. 6. Schematic for the subchannels producing WD + MS systems, which may lead to SNe Ia. For every subchannel, the binary system experiences a common envelope phase. (For color version, see page II-CP11.) Source: From Ref. 48.

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In addition to the models mentioned above, the double-detonation model (subChandrasekhar mass model) is also frequently discussed. In this model, the companion of the CO WD is a helium WD or a helium star, which fills its Roche lobe and initiates a stable mass transfer. If the mass-transfer rate is not high enough, the helium material may not burn stably, and as a consequence gradually accumulates on the CO WD. When the helium layer is massive enough, a detonation, which is a supersonic blast wave, is ignited at the bottom of the helium layer, in which the critical mass to ignite a detonation is generally dependent on the WD mass and the mass-transfer rate.64 At the moment of ignition, WD is generally less massive than the Chandrasekhar mass limit. The resulting detonation wave compresses the inner CO material, leading to a second detonation near the core of the CO WD.65–67 After the supernova explosion, a low mass helium star or a low mass helium WD may survive to form a hypervelocity star, whose velocity is high enough to exceed the escape velocity of its parent galaxy.68 Remarkable progress has been made on this model recently, though many problems still exist.69,70 If this model does indeed produce SNe Ia, it would contribute quite significantly to SNe Ia rates.71 Besides the SD and DD scenarios and the double detonation model, other similar models have been proposed to explain the diversity generally observed among SNe Ia, such as the super-Chandrasekhar mass model, the single star model, the spinup/spin-down model, the delayed dynamical instability model, the core-degenerate model, the model of a collision between two WDs, and the model of WDs near black holes, etc. (See Wang and Han48 for a review.) At present, it is premature to exclude any of these models conclusively, and no single model can explain all the properties of SNe Ia alone. It is possible that many of these models, or at least both the SD and DD scenarios, contribute to SNe Ia.48,49,72,73 4. Effect of SN Ia Populations on Their Brightness When applying SNe Ia as cosmological standard candles, it is usually assumed that their properties and the above calibrated relations are invariant with redshift, i.e. the demographics of SNe Ia does not evolve with redshift, or the difference between SNe Ia arising from different stellar populations is negligible. This assumption is crucial for precision cosmology, since it may lead to systematic errors in the measurements of cosmological parameters. One way to test this assumption is to search for correlations between the properties of SNe Ia and those of their environments, since the environments of SNe Ia can represent the host population which gives rise to SNe Ia to a great extent. By the merit of these studies, it is now known that SN lightcurves and peak luminosities are correlated with host galaxy star formation rate, host population age, galaxy mass and metallicity.74–77 Even before SNe Ia were applied to measure cosmological parameters, it was generally agreed upon that the brightest SNe Ia always occur in late-type galaxies.78 It should be noted, however, that both late- and early-type galaxies can host dimmer

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SNe Ia, and subluminous SNe Ia tend to be discovered in galaxies with a significant old population,79 which leads to a lower mean peak luminosity for SNe Ia in earlytype galaxies than in their late-type counterparts.78,80 In addition, the mean peak brightness of SNe Ia in a galaxy is more homogeneous in the outer than in the inner regions.81,82 Also, both the birth rate and luminosity of SNe Ia trace the star formation rate of host galaxies.74,77 All these observations seem to imply that age is one of the factors which affect the luminosity of SNe Ia, and that dimmer SNe Ia can arise from progenitors spanning a wide range of ages.75,76,83–85 Considering that the typical galactic star formation rates increase by a factor of ∼ 10 up to z = 1.5, one may expect that the luminosities of SNe Ia will generally increase with redshift, and it has been shown that the mean “stretch factor” increases as well.86,87 According to these observations, we may conclude that the average value of the maximum luminosities of SNe Ia, as well as the range over which they span, both decrease with their delay times. It can also be concluded that, should the maximum luminosity be determined by only one parameter, as postulated by the Phillips relation,10 it can be expected that the possible range of the parameter must decrease, and that its mean value must either increase or decrease with the age of SNe Ia. At present, although it has been widely accepted that the peak luminosity of SNe Ia is dictated by the quantity of 56 Ni produced during their explosion, it is still controversial which parameter physically determines the 56 Ni production.88 The dimmer brightness for an old SN Ia could be either due to a high ignition density for a long cooling time before accretion,89,90 or due to a lower carbon abundance requirement for a massive WD with a less massive secondary under the frame of the SD model.91–93 In addition, the total mass of the DD systems was suggested to the origin of the brightness variation of SNe Ia,37,94 but it seems not to fit the age constraints on the brightness of SNe Ia from observations.95 Metallicity is another factor affecting the luminosity of SNe Ia.75–77,96,97 Theoretically, high metallicity progenitors would produce subluminous SNe Ia: A high 22 Ne abundance in high metallicity WDs generates more neutrons to feed the explosion nucleosynthesis, yielding more stable 58 Ni, a process which consumes the radioactive 56 Ni powering the lightcurve of SNe Ia.98 The spectra of high-z SNe Ia, whose metallicity is on an average lower than that of local SNe Ia, contain less intermediate mass elements, which is consistent with the idea that these SNe Ia produce more 56 Ni to power their luminosity.87 Metallicity may lead to an ∼ 25% variation on the peak luminosity of SNe Ia, but observation has shown that metallicity might only have an ∼ 10% effect on this luminosity.75 Such a difference between theory and observation could be the result of a metallicity–age–carbon abundance degeneracy, or an unknown metallicity effect. A high metallicity may lead to a relatively shorter delay time,95,99,100 regardless of what the progenitor model is, and hence a slightly brighter SNe Ia. On the other hand, a star of given mass with a high metallicity will produce a relatively less massive WD42,43 that has a higher carbon abundance,92 which was suggested to produce a brighter SNe Ia.92,93 Therefore, the

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effect of metallicity on the peak luminosity of SNe Ia could be more complex than what one would imagine at first glance. The mass of a host galaxy may also significantly affect the properties of SNe Ia. For example, many observations have shown that the birth rate of SNe Ia depends heavily on the mass of their host galaxies.101–103 Compared with the age and metallicity, the effect of the mass of host galaxies could be more significant.77,84,96,104,105 More massive galaxies tend to give rise to dimmer SNe Ia,77,105 a trend which seems to be a combined result of the effects of age and metallicity on the brightness of SNe Ia, i.e. higher mass galaxies have the tendency to retain more metals due to gravitational effects, and at the same time, tend to have more old stars, supergiant elliptical galaxies in particular. The above facts imply that the peak luminosities of SNe Ia evolve with redshift. Despite this, if the absolute magnitude of SNe Ia corrected by their lightcurve shape and color does not change with redshift, it is not necessarily problematic to take SNe Ia as standard candles for cosmology. However, people have to face an embarrassing situation that the age, metallicity and host galaxy mass have a systemic effect on the brightness of SNe Ia even after their brightness was corrected by the lightcurve shape and color.77,84,96,97,104,105 For example, as shown in Fig. 7, Sullivan et al.77 found that SNe Ia originating in massive host galaxies and galaxies with low specific star-formation rates, of the

Fig. 7. Residuals from the best-fitting cosmology for a low-redshift sample as a function of host galaxy mass after correction for lightcurve shape and color. The red circles are the mean residuals in bins of host galaxy mass. The histograms on the right-hand the distribution of the residuals for SNe Ia in low (dashed histogram) and high (gray histogram) mass hosts, where the mass boundary is 1010 M
, and the horizontal lines show the average residuals for the subsamples in low and high mass hosts. (For color version, see page II-CP11.) Source: From Ref. 77.

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same lightcurve shape and color, are generally 0.08 magnitudes brighter than those found anywhere else. This result does not depend on any assumed cosmological model or SN lightcurve width. Such a trend may lead to a systematic error on the measurement of the EOSDE, if it is not corrected. Therefore, besides the lightcurve shape and the color, at least a third parameter to correct the brightness of SNe Ia, which is correlated to one or more properties of the host galaxies, must be incorporated to avoid systematic Hubble diagram residuals. 5. SN Ia’s Role in Cosmology The previous sections have discussed the properties of SNe Ia with an emphasis on the corrections that can be applied to compensate for the variations in the absolute luminosities and also in their spectra.a In this section, we focus on how Eq. (3) must be modified when the condition z
1 no longer holds. We emphasize how information concerning the cosmological model can be extracted from the observed redshift-apparent magnitude relation. To this end, we must first refine our definition of distance, which is complicated by the expansion of the universe. The “co-moving” distance dCM to an object at redshift z is given by the integral 
z  H0 dz dCM = cH0 −1 , (4) H(z  ) 0 where H is the Hubble constant, which is a function of z, and H0 is its present value. For a universe filled with radiation, matter and a cosmological constant: H0 = [ΩR (1 + z  )4 + ΩM (1 + z  )3 + Ωk (1 + z  )2 + ΩΛ ]−1/2 , H(z  )

(5)

where Ωk = (1 − ΩR − ΩM − ΩΛ ). In the case of a “quintessence” or “dark energy” component of constant w = p/ρc2 instead of a cosmological constant (with w = −1), we would replace the ΩΛ term with ΩDE (1 + z  )3(1+w) . For Ωk = 0 (a spatially flat universe), the “co-moving” distance as defined above coincides with the “angulardiameter” distance dAD , but in the presence of a nonflat spatial geometry,   H −1 (Ωk )−1/2 sinh[H0 (Ωk )1/2 dCM (z)], if Ωk > 0,   0 dAD (z) = dCM (z), (6) if Ωk = 0,   H −1 (−Ω )−1/2 sin[H (−Ω )1/2 d (z)], if Ω < 0. 0

k

0

k

CM

k

Finally, the “luminosity” distance dL is defined in terms of the “angular-diameter” distance according to dL (z) = (1 + z)dAD (z), a When

(7)

the apparent luminosities of objects at differing redshifts are being compared, care must be taken to correct for the redshifting of the spectral bands. This is known as the so-called “Kcorrection”.

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where the (1 + z) factor has been inserted in order to account for two effects: (i) As a photon propagates from redshift z to us today, its energy is redshifted, or diminished, by a factor of (1 + z)−1 , and (ii) the rate of emission of photons is lowered by a factor of (1 + z)−1 in terms of today’s time, rather than the time at the instant of emission. It follows that using the formula =

L , dL 2

(8)

where L is the absolute (bolometric) luminosity and  is the apparent (bolometric) luminosity holds for all redshifts when the luminosity distance dL as defined above is used. The object of cosmological supernova studies is to measure the function dL (z) beyond its first-order linear term whose coefficient is cH0 −1 . By measuring the higher-order corrections (quadratic order and beyond), we uncover the equation-ofstate of what is driving the present expansion of the universe, and through the cubic and higher terms discover how this equation-of-state evolved with redshift. The SN Ia’s role in cosmology has been reviewed by many authors and further details on this technique can be found in their reviews.38,39,106,107 Here, we summarize some historic events in this field. In 1927, Lemaˆıtre108 deduced that the velocity of recession of distant objects relative to an observer can be approximated to be proportional to distance, should Einstein’s general relativity equation be assumed to be true. In 1929, Hubble109 noticed that a linear relation actually exists between an object’s recession velocity and its distance, judging by 22 external galaxies. This relation was henceforth named Hubble’s law, and hallmarks the discovery of the expansion of the universe. After this discovery, two questions arise naturally, i.e. what is the future of the universe? and what are the energy sources driving the expansion of the universe? To answer these questions, one needs to measure the recession velocity variation as a function of time or redshift, i.e. the deceleration parameter of the expansion q0 . In modern cosmology, including Einstein’s cosmological constant,110 the cosmic deceleration parameter is defined as ΩM − ΩΛ , (9) 2 where ΩM is the ratio between the actual mass density of the universe and the critical density above which the universe would collapse again someday if ΩΛ = 0, which is only dependent on the actual mass density of the universe, and ΩΛ describes how the cosmological constant affects the expansion of the universe.107 For a universe of ΩΛ = 0, the behavior of the universe is determined by ΩM and ΩΛ together. Here, q0 = 0 leads to a constant expansion velocity, q0 < 0 leads to an accelerating expansion and q0 > 0 leads to a decelerating expansion. It can be clearly seen in the equation that the expansion of the universe is accelerating if ΩΛ > (ΩM /2). For a universe of ΩΛ = 0, q = (ΩM /2), and therefore, the universe is q0 =

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Fig. 8. Joint confidence level for (ΩM , ΩΛ ) from SNe Ia corrected by MLSC method. Source: From Ref. 17.

always decelerating, where q > 12 , q = 12 and q < 12 corresponds to the closed, flat and open universe, respectively (please see Fig. 8). Although it was known that SNe Ia are very good standard candles,4 significant progress on this subject had yet to be made until the 1990s, after the advent of CCD imagers.111,112 In the 1990s, two teams focused on the measurement of the distance to high-z supernovae, i.e. the Supernova Cosmology Project (SCP) and the High-z team (HZT). Both teams built up large samples, including those measured by the Hubble Space Telescope (HST), and then individually published their twin studies.17,20 In the twin studies, they adjusted the parameters of an (ΩM , ΩΛ ) universe to fit their data, as shown in Fig. 8 which is an example from Riess et al.,17 and obtained a similar conclusion. The surprising result is that high-z SNe Ia were observed to be fainter than expected from their low-z counterparts in

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a matter-dominated universe. A matter-dominated closed universe (i.e. ΩM = 1) was unambiguously ruled out at a confidence level of greater than 7σ. Neither does an open, Λ = 0 cosmology fit the data well; the cosmological constant is nonzero and positive, with a confidence level of P (Λ > 0) = 99%. For a flat universe (ΩM + ΩΛ =1), a positive ΩΛ was found to be required at a confidence level of 7σ. Actually, the probability of an accelerating expansion of the universe with (ΩM > 0, ΩΛ > 0, q0 < 0) was modest, typically 3σ or less, i.e. the confidence level for a Λ dominated accelerating expansion was less than 3σ. For a flat universe, the best-fit parameters are ΩM ≈ 0.3 and ΩΛ ≈ 0.7, which supported an accelerated expansion as derived from Eq. (9). The twin studies are regarded as the original evidences for the discovery of the accelerating-expansion universe, and in 2011, Saul Perlmutter, Brian Schmidt and Adam Riess were awarded the Nobel Prize in Physics for the discovery. It should be noticed that the expansion of the universe is actually evolving. At z ≈ 0.45, the density of matter and dark energy was equal, and matter dominated at early epochs, i.e. the expansion of the universe was decelerating at z > 0.45.39 The discovery of accelerating expansion today implies that our universe is dominated by some unknown form of energy, which is often characterized by the ratio of pressure to density, i.e. the dark energy equation-of-state, w = P/ρc2 . For some arbitrary length scale a, the density of the universe evolves as ρ ∝ a−3(1+w) .

(10)

From this equation, it is easily seen that w = 0 for normal matter, i.e. the density decreases as the universe expands. If w = −1, which corresponds to Einstein’s cosmological constant, its density is a constant, and does not dilute with the expansion of the universe. A value of w < −1 is permitted although it is debatable theoretically. In this situation, the dark energy density will increase with time, and may ultimately lead to the destruction of galaxies.37,113 If −1 < w < 0, the dark energy density will decrease with time, and it will become problematic whether the expansion of the universe will continue to accelerate if the value of w is large enough. If ΩM is assumed to be negligible, a value of w < − 31 for dark energy would be sufficient for the indefinite acceleration of its expansion. However, considering that it is very likely for ΩM to have a value of larger than 0.2,17,20 the dark energy has to have an even smaller value of w for this to happen. It is verified that the critical point for the expansion of the universe to continue accelerating is in fact w = − 12 .114 However, it is possible that w is not a constant, but rather a variable which also evolves with time in the general form w(z) = w0 + wa z/(1 + z), where w will be a constant w0 if wa = 0. Following the discovery of dark energy, the EOSDE and the evolution of EOSDE with time became burning issues. Although the measurement of w is still consistent with an accelerating expansion of the universe, the uncertainty of w is still very large in those early observations. The value of w was generally located between −0.7 and −1.6 at a typical confidence level of 95%.22,114,115 Today, more

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Fig. 9. Confidence level contours in the plane of cosmological parameters (ΩM , w0 , wa ). The horizontal dashed lines indicate a cosmological constant, i.e. w0 = −1 and wa = 0. Source: From Ref. 120.

and more evidences based on SNe Ia observations from SNe Ia groups and other observational constraints, such as those from Supernova Legacy Survey (SNLS), Harvard-Smithsonian Center for Astrophysics (CfA), Sloan Digital Sky Survey (SDSS) and SCP, support a flat universe with a constant w = −1, i.e. at least rapidly evolving forms of dark energy were ruled out although the current observational constraints on wa are weak.34–36,116–120 For example, in Fig. 9, combined with full Wilkinson Microwave Anisotropy Probe 7-year (WMAP7) power spectrum, SNLS3 and SDSS DR7 data, ΩM = 0.271 ± 0.015, w0 = −0.905 ± 0.196 +1.094 were obtained, which are consistent with a flat, w = −1 and wa = −0.984−1.097 120 For most of these studies, the HST played a vital role in determining universe. the cosmological parameters, and provides many high-quality, high-z observations which remain one of our most efficient tools for understanding the properties of dark energy.115,116,121 An event at z = 1.914 was even discovered recently by HST, whose magnitude is consistent with that expected from ΛCDM cosmology.122 6. Issues and Prospects SNe Ia are now applied to constrain the properties of the dark energy and the evolution of the properties with time. Although SNe Ia may provide more strict constraints than some other surveys, such as baryon acoustic oscillations (BAO) and weak gravitational lensing (WL), the combination of the results from more than one survey is often applied to constrain the EOSDE parameter.113,120 For dark energy, a change of 1% in w only leads to a change of 3% in dark energy

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density at redshift z = 2 and a change of 0.2% at z = 1−2.113 Therefore, to study the properties of the dark energy quantitatively, a precision of better than 2% to measure distance is required.123 However, even for the most well-observed sample, the present distance accuracy seems not to be good enough. At present, a wellobserved low redshift sample may fulfill the requirement of 2% precision, but the distance error increases to ∼ 5% at redshifts greater than 1.106,127 As we know, SNe Ia with redshifts greater than 1 provide the best constraints on the EOSDE. For precision cosmology, there are still several difficulties with regard to SNe Ia. First, not all SNe Ia may be taken to be distance indicators, such as 2002cx-like supernovae.124 Such SNe Ia conform to no relations which regular SNe Ia do, such as the Phillips relation, and therefore cannot be standardized. Another subclass of SNe Ia that do not adhere to the Phillips relation are the so-called superluminous SNe Ia.125 If these peculiar supernovae are not excluded from an SNe Ia sample, the confidence level of the resulting distance measurement would decrease greatly. Second, new lightcurve analysis methods which have been developed tend to be based on new intrinsic parameters, which are assumed either to be invariant with redshift, or at least to vary with redshift in such a way that does not introduce further systematic uncertainty. Despite the reduction of the luminosity uncertainty from 0.15 magnitudes to less than 0.10 magnitudes, or better than 5% in terms of distance uncertainty,33,126 which these methods serve to achieve, such assumptions ought to be checked carefully, since it has been shown that systematic-uncertaintyinducing evolution may exist,86,87 and besides the lightcurve shape and the color, a third parameter correlated to one or some properties of the host galaxy is necessary. Finally, it is crucial to increase the detection efficiency of SNe Ia at high redshifts for further studies, especially at redshifts greater than 1. Two factors may affect this detection efficiency. One is that only luminous SNe Ia are likely to be detected due to Malmquist bias, but luminous SNe Ia are relatively rare. The other is that most of the radiation may be shifted into infrared bands for the SNe Ia at high redshifts, which makes it difficult to detect SNe Ia by optical telescopes. Infrared space telescopes, in this context, may be an alternative for future SNe Ia science.38 For example, the recently installed Wide-Field Camera 3 on HST has granted SN surveys an unprecedented depth, i.e. it is possible to detect a typical SN Ia at z ≈ 2.5 by HST. In addition to the difficulties mentioned above, for high-z supernovae, the confidence level for classification by spectrum is not as high as that for low-z supernovae. For a high-z SNe Ia sample, although practical studies of photometric identification have developed methods which achieve a purity of ∼ 95% for SNe Ia,128–130 there exists a possibility that such samples may be contaminated by a few Type Ib/Ic supernovae.122,131 Therefore, techniques to remove such sample impurities need to be developed. However, if an appropriate observational strategy is adopted, the effect of time dilation could provide a great opportunity for high-z SN observation. If the cosmological redshift is derived from the expansion of the universe, the observed time interval of an event will be dilated by a factor of (1 + z), which means that

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the lightcurve of a high-z SN Ia will present a broadening peak and a more shallow slope for the later exponential decline phase. Therefore, the effect of time dilation increases the probability to detect the high-z SNe Ia, although it becomes relatively difficult to obtain a complete light curve. Acknowledgments We are very grateful to the anonymous referees for their kind comments that improved this manuscript greatly. This work was partly supported by NSFC (Grant Nos. 11473063, 11522327, 11390374 and 11521303), the Western Light Key Project of the Chinese Academy of Sciences and Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences. Z.H. thanks the support by the Strategic Priority Research Program “The Emergence of Cosmological Structures” of the Chinese Academy of Sciences, Grant No. XDB09010202, and Science and Technology Innovation Talent Programme of the Yunnan Province (Grant No. 2013HA005). References 1. 2. 3. 4. 5. 6. 7. 8. 9.

10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26.

K. Lundmark, K. Sven. Vetenskapsakad. Handl. 60 (1920) 1. R. Minkowski, Publ. Astron. Soc. Pac. 52 (1940) 206. A. V. Filippenko, Annu. Rev. Astron. Astrophys. 35 (1997) 309. D. Branch and G. A. Tammann, Annu. Rev. Astron. Astrophys. 30 (1992) 359. A. V. Filippenko et al., Astron. J. 104 (1992) 1543. B. Leibundgut et al., Astron. J. 105 (1993) 301. A. V. Filippenko et al., Astrophys. J. 384 (1992) L15. M. M. Phillips et al., Astron. J. 103 (1992) 1632. A. Kim, Stretched and non-stretched B-band Supernova lightcurves, LBNL Report No. LBNL-56164 Lawrence Berkeley National Laboratory, US Department of Energy, Berkeley, California (2004) M. M. Phillips, Astrophys. J. Lett. 413 (1993) L105. M. M. Phillips et al., Astron. J. 118 (1999) 1766. P. M. Garnavich et al., Astrophys. J. 613 (2004) 1120. B. W. Rust, The use of supernovae lightcurves for testing the expansion hypothesis and other cosmological relations, Ph.D. thesis, University of Illinois (1974). Y. P. Pskovskii, Sov. Astron. 21 (1977) 675. Y. P. Pskovskii, Sov. Astron. 28 (1984) 658. A. Riess et al., Astrophys. J. 473 (1996) 88. A. Riess et al., Astron. J. 116 (1998) 1009. A. Conley et al., Astrophys. J. 664 (2007) L13. S. Perlmutter et al., Astrophys. J. 483 (1997) 565. S. Perlmutter et al., Astrophys. J. 517 (1999) 565. G. Contardo et al., Astron. Astrophys. 359 (2000) 876. J. L. Tonry et al., Astrophys. J. 594 (2003) 1. S. Jha, A. G. Riess and R. P. Kirshner, Astrophys. J. 659 (2007) 122. J. Guy et al., Astron. Astrophys. 466 (2007) 11. A. Conley et al., Astrophys. J. 681 (2008) 482. S. Dado and A. Dar, Astrophys. J. 809 (2015) 39, arXiv:1301.3333 [astro-ph.HE].

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X. Meng, Y. Gao and Z. Han

27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46.

P. Nugent et al., Astrophys. J. Lett. 455 (1995) L147. S. Bailey et al., Astron. Astrophys. 500 (2011) L17. J. M. Silverman et al., Mon. Not. R. Astron. Soc. 425 (2012) 1889. X. F. Wang et al., Astrophys. J. Lett. 620 (2005) L87. X. F. Wang et al., Astrophys. J. 645 (2006) 488. K. Maeda et al., Mon. Not. R. Astron. Soc. 413 (2011) 3075. A. G. Kim et al., Astrophys. J. 766 (2013) 84. R. Kessler et al., Astrophys. J. Suppl. Ser. 185 (2009) 32. M. Hicken et al., Astrophys. J. 700 (2009) 1097. R. Amanullah et al., Astrophys. J. 716 (2010) 712. D. A. Howell, Nat. Commun. 2 (2011) 350. B. Leibundgut, Annu. Rev. Astron. Astrophys. 39 (2001) 67. P. Astier, Rep. Prog. Phys. 75 (2012) 116901. F. Hoyle and W. A. Fowler, Astrophys. J. 132 (1960) 565. P. E. Nugent et al., Nature 480 (2011) 344. H. Umeda et al., Astrophys. J. 513 (1999) 861. X. Meng, X. Chen and Z. Han, Astron. Astrophys. 487 (2008) 625. D. Branch, Nature 431 (2004) 1044. W. D. Arnett, Astrophys. J. 253 (1982) 785. M. Livio, in Type Ia Supernovae: Theory and Cosmology, eds. J. Truran and J. C. Niemeyer (Cambridge University Press, New york, 1999), p. 33. B. Leibundgut, Astron. Astrophys. Rev. 10 (2000) 179. B. Wang and Z. Han, New Astron. Rev. 56 (2012) 122. D. Maoz, F. Mannucci and G. Nelemans, Annu. Rev. Astron. Astrophys. 52 (2014) 107. J. Whelan and I. Iben, Astrophys. J. 186 (1973) 1007. K. Nomoto, F.-K. Thielemann and K. Yokoi, Astrophys. J. 286 (1984) 644. X. D. Li and E. P. J. van den Heuvel, Astron. Astrophys. 322 (1997) L9. I. Hachisu, M. Kato, K. Nomoto and H. Umeda, Astrophys. J. 519 (1999) 314. I. Hachisu, M. Kato and K. Nomoto, Astrophys. J. 522 (1999) 487. N. Langer et al., Astron. Astrophys. 362 (2000) 1046. Z. Han and Ph. Podsiadlowski, Mon. Not. R. Astron. Soc. 350 (2004) 1301. B. Wang, X. Meng, X. Chen and Z. Han, Mon. Not. R. Astron. Soc. 395 (2009) 847. I. Iben and A. V. Tutukov, Astrophys. J. Suppl. Ser. 54 (1984) 335. R. F. Webbink, Astrophys. J. 277 (1984) 355. R. Pakmor et al., Nature 463 (2010) 61. M. H. van Kerkwijk, P. Chang and S. Justham, Astrophys. J. 722 (2010) L157. H. Saio and K. Nomoto, Astron. Astrophys. 150 (1985) L21. H. Saio and K. Nomoto, Astrophys. J. 500 (1998) 388. I. Iben and A. V. Tutukov, Astrophys. J. 342 (1989) 430. S. E. Woosley and T. A. Weaver, Astrophys. J. 423 (1994) 371. E. Livne and D. Arnett, Astrophys. J. 452 (1995) 62. K. J. Shen, J. Guillochon and R. J. Foley, Astrophys. J. 770 (2013) L35. S. Geier et al., Science 347 (2015) 1126. K. J. Shen and L. Bildsten, Astrophys. J. 699 (2009) 1365. S. A. Sim et al., Mon. Not. R. Astron. Soc. 420 (2012) 3003. A. J. Ruiter et al., Mon. Not. R. Astron. Soc. 417 (2011) 408. P. Ruiz-Lapuente, New Astron. Rev. 62 (2014) 15. X. F. Wang et al., Science 340 (2013) 170. M. Sullivan et al., Astrophys. J. 648 (2006) 868.

47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74.

May 4, 2017 10:3 b2167-ch16

SNe Ia as a cosmological probe

75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120. 121.

II-171

D. A. Howell et al., Astrophys. J. 691 (2009) 661. J. S. Gallagher et al., Astrophys. J. 685 (2008) 752. M. Sullivan et al., Mon. Not. R. Astron. Soc. 406 (2010) 782. M. Hamuy et al., Astron. J. 112 (1996) 2391. D. A. Howell, Astrophys. J. 554 (2001) L193. T. D. Brandt et al., Astron. J. 140 (2010) 804. L. Wang, P. H¨ oflich and J. C. Wheeler, Astrophys. J. Lett. 483 (1997) L29. A. G. Riess et al., Astron. J. 117 (1999) 707. J. D. Neill et al., Astrophys. J. 707 (2009) 1449. R. R. Gupta et al., Astrophys. J. 740 (2011) 92. M. Rigault et al., Astrophys. J. 802 (2015) 20. D. A. Howell, M. Sullivan, A. Conley and R. Carlberg, Astrophys. J. Lett. 667 (2007) L37. M. Sullivan et al., Astrophys. J. Lett. 693 (2009) L76. Ph. Podsiadlowski et al., New Astron. Rev. 52 (2008) 381. B. K. Krueger et al., Astrophys. J. Lett. 719 (2010) L5. X. Chen, Z. Han and X. Meng, Mon. Not. R. Astron. Soc. 438 (2014) 3358. X. Meng and W. Yang, Astrophys. J. 710 (2010) 1310. H. Umeda et al., Astrophys. J. Lett. 522 (1999) L43. X. Meng and W. Yang, Astron. Astrophys. 531 (2011) A94. D. Maoz and F. Mannucci, Publ. Astron. Soc. Aust. 29 (2012) 447. X. Meng and W. Yang, Astron. Astrophys. 543 (2012) A137. M. Childress et al., Astrophys. J. 770 (2013) 108. B. T. Hayden et al., Astrophys. J. 764 (2013) 191. F. X. Timmes, E. F. Brown and J. W. Truran, Astrophys. J. Lett. 590 (2003) L83. X. Meng, X. Chen and Z. Han, Mon. Not. R. Astron. Soc. 395 (2009) 2103. X. Meng, Z. Li and W. Yang, Publ. Astron. Soc. Jpn. 63 (2011) L31. F. Mannucci, Chin. J. Astron. Astrophys. Suppl. 8 (2008) 143. W. Li et al., Mon. Not. R. Astron. Soc. 412 (2011) 1473. Y. Gao and C. Pritchet, Astron. J. 145 (2013) 83. J. Johansson et al., Mon. Not. R. Astron. Soc. 435 (2013) 1680. M. Rigault et al., Astron. Astrophys. 560 (2013) A66. A. G. Kim et al., Astropart. Phys. 63 (2015) 2. S. M. Carroll, W. H. Press and E. L. Turner, Annu. Rev. Astron. Astrophys. 30 (1992) 499. G. Lemaˆıtre, Ann. Soc. Sci. Brux. A 47 (1927) 49. E. Hubble, Contrib. Mt. Wilson Obs. 3 (1929) 23. A. Einstein, Sitz. der K¨ onigl. Preuß. Akad. der Wiss. (1917) 142. M. Hamuy et al., Astron. J. 112 (1996) 2408. M. Hamuy et al., Astron. J. 112 (1996) 2398. J. A. Frieman, M. S. Turner and D. Huterer, Annu. Rev. Astron. Astrophys. 46 (2008) 385. R. A. Knop et al., Astrophys. J. 598 (2003) 102. A. G. Riess et al., Astrophys. J. 607 (2004) 665. A. G. Riess et al., Astrophys. J. 659 (2007) 98. G. Miknaitis et al., Astrophys. J. 666 (2007) 674. W. M. Wood-Vasey et al., Astrophys. J. 666 (2007) 694. A. Conley et al., Astrophys. J. Suppl. Ser. 192 (2011) 1. M. Sullivan et al., Astrophys. J. 737 (2011) 102. P. M. Garnavich et al., Astrophys. J. Lett. 493 (1998) L53.

May 4, 2017 10:3 b2167-ch16

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122. 123. 124. 125. 126. 127. 128. 129. 130. 131.

X. Meng, Y. Gao and Z. Han

D. O. Jones et al., Astrophys. J. 768 (2013) 166. P. H¨ oflich et al., Astrophys. J. 710 (2010) 444. W. D. Li et al., Publ. Astron. Soc. Pac. 115 (2003) 453. D. A. Howell et al. Nature 443 (2006) 308. K. S. Mandel, G. Narayan and R. P. Kirshner, Astrophys. J. 731 (2011) 120. P. Astier et al., Astron. Astrophys. 572 (2014) A80. R. Kessler et al., Publ. Astron. Soc. Pac. 122 (2010) 1415. M. Sako et al., Astrophys. J. 738 (2011) 162. G. Bazin et al., Astron. Astrophys. 534 (2011) A43. S. A. Rodney et al., Astrophys. J. 746 (2012) 5.

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Chapter 17 Gravitational lensing in cosmology

Toshifumi Futamase Astronomical Institute, Tohoku University, Sendai 9808578, Japan [email protected]

Gravitational lensing is a unique and direct probe of mass in the universe. It depends only on the law of gravity and does not depend on the dynamical state nor the composition of matter. Thus it is used to study the distribution of the dark matter in the lensing object. Combined with the traditional observations such as optical and X-ray, it gives us useful information on the structure formation in the universe. The lensing observables depend also on the global geometry as well as on the large scale structure of the universe. Therefore it is possible to derive useful constraints on cosmological parameters once the distribution of lensing mass is accurately known. Since the first discovery of a lensing event by a galaxy in 1979, various kinds of lensing phenomena caused by stars, galaxies, clusters of galaxies and large scale structure have been observed and are used to study the mass distribution at various scales and cosmology. Gravitational lensing is now regarded as an indispensable research tool in observational cosmology. In this article we give a pedagogical introduction to gravitational lensing and its applications to cosmology. Keywords: Cosmology; gravitational lensing; strong lensing; weak lensing. PACS Number: 98.80.Es

1. Introduction and History Propagation of light rays from a distant source to the observer is governed by the gravitational field of the intervening mass distribution. This fact was realized even before Einstein formulated the general theory of relativity. It seems that Cavendish and Michell are the first who presented the idea that the bending of a light ray occurs near a massive body around 1780 (see Valls-Gabaud2 for the detailed history). Then German astronomer, Soldner, calculated the bending angle of a light ray around a massive object in 1801 using Newtonian gravity and obtained one half of the correct value.3 The correct value can be only calculated by knowing the fact that spacetime is curved by a massive object. General relativity provides a method to calculate the spatial curvature around a massive object. Einstein calculated the bending angle of a light ray grazing the surface of the sun correctly as 1.76 arcsec (Einstein, 1915). In 1919, a team led by A. Eddington succeeded to measure the change in position of stars around the sun during an eclipse and obtained results consistent with the prediction by general relativity. II-173

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Eddington explained the gravitational deflection of light by the analogy of lensing effect in both 1911 and 1915 predictions of the bending angle. In 1924 the Russian scientist Chwolson pointed out the possibility of producing fake double stars by gravitational lensing and furthermore without detailed calculation he pointed out a possible existence of perfect ring image if the source is located exactly on the line of sight of the lens.4 In 1936 a Czech amateur scientist, Mandl, visited Einstein at Princeton asking about the possible existence of multiple images of a distant star due to the gravitational force of a star. Stimulated by the discussion, Einstein wrote a paper on gravitational lensing by a star with detailed calculation and showed the existence of the so-called Einstein ring, predicted by Chwolson.5 He concluded however, that gravitational lensing by a star is unlikely to occur because of the extremely small probability of lensing. It should be noted that Link studied similar lensing event before Einstein. He calculated not only the position of the image but also magnification.6,7 Zwicky estimated the probability of lensing of one galaxy by another galaxy as ∼ 10−3 (Ref. 8) which is much larger than that of a star and tried to discover lensing phenomena in galaxy clusters. The attempt failed mainly because the lensed images are too faint to be detected by the apparatus available at that time. After that there were no active researches on gravitational lensing except some important papers. In 1960 Russian astronomers Idlis and Gridneva wrote a paper which is regarded as the precursor of the weak lensing idea. In 1964 Refsdal9 and Liebes10 argued that the gravitational lensing effect may be used to measure the mass of a lensing object. Refsdal also pointed out that if a source’s luminosity varies in time, there will be a time lag of variability between the lensed images which can be used to estimate the Hubble parameter. The first lensing system QSO 0957+561A, B was discovered in early 1979 by Walsh, Carswell, and Weymann.11 They made photometric observations of two images of a single quasar at z = 1.41. The lensing galaxy was also observed at redshift z = 0.355 in a cluster of galaxies. In 1997 the time lag of the variability was observed as 417 ± 3 days between images A and B.12 Since then, other systems with multiple images have been observed, and the list has grown to approximately 200 by now. Time lags have been observed in more than 10 lensed systems as well. This list is expected to be on the order of 100 in the coming decade. In the late 1980s, high resolution observations by the Hubble Space Telescope (HST) became available and discovered a very elongated image called a giant luminous arc (GLA) in the cluster Abell 370 in 1986.13 A multitude of arcs and distorted images in massive clusters have been described by Soucail et al.14 An Einstein ringlike image was also found in the system MG 1131+0456 in 1988.15 Gravitational lensing is classified into three regimes: strong, weak and micro lensing. Figure 1 shows a schematic representation of strong and weak lensing phenomena. Strong lensing results in multiple images and/or highly deformed arclike images, called giant luminous arcs. It is used not only to measure the spatial dark matter distribution in the lensing object (e.g. Kneib et al.16 ), but also to constrain

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the cosmological parameters. It has been shown that a good reconstruction of a lensing system may constrain the density parameter and the cosmological constant (Futamase and Yoshida, 1999; Yamamoto et al., 2001; Soucail, Kneib and Golse, 2004; Meneghetti et al., 2005; Jullo et al.17 ). An important application of strong lensing is to use it as a natural telescope. Strong lensing magnifies small and faint objects to make them observable, objects that are otherwise not observable. In particular, strong lensing by clusters has been used to study distant galaxies (Ellis et al.18 ; Kneib et al.19 ; Richard et al.20 ; Bouwens et al.21 ; Coe et al.22 ). For this study, a highly accurate determination of mass distribution is necessary, and this becomes possible using a large number of multiple-image and arcs discovered by the HST profitting from progress in lensmodeling techniques. It is claimed that the mass in cluster cores is measured to a few percent accuracy (e.g. Bradac et al.23,24 ; Jullo et al.25 ; Jullo and Kneib26 ). In the case of clusters observed by the Hubble Space Telescope Frontier Fields galaxies at z > 10 and as faint as 32 AB magnitude (about magnitudes fainter than the limit of Hubble Ultra Deep Field) are expected to be detected (Richard et al.27 ). Future prospects for this study seem very promising. Another application of strong lensing is lens statistics (Turner, Ostriker and Gott28 ). It has been shown that lens statistics are a good indicator of cosmological parameters, in particular, the cosmological constant (Turner29 ; Fukugita, Futamase and Kasai30 ). Weak lensing results only in weakly deformed images and the lensing signal is obtained only by averaging over background galaxies. While weak lensing is not spectacular compared with strong lensing, it is very important and useful. For example, strong lensing images are only observed in the central part of a cluster, which constitutes only a tiny part of the whole mass distribution of the cluster. Information from weak lensing is necessary to reveal the whole mass distribution of the cluster. The method of mass reconstruction by measuring the weak lensing effect was developed and applied to Abell 1689 and Cl 1409+53 by Tyson et al.31 More sophisticated methods of weak lensing mass reconstruction were developed by Kaiser and Squires32 and Kaiser, Squires and Broadhurst (now known as the KSB method33 ). This method and its refinements have been applied to many clusters of galaxies. Weak lensing by large scale structure known as cosmic shear depends on the growth rate of the structure and thus on the cosmic expansion rate as well as the law of gravity. Thus, the accurate measurement of cosmic shear is expected to constrain the nature of dark energy and the deviation from general relativity. The first detection of cosmic shear was made in 2000 by several groups,35–40 and is one of the most active area of research in gravitational lensing. Microlensing is a type of strong lensing by a stellar size object but whose deflection angle is on the order of micro-arcsec. Thus no separate images are observed, but only the magnification of the image is observed. It is used to discover MAssive Compact Halo Objects (MACHOs)41 and planets outside of the solar system. We will not consider microlensing in this review.

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For general review of gravitational lensing, we refer the reader to Schneider, Ehlers and Falco,42 Blandford and Narayan,43 Refsdal and Surdej44 and Narayan and Bartelmann.45 2. Basic Properties for Lens Equation The global geometry and overall evolution of the universe is well described as homogeneous and isotropic by the Friedman–Robertson–Walker (FRW) metric. However, the matter distribution is highly inhomogeneous, up to 100 Mpc as we observe. A light ray propagates through these inhomogeneities feeling the local gravitational potential. How the averaged FRW universe appears from the local inhomogeneous universe within the framework of general relativity is a complicated and difficult problem. Fortunately, the gravitational lensing phenomena considered in this review are caused by weak gravitational fields and we may regard the field as locally stationary on the time scale of the deflection of a light ray. This allows us to use a simple treatment of light ray propagation. We will derive the gravitational lensing equation in the cosmological situation starting from the geodesic equation and then summarize the basic concepts and properties of this equation. 2.1. Derivation of the cosmological lens equation To study the propagation of a light ray, we need to specify the background geometry. Since we can assume that the gravitational field is weak everywhere and locally quasi-stationary, it is sufficient to consider the following metric:
  (1) ds2 = a2 −(1 + 2Ψ)dη 2 + (1 − 2Ψ) dχ2 + r2 (χ)dΩ2 , where dΩ2 = dθ2 + sin2 θdφ2 is the line element on the 2-dimensional unit sphere t and η = dt
/a(t
) is the conformal time. r(χ) is the comoving angular diameter distance defined by

  χ  −1/2  K sin √ K >0    K   K =0 r(χ) = χ (2)  

    χ  (−K)−1/2 sinh √ K < 1. −K The curvature parameter K is expressed by the total density parameter at present Ω0 as K = (Ω0 − 1)H02 . Although the above metric is the first-order deviation from the FRW background, it may still apply in the situation with a fully nonlinear matter distribution. This can be understood by noticing that the Newtonian potential is determined by the cosmological Poisson equation. ∇2 Ψ = 4πGa2 ρb δm ,

(3)

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where ρb is the background density and δm is the matter density contrast defined as δm = (ρm − ρb )/ρb . This equation gives

−2 δm   1 |Ψ|, a Ωm,0 LH

(4)

where  and LH = H0−1 denote the characteristic comoving scale of the density fluctuation and the Hubble radius, respectively. Thus one can safely assume |Ψ|  1 even if δm  1 as long as   LH (Futamase46 ) which is always assumed in lensing studies. The propagation of a light ray is described by a null geodesic equation. dk µ + Γµαβ k α k β = 0, dλ

(5)

gµν k µ k ν = 0.

(6)

One can solve the above equations perturbatively with the 0th order being the solution in the homogeneous and isotropic background k µ (λ) = k (b)µ (λ) + δk µ (λ).

(7)

By choosing the origin of the coordinate at the observer (us) and assuming a small deflection angle everywhere, we can set the path of the background solution as follows: k (b)µ = (−1, 1, 0, 0),

(8)

where we choose polar coordinate (t, r, θ, φ) and carry out backward ray-tracing from the observer at λ = 0. The integral curve is chosen as x(b)µ = (−λ, λ, θI , 0),

(9)

where |θI |  1 is interpreted as the image position angle. Since we observe the projection effect on the sky, we are interested in the perpendicular components to the line of sight of the above equation. The perpendicular components of the first order solution may be written as δk a (λ) = −

2 2 r (λ)

λ

dλ
∂ a Ψ(λ
),

(10)

0

where a = θ, φ and ∂ a Φ = (Φ,θ , Φ,φ / sin2 θ). The Newtonian calculation gives the same expression with a factor of 1 instead of 2. The difference is due to the effect of spatial curvature. This may be integrated to give:

θa (λ) = θIa − 2 0

λ

dλ


r(λ
− λ) a ∂ Ψ(λ
). r(λ
)r(λ)

(11)

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Since lensing phenomena are always observed in a small patch of the sky, we can safely ignore the curvature of the sky (flat sky approximation) and use locally Cartesian coordinates θ = (θ1 , θ2 ) around the line of sight k (b)µ . Defining the source plane coordinate as β = θ(λs ) where λS is the location of the source and the image plane coordinate as θ = θ I , one can write the cosmological lens equation as follows: β = θ − α(θ),

(12)

where the projected bending angle is expressed by the Newtonian potential Ψ as follows:

χs r(λs − λ) (3) dλ α=2 ∇ Ψ(X(λ)), (13) r(λs ) 0 where ∇(3) is the 3-dimensional gradient operator and ∂x = ∂θ /r(λ). This suggests that the bending angle may be expressed by the 2-dimensional gradient ∇θ of a 2-dimensional potential ψ(θ1 , θ2 ) α = ∇θ ψ,

(14)

where ∇θ = (∂θ1 , ∂θ2 ). In particular, if the lensing object lies within a sufficiently small region between λL ± ∆λ (∆λ  λL ) such as clusters and galaxies compared with the relevant cosmological distances, one can assume that the light ray is deflected at λ = λL . The projected bending angle takes the form

DLS λL +∆λ α=2 dλ ∇(3) Ψ(x(λ)), (15) DS λL −∆λ where DS = a(λs )r(λs ) and DLS = a(λs )r(λs − λL ) are the angular diameter distances from the observer to the source, and from the lensing object to the source, respectively. This is called the thin lens approximation. Figure 3 shows the configuration of the lensing equation in the thin lens approximation. Using the above expression for α, one finds 2κ ≡ ∇θ · α = ∆θ ψ =

DL DLS 8πGΣ . DS c2

(16)

We also defined the convergence κ, which is the normalized surface mass density κ(θ) ≡

Σ(θ) , Σcr

where the critical surface mass density is given by

 dS c2 D S −1 = 1.16 × 10 h Σcr ≡ g/cm−2 , 4πGDLS DL dL dLS where dX denotes the angular diameter distance DX in units of c/H0 .

(17)

(18)

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Cluster of Galaxies

Background Galaxy

Non-Linear

Multiple Images

Arclets

Optical Path Wave Front

Weak Shear

Multiple Images Area

Linear

Fig. 1.

The illustration of the gravitational lensing mechanism (see Ref. 1).

A region of a mass distribution with Σ > Σcr can produce multiple images for some source positions and is referred to as being super-critical. One can write the bending angle in the following more tractable form using the surface mass density Σ, or the convergence κ: α=

4G DL DLS c2 D S

d2 θ
Σ(θ
)

θ − θ
1 = |θ − θ
|2 π

d2 θ
κ(θ
)

θ − θ
. |θ − θ
|2

(19)

2.2. Properties of lens mapping One may regard the lens equation as mapping from the source plane to the image plane. It is not a one-to-one mapping, in general. The Jacobian of the mapping characterizes the local distortion of the mapping: Aij =

∂βi = δij − αij = δij − ψ,ij . ∂θj

(20)

This 2 by 2 symmetric matrix may be decomposed into a trace and traceless part as follows:  A(θ) =

1 − κ − γ1

−γ2

−γ2

1 − κ + γ1



 = (1 − κ)

1

 0

0

1

 −

γ1

γ2

γ2

−γ1

 ,

(21)

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Fig. 2. Giant luminous arc in galaxy cluster Abell 383. Source: http://www.spacetelescope.org/images/heic1106a/.

Fig. 3.

Geometry of the gravitational lensing.

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where γ ≡ γ1 + iγ2 is the gravitational complex shear defined by 1 (ψ,11 − ψ,22 ), 2

(22)

γ2 = |γ| sin(2φ) = ψ,ij .

(23)

γ1 = |γ| cos(2φ) =

In general the quantity a is called a spin-s quantity when a changes as a
= eisφ a,

(24)

under the rotation of coordinates by an angle φ. The convergence and gravitational shear are then the spin-0 and spin-2 quantities, respectively. Using this property, it is convenient to define the tangential and radial components of the shear relative to some reference point θ 0 . Noticing that the tangential component of the shear is obtained from γ1 by rotating the angle π/2 − φ (where φ is the polar angle of θ − θ 0 ), we have the following: γt (θ; θ 0 ) = −Re(γe−2iφ ) = −γ1 cos 2φ + γ2 sin 2φ,

(25)

γ× (θ; θ0 ) = −Im(γe−2iφ ) = γ1 sin 2φ − γ2 cos 2φ,

(26)

where Re(z) and Im(z) denote the real and imaginary part of a complex number z, respectively. γt and γ× are referred to as tangential and cross shears, respectively (Fig. 4). Under the parity transformation, these change as γt → γt , γ× → −γ× . Tangential shear plays an important role in cluster mass reconstruction and cosmic shear, as shown later. The convergence and the shear are related in the following way:

1 (27) γ(θ) = d2 θ
D(θ − θ
)κ(θ
) π with D(θ) =

 2   1 ∂ ∂2 ∂2 θ2 − θ12 − 2iθ1 θ2 e2iφ − + i ln |θ| = 2 =− 2 , 2 2 4 2 ∂θ1 ∂θ2 ∂θ1 ∂θ2 |θ| θ

Fig. 4.

The relationship between γ1 , γ2 and γt , γ× .

(28)

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where we have introduced polar coordinates (θ, φ) such that θ = (θ cos φ, θ sin φ). Note that there exists a global transformation that leaves the shear γ(θ) invariant such that κ(θ) → κ(θ) + κ0 ,

(29)

where κ0 is an arbitrary constant. This degeneracy leads to an ambiguity in the mass reconstruction based only on the image distortion in weak lensing analysis. In practical calculations, it is more convenient to work in Fourier space because one can make use of the Fast Fourier transform. The kernel is expressed as follows in Fourier space: k 2 − k22 + 2ik1 k2 ˆ = πe2iφk , D(k) =π 1 k2

(30)

where k = (k, φk ) is the vector of the polar coordinates in Fourier space. Since D(k)D∗ (k) = π 2 , the inverse of (26) is straightforward: 1 ∗ D (k)ˆ γ (k), (31) π where κ ˆ and γˆ are the Fourier transforms of the corresponding quantities. The geometrical meaning of convergence and shear is easily seen by considering the deformation of a circular source. Let us consider a circle with radius dβ. Consider a small circle defined by the equation κ ˆ (k) =

dβ T dβ = 1,

(32)

where dβ is a position vector from the origin and dβ T is its transverse. Then the image mapped by the lensing satisfies the following equation: 1 = dθT AT Adθ.

(33)

The matrix A has the following eigenvalues: λ± = 1 − κ ± |γ| = (1 − κ)(1 ± |g|),

(34)

where g = γ/(1 − κ) is called the reduced shear. The associated eigenvectors are   π      cos φ − cos φ cos φ   2 V+ = =  (35) π  , V− = sin φ , − sin φ sin φ − 2 where φ = 12 tan−1 (g2 /g1 ) is the position angle of the eigenvector V− . Using the orthogonal matrix O which diagonalizes the matrix A, one finds that the unit circle is mapped to the following ellipse (see Fig. 5)     dΘ1 2 dΘ2 2  1  +  1  = 1, (36) λ− λ+ where dΘ = Odθ. The dΘ1 is the coordinate along the eigenvector V− .

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Fig. 5. The unit circle is mapped to an ellipse with the length of the major axis 1/|λ− | and the minor axis 1/|λ+ | for kappa < 1 and its inclination from dθ axis is φ.

Since 1/|λ− | > 1/|λ+ | for κ < 1, this is the equation of an ellipse elongated in the direction of φ (tangential direction) with the length of the major axis 1/|λ− | and minor axis |1/λ+ |. For κ > 1 the ellipse is elongated in the direction φ − π/2 (radial direction) with the length of the major axis 1/|λ+ | and the minor axis 1/|λ− |. It is interesting to note that the distortion disappears along the curve defined by κ(θ) = 1, which always lies in the odd-parity region. Moreover in the region defined by κ  1 and γ  0 the mapped image is highly magnified without deformation. These are referred to as GRAavitationally lensed yet Morphologically Regular images, or GRAMORs. Such an image was predicted in 1998 by Williams and Lewis47 and Futamase, Hattori and Hamana48 and was discovered in the galaxy cluster MACS J1149.5+2223 (Broadhurst et al.49 ). Thus the convergence expresses an isotropic magnification (κ > 0) or demagnification (κ < 0). The (reduced) shear g expresses a deformation without changing the area. The component g1 expresses the deformation along the θ1 (θ2 ) axis for g1 > 0 (g1 < 0). The component g2 expresses the deformation along the axis rotated 45◦ from the θ1 axis. 2.3. Caustic and critical curves According to Liouville’s theorem, surface brightness is conserved by gravitational lensing. This may be shown as follows. The energy in the interval dν radiated in the solid angle dΩ through the area ds in the time dt may be written as IcdtdsdνdΩ (I is the surface brightness). This may also be written using the photon distribution

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function f (x, p) as IcdtdsdνdΩdν = hνf d3 xd3 p = 4πh3 ν 3 f cdtdsdνdΩ.

(37)

Thus I ∝ ν 3 f . Liouville’s theorem tells us that f = constant and ν = constant. Because there is no energy exchange in gravitational lensing, the surface brightness is conserved. Thus the magnification of the flux due to lensing is given simply by ratio of the solid angle between image and source. This is given by the inverse of the determinant of the Jacobian matrix A: µ=

1 1 1 = = . 2 2 det A (1 − κ) (1 − |g| ) (1 − κ)2 − |γ|2

(38)

Images with det A > 0 (λ− > 0 or λ+ < 0) have the same parity as the source, while images with det A < 0 (λ+ > 0 and λ− < 0) have parity opposite to the source. The closed curves in the image plane on which the point image is magnified indefinitely are called “critical curves” which is defined by the condition λ± = 0. The corresponding curves in the source plane are called “caustics”. The critical curves separate the image plane into even- and odd-parity regions. In practice, realistic objects like galaxies have a finite size and det A = 0 applies to every part of the object to be magnified indefinitely. Thus the extended source is not magnified indefinitely on the critical curve, but is highly magnified. Such images are called luminous giant arcs. If the lensing object is sufficiently massive and satisfies κ > 1 in some region, then det A(θ) = 0 is satisfied somewhere in the image plane and a critical curve exists. When convergence is a decreasing function of the distance from the center of the lensing object, the curve defined by λ+ = 0 is called the inner critical curve and the curve defined by λ− = 0 is called the outer critical curve. Since κ > 1 on the inner critical curve, the image is elongated in the radial direction (radial arc), and thus the inner critical curve is sometimes called the radial critical curve. On the other hand κ < 1 on the outer critical curve, so the image on the outer critical curve is elongated in the tangential direction (tangential arc) and the outer critical curve is called the tangential critical curve. The configuration of caustics determines the number of images. This is explained by the lens model examples below. The point determined by the condition κ(θ) = 1, namely Σ(θ) = Σcr is the critical point in the absence of shear. Although the shear is non-zero in general, the region where κ(θ) ≥ 1 is referred to as the strong lensing region associated with multiple images, while the region where κ(θ)  1 is referred to as the weak lensing region. 2.3.1. Circular lenses Let us now consider simple examples of a lens model. First, we consider a lens with a spherically symmetric mass distribution. In this case, the bending angle at a point θ is determined by the projected mass M (θ) enclosed by a circle of radius θ = |θ|

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around the center of mass, and is given by α(θ) =

DLS 4GM (θ) θ=κ ¯(θ)θ, D L D S c2 θ 2

(39)

where κ ¯ is the average of κ inside the radius θ: κ ¯ (θ) =

1 M (θ) DLS 4GM (θ) = . 2 Σcr π(DL θ) D L D S c2 θ 2

(40)

The lens equation becomes the scalar equation: β(θ) = θ − κ ¯ (θ)θ.

(41)

This shows immediately that the image with θ ≥ 0 satisfies θ ≥ β and an image with θ < 0 (opposite side of the source projected on the image plane) lies in the region with κ ¯ (θ) > 1. The Jacobian matrix is   

 2 2 1 0 2θ1 θ2 θ d¯ κ κ (θ1 ) − (θ2 ) 1 d¯ A(θ) = 1 − κ ¯− − . (42) 2 dθ 2θ dθ 0 1 2θ1 θ2 −(θ1 )2 + (θ2 )2 Remembering that the mean convergence is a decreasing function of the distance from the center in general and γ = −θ/2d¯ κ/dθ, the eigenvalues are then calculated to be d dβ [θ¯ κ(θ)] = λ+ = 1 − κ + γ = 1 − , (43) dθ dθ β λ− = 1 − κ − γ = 1 − κ ¯(θ) = . (44) θ As explained above, the eigenvalue λ− describes the tangential deformation of the image due to a geometrical effect, and λ+ describes the radial deformation of the image due to the gravitational tidal effect which depends on the radial density profile of the lensing object. In the spherical lens case, the lens mapping may be understood by specifying a curve in the (β, θ) plane (see Fig. 6, lens mapping). The curve is characterized by a mass distribution. Figure 6 shows the mapping given by a point mass where there are only two images. If the lensing mass distribution is non-singular at the origin, the mapping is shown in Fig. 7. As seen in this figure, the caustic corresponding to the outer critical curve is the origin in the source plane (actually the line in this case). When a source of a finite size approaches the caustic from the inside, two images with opposite parity approach perpendicularly to the inner critical line and these two images disappear when the source passes across the caustics. The opposite phenomenon occurs when the source approaches the caustic from the outside. Thus, the source must be inside the caustics to make multiple images. This may be more easily seen in Fig. 8, where the critical lines and caustics are shown in the image plane and source plane, respectively. This is a general property for caustics which are locally regarded as a straight line. Such caustics are called “fold caustics”.

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Fig. 6.

Fig. 7.

Lens mapping given by a point mass.

Lens mapping given by a non-singular lens model.

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Caustics and critical curves for a spherical lens. (For color version, see page II-CP12.) Source: http://gravitationallensing.pbworks.com/w/page/15553257/Strong%20Lensing.

2.3.2. The Einstein radius and radial arcs From symmetry, the tangential caustics are given by β = 0. The image is a ring on the tangential critical curve, provided that the lens configuration is super-critical. Such a ring is known as an Einstein ring and the angular radius is called the Einstein radius. The Einstein radius is obtained by solving the lensing equation by taking β = 0:   1/2 1/2

1/2 4GM (θE ) DLS M (θE ) DLS = 16.5 θE = . (45) c2 DL DS 1014 h−1 M DL DS In actual lensing events, a source such as a galaxy is not perfectly located at β = 0. In that case, we observe a pair of highly magnified, stretched images (called arcs) along the tangential critical curve, with a shorter arc just inside and a longer arc just outside of the critical curve. Thus, the position θarc of the tangential arcs is used to roughly estimate the mass of a lensing object inside the circle traced by the arc as follows:



2 DL DS θE 14 −1 M (θarc )  M (θE ) = 3.31 × 10 h M . (46) DLS 30

We now consider radial arcs which are sometimes observed in galaxy clusters. As shown above, the condition needed to have a radial arc is λ+ = 0. Radial arcs are very useful to study the radial density profile. For example, let suppose that the convergence has a simple power law form as

n θ0 (47) κ(θ) = κ0 θ with 0 < n < 2. The mean convergence is then given by κ ¯ (θ) =

κ(θ) n. 1− 2

(48)

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The condition λ+ = 0, i.e. d[θ¯ κ]/dθ = 0 becomes

θr =

1 κ0 (1 − n)

1/n θ0 .

(49)

Thus the condition n < 1 is required for a lens to have a radial critical curve. As typical examples of a spherical lens, we consider two models. The first is the singular isothermal sphere (SIS) model. The 3-dimensional density is given by ρ(r) =

σv2 . 2πGr2

(50)

The convergence and the averaged convergence are given by  σ 2 D 1 v LS κ(r) = 2πG , c DS θ  σ 2 D 1 v LS κ ¯ (θ) = 4πG . c DS θ

(51) (52)

The next example is the so-called NFW profile, which is a numerically predicted universal profile for CDM halos given by Navarro, Frenk and White (NFW).175 The density is given by ρ(r) =

rcrit δc 

2 , r r 1+ rs rs

(53)

where ρcrit , ρc and rc represent the critical density of the universe, the characteristic overdensity of the CDM halo, and the scale radius, respectively. The convergence and the averaged convergence are given by Bartelmann50 and Wright et al.51

 θ κ(r) = κs f , (54) θs

 −2 θ θ κ ¯ (θ) = 2κs g , (55) θs θs where κs = 2δc ρcrit rs Σ−1 crit and θs = rs /DL . The function f (x) given by      1 2 1−x   −1 +  arctanh  2  1+x  (1 − x2 ) 1 − x    f (x) = 1  3          1 2 x − 1   arctan  x2 − 1 1 −  2 x+1 (x − 1)

and g(x) are then

x1

(56)

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  2 1−x    arctanh   2 1+x    (1 − x )  x  g(x) = ln + 1  2       2 x−1   arctan  2 x +1 (x − 1)

x1

2.3.3. Non-circular lenses There are no exactly circular lenses in nature except a star, which we do not treat here. Any lensing object is somehow non-circular. Usually rather than giving the explicit distribution for the surface mass density or 3-dimensional mass density, the gravitational potential is given. Several interesting models exist for non-circular potentials. As far as an elliptical potential is concerned, the general model contains at least five parameters (an indicator of the strength of the lens such as the Einstein angle in the circular case, core radius, ellipticity, the power law index of density falloff and the angle of the major axis from a fixed direction). For example, the Tilted Plummer Elliptical potential is given by α2 ψ(x, y) = E η



ωp2 + rep α2E

η/2 ,

(58)

2 = x2 (1 − p ) + y 2 (1 + p ), ωp is related to the core radius, p is the ellipwhere rep ticity and η is the power law index. η = 1 corresponds to the isothermal case. The reader interested in this model should consult the paper by Grogin and Narayan.52 Although one needs a specific potential for model fitting, general properties of elliptical models are similar for any known model. Figure 9 shows small circular sources in the source plane and the corresponding images in the image plane. In the elliptical case, there exist inner caustic and inner critical curves. The inner caustics have points where the derivative along the curve is not well defined. Such a point is called a cusp. Five images exist when the source is inside the inner caustic. Three images exist when the source is between the inner and outer caustics. As seen in Fig. 9, when a sufficiently small finite source approaches the cusp from the inside, three images (two images parallel along and outside the caustics, one perpendicular to and inside the caustics) approach a corresponding point on the critical curve and vanish when the source passes through the cusp. When the source approaches a cusp from the outside, the opposite phenomenon occurs and three images seem to appear from the corresponding point on the inner critical curve. See Blandford and Kochanek53 for a more systematic understanding of caustic and critical curves.

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Fig. 9. The caustics and critical curves of an elliptical lens with sufficiently small core. The upper and bottom panels are the source plane and image plane, respectively. The sources are circular with same size. The corresponding (same color with source) image positions and shapes are shown. The left part of figures show the case when the source is close to inner fold caustics. On the other hand, the right part of figures show the case when the source is close to inner cusp caustics (Hattori et al.54 ). (For color version, see page II-CP12.)

3. Strong Lensing 3.1. Methods of solving the lens equation: LTM and non-LTM In strong gravitational lensing, there are a number of observables such as the relative positions of images, relative fluxes of images and time delays of luminosity changes between lensed images. These observables depend not only on the mass distribution of the source but also on the cosmological parameters. There are essentially two methods for constructing mass models of the source. The first is called the parametric method, which assumes a simple but physically reasonable model of potential such as an isothermal sphere, NFW model, elliptical pseudo-isothermal model and so on with parameters that have clear physical meanings. Physically reasonable means that one infers the shape of the lensing potential from the image position as well as the lens position and shape assuming the existence of a strong correlation between light and mass. Thus, this method is now called the light traces mass (LTM) method. In the case of a galaxy lens, the relative position of the images and the existence of arcs gives us enough information to determine the form of the potential.

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The other method is called the non-parametric method, which is also referred to as the “grid-based” method or non-LTM method because one does not assume the potential shape from the beginning. This method has become both possible and popular since deep observations using 8–10 m telescopes and the HST reveal detailed shape information of images for galaxy lenses, and a multitude of arcs and images for cluster lenses. Such a wealth of information allows us to establish the grid by grid correspondence between the image plane and the source plane. The basic principle of this method is the conservation of the surface brightness. Namely, the surface brightness of two grids related by the lens equation is the same. We will not go into further details of these two methods, partly because there are many strong lens model software codes using both LTM and non-LTM methods, which are publicly available. Instead we shall focus on the basic properties of strong gravitational lensing. For more detailed treatment of strong lensing, see an extensive review by Kochanek.55 3.2. Image magnification In strong lensing with multiple images, the ratio of the luminosity between images is an important observable and is used to constrain the mass distribution on the local scale inside the lensing galaxy. To derive the flux ratio between images, it should be noted that gravitational lensing does not change the surface brightness. Therefore, the magnification of the image compared with the source is just the ratio of the size between the image and source, and thus the magnification of the image is the inverse of the determinant of the Jacobian. We also define the magnification matrix as the inverse matrix of the Jacobian M = A−1 .

(59)

The magnification of an image is not known because we do not know the luminosity of the source. However since there are multiple images, the ratio of magnification is observable and is an important constraint of the mass model of the lensing object. Suppose we have two images A and B, then using the magnification matrix, we have δθB = M (θB )ij δβ = M (θ B )M −1 (θA )δθ A ≡ MBA δθ A .

(60)

The flux ratio is very useful to constrain the lens model. Although a simple smooth lens potential is enough to predict image positions consistent with observation, this is not the case for the flux ratio between images. This may be regarded as an indication of the existence of dark matter substructure in the halo of the lensing galaxy.56,57 3.3. Time delays Another possible observable in strong lensing is time difference (or time delay) between arrival of light rays in multiple images. (Refsdal9 ). Since the photons from

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the source travel along different paths to make the different images, the time taken by the photon from the source to the observer differs from image to image, resulting in an observable time delay. Time delays have been observed for a number of systems, and were used as a method to measure the global Hubble parameter before WMAP measured the Hubble parameter to within several percent accuracy. The method is free from any empirical assumptions associated with the method of the distance ladder. However this method was actually not very powerful in the past because of poor time delay measurements, resulting from simplified and inaccurate assumptions about the lens mass distribution. Nonetheless, this method is potentially very useful if we have enough information to determine the lens mass distribution accurately enough. It has been shown that a single gravitational lens with well-measured time delays and a detailed lens model can be used to measure time delay distances defined as D∆ ≡ (1 + zL )DL DS /DLS (see below) to 6–7% total uncertainty (random and systematic Suyu et al.58 ). In fact, Suyu et al.59 have shown that in quadruple lens systems the lens model inaccuracy can be less important than the inaccuracy in the time delay measurements. Their result is based on detailed modelling of the gravitational lens RXJ1131-1231 with the spatially extended Einstein ring observed by the HST and well-measured time delays between its multiple images. On the order of 100 new quadruple lensed quasars are expected to be discovered in the near future with imaging surveys such as the Subaru Hyper SupremeCam Survey, Pan-STARRS-1,60 the Dark Energy Survey (DES).61 When their time delay data are available, it is expected that together with supernovae and cosmic microwave background information, we can improve the dark energy figure of merit by almost a factor of 5, and determine the matter density parameter Ωm,0 to 0.004, the Hubble parameter H0 to 0.7%, and the dark energy equation of state time variation parameter wa to ±0.26, systematics permitting (Linder62 ). Here we explain the basic theory of time delays. If a source produces a short burst of light, then the photons of the burst will arrive at a time   1 + zL dL dS 1 2 τ= (θ − β) − ψ(θ) , (61) H0 dLS 2 where dX is the angular diameter distance in unit of c/H0 . The first term is just the result of the increase in the path length and the second term is the result of a decrease in the speed of light near the gravitating object. The time delay between the two images A and B is given by ∆AB = τA − τB =

1 + zL dL dS H0 dLS



  1 (θA − β)2 − (θB − β)2 − (ψ(θA ) − ψ(θB )) . 2

(62)

For practical application of time delays, we have to take into account the fact that the lens galaxy is sometimes a member of a small group of galaxies or is

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under the influence of nearby galaxies. In that case, we consider these effects by the external potential in the following form:   1 ψc (θ)  ψc (0) + ∇ψc (0) · θ + κc (0) θx2 + θy2 2   1 + γ1c (0) θx2 − θy2 + γ2c (0)θx θy , 2

(63)

where we take the origin at the position of the lens galaxy and assume that the length scale of the external field is large compared with the separation of two images and thus neglect the higher order terms. The coefficients are given by κc =

1 (ψc,xx + ψc,yy ), 2

γ1c =

1 (ψc,xx − ψc,yy ), 2

γ2c = ψc,xy .

(64)

It is easy to show that the first two terms of the above expansion does not change any observables in the strong lensing system. The third term expresses the contribution from the uniform sheet. It does have any effect on the time delay. We now show that a uniform sheet will bring an ambiguity into the Hubble constant determination. We consider the transformation of the potential c ψ → (1 − c)ψ + (θx2 + θy2 ) 2

(65)

with a constant c. This changes the bending angle as follows: α → (1 − c)α + cθ.

(66)

The lensing equation tells us that θ does not change by changing the source position β → (1 − c)β. This will change the magnification matrix as M → M/(1 − c), but the relative magnification matrix MBA = MB MA−1 does not change. However the time delay equation (62) tells that the combination H0 ∆τBA changes as above H0 ∆τBA → (1 − c)H0 ∆τBA .

(67)

This means that the observed time delay does not change if we can change the distance scale at the same time as H0 → (1 − c)H0 .

(68)

Thus even if we determine the lens model by image positions, as well as the ratio of relative magnification between images and time delay, there remains an ambiguity in the measurement of Hubble parameter H0 = 100h(1 − κc (0)) [km/s/Mpc].

(69)

The ambiguity κc (0) is the projected surface mass density near the lens position. Weak lensing analysis in the region around the lens galaxy is a direct method to determine κc . In fact there have been attempts to measure κc (0) in QSO0957+561 A, B by weak lensing.63,64

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3.4. Comparison of lens model software Analysis of strong gravitational lensing relies on a software analysis of observational data. There have been many lens model software codes available and thus it is important to have a systematic comparison between them. A number of software codes used in research studies are not publicly available. We will review here several of the publicly available codes that have been used in previous studies. A comprehensive review is available in Lefor, Futamase and Akhlaghi.65 3.4.1. Non-light traces mass software These codes were formerly referred to as non-parametric, and include: • PixeLens: This program uses a non-parametric model with Bayesian statistics. Inputs to the program consist of model constants (redshifts, pixel size, etc.) and image data. The radius of the mass map in pixels and the redshifts for the lens and source must be given. The code reconstructs a pixellated mass map by generating large ensembles of models with a Metropolis algorithm (Saha and Williams66 ). • Lensview: Lensview is based on the LensMEM algorithm, which finds the best fitting lens model and source brightness using a maximum entropy constraint. Lensview is used to study lens inversion, obtaining a model of the source based on lensed images. Due to its comprehensive nature, it is possible to specify very complicated lens models based on one or more components (Wayth and Webster67 ). • LensPerfect: LensPerfect uses a parametric model but is also model-free as described by its developers, who further characterize it as non-LTM. LensPerfect solutions are given as sums of basis functions. While most parametric models are based on a physical object, the basis functions used by LensPerfect have no physical interpretation. Input to the program is via a text file and graphical output is shown immediately on the display (Coe et al.68 ). • GRALE: GRALE can be used to simulate gravitational lenses and to invert lensing systems. The GRALE algorithm uses a non-parametric technique to infer the mass distribution of a gravitational lens system with multiple strong lensed systems (Liesenborgs et al.69 ). 3.4.2. Light traces mass software These codes were formerly referred to as parametric codes, and include: • Lenstool: Lenstool was first described in 1993 and has been continually upgraded. The current version uses a Markov Chain Monte Carlo sampler to avoid local minima in likelihood functions. The software uses a text file for data input and describes the model using a wide array of parameters. Multiple lens potentials are supported, all at the same redshift. Lenstool is a comprehensive program for gravitational lens modeling (Jullo et al.25 ).

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• Glafic: Glafic allows both point and extended sources in the analysis of lensed images as well as a large catalog of lens potentials. A text file is used to describe the model and image data as well as priors. Glafic can accurately recover lens model parameters of known lensing systems (Oguri70 ). • Gravlens/Lensmodel: This software has a large catalog of mass models and uses a text file for commands and data input. The software uses a tiling algorithm to determine the number and position of lensed images associated with a given source, and uses a polar grid centered on the main galaxy (Keeton71 ). This list of software is by no means comprehensive but provides a list of some of the commonly used software for strong gravitational lens models available for public download. All the lens model software described here is available directly from, or using links provided by, the Astrophysical Source Code Library (http://www.ascl.net). 3.5. Lens statistics The statistics of strong lensing events have interesting applications to cosmology. There have been many studies of statistics using quasars, giant luminous arcs and recently GRAMORs. It has attracted much attention because the statistics are sensitive to the cosmological constant via the cosmological volume element (e.g. Turner72 ; Fukugita, Futamase and Kasai30). As shown below, a detailed and accurate knowledge of the properties of the source and lensing object is necessary to use lens statistics as a useful method to measure the cosmological parameters and the redshift evolution of the lensing galaxies (e.g. Kochanek73 ; Ofek et al.74 ; Chae and Mao75 ; Matsumoto and Futamase76 ; Chae77 ; Cao and Zhu82 ). Furthermore, it has been shown that sample incompleteness can bias the results significantly (Capelo and Natarajan78 ). Recent large scale galaxy surveys such as SDSS overcame these difficulties and made the statistics a useful tool in observational cosmology (see Oguri et al.79 ). Here we explain the basic formulation of quasar statistics as an example of lens statistics. Consider the probability that a quasar with redshift zs is lensed by a lensing object with redshift range zL ∼ zL + dzL . Writing the cross-section as σL , the differential probability may be written as follows. We have cdt dzL , dτ (zL ) = n(zL )σL (70) dzL where n(zL ) is the number density of lensing objects. The cross-section is given by solving the lens equation once we choose an appropriate lens potential, and depends on the redshifts of the source and the lens as well as the relative position of the source and lens.

 L

Φ µ d2 β , (71) σL = µΦ(L) S

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where the integration region S in the source plane is over the position where multiple images are produced. Here we include the magnification bias. This is necessary because some observed lensed quasars are magnified and become brighter than a limited magnitude. Thus we need a magnification factor µ for each source position and the quasar luminosity function Φ(L). The cosmological dependence is in the quantity cdt/dzL , which is calculated in FRW geometry to be cdt c 1  = , dzL H0 (1 + zL ) Ωm,0 (1 + zL )3 + (1 − Ωm,0 − ΩΛ,0 )(1 + z)2 + ΩΛ,0

(72)

where Ωm,0 is the matter density parameter and ΩΛ,0 = Λ/3H02 is the normalized cosmological constant. In practical applications, the number density of the lens object is calculated using the luminosity function Φ(L) = dn(L)/L or the velocity function ψ(σv ) = dn(σv )/dσv . For example, using the velocity function the differential probability of lensing per unit redshift with an image separation between θ and θ + d∆θ is given by d2 τ = d∆θdzL

0

zS

dσv φ(σv )(1 + zL )3

cdt dσL . dzL d∆θ

(73)

A recent study based on the lens sample from the Sloan Digital Sky Survey Quasar Lens Search (SQLS) (Oguri et al.79 ) used 19 lensed quasars selected from 50,836 source quasars where the velocity dispersion of the lensing galaxies suggested by SDSS data (Bernardi et al.80 ) and the quasar luminosity function for 0.4 < z < 2.3 measured by the combined analysis of SDSS and 2dF data (Croom et al.81 ) are used. It is shown that the SQLS sample constrains the cosmologi+0.06 +0.06 cal constant to ΩΛ,0 = 0.790.07 (stat.)0.06 (syst.) for a flat universe. It is also shown that the dark energy equation of state parameter is consistent with w = 1 when the SQLS data is combined with constraints from baryon acoustic oscillation (BAO) measurements or results from the Wilkinson Microwave Anisotropy Probe (WMAP). Furthermore, no redshift evolution of the galaxy velocity function at z ≤ 1 is found. 4. Weak Lensing Although strong lensing is an important and spectacular phenomenon, it is a rare event and most light rays from distant sources propagate through relatively weak gravitational fields such as present in the outskirts of clusters and regions sourced by the large scale structure. In this case the intervening gravitational fields only slightly distort the shape of background sources, resulting in a systematic distortion pattern of background source images known as weak gravitational lensing. In the past decade, weak lensing has become a powerful, reliable method to measure the distribution of matter in clusters, dominated by invisible dark matter (DM), without

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assumptions concerning the physical and dynamical state of the system.83–91 Not only clusters, but also weak lensing due to large scale structure, attracts much attention because it carries information regarding the growth of structure which is affected by the nature of dark energy. For a general review of weak gravitational lensing, see Bartelmann and Schneider92 and Umetsu.93,94

4.1. Basic method We shall explain one method of mass reconstruction using weak lensing mass observations in some detail. For clarity of the argument we treat the simplest situation in this subsection. The complexities are mentioned later. First we select a sample of background galaxies for each lensing object considered. Since the lensing signal depends on the redshifts of the sources, it is necessary to have redshift information for accurate determination of the mass distribution of the lensing object. For example, in a weak lensing analysis of galaxy clusters, the contamination by faint member galaxies dilutes the lensing signal, in particular in cluster central regions.95 In the case of cluster lensing, the background galaxies are selected by color information. Namely, their colors are redder than the colormagnitude sequence of cluster member galaxies due to large k-corrections (for more detail, see Medezinski et al.,96 Okabe et al.34 However, more detailed redshift information regarding the background galaxies is necessary for accurate determination of cosmic shear (see below). Once we define the sample of background objects, the next step is to measure the shape of each background object. Several shape measurement schemes have been developed. We concentrate here on the so-called moment method, which uses multipole moments of brightness distribution to characterize the shape of the background objects (KSB33 and Okura and Futamase97 ). Although this is not the optimal method as seen in Fig. 10 below, it is still the most commonly used method. In the moment method we measure the quadrupole moments of surface brightness for each background object

(obs) (74) Qij = d2 θ∆θi ∆θj I(θ), where I(θ) is the surface brightness distribution and ∆θi = θi − θ0,i with θ0,i the center of the image. Below we define the center as the point at which the dipole moment vanishes. Then we define a purely spin-2 quantity from the measured quadrupole moments   (obs) (obs) (obs) Q11 − Q22 2Q12 (obs) = , (obs) , (75)  obs) (obs) (obs) Q11 + Q22 Q11 + Q22 which is called the ellipticity. For another definition of ellipticity, see Bartelmann (s) and Schneider.92 Similarly we define the quadrupole moments Qij and ellipticity

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(s) for each of the corresponding sources which are not observable. Using the lens equation we find the relationship between the ellipticity of the observed image and of the intrinsic source (s) =

(obs) − 2g + g 2 (obs) . 1 + |g|2 − 2Re(g(obs) ∗ )

(76)

(s)  (obs) − 2g.

(77)

In the weak lensing limit

Since we can expect that the intrinsic ellipticities of nearby galaxies on the sky do not correlate with each other, we can average over a certain number of background galaxies to obtain

 σe obs  2g + O √ , (78) N where σe  0.3 − 0.4 is the dispersion of the intrinsic ellipticity of galaxies, and N is the number of background galaxies averaged over. The more galaxies we average over, the more accurate the estimation of the gravitational shear we have. In the best seeing condition at the summit of Mauna Kea we may have N ≥ 50 per arcmin2 . In a realistic situation, the number is less than 30 or so. In the case of cluster lensing, the lensing signal is on the order of 0.1 and thus we can safely detect the lensing signal by clusters. Once we have the averaged shear field we can convert it to a convergence map which is the normalized surface mass density using the relationship between the convergence and the shear. Although this is the essence of weak lensing mass reconstruction, things are much more complicated for real observations.

Fig. 10. Stellar ellipticity distributions before and after PSF anisotropy correction. The left panel shows the observed ellipticity components (e∗1 , e∗2 ) before PSF correction, and the right panel shows the residual ellipticity components (δe∗1 , δe∗2 ) after PSF anisotropy correction (Umetsu and Broadhurst101 ).

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4.1.1. Shape measurements In practical observations of weak lensing, the observed image is not the lensed image but rather the image smeared by various sources of noise such as the distortion by atmospheric turbulence, imperfect optics, pixelization by the CCD chip, and photon noise. Thus we have to correct these effects before applying a mass reconstruction method. It is supposed that these effects are expressed by the point spread function (PSF) P (θ) as follows:

(obs) (θ) = d2 ψI(ψ)P (θ − ψ). (79) I In order to improve the accuracy of weak lensing analysis, we need to reduce various sources of noise and to correct the PSF effects without introducing other errors. In this review we focus our attention on PSF correction in some detail because this has the most important effect in weak lensing analysis. There are several methods for PSF correction available today. Here we employ the moment-based approach (see Kaiser, Squires and Broadhurst33; hereafter KSB). In realistic observations, since the observed shapes of background galaxies are rather noisy, particularly for small faint galaxies, we need to introduce a weight function W to reduce the random noise in the outer part of the image, so that



θ (obs) , (80) Qij = d2 θθi θj I (obs) (θ)(θ)W σ2 where σ is a typical scale for the galaxy. The simple choice is the spherical Gaussian shape:

 |θ|2 θ (81) W ∝ e− σ 2 . σ It is known that this choice introduces a systematic error in measuring the shape that overestimates the shear for images with large ellipticities. The error is overcome by introducing an elliptical window function (see Okura and Futamase98–100 ). The PSF correction proceeds in three steps. First we decompose the PSF function into isotropic and anisotropic part as follows:

(82) P (θ) = d2 θ
q(θ
)P (iso) (θ − θ
), where q, P (iso) are normalized and have vanishing 1-order moments.

dθq(θ) = dθP (iso) (θ) = 1,

(83)

dθθi q(θ) =

dθθi P (iso) (θ) = 0.

(84)

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(i) Anisotropic PSF correction We define

d2 θ
I(θ
)P (iso) (θ − θ
).

(85)

Then the observed intensity may be written as follows:

(obs) (θ) = dθ
q(θ − θ
)I (iso) (θ
). I

(86)

I (iso) (θ) =

Assuming a nearly spherical shape of the anisotropic PSF function, we only consider the quadrupole moment of the anisotropic PSF function

(87) qij = d2 θq(θ)qi qj . Then the quadrupole moment Q(iso) calculated from I (iso) is related to the observed quadrupole moment as follows: 1 Q(iso) = Q(obs) − Zijk qk , 2 where q1 = q11 − q22 , q2 = 2q12 and

Zijk = d2 θI (obs) (θ)

2   |θ| ∂2 θi θj W . ∂θi ∂θj σ2

(88)

(89)

Using the definition of ellipticity and the above expression, we find the ellipticity corrected by the anisotropic PSF effect is (iso) = e(obs) − Pijsm qj ,

(90)

where P sm is called the smear polarizability which expresses the response of the ellipticity against the anisotropic PSF effect (a detailed expression may be found in the original paper by KSB and the review by Bartelmann and Schneider). From dimensional arguments P sm ∝ rg−2 (where rg is the Gaussian scale of the object), and thus the PSF anisotropic effect becomes important for objects with small S/N . The above correction requires a knowledge of qi which is given by applying the equation to stars. Since stars do not suffer lensing and their images have no ellipticity after isotropic smearing, e(∗,iso) = 0 and thus we have ∗,(obs)

qi∗ = (P ∗,sm )−1 ij ej

.

(91)

The quantities with an asterisk denote quantities associated with stars. To obtain the smooth map of qi used in Eq. (97), the co-added mosaic image is divided into small chunks with scales determined by the typical coherent scale of PSF anisotropy patterns. By doing this, PSF anisotropy in individual chunks is well described by fairly low-order polynomials. Typically the rms value of stellar ellipticities is reduced from a few % to (4 − 8) × 10−3 after anisotropic PSF correction.

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Figure 10 shows the distribution of stellar ellipticity before and after the PSF anisotropy correction using the Subaru data for the cluster Abell 1689 (Umetsu and Broadhurst101). (ii) Isotropic PSF correction Next, we have to correct for magnification due to the PSF effect. For this purpose ˆ we consider the intensity I(β) in the source plane corresponding to I (iso) (θ). If we ˜ then we can write write the lensing equation as β = Aθ, ˆ Aθ). ˜ I (iso) (θ) = I(

(92)

The inverse relationship may be written by introducing the PSF in the source plane and may be written as

ˆ (93) I(β) = d2 β
I s (β
)Pˆ (β − β
), where we have introduced the PSF in the source plane corresponding to the isotropic part of the PSF in the image plane, so that Pˆ (β) =

1 P (iso) (A˜−1 θ). det A˜

(94)

This function is normalized and has a zero mean. Note that Pˆ does have an anisotropic part from the lensing effect. Thus we decompose into isotropic and anisotropic parts, so that

q (β − β
). (95) Pˆ (β) = d2 β
Pˆ (iso) (β
)ˆ Then the same argument in the image plane applies here to give eˆ0i = eˆi − Pijsm qˆj , where eˆ0 is the ellipticity of the intensity Iˆ0 defined by

ˆ I(β) = d2 β
Iˆ0 qˆ(β − β
).

(96)

(97)

We have used I (obs) instead of Iˆ in P sm as before. The final step is to relate the artificial ellipticity eˆ0 to the isotropically smeared ellipticity. This is simply obtained by using the relationship between I (iso) and ˆ which is defined above. Using this relationship, the quadrupole moment of the I, artificial source Iˆ is calculated to be

2

|β| 2 ˆ ˆ Qij = d ββi βj I(β)W (98) σ ˆ2

2 

|θ| − δm ηm (θ) 2 (iso) ˜ ˜ ˜ (θ)W = (det A)Aik Aj d θθk θ I , (99) σ2

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where σ ˆ 2 = (1 − κ)2 (1 + |g|2 )σ 2 and δi = 2gi /(1 + |g|2 ). Under the assumption that qˆ is small, we can replace I (iso) in the observed image I (obs) , and then by expanding the window function in the first order of δi , we obtain (iso)

ei

= eˆi + Pijsh gj ,

(100)

where P sh is called shear polarizability and expresses the response of the ellipticity against the shear. The detailed expression may be found in the original paper which describes KSB. Combining these two relationships, we have (iso)

ei

= eˆ0i + Pijsh gj + Pijsm qˆj .

(101)

By using star images, we have ∗,sh qˆi = −(P ∗,sm )−1 ik Pkj gj .

e

(102)

Finally, by using the relationship between e(iso) and the observed ellipticity , we finally have

(obs)

eˆ0 = e(obs) − Pijsm qj − Pijg gj ,

(103)

∗,sh sm (P ∗,sm )−1 Pijg = Pijsh − Pik k P j .

(104)

where we have defined

Since the intrinsic ellipticity of background galaxies do not have a correlation, averaging gives us gi = (P g )−1 ij (ej

(obs)

− Pijsm qj ) ,

(105)

∗,(obs)

with qi = (P ∗,sm )−1 . ij ej As seen from the above argument, one has to make some assumptions such as a small anisotropic part q and the use of the observed image I (obs) in the evaluation of smear polarizability and shear polarizability. These assumptions caused systematic errors in the original KSB weak lensing analysis. There have been some proposals to improve the accuracy of the analysis. Erben et al. have proposed using the trace part in P g to reduce the noise which creates non-diagonal components in P g . g Ps,ij =

1 T r(P g )δij ≡ Psg δij . 2

(106)

(Hudson et al.102 ; Hoekstra et al.103 ; Erben et al.104 ; Hetterscheidt et al.105 ). The diagonal parts are measured for individual galaxies, but they are very noisy for small and faint galaxies. To avoid the noisy part, an averaging in object-parameter space was introduced by Van Waerbeke et al.35 ; see also Erben et al.104 ; Hamana et al.106

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Although in the KSB method the result in principle should not depend on the choice of the scale in the weight function, this is not the case for real observations. Hoekstra et al.103 have shown that the scale used in the weight function of galaxies gives a better result. In practice, the averaged shear over the observed region is estimated by introducing the weight function ug,I such that ug,I =

1 2 + α2 , σg,I

(107)

where I labels the object in the region, the constant α is the softening parameter and σg is the rms error of the complex distortion measurement. Usually α  σg2  0.4 is used, which is a typical value of the mean rms σ ¯g over the background sample. Then the averaged shear is given by  ωg (θ − θI )ug,I gi,I gi (θ) = I , (108) I ωg (θ − θ I )uI where gi,I is the reduced shear estimated of the√Ith galaxy at an angular position θI , and ωg (θ) ∝ exp(−θ2 /θg2 ) with θg = FWHM/ 4 ln 2 which pixelizes the distortion data into a regular grid of pixels. The error variance for the above smoothed shear is given as  2 2 ωg,I u2g,I σi,I 2 (θ) = I σg 

2 ,

(109)

ωg,I uI

I 2 δij δIJ has been where ωg,I = ωg (θ − θ I ) and the relationship gi,I gj,J = (1/2)σg,I used. The smoothing scale is chosen to optimize the weak lensing detection of target mass structures, depending on both the size of the structure and the strength of noise power (∝ σ ¯g2 /ng ). Once the smoothed shear field is thus obtained, it can be converted to the convergence field using the relation κ(k) = πD∗ (k)γ(k). Actually this inversion method has to be applied in an infinite space. For a finite space one need to use the finite-field solution of the inversion problem for the reconstruction kernel. Such method has been developed by Seitz and Schneider.107 For a sufficiently wide field where the data field is not dominated by a positively or negatively biased density field, both methods give the almost same convergence field. This is not the case for a nearby cluster for which we have to use a finite field method.

4.2. E/B decomposition Observations and data analysis suffer from errors and noise. In particular, weak lensing analysis needs many corrections such as for the PSF distortion to isolate

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a physical signal of a weak tidal gravitational field. If the corrections are not perfect, they may systematically change the signal. Thus it is critically important to characterize the effect of systematic errors on the final results. Here we consider a method for the evaluation of systematic errors called E/B decomposition. This method is based on the fact that gravitational shear is generated by a Newtonian potential in the lowest order, and thus the gravitational field is rotation-free. Let us first define a new 2D vector field u from the reconstructed convergence κ by   γ1,1 + γ2,2 . (110) u ≡ ∇κ = γ2,1 − γ1,2 Using the vector field, we define the E-mode κ and B-mode κ as follows: ∇2 κE ≡ ∇ · u,

(111)

∇2 κB ≡ ∇ × u = u2,1 − u1,2 ,

(112)

which gives both E-mode and B-mode potentials ∇2 φE,B = 2κE,B .

(113)

Note that the gravitational effect is given by the Newtonian potential Ψ in the lowest order so that φE = Ψ and φB = 0, but in practical observations, φE = Ψ and φB = 0. The complex shear may be expressed by the complex potential φ = φE +iφB as follows:       1 E 1 B B E B γ= + + i φ − φ − φ φ,11 − φE (114) φ ,22 ,12 ,12 ,22 . 2 2 ,11 This expression shows that the transformation γ
(θ) = iγ(θ) is equivalent to an interchange of the E- and B-modes of the original maps. By noticing that the shear transforms as γ
= γe2iφ under the rotation of an angle φ, this operation is a rotation of each ellipticity by π/4 with each potential being fixed. Figure 11 shows the E and B mode shear. There are several sources of B-mode shear. One artificial source is PSF residuals. Since the PSF correction so far proposed adopts some sort of approximation, it gives rise to B-mode shear. In addition to artificial B-modes, there are also physical B-modes. Intrinsic ellipticities of background galaxies contribute to the shear estimate. By assuming randomness of orientation of the intrinsic ellipticities, such uncorrelated ellipticities yield statistically identical contributions to the E- and B-modes. By increasing the number density of background galaxies, these modes may be reduced. Another physical B-mode is due to the existence of intrinsic alignments,

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E mode

B mode

Fig. 11.

Shear patterns of E- and B-modes (Van Waerbeke and Mellier108 ).

which may be generated by tidal forces between halos of galaxies. There are two kinds of intrinsic alignments. One is called II correlation. Galaxies around a large halo may have some correlation of the orientation of ellipticity by a tidal gravitational force. This effect creates both E- and B-modes. It is reasonable to assume that the effect only applies to nearby galaxies, and thus the effect is eliminated by excluding nearby galaxy pairs from the background sample.109 Another effect is called GI correlation, which is the correlation between the intrinsic ellipticities and large scale structure.110 This is because the ellipticites are influenced by a local density field and the same density field causes weak lensing for high-z galaxies (cosmic shear). Thus the B-mode signals serve as a useful null check for systematic effects. Finally we highlight an ambiguity in weak lensing analyses. As pointed out above, the relationship between the convergence and the shear is not unique. Any constant mass sheet does not change the shear field which is obtained by the weak lensing limit. This is called the mass-sheet degeneracy. Actually as shown above, the observable is not shear itself but the reduced shear γ(θ) . (115) g(θ) = 1 − κ(θ) This field is invariant under the following global transformation: κ(θ) → λκ(θ) + 1 − λ,

γ(θ) → λγ(θ),

(116)

where λ = 0 is an arbitrary constant. There is a method to break this degeneracy once one realizes that the above transformation is equivalent to the following scaling

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for the Jacobian matrix: A(θ) → λA(θ).

(117)

One can exploit magnification effects since the inverse of the determinant of the Jacobian is the magnification. 4.3. Magnification bias Recent advances in weak lensing observations made the measurement of the magnification bias possible by measuring the depletion of the surface density of background objects. There are two effects as a result of lensing magnification. One is the expansion of the area of sky. The other is the amplification of the apparent magnitude which is given by mobs = m + 2.5 log µ,

(118)

where m is the apparent magnitude without lensing. These effects change the magnitude-limited number density n(< mlim ) of the background objects, so that n(< mlim ) =

1 n0 (< mlim + 2.5 log µ), µ

(119)

where mlim is the limiting magnitude and n0 is the unlensed number density count per solid angler. Let us suppose that n0 near the limited magnitude obeys a power law, α=

d log n0 (< m) , dm

(120)

where α is the power law index around m = mrmlim . Then we have n(< mlim ) = n0 (< mlim )µ2.5α−1 .

(121)

Since µ > 1, n(< mlim ) < n0 (< mlim ) if α < 0.4. The ratio is b=

n < mlim ) = µ2.5α−1  1 + (5α − 2), n0 (< mlim )

(122)

where we have used the weak lensing limit in the final step. For red background galaxies at a median redshift ∼ 1, the intrinsic power law index at faint magnitudes is observed to be relatively flat α  0.1 and thus causes depletion of the number count. This depletion has been observed in several massive clusters (Broadhurst et al.95 ; Umetsu and Broadhurst101; Umetsu et al.111 ). 4.3.1. Simulation test Although many improvements in weak lensing analysis have taken place, systematic errors in shape measurement must be reduced by a factor of 5–10 by the time

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the next generation large scale galaxy surveys are conducted. It is important to test the accuracy of shear measurement methods. This is done by using simulated data with realistic complicated galaxies and PSF effects such as the GREAT08 and GREAT10 challenges (Bridle et al.112,113 ; Kitching et al.114,116,119 ) and STEP (Heymans et al.117 ; Massey et al.118 ). The accuracy of these methods is measured by the two parameters m and c, defined by γ obs − γ true = mγ true + c,

(123)

where m is called the calibration bias and c is called PSF residual. Figure 12 shows the results using various methods of weak lensing analysis.119 4.3.2. Higher-order weak lensing-flexion and HOLICs In the usual treatment of weak lensing, the quadrupole moments of the shape are measured to evaluate the ellipticity which are spin-2 quantities. However, the quadrupole moments are not enough to characterize the shape of galaxies in general. In fact, there are many background galaxies, in particular small faint galaxies with shapes not well described by simple ellipses. It would be very hard to describe general complicated shapes mathematically, but if the shape deviates slightly from an ellipse, higher-order moments such as the octopole moments may be useful. Remembering that the spin-2 combination is constructed by a combination of the quadrupole moments, the quantities with definite values of spin should be constructed by appropriate combinations of higher order moments (Okura, Umetsu and Futamase121 ). Such quantities are called Higher Order Lensing Image’s Characteristics (HOLICs). Here we concentrate on the octoHOLICs constructed from octopole moments.

Fig. 12. Left: The multiplicative m and additive c biases of various measurement methods are shown. The symbols indicate the method (shown in the right panel). The central figure expands the x- and y-axes to show the best performing methods. The figures are cited from Kitching et al.119 The detailed description and references of the methods are also found there.

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The following quantities are the octo-HOLICs: ζ≡

(Q111 + Q122 ) + i(Q112 + Q222 ) , ξ

(124)

δ≡

(Q111 − 3Q122 ) + i(3Q112 − Q222 ) , ξ

(125)

where Qijk and Qijk are the octopole and 16th pole moments, respectively and ξ is the normalization factor defined as ξ ≡ Q1111 + 2Q1122 + Q2222 .

(126)

The first HOLICs ζ has spin-1, and the second HOLICs δ has spin-3. These HOLICs have dimensions of [angle]−1 . As the spin-2 ellipticity is related to gravitational shear which is the second derivative of the lens potential, the above HOLICs are related to the third derivative of the lens potential which is called flexion (Goldberg and Bacon, 2005; Bacon et al., 2006). Higher-order shape change is calculated as 1 dβi = Aij dθj + Dijk dθj dθk + O(θ3 ), 2

(127)

where Dijk = Aij,k = −ψ,ijk (Bacon, Rowe & Taylo, 2005). Using this equation, we can show a relationship between the intrinsic HOLICs and the observed HOLICs. By linearizing the relationship we have ζ (s)  ζ −

9 F , 41−κ

(128)

δ (s)  δ −

3 G , 41−κ

(129)

F = ∂κ,

G = ∂γ,

(130)

where F and G are defined as

where ∂ = ∂1 + i∂2 is the complex gradient operator. Thus F has spin-1 and is called the first flexion. G has spin-3 and is called the second flexion. By averaging over the background galaxies, we know that the HOLICs are an estimator of the flexion. So far, the spin-1 HOLICs ζ has been measured in Abell 1689 and used for mass reconstruction (Leonard et al.122 ; Okura, Umetsu and Futamase121 ; Leonard et al.123 ; Cain, Schechter and Bautz124 ). For recent development and other approach for Flexion analysis, see, for example, Viola, Melchior and Bartelmann,125 Rowe et al.126 and Levinson.127 4.4. Cluster mass reconstruction Since weak lensing analysis cannot resolve mass structure smaller than a finite scale on the order of 1 arcmin, it is applied mainly to clusters of galaxies and to the large

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scale structure of the universe. Clusters of galaxy are the largest self-gravitating objects in the universe, and thus their statistical properties contain indispensable informations on the initial density fluctuations and their nonlinear growth. The most important such information will be their mass distribution. On the other hand, a standard structure formation scenario based on CDM and a cosmological constant predicts that larger structures are the result of assembly of smaller structures and that the spherically averaged CDM halo is well described by the so-called NFW profile.175 Weak lensing thus plays an important role to study cluster properties. In this subsection, we explain frequently used techniques for measuring the cluster mass profile and then give two examples of weak lensing observations. For a more comprehensive treatment of cluster weak lensing, see the reviews by Umetsu93 and by Kneib and Natarajan.176 It is convenient to write the critical mass density in the form Σcrit =

c2 β −1 , 4πGD

β(z , zs ) ≡

D s (zs ) , Ds (zs )

(131)

where β(zs ) is a function of the source redshift zs with fixed z and represents the geometrical strength of lensing for a source with redshift zs , and thus is called the lensing depth. If the source redshift distribution is given by N (z), the mean lensing depth is

β ≡

∞

0

dzN (z)β(z) ∞

.

(132)

dzN (z)

0

It is convenient to use the normalized mean lensing depth defined by177 ω=

β . β(zs → ∞)

(133)

The convergence and shear may then be written in terms of the corresponding quantities for a fiducial source at infinite redshift as κ(θ) = ωκ∞ (θ) and γ(θ) = ωγ∞ (θ). In this way, the redshift information of background galaxies is taken into account. Actually the observable is the reduced shear g = γ/(1 − κ) which is nonlinear in κ. The averaging by source redshift is complicated operation. The detailed treatment of the averaging is given by Seitz and Schneider177 and by Hoekstra et al.178 As explained above, the tangential shear g+ is a coordinate independent quantity for a given reference point. For cluster lensing, the reference point is naturally the center of the cluster, which can be determined from the symmetry of the strong lensing pattern, the X-ray centroid position, or the position of brightest cD galaxy. It is also convenient that the cross-shear g× may be used as an estimator of the systematic error since g× is the divergence-free, curl-type distortion pattern

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of background images. The reason to use tangential shear for measuring the mass profile is that the azimuthal average of the tangential shear allows us to directly compare with the theoretical model by using the relationship ¯ (< θ, θ0 ) − κ (θ, θ 0 ), γt (θ; θ 0 ) = κ

(134)

where ·· is the azimuthal average and κ ¯ (< θ, θ0 ) is the average of the convergence within the radius θ. In the following, we understand that θ is the radius from the origin θ 0 and will not write the origin. The above relationship is obtained by the definition of κ ¯ in the following form:

θ

θ ∂φ ¯ (< θ) = 2 dθ
θ
κ(θ) = θ2 κ (135) dψ , 2π ∂θ 0 where we have used the fact that 2κ = ∆φ, and the 2D version of Gauss’s theorem. By taking the derivative with respect to θ and using the relationship ∂2φ = κ − γt , ∂θ2

(136)

dθ2 κ ¯ (< θ) = θ¯ κ(< θ) + θ[ κ (θ) − γt (θ)]. dθ

(137)

we have

This is the required relationship by noting d(θ2 κ ¯ /)dθ = 2 κ . It is also useful to define the following quantity, over the annulus bounded by θ and θout located outside the mass to be measured: ¯ (< θ) − κ ¯ (θ < θ < θout ), ζ(θ, θout ) ≡ κ

(138)

where θ and θout are the inner and outer radii of the annulus centered on some position θ0 usually taken at the position of the brightest galaxy in the cluster. This is referred to as ζ-statistics (Fahlman et al.128 ). Using the above equations, we can derive

θout 2 ζ(θ, θout ) = d ln θ γt (θ) . (139) θ2 θ 1− 2 θout Thus ζ is directly obtained by observing the image distortion of background galaxies in the weak lensing regime where |κ|, |γ|  1. Since κ ¯ (θ < θ < θout ) is positive definite, κ ¯ (< θ) ≥ ζ(θ, θout ). Thus we can define the lowest projected mass contained within the radius θ as Mζ (< θ) = π(Dd θ)2 Σcr ζ(θ, θout ).

(140)

By fixing the outer radius θout sufficiently large (but not so large to contain neighboring clusters), ζ(θ, θout ) as a function of θ is called the radial profile. This profile does not depend upon the invariance transformation. We can fit the observed radial profile with some theoretical models.

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Using the technique explained above one can observe the mass distribution of galaxy clusters. The structure formation scenario based on CDM makes particular predictions for the averaged density profile of the main halo and statistical properties of dark matter subhalos. Thus the measurement of the mass distribution by weak lensing will give us a direct test of the CDM structure formation scenario. There are several on-going projects aiming at an accurate measurement of the cluster mass profile.129–133 Here we give two examples of weak lensing observations of clusters performed using the Subaru telescope. If many lensed images with redshift information are observed in the central region of a cluster, a very accurate determination of the mass distribution in the central region is possible. Also the magnification bias is observed in some massive clusters. One can then combine all information to acquire an accurate model independent determination of mass distribution over a wide range of cluster radii. The CRASH survey179 is designed for this purpose, combining high-resolution 16-band HST imaging with wide-field Subaru observations for a sample of 20 X-ray selected relaxed clusters (Zitrin et al.180 ; Coe et al.181 ; Umetsu et al. 2012). The Bayesian approach to combine this information in a joint mass-profile analysis has been proposed by Umetsu et al.182,183 For another approach to combine the strong lensing and weak lensing, see Merten et al.184 4.4.1. Density profile It has been known that the density profile of a collapsed object is well described by the NFW profile regardless of scale, as the result of numerical simulations.175 Here we show the result of our measurement of the mean density profile by stacking the weak lensing signals of approximately 50 mass-selected clusters in the redshift range 0.15 < z < 0.3 (Okabe et al.34 ). These are selected from the ROSAT All Sky Survey catalogs (Ebeling et al.134,135 ; Boehringer et al.136 ) that satisfy LX [0.1 − 2.4 keV]/E(z)2.7 ≥ 4.2 × 1044 ergs−1 , 0.15 ≤ z ≤ 0.30, nH < 7 × 1020 cm−2 , and −25◦ < δ < +65◦, where E(z) = H(z)/H0 is the normalized Hubble parameter. As mentioned above, the selection of background galaxies is important for an accurate mass reconstruction. The sample of background galaxies basically consists of red galaxies satisfying the condition ∆C ≡ (V − i
) − (V − i
)ES0 > 0 for each cluster. We made a positive color cut ∆C > 0.475 to eliminate contamination by faint red cluster members due to statistical errors and possible intrinsic scatter in galaxy color (for details, see Okabe et al.34 ). This results in just 1% contamination of the sample by foreground and cluster members. Each individual cluster is detected at a typical peak signal-to-noise ratio (S/N )  4 using the 2D Kaiser–Squires mass reconstructions. The signal is stacked by the shear catalogs in physical length units centered on the respective brightest cluster galaxies (BCGs) and the average cluster mass distribution for the full sample, with a peak S/N of 28. The stacked lensing mass measurement is less sensitive to cluster internal structures and 3-dimensional halo orientation.

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γ Fig. 13. Stacked tangential shear profile (∆Σ+
Σcr 1−κ ) of 50 clusters in units of projected mass density, where different cluster and background redshift galaxies are weighted by the lensing kernel. The projected radius is computed from the weighted mean cluster redshift. The solid, dashed, dotted and dashed-dotted curves are the best-fit NFW, SIS, gNFW, and Einasto profiles, respectively. The lower portion shows the results of the 45◦ test for systematic errors (Okabe et al.34 ). (For color version, see page II-CP12.)

Thus it enables us to measure an averaged cluster mass profile for the sample. When the signals from 50 clusters are combined, the number density of background galaxies becomes 266.3 arcmin−2 . The B-mode signal is smaller by at least an order of magnitude than the E-mode signal. Figure 13 shows the stacked shear profile over the radial range 100 h−1 kpc < r < 2.8 h−1 Mpac. This shows that the profile is well described by the predicted NFW profile with parameters +0.53 +0.49 × 1014 h−1 M and cvir = 5.41−0.45 with less than 10% statistical Mvir = 7.19−0.50 error. The concentration parameter obtained from stacked lensing profile is broadly in line with theoretical predictions (e.g. Bhattacharya et al.137 ). No evidence of departures from the NFW profile is found. 4.4.2. Dark matter subhalos in the coma cluster According to the CDM structure formation scenario, less massive halos are accreted into more massive halos, which are then subsequently eroded by effects combined with tidal stripping and dynamical friction of the host halo, eventually becoming a smooth component. In this process the central regions of subhalos have survived under the overdensity field, and constitute their population. Numerical simulations of, and analytic approaches to CDM based structure formation scenario predict that subhalo mass functions at the intermediate and low mass scales follow a power

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−α law, dn/dlnMsub ∝ Msub with α ∼ 0.9−1.0 (e.g. Taylor and Babul138–140 ; Oguri 141 and Lee ; van den Bosch et al.142 ; Diemand et al.143 ; De Lucia et al.144 ; Gao et al.;145 Shaw et al.146 ; Angulo et al.147 ; Giocoli et al.;148 Klypin et al.149 ; Gao et al.150 ; Wu et al.151 ). Thus the statistical properties of cluster subhalos such as mass function and spatial distribution, provide us with useful information about the mass assembly history. Their measurement is the most stringent test of CDM predictions on scales of less than several Mpc. A characteristic feature of the subhalo mass function is also critically important to constrain the nature of dark matter, because it depends on the particle mass of dark matter. Furthermore, measurement of the correlation between member galaxies and subhalo masses sheds important insight on the physics of galaxy evolution associated with dark matter. Thus it is important to measure the mass function directly from observations without assuming a relationship between dark matter and luminous matter and the dynamical state of the system. Weak lensing observations of very nearby clusters (z ≤ 0.05) are ideal for this purpose because their large apparent size enables us to easily resolve less massive subhalos inside the clusters and provides a correspondingly large number of background galaxies, which leads to low statistical errors and compensates for the low lensing efficiency to achieve a high S/N. We show the results of a 4.1 deg2 weak gravitational lensing survey of subhalos in the very nearby Coma cluster at redshift z = 0.0236 using the Subaru/Suprime-Cam (Okabe et al., 2012). The observed area is about 80% within r200 inside of which the mean density is 200 times the critical density at the cluster redshift. Note that one arcmin corresponds to 20 h−1 kpc for the cosmology with Ωm,0 = 0.27, ΩΛ,0 = 0.73 and H0 = 100 hkms−1Mpc−1 . Figure 14 shows the projected mass distribution with a smoothing scale of FWHM= 4
. and units of significance of ν = κ/σκ . As shown in this figure, 32 subhalos were detected and their masses are measured in a model-independent manner, down to the order of 103 of the virial mass of the cluster. All of them are associated with a small group of cluster members. The mean distortion profiles stacked over subhalos show a sharply truncated feature well-fitted by a Navarro– Frenk–White (NFW) mass model with the truncation radius, as expected due to tidal destruction by the main cluster. It is found that subhalo masses, truncation radii, and mass-to-light ratios decrease toward the cluster center. Figure 15 shows the subhalo mass function dn/dlnMsub constructed by weak lensing mass measurements. It covers a range of two orders of magnitude in mass and is well described by a single power law

dn −α ∝ Msub , d ln Msub +0.42 . with α = 1.09−0.32

(141)

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Fig. 14. Projected mass distribution with a smoothing scale of FWHM= 4 and units of significance of ν = κ/σκ . The shear is used without taking into account the LSS lensing effect. The contours of significance start at 1σ with a step value of 1σ (Okabe et al.158 ). (For color version, see page II-CP13.)

This agrees with the predicted value of ∼ 0.9–1.0 from an N-body simulation based on the CDM scenario. 4.5. Cosmic shear The most challenging observation in weak lensing is cosmic shear, which is weak lensing due to the large scale structure of the universe. It attracted much attention because its signal depends on the expansion rate of the universe and the growth rate of structure. It therefore carries useful information about dark energy and the theory of gravity. Cosmic shear has already been detected by several groups.35–38 Subsequent detections have put useful constraints on the matter density parameter Ωm,0 and on the matter density spectrum normalization σ8 (Bacon et al.152 ; Hoekstra, Yee and Gladders153 ; Bacon et al.154 ; Hamana et al.106 ). More recent results are found, for example, in Fu et al.155 for the CFHTLS third data release; Schrabback et al.156

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Fig. 15. Subhalo mass function spanning two orders of magnitude of subhalo masses. The red solid and blue dotted lines are the best-fit power-law model and Schechter function, respectively. The best-fit powers are in remarkable agreement with CDM predictions. Green dashed lines are the mass function for spurious peaks. The thick and thin dashed lines are the best-fit and the 68% C.L. uncertainty, respectively (Okabe et al.158 ). (For color version, see page II-CP13.)

-5

-1.0 10

Fig.E-mode, are 16. Theand variance black open of thepoints aperture are mass B-mode. measured The signal by the in CFHTLS full angular Wide. scaleRed is shown filled points in the enlargement in the panel. It shows a very small B-mode on very large scales (120  –230 ).155

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for the Hubble Space Telescope COSMOS survey; Benjamin et al.157 and Kilbinger et al.159 for CFHTLens; and Jee et al.160 for the Deep Lens Survey. In particular, the Canada–France–Hawaii Telescope Lensing Survey (CFHTLenS) covers 154 deg2 in five optical bands where the shapes of 4.2 million galaxies with 0.2 < z < 1.3 (photometric redshift) are measured. Combining with WMAP7, BOSS and an HST distance-ladder prior on the Hubble parameter, it is found that Ωm,0 = 0.283±0.010 and σ8 = 0.813 ± 0.04 for a flat ΛCDM model. On-going and planned galaxy surveys aiming to observe cosmic shear including the Panoramic Survey Telescope & Rapid Response System (Pan-STARRS),60 the Subaru Hyper Suprime Cam (HSC) Survey,161 the Dark Energy Survey (DES),162 the Large Synoptic Survey Telescope (LSST),163 the KIlo-Degree Survey (KIDS)164 and Euclid.165 However the signal is extremely small and contains many systematic errors such as inaccurate shape measurement, and inaccurate determination of redshifts. Also, we do not yet have a complete theoretical understanding of small scale clustering. Thus it is essential to reduce these systematic errors down to a required level in order to place a useful constraint on the cosmological information (Huterer et al.166 ; Cropper et al.167 ; Massey et al.168 ). This is an ongoing and actively developing research field. Here we give an introduction to understand the physics of cosmic shear. Cosmic shear observations measure the coherent distortion of background galaxies. This is related to the convergence of the matter density field and thus the matter power spectrum. In fact, there is a relationship between the shear and convergence: ˆ γ (k)ˆ γ (k) = ˆ κ(k)ˆ κ(k
) = (2π)2 C(k)δD (k − k
),

(142)

where the quantities with a “hat” are Fourier transforms of the corresponding quantities, and C(k) is the lensing power spectrum of the shear (convergence). As easily imagined, the power spectrum is related to the matter power spectrum, and thus the observation of the cosmic shear provides a method to measure the evolution of the matter power spectrum. The details are as follows. First, the cosmological lens equation may be written as the equation for the convergence field by the matter distribution at the comoving distance λ(z) as follows:

3Ωm H02 λH
(λ − λ
)λ
dχ δ(x
, λ), (143) κ(θ, λ) = 2 a(λ
)λ 0 where δ is the matter density contrast and the Limber approximation has been used. We have neglected the contribution from integration along the line of sight. As seen from the above equation, the dependence of the cosmological parameters appears in two parts, in the comoving distance and in the evolution of the density contrast. The latter depends not only on the global geometry of the universe but also on the local gravitational force, and thus cosmic shear is in principle able to distinguish between dark energy and an alternative theory of gravity as the origin of the accelerated expansion.

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The power spectrum for shear (convergence) may be written in terms of the matter power spectrum Pδ as follows:



9H04 Ωm λH g 2 (λ) k C(k) = dλ 2 Pδ ;χ . (144) 4 a (λ) λ 0 This is the desired relationship between the power spectrum of shear and the matter power spectrum. Here g(χ) is defined as

λH λ
− λ dλ
p(λ) g(λ) = , (145) λ
0 with p(χ) being the observed probability distribution of galaxies for the survey. A precise knowledge of p(χ) is only obtained by spectroscopic observation, which is very time consuming for a large survey. Assuming a close correlation between the dark matter density contrast δm (z) and the number density of galaxies ng (z), the sum of a huge number of galaxies makes the convergence observable. Although large and deep redshift surveys are planned in the future, a photometric determination of the redshift (photo-z method) is used for existing surveys and those planned in the near future. One of the errors in the observation of cosmic shear arises from the error in the photo-z method. 4.5.1. How to measure the cosmic density field In practice, the following two point correlation functions are observed: ξ± (θ) = γt γt (θ) ± γ× γ× (θ),

ξ× (θ) = γt γ× (θ).

(146)

Since γt → γt nad γ× → −γ× under the parity transformation, ξ× should vanish. These correlation functions may be expressed in terms of the power spectrum. For this purpose, it is convenient to choose the coordinate as θ = (θ, 0) and then γ1 = −γt , γ2 = −γ× . Thus

d ξ+ = γ(0)γ(θ) = J0 (θ)C(). (147) 2π Similarly

ξ− =

d J4 (θ)C(), 2π

(148)

where Jn (x) is the nth Bessel function. However, the power spectrum is expressed in terms of these two point functions

(149) C() = 2π dθθξ+ (θ)J0 (θ) = 2π dθθξ− (θ)J4 (θ), where we use the orthogonality of the Bessel functions

∞ 1 dxxJn (ux)Jn (vx) = δD (u − v). u 0

(150)

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In practice, the averaged tangential shear is used. For this purpose, first we define the window function U , which satisfies the property

(151) dθθU (θ; θr ) = 0, where θr characterizes the radius of the averaging. The filtered convergence is then defined as follows:

(152) F (θr , ; θ0 ) ≡ d2 θ
U (|θ
− θ 0 |; θr )κ(θ
), where θ 0 is the center of the region averaged over. One can easily confirm that the above may be written as the integral of the tangential shear

(153) F (θr ; θ 0 ) ≡ d2 θ
Q(|θ
− θ0 |; θr )γt (θ
), where

θ 2 dθ
θ
U (θ
; θr ) − U (θ; θr ). θ2 0 Consequently, the ensemble average is given by

2

2 dθθU (θ, θr )J0 (θ) . F (θr ; θ 0 ) = 2π dC()

Q(θ; θr ) =

(154)

(155)

One can obtain information regarding the power spectrum (which depends on the cosmological parameters) by observing the dispersion of the aperture mass which is constructed by measuring the tangential shear. There are several choices for the window function. Here we use one proposed by Schneider et al. (2002): 



 θ2 1 θ2   9 1 − − θ < θr 2 θr2 3 θr2 U (θ) = πθr , (156)   0 θ > θr

  2 2   6 θ 1− θ θ < θr 2 2 θr2 Q(θ) = πθr θr . (157)   0 θ > θr Consequently, the ensemble average is called the aperture mass and is written as follows:

2 (158) Map (θr ) = 2π dC()I(θr ) with

 I(θr ) =

2 12 J (θ ) . 4 r π(θr )2

(159)

The functional form of the kernel I(θr ) for the aperture mass shows that it plays a role of picking up modes θr ∼ 5 in the power spectrum, and is considered a good statistical quantity to evaluate the power spectrum.

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Since the power spectrum may be expressed by the correlation functions ξ±  

 

1 2θr θdθ θ θ 2 Map (θr ) = T+ ξ+ (θ) + T− (160) ξ− (θ) , 2 0 θr2 θr θr where

∞

T+ (x) = 576 0

∞

T− (x) = 576 0

we can also define

dt J0 (xt) [J4 (t)]2 , t3

(161)

dt 2 J4 (xt) [J4 (t)] , t3

(162)

M× =

d2 θQ(θ)γ× (θ).

(163)

2 This quantity vanishes identically as long as the convergence is real. Thus M×

may be used for the evaluation of noise:  

 

1 ∞ θdθ θ θ 2 M× (θr ) = T+ ξ+ (θ) − T− (164) ξ− (θ) . 2 0 θr2 θr θr 2 In an ideal situation, M×

should vanish identically because ξ× = 0. This does not happen in practical observations due to the existence of noise. Actually it may 2

is the be shown by using the E/B decomposition of the convergence that M× B-mode part of the aperture mass. However the observed region is finite and the number density of background galaxies is finite in practical observations. Thus we cannot take a correlation √ between two galaxies separated more than some θmax and less than θmin ∼ 1/ ng . Thus E- and B-mode decomposition is not perfect and there exists leakage from the E-mode to the B-mode on small scales if θr ≤ 2θmax . To avoid such leakage it is possible to choose a window function to define a filtered convergence that it can be calculated using only the two point correlation function with a finite separation.170 In practical measurements of the two-point correlation function, we need to construct an unbiased estimator and a covariance matrix in order to compare observables with theory. We ask the reader to consult relevant references such as Schneider et al.171 and Semboloni et al.172 for detail.

5. Conclusion and Future At 100 years after its birth, general relativity has become an indispensable tool for understanding our universe. Gravitational lensing is not the only example, but is certainly one of the most actively investigated examples, which was brought about by rapid advances in observational apparatus and techniques since the 1980s. This tendency will continue in the future with ongoing and planned weak lensing surveys such as the Subaru HSC survey, Pan-STARR, DES, KIDS and Euclid. The role of gravitational lensing will become increasingly important in observational cosmology.

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In this review, we have provided a pedagogical introduction to the basics of gravitational lensing and highlighted some of the recent achievements. The research area where gravitational lensing plays a role is very wide. In fact, any cosmological observation suffers from the effect of gravitational lensing to some extent. For example, the magnitude–redshift (m–z) relationship in Type Ia Supernovae (SNIa) was used to confirm the acceleration of the universe173,174 and it is expected that future supernovae surveys will determine the nature of dark energy. Since the m–z relation is influenced by the lensing by large scale structure, this program is possible only with an accurate estimate of the lensing effect. Lensing effects by large scale structure in cosmological observables have been regarded as noise from an observational point of view, but their measurement may turn out to yield important information on structure formation and cosmology if we have an accurate theoretical understanding of its origin. Acknowledgments I would like to thank my collaborators, Alan Kawarai Lefor, Nobuhiro Okabe, Yuki Okura, Masahiro Takada, and Keiichi Umetsu. Parts of this review are based on the work done with them. Unfortunately I am only able to cover a part of this broad research field in this review, and there are many other interesting applications, such as CMB lensing. We ask the reader to consult other reviews on gravitational lensing for a more comprehensive treatment. I thank Masayo Morioka for preparing the figures in this review and Alan Kawarai Lefor for improving the English usage. The author would like to thank Prof. Wei-Tou Ni for the opportunity to contribute to the Centennial General Relativity book. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

J.-P. Kneib, Ph.D. Thesis (1993). D. Valls-Gabaud, Albert Einstein Century Int. Conf. 861 (2006) 1163. J. Soldner, Astronomishes Jahrbuch fur 1804 (1801) 161. O. Chwolson, Astronomische Nachrichten 221 (1924) 329. A. Einstein, Science 84 (1936) 506. F. Link, Comptes Rendus Acad. Sci. 202 (1936) 917. F. Link, Bull. Astron. 10 (1937) 73. F. Zwicky, PR 51 (1937) 290. S. Refsdal, Mon. Not. R. Astron. Sco. 128 (1964) 307. S. Liebes, Phys. Rev. 133 (1964) 835. D. Walsh, R. F. Carswell and R. J. Weymann, Nature 279 (1979) 381. T. Kundi´c et al., Astrophys. J. 482 (1997) 75. R. Lynds and V. Petrosian, Bull. Am. Astron. Soc. 18 (1986) 1014. G. Soucail, B. Fort, Y. Mellier and J. P. Picat, AA 172 (1987) L14. J. N. Hewitt, E. L. Turner, D. P. Schneider, B. F. Burke and G. I. Langston, Nature 333 (1988) 537. 16. J.-P. Kneib, R. S. Ellis, I. Smail, W. J. Couch and R. M. Sharples, Astrophys. J. 471 (1996) 643.

May 4, 2017 9:54 b2167-ch17

Gravitational lensing in cosmology

17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61.

II-221

E. Jullo et al., Science 329 (2010) 924. R. S. Ellis, arXiv:0109294. J.-P. Kneib, R. S. Ellis, M. R. Santos and J. Richard, Astrophys. J. 607 (2004) 697. J. Richard et al., Astrophys. J. 685 (2008) 705. R. J. Bouwens et al., Astrophys. J. 690 (2009) 1764. D. Coe et al., Astrophys. J. 762 (2013) 32. M. Bradaˇc et al., Astrophys. J. 652 (2006) 937. M. Bradaˇc et al., Astrophys. J. 706 (2009) 1201. E. Jullo et al., New J. Phys. 9 (2007) 447. E. Jullo and J.-P. Kneib, Mon. Not. R. Astron. Soc. 395 (2009) 1319. J. Richard et al., arXiv:1405.3303. E. L. Turner, J. P. Ostriker and J. R. Gott, Astrophys. J. 284 (1984) 1. E. Turner, Astrophys. J. 365 (1990) L43. M. Fukugita, T. Futamase and M. Kasai, Mon. Not. R. Astron. Soc. 246 (1990) 24P. J. A. Tyson, R. A. Wenk and F. Valdes, ApJL 349 (1990) L1. N. Kaiser and G. Squires, Astrophys. J. 404 (1993) 441. N. Kaiser, G. Squires and T. Broadhurst, Astrophys. J. 449 (1995) 460. N. Okabe, G. P. Smith, K. Umetsu, M. Takada and T. Futamase, ApJL 769 (2013) L35. L. van Waerbeke et al., A&A 358 (2000) 30. D. J. Bacon, A. R. Refregier and R. S. Ellis, Mon. Not. R. Astron. Soc. 318 (2000) 625. N. Kaiser, G. Wilson and G. A. Luppino, arXiv:astro-ph/0003338. R. Maoli et al., A&A 368 (2001) 766. L. Van Waerbeke et al., A&A 393 (2002) 369. D. Wittman et al., Nature 405 (2000) 143. B. Paczynski et al., Astrophys. J. 435 (1994) L113. P. Schneider, J. Ehlers and E. E. Falco, Gravitational Lenses, Vol. XIV (SpringerVerlag, Berlin, 1992). R. D. Blandford and R. Narayan, 30 (1992) 311. S. Refsdal and J. Surdej, Rep. Prog. Phys. 57 (1994) 117. R. Narayan and M. Bartelmann, arXiv:astro-ph/9606001. T. Futamase, Mon. Not. R. Astron. Soc. 237 (1989) 187. L. L. R. Williams and G. F. Lewis, Mon. Not. R. Astron. Soc. 294 (1998) 299. T. Futamase, M. Hattori and T. Hamana, ApJL 508 (1998) L47. A. Zitrin and T. Broadhurst, ApJL 703 (2009) L132. M. Bartelmann, AA 303 (1995) 643. C. O. Wright and T. G. Brainerd, arXiv:9908213. N. A. Grogin and R. Narayan, Astrophys. J. 464 (1996) 92. R. D. Blandford and C. S. Kochanek, Dark Matter in the Universe, J. Bahcall, T. Piran and S. Weinberg, (eds.) (World Scientific, Springer), p. 133. M. Hattori, J. Kneib and N. Makino, Progress Theor. Phys. Suppl. 133 (1999) 1. C. S. Kochanek, in Proc. 33rd Saas-Fee Advanced Course, G. Meylan, P. Jetzer and North P. (eds.) (Springer-Verlag, Berlin). S. Mao and P. Schneider, Mon. Not. R. Astron. Soc. 295 (1998) 587. M. Chiba, Astrophys. J. 565 (2002) 17. S. H. Suyu et al., Astrophys. J. 766 (2013) 70. S. H. Suyu et al., ApJL 788 (2014) L35. N. Kaiser, W. Burgett and K. Chambers et al., Proc. SPIE 7733 (2010) 77330. M. Oguri and P. J. Marshall, MNRAS 405 (2010) 2579.

May 4, 2017 9:54 b2167-ch17

II-222

T. Futamase

62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88.

E. V. Linder, Phys. Rev. D 84 (2011) 123529. H. Dahle, S. J. Maddox and P. B. Lilje, Astrophys. J. 435 (1994) L79. P. Fisher, G. Bernstein, G. Rhee and J. A. Tyson, AJ 113 (1997) 521. A. T. Lefor, T. Futamase and M. Akhlaghi, NewAR 57 (2013) 1. P. Saha and L. L. R. Williams, AJ 127 (2004) 2604. R. B. Wayth and R. L. Webster, Mon. Not. R. Astron. Soc. 372 (2006) 1187. D. Coe et al., Astrophys. J. 681 (2008) 814. J. Liesenborgs et al., Mon. Not. R. Astron. Soc. 386 (2008) 307. M. Oguri, PASJ 62 (2010) 1017. C. R. Keeton, arXiv:astro-ph/0102340. E. L. Turner, ApJL 365 (1990) L43. C. S. Kochanek, Astrophys. J. 466 (1996) 638. E. O. Ofek, H.-W. Rix and D. Maoz, Mon. Not. R. Astron. Soc. 343 (2003) 639. K.-H. Chae and S. Mao, ApJL 599 (2003) L61. A. Matsumoto and T. Futamase, Mon. Not. R. Astron. Soc. 384 (2008) 843. K.-H. Chae, Mon. Not. R. Astron. Soc. 402 (2010) 2031. P. R. Capelo and P. Natarajan, New J. Phys. 9 (2007) 445. M. Oguri et al., AJ 143 (2012) 120. M. Bernardi et al., Mon. Not. R. Astron. Soc. 404 (2010) 2087. S. M. Croom et al., Mon. Not. R. Astron. Soc. 399 (2009) 1755. S. Cao and Z.-H. Zhu, A&A 538 (2012) A43. J. Tyson, R. A. Wenk and F. Valdes, Astrophys. J. 349 (1990) L1. N. Kaiser and G. Squires, Astrophys. J. 404 (1993) 441. P. Schneider and C. Seitz, AA 294 (1995) 411. T. Brainerd, R. D. Blandford and I. Smail, Astrophys. J. 466 (1996) 623. M. J. Huudson, S. D. Gwyn, H. Dahle and N. Kaiser, Astrophys. J. 503 (1998) 531. D. Clowe, G. A. Luppino, N. Kaiser, J. P. Henry and I. M. Gioia, Astrophys. J. 498 (1998) L61. H. Hoekstra, M. Franx, K. Kuijken and P. G. van Dokkum, Mon. Not. R. Astron. Soc. 333 (2002) 911. H. Dahle et al., ApJS 139 (2002) 313. E. S. Cypriano, L. Sordre, J.-P. Kneib and L. E. Campusano, Astrophys. J. 613 (2004) 95. M. Bartelmann and P. Schneider, Phys. Rep. 340 (2001) 291. K. Umetsu, arXiv:1002.3952. K. Umetsu, M. Tada and T. Futamase, PTP Suppl. 133 (1999) 53. T. Broadhurst et al., Astrophys. J. 619 (2005) L143. E. Medezinski et al., Astrophys. J. 663 (2007) 717. Y. Okura and T. Futamase, Astrophys. J. 699 (2009) 143. Y. Okura and T. Futamase, Astrophys. J. 730 (2011) 9. Y. Okura and T. Futamase, Astrophys. J. 748 (2012) 112. Y. Okura and T. Futamase, Astrophys. J. 771 (2012) 37. K. Umetsu and T. Broadhurst, Astrophys. J. 684 (2008) 177. M. J. Hudson, S. D. J. Gwyn, H. Dahle and N. Kaiser, Astrophys. J. 503 (1998) 531. H. Hoekstra, M. Franx, K. Kuijken and G. Squires, Astrophys. J. 504 (1998) 636. T. Erben, L. van Waerbeke, E. Bertin, Y. Mellier and P. Schneider, A&A 366 (2001) 717. M. Hetterscheidt et al., A&A 468 (2007) 859. T. Hamana et al., Astrophys. J. 597 (2003) 98.

89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106.

May 4, 2017 9:54 b2167-ch17

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107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120. 121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149.

II-223

S. Seitz and P. Schneider, AA 305 (1998) 383. L. van Waerbeke and Y. Mellier, arXiv:astro-ph/0305089. M. Takada and M. White, Astrophys. J. 601 (2004) L1. C. M. Hirata and U. Seljak, PRD 70 (2004) 063526. K. Umetsu et al., arXiv:1404.1375U. S. Bridle et al., Ann. Appl. Stat. 3 (2009) 6. S. Bridle et al., Mon. Not. R. Astron. Soc. 405 (2010) 2044. T. D. Kitching, A. F. Heavens and L. Miller, Mon. Not. R. Astron. Soc. 413 (2011) 2923. T. D. Kitching et al., Mon. Not. R. Astron. Soc. 423 (2012) 3163. T. D. Kitching et al., ApJS 205 (2013) 12. C. Heymans et al., Mon. Not. R. Astron. Soc. 368 (2006) 1323. R. Massey et al., Mon. Not. R. Astron. Soc. 376 (2007) 13. T. D. Kitching et al., Mon. Not. R. Astron. Soc. 423 (2012) 3163. Y. Okura, K. Umetsu and T. Futamase, Astrophys. J. 660 (2007) 995. Y. Okura, K. Umetsu and T. Futamase, Astrophys. J. 680 (2008) 1. A. Leonard, D. M. Goldberg, J. L. Haaga and R. Massey, Astrophys. J. 666 (2007) 51. A. Leonard, L. J. King and D. M. Goldberg, Mon. Not. R. Astron. Soc. 413 (2011) 789. B. Cain, P. L. Schechter and M. W. Bautz, Astrophys. J. 736 (2011) 43. M. Viola, P. Melchior and M. Bartelmann, arXiv:1107.3920. B. Rowe et al., arXiv:1211.0966. R. S. Levinson, Publ. Astron. Soc. Jpn. 125 (2013) 1474. G. Fahlman, N. Kaiser, G. Squires and D. Woods, Astrophys. J. 437 (1994) 56. D. E. Applegate et al., Mon. Not. R. Astron. Soc. 439 (2014) 48. H. Hoekstra et al., Mon. Not. R. Astron. Soc. 427 (2012) 1298. M. Oguri et al., Mon. Not. R. Astron. Soc. 420 (2012) 3214. N. Okabe and K. Umetsu, Publ. Astron. Soc. Jpn. 60 (2008) 345. A. B. Newman et al., Astrophys. J. 765 (2013) 24. H. Ebeling et al., Mon. Not. R. Astron. Soc. 301 (1998) 881. H. Ebeling et al., VizieR Online Data Catalog 730 (2000) 10881. H. B¨ ohringer et al., Astron. Astrophys. 425 (2004) 367. A. Bhattacharya et al., Astrophys. J. 766 (2013) 32. J. E. Taylor and A. Babul, Mon. Not. R. Astron. Soc. 348 (2004) 811. J. E. Taylor and A. Babul, Mon. Not. R. Astron. Soc. 364 (2005) 515. J. E. Taylor and A. Babul, Mon. Not. R. Astron. Soc. 364 (2005) 535. M. Oguri and J. Lee, Mon. Not. R. Astron. Soc. 355 (2004) 120. F. C. van den Bosch, X. Yang and H. J. Mo, Mon. Not. R. Astron. Soc. 340 (2003) 771. J. Diemand, B. Moore and J. Stadel, Mon. Not. R. Astron. Soc. 352 (2004) 535. G. De Lucia et al., Mon. Not. R. Astron. Soc. 348 (2004) 333. L. Gao, S. D. M. White, A. Jenkins, F. Stoehr and V. Springel, Mon. Not. R. Astron. Soc. 355 (2004) 819. L. D. Shaw, J. Weller, J. P. Ostriker and P. Bode, Astrophys. J. 646 (2006) 815. R. E. Angulo, C. G. Lacey, C. M. Baugh and C. S. Frenk, Mon. Not. R. Astron. Soc. 399 (2009) 983. C. Giocoli, G. Tormen, R. K. Sheth and F. C. van den Bosch, Mon. Not. R. Astron. Soc. 404 (2010) 502. A. A. Klypin, S. Trujillo-Gomez and J. Primack, Astrophys. J. 740 (2011) 102.

May 4, 2017 9:54 b2167-ch17

II-224

T. Futamase

150. L. Gao et al., Mon. Not. R. Astron. Soc. 425 (2012) 2169. 151. H.-Y. Wu, O. Hahn, R. H. Wechsler, P. S. Behroozi and Y.-Y. Mao, Astrophys. J. 767 (2013) 23. 152. D. J. Bacon, A. Refregier, D. Clowe and R. S. Ellis, Mon. Not. R. Astron. Soc. 325 (2001) 1065. 153. H. Hoekstra, H. K. C. Yee and M. D. Gladders, Astrophys. J. 577 (2002) 595. 154. D. J. Bacon, R. J. Massey, A. R. Refregier and R. S. Ellis, Mon. Not. R. Astron. Soc. 344 (2003) 673. 155. L. Fu et al., Astron. Astrophys. 479 (2008) 9. 156. T. Schrabback et al., Dark Matter Astropart. Part. Phys., Proceedings of the 6th International Heidelberg Conference, ed. H. V. Klapdor-Kleingrothaus of Geraint F. Lewis (World Scientific Publishing Co) (2008), pp. 260–273. 157. J. Benjamin et al., Mon. Not. R. Astron. Soc. 431 (2013) 1547. 158. N. Okabe, T. Futamase, M. Kajisawa and R. Kuroshima, Astrophys. J. 784 (2014) 90. 159. M. Kilbinger et al., Mon. Not. R. Astron. Soc. 430 (2013) 2200. 160. M. J. Jee et al., Astrophys. J. 765 (2013) 74. 161. The HSC collab., http://member.ip,u.jp/masahiro.takada/proposal rv.pdf. 162. D. L. DePoy et al., SPIE 7014 (2008) 70140E. 163. J. A. Tyson, D. M. Wittman, J. F. Hennawi and D. N. Spergel, in Proc. 5th Int. UCLA Symp. (2002). 164. G. Verdoes Kleijn et al., arXiv:1112.0886. 165. Euclid Science Study, arXiv:0912.0914. 166. D. Huterer, M. Takada, G. Bernstein and B. Jain, Mon. Not. R. Astron. Soc. 366 (2006) 101. 167. M. Cropper et al., Mon. Not. R. Astron. Soc. 431 (2013) 3103. 168. R. Massey et al., Mon. Not. R. Astron. Soc. 429 (2013) 661. 169. J. A. Peacock and S. J. Dodds, Mon. Not. R. Astron. Soc. 280 (1996) L19. 170. P. Schneider and M. Kilbinger, Astron. Astrophys. 462 (2007) 841. 171. P. Schneider, L. van Waerbeke, M. Kilbinger and Y. Mellier, Acta Astronom. 396 (2002) 1. 172. E. Semboloni et al., Mon. Not. R. Astron. Soc. 375 (2007) L6. 173. M. Schmidt et al., Astrophys. J. 507 (1998) 46. 174. S. Perlmutter et al., Nature 391 (1998) 51. 175. J. F. Navarro, C. S. Frenk and S. D. White, Astrophys. J. 490 (1997) 493. 176. J.-P. Kneib and P. Natarajan, Astron. Astrophys. Rev. 19 (2011) 47. 177. C. Seitz and P. Schneider, Astron. Astrophys. 318 (1997) 687. 178. H. Hoekstra, M. Franx and K. Kuijken, Astrophys. J. 532 (2000) 88. 179. M. Postman et al., ApJCS 199 (2012) 25. 180. A. Zitrin et al., Astrophys. J. 742 (2011) 117. 181. D. Coe et al., Astrophys. J. 757 (2012) 22. 182. K. Umetsu, Astrophys. J. 795 (2014) 163. 183. K. Umetsu, T. Broadhurst, A. Zitrin, E. Medezinski and L. Hsu, Astrophys. J. 729 (2011) 127. 184. J. Merten, M. Cacciato, M. Meneghetti, C. Mignone and M. Bastelmann, Astron. Astrophys. 500 (2009) 681.

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Chapter 18 Inflationary cosmology: First 30+ years

Katsuhiko Sato National Institutes of Natural Sciences, 2F Hulic Kamiyacho Building, 4-3-13 Toranomon, Minato-ku, Tokyo 105-0001, Japan Department of Physics, School of Science and Engineering, Meisei University, 2-1-1 Hodokubo, Hino-shi, Tokyo 191-8506, Japan Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), UTIAS, WPI, The University of Tokyo, Kashiwanoha, Kashiwa, Chiba 277-8568, Japan [email protected] Jun’ichi Yokoyama Research Center for the Early Universe (RESCEU), Department of Physics, School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), UTIAS, WPI, The University of Tokyo, Kashiwanoha, Kashiwa, Chiba 277-8568, Japan [email protected]

Starting with an account of historical developments in Japan and Russia, we review inflationary cosmology and its basic predictions in a pedagogical manner. We also introduce the generalized G-inflation model, in terms of which all the known single-field inflation models may be described. This formalism allows us to analyze and compare the many inflationary models that have been proposed simultaneously and within a common framework. Finally, current observational constraints on inflation are reviewed, with particular emphasis on the sensitivity of the inferred constraints to the choice of datasets used. Keywords: Cosmology; inflation; cosmic microwave background.

1. Introduction The classical Big Bang Cosmology1 was the first theory of the universe starting from physical principles that succeeded in interpreting in a unified manner the three fundamental cosmological observations of the time: The homogeneous cosmic expansion, originally discovered by Hubble2 and Lemaˆıtre3; the cosmic microwave II-225

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background radiation (CMB) discovered by Penzias and Wilson4 ; and the abundance of light elements.5,6 The classical Big Bang cosmology, however, suffered from several fundamental difficulties such as the horizon problem, the flatness problem,7 and the initial singularity problem. All these problems relate to the initial conditions for the universe. Because reliable theories to describe the very early universe were lacking, these problems were not studied seriously until quite long after the confirmation of the classical Big Bang cosmology by the discovery of CMB. Such a situation was somewhat changed by the introduction of grand unified theories (GUTs) by Georgi and Glashow.8 Since the unification scale of GUTs, MGUT = 1015−16 GeV is on one hand too big to be experimentally accessible but on the other hand is just below the Planckian scale, MPl = (
c5 /G)1/2 = 1.2 × 1019 GeV, where quantum gravitational effects become important, there emerged a trend to use the high temperature and density regime in the early universe as an arena for its verification, stimulating serious study of the birth of the universe. At the same time, however, GUT theories challenged the classical Big Bang cosmology by predicting a phase transition9 which overproduces10 magnetic monopoles11,12 that would dominate the matter content of the universe, leaving no room for ordinary baryonic matter, with Ωmonopole /Ωb
1015 . Fortunately, however, the path toward resolving this serious problem was already present in the GUT itself, as observed independently by Sato13–15 and Guth.16 Inflationary cosmology thus came into being. These pioneers observed that in the symmetric state of the Higgs field, which is realized thermally shortly after the Big Bang, the field has a large potential energy 4 , and that if this state is metastable with a long enough density of order of MGUT lifetime, the universe would be dominated by the false vacuum energy density which would not decrease as long as the Higgs field remains in that state no matter how much the universe expands. As a result, an exponential cosmic expansion is realized which was referred to as cosmic inflation. Since both the particle horizon and the curvature radius are exponentially stretched during inflation, it can in principle provide solutions to the horizon and the flatness problems. Apart from the GUT-based approach mentioned above, there was a longer tradition in the Russian school to pursue theories of the very early universe. Although many were speculative, a number of important ideas were proposed. An important such idea was Starobinsky’s curvature square theory which realized exponential cosmic expansion followed by Friedmann regime.17 Thus nowadays these three authors are regarded as the pioneers of inflationary cosmology18 before the slow-roll scenario was proposed.19,20 Before starting to review standard inflation updating Ref. 21, we first describe early history of inflation. Since much has already been written about the developments in America (see e.g. Refs. 22 and 23), we emphasize those in Japan and Russia, which followed remarkably different paths and have not been discussed in much detail in the English language literature. We take units c =
= 1.

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1.1. Developments in Japan In Japan theories of the early universe were developed in close connection with unified theories of elementary interactions. Compact astrophysical objects were also used as their testing ground. Unification of elementary interactions in nature has been a long-standing issue of theoretical physics which is also called the Einstein’s dream. As the first step, the theory to unify the weak interaction with the electromagnetic force was proposed by Weinberg24 and Salam25 in 1967, but it was not immediately widely accepted in the community. Only after 1973 when the Gargamelle experiment at CERN discovered the neutral current did this theory become popular. The importance of the neutral current interaction in the neutrino diffusion in supernovae explosions was shown in Ref. 26, which was confirmed by the Nobel-prize winning discovery of the neutrino bursts from SN1987A.27 The mass of the Higgs boson in the electroweak theory was constrained by CMB28 and stellar evolution.29 Symmetry restoration and further breakdown in high density matter was also pointed out in Refs. 30 and 31 in the electroweak theory. Further progress toward a unified theory was made by Georgi and Glashow, who proposed the grand unified theories (GUTs).8 They argued that the three fundamental interactions unify at an extremely high energy scale around MGUT = 1015−16 GeV. At this high energy scale baryon nonconserving interactions characteristic of GUTs are expected to have occurred without suppression. Yoshimura32 proposed a model of baryogenesis in the early universe making use of these interactions. Applying studies of high-temperature behavior of unified gauge theories33–36 to the hot big bang universe, Sato and Sato proposed that all the four elementary interactions were unified above the Planckian scale when gravity bifurcated, and at the GUT phase transition, electroweak and strong interactions were separated from each other, followed by the Weinberg–Salam phase transition. Then the cosmic history of the elementary interactions can be expressed by a tree diagram as shown in Fig. 1. Sato also realized that, if the symmetry-breaking phase transition of the GUT Higgs field in the early universe is first-order, the universe would expand exponentially in a supercooled stage when the false vacuum energy dominates the universe, and that a number of interesting cosmological consequences would be obtained. In the first paper of the series, Sato showed that the particle horizon would be exponentially stretched, so that even if CP symmetry is softly broken in baryogenesis, one may realize a large enough domain with positive baryon number instead of a mixture of small domains with positive and negative baryon numbers.14 He then showed that fluctuations associated with the phase transition are exponentially stretched and may account for the origin of seed fluctuations for large-scale structures.13 Although the current paradigm of cosmological structure formation attributes the origin of seed fluctuations to quantum fluctuations of a nearly massless scalar field which induces inflation, the essential features of

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Fig. 1. Evolutionism of interactions. The original picture was first published in an article by Humitaka Sato and Katsuhiko Sato in June 1976 in a Japanese science magazine “Shizen” (meaning Nature) published by Chuo-Koron-sha.

the generation of superhorizon fluctuations and its reentry to the Hubble radius in the subsequent Friedmann stage were already there.13 Finally, the monopole problem was also shown to be solved as the exponential expansion can push it away from the observable domain.15 Monopoles are diluted by subsequent entropy production. Unfortunately, however, this type of scenario making use of a first-order phase transition, now referred to as old inflation, was not successful because inflation could not be terminated to realize a hot Friedmann universe over the comoving scale large enough to correspond to the current Hubble horizon. On the other hand, Sato and collaborators seriously considered geometrical outcomes after the phase transition and reached the revolutionary view that our universe is not unique but there can be multiple universes that are causally disconnected from each other.37,38 Nowadays the multiverse is very popular in unified theories such as string theories and even the probability that a universe like ours is created is seriously considered. One should remember that these papers were first to discuss multiverse as an outcome of modern unified theories. 1.2. Developments in Russia In former Soviet Union, the possibility of exponential cosmic expansion was first suggested by Gliner39 in 1965, who classified the possible algebraic structures of

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the energy–momentum tensor. Among them he found a form of matter called the µ-vacuum, which has an equation of state (EOS) P = −µ with the energy density ρ = µ. This state has the same invariance as the vacuum and with µ > 0 the metric takes the de Sitter form. This proposal, however, cannot be considered as the birth of inflationary cosmology since a discussion of the transition to the Friedmann regime is absent. In fact, the observation that the matter with EOS P = −ρ induces exponential expansion was already recognized by Lemaˆıtre.40 This idea was pushed forward by Gurevich41 to solve the problem of causal connection of galaxies. Although he made some thermodynamic arguments of a “Λ” field (he used the character Λ instead of µ), its transition to normal matter remained unclarified. Almost at the same time Gliner and Dymnikova42 considered a cosmology with EOS P = −ρ only in the very early universe to avoid the initial singularity. In this scenario the EOS was supposed to change to that of normal radiation or matter in the course of expansion, but again its physical mechanism was left unexplored. A paper that was very interesting from modern point of view but not recognized as such at that the time was the paper43 by Starobinsky in 1978 on a nonsingular isotropic cosmological model whose material content is a massive scalar field. Although his objective was to find a viable bouncing solution to avoid the initial singularity, all the currently well-known field equations and their slow-roll solutions that are used in chaotic inflation44 had already been given there. As noted by Starobinsky himself in his talk at the Tomalla Prize ceremony in 2009, the decisive step for Linde to propose chaotic inflation was to throw away the contraction phase (and replace it with large quantum fluctuations in the Planckian regime to establish natural initial conditions). Thus in 1970s much work was done in Russia about the birth and the very early stage of the universe, but many of the papers were speculative and not based on concrete models, and moreover did not make contact with observation. One should note that gems of truly novel ideas often emerge from such speculation. One such novel idea is the calculation of the primordial tensor perturbations in the early de Sitter stage which was first elaborated by Starobinsky45 in 1979, even before the concept of inflationary cosmology was introduced. His original motivation was physically sound with the desire to probe the initial state of the universe. He had quantum creation of the universe in mind and postulated that the initial state of the universe should be highly symmetric. Therefore, he considered de Sitter spacetime at the outset. He then considered the quantum mechanical behavior of gravitons to probe the initial state of the universe before the big bang, when other particles with stronger interactions were likely thermalized thus losing the memory of the initial state. In this paper again, however, the physical discussion on the transition to the Friedmann regime was absent. The sound physical mechanism to realize de Sitter expansion as well as subsequent transition to the Friedmann regime was finally given by Starobinsky by incorporating higher-order curvature terms in the action

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based on quantum corrections.17 Mukhanov and Chibisov46 calculated quantum fluctuations generated in this model and pointed out the possibility that they may account for the origin of seed fluctuations for galaxy formation. Meanwhile, apart from these mostly geometrical considerations, Kirzhnits and Linde33,34 investigated the high temperature behavior of unified gauge theories and phase transitions which may have occurred in the hot or dense early universe. Their study formed a basis of the inflationary cosmology. Indeed soon after the old inflation model was proposed and its problems were elucidated, Linde19 proposed the new inflation model based on a Coleman–Weinberg type potential for the GUT Higgs field, in which inflation occurs as the scalar field induces a slow-roll over phase transition toward its zero-temperature minimum after thermal correction of the potential has disappeared due to cosmic expansion.19,20 This was the first slow-roll inflation model and the first small-field inflation model using a concave potential which is sometimes called hill-top inflation.47 Unfortunately, however, the original new inflation predicted a too short epoch of inflation with too large density fluctuations,48 and thus could not reproduce our universe. Furthermore, as realized by Linde himself, the universe would have been far from a thermal state at the GUT era, hence one cannot expect thermal symmetry restoration to set the appropriate initial condition for new inflation. These consideration led to chaotic inflation.44 1.3. Inflation paradigm Now the standard paradigm of inflation is that the accelerated expansion is realized when a scalar field, dubbed as the inflaton, slowly rolls down its potential toward a global minimum in a time scale longer than the Hubble time.19,20,44 The remarkable feature of the slow-roll inflation is that it cannot only explain the global properties of the observed universe, but also provide seeds of density and curvature fluctuations that evolve into large-scale structures.46,48–50 The temperature (and polarization) anisotropies generated on CMB at the same time provide indirect cosmological tests of the inflation paradigm (see e.g. Refs. 51–53 among others). In fact, the slow-roll inflation driven by a potential is not the only possibility. There are presently two alternatives. One class of models realizes inflation without introducing any inflaton field but modifies gravity from the Einstein’s general relativity. As mentioned in Sec. 1.2 Starobinsky was the first to show that quasiexponential expansion and subsequent transition to power-law expansion could be realized by incorporating higher-order curvature terms in the action based on quantum corrections.17 Nowadays its simpler version including square scalar curvature term besides the Einstein action is referred to as the Starobinsky model. The other alternative is comprised of models which make use of a scalar field with a higherorder kinetic function. If we can realize a state with a constant canonical kinetic function its energy density can have the same equation of state as the cosmological constant to drive exponential inflation. Such models are referred to as k-inflation.54

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All these models can be regarded as a subclass of the generalized G-inflation model,55,56 which is the most general theory of single-field inflation with its field equations given by second-order differential equations as with most of the other theories of fundamental physics. 2. Resolution of Fundamental Problems Although the original inflation models of Sato and Guth realized exactly exponential cosmic expansion, it is not a necessary condition to resolve horizon and flatness problems. All we need is to realize a sufficiently long period of accelerated expansion, when the particle horizon increases more rapidly than t and the Hubble horizon, so the horizon problem can be solved. From the Einstein equations for the cosmic scale factor a(t)
2 K 8πG a˙ + 2 = ρ, (1) a a 3 a ¨ 4πG =− (ρ + 3P ), (2) a 3 we see that such an expansion law is possible if cosmic energy density ρ and pressure P satisfy ρ + 3P < 0. Then the energy density decreases less rapidly than a−2 (t) so that the curvature term in the Friedmann equation (1) K/a2 (t) becomes unimportant. As a result, the flatness problem is also solved simultaneously. Of course, such a stage dominated by a new energy component should not last forever but the universe must be converted to a state dominated by radiation to realize the initial state of the hot Big Bang cosmology, since we do not wish to demolish its success. The accelerated or inflationary expansion must be followed by creation of radiation and entropy. In fact it is this epoch rather than the inflationary expansion itself that the monopole and other unwanted relic problems such as domain walls, gravitinos, etc. are solved. Since inflation exponentially dilutes both monopole density and entropy density in the same way, the current universe with a huge entropy with a negligible amount of monopoles would never be realized without entropy production at the reheating. Let us consider quantitatively how much inflationary expansion is required to solve the horizon and flatness problems in terms of entropy consideration. First the entropy contained in the current Hubble radius H0−1 = 4.2 × 103 Mpc is given by that of the CMB photon and neutrino background with the effective temperature Tν = 1.95K as S0 = 2.6×1088. We consider the condition that the comoving Hubble −3 at t = ti , contains more entropy than volume at the beginning of inflation, 4π 3 H S0 after the reheating, so that currently observable region is well inside the initial Hubble radius at the onset of inflation. For simplicity, suppose that the energy density takes a constant value ρinf during inflation and it continues from t = ti to tf to stretch the scale factor by af /ai ≡ eN , and that after inflation the universe is dominated by a component with equationof-state P = wρ until the universe becomes radiation dominant with the reheating

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temperature TR . At this time the initial Hubble radius has been stretched to 1
2 − 3(1+w) π g∗ TR4 −1 N H e ≡ rH . 30ρinf

(3)

The corresponding volume contains the entropy
 1
− 1+3w
 −1+3w 1+w 1+w 4π 2 g∗ 3 4π 3 16π 3 e3N 45g∗w 1+w H TR S= TR × rH = , 90 3 270 4π 3 MPl MPl (4) where g∗ is number of relativistic degrees of freedom at reheating. Requiring this to be larger than S0 , we find  g   r  12.5 − 10.8w w 1 + 3w ∗ N > 67.7 − − ln + ln 1+w 3 + 3w 106.75 6(1 + w) 0.01

 2 TR 1 − 3w H + ln ≡ Nmin , r ≡ 0.01 , (5) 3 + 3w 108 GeV 2.4 × 1013 GeV where r is the tensor-to-scalar ratio to be defined later. As mentioned above, this is the condition that the initial Hubble radius at the onset of inflation is stretched to be larger than the current Hubble radius. However, we must note that the observed density perturbation on this scale is observed to be 10−5 . We expect that the initial amplitude of fluctuation on this scale can be close to unity just to initiate inflation. Thus it is not sufficient to have N = Nmin but 1 we need more inflation by a factor of (10−5 )− 2 ∼ 500 to suppress the amplitude of fluctuations to an acceptable level.57 Thus the minimal condition to solve the horizon problem is actually N > Nmin + ln 500 = Nmin + 6.2. Taking this extra number of e-folds into account we find
 1  r  1 TR N > 55 + ln + ln (6) 6 0.01 3 108 GeV for w = 0, and


 1  g∗  1  r  1 TR N > 67 − ln + ln − ln 6 106.75 3 0.01 3 108 GeV

(7)

for w = 1. Next let us consider the evolution of the total density parameter Ωtot using K = H 2 (Ωtot − 1). a2 We find Ωtot (t0 ) − 1 = Ωtot (ti ) − 1




a(ti )Hinf a 0 H0

2

= e−2(N −Nmin ) < 500−2 = 4 × 10−6 .

(8)

(9)

Thus, if the horizon problem is solved by inflation, the flatness problem is automatically solved and such inflation models predict the total density parameter today is equal to unity with four to five digit accuracy.

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3. Realization of Inflation 3.1. Three mechanisms As mentioned in the introduction one can classify realization mechanisms of inflation into three broad categories. The standard one is the slow-roll inflation models. Consider a scalar field φ, the inflaton, with a canonical kinetic term X ≡ − 21 g µν ∂µ φ∂ν φ and a potential V [φ]. Taking the variation of the action   √ √ S = L −gd4 x = (X − V [φ]) −gd4 x (10) with respect to the metric tensor, one finds the energy–momentum tensor 2 δS Tµν = − √ = ∂µ φ∂ν φ + Lgµν . −g δg µν

(11)

For homogeneous field configuration, the energy density, ρ, and the pressure, P , are given by ρ=

1 ˙2 φ + V [φ], 2

P =

1 ˙2 φ − V [φ], 2

(12)

respectively. Hence if the potential is so flat that V [φ] > φ˙ 2 is satisfied for a long enough period, the equation-of-state parameter w ≡ P/ρ is smaller than −1/3 and an accelerated expansion can be realized. Next, in order to see how a scalar field with a noncanonical kinetic term may affect the dynamics of the universe, let us rewrite the Lagrangian in more general form as L = K(X, φ). Then calculating the energy–momentum tensor in the same way as (11), we find ρ = 2XK X − K,

P =K

(13)

in this case. Thus an accelerated expansion is possible even without any potential, if XK X < −K is satisfied. In particular, in the case KX = ∂K/∂X = 0 holds we find de Sitter expansion. This is what is called k-inflation. If we expand the Lagrangian as a power series of X like K(X, φ) = K1 (φ)X + K2 (φ)X 2 . . .

(14)

we immediately find that K1 (φ) and K2 (φ) should have an opposite sign in order to realize inflation. So far we have implicitly assumed the gravity sector is expressed by the Einstein– Hilbert action  √ 1 (15) Sg = 2 R −gd4 x 2κ

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and considered a single scalar field as a matter ingredient. We may realize inflation even without any scalar field matter by modifying gravity as  √ 1 Sg = 2 f (R) −gd4 x, (16) 2κ where f (R) is a function of scalar curvature R. The trace of field equation is given by 3f (R) + f  (R)R − 2f (R) = 0.

(17)

One can therefore find a de Sitter solution, R = 12H =const., if f (R) satisfies f  (R) = 2f (R), namely, f (R) ∝ R2 . A pure R2 model cannot terminate inflation nor has the proper Einstein limit, so we adopt 2

R2 . (18) 6M 2 This theory is a nonperturbative extension of the Einstein gravity, which contains additional scalar degree of freedom called scalaron. These mechanisms may not work in a mutually isolated manner but the actual inflation may have been induced by their combinations58 with possible higher-order derivative terms.56 f (R) = R +

3.2. Inflation scenario Let us describe cosmic evolution using a slow-roll inflation model as a prototype. Once inflationary expansion sets in, the universe rapidly becomes homogeneous, isotropic, and spatially flat, and the energy density other than the inflaton is soon diluted away. Hence we may consider a spatially flat FLRW universe from the beginning except when we discuss the feasibility of inflation in a generic inhomogeneous and anisotropic spacetime. Thus the scalar field equation and the Einstein equation read (19) φ¨ + 3H φ˙ + V  [φ] = 0,
2 8πρφ a˙ ρφ 1 = H2 = = , ρφ = φ˙ 2 + V [φ], (20) 2 2 a 3MPl 3MG 2 √ √ 1 respectively, where MG = MPl / 8π = 1/ 8πG = (
c5 /8πG) 2 = 2.4 × 1018 GeV is the reduced Planck mass. To realize successful inflation the potential V [φ] must remain the dominant contribution to ρφ for a sufficiently long time, that is, the scalar field must remain practically constant over the cosmic expansion time scale. For this purpose φ¨ must contribute negligibly in the equation of motion. Then field equations read (21) 3H φ˙ + V  [φ] = 0,
2 a˙ 8πV [φ] V [φ] = H2 = = 2 2, a 3MPl 3MG which are called the slow-roll equations of motion.

(22)

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For these approximate equations to hold, the potential must satisfy
 2 2  MG V [φ] 2 V [φ] V ≡  1, ηV ≡ MG , |ηV |  1. 2 V [φ] V [φ]

(23)

Here V and ηV are called the (potential) slow-roll parameters. Conversely, if these slow-roll conditions (23) are satisfied, inflationary expansion is likely to start as soon as the potential energy dominates the cosmic energy density. Inflation or accelerated ˙ becomes expansion is terminated when φ˙ 2 becomes larger than V [φ] or when |H| 2 larger than H . Under the slow-roll approximation, (21) and (22), the former occurs at V = 3/2 and the latter at V = 1. The discrepancy is due to the invalidity of the slow-roll approximation in such a regime. Under the slow-roll approximation, the number of e-folds of inflation after φ has crossed a value φN is given by     φf dφ 1  φf dφ  √ H N = Hdt = = (24)  , M G  φN 2V  φ˙ φN with φf being the value at the end of inflation. Figure 2 shows typical shape of the inflation potential, which has two possible field ranges to realize inflation. Near the local maximum at φ = 0 the potential is so flat that inflation may be possible. Models of such class are called small-field models. On the other hand, if the potential increases at most with a power-law, one can see that the slow-roll conditions (23) are satisfied for super-Planckian field values, so that inflation is also realized there. Such a model is called a large-field model. In both cases the slow-roll conditions no longer holds once the field approaches the global minimum at φ = v, and it starts to oscillate around it. Thus the potential energy density is transferred to field oscillation energy, which will eventually decay V[φ] Large field model Small field model

v Fig. 2.

φ

Scalar field potential for single-field slow-roll inflation.

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into particles interacting with φ. The universe will then be heated up to realize the initial condition of the big bang universe physically in the modern context. In fact, Starobinsky’s R2 inflation can be regarded as a variant of a largefield model because the modified gravity action can be converted to the Einstein gravity plus a scalar field with a potential. Specifically, in terms of conformal transformation,
 R (25) g¯µν ≡ 1 + gµν , 3M 2 we find the action is equivalent with
   √ √ 1 1 µν 4 ¯ 4 Sg = −¯ gd xR + −¯ gd x g¯ ∂µ φ∂ν φ − V [φ] , 2κ 2 
 √ 3 R 3 2 2 − 23 κφ 2 κφ ≡ ln 1 + ) . , V [φ] ≡ M M (1 − e G 2 3M 2 4

(26)

(27)

Thus the system is equivalent with a scalar field matter in the Einstein gravity. 2 For φ  MG = κ−1 , the potential has a plateau with a height 3MG M 2 /4 where inflation can occur. Inflation is followed by scalar field oscillation around the origin and reheating proceeds through gravitational decay of the inflaton φ. Finally, k-inflation of the type (14) is terminated when both K1 (φ) and K2 (φ) become positive. Then X starts to decrease rapidly and only the first term becomes relevant. Now the universe is dominated by the kinetic energy of a free scalar field whose energy density dissipates in proportion to a−6 (t) with the equationof-state parameter w = 1. This abrupt change from de Sitter to a power-law a(t) ∝ t1/3 induces gravitational particle production to reheat the universe in this model. 4. Slow-Roll Inflation Models Let us focus on specific models of slow-roll inflation with a canonical kinetic term. 4.1. Large-field models This model was originally proposed by Linde under the name “chaotic inflation”,44 since it makes use of the chaotic initial condition of the universe at the Planck time tPl = (
G/c5 )1/2 = 5.4 × 10−44 sec, when we expect large quantum fluctuations to dominate. Let us consider the following simple Lagrangian as an example assuming that Einstein gravity starts to hold at that epoch. 1 Lφ = − (∂φ)2 − V [φ], 2

V [φ] =

1 2 2 m φ . 2

(28)

By virtue of the uncertainty principle, at the Planck time we expect that both gradient energy and potential energy densities were fluctuating with the Planckian

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magnitude, so that 1 1 2 2 4 4 − (∂φ)2  MPl , m φ  MPl , (29) 2 2 in each coherent domain. Taking m  MPl we find φ can take a large value up 2 /m  MPl . If the magnitude of φ is saturated, from the gradient to φ ∼ MPl energy constraint we find the coherent length of the field, L, satisfies 12 (φ/L)2 ∼ −1 4 4 /(Lm)2  MPl , or L
m−1  MPl . That is, φ is homogeneous over the MPl Compton wavelength well above the horizon scale at the Planckian regime. Then over the horizon scale one may regard the field as homogeneous and apply the homogeneous field equations derived in the previous section. Solving the field equation (19) and the Einstein equation (20), one finds mMPl (30) φ(t) = φi − √ (t − ti ), 2 3π  4π m m2 a(t) = ai exp φi (t − ti ) − (t − ti )2 3 MPl 6

 1 φ2 (t) = af exp − 2π 2 . (31) 2 MPl √ Inflationary expansion is terminated around φ  MPl / 4π when V becomes ˙ unity and |φ/φ| becomes as large as H. Therefore, in order to solve the horizon problem one needs an initial amplitude φi
3MPl . The most stringent constraint on this model comes from the amplitude of density and curvature perturbation generated during inflation, which sets the mass m ∼ = 1013 GeV and constrains the self-coupling of the form λφ4 /4 to λ  10−12 (see Sec. 13.1).59 4.2. Small-field model As mentioned in Sec. 1.2, the first small-field model is Linde’s new inflation model19 which actually has a problem in realizing the required initial condition. There is another class of small-field inflation models called topological inflation60,61 in which this problem is solved for topological reason without resorting to thermal symmetry restoration. Consider a simple Lagrangian 1 λ L = − (∂φ)2 − V [φ], where V [φ] = (φ2 − v 2 )2 , (32) 2 4 which has a local maximum at φ = 0. Since φ is a real scalar field, this model admits a domain wall solution connecting two vacuum states φ = ±v. Neglecting gravitational effect for the moment, we find a solution 

λ vz (33) φ(x) = v tanh 2

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that describes a domain wall on the xy-plane. Its thickness d0 can be estimated by −1 2 equating (∇φ)2 ∼ (v/d0 ) and V [0] ≡ Vc as d0 ≈ vVc 2 . On the other hand, the Hubble horizon scale corresponding to the vacuum energy density Vc is given by

− 12  12 8πG 3 −1 Hc = = MPl Vc , (34) 3 8πVc which is smaller than the thickness of the wall if v
MPl . In this case one can find a region with a large potential energy density V ∼ Vc whose dimension is larger than the Hubble scale near the center of the domain wall. Such a field configuration provides a sufficient condition to initiate inflation there without being affected by the configuration outside the Hubble horizon. In this model, an initially random field configuration in the global spacetime naturally creates domain walls inside of which inflation sets in near the central core. In this sense this model provides a natural mechanism of small-field inflation provided that the universe continues to expand until the energy density of the domain wall is locally dominant. We can also show that inflation ends after a finite time except for the locus with φ = 0, and the reheating process proceeds just as in the chaotic inflation model. 4.3. Hybrid inflation This is a model to induce inflation by a false vacuum energy density through a nonthermal symmetry restoration by virtue of an extra scalar field.62 The simplest model of hybrid inflation consists of a real scalar field (φ) and a complex scalar field (χ) with a Lagrangian 1 L = −(∂χ)† (∂χ) − (∂φ)2 − V [χ, φ], 2 where V [χ, φ] =

λ 1 (|χ|2 − v 2 )2 + g 2 φ2 |χ|2 + m2 φ2 . 2 2

(35)

From ∂V = λ(|χ|2 − v 2 )χ + g 2 φ2 χ, ∂χ† ∂2V ≡ Mχ2 = λ(2|χ|2 − v 2 ) + g 2 φ2 , ∂χ∂χ†

(36)

we find that χ = 0 is a potential minimum if the inequality g 2 φ2 > λv 2 is satisfied initially. Then the potential reads λ 4 1 2 2 v + m φ . (37) 2 2 If the first term dominates √ the energy density of the universe, inflation takes place. It is terminated at φ < gλv when χ induces a phase transition, which occurs within V [χ = 0, φ] =

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one Hubble time typically unless fine-tuning of parameters is applied. This model is attractive from particle physics viewpoint because enough inflation is possible even if φ is much smaller than the Planck scale63 and neither λ nor g requires fine-tuning. Instead the initial fields configuration of the two scalar fields do need tuning to realize inflation.64,65 In this model, since χ is a complex scalar field, a network of strings will be formed after the phase transition. Where χ a real scalar field, a network of domain walls would be formed and dominate the energy density. Other mechanisms of formation for the topological defects have been proposed in Refs. 66–70 near or at the end of inflation. Hence there may well remain some relic defects even if their energy scale is higher than the maximum temperature after inflation so that symmetry restoration by thermal effects is impossible. 5. Reheating The entropy production process required after inflation to realize the initial condition of hot Big Bang cosmology is called reheating, although in modern inflationary cosmology the universe had never previous been dominated by radiation. As is clear from the discussion in the previous section, the universe is dominated by coherent scalar field oscillation after potential-driven slow-roll inflation. Let us take the large-field√massive scalar model as an example. As the inflaton starts to satisfy |φ|  MPl / 6π the scalar field starts to oscillate around the origin with a period 2π/m. Such field oscillations are equivalent to the condensation of the homogeneous zero-mode of the scalar field which decreases its amplitude through cosmic expansion, to decay into radiation finally. In the initial oscillatory regime when its amplitude is large, some nonperturbative particle production also takes place efficiently to some extent. This is called preheating.71–73 But the final reheating stage is always dominated by the perturbative decay of the scalar field, which is accounted for by incorporating a decay term Γφ φ˙ in its equation of motion.74–76 Here Γφ is the decay rate of a φ particle which is much smaller than m since the inflation must be weakly coupled to other fields to suppress quantum fluctuations to an acceptable level as discussed in the following sections. Multiplying φ˙ to (19) after introducing the above mentioned dissipation term, we find
 d 1 ˙2 1 2 2 φ + m φ = −(3H + Γφ )φ˙ 2 . (38) dt 2 2 Since φ is rapidly oscillating over the cosmic expansion time scale, we can replace 2 φ˙ 2 in the right-hand side by an average over the oscillation period, φ˙ , which is identical to the total energy density of φ, ρφ , thanks to the virial theorem.77 Thus, we find a Boltzmann equation dρφ = −(3H + Γφ )ρφ , dt

(39)

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which is associated with that for radiation energy density ρr dρr = −4Hρr + Γφ ρφ . dt These two equations can be solved as

−3 a(t) ρφ (t) = ρφ (tf ) exp[−Γφ (t − tf )], a(tf ) −4  t

a(t) ρr (t) = Γφ ρφ (τ )dτ. tf a(τ )

(40)

(41)

(42)

Thus the universe starts to be dominated by radiation at t Γ−1 φ , when the radiation temperature TR is given by



 12 1 8π 8π π 2 g∗ 4 2 ∼ 1 ∼ 1 = ρ T = = Γφ , R 2 r 2 30 3MPl 2t 3MPl 2
1
1
 12 200 4  200 4 Γφ 11 ∼ 0.1 TR ∼ M Γ 10 GeV. = Pl φ = g∗ g∗ 105 GeV

H=

(43)

(44)

Such is the case inflation is followed by scalar field oscillation. If inflation is induced purely by kinetic terms as in the case of k-inflation54 and G-inflation,55 the entropy after inflation is attributed to gravitational particle production78 associated with the change of the cosmic expansion law. Here we describe this process following Ref. 79. Let us consider the case where the cosmic expansion law changes from de Sitter with the Hubble parameter Hinf to a power law with the index t1/3 corresponding to a universe dominated by the kinetic energy of a free massless homogeneous scalar field which is called the kination regime. From the standard calculation of particle creation,78,80,81 we consider the creation of massless minimally coupled scalar particles. The relevant Bogolubov coefficient βω is given by  ∞ i βω = e−2iωη V (η)dη, (45) 2ω −∞ with V (η) = 16 a2 (η)R(η), which yields expressions for the number density and energy density  ∞ 1 nr = |βω |2 ω 2 dω, (46) 2π 2 a3 0  ∞ 1 ρr = |βω |2 ω 3 dω. (47) 2π 2 a4 0 If we approximate the transition from de Sitter to the power law expansion occurred abruptly we find the resultant radiation energy density diverges. We must adopt

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a model where the scale factor evolves smoothly up to the second time derivative. We consider the following model where the scale factor a2 (η) ≡ f (Hinf η) evolves as   1   (x < −1),   x2 f (x) = (48)  a0 + a1 x + a2 x2 + a3 x3 (−1 < x < x0 − 1),     b0 (x + b1 ) (x0 − 1 < x) requiring f (x), f  (x), f  (x) to be continuous at x = −1 and x = x0 − 1, so that the scalar curvature is continuous. Since the transition occurs within the Hubble time x0 is smaller than unity. The coefficients are determined as (up to O((x0 )0 )), a0 = 6 −

1 , x0

a1 = 8 −

a3 = −

1 , x0

3 , x0

b0 = 2,

a2 = 3 − b1 =

3 , x0

3 . 2

(49) (50)

Then the energy density of created particles is given by ρr = where

 I =−



x

−∞

dx1

x

−∞

4 Hinf I, 128π 2 a4

dx2 ln(|x1 − x2 |)V˜  (x1 )V˜  (x2 ),

1 f  f − (f  )2 2 V˜ (x) = . f2

(51)

(52)

(53)

The upper limit of integration x = Hη is the time when V˜ falls well below unity so that the notion of the particle is well-defined. We find I −36 ln x0 , so that the energy density in the beginning of the kination regime is given by
 4 1 9Hinf ln ρr = , (54) 32π 2 a4 Hinf ∆t where ∆t is the time elapsed during the transition from de Sitter to kination. We will ignore the logarithmic dependence on ∆t and take ln(1/(H∆t)) ∼ 1. If there are N modes of species created this way, (54) should be multiplied by N . Since the energy density of the scalar field in the kination regime behaves as 2 2 Hinf a−6 , ρφ = 3MG

(55)

the scale factor when the universe becomes radiation dominant is given by a2R =

2 32π 2 MG , 3NH 2inf

(56)

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so the energy density of the relativistic particles is ρr |R =

8 81N 3 Hinf 4. (32π 2 )3 MG

(57)

Thus the reheating temperature is given in terms of the tensor-to-scalar ratio r by
1 2 3  g − 14  r  3N 4 30 4 Hinf 3 ∗ 6 4 TR = GeV. (58) 3.9 × 10 N 3 2g 2 106.75 0.01 M π 4 (32π ) G ∗ 6. Generation of Quantum Fluctuations that Eventually Behave Classically Thanks to the smallness of the slow-roll parameters, quantum field theoretic properties of the inflaton during inflation is quite similar to that of a massless minimally coupled scalar field ϕ(x, t) in de Sitter space ds2 = −dt2 + e2Ht dx2 .

(59)

Such a scalar field is known to exhibit anomalous growth of the square vacuum expectation value according to82–84
2 H 2

ϕ(x, t)  = Ht. (60) 2π Decomposing the scalar field as  d3 k ϕ(x, t) = ˆ†k ϕ∗k (t)e−ik ·x ) (ˆ ak ϕk (t)eik ·x + a (2π)3/2  d3 k ≡ ϕˆk (t)eik ·x , (2π)3/2

(61)

˙ t), hence we find the momentum conjugate to ϕ(x, t) reads π(x, t) = a3 (t)ϕ(x, the canonical commutation relation [ϕ(x, t), π(x
, t)] = iδ(x − x
) is equivalent to ˆ†k
] = δ (3) (k − k ) if we impose the normalization condition [ˆ ak , a ϕk (t)ϕ˙ ∗k (t) − ϕ˙ k (t)ϕ∗k (t) =

i a3 (t)

.

(62)

Thus a ˆ†k and a ˆk satisfy the usual commutation relations and act as creation and annihilation operators of a k-mode, respectively. The mode function ϕk (t) satisfies

2  d k2 d + 2Ht ϕk (t) = 0, + 3H (63) dt2 dt e as derived from the Klein–Gordon equation in de Sitter space, which is solved as  3 iH π (1) ϕk (t) = H(−η) 2 H 3 (−kη) = √ (1 + ikη)e−ikη , (64) 2 4 2k 3

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under the condition (62). Here η is the conformal time defined by 

t

η≡

dt = a(t)



t

dt 1 =− . eHt HeHt

(65)

Note that since (63) is a second-order differential equation there are two inde(2) (1) pendent solutions proportional to H 3 (−kη) and H 3 (−kη). We have adopted only 2 2 (64) which coincides with the positive-frequency mode in the Minkowski space at large wavenumber limit where the effect of cosmic expansion is negligible. k Note also that −kη = Ha(t) is nothing but the ratio of physical wavenumber to the Hubble parameter. In the superhorizon limit k  a(t)H, we find iH iH ϕk = √ (1 + ikη)e−ikη = √ 3 2k 2k 3 
2 iH k → √ 1+O , Ha 2k 3


1−

ik Ha



k

ei Ha (66)

so that we can put ϕ∗k (t) = −ϕk (t) and regard ϕˆk (t) = ϕk (t)(ˆ ak − a ˆ†−k ) holds when the wavelength corresponding to k is much larger than the Hubble radius. Then the ˆk (t) = a(t)3 ϕ˙ k (t)(ˆ ak − a ˆ†−k ), which means momentum conjugate to ϕˆk is given by π ˆk have the same operator dependence in the superhorizon regime. that both ϕˆk and π Thus in this regime, where the decaying mode is negligible, the quantum operˆk commute with each other, so that quantum fluctuations behave as ators ϕˆk and π classical statistical fluctuations.85 Furthermore, the absolute square of mode function behaves as |ϕk (t)|2 =

 H2  H2 1 + (kη)2 → 3 , 3 2k 2k

when

k → 0, Ha(t)

(67)

that is, it takes a constant value proportional to k −3 . Multiplying the phase space 4πk2 density over a unit logarithmic frequency interval, (2π) 3 dk with dk = kd ln k = k, we find the dispersion takes a constant over each logarithmic frequency interval. One can immediately see that (60) can be reproduced from this mode function by summing up only superhorizon modes only as 

HeHt

ϕ (x, t) 2

d3 k |ϕk (t)| = (2π)3 2

H




H 2π

2

Ht,

(68)

where the infrared cutoff is identified with the mode that left the Hubble radius at the beginning of inflation set to t = 0 here.86 The behavior that the square expectation value increases in proportion to time is the same as that of Brownian motion with a step ±H/(2π) at each time interval

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H −1 . Thus one can describe the quantum fluctuations generated during inflation as follows. H For each Hubble time H −1 , a quantum fluctuation with amplitude 2π , which is the Hawking temperature of de Sitter space, is continuously generated on the scale of Hubble radius, and it is stretched by subsequent cosmic expansion to form the long wavelength fluctuations.

Finally as a preparation for the calculations of curvature and tensor perturbations, let us write down the action of a massive scalar field ϕ using conformal time with the line element ds2 = a2 (η)(−dη 2 + dx2 ).

  √ 1 µν 1 2 2 4 S= −gd x − g ∂µ φ∂ν φ − m φ 2 2      1 = dηd3 x a2 φ2 − (∇φ)2 − a4 m2 φ2 , (69) 2 where a prime denotes differentiation with respect to conformal time. Introducing a rescaled variable χ ≡ aφ, we find


   a 1 3 2 2 2 2 S= dηd x χ − (∇χ) − a m − (70) χ2 2 a which is equivalent to the action of a scalar field with a time-dependent mass term in Minkowski space. In de Sitter space with a(η) = −1/(Hη), the mode function satisfies χk + k 2 χk −

2 χk = 0, (−η)2

(71)

where we have taken m = 0. Its solution may be expressed as  πη  12 ϕk (t) (1) χk (η) = − H 3 (−kη) = 2 4 a(t)

(72)


in agreement with (64) with the normalization condition χ χ∗ − χχ∗ = i equivalent to (62). 7. Cosmological Perturbation We now superimpose linearized metric perturbations onto the spatially flat homogeneous and isotropic background spacetime ds2 = −dt2 + a(t)2 dx2 ,

(73)

obtaining the perturbed spacetime ds2 = −(1 + 2A)dt2 − 2aBj dtdxj + a2 (δij + 2HL δij + 2HT ij )dxi dxj ,

(74)

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where all new variables are functions of time and position87–89 with Tr HT ij = 0. As is well known, a spatial vector Bj can be decomposed to rotation-free component and divergence-free component as j , Bj = ∂j B + B

j = 0. where ∂j B

(75)

Similarly, the spatial tensor HT ij can be decomposed as
 δij 2  T j + ∂j H  T i + HT T ij , HT ij = ∂i ∂j − ∇ HT + ∂i H 3  T j = 0, where ∂j H

∂j HT T k j = 0,

HT T j j = 0.

(76)

Here HT T ij is the transverse-traceless component corresponding to gravitational waves. Thus perturbation variables are classified into scalar, vector, and tensor variables according to the spatial transformation law. Variables of different types are not mixed in linear perturbation theory and we can treat them independently. Since the vector perturbation has only a decaying mode under normal circumstances in the linear perturbation theory,87,88 we do not consider it hereafter and focus on the scalar and tensor perturbations. First let us consider scalar perturbations which are related with the density and curvature fluctuations, keeping only A, B, HL and HT . In fact, not all of these variables are geometrical quantities but gauge modes are included as well. They arise from the arbitrariness of the choice of the background spacetime with respect to which we define the perturbation variables. One should remember that the real physical entity is an inhomogeneous and anisotropic spacetime, and that it may be regarded as homogeneous and isotropic only after taking some average whose definition is not unique in general. To characterize the gauge dependence, let us consider two background spacetimes with coordinates xµ and xµ , and assume that these two coordinates are mutually related by the coordinate transformation x0 = x0 + δx0 ≡ x0 + T,

xi = xi + δxi ≡ xi + ∂ iL,

(77)

where δxµ are small quantities whose relative magnitudes are of the same order as the perturbation variables. Here T and L are functions of the space and time coordinates. Under this coordinate transformation, we find that the scalar perturbation variables at the same coordinate values in two different backgrounds are related according to A = A − T˙ , 1 B = B + aL˙ + T, a H L = HL −

∇2 L − HT, 3

H T = HT + L,

(78)

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respectively. Here T and L represent gauge degrees of freedom on scalar perturbation. In order to avoid their appearance, one may either find combinations of perturbation variables that remain unchanged after gauge transformation, or fix the gauge completely. For example, if we set H T = 0, L is fixed. If we further set B = 0, T is also fixed and there remains no gauge mode. In this case the perturbed metric is given by ds2 = −(1 + 2Ψ)dt2 + a2 (1 + 2Φ)dx2 ,

(79)

where we have rewritten A = Ψ and HL = Φ to stress their gauge invariance. This is called the longitudinal gauge. In order to fix the gauge, one may combine transformation properties of matter variables such as a scalar field, which transforms as ˙ φ(t, x) = φ(t − T, xj − ∂ j L) = φ(t − T ) + δφ = φ(t) − φ(t)T + δφ,

(80)

˙ δφ = δφ − φT.

(81)

that is,

8. Generation of Curvature Fluctuations in Inflationary Cosmology Here, we consider the generation of curvature perturbations in single-field inflation models that include not only potential-driven slow-roll models but also k-inflation. Further generalization to the case of the most general single-field inflation model with second-order field equations will be carried out in Sec. 10 based on Ref. 56. We start with the action

2   MG 4 √ R + K(X, φ) . (82) S = d x −g 2 In the spatially flat homogeneous and isotropic spacetime (73) the background equations read 2 3MG H 2 = ρ = 2XKX − K,

2 ˙ 2 H + 3MG 2MG H 2 = −P = −K,

(83)

together with the scalar field equation of motion KXφ 2 ˙ 2 Kφ 2 φ¨ + 3Hc2s φ˙ + c φ − c = 0, KX s KX s

c2s ≡

PX KX = . ρX KX + 2XKXX

(84)

In order to derive the second-order action with respect to the curvature perturbation in the comoving gauge, R, it is convenient to use (3 + 1) formalism ds2 = −N 2 dt2 + hij (dxi + N i dt)(dxj + N j dt),

hij ≡ a2 (t)e2R δij ,

(85)

in which the constraint equations for the perturbation variables are easily obtained.90 This coordinate choice still has a gauge degree of freedom with respect to time translation by which the scalar field fluctuation is transformed as (81). Setting δφ = 0 one can fix this gauge freedom.

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Then writing the action (82) in this gauge, we find   √ √ 1 M2 2 (3) S= d4 x hN (MG R + 2K) + G d4 x hN −1 (Eij E ij − E 2 ) + · · · , 2 2 (86) where 1 ˙ (hij − Ni|j − Nj|i ), E ≡ Tr E, (87) 2 where terms replaced by · · · are all total derivative terms, so they do not affect the dynamics of the system. Here R(3) is scalar curvature of 3-space and | denotes covariant derivative with respect to hij . Since we are concerned with scalar-type fluctuations, we set N = 1 + α and Ni = ∂i ψ, where α and ψ are perturbation variables of the same order as R. Differentiating the action (86) with respect to N , we find the Hamiltonian constraint Eij ≡

R(3) + 2

K KX 1 ij 2 2 − 4X M 2 − N 2 (Eij E − E ) = 0, MG G

(88)

which yields H 2 1 2 ∇ ψ = − 2 ∇2 R + Σα, MG Σ ≡ XKX + 2X 2 KXX . (89) 2 a a The momentum constraint obtained by differentiation with respect to Ni reads

 1 j j (E − Eδi ) = 2Hα,i − 2R˙ ,i = 0, (90) N i |j ˙ which gives α = R/H. Inserting these relations into the action, we obtain an action for R only as

  Σ ˙2 H˙ (∇R)2 2 3 3 R S2 = MG dtd xa −  ≡ − ,  . (91) H H H2 a2 H2 1

1

Introducing new variables z ≡ a(2Σ) 2 /H = a(2H ) 2 /cs , v ≡ MG zR and using conformal time, we find

  1 z  2 3 2 2 2 dηd x v − cs (∇v) + v , S2 = (92) 2 z which like (70) is an action of a free scalar field with a time-dependent mass term

  z  ηH   ηH  s˙ η˙ H 2 2 − H − s + = (aH) 1−s+ − + z 2 2 H 2H (93) ≡ (aH)2 (2 + q), ˙H c˙s ηH ≡ , s≡ , HH Hcs with the important difference that the “sound velocity” cs may not be identical to unity. Still, one can quantize v in the same way as a free scalar field if the time variation of cs is negligible. q is a small number consisting of slow-roll parameters.

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Using the de Sitter scale factor a = −1/(Hη) one can obtain the mode function in the same way as in Eq. (72), so that
  πη  12 1 i (1) ∼ vk = − Hν (−kcs η) = √ 1− e−ikcs η , 4 kcs η 2kcs (94) 1
3 4 2 ∼ 3 ν= 1+ q = , 2 9 2 which approaches to a constant in the long wave limit. Thus the power spectrum of curvature perturbations multiplied by the phase space density reads  2 4πk 3 4πk 3  vk  H2 2 PR (k) ≡ |R | = = (95) k 2c  .   3 3 2 (2π) (2π) MG z 8π MG s H Since each quantity in the right-hand side has a weak time dependence, it is evaluated at the time k-mode left the sound horizon when |kcs η| = 1 was first satisfied. The validity of this approximation relies on the fact that the mode amplitude remains constant after the mode has exited the horizon. The spectral index of curvature perturbation is defined and given by d ln PR (k) (96) = −2H − ηH − s. d ln k In case the scalar field has a canonical kinetic term as in the standard theory of 2 H 2 ). Thus we find particle physics, we find cs = 1 and H = φ˙ 2 /(2MG
2 2
2 H δϕ PR (k) = = H . (97) 2π φ˙ φ˙ ns − 1 ≡

Here δϕ = H/(2π) is the amplitude of scalar field fluctuation. This equality is intuitively understandable since it represents relative fluctuation of local scale factor δa δϕ = δN = Hδt = H , a φ˙

(98)

where N ≡ ln a. Since this δN gives appropriate expression for long wave curvature fluctuations if we take an appropriate gauge, this δN method91,92 is often used in the literature. For slow-roll inflation models with a canonical kinetic term, from (21) and (22) we find H = V and ηH = −2ηV + 4V , so the spectral index is given by ns − 1 = −6V + 2ηV .

(99)

Furthermore the scale dependence of the spectral index, which is called the running spectral index, is given by dns = 16V ηV − 242V − 2ξV , d ln k

4 ξV ≡ MG

V  (φ)V  (φ) . V 2 (φ)

(100)

These quantities are observationally measured using CMB, as we shall see in Sec. 12.

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9. Tensor Perturbation Next we consider tensor perturbations generated during inflation as quantum gravitational waves.45,93,94 In order to derive field equation for the graviton, first we superimpose a small metric perturbation hµν onto Minkowski space background as gµν = ηµν + hµν . The indices of hµν is raised by the Minkowski tensor. We again need to calculate the action up to second-order in the perturbed variables, so we should use g µν = η µν − hµν + hµα hνα . Taking the longitudinal and traceless gauge, ,j i we find h00 = h0i = 0, hα α = hi = 0, hij = 0, so that the Ricci scalar is given by 3 ij,µ 1 R = hij h,µ hij,µ − hij,l hjl,i . ij,µ + h 4 2

(101)

Here the Greek indices runs from 0 to 3 while the Latin indices from 1 to 3. Since what we need is a perturbed action in cosmological background with the metric   (102) ds2 = a2 (η) −dη 2 + (δij + hij )dxi dxj ≡ g˜µν dxµ dxν . We make use of a conformal transformation g˜µν = Ω2 gµν to calculate the desired quantity. The Ricci tensor transforms as ˜ µν = Rµν − 2(ln Ω);µν − gµν g στ (ln Ω);τ σ R + 2(ln Ω);µ (ln Ω);ν − gµν g στ (ln Ω);σ (ln Ω);τ . Taking Ω = a(η), we find the Ricci scalar in the metric (102) is given by
   ˜ = a−2 R + 6 a − 3 a hij hij . R a a

(103)

(104)

Since we wish to calculate properties of quantum gravitational waves generated during inflation, let us assume inflation is driven by constant vacuum energy density, which is equivalent to a cosmological constant Λ, and calculate the second-order action for hij , which reads  2    MG ˜ − 2Λ) −˜ S2T = gd4 x (R 2 second-order 2 

M = G dηd3 xa2 (hij hji − hij,l hj,l (105) i ), 8 where we have used Λa4 = 2aa − a2 . This action has the same form as a free scalar field again, so we can quantize it in the same way as before. Indeed taking new variables as zT ≡ a/2, uij ≡ MG zT hij , we find

  1 a 2 3 2 2 u , S2T = (106) dηd x uij − (∇uij ) + 2 a ij which is indeed of the same form as (70).

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For exact de Sitter space with H = 0 we have a = −1/(Hη), whereas in case H = 0, one can express the scale factor around an arbitrary time η∗ using the Hubble parameter at that time H∗ as 
 1 −ν 1 1 31− 3 −η 2 T a=− , νT = . (107) H∗ η∗ 1 −  −η∗ 2 1− Therefore, we find the normalized mode function  πη  12 uA = − Hν(1) (−kη)eA ij ij (k), T 4

A = +, ×,

(108)

A ∗ijB where eA (k) = δ AB . ij (k) is a polarization tensor that satisfies eij (k)e Thus, the power spectrum of long-wave tensor perturbation is given by

PT (k) ≡

∗ijA 4πk 3 4πk 3 uA 2H 2 ij u ∗ij h h = = ij 2 2 2 (2π)3 (2π)3 MG zT π 2 MG

(109)

with the spectral index d ln PT (k) = −2H . d ln k Finally the ratio of this power spectrum to the curvature perturbation, nt ≡

r≡

PT (k) = 16cs H = −8cs nt PR (k)

(110)

(111)

is called the tensor-to-scalar ratio. It is one of the most important quantity subject to ongoing and future observational tests of inflation. In Sec. 11, we present the relation between r and nt in more generic theories. 10. The Most General Single-Field Inflation So far we have considered a rather limited class of inflationary universe models including only potential-driven slow-roll inflation and k-inflation. But theories consisting of tensor and a single scalar degrees of freedom have much more functional degrees of freedom under the conditions that they are ghost free and stable. In fact, the most general single-field action with gravity which yields second-order field equations was formulated by Horndeski95 back in 1974. A theory constructed with the same objective but totally different appearance was also proposed by covariantizing theories with Galilean invariance96 ∂µ φ → ∂µ φ + bµ and referred to as generalized Galileon.97 The equivalence of two theories was shown in Ref. 56. Here, making use of this theory, we provide a comprehensive and thorough study of the most general noncanonical and nonminimally coupled single-field inflation models yielding second-order field equations based on Ref. 56, which is called generalized G-inflation since it is a natural and comprehensive extension of G-inflation model,55 which was the first inflation model using Galileon-like higher derivative theory.

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We use the modern notation of Ref. 97, which contains four arbitrary functions of φ and the canonical kinetic function X ≡ −∂µ φ∂ µ φ/2 as L2 = K(φ, X),

(112)

L3 = −G3 (φ, X)φ,

(113)

L4 = G4 (φ, X)R + G4X (φ, X)[(φ)2 − (∇µ ∇ν φ)2 ],

(114)

L5 = G5 (φ, X)Gµν ∇ ∇ φ µ

ν

1 − G5X (φ, X)[(φ)3 − 3(φ)(∇µ ∇ν φ)2 + 2(∇µ ∇ν φ)3 ], 6

(115)

where Gµν is the Einstein tensor, (∇µ ∇ν φ)2 = (∇µ ∇ν φ)(∇µ ∇ν φ), (∇µ ∇ν φ)3 = (∇µ ∇ν φ)(∇ν ∇λ φ)(∇λ ∇µ φ), and GiX = ∂Gi /∂X. The Lagrangian of the Einstein gravity may be automatically included by taking 2 /2, and the extension to a nonminimally coupled scalar field is also G4 = MG immediate. Thus, the total action we consider is given by 5   √ d4 x −gLi , S= (116) i=2

which is the most general single scalar theory resulting in equations of motion containing derivatives up to second-order. Then the standard slow-roll inflation can be achieved by K = X − V [φ] with all the other terms except for the abovementioned Einstein term equal to zero. It can also realize other inflation models easily; k-inflation54 is realized by K(φ, X), G-inflation55 by K and G3 terms, and even New Higgs inflation,98 which couples the Einstein tensor with the scalar kinetic term, can be realized taking an appropriate form of G5 . We also note that the curvature-square inflation17 or more general f (R) inflation can also be recast in the present form by defining a new field as φ = df/dR. Below we describe field equations for the homogeneous background and present two examples based on Refs. 56 and 99. 10.1. Homogeneous background equations The background equations of motion can be derived from (116) taking φ = φ(t) and the spacetime metric ds2 = −N 2 (t)dt2 + a2 (t)dx2 . Variation with respect to N (t) gives the constraint equation corresponding to the Friedmann equation, which can be written as 5 

Ei = 0,

(117)

i=2

where E2 = 2XK X − K,

(118)

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˙ E3 = 6X φHG 3X − 2XG3φ ,

(119)

˙ 4φX − 6H φG ˙ 4φ , E4 = −6H 2 G4 + 24H 2 X(G4X + XG4XX ) − 12HX φG

(120)

2 ˙ E5 = 2H 3 X φ(5G 5X + 2XG5XX ) − 6H X(3G5φ + 2XG5φX ).

(121)

Being the constraint equation, the above terms contain derivatives of the metric and the scalar field only up to first-order. The evolution equation is obtained by differentiation with respect to a(t), which reads 5  Pi = 0, (122) i=2

where P2 = K,

(123)

¨ 3X ), P3 = −2X(G3φ + φG

(124)

˙ 4 − 12H 2 XG4X − 4H XG ˙ 4X − 8HXG ˙ ˙ P4 = 2(3H 2 + 2H)G 4X − 8HX XG4XX ˙ 4φ + 4XG4φφ + 4X(φ¨ − 2H φ)G ˙ 4φX , + 2(φ¨ + 2H φ)G

(125)

¨ 5X − 4H 2 X 2 φG ¨ 5XX P5 = −2X(2H 3 φ˙ + 2H H˙ φ˙ + 3H 2 φ)G ˙ 5φφ . + 4HX (X˙ − HX )G5φX + 2[2(HX ) + 3H 2 X]G5φ + 4HX φG

(126)

Although the background quantities Ei and Pi are defined in an analogous way in which the energy density and the isotropic pressure of a usual scalar field are defined, the partition between the gravitational and scalar-field parts of the Lagrangian is ambiguous, because the gravitational contributions are treated with the matter contributions on an equal footing in the above expressions. Indeed, one can see 2 /2, E4 and P4 reduce to the Einstein tensor multiplied by −1: that, for G4 = MG 2 2 2 ˙ (3H 2 + 2H). E4 = −3MG H and P4 = MG Finally, variation with respect to φ(t) gives the scalar-field equation of motion, 1 d 3 (a J) = Pφ , a3 dt

(127)

where ˙ X + 6HX G3X − 2φG ˙ 3φ + 6H 2 φ(G ˙ 4X + 2XG4XX ) − 12HX G4φX J = φK ˙ 5φ + XG5φX ) + 2H 3 X(3G5X + 2XG5XX ) − 6H 2 φ(G

(128)

and ¨ 3φX ) + 6(2H 2 + H)G ˙ 4φ + 6H(X˙ + 2HX )G4φX Pφ = Kφ − 2X(G3φφ + φG ˙ 5φX . − 6H 2 XG5φφ + 2H 3 X φG

(129)

The solutions of the above equations admitting accelerated cosmic expansion constitute generalized G-inflation. We present two examples below.

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10.2. Kinetically driven G-inflation If we impose shift symmetry, φ → φ + c, the scalar field has no potential. In this case, the field equations are 5 

˙ − K − 6H 2 (G4 − 2XG4X ) + 4H 3 X φG ˙ 5X = 0, Ei = φJ

(130)

i=2 5 

˙ − 2X φG ¨ 3X + 2 d [2H(G4 − 2XG4X ) − H 2 X φG ˙ 5X ] = 0, (Ei + Pi ) = φJ dt i=2 (131) d 3 (a J) = 0. dt

(132)

From Eq. (132), one immediately finds that J ∝ a−3 → 0. Shift-symmetric models thus have an attractor, J = 0, along which H = const., φ˙ = const., satisfying ˙ X + 6HXG3X + 6H 2 φ˙ (G4X + 2XG4XX ) φK + 2H 3 X (3G5X + 2XG5XX ) = 0, ˙ 5X = 0. K + 6H 2 (G4 − 2XG4X ) − 4H 3 X φG

(133) (134)

Provided that Eqs. (133) and (134) have a nontrivial root, H = 0, φ˙ = 0, we obtain inflation driven by φ’s kinetic energy. As a simple example, let us take K = −X +

X2 , 2M 3 µ

G3 =

X , M3

(135)

with M and µ being constants with unit mass dimension. Writing the field equations ˙ for this Lagrangian, we can easily find a de Sitter solution with a constant φ, X = M 3 µx, H2 =

M 3 (1 − x)2 , 18µ x

(136)

 √ where x (0 < x < 1) is a constant satisfying (1 − x)/x 1 − x/2 = 6µ/MG . For √ µ  MG , it can be seen that x 1 − 3µ/MG and hence the Hubble rate during 2 ). We can show that inflation is given in terms of M and µ as H 2 M 3 µ/(6MG this inflationary solution is an attractor of the theory. Inflation can end if the shift symmetry is broken at some φ where the first term of K changes sign. Then the universe will soon become dominated by the kinetic energy of the scalar field, which decreases in proportion to a−6 after the higherorder term in K becomes negligible compared with the first term. Since the shift symmetry of the original Lagrangian prevents direct interaction between φ and standard-model sector, reheating proceeds only through gravitational particle production.78,79 The reheating temperature in such a case is given by (58).

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10.3. Potential-driven slow-roll G-inflation Next we consider the opposite extreme. Suppose that the functions in the Lagrangian can be expanded in terms of X as K(φ, X) = −V (φ) + K(φ)X + · · · ,

(137)

Gi (φ, X) = gi (φ) + hi (φ)X + · · · ,

(138)

and consider the case in which the inflaton field value φ(t) changes very slowly. In this case, the potential term manifestly breaks the shift symmetry, so that the model is capable of a graceful exit from inflation.100 Neglecting all the terms multiplied by φ˙ in the gravitational field equations, we obtain 5 

Pi −

i=2

5 

Ei −V (φ) + 6g4 (φ)H 2 ,

(139)

i=2

where we have assumed ˙  H2 |H|

¨  |H φ|. ˙ and |φ|

(140)

We may thus have slow-roll inflation with H2

V . 6g4

(141)

During slow-roll, we approximate ˙  |HJ|, |J|

|g˙ i |  |Hgi |,

|h˙ i |  |Hhi |.

(142)

Under the above approximation, we have the slow-roll equation of motion for φ, 3HJ −Vφ + 12H 2 g4φ ,

(143)

J Kφ˙ − 2g3φ φ˙ + 6(Hh3 X + H 2 h4 φ˙ − H 2 g5φ φ˙ + H 3 h5 X).

(144)

with

Which term is dominant in Eq. (144) depends on the magnitude of the coefficients hi (φ) of X. Note here that we can set g3 = 0 and g5 = 0 without loss of generality, because g3φ can be absorbed into a redefinition of K and g5φ into h4 , that is, K − 2g3φ → K, h4 − g5φ → h4 . 2 /2 with all the other coefficients equal to zero, If we take K = 1 and g4 = MG we recover the standard slow-roll inflation in the Einstein gravity driven by the potential V (φ). Identifying φ with the real neutral component of the standard Higgs field, we can reproduce various types of Higgs inflation by taking each of the remaining functions as follows: K = 1 + κφ2n ,

running kinetic inflation,101

(145)

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h3 (φ) =

φ , M4

2 MG ξ − φ2 , 2 2 1 h4 (φ) = , 2µ2

g4 (φ) =

II-255

Higgs G-inflation,102

(146)

original Higgs inflation,103–105

(147)

new Higgs inflation,98

(148)

φ , running Einstein inflation.99 (149) M6 Thus all types of Higgs inflation models can be expressed in the framework of generalized G-inflation. Furthermore, combinations of these models can also be analyzed coherently in this context.99 h5 (φ) =

11. Power Spectrum of Perturbations in Generalized G-inflation Next we incorporate scalar and tensor perturbations into the homogeneous solution again based on Ref. 56. To do this we consider the perturbed metric ds2 = −N 2 dt2 + γij (dxi + N i dt)(dxj + N j dt), in the unitary gauge φ = φ(t), where N = 1 + α,

Ni = ∂i β,

γij = a (t)e 2

2ζ


 1 δij + hij + hik hkj . 2

(150)

(151)

Here α, β, and ζ are scalar perturbations and hij is a tensor perturbation satisfying √ hii = 0 = hij,j . With the above definition of the perturbed metric, −g does not contain hij up to second-order, and the coefficients of ζ 2 and αζ vanish, thanks to the background equations. 11.1. Tensor perturbations The quadratic action for the tensor perturbations is found to be

  1 FT (2) ST = dtd3 x a3 GT h˙ 2ij − 2 (∇hij )2 , 8 a

(152)

where ¨ 5X + G5φ )], FT = 2[G4 − X(φG

(153)

˙ 5X − G5φ )]. GT = 2[G4 − 2XG4X − X(H φG

(154)

One may notice that GT can also be expressed as 1  ∂Pi . 2 i=2 ∂ H˙

(155)

FT . GT

(156)

5

GT = The squared sound speed is given by

c2T =

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One sees from the action (152) that ghost and gradient instabilities are avoided provided that FT > 0,

GT > 0.

(157)

Note that c2T is not necessarily unity in the general case, contrary to the standard inflation models. To canonically normalize the tensor perturbation, we define cT a dyT = dt, zT = (FT GT )1/4 , vij = zT hij , (158) a 2 and then the quadratic action is written as

  1 zT 2 (2) 3  2 2 ST = dyT d x (vij ) − (∇vij ) + v , 2 zT ij

(159)

where a prime denotes differentiation with respect to yT . In terms of the Fourier wavenumber k, sound horizon crossing occurs when k 2 = zT /zT ∼ 1/yT2 for each mode. On superhorizon scales, the two independent solutions to the perturbation equation that follows from the action (159) are  dyT . (160) vij ∝ zT and zT zT2 In terms of the original variables, the two independent solutions on superhorizon scales are given by  t dt hij = const. and . (161) a3 GT The second solution corresponds to a decaying mode. 2 ˙ To evaluate the primordial power spectrum, let us assume that  = −H/H const., fT =

F˙ T G˙T const. and gT = const. HFT HGT

(162)

We also define the variation parameter of the sound velocity of tensor perturbations as sT =

c˙T 1 = (fT − gT ). HcT 2

(163)

Clearly only two of the three parameters are independent. We additionally impose conditions gT fT + > 0, 2 2

(164)

3 −  + gT > 0.

(165)

1−−

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The former (equivalent to 1 >  + sT ) guarantees that the time coordinate yT runs from −∞ to 0 as the universe expands. The latter implies that the second solution in (161) indeed decays. We see that zT can be written as

1/2−νT (−yT ) 3/4 F ∗ 1 (−yT ∗ ) zT = T1/4 , fT gT H (−y ) T∗ 2GT ∗ ∗ 1−− + 2 2

(166)

where the quantities with ∗ are those evaluated at some reference time yT = yT ∗ . The normalized mode solution to the perturbation equation is given in terms of the Hankel function: √ π√ vij = −yHν(1) (−kyT ) eij , (167) T 2 where νT =

3 −  + gT , 2 − 2 − fT + gT

(168)

and eij is a polarization tensor. Notice that the conditions (164) and (165) guarantee the positivity of νT . On superhorizon scales, −kyT  1, we obtain Γ(νT ) (−yT )1/2−νT 3/2−νT k 3/2 hij ≈ 2νT −2
 k eij . 3 zT Γ 2 Thus, we find the power spectrum of the primordial tensor perturbation:  1/2 GT H 2  PT = 8γT 3/2 2  , FT 4π −kyT =1

(169)

(170)

where γT = 22νT −3 |Γ(νT )/Γ(3/2)|2 (1 −  − fT /2 + gT /2). The tensor spectral tilt is given by nT = 3 − 2νT .

(171)

Contrary to the predictions of the conventional inflation models, a blue spectrum nT > 0 can be obtained if the following condition is satisfied: 4 + 3fT − gT < 0.

(172)

This condition is easily compatible with conditions (164) and (165). Thus, positive and large gT compared with  and fT can lead to a blue spectrum of tensor perturbations. In deriving the above formulas, we only assumed that , fT , and gT are constant. These parameters may not necessarily be very small as long as the inequalities (164) and (165) are satisfied under a sensible background solution.

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11.2. Scalar perturbations We now focus on scalar fluctuations setting hij = 0. Plugging the perturbed metric into the action and expanding it to second-order, we obtain

 FT (2) 3 3 SS = dtd xa −3GT ζ˙2 + 2 (∇ζ)2 + Σα2 a  ∇2 ∇2 ∇2 ˙ ˙ − 2Θα 2 β + 2GT ζ 2 β + 6Θαζ − 2GT α 2 ζ , (173) a a a where ˙ Σ = XKX + 2X 2 KXX + 12H φXG 3X ˙ 2 G3XX − 2XG3φ − 2X 2 G3φX − 6H 2 G4 + 6H φX + 6[H 2 (7XG4X + 16X 2 G4XX + 4X 3 G4XXX ) ˙ 4φ + 5XG4φX + 2X 2 G4φXX )] − H φ(G 3˙ 2 ˙ + 30H 3 φXG 5X + 26H φX G5XX

˙ 3 G5XXX − 6H 2 X(6G5φ + 9XG5φX + 2X 2 G5φXX ), + 4H 3 φX

(174)

2 ˙ ˙ ˙ Θ = −φXG 3X + 2HG4 − 8HX G4X − 8HX G4XX + φG4φ + 2X φG4φX 2 ˙ − H 2 φ(5XG 5X + 2X G5XX ) + 2HX (3G5φ + 2XG5φX ).

(175)

It is interesting to see that even in the most generic case, some of the coefficients are given by FT and GT , i.e. the functions characterizing the tensor perturbation, and only two new functions show up in the scalar quadratic action. Note that the following relations hold: Σ=X

5  ∂Ei i=2

1  ∂Ei + H , ∂X 2 i=2 ∂H 5

1  ∂Ei , 6 i=2 ∂H

(176)

5

Θ=−

(177)

which compactify the above lengthy expressions. Varying the action (173) with respect to α and β, we obtain the constraint equations Σα − Θ

2 ∇2 ˙ − GT ∇ ζ = 0, β + 3Θ ζ a2 a2 Θα − GT ζ˙ = 0.

(178) (179)

Using the constraint equations, we eliminate α and β from the action (173) and finally arrive at

  FS (2) 3 3 2 2 ˙ SS = dtd x a GS ζ − 2 (∇ζ) , (180) a

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where 1 d  a 2 G − FT , a dt Θ T Σ GS = 2 GT2 + 3GT . Θ

FS =

(181) (182)

The analysis of the curvature perturbation hereafter is completely parallel to that of the tensor perturbation. The squared sound speed is given by c2S = FS /GS , and ghost and gradient instabilities are avoided as long as FS > 0,

GS > 0.

(183)

In the case of standard potential driven inflation and k-inflation, where G3 = 0 = 2 2 /2, we find FS = MG . This means that positive H˙ is prohibited G5 and G4 = MG 106 In the generalized G-inflation, however, we may by the stability requirement. realize positive H˙ without violating the stability condition. This point was already clear in G-inflation and kinetic gravity braiding for which G3 = 0.55,107 Stable cosmological solutions with H˙ > 0 offer a radical and very interesting scenario of the earliest universe.108–111 Using the new variables dyS =

cS dt, a

zS =

√ 1/4 2a (FS GS ) ,

u = zS ζ,

(184)

the curvature perturbation is canonically normalized and the action is now given by

  1 z  (2) SS = dyS d3 x (u )2 − (∇u)2 + S u2 , (185) 2 zS where a prime denotes differentiation with respect to yS . Each perturbation mode exits the sound horizon when k 2 = zS /zS ∼ 1/yS2 , where k is the Fourier wavenumber. The two independent solutions on superhorizon scales are  ζ = const. and

t

dt . a3 GS

(186)

During inflation, it may be assumed that GS is slowly varying. In this case, the second solution decays rapidly. Note however that the nontrivial dynamics of the scalar field can induce a temporal rapid evolution of GS , which would affect the superhorizon behavior of the curvature perturbation through the contamination of the second mode in the same way as in Refs. 112–115. Given the specific background dynamics, one can evaluate such an effect using our general formulas. Closely following the procedure above for the tensor perturbations, we now evaluate the power spectrum of the primordial curvature perturbations. We assume

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that  const.,a fS =

F˙ S const., HFS

gS =

G˙S const. HGS

(187)

and then define νS =

3 −  + gS . 2 − 2 − fS + gS

(188)

The power spectrum is given by

 1/2 γS GS H 2  Pζ = ,  2 F 3/2 4π 2  S −kyS =1

(189)

where γS = 22νS −3 |Γ(νS )/Γ(3/2)|2 (1 −  − fS /2 + gS /2). The spectral index is ns − 1 = 3 − 2νS .

(190)

An exactly scale-invariant spectrum is obtained if 3 1  + fS − gS = 0. (191) 4 4 Here again, , fS , and gS are not necessarily very small (as long as ns − 1 0). Taking now the limit , fT , gT , fS , gS  1, the tensor-to-scalar ratio is given by
3/2
−1/2 FS GS F S cS r = 16 = 16 . (192) FT GT F T cT In the case of potential-driven slow-roll inflation considered in Sec. 10.3, we need have to leading order in slow-roll ˙ X 4φX (K + 6H 2 h4 ) + (h3 + H 2 h5 ), 2 H H ˙ X 6φX GS 2 (K + 6H 2 h4 ) + (h3 + H 2 h5 ) H H

FS

(193) (194)

2 ˙ and FT GT 2g4 , where we used the slow-roll equation 2g4  + g˙ 4 /H φJ/2H . 2 In this case, we have cT 1 and nT −(2+gT ) with fT gT g˙ 4 /(Hg4 ). If the K 2 ˙ ) g4 (2+gT ), or h4 term dominates in J, we have c2S 1 and FS GS J φ/(2H which yields the standard consistency relation:

r −8nT .

(195)

It is often argued that a negative nT is a generic prediction of inflationary cosmology, and that if observationally a positive nT was confirmed, it would falsify inflation. It is clear from the above discussion that such a statement is incorrect. The signature of tensor spectral index is not a discriminant of inflation and exotic cosmology but simply that of standard inflation and generalized G-inflation. a By

defining the variation parameter of the sound velocity of scalar perturbations as sS = c˙ S /(HcS ) = (fS − gS )/2, the formulae with fS and/or gS can be rewritten in terms of cS .

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12. Inflationary Cosmology and Observations Various predictions of inflationary cosmology are now confronting observational tests using CMB anisotropy and large-scale structures. Since the first-year WMAP results were disclosed,52,116,117 the cosmological parameters of homogeneous and isotropic universe as well as properties of fluctuations have been determined quite accurately with well chaaracterized error bars. Indeed, until the WMAP papers were published, although we had the concordance cosmology, we did not have any reliable error estimates because the concordance values of the cosmological parameters had been obtained by combining a number of different observational results and we did not have means to calculate error bars to make a sensible weighted average. Current precision cosmology makes use of the Markov Chain Monte Carlo (MCMC) method to constrain values of cosmological parameters as well as the amplitude and spectrum of primordial fluctuations using temperature and E-mode polarization data of CMB.116 Note, however, that the final values of these parameters as well as the magnitude of errors obtained by these procedures change considerably depending on which observational datasets one uses as well as which parameters are included in the model. Recent cosmological data as a whole show good agreement with spatially flat universe with a nonvanishing cosmological constant (Λ) and cold dark matter (CDM) based on simple single-field inflation paradigm that predicts almost scaleinvariant adiabatic fluctuations. Thus it makes sense to estimate various parameters of ΛCDM model using MCMC techniques. Below we quote results from the seven- and nine-year WMAP118,119 denoted respectively by W7 and W9 as well as Planck 201353,120 (P13) and 2015121,122 (P15) results to give an idea about what constraints are likely to survive in future independent of the subtleties of each experiment and accumulation of the data. Abbreviations in parentheses after each quoted number refers to the dataset used, whose meaning is self-explanatory. First as the most important test of the global spacetime, the spatial curvature is constrained as −0.0133 < ΩK0 ≡ −

K a20 H02

−0.079 < ΩK0 < 0.007 −0.0065 < ΩK0 < 0.0012 −0.09 < ΩK0 < 0.001 −0.0075 < ΩK0 < 0.0052 −0.019 < ΩK0 < 0.011 −0.0039 < ΩK0 < 0.0048

< 0.0084 (W7HSTBAO 95%CL), (W9 68%CL), (W9eCMBBAOH0 68%CL), (P13WPhighL 95%CL), (P13lensWPhighLBAO 95%CL), (P15TTTEEElens 95%CL), (P15TTTEEElensExt 95%CL),

which is consistent with (9). Hence from now on we fix ΩK0 = 0.

(196)

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The amplitude and the spectral index of curvature perturbations measured by WMAP read PR (k0 ) = (2.43 ± 0.11) × 10−9 ,

ns = 0.967 ± 0.014 (W7 68%CL),

PR (k0 ) = (2.430 ± 0.091) × 10−9 , ns = 0.968 ± 0.012 (W7HSTBAO 68%CL), (197) PR (k0 ) = (2.41 ± 0.10) × 10−9 , ns = 0.972 ± 0.013 (W9 68%CL), +0.078 ) × 10−9 , ns = 0.971 ± 0.010 (W9BAOH0 68%CL), PR (k0 ) = (2.427−0.079

(198) where the pivot scale is taken as k0 = 0.002 Mpc−1 . Planck, on the other hand, takes the pivot scale at k0 = 0.05 Mpc−1 and finds +0.053 PR (k0 ) = (2.195−0.059 ) × 10−9 ,

ns = 0.9603 ± 0.0073 (P13WP 68%CL),

+0.056 ) × 10−9 , PR (k0 ) = (2.200−0.054

ns = 0.9608 ± 0.0054 (P13WPhighLBAO 68%CL), (199)

PR (k0 )

=

+0.076 (2.207−0.073 )

× 10

−9

,

ns = 0.9645 ± 0.0049 (P15TTTEEElowP 68%CL),

+0.076 ) × 10−9 , PR (k0 ) = (2.204−0.074

ns = 0.9673 ± 0.0045 (P15TTlowPBAO 68%CL). (200)

If we allow scale-dependence of the spectral index, the values change considerably as +0.050 , ns = 1.027−0.051

ns = 1.008 ± 0.042, ns = 1.009 ± 0.049, ns = 1.020 ± 0.029,

dns = −0.034 ± 0.026 (W7 68%CL), d ln k dns = −0.022 ± 0.020 (W7HSTBAO 68%CL), d ln k

(201)

dns = −0.019 ± 0.025 (W9 68%CL), d ln k dns = −0.023 ± 0.011 (W9eCMBBAOH0 68%CL). d ln k (202)

These numbers refer to the values at k0 = 0.002 Mpc−1 . Planck finds dns = −0.013 ± 0.009 (P13WP 68%CL), d ln k dns = −0.011 ± 0.008 (P13lensWPhighL 68%CL), d ln k

(203)

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dns = −0.0057 ± 0.0071 (P15TTTEEElowP 68%CL), d ln k dns = −0.0033 ± 0.0074 (P15lowPlens 68%CL), d ln k

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(204)

on their pivot scale k0 . Thus there is no need to consider inflation models with appreciable running any more.123–125 The amplitude of tensor perturbations, which tells us the energy scale of inflation, is also an important quantity to distinguish inflation models. If we incorporate tensor perturbations in power-law ΛCDM model the constraints are r < 0.36 (W7 95%; no running), r < 0.24 (W7BAOH0 95%; no running), r < 0.38 (W9 95%; no running), r < 0.13 (W9eCMBBAOH0 95%; no running),

(205)

r < 0.11 (P13WPhighL 95%; no running), r < 0.11 (P15TTlowPlens 95%; no running), r < 0.10 (P15TTTEEElowP 95%; no running), using the pivot scale k0 = 0.002 Mpc−1 , whereas if we allow a running spectral index the above constraint relax substantially to r < 0.49 (W7 95%; with running), r < 0.49 (W7BAOH0 95%; with running), r < 0.50 (W9 95%; with running), r < 0.47 (W9eCMBBAOH0 95%; with running),

(206)

r < 0.26 (P13WPhighL 95%; with running), r < 0.18 (P15TTlowP 95%; with running), r < 0.15 (P15TTTEEElowP 95%; with running). Thus the above constraints change significantly if we allow running spectral index of curvature perturbation or not, since they are indirectly obtained through the temperature anisotropy and the E-mode polarization. Because of this sensitivity to the nobel assumptions, it is important to constrain r directly through B-mode polarization of CMB, which is not generated by scalar perturbations so that the result does not rely on assumptions concerning the spectrum of curvature perturbations. In March 2014, BICEP2 collaboration announced possible detection of primordial tensor perturbations through their observation of B-mode polarization of CMB at the South Pole,126 favoring a fairly large value of

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r ∼ 0.2, namely high-scale inflation. It turned out, however, that their result was significantly contaminated by the galactic dust and large r is no longer favored. The joint analysis of BICEP2/Keck Array data and Planck 2015 data was performed obtaining the upper bound, r0.05 < 0.12 at 95% CL.127 Thus the current constraint from B-mode polarization is at the same level as those obtained indirectly from temperature anisotropy and E-mode polarization with the power-law model. These constraints are often graphically displayed in a (ns , r) plane128 with oneand two-sigma contours and compared with various inflation models. We do not do the same here for two reasons. First constraints vary appreciably depending on what dataset one uses and what underlying model one adopts as seen above. Second occupying center of the likelihood contour at the current level of observations does not necessarily mean that such a model is the right one, and we should be open minded for further development of the observations. Below we instead calculate these observables in each class of models discussed in Sec. 4.

12.1. Large-field models Consider a massive scalar model as an example of large-field models. The number of e-folds of inflation after φ crossed φN is given from (31) as 1 N∼ = 4




φN MG

2 .

(207)

Hence the slow-roll parameters are
V = ηV = 2

MG φN

2 =

1 , 2N

ξV = 0

(208)

which gives the amplitude of curvature perturbation, (95), PR (k) =

1 6π 2




mN MG

2 .

(209)

Setting N = 55 at the pivot scale tentatively, we find m = 1.6 × 1013 GeV and the quartic coupling is constrained as λ  8 × 10−13 . The spectral index and its running are given by ns = 1 −

2 = 0.964, N

dns 2 = − 2 = −6.6 × 10−4 , d ln k N

(210)

in agreement with the observations. The amplitude of tensor perturbation, r = 16 = 0.15, is so large that it is now strongly disfavored by observations.127 It is remarkable that the simplest and most attractive model of inflation is now about to be ruled out by the observation.

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Let us also consider R2 inflation in the Einstein frame as a kind of large-field models. From (23), (24) and (27), we find 3 √2 4 1 3 ∼ √2 N∼ , = e 3 κφN , V = = κφ 4 3 (e 3 N − 1)2 4N 2 √2 (211) 4 e 3 κφN − 2 ∼ 1 √ ηV = − =− , 3 (e 23 κφN − 1)2 N which yield ns = 1 −

2 = 0.964, N

r=

12 = 4.0 × 10−3 , N2

(212)

for N = 55. 12.2. Small-field model Next we consider a small field model (32). From V [φ] = parameters are given by V =

2 2 φ 8MG , 2 (φ − v 2 )2

ηV =

2 (3φ2 − v 2 ) 4MG , 2 (φ − v 2 )2

λ 2 4 (φ

ξV =

− v 2 )2 the slow-roll

4 2 φ 96MG . 2 (φ − v 2 )3

(213)

Since inflation ends when H = V = 1, the field value, φf , at that time is given by  √ 2 2 + 8M 2 v 2 ∼ (β 2 − 2 2β)M 2 , φ2f = v 2 + 4MG − 16MG (214) = G G where we have set v ≡ βMG and the last approximate equality holds when β
10. The number for e-folds from φ = φN to φf reads
  φf dφ β2 φf 1 β2 1 φf 2 2 N= H = ln − − ln . (215) 2 (φf − φN ) 4 4 φN 8MG φN 2 φ˙ φN The last approximation is valid when β 10 − 20. The amplitude of curvature perturbation is given by
2
 λ(φ2N − v 2 )4 MG λβ 8 8N λβ 8 √ PR (k0 ) = exp + 1 , 6 φ2 768π 2 MG 768π 2 φN β2 768π 2 (β − 2)2 N (216) where the second and third approximations are based on φN  v and φf (β − √ 2)MG , respectively. The spectral index and its running are given by ns − 1 = −

2 8(3φ2N + v 2 )MG 8 − 2, 2 2 2 (φN − v ) β

4 dns (320v 2 φ2N + 192φ4N )MG 320 =− − 6 2 2 4 d ln k (φN − v ) β




(217) φN MG

2 .

(218)

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For β = 15 and N = 55, for example, we find ns = 0.964 with negligibly small running, and the self-coupling is given by λ 7 × 10−14 . 12.3. Hybrid inflation model Here, we consider the case where vacuum energy drives inflation, which is distinct from the above two classes. The potential in this regime is approximated as V [φ] = V0 +

m2 2 φ 2

and so the slow-roll parameters are given by
2 1  m 4 φ m2 V = , ηV = , 18 H MG 3H 2

(219)

ξV = 0.

(220)

In this case one cannot determine the energy scale of inflation from the amplitude of curvature fluctuations alone. The spectral index is usually determined by ηV as ns − 1 2ηV =

2m2 > 1, 3H 2

(221)

which is inconsistent with current observations. The running spectral index takes a small value
3
2
2 dns 2m2 φ φ 3 16V ηV = (ns − 1) (222) d ln k 3H 2 MG MG so does the tensor-to-scalar ratio,
2
2
2 2m2 φ φ 2 r=2 (n − 1) . s 3H 2 MG MG

(223)

One should note that if the second term dominates over the vacuum energy in (219), the prediction becomes closer to a large-field model and the spectral index can be smaller than unity. 12.4. Noncanonical models and multi-field models So far we have focused on the amplitude and spectral index of primordial curvature and tensor perturbations. Indeed in single-field slow-roll inflation there is only one fluctuating degree of freedom, so only adiabatic fluctuations are generated with no isocurvature counterparts. Furthermore, the curvature perturbation generated in slow-roll inflation behaves as a practically free massless scalar field during inflation, and so its vacuum fluctuation is Gaussian distributed. This is why the power spectrum fully quantifies the properties of fluctuations. In some classes of noncanonical inflation models including DBI inflation,129,130 ghost inflation,131 k-inflation54 and G-inflation,55 the inflaton may change rapidly and its interaction may not be negligible. In such circumstances deviation from

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Gaussian statistics may be significant.90,132–135 Furthermore, in models with multiple fluctuating fields besides the inflaton, one may also expect large deviation from Gaussianity.136–138 Much work was done in recent years to quantify the skewness or the bispectrum of the temperature anisotropy, and a number of contrived models were proposed that predicted an observationally falsifiable magnitude of the deviation from Gaussianity by Planck.139 The constraints on non-Gaussianity provided by analyzing the Planck data140 falsified most of these models that had been proposed predicting large bispectral non-Gaussianity; therefore, we do not discuss this issue in more detail. Another possible signature of multi-field models would be the presence of isocurvature modes. But again CMB observations do not need any isocurvature components at least on relevant scales.119 13. Conclusion The fact that we live in a big and old universe with a large amount of entropy can be regarded as a primary consequence of inflation in the early universe. Any attempt to avoid inflation by providing alternative explanations to the other predictions of inflationary cosmology, namely, generation of adiabatic density perturbation with nearly scale-invariant spectrum as well as tensor perturbation should also have sound explanation to the aforementioned global properties of the spacetime. Some people, however, try to refute this viewpoint claiming that the global properties of the universe, namely the homogeneity, isotropy, and longevity had been known long before the first inflation models13,16,17 were proposed so that they cannot be regarded as the predictions of inflationary cosmology. While this statement is correct, one should also recognize that theories in physics have a double role. The first is to make new prediction which may be confirmed or falsified by experiment and observation. The second is to provide sensible explanations to known mysteries of the nature. Inflationary cosmology certainly provides a simple mechanism to explain fundamental properties of the global structure of the space, playing the latter role of theoretical physics. Thus the above mentioned criticism is unjustified. Furthermore, the simplest class of slow-roll inflation models predicted nearly scale-invariant, adiabatic, and Gaussian curvature and density fluctuations, all of which are now being confirmed by dedicated CMB experiments. In particular, the observed angular correlation between temperature anisotropy and E-mode polarization clearly indicates that the observed fluctuations were generated as super-Hubble fluctuations as predicted by inflation. We are tempted to interpret the fact that Planck did not confirm any sizable non-Gaussianity140 despite the recent efforts of many theorists to make models of non-Gaussian fluctuations as indicating that nature preferred the simplest inflation models. This situation, on the other hand, highlights the difficulty of singling out the right model responsible for our universe by exploiting the small numbers of

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observational clues available. For example, among the models occupying central regions of the likelihood contours in (ns , r) plane now, it is very difficult to distinguish between R2 model17,141,142 and a Higgs inflation model with a large and negative nonminimal coupling103–105 using observations of perturbation variables alone. We should combine with the analysis of the reheating processes of these models for more detailed comparison, which may be observationally probed by future space-based laser interferometers such as DECIGO.143,144 On the other hand, the latest observational upper limits on B-mode polarization of CMB is ruling out the simplest inflation model, namely, chaotic inflation with a massive scalar field.127 This is somehow a shocking result since we also have sensible particle physics models for this inflation mechanism. This may indicate that our understanding of the gravity sector at such high energy scale may be wrong, as there is no reason believe that the Planck scale remains intact up to the Planck scale. Acknowledgments Part of the present chapter is an update of the review paper written by JY21 reproduced here through the Creative Commons license. This paper is also partially based on a Japanese text book written by JY and edited by KS.145 JY is grateful to Alexei A. Starobinsky for recounting the history of developments in Russia and to Tsutomu Kobayashi and Masahide Yamaguchi for fruitful collaboration on generalized G-inflation. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

G. Gamow, Phys. Rev. 70 (1946) 572. E. Hubble, Proc. Natl. Acad. Sci. 15 (1929) 168. G. Lemaˆıtre, Ann. Societe Sci. Brux. 47 (1927) 49. A. A. Penzias and R. W. Wilson, Astrophys. J. 142 (1965) 419. R. A. Alpher, H. Bethe and G. Gamow, Phys. Rev. 73 (1948) 803. C. Hayashi, Prog. Theor. Phys. 5 (1950) 224. R. H. Dicke and P. J. E. Peebles, The big bang cosmology: Enigmas and nostrums, in General Relativity, An Einstein Centenary Survey, Chap. 9 (Cambridge, 1979). H. Georgi and S. L. Glashow, Phys. Rev. Lett. 32 (1974) 438. T. W. B. Kibble, J. Phys. A 9 (1976) 1387. J. Preskill, Phys. Rev. Lett. 43 (1365) 1979. G. ’t Hooft, Nucl. Phys. B 79 (1974) 276. A. M. Polyakov, JETP Lett. 20 (1974) 194. K. Sato, Mon. Not. R. Astron. Soc. 195 (1981) 467. K. Sato, Phys. Lett. B 99 (1981) 66. M. B. Einhorn and K. Sato, Nucl. Phys. B180 (1981) 385. A. H. Guth, Phys. Rev. D 23 (1981) 347. A. A. Starobinsky, Phys. Lett. B 91 (1980) 99. Class for Physics of the Royal Swedish Academy of Sciences, AAPPS Bull. 11 (2011). A. D. Linde, Phys. Lett. B 108 (1982) 389. A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. 48 (1982) 1220.

May 4, 2017 9:54 b2167-ch18

Inflationary cosmology: first 30+ years

21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69.

II-269

J. Yokoyama, Inflation: 1980-201X, PTEP 2014(6) (2014) 06B103. S. K. Blau and A. H. Guth, Inflationary Cosmology (1986). A. H. Guth, Inflation and cosmological perturbations (2003), 725. S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264. A. Salam, Conf. Proc. C 680519 (1968) 367. K. Sato, Prog. Theor. Phys. 54 (1975) 1325. K. Hirata et al., Phys. Rev. Lett. 58 (1987) 1490. K. Sato and H. Sato, Prog. Theor. Phys. 54 (1975) 912. K. Sato and H. Sato, Prog. Theor. Phys. 54 (1975) 1564. K. Sato and T. Nakamura, Prog. Theor. Phys. 55 (1976) 1133. K. Sato and T. Nakamura, Prog. Theor. Phys. 55 (1976) 978. M. Yoshimura, Phys. Rev. Lett. 41 (1978) 281. D. A. Kirzhnits and A. D. Linde, Phys. Lett. B 42 (1972) 471. D. A. Kirzhnits and A. D. Linde, Sov. Phys. JETP 40 (628) 1975. L. Dolan and R. Jackiw, Phys. Rev. D 9 (1974) 3320. S. Weinberg, Phys. Rev. D 9 (1974) 3357. K. Sato, M. Sasaki, H. Kodama and K. Maeda, Prog. Theor. Phys. 65 (1443) 1981. K. Sato, H. Kodama, M. Sasaki and K. Maeda, Phys. Lett. B 108 (1982) 103. ´ B. Gliner, Sov. J. Exp. Theor. Phys. 22 (1966) 378. E. G. Lemaˆıtre, Proc. Natl. Acad. Sci. 20 (1934) 12. L. E. Gurevich, Astrophys. Space Sci. 38 (1975) 67. ´ B. Gliner and I. G. Dymnikova, Soviet Astron. Lett. 1 (93) 1975. E. A. A. Starobinskii, Sov. Astron. Lett. 4 (1978) 82. A. D. Linde, Phys. Lett. B 129 (1983) 177. A. A. Starobinsky, JETP Lett. 30 (1979) 682 (in Russian). V. F. Mukhanov and G. V. Chibisov, JETP Lett. 33 (1981) 532 (in Russian). L. Boubekeur and D. H. Lyth, J. Cosmol. Astropart. Phys. 0507 (010) 2005. A. A. Starobinsky, Phys. Lett. B 117 (1982) 175. S. W. Hawking, Phys. Lett. B 115 (1982) 295. A. H. Guth and S. Y. Pi, Phys. Rev. Lett. 49 (1982) 1110. G. F. Smoot et al., Astrophys. J. 396 (1992) L1. H. V. Peiris et al., Astrophys. J. Suppl. 148 (213) 2003. P. A. R. Ade et al., Planck 2013 results. XXII. Constraints on inflation (2013). C. Armendariz-Picon, T. Damour and V. F. Mukhanov, Phys. Lett. B 458 (1999) 209. T. Kobayashi, M. Yamaguchi and J. Yokoyama, Phys. Rev. Lett. 105 (2010) 231302. T. Kobayashi, M. Yamaguchi and J. Yokoyama, Prog. Theor. Phys. 126 (2011) 511. A. Kashlinsky, I. I. Tkachev and J. Frieman, Phys. Rev. Lett. 73 (1994) 1582. L. A. Kofman, A. D. Linde and A. A. Starobinsky, Phys. Lett. B 157 (1985) 361. D. S. Salopek, Phys. Rev. Lett. 69 (1992) 3602. A. Vilenkin, Phys. Rev. Lett. 72 (1994) 3137. A. D. Linde, Phys. Lett. B 327 (1994) 208. A. D. Linde, Phys. Rev. D 49 (1994) 748. D. H. Lyth, Phys. Rev. Lett. 78 (1997) 1861. N. Tetradis, Phys. Rev. D 57 (1998) 5997. L. E. Mendes and A. R. Liddle, Phys. Rev. D 62 (2000) 103511. J. Yokoyama, Phys. Lett. B 212 (1988) 273. J. Yokoyama, Phys. Lett. B 231 (1989) 49. J. Yokoyama, Phys. Rev. Lett. 63 (1989) 712. K. Kamada, K. Nakayama and J. Yokoyama, Phys. Rev. D 85 (2012) 043503.

May 4, 2017 9:54 b2167-ch18

II-270

K. Sato and J. Yokoyama

70. K. Kamada, Y. Miyamoto and J. Yokoyama, J. Cosmol. Astropart. Phys. 1210 (2012) 023. 71. J. H. Traschen and R. H. Brandenberger, Phys. Rev. D 42 (1990) 2491. 72. L. Kofman, A. D. Linde and A. A. Starobinsky, Phys. Rev. Lett. 73 (1994) 3195. 73. L. Kofman, A. D. Linde and A. A. Starobinsky, Phys. Rev. D 56 (1997) 3258. 74. L. F. Abbott, E. Farhi and M. B. Wise, Phys. Lett. B 117 (1982) 29. 75. A. Albrecht, P. J. Steinhardt, M. S. Turner and F. Wilczek, Phys. Rev. Lett. 48 (1437) 1982. 76. J. Yokoyama, Phys. Rev. D 70 (2004) 103511. 77. M. S. Turner, Phys. Rev. D 28 (1983) 1243. 78. L. H. Ford, Phys. Rev. D 35 (1987) 2955. 79. T. Kunimitsu and J. Yokoyama, Phys. Rev. D 86 (2012) 083541. 80. L. Parker, Phys. Rev. 183 (1969) 1057. 81. Ya. B. Zeldovich and A. A. Starobinsky, Sov. Phys. JETP 34 (1972) 1159. 82. T. S. Bunch and P. C. W. Davies, Proc. R. Soc. Lond. A 360 (1978) 117. 83. A. Vilenkin and L. H. Ford, Phys. Rev. D 26 (1982) 1231. 84. A. D. Linde, Phys. Lett. B 116 (1982) 335. 85. D. Polarski and A. A. Starobinsky, Class. Quantum Grav. 13 (1996) 377. 86. A. Vilenkin, Nucl. Phys. B 226 (1983) 527. 87. J. M. Bardeen, Phys. Rev. D 22 (1980) 1882. 88. H. Kodama and M. Sasaki, Prog. Theor. Phys. Suppl. 78 (1984) 1. 89. V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger, Phys. Rep. 215 (1992) 203. 90. D. Seery and J. E. Lidsey, J. Cosmol. Astropart. Phys. 0506 (2005) 003. 91. A. A. Starobinsky, JETP Lett. 42 (1985) 152. 92. M. Sasaki and E. D. Stewart, Prog. Theor. Phys. 95 (1996) 71. 93. V. A. Rubakov, M. V. Sazhin and A. V. Veryaskin, Phys. Lett. B 115 (1982) 189. 94. L. F. Abbott and M. B. Wise, Nucl. Phys. B 244 (1984) 541. 95. G. W. Horndeski, Int. J. Theor. Phys. 10 (1974) 363. 96. A. Nicolis, R. Rattazzi and E. Trincherini, Phys. Rev. D 79 (2009) 064036. 97. C. Deffayet, X. Gao, D. A. Steer and G. Zahariade, Phys. Rev. D 84 (2011) 064039. 98. C. Germani and A. Kehagias, Phys. Rev. Lett. 105 (2010) 011302. 99. K. Kamada, T. Kobayashi, T. Takahashi, M. Yamaguchi and J. Yokoyama, Phys. Rev. D 86 (2012) 023504. 100. M. Kawasaki, M. Yamaguchi and T. Yanagida, Phys. Rev. Lett. 85 (2000) 3572. 101. K. Nakayama and F. Takahashi, J. Cosmol. Astropart. Phys. 1011 (2010) 009. 102. K. Kamada, T. Kobayashi, M. Yamaguchi and J. Yokoyama, Phys. Rev. D 83 (2011) 083515. 103. J. L. Cervantes-Cota and H. Dehnen, Nucl. Phys. B 442 (1995) 391. 104. F. L. Bezrukov and M. Shaposhnikov, Phys. Lett. B 659 (2008) 703. 105. A. O. Barvinsky, A. Yu. Kamenshchik and A. A. Starobinsky, J. Cosmol. Astropart. Phys. 0811 (2008) 021. 106. J. Garriga and V. F. Mukhanov, Phys. Lett. B 458 (1999) 219. 107. C. Deffayet, O. Pujolas, I. Sawicki and A. Vikman, J. Cosmol. Astropart. Phys. 1010 (2010) 026. 108. P. Creminelli, M. A. Luty, A. Nicolis and L. Senatore, J. High Energy Phys. 0612 (080) 2006. 109. P. Creminelli, A. Nicolis and E. Trincherini, J. Cosmol. Astropart. Phys. 1011 (2010) 021. 110. D. Pirtskhalava and L. Santoni, J. High Energy Phys. 1412 (2014) 151.

May 4, 2017 9:54 b2167-ch18

Inflationary cosmology: first 30+ years

II-271

111. T. Kobayashi, M. Yamaguchi and J. Yokoyama, Galilean Creation of the Inflationary Universe (2015). 112. O. Seto, J. Yokoyama and H. Kodama, Phys. Rev. D 61 (2000) 103504. 113. S. M. Leach and A. R. Liddle, Phys. Rev. D 63 (2001) 043508. 114. S. Inoue and J. Yokoyama, Phys. Lett. B 524 (2002) 15. 115. Y. Takamizu and J. Yokoyama, Phys. Rev. D 83 (2011) 043504. 116. C. L. Bennett et al., Astrophys. J. Suppl. 148 (2003) 1. 117. D. N. Spergel et al., Astrophys. J. Suppl. 148 (2003) 175. 118. E. Komatsu et al., Astrophys. J. Suppl. 192 (2011) 18. 119. G. Hinshaw et al., Astrophys. J. Suppl. 208 (2013) 19. 120. P. A. R. Ade et al., Planck 2013 results. XVI. Cosmological parameters, A and AP (2013). 121. P. A. R. Ade et al., Planck 2015 results. XX. Constraints on inflation, A and AP (2015). 122. P. A. R. Ade et al., Planck 2015 results. XIII. Cosmological parameters. (2015). 123. M. Kawasaki, M. Yamaguchi and J. Yokoyama, Phys. Rev. D 68 (2003) 023508. 124. M. Yamaguchi and J. Yokoyama, Phys. Rev. D 68 (2003) 123520. 125. M. Yamaguchi and J. Yokoyama, Phys. Rev. D 70 (2004) 023513. 126. P. A. R. Ade et al., BICEP2 I: Detection of B-mode Polarization at degree angular scales (2014). 127. P. A. R. Ade et al., Phys. Rev. Lett. 114(10) (2015) 101301. 128. S. Dodelson, W. H. Kinney and E. W. Kolb, Phys. Rev. D 56 (1997) 3207. 129. M. Alishahiha, E. Silverstein and D. Tong, Phys. Rev. D 70 (2004) 123505. 130. E. Silverstein and D. Tong, Phys. Rev. D 70 (2004) 103505. 131. N. Arkani-Hamed, P. Creminelli, S. Mukohyama and M. Zaldarriaga, J. Cosmol. Astropart. Phys. 0404 (2004) 001. 132. D. Langlois, S. Renaux-Petel, D. A. Steer and T. Tanaka, Phys. Rev. Lett. 101 (2008) 061301. 133. D. Babich, P. Creminelli and M. Zaldarriaga, J. Cosmol. Astropart. Phys. 0408 (2004) 009. 134. T. Kobayashi, M. Yamaguchi and J. Yokoyama, Phys. Rev. D 83 (2011) 103524. 135. A. De Felice and S. Tsujikawa, Phys. Rev. D 84 (2011) 083504. 136. N. Bartolo, S. Matarrese and A. Riotto, Phys. Rev. D 69 (2004) 043503. 137. T. Suyama and M. Yamaguchi, Phys. Rev. D 77 (2008) 023505. 138. D. H. Lyth and Y. Rodriguez, Phys. Rev. Lett. 95 (2005) 121302. 139. N. Bartolo, E. Komatsu, S. Matarrese and A. Riotto, Phys. Rep. 402 (2004) 103. 140. P. A. R. Ade et al., Planck 2013 Results. XXIV. Constraints on primordial nonGaussianity (2013). 141. A. Vilenkin, Phys. Rev. D 32 (1985) 2511. 142. M. B. Mijic, M. S. Morris and W.-M. Suen, Phys. Rev. D 34 (1986) 2934. 143. N. Seto, S. Kawamura and T. Nakamura, Phys. Rev. Lett. 87 (2001) 221103. 144. S. Kawamura et al., Class. Quantum Grav. 23 (2006) S125. 145. J. Yokoyama, Utyuron I, 2nd edn. Modern Astronomy Series, Vol. 2 Nihon Hyoron-sha (2012).

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Chapter 19 Inflation, string theory and cosmic strings

David F. Chernoff Department of Astronomy, Cornell University, Ithaca, New York, USA chernoff@astro.cornell.edu S.-H. Henry Tye Jockey Club Institute for Advanced Study and Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong Department of Physics, Cornell University, Ithaca, New York, USA [email protected]

At its very beginning, the universe is believed to have grown exponentially in size via the mechanism of inflation. The almost scale-invariant density perturbation spectrum predicted by inflation is strongly supported by cosmological observations, in particular the cosmic microwave background (MB) radiation. However, the universe’s precise inflationary scenario remains a profound problem for cosmology and for fundamental physics. String theory, the most-studied theory as the final physical theory of nature, should provide an answer to this question. Some of the proposals on how inflation is realized in string theory are reviewed. Since everything is made of strings, some string loops of cosmological sizes are likely to survive in the hot big bang that followed inflation. They appear as cosmic strings, which can have intricate properties. Because of the warped geometry in flux compactification of the extra spatial dimensions in string theory, some of the cosmic strings may have tensions substantially below the Planck or string scale. Such strings cluster in a manner similar to dark matter leading to hugely enhanced densities. As a result, numerous fossil remnants of the low tension cosmic strings may exist within the galaxy. They can be revealed through the optical lensing of background stars in the near future and studied in detail through gravitational wave emission. We anticipate that these cosmic strings will permit us to address central questions about the properties of string theory as well as the birth of our universe. Keywords: Inflation; string theory and cosmology; cosmic strings; early universe; gravitational microlensing. PACS Number(s): 98.80.Bp, 98.80.Cq, 98.80.Es, 11.25.−w, 11.27.+d

1. Introduction By 1930, quantum mechanics, general relativity and the Hubble expansion of our universe were generally accepted by the scientific community as three triumphs of modern physical science. Particles and fields obey quantum mechanical rules and II-273

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spacetime bends and warps according to Einstein’s classical description of gravity. The dynamical arena for particles, fields and spacetime is the universe and modern cosmology is born as the application of these physical laws to the universe itself. The path forward seemed clear but in the 1950s and 1960s physicists first began to appreciate that trying to quantize the classical gravitational field led to severe inconsistencies. This nonrenormalizability problem was further sharpened once the quantization of gauge (vector) fields was well understood in early 1970s. At about the same time, string theory — a quantum mechanical theory of onedimensional objects — was found to contain a graviton (a massless spin-2 particle). The promise of a self-consistent theory of quantum gravity has attracted huge attention since the 1980s. Tremendous progress has been made in the past 30 years in understanding many aspects of string theory; yet today we are no closer to knowing which of many theoretical possibilities might describe our universe and we still lack experimental evidence that the theory describes nature. By now, string theory is such a large topic with so many research directions that we have to choose a specific area to discuss. The centenary of general relativity prompts us to reflect on two very amazing manifestations of general relativity: namely black holes and cosmology. Given the tremendous progress in cosmology in the past decades, both in observation and in theory, we shall focus here on the intersection of string theory and cosmology. Our fundamental understanding of low energy physics is grounded in quantum field theory and gravity but if we hope to work at the highest energies then we cannot escape the need for an ultraviolet completion to these theories. String the√ α
is the ory provides a suitable framework. The string scale, MS = 1/ α
where √ Regge slope, is expected to be below the reduced Planck scale MPl = 1/ 8πG  2.4 × 1018 GeV (where G is the Newton’s constant) but not too far from the grand unified scale MGUT  1016 GeV (where the proton mass is about 1 GeV, in units where c =
= 1). Particle physics properties at scales much lower than MS are hard to calculate within our present understanding of string theory. High energy experiments can reach only multi-TeV scales, orders of magnitude below the expected MS . The difficulty of finding a common arena to compare theory and observation is hardly a new dilemma: although standard quantum mechanics works well at and above eV energy scales, it is woefully inadequate to describe properties of proteins which typically involve milli-eV scales and lower. Where should we look? One place to look for stringy effects is in the early universe. If the energy scale of the early universe reaches MS , an abundance of physical effects will provide a more direct path to test our ideas than currently feasible in high energy experiments. In fact, attempts to interpret observations will tell us whether our present understanding of physics and the universe is advanced enough to permit us to ask and answer sensible questions about these remote physical regimes. We shall focus on the inflationary universe scenario,1–3 which reaches energy scales sufficiently high to create ample remnants. The hot big bang’s nucleosynthetic epoch has direct observational support at and below MeV scales whereas the energies of interest to

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us are ∼ MS . We are most fortunate that precision cosmology has begun to let us address questions relevant to these extreme conditions. Even in this rather specialized topic of inflationary cosmology and string theory, a comprehensive review by Baumann and McAllister4 appeared recently. So we shall take the opportunity here to present a brief introduction and share within this framework some of our own views on a particularly interesting topic, namely, cosmic strings. The subject of cosmic strings has been extensively studied too,5 so here we shall focus on low tension cosmic strings, which may appear naturally in string theory. Since current gravitational observations are insensitive to quantum effects, one may wonder why an ultraviolet completion of the gravity theory is needed if inflation, in fact, occurs several orders of magnitude below the Planck scale. To partly answer this, we like to recall that in 1920s, there was no obvious necessity for a quantum theory consistent with special relativity given that electrons in atoms and materials move at nonrelativistic speed only. However, the Dirac theory predicted anti-matter, the value of the electron’s gyromagnetic moment and a host of interesting properties (e.g. the Lamb shift) that are well within reach of the nonrelativistic study of electrons and atoms. The standard electroweak model, the ultraviolet completion of the Fermi’s weak interaction model, is another example. Among other successes, the standard electroweak model predicted the presence of the neutral current as well as explains the π 0 decay to two photons via anomaly and the basis of the quark–lepton family via the anomaly free constraints. All these phenomena are manifest at energy scales far below the electroweak scale. In the current context, we need to search for stringy effects that are unexpected in classical general relativity and might be observed in the near future at energy scales far below the Planck scale. There are many specific ways to realize inflation within string theory. These may be roughly grouped into models with small and large field ranges. They have rather different properties and predictions. Recall that 6 of the 9 spatial dimensions must be compactified into a Calabi–Yau like manifold with volume V6 , where
2 MPl ∼ MS6 V6  1. (1) MS During inflation the inflaton field φ moves from an initial to final position within the manifold, covering a path or field range ∆φ. Now, the distance scale inside the compactified manifold is bounded by MPl . The case ∆φ < MPl is known as small field inflation. The opposite limit when the field range far exceeds the typical scale of the manifold, ∆φ  MPl , is large field inflation. In the latter case the inflaton wanders inside the compactified manifold for a while during inflation. This scenario may occur if the inflaton is an axion-like field and executes a helical-like motion. Flux compactification of the six spatial dimensions in string theory typically introduces many axions and naturally sets the stage for this possibility. The recent B-mode polarization measurement in the cosmic microwave background (CMB) radiation6 by BICEP2 is most interesting. Primordial B-mode

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polarization is sourced by tensor perturbations during inflation and measurable primordial B-mode polarization implies that ∆φ  MPl . This is a prototypical example of how it is now possible to probe high energies and early moments of the universe’s history by means of cosmological observations. However, since dust can contaminate today’s observed signal,7,8 it remains to be seen whether the BICEP2 observations definitively imply that large field inflation takes place. We believe this issue should be settled soon. In the meantime, we shall entertain both possibilities here. In both small and large field ranges macroscopic, one-dimensional objects, hereafter cosmic strings, can appear rather naturally. In the former case, the cosmic strings can be the fundamental superstrings themselves, while in the latter case, they can be fundamental strings and/or vortices resulting from the Higgs like mechanism. Cosmic strings with tension above the inflation scale will not be produced after inflation and we should expect only relatively low tension strings to be produced. Because of the warped geometry and the presence of throats in the compactification in string theory, some cosmic strings can acquire very low tensions. In flux compactification such strings tend to have nontrivial tension spectrum (and maybe even with beads). If the cosmic string tension is close to the present observational bound (about Gµ < 10−7 ), strings might contribute to the B-mode polarization at large  multipoles independent of whether or not BICEP2 has detected primordial B mode signals. The string contribution to the CMB power spectrum of temperature fluctuations is limited to be no more than ∼ 10% and this constrains any string B-mode contribution. Because the primordial tensor perturbation from inflation decreases quickly for large  multipoles one very informative possibility is that cosmic strings are detected at large  and other astrophysical contributions are sub-dominant. On the other hand, if tension is low (Gµ  10−7 ) then the cosmic string effect for all  is negligible compared to known sources and foreground contributions. Other signatures must be sought. For small Gµ string loops are long lived and the oldest (and smallest) tend to cluster in our galaxy, resulting in an enhancement of ∼ 105 in the local string density. This enhancement opens up particularly promising avenues for detecting strings by microlensing of stars within the galaxy and by gravitational wave emission in the tension range 10−14 < Gµ  10−7 . As a cosmic string passes in front of a star, its brightness typically doubles, as the two images cannot be resolved. As a string loop oscillates in front of a star, it generates a unique signature of repeated achromatic brightness doubling. Here, we briefly review the various observational bounds on cosmic strings and estimate the low tension cosmic string density within our galaxy as well as the likelihood of their detection in the upcoming observational searches. It is encouraging that searches of extrasolar planets and variables stars also offer a chance to detect the micro-lensing of stars by cosmic strings. Once a location is identified by such a detection, a search for gravitational wave signals should follow.

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Detection of cosmic strings followed by the measurement of their possible different tensions will go a long way in probing superstring theory. Although a single string tension can easily originate from standard field theory, a string tension spectrum should be considered as a distinct signature of string theory. It is even possible that some cosmic strings will move in the compactified dimensions with warped geometry, which can show up observationally as strings with varying tension, both along their lengths as well as in time. In summary, cosmic strings probably offer the best chance of finding distinct observational support for the string theory.

2. The Inflationary Universe So far, observational data agrees well with the simplest version of the slow-roll inflationary universe scenario, i.e. a single, almost homogeneous and isotropic, scalar inflaton field φ = φ(t) subject to potential V (φ). For a Friedmann–Lemaˆıtre– Robertson–Walker metric, general relativity yields schematic simple equations for the cosmic scale factor a(t) and for φ,  
2 a˙ 1 φ˙ 2 ρc (0) ρm (0) ρr (0) = V (φ) + + 2 + H = + + ···, 2 a 3MPl 2 a a3 a4 2

dV φ¨ + 3H φ˙ = − = −V
(φ), dφ

(2)

(3)

where the Hubble parameter H = a/a ˙ (a(t) ˙ = da/dt) measures the rate of expansion of the universe, and ρc (0), ρm (0) and ρr (0) are the curvature, the nonrelativistic matter and the radiation densities at time t = 0.a In an expanding universe, a(t) grows and the curvature, the matter and the radiation terms diminish. The energy and pressure densities for the inflaton are ρφ =

φ˙ 2 + V (φ), 2

pφ =

φ˙ 2 − V (φ), 2

(4)

where a canonical kinetic term for φ is assumed. If V (φ) is sufficiently flat and if φ˙ is initially small then Eq. (3) implies that φ moves slowly in the sense that its kinetic energy remains small compared to the potential energy, φ˙ 2  V (φ). In the limit that V (φ) is exactly constant and dominates in Eq. (2) then H is constant and a(t) ∝ eHt . More precisely one may define the inflationary epoch as that period when the expansion of the universe is accelerating, i.e. a ¨ > 0, or, in a spatially flat energy density has been divided into ρm and ρr . The ultra-relativistic massive particles are lumped into ρr and the marginally relativistic massive particles included in ρm . This division is epoch-dependent and schematic. a The

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universe, 2 ˙ H = −(ρ + p), 2MPl

=−


 H˙ 3 p = 1 + H2 2 ρ

(5)

< 1. Slow-roll inflation means small , H nearly constant and a(t) ∼ a(0)eHt . In a typical slow-roll inflationary model inflation ends at t = tend when the inflaton encounters a steeper part of the potential and  > 1. The number of e-folds of inflation is Ne  Htend , a key parameter. The energy released from the potential V (φ) heats the universe at the end of inflation and starts the hot big bang. The period of slow-roll must last at least Ne > 50 e-folds to explain three important observations about our universe that are otherwise unaccounted for in the normal big bang cosmology: flatness, lack of defects and homogeneity. For any reasonable initial curvature density ρc (0), the final curvature density ρc (tend ) < ρc (0)e−150 will be totally negligible, thus yielding a flat universe. This is how inflation solves the flatness problem. Any defect density (probably included in ρm (0)) present in the universe before the inflationary epoch will also be inflated away, thus solving the so-called “monopole” or defect problem. Since the cosmic scale factor grows by a huge factor, the universe we inhabit today came from a tiny patch of the universe before inflation. Any original inhomogeneity will be inflated away. Inflation explains the high degree of homogeneity of the universe. In summary, if Ne is sufficiently large then inflation accounts for the universe’s observed flatness, defect density and homogeneity. What is amazing is that inflation also automatically provides a mechanism to create primordial inhomogeneities that ultimately lead to structure formation in our universe. As the inflaton slowly rolls down the potential in the classical sense, quantum fluctuations yield slightly different ending times tend and so slightly different densities in different regions. The scalar and the metric fluctuations may be treated perturbatively, and one obtains the dimensionless power spectra in terms of H, ∆2S (k) =

1 H2 2 || , 8π 2 MPl

∆2T (k) =

2 H2 2 2 = r∆S (k), π 2 MPl

(6)

r = 16, where ∆2S and ∆2T are the scalar and the tensor modes, respectively. A scalar scaleinvariant power spectrum corresponds to constant ∆2S , which occurs when a space expands in a nearly de-Sitter fashion for a finite length of time. The scalar mode is related to the temperature fluctuations first measured by COBE.9 Measurements and modeling have been refined over the past two decades. Since φ rolls in the

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nonflat potential in the inflationary scenario, ∆2S will have a slight k dependence. This is usually parametrized with respect to the pivot wave number kp in the form
∆2S (k) = ∆2S (kp )
∆2T (k)

=

∆2T (kp )

k kp k kp

ns −1+(dns /d ln k) ln(k/kp )/2+···, (7) nt +(dnt /d ln k) ln(k/kp )/2+···, (8)

where the PLANCK and WMAP best fit values for ΛCDM quoted in Ref. 10 for kp = 0.05 Mpc−1 are ns = 0.9603 ± 0.0073, +0.58 × 10−9 . ∆2S (kp ) = 2.19−0.53

(9) (10)

These do not change appreciably when a possible tensor component is included and PLANCK constraints on r are given at pivot 0.002 Mpc−1 . In the slow-roll approximation, where φ¨ is negligible in Eq. (3), the deviation from scale-invariance is quantified by the spectral tilt ns − 1 =

d ln ∆2S (k) = −6 + 2η, d ln k

dns = −16η + 242 + 2ξ 2 , d ln k

(11)

where the parameters that measure the deviation from flatness of the potential are
2 M2 V
 = Pl , (12) 2 V η=

2 V

MPl , V

ξ2 =

4 V
V


MPl . V2

(13)

where  and η are the known as the slow-roll parameters which take small values during the inflationary epoch. The tensor mode comes from the quantum fluctuation of the gravitational wave. Its corresponding tilt is nt = −2.

(14)

For a small field range ∆φ, the constraint on Ne dictates a rather flat potential and small , which in turn implies a very small r. This relation can be quantified by the Lyth bound,11  ∆φ r ≥ Ne . (15) MPl 8 If 60 ≥ Ne ≥ 40, we find that typical values of r satisfies r < 0.005 for ∆φ < MPl . This is much smaller than r  0.2 reported by BICEP2.6

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While ∆2S is a combination of H 2 and 1/, ∆2T provides a direct measurement of H 2 and hence the magnitude of the inflaton potential during inflation:  r  V  (2.2 × 1016 GeV)4 . (16) 0.2 The imprint of tensor fluctuations are present in the CMB but ∆2T  ∆2S so it is not possible to measure ∆2T directly from total temperature fluctuations at small .b Fortunately, the CMB is linearly polarized and can be separated into E-mode and B-mode polarizations. It happens that ∆2T contributes to both modes while ∆2S contributes only to the E-mode. So a measurement of the primordial B-mode polarized CMB radiation is a direct measurement of ∆2T . Searching for the B-mode CMB is very important. Besides the intrinsic smallness of the B-mode signal, which makes detection a major challenge, the primordial fluctuation may be masked by the interstellar dust. It is likely that the uncertainty from dust would have been resolved by the time this paper appears. So we may consider this as a snapshot after the announcement of the BICEP2 data and before a full understanding of the impact of dust on the reported detection. Due to this uncertainty, we shall consider both a negligibly small r (say r < 0.002) and an observable primordial r ≤ 0.2. 3. String Theory and Inflation By now, string theory is a huge research subject. So far, we have not yet identified the corner where a specific string theory solution fits what happens in nature. It is controversial to state whether we are close to finding that solution or we are still way off. By emphasizing the cosmological epoch, we hope to avoid the details but try to find generic stringy features that may show up in cosmological observations. Here we give a lightning pictorial summary of some of the key features of Type IIB superstring theory so readers can have at least a sketchy picture of how string theory may be tested. Because of compactification, we expect modes such as Kaluza–Klein modes to appear in the effective four-dimensional theory. Their presence will alter the relation between the inflaton potential scale and r (16) to: 1  r  (2.2 × 1016 GeV)4 , (17) V  ˆ 2 0.2 N tensor and scalar perturbations have different  dependencies experimentalists deduce the scalar spectrum from high  measurements where scalar power will dominate any tensor contribution. Then they observe the total temperature fluctations at low  where tensors should make their largest impact. If the theoretical relation between low and high  fluctuations is known (“no running” being the simplest possibility) then the total observed low  power limits whatever extra contribution might arise from the tensor modes. The WMAP12 and Planck13 collaborations (incorporating data from the South Pole Telescope (SPT) and Atacama Cosmology Telescope (ACT) microwave background experiments and baryon acoustic oscillation observations) placed limits on r, r < 0.13 and r < 0.11, respectively. These upper limits on r are somewhat less than the r values reported by BICEP2 and under investigation.14 b Since

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ˆ effectively counts the number of universally coupled (at one-loop) degrees where N of freedom below the energy scale of interest here.15 Present experimental bound ˆ is very loose. Surprisingly, a value as big as N ˆ ∼ 1025 is not ruled out. on N 3.1. String theory and flux compactification Here is a brief description on how all moduli are dynamically stabilized in flux compactification. Recall that 2-form electric and magnetic field strengths follow from the 1-form field Aµ in the electromagnetic theory, under which point-like particles (such as electrons) are charged. In analogy to this, fundamental strings in string theory are charged under a 2-form field Bµν which yields 3-form field strengths. In general, other dimensional objects are also present in string theory, which are known as branes. A p-brane spans p spatial dimensions, so a membrane is a 2-brane while a 1-brane is string like; that is a 1-brane is really a string. In the Type IIB superstring theory, there are a special type of Dp-branes where p is an odd integer. Among other properties of Dp-branes, we like to mention two particularly relevant ones here : (1) each end of an open string must end on a Dp-brane, and (2) with the presence of D1-string (or -brane), there exists another 2-form field Cµν under which a D1-string is charged. Self-consistency of Type IIB superstring theory requires it to have nine spatial dimensions. Since only three of them describe our observable world, the other six must be compactified. Consider a stack of D3-branes of cosmological size. Such a stack appear as a point in the six compactified dimensions. In the brane world scenario, all standard model particles (electrons, quarks, gluons, photons, etc.) are light open string modes whose ends can move freely inside the D3-branes, but cannot move outside the branes, while the graviton, being a closed string mode, can move freely in the branes as well as outside, i.e. the bulk region of the six compactified dimensions. In this sense, the D3-branes span our observable universe. We can easily replace the D3-branes by D7-branes, with 4 of their dimensions wrapping a 4-cycle inside the compactified six-dimensional space. Dark matter may come from unknown particles inside the same stack, or from open string modes sitting in another stack sitting somewhere else in the compactified space. As a self-consistent theory, the six extra dimensions must be dynamically compactified (and stabilized). This is a highly nontrivial problem. Fortunately, the 3-form field strengths of both the NS–NS field Bµν and the R–R field Cµν are quantized in string theory. Wrapping 3-cycles in the compactified dimensions, these fluxes contribute to the effective potential V in the low energy approximation. Their presence can render the shape and the size of the compactified six-dimensional space to be dynamically stabilized. This is known as flux compactification. Here the matter content and the forces of nature are dictated by the specific flux compactification.16,17 Warped internal space appears naturally. This warped geometry will come to play an important role in cosmology.

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To describe nature, a Calabi–Yau like manifold is expected, with branes and orientifold planes. At low energy, a particular manifold may be described by a set of dynamically stabilized scalar fields, or moduli. One or more K¨ ahler moduli parameterize the volume while the shape is described by the complex structure moduli, whose number may reach hundreds. A typical flux compactification involves many moduli and three-form field strengths with quantized fluxes (see Ref. 18). With such a large set of dynamical ingredients, we expect many possible vacuum solutions; collectively, this is the string theory landscape, or the so-called cosmic landscape. Here, a modulus is a complex scalar field. Written in polar coordinate, we shall refer to the phase degree of freedom as an axion. For a given Calabi–Yau like manifold, we can, at least in principle, determine the four-dimensional low energy supergravity effective potential V for the vacua. To be specific, let us consider only 3-form field strengths F3i wrapping the threecycles inside the manifold. (Note that these are dual to the four-form field strengths in four-dimensional spacetime.) We have V (F3i , φj ) → V (ni , φj ), (i = 1, 2, . . . , N , j = 1, 2, . . . , K) where the flux quantization property of the 3-form field strengths F3i allow us to rewrite V as a function of the quantized values ni of the fluxes present and φj are the complex moduli describing the size and shape of the compactified manifold as well as the couplings. There are barriers between different sets of flux values. For example, there is a (finite height) barrier between n1 and n1 − 1, where tunneling between V (n1 , n2 , . . . , nN , φj ) and V (n1 − 1, n2 , . . . , nN , φj ) may be achieved by brane-flux annihilation.19 For a given set of ni , we can locate the meta-stable (classically stable) vacuum solutions V (ni , φj ) by varying φj . We sift through these local minima which satisfy the following criteria: they have vanishingly small vacuum energies because that is what is observed in today’s universe, and long decay lifetimes to lower energy states because our universe is long-lived. These criteria restrict the manifolds, flux values and minima of interest; nonetheless, within the rather crude approximation we are studying, there still remain many solutions; and statistically, it seems that a very small cosmological constant is preferred.20 3.2. Inflation in string theory When the universe was first created (say, a bubble created via tunneling from nothing), φj are typically not sitting at φj,min , so they tend to roll toward their stabilized values. Heavier moduli with steeper gradients probably reached their respective minima relatively quickly. The ones with less steep directions took longer. The last ones to reach their stabilized values typically would move along relatively flat directions, and they can play the role of inflatons. So the vacuum energy that drives inflation is roughly given by the potential when all moduli except the inflaton have already reached bottom. Since the flux compactification in string theory introduces dozens or hundreds of moduli (each is a complex scalar field in the low energy effective theory approximation), one anticipates that there are many

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candidates for inflation. Even if we assume that just one field is responsible for the inflationary epoch in our own universe it seems likely that string theory can choose among many possibilities to realize single field inflation in many different ways. In fact, the picture from string theory may offer additional interesting possibilities beyond field theory. For example, suppose there was a pair of brane–anti-brane present in the early universe whose tensions drive the inflation. As we shall see, the inflaton field happens to be the distance between them. After their annihilation that ends inflation, the inflaton field no longer exists as a degree of freedom in the low energy effective field theory. Some inflationary scenarios in string theory generate unobservably small primordial r while others give observably large primordial r. Small field range implies small r, but large field range does not necessary imply large r, which happens when the Lyth bound (15) is not saturated. To obtain enough e-folds, we need a flat enough potential, so it is natural to consider an axion as the inflaton, as first proposed in natural inflation.21,22 The axion has a natural shift symmetry φ → φ + any constant. This continuous symmetry can easily be broken, via nonperturbative effects, to a discrete symmetry, resulting in a periodic potential that can drive inflation. The typical field range is sub-Planckian implying a small r. We shall see shortly that there are many string theory models built out of the axion that permit large field ranges and some of these can yield large r. Here, we shall describe the various string theory realizations of inflation within the framework of Type IIB string theory. The pictorially simplest models occur in the brane world scenario. We shall present a few sample models to give a picture of the type of models that have been put forward. Readers can find a more complete list in Ref. 4. If r turns out to be large, it will be interesting to see whether some of the small r models can be adapted or modified to have a large r. For example, one can try to modify a brane inflationary scenario into a warm inflationary scenario with a relatively large r. 4. Small r Scenarios Besides fundamental superstrings, Type IIB string theory has Dp-branes where the number of spatial dimensions p is odd. Furthermore, supersymmetry can be maintained if there are only D3- and D7-branes; so, unless specified otherwise, we shall restrict ourselves to this case. String theory has nine spatial dimensions. Since our observable universe has only three spatial dimensions, the other six spatial dimensions have to be compacted in a manifold with volume V6 . The resulting (i.e. dynamically derived) MPl is related to the string scale MS via Eq. (1) and the typical field range ∆φ is likely to be limited by MPl > MS . For such a small field range, the potential has to be flat enough to allow 50 or more e-folds and the Lyth bound (15) for 60 ≥ Ne ≥ 40 implies r < 0.005. We shall refer to these models as small r scenarios.

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4.1. Brane inflation The discovery of branes in string theory demonstrated that the theory encompasses a multiplicity of higher-dimensional, extended objects and not just strings. In the brane world scenario, our visible universe lies inside a stack of D3-branes, or a stack of D7-branes. Here, 6 of the 9 spatial dimensions are dynamically compactified while the three spatial dimensions of the D3-branes (or 3 of the D7-branes) are cosmologically large. The six small dimensions are stabilized via flux compactification16,17 ; the region outside the branes is referred to as the bulk region. The presence of RR and NS–NS fluxes introduces intrinsic torsion and warped geometry, so there are regions in the bulk with warped throats (Fig. 1). Since each end of an open string must end on a brane, only closed strings are present in the bulk away from branes. There are numerous such solutions in string theory, some with a small positive vacuum energy (cosmological constant). Presumably the standard model particles are open string modes; they can live either on D7-branes wrapping a 4-cycle in the bulk or (anti-)D3-branes at the bottom of a warped throat (Fig. 1). The (relative) position of a brane is an open string mode. In brane inflation,23 one of these modes is identified as the inflaton φ, while the inflaton potential is generated by the classical exchange of closed string modes including the graviton. From the open string perspective, this exchange of a closed string mode can be viewed as a quantum loop effect of the open string modes.

Fig. 1. A pictorial sketch of a generic flux compactified six-dimensional bulk, with a number of warped throats (4 of which are shown here). Besides the warped throats, there are D7-branes ¯ wrapping 4-cycles. The blue dots stand for mobile D3-branes while the red dots are D3-branes ¯ sitting at the bottoms of throats. In the D3-D3-brane inflationary scenario, the tension of the brane pairs provide the vacuum energy that drives inflation as the D3-brane moves down A¯ throat. Inflation ends as the D3-brane annihilates with the D3-brane in A-throat. The standard model branes may live in A-throat or S-throat. As an alternative, inflation may take place while branes are moving out of B-throat. (For color version, see page II-CP14.)

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¯ 4.1.1. D3-D3-brane inflation In the early universe, besides all the branes that are present today, there is an extra 24,25 ¯ ¯ pair of D3-D3-branes. Due to the attractive forces present, the D3-brane is expected to sit at the bottom of a throat. Here again, inflation takes place as the D3¯ brane moves down the throat toward the D3-brane, with their separation distance as the inflaton, and inflation ends when they collide and annihilate each other, allowing the universe to settle down to the string vacuum state that describes our universe today. Although the original version encounters some fine-tuning problems, the scenario becomes substantially better as we make it more realistic with the introduction of warped geometry.26 Because of the warped geometry, a consequence of flux compactification, a mass M in the bulk becomes hA M at the bottom of a warped throat, where hA  1 is the warped factor (Fig. 1). This warped geometry tends to flatten, by orders of ¯ magnitude, the inflaton potential V (φ), so the attractive D3-D3-brane potential is rendered exponentially weak in the warped throat. The attractive gravitational (plus RR) potential together with the brane tensions takes the form
 1 φ4A , (18) V (φ) = 2T3 h4A 1 − NA φ4 where T3 is the D3-brane tension and the effective tension is warped to a very small value T3 h4A . The warp factor hA depends on the details of the throat. Crudely, h(φ) ∼ φ/φedge , where φ = φedge when the D3-brane is at the edge of the throat, so h(φedge )  1. At the bottom of the throat, where φ = φA  φedge , hA = h(φA ) = φA /φedge . The potential is further warped because NA  1 is the D3 charge of the ¯ throat. The D3-D3-brane pair annihilates at φ = φA . In terms of the potential (18), a tachyon appears as φ → φA so inflation ends as in hybrid inflation. The energy released by the brane pair annihilation heats up the universe to start the hot big bang. To fit data, hA ∼ 10−2 . If the last 60 e-folds of inflation takes place inside the throat, then φedge ≥ φ ≥ φA during this period of inflation. This simple model yields ns  0.97, r < 10−5 and vanishing non-Gaussianity. The above model is very simple and well motivated. There are a number of interesting variations that one may consider. Within the inflaton potential, we can add new features to it. In general, we may expect an additional term in V (φ) of the form βH 2 φ2 , 2 where H is the Hubble parameter so this interaction term behaves like a conformal coupling. Such a term can emerge in a number of different ways: • Contributions from the K¨ ahler potential and various interactions in the superpo26 tential as well as possible D-terms,27 so β may probe the structure of the flux compactification.28,29 • It can come from the finite temperature effect. Recall that finite temperature T induces a term of the form T 2 φ2 in finite temperature field theory. In a de-Sitter

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universe, there is a Hawking–Gibbons temperature of order H thus inducing a term of the form H 2 φ2 . ¯ • The D3-brane is attracted to the D3-brane because of the RR charge and the gravitational force. Since the compactified manifold has no boundary, the total RR charge inside must be zero, and so is the gravitational “charge.” A term of the above form appears if we introduce a smooth background “charge” to cancel the gravitational “charges” of the branes.30 On the other hand, negative tension is introduced via orientifold planes in the brane world scenario. If the throat is far enough away from the orientifold planes, it is reasonable to ignore this effect. Overall, β is a free parameter and so is naively expected to be of order unity, β ∼ 1. However, the above potential yields enough inflation only if β is small enough, β < 1/5.31 The present PLANCK data implies that β is essentially zero or even slightly negative. Another variation is to notice that the six-dimensional throat can be quite nontrivial. In particular, if one treats the throat geometry as a Klebanov–Strassler deformed conifold, gauge-gravity duality leads to the expectation that φ will encounter steps as the D3-brane moves down the throat.32 Such steps, though small, may be observed in the CMB power spectrum. 4.1.2. Inflection point inflation Since the six-dimensional throat has one radial and five angular modes, V (φ) gets corrections that can have angular dependencies. As a result, its motion toward the bottom of the throat may follow a nontrivial path. One can easily imagine a situation where it will pass over an inflection point. Around the inflection point, V (φ) may take a generic simple form V (φ)  V0 + Aφ +

Bφ2 Cφ3 + , 2 3

where η = 0 at the inflection point B + 2Cφ = 0. As shown in Refs. 29 and 33, √ given that V (φ) is flat only around the inflection point, Ne ∼ 1/ , so  must be very small, resulting in a very small r. Here ns − 1 = 2η so we expect ns to be very close to unity, with a slight red or even blue tilt. Although motivated in brane inflation, an inflection point may be encountered in other scenarios of the inflationary universe, so it should not be considered as a stringy feature. 4.1.3. DBI model Instead of modifying the potential, string theory suggests that the kinetic term for a D-brane should take the Dirac–Born–Infeld form34,35  1 µ 1 1 ∂ φ∂µ φ → − , (19) 1 − f (φ)∂ µ φ∂µ φ + 2 f (φ) f (φ)

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where f (φ) ∼ h(φ)−4 ∼ A/φ4 is the warp factor of a throat. This DBI property is intrinsically a stringy feature. Here, the warp factor plays the role of a brake that slows down the motion of the D-brane as it moves down the throat, f (φ)(∂t φ)2 < 1. This braking mechanism is insensitive to the form of V (φ), so many e-folds are assured as φ → 0. It produces a negligibly small r but a large non-Gaussianity in the equilateral bi-spectrum as the sound speed becomes very small. The CMB data has an upper bound on the non-Gaussianity that clearly rules out the dominance of this stringy DBI effect. Nevertheless, it is a clear example that stringy features of inflationary scenarios may be tested directly by cosmological observation. Instead of moving down a throat, one may also consider inflation while a brane is moving out of a throat.36,37 In this case, the predictions are not too different from that of the inflection point case. 4.1.4. D3-D7-brane inflation Here, the inflaton is the position of a D3-brane moving relative to the position of a higher-dimensional D7-brane.38 Since the presence of both D3 and D7-branes preserves supersymmetry (as opposed to the presence of D5-branes), inflation can be driven by a D-term and a potential of the form like V (φ) = V0 + a ln φ − bφ2 + cφ4 + · · · may be generated. Such a potential yields ns  0.98, which may be a bit too big. One may lower the value a little by considering variations of the scenario. In any case, one ends up with a very small r. 4.2. K¨ ahler moduli inflation In flux compactification, the K¨ ahler moduli are typically lighter than the complex structure moduli. Intuitively, this implies their effective potentials are flatter than those for the complex structure moduli. For a Swiss-cheese like compactification, besides the modulus for the overall volume, we can also have moduli describing the sizes of the holes inside the manifold. So it is reasonable to find situations where a K¨ ahler modulus plays the role of the inflaton. A potential is typically generated by nonperturbative effects, so examples may take the form39 V  V0 (1 − Ae−kφ ),

(20) 2 −kφ

< where both A and k are positive and of order unity or bigger. Here, η  −Ak e 0 and   η 2 /2k 2 , so the potential is very flat for large enough φ. If the inflaton measures the volume of a blow-up mode corresponding to the size of a 4-cycle √ in a Swiss-Cheese compactification in the large volume scenario, we have k ∼ V ln V, where the compactification volume is of order V ∼ 106 in string units, so k is huge and r  2(ns − 1)2 /k 2 ∼ 10−10 . Models of this type are not close to saturating

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the Lyth bound (15). Other scenarios40 of this type again have small r. It will be interesting whether this type of models can be modified to have a larger r. 5. Large r Scenarios The Lyth bound (15) implies that large r requires large inflaton field range ∆φ  MPl . A phenomenological model with relatively flat potential and large field range is not too hard to write down, e.g. chaotic inflation. However, the range of a typical modulus in string theory is limited by the size of compactification, ∆φ < MPl . Even if we could extend the field range (e.g. by considering an irregular shaped manifold), the corrections to a generic potential may grow large as φ explores a large range. Essentially, we lose control of the approximate description of the potential. Axions allow one to maintain control of the approximation used while exploring large ranges. Here we shall briefly review two ideas, namely the Kim–Nilles–Peloso Mechanism41 and the axion monodromy.42,43 In both cases, an axion moves in a helicallike path. The flatness of the potential and the large field range appear naturally. 5.1. The Kim–Nilles–Peloso mechanism Let us start with a simple model and then build up to a model that is relatively satisfactory. 5.1.1. Natural inflation It was noticed long ago that axion fields may be ideal inflaton candidates because an axion field φ, a pseudo-scalar mode, has a shift symmetry, φ → φ + const. This symmetry is broken to a discrete symmetry by some nonperturbative effect so that a periodic potential is generated,21,22

 φ m2 2 A 2 V (φ) = A 1 − cos φ + · · · ∼ φ , → (21) f 2f 2 2 where we have set the minima of the potential at φ = 0 to zero vacuum energy. With suitable choice of A and f the resulting axion potential can be relatively small and flat, an important property for inflation. As f becomes large, this model approaches the (quadratic form) chaotic inflation,44 which is well studied. To have enough e-folds, we may need a large field range, say, ∆φ > 14 MPl, which is possible here only if the decay constant f > ∆φ  MPl . This requires a certain degree of fine tuning since a typical f is expected to satisfy f < MPl (see, for example, Fig. 2 in Ref. 45). To fit the scalar mode perturbations of COBE m  7 × 10−6 MPl . The range of predictions45 in the r versus ns plot is shown in Fig. 2. 5.1.2. N-flation There are ways to get around this to generate enough e-folds. One example is to extend the model to include N different axions (similar to the idea of using many

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Fig. 2. The tensor-to-scalar ratio r versus the primordial tilt ns plot for pivot scale 0.002 Mpc−1 . Pairs of solid and dashed semi-circular lines encompass 68% and 95% confidence limits for r and ns as given in Ref. 10. The regions are based on PLANCK data combined with: WMAP [large scale polarization] (gray), WMAP plus BAO [baryon acoustic oscillations in galaxy surveys] (blue) and WMAP plus higher  CMB data [ACT and SPT] (red) as described in detail in Ref. 10. All remaining lines, points and arrows describe theoretical models. Convex potentials lie above the straight, solid black line; concave potentials below it. Power law inflation generates results along the straight, dashed line. String-related models include the short yellow segment for a linear inflaton potential V ∼ φ, the short red segment for V ∼ φ2/3 and the short black segment (chaotic inflation) for V ∼ φ2 . Each small (large) dot stands for 50 (60) e-folds. Two purple lines moving to the left show the full range of predictions for natural inflation. All small field models and ˆ some large field ones suppress r (these are not plotted). The black arrow is the vertical shift ∆, illustrating how the results for cosine potential differ from those of the φ2 potential. (For color version, see page II-CP14.)

scalar fields46,47 ) each with a term of the form in Eq. (21), with a different decay constant fi < MPl . This N-flation model48 is a string theory inspired scenario since flux compactification results in the presence of many axions. In the approximation that the axions are independent of each other (that is, their couplings with each other are negligible), the analysis is quite straightforward and interesting.49 The predictions are similar to that of a large field model, that is, 0.93 < ns < 0.95 but r ≤ 10−3 . In general, the axions do couple to each other and the situation can be quite complicated. For a particularly simple case, inspired by string theory and supergravity, a statistical analysis has been carried out and clear predictions can be made.49 For a large number of axions, one has

 φi φi φj αi cos + V (φi ) = V0 + βij cos − . (22) fi fi fj i i,j The statistical distribution of results yields small values for r quite similar to that of the N-flation model.

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5.1.3. Helical inflation Let us consider a particularly interesting case of a 2-axion model of the form41


 
 φ1 φ1 φ2 V (φ1 , φ2 ) = V1 + V2 = V0 1 − cos + A 1 − cos − , (23) f1 f1
f2 where we take f1
 f1  MPl . Note that the second term in V (23), namely V2 , vanishes at φ1 φ2 − = 0. f1
f2

(24)

This is the minimum of V2 and the bottom of the trough of the potential V . For f1 large enough A, inflaton will follow the trough as it rolls along its path. Now 2πf
1 measures the number of cycles that φ2 can travel for 0 < φ1 < f1 . Although φ2 has a shift symmetry φ2 → φ2 + 2πf2 , the path does not return to the same configuration after φ2 has traveled for one period because φ1 has also moved. Thus, instead of a shift symmetry, the system has a helical symmetry. Moving along the path of the trough (24), we see that the second term in the potential (23) vanishes and so the potential V reduces to that with only the first term and the range of the inflaton field φ2 can easily be super-Planckian. To properly normalize the fields, one defines two normalized orthogonal directions, f1
φ1 + f2 φ2 X=  , f1
2 + f22

f2 φ1 − f
φ2 Y = 
2 1 2 , f1 + f2

(25)

the inflaton will roll along the X-direction while Y -direction is a heavy mode which can be integrated out. Note that the system is insensitive to the magnitude of A as long as it is greater than O(1) such that Y -direction is heavy enough. For instance, the slow roll parameters  and η are not affected which means that the observables ns and r are insensitive to A. Since the X path is already at the minimum of the second term, at Y = 0, the effective potential along X is determined by V1 ,


 X cos θ 1 V0 X 2 V (X) = V0 1 − cos ≈ cos2 θ, (26) f1 2 f12 cos θ ≡ 

f1
f1
2 + f22

.

(27)

So this helical model is reduced to the single cosine model (21) where
 f1 f1 f2 f= f1
2 + f22 
f1
f1 can be bigger than MPl even if all the individual fi < MPl . So it is not difficult to come up with stringy models that can fit the existing data. In view of the large B-mode reported by BICEP2,6 this model was revisited by a number of groups.50–57 In fact, one can consider a more general form for V1 (φ1 ),58 which can lead to a large field range model with somewhat different predictions.

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Since the shift symmetry of an axion is typically broken down to some discrete symmetry, the above cosine model is quite natural. The generalization to more than two axions is straightforward, and one can pile additional helical motions on top of this one, increasing the value of the effective decay constant f by additional big factors. All models in this class reduce to a model with a single cosine potential, which in turn resembles the quadratic version of chaotic inflation. This limiting behavior is quite natural in a large class of axionic models in the supergravity framework.59 Because of the periodic nature of an axionic potential, this cosine model can have a smaller value of r than chaotic inflation.44 The deviations from chaotic inflation occur in a well-defined fashion. Let 2 ˆ = 16∆ = r + 4(ns − 1) = −4MPl , ∆ (28) f2 ˆ = 0 while the cosine model has ∆ ˆ < 0. All other where φ2 chaotic inflation has ∆ quantities such as runnings of spectral indices have very simple dependencies on ˆ As shown in Table 1, this deviation is quite distinctive of the periodic nature of ∆. the inflaton potential. For ns = 0.96, r = 0.16 in the φ2 model, while r can be as ˆ = −0.12) in the cosine model. As data improves, a negative small as r = 0.04 (or ∆ ˆ can provide a distinctive signature for a periodic axionic potential for value of ∆ inflation. 5.2. Axion monodromy The above idea of extending the effective axion range can be carried out with a single axion, where the axion itself executes a helical motion. This is the axion Table 1. Comparison between φ2 chaotic inflation and the cosine model for the various physically measurable quantities51 where ∆ (28) measures the difference. Here MPl = 1.

V (φ) =

1 (V
/V )2 2

η = V

/V

1 2 2 m φ 2

)] V0 [1 − cos( φ f

1 r 16

1 r 16

1 r 16

1 r 16

+ 2∆

ξ 2 = V
V


/V 2

0

ω 3 = V
2 V



/V 3

0

ns − 1 = 2η − 6

− 41 r

− 14 r + 4∆

nt = −2

− 81 r

− 18 r

dns d ln k

= −16η + 242 + 2ξ 2

1 2 r 32

dnt d ln k

= −4η + 82

1 2 r 64

d2 ns d ln k2

= −1923 + 1922 η − 32η2

−24ξ 2 + 2ηξ 2 + 2ω 3

1 3 − 128 r

1 r∆ 2

1 2 r ∆ 32

1 2 r 32

1 2 r 64

+ r∆2

− r∆

− 12 r∆

1 3 − 128 r + 38 r 2 ∆ − 4r∆2

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monodromy model.42,43 In this scenario, inflation can persist through many periods around the configuration space, thus generating an effectively large field range with an observable r. Recall that a gauge field, or one-form field (i.e. with one spacetime index), is sourced by charged point-like fields, while a 2-form (anti-symmetric tensor) field is sourced by strings. So we see that there will be at least one 2-form field in string theory. Consider a 5-brane that fills our four-dimensional spacetime and wraps a 2-cycle inside the compactified manifold. The axion is the integral of a 2form field over the 2-cycle. Integrating over this 2-cycle, the six-dimensional brane action produces a potential for the axion field in the resulting four-dimensional effective theory. Here, the presence of the brane breaks the axion shift symmetry and generates a monodromy for the axion. For NS5-branes, a typical form of the axion potential (coming from the DBI action) is  (29) V (φ) = A b2 + φ2 . For small parameter b, V (φ)  Aφ. The prediction of such a linear potential is shown in Fig. 2. One can consider D5-branes instead. D7-branes wrapping 4-cycles in the compactified manifold is another possibility. For large values of φ, a good axion monodromy model requires that there is no uncontrollable higher-order stringy or quantum corrections that would spoil the above interesting properties.60–62 Variants of this picture may allow a more general form of the potential, say V (φ) ∼ φp where p can take values such as p = 2/3, 4/3, 2, 3, thus generating r  0.04, 0.09, 0.13 and 0.2, respectively. The p = 2 case predicts a ns value closest to r ∼ 0.16 reported by BICEP2 after accounting for dust. The observational bounds are currently under scrutiny. Readers are referred to Ref. 4 for more details. 5.3. Discussions If BICEP2’s detection of r is confirmed, it does not necessarily invalidate completely all the small r models discussed above. It may be possible to modify some of them ¯ to generate a large enough r. As an example, if one is willing to embed the D3-D3brane inflation into a warm inflationary model, a r  0.2 may be viable.63 It will be interesting to re-examine all the small r string theory models and see whether and how any of them may be modified to produce a large r. As string theory has numerous solutions, it is not surprising that there are multiple ways to realize the inflationary universe scenario. With cosmological data available today, theoretical predictions and contact with observations are so far quite limited. As a consequence, it is rather difficult to distinguish many string theory inspired predictions from those coming from ordinary field theory (or even supergravity models). There are exceptions, as pointed out earlier. For example, the DBI inflation prediction of an equilateral bi-spectrum in the non-Gaussianity, or the determined spacing of steps in the power spectrum itself, may be considered to be distinct enough that if either one is observed, some of us may be convinced that

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it is a smoking gun of string theory. Although searches for these phenomena should and would continue, so far, we have not been lucky enough to see any hint of them. Here we like to emphasize that there is another plausible signature to search for. Since all fundamental objects are made of superstrings (we include D1-strings here), and the universe is reheated to produce a hot big bang after inflation, it is likely that, besides strings in their lowest modes which appear as ordinary particles, some relatively long strings will also be produced, either via the Kibble mechanism or some other mechanism. They will appear as cosmic strings. When there are many axions, or axion-like fields, with a variety of plausible potentials, the possibilities may be quite numerous and so predictions may be somewhat imprecise. For any axion or would-be-Goldstone bosons, with a continuous U (1) symmetry, we expect a string-like defect which can end up as cosmic strings if generated in early universe. In field theory, a vortex simply follows from the Higgs mechanism where the axion a appears as the phase of a complex scalar field, Φ = ρeia/f . In string theory, we note that such an axion a is dual to a 2-form tensor field Cλκ , i.e. ∂ µ a = µνλκ ∂ν Cλκ . As pointed out earlier, such a 2-form field is sourced by a string. So the presence of axions would easily lead to cosmic strings (i.e. vortices, fundamental strings and D1-strings) and these may provide signatures of string theory scenarios for the inflationary universe. This is especially relevant if cosmic strings come in a variety of types with different tensions and maybe even with junctions. Following Fig. 1, we see that besides the axions responsible for inflation, there may be other axions. Some of them may have mass scales warped to very small values. If a potential of the form (21) is generated, a closed string loop becomes the boundary of a domain wall, or membrane. It will be interesting to study the effect of the membrane on the evolution of a cosmic string loop. The tension of the membrane and the axion mass are of order √ A σ ∼ Af, m2 = 2 . f It is interesting to entertain the possibility that this axion can contribute substantially to the dark matter of the universe. If so, its contribution to the energy density is roughly given by A while its mass is estimated to be m  10−22 eV.64 This yields √ −118 4 3 A  10 MPl , f  A/m  10−10 MPl and σ  10−69 MPl  10−14 GeV3 . For such a small membrane tension, the evolution of the corresponding cosmic string is probably not much changed. 6. Relics: Low Tension Cosmic Strings Witten’s65 original consideration of macroscopic cosmic strings was highly influential. He argued that the fundamental strings in heterotic string theory had tensions too large to be consistent with the isotropy of the COBE observations. Had they been produced they would be inflated away. It is not even clear how inflation might be realized within the heterotic string theory.

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However, with the discovery of D-branes,66 the introduction of warped geometries67 and the development of specific, string-based inflationary scenarios, the picture has changed substantially. Open fundamental strings must end on branes; so both open and closed strings are present in the brane world. Closed string loops inside a brane will break up into pieces of open strings, so only vortices (which may be only metastable) can survive inside branes. Here, D1-strings (i.e. D1-branes) may be treated as vortices inside branes and survive long enough to be cosmologically interesting.68 The warped geometry will gravitationally redshift the string tensions to low values and, consequently, strings can be produced after inflation. A string with tension T in the bulk will be warped to µ = h2 T with h being the warp factor, which can be very small when the string is sitting at the bottom of a throat. (It is h = hA in Eq. (18) in throat A.) That is important because relics produced during inflation are rapidly diluted by expansion. Only those generated after (or very near the end of) inflation are potentially found within the visible universe. Since the Type IIB model has neither D0-branes nor D2-branes, the well-justified conclusion that our universe is dominated neither by monopoles nor by domain walls follows automatically. Scenarios that incorporate string-like relics may prove to be consistent with all observations. Such relics appear to be natural outcomes of today’s best understood string theory scenarios. The physical details of the strings in Type IIB model can be quite nontrivial, including the types of different species present and the range of string tensions. Away from the branes, p fundamental F1-strings and q D1-strings can form a (p, q) bound state. Junctions of strings will be present automatically. If they live at the bottom of a throat, there can be beads at the junctions as well. To be specific, let us consider the case of the Klebanov–Strassler warped throat,69 whose properties are relatively well understood. On the gravity side, this is a warped deformed conifold. Inside the throat, the geometry is a shrinking S 2 fibered over a S 3 . The tensions of the bound state of p F1-strings and that of q D1-strings were individually computed.70 The tension formula for the (p, q) bound states is given by Ref. 71 in terms of the warp factor h, the strings scale MS and the string coupling gs , 
2   h2 MS2 q 2 bM 2 πp Tp,q  + sin , (30) 2π gs2 π M where b = 0.93 numerically and M is the number of fractional D3-branes (that is, the units of 3-form RR flux F3 through the S 3 ). Interestingly, the F 1-strings are charged in ZM and are non-BPS. The D-string on the other hand is charged in Z and is BPS with respect to each other. Because p is ZM -charged with nonzero binding energy, binding can take place even if (p, q) are not coprime. Since it is a convex function, i.e. Tp+p
< Tp + Tp
, the p-string will not decay into strings with

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smaller p. M fundamental strings can terminate to a point-like bead with mass70 
3/2 gs bM h Mbead = MS 4π π 3 irrespective of the number of D-strings around. Inside a D-brane, F1-strings break into pieces of open strings so we are left with D1-strings only, in which case they resemble the usual vortices in field theory. For M → ∞ and b = h = 1, the tension (30) reduces to that for flat internal space.72 Cosmological properties of the beads have also been studied.73 Following from gauge-gravity duality, the interpretation of these strings in the gauge theory dual is known. The F 1-string is dual to a confining string while the D1string is dual to an axionic string. Here the gauge theory is strongly interacting and the bead plays the role of a “baryon”. It is likely that different throats have different tension spectra similar to that for the Klebanov–Strassler throat. Finally, there is some evidence that strings can move in both internal and external dimensions and are not necessarily confined to the tips of the throat.74 This behavior can show up as a cosmic string with variable tension. Following gauge-gravity duality, one may argue that one can also obtain these types of strings within a strongly interacting gauge theory; however, so far we are unable to see how inflation can emerge from such a nonperturbative gauge theory description. As we will describe in more detail, all indications suggest that, once produced after inflation, a scaling cosmic string network will emerge for the stable strings. Before string theory’s application to cosmology, the typical cosmic string tension was presumed to be set by the grand unified theory’s (GUTs) energy scale. Such strings have been ruled out by observations. We will review the current limits shortly. Warped geometry typically allows strings with tensions that are substantially smaller, avoiding the observational constraints on the one hand and frustrating easy detection on the other. The universe’s expansion inevitably generates sub-horizon string loops and if the tension is small enough the loops will survive so long that their peculiar motions are damped and they will cluster in the manner of cold dark matter. This results in a huge enhancement of cosmic string loops within our galaxy (about 105 times larger at the Sun’s position than the mean throughout the universe) and makes detection of the local population a realistic experimental goal in the near future. We will focus on the path to detection by means of microlensing. Elsewhere, we will discuss blind, gravitational wave searches. A microlensing detection will be very distinctive. The nature of a microlensing loop can be further confirmed and studied through its unique gravitational wave signature involving emission of multiple harmonics of the fundamental loop period from the precise microlensing direction. Since these loops are essentially the same type of string that makes up all forms of microscopic matter in the universe, their detection will be of fundamental importance in our understanding of nature.

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6.1. Strings in brane world cosmology String-like defects or fundamental strings are expected whenever reheating produces some closed strings toward the end of an inflationary epoch. Once inflation ends and the radiation dominated epoch begins, ever larger sections of this cosmic string network re-enter the horizon. The strings move at relativistic speeds and long lengths collide and break off sub-horizon loops. Loops shrink and evaporate by emitting gravitational waves in a characteristic time τ = l/(ΓGµ) where l is the invariant loop size, µ is the string tension while Γ is numerically determined and Γ ∼ 50 for strings coupled only to gravity.5 To ease discussion, we shall adopt this value for Γ. String tension is the primary parameter that controls the cosmic string network evolution, first explored in the context of phase transitions in grand unified field theories (GUTs), which may be tied to the string scale. Assuming that the inflation scale is comparable to the GUT scale, inflation generated horizoncrossing defects whose tension is set by the characteristic grand unification energy.5 These GUT strings with Gµ ∼ 10−6 would have seeded the density fluctuations for galaxies and clusters but have long been ruled out by observations of the CMB.9,75,76 Here, string theory comes to the rescue. Six of the string theory’s 9 spatial dimensions are stably compactified. The flux compactification involves manifolds possessing warped throat-like structures which redshift all characteristic energy scales compared to those in the bulk space. In this context, cosmic strings produced after inflation living in or near the bottoms of the throats can have different small tensions.77–81 The quantum theory of one-dimensional objects includes a host of effectively one-dimensional objects collectively referred to here as superstrings. For example, a single D3-brane has a U (1) symmetry that is expected to be broken, thus generating a string-like defect. (For a stack of n branes, the U (n) ⊃ U (1) symmetry is generic.) Note that fundamental superstring loops can exist only away from branes. Any superstring we observe will have tension µ exponentially diminished from that of the Planck scale by virtue of its location at the bottom of the throat. Values like Gµ < 10−14 (i.e. energies < 1012 GeV) are entirely possible. A typical manifold will have many throats and we expect a distribution of µ, presumably with some Gµ > 10−14 . In addition to their reduced tensions, superstrings should differ from standard field theory strings (i.e. vortices) in other important ways: long-lived excited states with junctions and beads may exist, multiple noninteracting species of superstrings may coexist and, finally, the probability for breaking and rejoining colliding segments (intercommutation) can be much smaller than unity.82 Furthermore a closed string loop may move inside the compacted volume as well. Because of the warped geometry there, such motion may be observed as a variable string tension both along the string length and in time.

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6.2. Current bounds on string tension Gµ and probability of intercommutation p Empirical upper bounds on Gµ have been derived from null results for experiments involving lensing,89–91 gravitational wave background and bursts,92–105 pulsar timing84,106–111 and CMB radiation.9,10,75,76,112–124 We will briefly review some recent results but see Refs. 125 and 126 for more comprehensive treatments. All bounds on string tension depend upon uncertain aspects of string physics and of network modeling. The most important factors include: • The range of loop sizes generated by network evolution. “Large” means comparable to the Hubble scale, “small” can be as small as the core width of the string. Large loops take longer to evaporate by emission of gravitational radiation. • The probability of intercommutation p is the probability that two crossing strings break and reconnect to form new continuous segments. Field theory strings have p ∼ 1 but superstrings may have p as small as 10−3 , depending on the crossing angle and the relative speed. The effect of lowering p is to increase the network density of strings to maintain scaling. • The physical structure of the strings. F1 strings are one-dimensional, obeying Nambu–Goto equations of motion. D1 strings and field theory strings are vortices with finite cores. • The mathematical description of string dynamics used in simulations and calculations. The Abelian Higgs model is the simplest vortex description but the core size in calculations is not set to realistic physical values. Abelian Higgs and Nambu–Goto descriptions yield different string dynamics on small scales. • The number of stable string species. Superstrings have more possibilities than simple field theory strings. These include bound states of F1 and D1 strings with beads at the junctions and noninteracting strings from different warped throats. • The charges and/or fluxes carried by the string. Many effectively one-dimensional objects in string theory can experience nongravitational interactions. • The character of the discontinuities on a typical loop. The number of cusps and/or the number of kinks governs the emitted gravitational wave spectrum. While there has been tremendous progress, all of these areas are under active study. When the model-related theoretical factors are fixed each astrophysical experiment probes a subset of the string content of spacetime. For example, CMB power spectrum fits rely on well-established gross properties of large-scale string networks which are relatively secure but do not probe small sized loops which may dominate the total energy density and which would show up only at large . Analysis of combined PLANCK, WMAP, SPT and ACT data124 implies Gµ < 1.3 × 10−7 for Nambu–Goto strings and Gµ < 3.0 × 10−7 for field theory strings. Limits from optical lensing in fields of background galaxies rely on the theoretically wellunderstood deficit angle geometry of a string in spacetime but require a precise

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accounting for observational selection effects. Analysis of the GOODS and COSMOS optical surveys90,91 yields Gµ < 3 × 10−7 . Taken together these observations −7 imply Gµ < ∼ 1–3 × 10 . There is a well-established bound on the gravitational energy density at the time of Big Bang Nucleosynthesis because an altered expansion rate impacts light element yields (e.g. see Ref. 127). The gravitational radiation generated by any string network cannot exceed the bounds. If a network forms large Nambu–Goto loops of one type of string with intercommutation probability p an estimate of the −7 2 BBN constraint is Gµ < ∼ 5 × 10 p (Ref. 128) (this depends implicitly on the loop formation size and a still-emerging understanding of how network densities vary with p). LIGO’s experimental bound on the stochastic background gravitational radiation104 from a similar network implies a limit on Gµ of the same general form as the BBN limit but weaker. Advanced LIGO is projected to reach Gµ ∼ 10−12 for p = 1.128 The same LIGO results can be used to place a reliable, conservative bound of Gµ < 2.6 × 10−4 over a much wider range of possible models, almost independent of loop size and of frequency scaling of the emission.126 More stringent bounds rely on additional assumptions. Consider a specific set of choices for the secondary parameters of strings: one string species, Nambu–Goto

Fig. 3. Observational constraints limit string tension and intercommutation probability if the network forms large loops. Each line is a particular observational constraint. The region below and to the right of a line is disfavored. Smaller p increases the number density of loops and larger Gµ increases gravitational signal amplitude. Conversely, the region to the left and above a line remains consistent with observational limits. The illustrated constraints are: Big Bang Nucleosynthesis (orange line) and CMB (yellow line) from the gravitational wave spectra in Ref. 128, pulsar time of arrival for the Parkes Pulsar Timing Array (blue line, based on the analytic form in Ref. 109) and for NANOGrav (red dotted line) in Ref. 111. All lines depend upon theoretical modeling of the network, especially the fraction of large loops formed. For example, the initials BP (green)132 are an example in which this fraction has been inferred from simulations and the bound on Gµ at p = 1 indicated (the line has not been calculated). This result is approximately ∼ 30 times less restrictive on Gµ than an equivalent analysis in which all loops are born at a single large size as has been assumed in all the other analyses. (For color version, see page II-CP14.)

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dynamics, only gravitational interactions, large loops (α = 0.1), each with a cusp. The time of arrival of pulses emitted by a pulsar vary on account of the gravitational wave background. When the sequence is observed to be regular the perturbing −9 amplitude of the waves is limited. The current limit is Gµ < ∼ 10 for p = 1 and −12 −3 111 < for p = 10 . Figure 3 is a graphical summary that illustrates some of ∼ 10 the bounds discussed. The BBN (orange line) and CMB (yellow line) constraints are shown as a function of string tension Gµ and intercommutation probability p. The Parkes Pulsar Timing Array limit (blue line)109 and the NANOGrav limit (red dotted line)111 are based on radiating cusp models. Each constraint rules out the area below and to the right of a line. Most analyses do not account for the fact that only a small fraction of horizon-crossing string actually form large loops which are ultimately responsible for variation in arrival times. In this sense the lines may be over-optimistic (see figure caption). Superconducting strings have also been proposed.129 The bound on supercon−10 130 Interestingly, it was pointed out ducting cosmic strings is about Gµ < ∼ 10 . that the recently observed fast radio bursts can be consistent with being produced by superconducting cosmic strings.131 In short, cosmic superstrings are generically produced toward the end of inflation and observations imply tensions substantially less than the original GUT-inspired strings. Multiple, overlapping approaches are needed to minimize physical uncertainties and model-dependent aspects. There is no known theoretical impediment to the magnitude of Gµ being either comparable to or much lower than the current observational upper limits. 7. Scaling, Slowing, Clustering and Evaporating (see Ref. 133) Simulated cosmological string networks converge to self-similar scaling solutions.5,82,114 Consequently Ωlong (Ωloop ), the fraction of the critical density contributed by horizon-crossing strings (loops), is independent of time while the characteristic size of a loop formed at time t scales with the size of the horizon: l = αt for some fixed α. To achieve scaling, long strings that enter the horizon must be chopped into loops sufficiently rapidly — if not, the density of long strings increases and over-closes the universe. In addition, loops must be removed so that Ωloop stabilizes — if not, loops would come to dominate the contribution of normal matter. Scaling of the string network is an attractor solution in many well-studied models and the universe escapes the jaws of both Scylla and Charybdis. The intercommutation probability determines the efficiency of chopping and µ determines the rate of loop evaporation. Studies of GUT strings5 took Gµ ∼ 10−6 (large enough to generate perturbations of interest at matter-radiation equality) and α small (set by early estimates of gravitational wave damping on the long strings) with the consequence Hτ ∼ α/ΓGµ  1 where H is the Hubble constant. Newly formed GUT loops decay quickly. The superstrings of interest here have smaller Gµ so that loops of

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a given size live longer. In addition, recent simulations134–138 produce a range of < large loops: 10−4 < ∼ α ∼ 0.25. The best current understanding is that ∼ 10–20% of the long string length that is cut up goes into loops comparable to the scale of the horizon (α ∼ 0.1) while the remaining ∼ 80–90% fragments to much smaller size scales.139,140 The newly formed, large loops are the most important contribution for determining today’s loop population. Cosmic expansion strongly damps the initial relativistic center of mass motions of the loops and promotes clustering of the loops as matter perturbations grow.141,142 Clustering was irrelevant for the GUT-inspired loops. They moved rapidly at birth, damped briefly by cosmic expansion and were re-accelerated to mildly relativistic velocities by the momentum recoil of anisotropic gravitational wave emission (the rocket effect) before fully evaporating.93,143,144 GUT loops were homogeneously distributed throughout space. By contrast, below a critical tension Gµ ∼ 10−9 all superstring loops accrete along with the cold dark matter.142 Loops of size l = lg ≡ ΓGµt0 are just now evaporating where t0 is the age of the universe. The mean number density of such loops is dominated by network fragmentation when the universe was most dense, i.e. at early times. When they were born they came from the large end of the size spectrum, i.e. a substantial fraction of the scale of the horizon. The epoch of birth is ti = lg /α = ΓGµt0 /α. For Gµ < 7 × 10−9 (α/0.1)(50/Γ) loops are born before equipartition in ΛCDM, i.e. ti < teq . The smallest loops today have lg ≈ 40pc(Gµ/2×10−10) with characteristic mass scale Mg = 1.7 × 105 M (Gµ/2 × 10−10 )2 for Γ = 50. Loop number and energy densities today are dominated by the scale of the gravitational cutoff, the smallest loops that have not yet evaporated. The characteristic number density dn/d log l ∝ (ΓGµ)−3/2 (αteq /t0 )1/2 /t30 and  the energy density dρloop /d log l ∼ ΓGµt0 dn/d log l. The latter implies Ωloops ∝ αGµ/Γ whereas Ωlong ∝ ΓGµ. In scenarios with small µ it’s the loops that dominate long, horizoncrossing strings in various observable contexts. The probability and rate of local lensing are proportional to the energy density. Loops are accreted as the galaxy forms. Figure 4 illustrates schematically the constraints for a loop with typical initial peculiar velocity to be captured during galaxy formation and to remain bound today. The loop must lie within the inner triangular region which delimits small enough tension and early enough time of formation. Above the horizontal line capture is impossible; below the diagonal line detachment by the rocket effect has already occurred. The critical tension for loop clustering in the galaxy is set by the right-hand corner of the allowed region. To briefly summarize: F , the enhancement of the galactic loop number density over the homogeneous mean, simply traces E, the enhancement of cold dark matter over ΩDM ρc , for critical density ρc . This encapsulates conclusions of a study of the growth of a galactic matter perturbation and the simultaneous capture and escape of network-generated loops.142 At a typical galactocentric distance of 11 kpc log10 F¯ = log10 E¯+f (y) where log10 E¯ = 5.5 and f (y) = −0.337−0.064y 2 −g(y) and

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Fig. 4. Bounds on formation time and string tension for a loop with initial velocity vi = 0.1 to be captured at physical radius 30 kpc by galaxy formation. (1) Upper bounds on the formation time ti /t0 are given by the horizontal lines. The condition that cosmic drag lower the velocity to less than the circular rotation velocity today is given by the red line. The more stringent condition that capture occur at 30 kpc is given by the turquoise line. (2) Upper bounds on the string tension Gµ are given by the diagonal lines. The condition that the loop be younger than its gravitational wave decay timescale is given by the red line. The more stringent condition that the loop not be accelerated out of the galaxy by today is given by the green line. (3) The shaded region encompasses string tensions and formation times giving bound loops at 30 kpc for vi = 0.1 and α = 0.1. The critical value of Gµ below which clustering is possible is determined by the upper right hand corner of the green and turquoise lines. Lowering vi raises the limit on ti /t0 (horizontal lines moves upward); lowering α shifts the bound to smaller Gµ (diagonal lines moves leftward). Shifting the loop orbital scale to smaller values (say the solar position) requires earlier formation times (horizontal lines shifts down) and allows larger Gµ (green line shifts to the right but is limited by the red line which is fixed). (4) The geometric symbols illustrate the sensitivity of the rocket effect. These are numerical experiments examining the outcome today (t = t0 ) for groups of 10–20 loops captured at 30 kpc with slightly different string tensions: stars = all loops bound, boxes = all loops ejected, triangles = some bound and some ejected (for clarity the points are slightly offset in the vertical but not the horizontal direction). (For color version, see page II-CP15.)

g(y) = 5(1 + tanh(y − 7)). The tension-dependent deviation from loops as passive tracers of cold dark matter is f (y), y = log10 (Gµ/10−15 ), fit for 0 ≤ y ≤ 5 from the capture study. Clustering saturates for small tensions: f (y) = f (0) for y < 0. The over-bar indicates quantities averaged over the spherical volume and, in the case of the loops, a weighting by loop length in the capture study. The extra piece g(y) plays a role for Gµ > 10−10 where it describes the suppression in clustering as one approaches the upper right corner of the triangle in Fig. 4. Numerical simulations have not yet accurately determined it. A single f (y) fits a range of radii as long as E¯  1 (typically, galactic distances less than ∼ 100 kpc). So we can easily have an enhancement F ∼ 105 for Gµ ∼ 10−14 at the Sun’s position. By comparison, GUT loops would have F = 1 for all positions and tensions.

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The model accounts for the local population of strings and we have used it to estimate microlensing rates, and LISA-like, LIGO-like and NANOGrav-like burst rates. 7.1. Large-scale string distribution We will start with a “baseline” description (loops from a network of a single, gravitationally interacting, Nambu–Goto string species with reconnection probability p = 1). The detailed description142 was motivated by analytic arguments139,140 that roughly 80% of the network invariant length was chopped into strings with very small loop size (α ∼ Gµ) and by the numerical result134–138 that the remaining 20% formed large, long-lived loops (α = 0.1). At a given epoch loops are created with a range of sizes but only the “large” ones are of interest for the local population. The baseline description is supposed to be directly comparable to numerical simulations which generally take p = 1. The most recent simulations132 are qualitatively consistent. Next, we parameterize the actual “homogeneous” distribution in the universe when string theory introduces a multiplicity of string species and the reconnection probability p may be less than 1. And finally we will form the “local” distribution which accounts for the clustering of the homogeneous distribution. In a physical volume V with a network of long, horizon-crossing strings of tension µ with persistence length L there are V /L3 segments of length L. The physical energy density is ρ∞ = µL/V = µ/L2 . The persistence length evolves as the universe expands. A scaling solution demands L ∝ t during power law phases. There are also loops within the horizon; their energy is not included in ρ∞ and the aim of the model is to infer the number density of loops of a given invariant size. Kibble145 developed a model for the network evolution for the long strings and loops in cosmology. It accounted for the stretching of strings and collisional intercommutation (long string segments that break off and form loops; loops that reconnect to long string segments). A variety of models of differing degrees of realism have been studied since then, guided by ever more realistic numerical simulations of the network. As a simple approximate description we focus on the Velocity One Scale model146 in a recently elaborated form.147,148 The reattachment of loops to the network turns out to be a rather small effect and is ignored. We extended existing treatments by numerically evaluating the total loop creation rate in flat ΛCDM cosmology. The loop energy in a comoving volume varies like E˙ l = Cρ∞ pva3 /L where C is the chopping efficiency, p is the intercommutation probability and v is the string velocity. All quantities on the right-hand side except p vary in time; C is a parameterized fit in matter and radiation eras. By integrating the model from large redshift to the current epoch one evaluates the fraction of the network that is lost to loop formation. The rate at which the loop energy changes is E˙ l = Aµa3 /(p2 t3 ) where A is a slowly varying function of redshift z and p shown in Fig. 5. The plotted variation of A with redshift indicates

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a 3 dE loop dt

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Fig. 5. The redshift-dependent rate of energy loss to loops experienced by the string network in ΛCDM cosmology for a given intercommutation probability p. This is an important input for predicting properties of today’s loops. The ordinate is the dimensionless measure of the rate of energy loss to loops, A = E˙ l /(µa3 /(p2 t3 )). If the scale factor a(t) were a power law in time then A would be constant with respect to z. This is not the case in the ΛCDM cosmology with transitions from radiation-to-matter-to-Λ dominated regimes. Different lines show results for p = 1 (top) to p = 10−3 (bottom) in powers of 10. If network density varied as 1/p2 then all these lines would overlap. This is approximately true for small p but not for p ≥ 0.1.

departures from a pure scaling solution (consequences of the radiation to matter transition for a(t) and the implicit variation of chopping efficiency C). Knowing A as a function of redshift and p is the input needed to evaluate of the loop formation rate. Loops form with a range of sizes at each epoch. Let us call the size at the time of formation αt. A very important feature predicted by theoretical analyses is that a large fraction of the invariant length goes into loops with α ∼ Gµ or smaller. The limiting behavior of numerical simulations as larger and larger spacetime volumes are modeled suggests ∼ 80% of the chopped up long string bears that fate. Such loops evaporate rapidly without contributing to the long-lived local loop population. The remaining 20% goes into large loops with α ∼ 0.1. Let us call the fraction of large loops f = 0.2. The birth rate density for loops born with size αtb is
 dn fA δ(l − αtb ). (31) = dtdl αp2 t4b A loop formed at tb with length lb shrinks by gravitational wave emission. Its size is l = lb − ΓGµ(t − tb )

(32)

at time t (Γ ∼ 50). The number density of loops of size l at time t is the integral of the birth rate density over loops of all length created in the past. For l < αt or, equivalently,

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tb < t we have dn (l, t) = dl tb =




f Aα2 p2




a(tb ) a

3

Φ3 , (l + ΓGµt)4

(33)

l + ΓGµt , αΦ

(34)

ΓGµ . α

(35)

Φ = 1+

The scaling dn/dl ∝ (l + ΓGµt)−4 was already noted by Kibble.145 The loop number distribution peaks at zero length but the quantity of interest in lensing is number weighted by loop length, ldn/dl. The characteristic scale at time t is lg = ΓGµt, i.e. roughly the size of a newly born loop that would evaporate in total time t. The distribution ldn/dl peaks at l = (2/3)lg . For Gµ < 7 × 10−9 (α/0.1)(50/Γ) the loops near lg today were born before equipartition, teq . We use a ∝ t1/2 to simplify the expression to give l

dn x = 5/2 dl (1 + x)




fA p2 t30



(ΓGµ)−3/2




αteq t0

1/2 ,

(36)

where t0 is today and x ≡ l/lg . The numerical results are A ∼ 7.68 for p = 1, f = 0.2, α = 0.1 and Γ = 50 (and from ΛCDM teq = 4.7 × 104 yr and t0 = 4.25 × 1017 s). These give
 dn x −3/2 l = 1.15 × 10−6 µ kpc−3 , (37) dl baseline (1 + x)5/2 −13 lg = 0.0206 µ−13 pc,

(38)

Mg = 0.043 µ2−13 M .

(39)

Here, µ−13 ≡ Gµ/c2 /10−13 is an abbreviation for the dimensionless string tension in units of 10−13 . The baseline distribution lies below147,148 on account of loss of a significant fraction of the network invariant length to small strings and of adoption the ΛCDM cosmology. Next, string theory modifications to the baseline are lumped into a common factor G

 dn dn =G (40) dl homog dl baseline to give the description of the actual homogeneous loop distribution. Prominent among expected modifications is the intercommutation factor p. Large scale string simulations have not reached a full understanding of the impact of p < 1 though there is no question that G increases as a result. The numerical treatment of the Velocity One Scale model implies that A is a weak function of p for small p and,

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in that limit, dn/dl ∝ 1/p2 . Ultimately, this matter will be fully settled via network simulations with p < 1. String theory calculations of the intercommutation probability suggests p = 10−1 – 10−3 implying G = 102 – 106. The number of populated, noninteracting throats that contain other types of superstrings is a known unknown and is unexplored. In our opinion, there could easily be 100’s of such throats for the complicated bulk spaces of interest. In the string theory scenarios G = 1 is a lower limit. For the purposes of numerical estimates in this paper we adopt G = 102 (with a given tension) as the most reasonable lower limit. Much larger G are not improbable while lower G are unlikely. We should emphasize that strings in different throats would have different tensions, so adopting a single tension here yields only a crude estimate. 7.2. Local string distribution If a loop is formed at time t with length l = αt then its evaporation time τ = l/ΓGµ. For Hubble constant H at t the dimensionless combination Hτ = α/(ΓGµ) is a measure of lifetime in terms of the universe’s age. Superstring loops with α = 0.1 and small µ live many characteristic Hubble times. New loops are born with relativistic velocity. The peculiar center of mass motion is damped by the universe’s expansion. A detailed study of the competing effects (formation time, damping, evaporation, efficacy of anisotropic emission of gravitational radiation) in the context of a simple formation model for the galaxy shows that loops accrete when µ is small. The degree of loop clustering relative to dark matter clustering is a function of µ and approximately independent of l. Smaller µ means older, more slowly moving loops and hence more clustering. Below we give a simple fit to the numerical simulations to quantify this effect. The spatially dependent dark matter enhancement in the galaxy is E(r) =

ρDM (r) , ΩDM ρc

(41)

where ρDM (r) is the local galactic dark matter density and ΩDM ρc is average dark matter density in the universe. Low tension string loops track the dark matter with a certain efficiency as the dark matter forms gravitationally bound structures.142 We fit the numerical results by writing the spatially dependent string enhancement to the homogeneous distribution as equal to the dark matter enhancement times an efficiency factor F (r) = E(r)10f (y) ,  −0.337 − 0.064y 2 − 5(1 + tanh(y − 7)) f (y) = −0.337 y = 2 + log10 µ−13

(42) for 0 ≤ y, for y < 0,

(43) (44)

−13

and µ−13 is the dimensionless tension in units of 10 . The clustering is never 100% effective because the string loops eventually evaporate. The efficiency saturates at

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y = 0 or Gµ/c2 = 10−15 . The local string population is enhanced by the factor F with respect to the homogeneous distribution


 dn dn dn (r) = F (r) = F (r)G . (45) dl local dl homog dl baseline To complete the description of the local loop population we adopt empirical fits to the galaxy’s dark matter halo.149 Model I is a “cored galaxy” center in which the central, limiting dark matter density is zero. Since winds from stars and ejection/heating by supernovae lift baryons out of star-forming regions, the center, they reduce the gravitational potential and tend to lower the dark matter density. Model II is a “cusped galaxy” center in which the density formally diverges. Many N-body simulations show that collisionless structure formation in ΛCDM yields such profiles. The two models are thought to bracket the range of physical possibilities in the central regions and are in general agreement on large scales. Let us define the effective number density of loops as the energy density in the distribution of loops of all size, ρloop , divided by energy of a loop of size equal to the characteristic gravitational cutoff ρloop µlg  dn l = dl . dl lg

n ¯=

G c2 10 Log10 n kpc

15

10

(46) (47)

8

3

5

R kpc 5

10

15

20

25

5

10

Fig. 6. The effective number density of string loops in two galactic models for a range of string tensions. The dotted red lines show results for a galactic model with a cusp at the center, the solid blue lines for a model with a core. The tension is Gµ/c2 = 10−15 for the uppermost curve and increases by powers of 10 until reaching 10−8 for the lowest curve. The effective number density depends upon the clustering of loops, the dark matter distribution within the galaxy, the network scaling and loop distribution and G the enhancement of string theory strings compared to field theory strings. (For color version, see page II-CP15.)

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Figure 6 plots the effective number density of string loops for two descriptions of the galaxy’s dark matter distribution (core and cusp) as a function of galactocentric radius. The different lines show a range of string tensions, all for G = 102 . The quantity n ¯ depends upon the efficacy of loop clustering, the dark matter distribution within our galaxy and the network scaling model. There are two curves plotted for each tension. These agree at large radii but differ near the galactic center where the detailed form of the dark matter distribution is uncertain. For many experiments the rate of detection scales as n ¯ weighted by powers of lg . The microlensing rate, for example, is proportional to the product of two factors: the number of string loops along a given line of sight ∝ n ¯ and the length of an individual ¯ lg ∝ n ¯ Gµ. At the Sun’s position, the string loop ∝ lg . The event rate scales like n effective number density increase by ∼ 15 orders of magnitude as Gµ/c2 drops from 10−8 to 10−15 . The microlensing event rate increases by ∼ 8 orders of magnitude. As we will discuss shortly, for the same variation in tension the timescale for an individual microlensing event decreases by ∼ 8 orders of magnitude. This creates a huge range of timescales of interest for the duration and rate of microlensing events. 8. Detection 8.1. Detection via Microlensing Consider a straight segment of string oriented perpendicular to the observer’s line of sight with respect to a background source as shown in Fig. 7. Let two photons from the source travel toward the string. The photons do not suffer any relative

Fig. 7. (Left) The conical spacetime geometry near a short segment of string (red line) permits photons to travel two paths (black arrows) from source (yellow ball) to observer when source, string and observer are nearly aligned. The ideal observer perceives two images of the source split by a small angle (black arc) proportional in size to string tension. For low string tension, the observer cannot resolve the separate images; however, the flux may be easily measured to be twice that of a single image (in the point source limit) at each wavelength (negligible Kaiser–Stebbins effect). (Right) The microlensing amplification of flux for a point source as a function of time for a string (green) is compared to a Newtonian point masses (blue; the ratio of impact parameter to Einstein radius is 0.31). No other known astrophysical sources produce this digital microlensing. (For color version, see page II-CP15.)

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deflection during the fly-by as long as they pass around the string in the same sense. However, there is a small angular region about the string with two paths from the source to the observer. Background sources within angle ΘE = 8πGµ = 1.04 × 10−3(Gµ/2 × 10−10) arcsec form two images. Unlike the case of a point mass, shear and distortion are absent. The angular size of a sun-like star at distance R is Θ /ΘE = 4.5 × 10−4 (2 × 10−10 /Gµ)(10 kpc/R) so galactic stellar sources generally −13 appear point-like for Gµ > ∼ 10 . Compact halo objects lens background sources150,151 and unresolved, lensed sources will appear to fluctuate achromatically in brightness. This is microlensing. Experimental efforts to detect microlensing phenomena have borne considerable fruit.152 Likewise, loops of superstring microlens background sources but have a special property: the source brightness varies by a factor of 2 as the angular region associated with the string passes across the observer-source line of sight. The amplitude variation, schematically compared in Fig. 7, is quite distinctive for point sources. The internal motions of a loop are relativistic and generally dominate the motion of the source and the observer. Numerical calculations have established that microlensing occurs when light passes near a relativistic, oscillating string loop153 ; the effect is not limited to a stationary string. of loops dn/dl is proThe total rate of lensing RL implied for a distribution √ 2 portional to the solid angle swept out per time cl/ 3r for loop l at distance r, or √  √ RL = dldrr2 dn/dl(cl/ 3r2 ). Small tensions give large lensing rates, RL ∝ 1/ Gµ because the integral over dn/dl ∝ l−2.5 is dominated by l ∼ lg , the gravitational cutoff. Figure 8 illustrates the hierarchy of relevant timescales. The characteristic duration of the event is δt = RΘE /c ∼ 630 s(R/10 kpc)(Gµ/2 × 10−10 ). Loops bound to the galaxy have center-of-mass motions vh ∼ 220 km s−1 ; microlensing of a given source will repeat ∼ c/vh times; new sources are lensed at rate ∼ (vh /c)RL . The characteristic loop oscillation timescale which governs the intervals in repetitive microlensing of a single source is tosc ∼ lg /c = 135 yr (Gµ/2 × 10−10 ). The unique fingerprint of loop lensing is repeated achromatic flux doubling (digital microlensing). Detection efficiency for digital lensing depends upon the string tension, the magnitude and angular size of the background stars and the time sampling of the observations. Estimates can be made for any experiment which repeatedly measures the flux of stellar sources. The first string microlensing search154 was recently completed using photometric data from space-based missions CoRoT155,c and RXTE.156,d The methodology was potentially capable of detecting strings with tensions 10−16 < Gµ < 10−11 though the expected number of detections was limited by the available lines of sight studied c The

2006 mission was developed and operated by the CNES, with the contribution of Austria, Belgium, Brazil, ESA (RSSD and Science Program), Germany and Spain. d The 1995 mission was developed and operated by NASA.

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Flux

Time

Fig. 8. A Schematic Pattern. Digital microlensing doubles the flux over the time period that source, loop and observer are aligned within the deficit angle created by the string (red line). The repetition interval for lensing a particular source is the loop oscillation timescale (green line) and ∼ 103 repetitions in total occur (yellow line). The timescale for a new source to be lensed by the original loop (or the original source by a new loop) is much longer (black line). (For color version, see page II-CP16.)

in the course of the missions. In principle, any photometry experiment that makes repeated flux measurements of an intrinsically stable astronomical source has the power to limit a combination of the number density of loops and string tension. One interesting possibility is the satellite GAIA launched by the European Space Agency in December 2013 designed for astrometry. It is monitoring each of about 1 billion stars about 70 times over a period of 5 years. Another is the Large Synoptic Survey Telescope (LSST) to be sited in Chile within the decade to photograph the entire observable sky every few days. Previous estimates for the rate of detections for GAIA141 and LSST157 were encouraging. Now, detailed calculations for WFIRST are also available. 8.2. WFIRST microlensing rates (see Ref. 158) The Wide-Field Infrared Survey Telescopee (WFIRST) is a NASA space observatory designed to perform wide-field imaging and slitless spectroscopic surveys of the near infrared sky. WFIRST will carry out a microlensing survey program for exo-planet detection in the direction of the galactic bulge by observing a total of 2.8 square degrees for 1.2 years primarily in a broad long wavelength filter (Wband 0.927–2.0 µm). Repeated, short exposures (∼ 1 minute) of the same fields are the key to searching for and to monitoring the amplification of bulge sources by star–planet systems along the line of sight. Fortuitously, the WFIRST experiment e http://wfirst.gsfc.nasa.gov/.

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is also sensitive to cosmic strings. We have evaluated the expected microlensing rate by cosmic strings for a realistic distribution of stars (stellar types, distances, velocities, etc.), dust obscuration and survey parameters (flux sensitivity, time of exposure and angular scale of stars). The lensing rate is split into digital events (the flux doubles), analog events (all potentially measurable flux enhancements given the signal-to-noise of the observations) and total events (all geometric configurations that can lens, whether detectable or not) evaluated for two galactic dark matter models (with cusp and with core at galactic center). Figure 9 shows the string lensing rate per square degree as a function of string tension Gµ/c2 for G = 102 . The lensing rate for digital and analog events exceeds 10 per square degree per year for 10−14 < Gµ/c2 < 10−10 for the cusp model and for 10−13 < Gµ/c2 < 10−11 for the core model. The results for the cusp and core models should be regarded as establishing the range of astrophysical possibilities for the loop density distribution within the experiment. The range of string theory possibilities (the variation of G) remains substantial. Digital and spatially resolved analog events can be reliably detected. Blending of sources in analog events needs to be simulated to fully characterize the fraction of such events that will be detectable. In any case, the decrease from blending is limited: sources brighter than magnitude 23 in the J-band (1.131– 1.454 µm) contribute at least 10% of the total analog rate, are well-separated and produce light curves detectable at high signal-to-noise. These results show that WFIRST microlensing searches can probe hitherto unexplored ranges of string tension. Future surveys of wider areas of the bulge and/or surveys lasting for longer periods of time can potentially scale up the total Log10 4

2

0

14

12

10

8

Log10

2

4

Fig. 9. The intrinsic string lensing rate (events per sq degree per year) for WFIRST’s study of the bulge for G = 102 . The three solid lines show digital (red), analog (green) and total (blue) event rates for the dark matter model with a cusp. The three dashed lines (same meaning for the colors) give the event rate for the model with a core. The analog rate does not account for blending which may diminish the detectable rate by no more than a factor of 10. (For color version, see page II-CP16.)

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expected rate of detection by as much as 100. Surveys with shorter exposures times are able to discern larger numbers of digital events. 8.3. Gravitational waves Besides uncovering a relic from the earliest moments of the universe’s formation, the observation of a superstring microlensing event provides some immediate information. If the event is resolved in time, the characteristic string tension is inferred. If repetitions are seen, the characteristic loop size is constrained. And any detection provides a precise sky location for follow up. Some classes of strings create electromagnetic signatures but all generate gravitational radiation possibly observable by LISA-like (hereafter, LISA) and LIGO-like instruments. The scenario that LISA detects the distinctive signature of local loop emission has been contemplated for a blind, all-sky search.159 They relied on the enhancement from local clustering141 and concluded that strings with 10−19 < Gµ < 10−11 were potentially detectable via fundamental and low-order harmonics. They also estimated the background from the galaxy and from the universe as a whole. Therefore, it is no surprise that a loop microlensing event offers some exciting possibilities. The key considerations are: (1) The precise location afforded by microlensing dramatically reduces the signal search space on the sky. (2) A generic loop radiates a full range of harmonics with frequency fn = nf1 = 2cn/l because the equations of motion are intrinsically nonlinear. (3) Many string harmonics fall near LISA’s peak sensitivity n∗ = f∗ /f1  1 (hLISA ∼ 10−23.8 at f∗ ∼ 10−2.3 Hz for 1 year periodic variation160 ; tension µcrit = 2.9 × 10−12 corresponds to a 1 year period; n∗ = 1.6 × 105(µ/µcrit )). (4) Galactic binary interference becomes problematic only at f < fWD where fWD ∼ 10−2.6 Hz.161 LISA’s verification binaries have known positions on the sky, orbital frequencies, masses, distance limits etc. determined by optical measurements and other means.160 A long-lived superstring loop detected by microlensing shares similar observational advantages: it has known position on the sky, distance upper limit estimated based on the microlensed star and emits a distinctive signal. This creates the ideal observational situation in which much of the data analysis can be conducted with a single fast Fourier transform. Different types of loops generate different gravitational wave signatures. The solution for a Nambu–Goto loop’s motion is the sum of left- and right-moving onedimensional waves subject to nonlinear constraints. When both modes are smooth (no derivative discontinuities) the dynamical solution can form a cusp. At that instant a bit of the string reaches the speed of light. When one mode is smooth but the other has a discontinuity the loop’s dynamical solution includes a kink, a discontinuity in the string’s tangent vector that moves along the string at the speed of light in one direction. When both modes have discontinuities we call the solution a k-kink. The presence of cusps, kinks and k-kinks may be inferred from the paths of the tangent vectors of the left and right moving modes which are constrained to

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lie on the surface of a sphere. These distinctions are relevant for gravitational wave emission: the cusp generates bursts of beamed radiation, the kink has a pulsar-like beam that generically sweeps across a substantial angle in the sky and the k-kink emits into a substantial fraction of the sky. The tendency for string loops to explore part of the internal space of a typical compactification during oscillations74 alters the cusp and kink beaming patterns with potentially important repercussions for rates of detection in gravitational wave experiments.162–164 We model the beaming anisotropy, the harmonic power and phase of the gravitational wave signal (qualitatively and quantitatively extending previous analyses159 ). Standard signal detection theory characterizes the strength of detection and the precision with which measurements can be made by comparing the putative signal to noise sources. Here, we consider only detector noise described as an additive, stationary, random Gaussian process with one-sided spectral noise at frequency f  is S(f ). The strength of signal h(t) is ρSNR = 2h, h where the symmetric ∞ ˜ ∗ (f ) )h inner product is g, h = −∞ df g˜(f ˜(f ) is the Fourier transform of g(t) S(f ) and g (the factor of 2 stems from one-sided noise versus two-sided signal). Figure 10 summarizes ρSNR for detecting a single source with beamed cusp, beamed kink and generic, unbeamed emission. Upper and lower line pairs show the effect of ignoring and accounting for the galactic white dwarf interference. Confusion is minimal because the string signal extends to frequencies where the binary density is small and because the signal overlap between string and binary sources is small. Log10 Ρ 8

6

4

2

14

13

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9

Log10 G

c2

Fig. 10. The log of LISA’s signal strength (log10 ρSNR ) for 1 year of observation of cusp, kink and k-kink (top to bottom) at a distance of 1 kpc with a known position on the sky as a function of string tension. The orientation of the cusp and kink is for the maximum signal. Upper/lower line pairs show the effect of ignoring/accounting for white dwarf interference. For tensions with Gµ/c2 > 3 × 10−12 the fundamental loop period exceeds the length of the observation so excess power is seen but the frequency resolution is insufficient to measure individual harmonics.

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The time-averaged microlensing rate is roughly proportional to the loop’s maximum projected area divided by its period of oscillation but the gravitational wave emissivity is dominated by specific loop configurations having rapidly moving segments when the loop is most contracted. Microlensing and beamed, gravitational wave emission generally will not and need not be simultaneous to be informative. We elaborate on this important point. Generic cusps and kinks have characteristic beaming patterns described above and undergo fully three-dimensional motions that yield time-averaged microlensing rates that are not particularly angle-dependent. Microlensing is a favorable means to locate such loops but the high frequency, directed gravitational wave beams they emit are visible only from special directions. A microlensing detection does not significantly influence the probability of intercepting beamed emission in experiments with duration exceeding the fundamental loop period. Instead, there is a roughly constant, low probability that a cusp that is present will beam in the observer’s direction and modest probability that a kink will (the beam’s angular size is smaller at higher frequencies of emission). The most likely configuration for k-kinks is a flat, degenerate box-like orbit with a well-defined plane. Special observers who lie in that plane will not see microlensing but most others are sensitive to the loop’s projected motion and see the effect if a suitable background source exists. The gravitational wave emission of galactic k-kinks is not strongly beamed but, as the signal estimates show, is still potentially detectable. Unbeamed emission from loops that happen to contain cusps and kinks should also be detectable whether or not the high frequency beam impinges on the observer. To summarize, there are excellent chances to detect some of this emission once the precise location of the source is known. If the fundamental string period is less than the duration of the experiment then the string’s frequency comb can be resolved and fit. Otherwise, there will be excess frequency-dependent power seen in the microlensing direction. The issue of confusion with local and cosmological string loops will be addressed in future studies. While nondetection by LISA would place bounds on tension and loop size the most informative scenario will be measuring the individual harmonics of gravitational wave emission. Cusps have almost smooth spectral amplitude and phase distributions, kinks show some mode-to-mode variation and k-kinks progressively more. The variation with harmonic order provides immediate information regarding derivative discontinuities, e.g. an experimental determination of cusp and kink content. This is directly applicable for understanding the string network in a cosmological context. Specifically, the discontinuities are created when horizon-crossing strings are chopped into loops but are modified as the strings are stretched out by the expanding universe. It is important to understand if the loops exclusively contain discontinuities or, also, cusps. This relates to anticipated cosmologically distant string sources that contribute to the gravitational wave background currently sought.

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A measurement of the fundamental frequency is equivalent to a precise measurement of the invariant string length. It can be determined in 1 year to relative accuracy at least as good as ∆f1 /f1 ∼ 1/ρSNR . The gravitational decay time for the typical string loop is likely to be of order the age of universe. A study of Fig. 10 suggests that the expected decay in a year ∆f1 /f1 ∼ 10−10 is too small to measure directly unless the loop is unexpectedly close to the end of its life or very nearby. Instead, first, one will measure changes in frequency from center of mass motion and acceleration within the galaxy and place upper limits on ∆f1 /f1 . The latter will constrain string couplings to axion-like fields and astrophysical effects of nearby matter. Since loops are long-lived sources emitting over a wide range of harmonics, eventually, multiple detectors will be brought to bear and the loop’s decay measured in a direct fashion. This program can yield a precise determination of the string tension.

9. Summary String theory contains a consistent quantum gravity sector and possesses a deep mathematical and physical structure. It has become apparent that the theory’s richness enables it to describe a huge multiplicity of solutions and that testing whether string theory is the theory of nature is no easy task. Fortunately and fundamentally, it possesses an unambiguous property, namely, the presence of strings. If the evolution of the early universe is able to produce stable or metastable closed strings that stretch across the sky, then these unusual objects may provide distinctive evidence for the theory. As more and more observational data have been collected and analyzed, most cosmologists now conclude that the universe started with an inflationary epoch. This conclusion quite naturally motivates investigations of the following interrelated issues: how inflation is realized, how and what type of cosmic strings are produced, how to search for and detect these objects and how to measure their properties. Here, we review how inflation may be realized within string theory and whether there are distinctive stringy features in string theory-motivated inflationary scenarios. In lieu of presenting a full review we sample some of the scenarios proposed already to give readers a taste of what can be achieved. As there are many ways to realize inflation within the string theory framework, we hope better data will eventually indicate how inflation happens in string theory. As we have pointed out, some stringy inflationary scenarios can have interesting distinct features to be detected. We are also encouraged to note that cosmic superstrings are naturally produced after inflation. Next, we review and discuss cosmic superstring features and production. One underlying message is that, because of the intricacies of flux compactification, multiple types of cosmic superstrings with a wide range of properties may be produced. Such superstrings are quite different from the vortices well studied within field theory. The tension may be drawn from a discrete spectrum and different string states

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can cooperate to form junctions and beads. A future experimental observation of any of these properties can legitimately be taken as a smoking gun for string theory. We review ongoing searches for strings in cosmology. Although specific experimental results are model-dependent the diversity of approaches has jointly lowered the upper limit for string tension by about two or more orders of magnitude below GUT scales. This is an important progress. However, well-motivated string models with warped geometries can yield characteristic string tensions that are far smaller than current upper limits, perhaps all the way to the general scales of the Standard Model itself. As Feynman famously said about miniaturization, “there’s plenty of room at the bottom”. Likewise, for string theory, there is a huge range of plausible tensions yet to be explored. We review the new physics and cosmology of these relatively low tension cosmic strings, particularly the tendency for string loops to track structure formation in the matter-dominated eras. As low tension strings decay slow enough, their clustering happens like dark matter, with a 105 order of magnitude enhancement in density within the galaxy. This opens up intriguing possibilities to seek strings having tensions many orders of magnitude less than current upper limits. We review one promising possibility that, for tension 10−14 ≤ Gµ < 10−7 , string loops within the Galaxy microlens stars within the Galaxy. This type of microlensing qualitatively differs from the well-studied microlensing effect of compact objects in the Galactic halo. Ongoing (e.g. GAIA of ESA) as well as planned (e.g. LSST, WFIRST) searches for variable stars and/or exo-planets are sensitive to cosmic superstring microlensing. This gives us hope that much improved upper limits on string tension and/or actual detections will emerge in the coming years. We outline the important role that gravitational wave observations might play if and when a string loop is detected. Acknowledgments We thank Jolyon Bloomfield, Tom Broadhurst, Jim Cordes, John Ellis, Eanna Flanagan, Romain Graziani, Mark Hindmarsh, Renata Kallosh, Andrei Linde, Liam McAllister, Levon Pogosian, Ben Shlaer, Yoske Sumitomo, Alex Vilenkin, Barry Wardell, Ira Wasserman and Sam Wong for discussions. We gratefully acknowledge the support of the John Templeton Foundation (University of Chicago 37426Cornell FP050136-B). DFC is supported by NSF Physics (Astrophysics and Cosmology Theory, No. 1417132); SHHT is supported by the CRF Grants of the Government of the Hong Kong SAR under HUKST4/CRF/13G. References 1. A. H. Guth, Phys. Rev. D 23 (1981) 347, http://adsabs.harvard.edu/abs/ 1981PhRvD..23..347G. 2. A. D. Linde, Phys. Lett. B 108 (1982) 389, http://adsabs.harvard.edu/abs/ 1982PhLB..108..389L.

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3. A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. 48 (1982) 1220, http:// adsabs.harvard.edu/abs/1982PhRvL..48.1220A. 4. D. Baumann and L. McAllister, Inflation and string theory, arXiv:1404.2601. 5. A. Vilenkin and E. P. S. Shellard, Cosmic strings and other topological defects (2000), http://adsabs.harvard.edu/abs/2000csot.book.....V. 6. P. Ade et al., Phys. Rev. Lett. 112 (2014) 241101, doi: 10.1103/PhysRevLett.112. 241101. 7. R. Flauger, J. C. Hill and D. N. Spergel, Toward an understanding of foreground emission in the BICEP2 region, arXiv:1405.7351. 8. M. J. Mortonson and U. Seljak, A joint analysis of Planck and BICEP2 B modes including dust polarization uncertainty, arXiv:1405.5857. 9. G. F. Smoot, C. L. Bennett, A. Kogut, E. L. Wright, J. Aymon, N. W. Boggess, E. S. Cheng, G. de Amici, S. Gulkis, M. G. Hauser, G. Hinshaw, P. D. Jackson, M. Janssen, E. Kaita, T. Kelsall, P. Keegstra, C. Lineweaver, K. Loewenstein, P. Lubin, J. Mather, S. S. Meyer, S. H. Moseley, T. Murdock, L. Rokke, R. F. Silverberg, L. Tenorio, R. Weiss and D. T. Wilkinson, Astrophys. J. 396 (1992) L1, http://adsabs.harvard.edu/abs/1992ApJ...396L...1S. 10. Planck Collab. (P. A. R. Ade et al.), Planck 2013 results. XXII. Constraints on inflation, arXiv:1303.5082. 11. D. H. Lyth, Phys. Rev. Lett. 78 (1997) 1861, doi: 10.1103/PhysRevLett.78.1861. 12. G. Hinshaw, D. Larson, E. Komatsu, D. N. Spergel, C. L. Bennett, J. Dunkley, M. R. Nolta, M. Halpern, R. S. Hill, N. Odegard, L. Page, K. M. Smith, J. L. Weiland, B. Gold, N. Jarosik, A. Kogut, M. Limon, S. S. Meyer, G. S. Tucker, E. Wollack and E. L. Wright, Astrophys. J. Suppl. Series 208 (2013) 19, doi: 10.1088/00670049/208/2/19. 13. Planck Collab. (P. A. R. Ade et al.), Astron. Astrophys. 571 (2014) A16, doi: 10.1051/0004-6361/201321591. 14. K. M. Smith, C. Dvorkin, L. Boyle, N. Turok, M. Halpern, G. Hinshaw and B. Gold, Phys. Rev. Lett. 113 (2014) 031301, doi: 10.1103/PhysRevLett.113.031301. 15. I. Antoniadis and S. P. Patil, The effective planck mass and the scale of inflation, arXiv:1410.8845. 16. S. B. Giddings, S. Kachru and J. Polchinski, Phys. Rev. D 66 (2002) 106006, doi: 10.1103/PhysRevD.66.106006. 17. S. Kachru, R. Kallosh, A. D. Linde and S. P. Trivedi, Phys. Rev. D 68 (2003) 046005, doi: 10.1103/PhysRevD.68.046005. 18. M. R. Douglas and S. Kachru, Rev. Mod. Phys. 79 (2007) 733, doi: 10.1103/RevModPhys.79.733. 19. R. Bousso and J. Polchinski, J. High Energy Phys. 0006 (2000) 006, doi: 10.1088/1126-6708/2000/06/006. 20. Y. Sumitomo and S.-H. H. Tye, J. Cosmol. Astropart. Phys. 2 (2013) 006, doi: 10.1088/1475-7516/2013/02/006. 21. K. Freese, J. A. Frieman and A. V. Olinto, Phys. Rev. Lett. 65 (1990) 3233, doi: 10.1103/PhysRevLett.65.3233. 22. F. C. Adams, J. R. Bond, K. Freese, J. A. Frieman and A. V. Olinto, Phys. Rev. D 47 (1993) 426, doi: 10.1103/PhysRevD.47.426. 23. G. Dvali and S. H. Tye, Phys. Lett. B 450 (1999) 72, http://adsabs.harvard. edu/abs/1999PhLB..450...72D. 24. C. P. Burgess, M. Majumdar, D. Nolte, F. Quevedo, G. Rajesh and R.-J. Zhang, J. High Energy Phys. 7 (2001) 047, doi: 10.1088/1126-6708/2001/07/047.

May 4, 2017 9:54 b2167-ch19

Inflation, string theory and cosmic strings

II-317

25. G. Dvali, Q. Shafi and S. Solganik, D-brane inflation, arXiv:hep/th 0105203, http://adsabs.harvard.edu/abs/2001hep.th....5203D. 26. S. Kachru et al., J. Cosmol. Astropart. Phys. 0310 (2003) 013, doi: 10.1088/14757516/2003/10/013. 27. C. Burgess, R. Kallosh and F. Quevedo, J. High Energy Phys. 0310 (2003) 056. 28. M. Berg, M. Haack and B. Kors, On the moduli dependence of nonperturbative superpotentials in brane inflation, arXiv:hep-th/0409282. 29. D. Baumann et al., J. High Energy Phys. 0611 (2006) 031, doi: 10.1088/11266708/2006/11/031. 30. S. Buchan, B. Shlaer, H. Stoica and S. H. Tye, J. Cosmol. Astropart. Phys. 02 (2004) 013, http://adsabs.harvard.edu/abs/2004JCAP...02..013B. 31. H. Firouzjahi and S.-H. H. Tye, J. Cosmol. Astropart. Phys. 0503 (2005) 009, doi: 10.1088/1475-7516/2005/03/009. 32. R. Bean, X. Chen, G. Hailu, S.-H. H. Tye and J. Xu, J. Cosmol. Astropart. Phys. 0803 (2008) 026, doi: 10.1088/1475-7516/2008/03/026. 33. A. Krause and E. Pajer, J. Cosmol. Astropart. Phys. 7 (2008) 023, doi: 10.1088/14757516/2008/07/023. 34. E. Silverstein and D. Tong, Phys. Rev. D 70 (2004) 103505, doi: 10.1103/PhysRevD.70.103505. 35. M. Alishahiha, E. Silverstein and D. Tong, Phys. Rev. D 70 (2004) 123505, doi: 10.1103/PhysRevD.70.123505. 36. X. Chen, Phys. Rev. D 71 (2005) 063506, doi: 10.1103/PhysRevD.71.063506. 37. X. Chen, J. High Energy Phys. 8 (2005) 045, doi: 10.1088/1126-6708/2005/08/045. 38. K. Dasgupta, C. Herdeiro, S. Hirano and R. Kallosh, Phys. Rev. D 65 (2002) 126002, doi: 10.1103/PhysRevD.65.126002. 39. J. P. Conlon and F. Quevedo, J. High Energy Phys. 1 (2006) 146, doi: 10.1088/11266708/2006/01/146. 40. C. P. Burgess, M. Cicoli and F. Quevedo, J. Cosmol. Astropart. Phys. 11 (2013) 003, doi: 10.1088/1475-7516/2013/11/003. 41. J. E. Kim, H. P. Nilles and M. Peloso, J. Cosmol. Astropart. Phys. 0501 (2005) 005, doi: 10.1088/1475-7516/2005/01/005. 42. E. Silverstein and A. Westphal, Phys. Rev. D 78 (2008) 106003, doi: 10.1103/PhysRevD.78.106003. 43. L. McAllister, E. Silverstein and A. Westphal, Phys. Rev. D 82 (2010) 046003, doi: 10.1103/PhysRevD.82.046003. 44. A. D. Linde, Phys. Lett. B 129 (1983) 177, doi: 10.1016/0370-2693(83)90837-7. 45. K. Freese and W. H. Kinney, Natural Inflation: Consistency with Cosmic Microwave Background Observations of Planck and BICEP2, arXiv:1403.5277. 46. A. R. Liddle, A. Mazumdar and F. E. Schunck, Phys. Rev. D 58 (1998) 061301, doi: 10.1103/PhysRevD.58.061301. 47. A. Mazumdar, S. Panda and A. P´erez-Lorenzana, Nucl. Phys. B 614 (2001) 101, doi: 10.1016/S0550-3213(01)00410-2. 48. S. Dimopoulos, S. Kachru, J. McGreevy and J. G. Wacker, J. Cosmol. Astropart. Phys. 8 (2008) 003, doi: 10.1088/1475-7516/2008/08/003. 49. R. Easther and L. McAllister, J. Cosmol. Astropart. Phys. 5 (2006) 018, doi: 10.1088/1475-7516/2006/05/018. 50. K. Choi, H. Kim and S. Yun, Phys. Rev. 90 (2014) 023545, doi: 10.1103/PhysRevD.90.023545. 51. S.-H. H. Tye and S. S. C. Wong, Helical inflation and cosmic strings, arXiv:1404.6988.

May 4, 2017 9:54 b2167-ch19

II-318

D. F. Chernoff and S.-H. H. Tye

52. R. Kappl, S. Krippendorf and H. P. Nilles, Aligned natural inflation: Monodromies of two axions, arXiv:1404.7127. 53. I. Ben-Dayan, F. Gil Pedro and A. Westphal, Hierarchical Axion Inflation, arXiv:1404.7773. 54. C. Long, L. McAllister and P. McGuirk, Phys. Rev. 90 (2014) 023501, doi: 10.1103/PhysRevD.90.023501. 55. K. Harigaya and M. Ibe, Inflaton potential on a Riemann surface, arXiv:1404.3511. 56. T. Higaki and F. Takahashi, Natural and multi-natural inflation in axion landscape, arXiv:1404.6923. 57. T. C. Bachlechner, M. Dias, J. Frazer and L. McAllister, A new angle on chaotic inflation, arXiv:1404.7496. 58. M. Berg, E. Pajer and S. Sj¨ ors, Phys. Rev. D 81 (2010) 103535, doi: 10.1103/PhysRevD.81.103535. 59. R. Kallosh, A. Linde and B. Vercnocke, Natural inflation in supergravity and beyond, arXiv:1404.6244. 60. F. Marchesano, G. Shiu and A. M. Uranga, F-term axion monodromy inflation, arXiv:1404.3040. 61. L. McAllister, E. Silverstein, A. Westphal and T. Wrase, The powers of monodromy, arXiv:1405.3652. 62. M. Arends, A. Hebecker, K. Heimpel, S. C. Kraus, D. Lust, C. Mayrhofer, C. Schick and T. Weigand, D7-Brane moduli space in axion monodromy and fluxbrane inflation, arXiv:1405.0283. 63. M. R. Setare, A. Sepehri and V. Kamali, Phys. Lett. B 735 (2014) 84, doi: 10.1016/ j.physletb.2014.05.081. 64. H.-Y. Schive, T. Chiueh and T. Broadhurst, Nat. Phys. 10 (2014) 496, doi: 10.1038/ nphys2996. 65. E. Witten, Phys. Lett. B 153 (1985) 243, http://adsabs.harvard.edu/abs/ 1985PhLB..153..243W. 66. J. Polchinski, Phys. Rev. Lett. 75 (1995) 4724, doi: 10.1103/PhysRevLett.75.4724. 67. L. Randall and R. Sundrum, Phys. Rev. Lett. 83 (1999) 3370, doi: 10.1103/PhysRevLett.83.3370. 68. L. Leblond and S.-H. H. Tye, J. High Energy Phys. 3 (2004) 055, doi: 10.1088/11266708/2004/03/055. 69. I. R. Klebanov and M. J. Strassler, J. High Energy Phys. 8 (2000) 052, doi: 10.1088/1126-6708/2000/08/052. 70. S. S. Gubser, C. P. Herzog and I. R. Klebanov, J. High Energy Phys. 9 (2004) 036, doi: 10.1088/1126-6708/2004/09/036. 71. H. Firouzjahi, L. Leblond and S.-H. H. Tye, J. High Energy Phys. 5 (2006) 047, doi: 10.1088/1126-6708/2006/05/047. 72. E. J. Copeland, R. C. Myers and J. Polchinski, J. High Energy Phys. 06 (2004) 013, http://adsabs.harvard.edu/abs/2004JHEP...06..013C. 73. L. Leblond and M. Wyman, Phys. Rev. D 75 (2007) 123522, doi: 10.1103/PhysRevD.75.123522. 74. A. Avgoustidis, S. Chadburn and R. Gregory, Phys. Rev. D 86 (2012) 063516, doi: 10.1103/PhysRevD.86.063516. 75. C. L. Bennett, A. J. Banday, K. M. Gorski, G. Hinshaw, P. Jackson, P. Keegstra, A. Kogut, G. F. Smoot, D. T. Wilkinson and E. L. Wright, Astrophys. J. 464 (1996) L1, http://adsabs.harvard.edu/abs/1996ApJ...464L...1B. 76. D. N. Spergel, R. Bean, O. Dor´e, M. R. Nolta, C. L. Bennett, J. Dunkley, G. Hinshaw, N. Jarosik, E. Komatsu, L. Page, H. V. Peiris, L. Verde, M. Halpern, R. S.

May 4, 2017 9:54 b2167-ch19

Inflation, string theory and cosmic strings

77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88.

89.

90. 91. 92. 93. 94. 95.

96. 97.

II-319

Hill, A. Kogut, M. Limon, S. S. Meyer, N. Odegard, G. S. Tucker, J. L. Weiland, E. Wollack and E. L. Wright, Astrophys. J. Suppl. Series 170 (2007) 377, http://adsabs.harvard.edu/abs/2007ApJS..170..377S. N. Jones, H. Stoica and S. H. Tye, J. High Energy Phys. 07 (2002) 051, http:// adsabs.harvard.edu/abs/2002JHEP...07..051J. S. Sarangi and S. H. Tye, Phys. Lett. B 536 (2002) 185, http://adsabs.harvard. edu/abs/2002PhLB..536..185S. N. T. Jones, H. Stoica and S. H. Tye, Phys. Lett. B 563 (2003) 6, http://adsabs. harvard.edu/abs/2003PhLB..563....6J. H. Firouzjahi and S. H. Tye, J. Cosmol. Astropart. Phys. 03 (2005) 009, http:// adsabs.harvard.edu/abs/2005JCAP...03..009F. S. E. Shandera and S. H. Tye, J. Cosmol. Astropart. Phys. 05 (2006) 007, http://adsabs.harvard.edu/abs/2006JCAP...05..007S. M. G. Jackson, N. T. Jones and J. Polchinski, J. High Energy Phys. 10 (2005) 013, http://adsabs.harvard.edu/abs/2005JHEP...10..013J. A. Vilenkin, Phys. Lett. B 133 (1983) 177, http://adsabs.harvard.edu/abs/ 1983PhLB..133..177V. C. Hogan and R. Narayan, Mon. Not. R. Astron. Soc. 211 (1984) 575, http:// adsabs.harvard.edu/abs/1984MNRAS.211..575H. A. Vilenkin, Astrophys. J. 282 (1984) L51, http://adsabs.harvard.edu/abs/ 1984ApJ...282L..51V. A. A. de Laix, Phys. Rev. D 56 (1997) 6193, http://adsabs.harvard.edu/abs/ 1997PhRvD..56.6193D. F. Bernardeau and J. Uzan, Phys. Rev. D 63 (2001) 23005, http://adsabs. harvard.edu/abs/2001PhRvD..63b3005B. M. Sazhin, G. Longo, M. Capaccioli, J. M. Alcal, R. Silvotti, G. Covone, O. Khovanskaya, M. Pavlov, M. Pannella, M. Radovich and V. Testa, Mon. Not. R. Astron. Soc. 343 (2003) 353, http://adsabs.harvard.edu/abs/2003MNRAS. 343..353S. M. V. Sazhin, M. Capaccioli, G. Longo, M. Paolillo, O. S. Khovanskaya, N. A. Grogin, E. J. Schreier and G. Covone, arXiv:astro-ph/0601494, http://adsabs.harvard.edu/abs/2006astro.ph..1494S. J. L. Christiansen, E. Albin, K. A. James, J. Goldman, D. Maruyama and G. F. Smoot, Phys. Rev. D 77 (2008) 123509, doi: 10.1103/PhysRevD.77.123509. J. L. Christiansen, E. Albin, T. Fletcher, J. Goldman, I. P. W. Teng, M. Foley and G. F. Smoot, Phys. Rev. D 83 (2011) 122004, doi: 10.1103/PhysRevD.83.122004. C. Hogan and R. Narayan, Mon. Not. R. Astron. Soc. 211 (1984) 575, http://adsabs.harvard.edu/abs/1984MNRAS.211..575H. T. Vachaspati and A. Vilenkin, Phys. Rev. D 31 (1985) 3052, http://adsabs. harvard.edu/abs/1985PhRvD..31.3052V. A. Economou, D. Harari and M. Sakellariadou, Phys. Rev. D 45 (1992) 433, http://adsabs.harvard.edu/abs/1992PhRvD..45..433E. R. A. Battye, R. R. Caldwell and E. P. S. Shellard, Gravitational waves from cosmic strings, in Topological Defects in Cosmology, Proc of a conference held in Rome, Italy, 10–12 October, 1996. World Scientific Publishers, eds. M. Signore and F. Melchiorri (1998), p. 11. T. Damour and A. Vilenkin, Phys. Rev. Lett. 85 (2000) 3761, http://adsabs. harvard.edu/abs/2000PhRvL..85.3761D. T. Damour and A. Vilenkin, Phys. Rev. D 64 (2001) 64008, http://adsabs. harvard.edu/abs/2001PhRvD..64f4008D.

May 4, 2017 9:54 b2167-ch19

II-320

D. F. Chernoff and S.-H. H. Tye

98. T. Damour and A. Vilenkin, Phys. Rev. D 71 (2005) 63510, http://adsabs. harvard.edu/abs/2005PhRvD..71f3510D. 99. X. Siemens, J. Creighton, I. Maor, S. R. Majumder, K. Cannon and J. Read, Phys. Rev. D 73 (2006) 105001, http://adsabs.harvard.edu/abs/2006PhRvD.. 73j5001S. 100. C. J. Hogan, Phys. Rev. D 74 (2006) 43526, http://adsabs.harvard.edu/abs/ 2006PhRvD..74d3526H. 101. X. Siemens, V. Mandic and J. Creighton, Phys. Rev. Lett. 98 (2007) 111101, http://adsabs.harvard.edu/abs/2007PhRvL..98k1101S. 102. B. Abbott et al., Phys. Rev. D 76 (2007) 082001, http://adsabs.harvard.edu/ abs/2007PhRvD..76h2001A. 103. B. P. Abbott et al., Phys. Rev. D 80 (2009) 062002, doi: 10.1103/PhysRevD.80. 062002. 104. The LIGO and Virgo Coallab. (B. P. Abbott et al.), Nature 460 (2009) 990, http://adsabs.harvard.edu/abs/2009Natur.460..990A. 105. J. Aasi et al., Phys. Rev. Lett. 112 (2014) 131101, doi: 10.1103/PhysRevLett.112. 131101. 106. F. R. Bouchet and D. P. Bennett, Phys. Rev. D 41 (1990) 720, http://adsabs. harvard.edu/abs/1990PhRvD..41..720B. 107. R. R. Caldwell and B. Allen, Phys. Rev. D 45 (1992) 3447, http://adsabs. harvard.edu/abs/1992PhRvD..45.3447C. 108. V. M. Kaspi, J. H. Taylor and M. F. Ryba, Astrophys. J. 428 (1994) 713, http://adsabs.harvard.edu/abs/1994ApJ...428..713K. 109. F. A. Jenet, G. B. Hobbs, W. van Straten, R. N. Manchester, M. Bailes, J. P. W. Verbiest, R. T. Edwards, A. W. Hotan, J. M. Sarkissian and S. M. Ord, Astrophys. J. 653 (2006) 1571, http://adsabs.harvard.edu/abs/2006ApJ...653.1571J. 110. M. R. Depies and C. J. Hogan, Phys. Rev. D 75 (2007) 125006, http://adsabs.harvard.edu/abs/2007PhRvD..75l5006D. 111. P. B. Demorest, R. D. Ferdman, M. E. Gonzalez, D. Nice, S. Ransom, I. H. Stairs, Z. Arzoumanian, A. Brazier, S. Burke-Spolaor, S. J. Chamberlin, J. M. Cordes, J. Ellis, L. S. Finn, P. Freire, S. Giampanis, F. Jenet, V. M. Kaspi, J. Lazio, A. N. Lommen, M. McLaughlin, N. Palliyaguru, D. Perrodin, R. M. Shannon, X. Siemens, D. Stinebring, J. Swiggum and W. W. Zhu, Astrophys. J. 762 (2013) 94, doi: 10.1088/0004-637X/762/2/94. 112. L. Pogosian, S. H. Tye, I. Wasserman and M. Wyman, Phys. Rev. D 68 (2003) 23506, http://adsabs.harvard.edu/abs/2003PhRvD..68b3506P. 113. L. Pogosian, M. Wyman and I. Wasserman, J. Cosmol. Astropart. Phys. 09 (2004) 008, http://adsabs.harvard.edu/abs/2004JCAP...09..008P. 114. S. H. Tye, I. Wasserman and M. Wyman, Phys. Rev. D 71 (2005) 103508, http://adsabs.harvard.edu/abs/2005PhRvD..71j3508T. 115. M. Wyman, L. Pogosian and I. Wasserman, Phys. Rev. D 72 (2005) 23513, http://adsabs.harvard.edu/abs/2005PhRvD..72b3513W. 116. L. Pogosian, I. Wasserman and M. Wyman, On vector mode contribution to CMB temperature and polarization from local strings (2006), arXiv: astro-ph/0604141. 117. U. Seljak, A. Slosar and P. McDonald, J. Cosmol. Astropart. Phys. 10 (2006) 014, http://adsabs.harvard.edu/abs/2006JCAP...10..014S. 118. N. Bevis, M. Hindmarsh, M. Kunz and J. Urrestilla, Phys. Rev. D 75 (2007) 65015, http://adsabs.harvard.edu/abs/2007PhRvD..75f5015B. 119. A. A. Fraisse, J. Cosmol. Astropart. Phys. 03 (2007) 008, http://adsabs. harvard.edu/abs/2007JCAP...03..008F.

May 4, 2017 9:54 b2167-ch19

Inflation, string theory and cosmic strings

II-321

120. N. Bevis and P. M. Saffin, Phys. Rev. D 78 (2008) 23503, http://adsabs.harvard. edu/abs/2008PhRvD..78b3503B. 121. L. Pogosian, S. H. Tye, I. Wasserman and M. Wyman, J. Cosmol. Astropart. Phys. 02 (2009) 013, http://adsabs.harvard.edu/abs/2009JCAP...02..013P. 122. R. Battye and A. Moss, Phys. Rev. D 82 (2010) 023521, doi: 10.1103/PhysRevD.82.023521. 123. J. Dunkley et al., Astrophys. J. 739 (2011) 52, doi: 10.1088/0004-637X/739/1/52. 124. Planck Collab. (P. A. R. Ade et al.), Planck 2013 results. XXV. Searches for cosmic strings and other topological defects, arXiv:1303.5085. 125. E. J. Copeland, L. Pogosian and T. Vachaspati, Class. Quantum Grav. 28 (2011) 204009, doi: 10.1088/0264-9381/28/20/204009. 126. S. A. Sanidas, R. A. Battye and B. W. Stappers, Phys. Rev. D 85 (2012) 122003, doi: 10.1103/PhysRevD.85.122003. 127. A. Buonanno, TASI lectures on gravitational waves from the early universe, arXiv:grqc/0303085. ¨ 128. S. Olmez, V. Mandic and X. Siemens, Phys. Rev. D 81 (2010) 104028, http://adsabs.harvard.edu/abs/2010PhRvD..81j4028O. 129. E. Witten, Nucl. Phys. B 249 (1985) 557, doi: 10.1016/0550-3213(85)90022-7. 130. K. Miyamoto and K. Nakayama, J. Cosmol. Astropart. Phys. 7 (2013) 012, doi: 10.1088/1475-7516/2013/07/012. 131. Y.-W. Yu, K.-S. Cheng, G. Shiu and H. Tye, J. Cosmol. Astropart. Phys. 11 (2014) 040, doi: 10.1088/1475-7516/2014/11/040. 132. J. J. Blanco-Pillado, K. D. Olum and B. Shlaer, Phys. Rev. D 89 (2014) 023512, doi: 10.1103/PhysRevD.89.023512. 133. D. F. Chernoff and S. H. Tye, in preparation. 134. V. Vanchurin, K. Olum and A. Vilenkin, Phys. Rev. D 72 (2005) 63514, http:// adsabs.harvard.edu/abs/2005PhRvD..72f3514V. 135. M. Sakellariadou, J. Cosmol. Astropart. Phys. 04 (2005) 003, http://adsabs. harvard.edu/abs/2005JCAP...04..003S. 136. C. J. Martins and E. P. Shellard, Phys. Rev. D 73 (2006) 43515, http://adsabs. harvard.edu/abs/2006PhRvD..73d3515M. 137. A. Avgoustidis and E. P. Shellard, Phys. Rev. D 73 (2006) 41301, http://adsabs. harvard.edu/abs/2006PhRvD..73d1301A. 138. C. Ringeval, M. Sakellariadou and F. R. Bouchet, J. Cosmol. Astropart. Phys. 02 (2007) 023, http://adsabs.harvard.edu/abs/2007JCAP...02..023R. 139. J. Polchinski and J. V. Rocha, Phys. Rev. D 74 (2006) 83504, http://adsabs. harvard.edu/abs/2006PhRvD..74h3504P. 140. J. Polchinski and J. V. Rocha, Phys. Rev. D 75 (2007) 123503, http://adsabs. harvard.edu/abs/2007PhRvD..75l3503P. 141. D. F. Chernoff and S. H. H. Tye, Cosmic string detection via microlensing of stars, arXiv:0709.1139, http://arXiv.org/abs/0709.1139. 142. D. F. Chernoff, Clustering of superstring loops, arXiv:0908.4077, http://arxiv. org/abs/0908.4077. 143. C. J. Hogan, Nature 326 (1987) 853, http://adsabs.harvard.edu/abs/ 1987Natur.326..853H. 144. C. J. Hogan and M. J. Rees, Nature 311 (1984) 109, http://adsabs.harvard.edu/ abs/1984Natur.311..109H. 145. T. W. B. Kibble, Nucl. Phys. B 252 (1985) 227, http://adsabs.harvard.edu/ abs/1985NuPhB.252..227K.

May 4, 2017 9:54 b2167-ch19

II-322

D. F. Chernoff and S.-H. H. Tye

146. C. J. A. P. Martins and E. P. S. Shellard, Phys. Rev. D 53 (1996) 575, doi: 10.1103/PhysRevD.53.R575. 147. S. Kuroyanagi, K. Miyamoto, T. Sekiguchi, K. Takahashi and J. Silk, Phys. Rev. D 86 (2012) 023503, doi: 10.1103/PhysRevD.86.023503. 148. S. Kuroyanagi, K. Miyamoto, T. Sekiguchi, K. Takahashi and J. Silk, Phys. Rev. D 87 (2013) 023522, doi: 10.1103/PhysRevD.87.023522. 149. J. Binney and S. Tremaine, Galactic Dynamics, 2nd edn. (Princeton University Press, 2008). 150. S. Liebes, Phys. Rev. 133 (1964) 835, http://adsabs.harvard.edu/abs/ 1964PhRv..133..835L. 151. B. Paczynski, Astrophys. J. 304 (1986) 1, http://adsabs.harvard.edu/abs/ 1986ApJ...304....1P. 152. S. Mao, Res. Astron. Astrophys. 12 (2012) 947, doi: 10.1088/1674-4527/12/8/005. 153. J. K. Bloomfield and D. F. Chernoff, Phys. Rev. D 89 (2014) 124003, doi: 10.1103/ PhysRevD.89.124003. 154. M. S. Pshirkov and A. V. Tuntsov, Phys. Rev. D 81 (2010) 083519, doi: 10.1103/ PhysRevD.81.083519. 155. COROT: COnvection, ROtation & planetary Transits. http://smsc.cnes.fr/COROT (2014), http://smsc.cnes.fr/COROT. 156. Rossi X-Ray Timing Explorer (RXTE): December 1995 — January 2012. http:// heasarc.gsfc.nasa.gov/docs/xte/XTE.html (Sept. 2014). http://heasarc. gsfc.nasa.gov/docs/xte/XTE.html. 157. D. F. Chernoff, White paper: Microlensing of superstring loops. unpublished document contributed to LSST (2009). 158. D. F. Chernoff, R. Graziani, A. Kwok and T. Morris, to appear. 159. M. R. DePies and C. J. Hogan, Harmonic gravitational wave spectra of cosmic string loops in the galaxy, http://adsabs.harvard.edu/abs/2009arXiv0904.1052D. 160. K. Danzmann and T. A. Prince, Mission documents [LISA International Science Team]. http://list.caltech.edu/mission documents (2011). 161. A. J. Farmer and E. S. Phinney, Mon. Not. R. Astron. Soc. 346 (2003) 1197, http://adsabs.harvard.edu/abs/2003MNRAS.346.1197F. 162. E. O’Callaghan and R. Gregory, J. Cosmol. Astropart. Phys. 3 (2011) 004, doi: 10.1088/1475-7516/2011/03/004. 163. E. O’Callaghan, S. Chadburn, G. Geshnizjani, R. Gregory and I. Zavala, J. Cosmol. Astropart. Phys. 9 (2010) 013, doi: 10.1088/1475-7516/2010/09/013. 164. E. O’Callaghan, S. Chadburn, G. Geshnizjani, R. Gregory and I. Zavala, Phys. Rev. Lett. 105 (2010) 081602, doi: 10.1103/PhysRevLett.105.081602.

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Chapter 20 Quantum gravity: A brief history of ideas and some prospects

Steven Carlip†,∗∗ , Dah-Wei Chiou‡,§,†† , Wei-Tou Ni¶,‡‡ and Richard Woodard,§§ †Department of Physics, University of California Davis, Davis, California 95616, USA ‡Department of Physics, National Taiwan Normal University, Taipei 11677, Taiwan §Department of Physics and Center for Condensed Matter Sciences, National Taiwan University, Taipei 10617, Taiwan ¶Center for Gravitation and Cosmology, Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan Department

of Physics, University of Florida, Gainesville, Florida 32611, USA ∗∗[email protected] ††[email protected] ‡‡[email protected] §§ [email protected]

We present a bird’s-eye survey on the development of fundamental ideas of quantum gravity, placing emphasis on perturbative approaches, string theory, loop quantum gravity (LQG) and black hole thermodynamics. The early ideas at the dawn of quantum gravity as well as the possible observations of quantum gravitational effects in the foreseeable future are also briefly discussed. Keywords: Perturbative quantum gravity; string theory; loop quantum gravity; black hole thermodynamics; quantum gravity phenomenology. PACS Number: 04.60.−m

1. Prelude Quantum gravity is the research that seeks a consistent unification of the two foundational pillars of modern physics — quantum theory and Einstein’s theory of general relativity. It is commonly considered as the paramount open problem of theoretical physics, and many fundamental issues — such as the microscopic structure of space and time, the origin of the universe, the resolution of spacetime singularities, etc. — relies on a better understanding of quantum gravity. II-325

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The quest for a satisfactory quantum description of gravity began very early. Einstein after proposing general relativity thought that quantum effects must modify general relativity in his first paper on gravitational waves in 19161 (although he switched to a different point of view working on the unification of electromagnetism and gravitation in the 1930s). Klein argued that the quantum theory must ultimately modify the role of spatiotemporal concepts in fundamental physics2–4 and his ideas were developed by Deser.5 With the interests and developments of Rosenfeld, Pauli, Blokhintsev, Heisenberg, Gal’perin, Bronstein, Frenkel, van Dantzig, Solomon, Fierz and many other researchers, three approaches to quantum gravity after World War II had already been enunciated in 1930s pre-World War II as summarized by Stachel6 : (i) Quantum gravity should be formulated by analogy with quantum electrodynamics (Rosenfeld, Pauli and Fierz). (ii) The unique features of gravitation will require special treatment — the full theory with its nonlinear field equations must be quantized and generalized as to be applicable in the absence of a background metric (Bronstein and Solomon). (iii) General relativity is essentially a macroscopic theory, e.g. a sort of thermodynamics limit of a deeper, underlying theory of interactions between particles (Frenkel and van Dantzig). Many ideas of the early time continue to provide valuable insight about the nature of quantum gravity even today. For example, the 1939 work on linearized general relativity as a spin-2 field by Fierz and Pauli7 inspired a recent development on massive gravity, bimetric gravity, etc. Modern work on quantum gravity, however, did not really start-off until the development of a canonical formalism in 1959–1961 by Arnowitt, Deser and Misner (ADM) for the case of asymptotically flat boundary conditions8–18 (for a review, see Ref. 19). This served as the basis for Schoen and Yau’s proof that the classical energy is bounded below20,21 and is the starting point for numerical simulations of classical general relativity. Of course the ADM formalism also provides the Hamiltonian and canonical variables whose quantization defines quantum general relativity. However, the sacrifice of manifest covariance made perturbative computations prohibitively difficult, although many strategies to tackle this difficulty have been suggested, e.g. in loop quantum gravity (LQG). Ninety-nine years passed since the very first conception of Einstein. Today, quantum gravity has grown into a vast area of research in many different (both perturbative and nonperturbative) approaches. Many directions have led to significant advances with various appealing and ingenious ideas; these include causal sets, dynamical triangulation, emergent gravity, H-space theory, LQG, noncommutative geometry, string theory, supergravity, thermogravity, twistor theory and much more. (For surveys and references on various approaches of quantum gravity,

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see Refs. 22–25; for detailed accounts of the history and development, see Refs. 6 and 26; for a popular science account of quantum gravity, see the book Ref. 27.)

2. Perturbative Quantum Gravity The quantization of general relativity on the perturbative level (with or without matter) was the life work of DeWitt, for which he was awarded the 1987 Dirac Medal and the 2005 Einstein Prize.28–30 His program eventually succeeded — against the expectation of most expert opinion during the 1960s — in generalizing Feynman’s covariant quantization of quantum electrodynamics so that it could be applied as well to theories such as general relativity and Yang–Mills that are based on nonAbelian gauge symmetries. There were three key steps: (i) The introduction of the background field method for representing the effective action as a gauge invariant functional of the fields.29,31 This made clear the connection between invariant counterterms and all possible ultraviolet divergences of scattering amplitudes at a fixed order in the loop expansion. (ii) The realization that non-Abelian gauge symmetries require the inclusion of opposite-normed ghost fields to compensate the effect of unphysical polarizations in loop corrections.29,32 (iii) The development of invariant regularization techniques, the first of which was dimensional regularization.33,34 This permitted efficient computations to be made without the need for noninvariant counterterms. The first application of these techniques was made in 1974 by ’t Hooft and Veltman.35 They showed that general relativity without matter has a finite S-matrix at one-loop order, but that the addition of a scalar field leads to the need for higher curvature counterterms that would destabilize the universe. Very shortly thereafter Deser and van Nieuwenhuizen showed that similar unacceptable counterterms are required at one-loop for general relativity plus Maxwell’s electrodynamics,36,37 for general relativity plus a Dirac fermion38 and (with Tsao) for general relativity plus Yang–Mills.39 The fate of pure general relativity was settled in 1985 when Goroff and Sagnotti showed that higher curvature counterterms are required at two-loop order.40–42 These results show that quantum general relativity is not fundamentally consistent as a perturbative quantum field theory. Opinion is divided as to whether this means that general relativity must be abandoned as the fundamental theory of gravity or whether quantum general relativity might make sense nonperturbatively. String theory discards general relativity, and does not even employ the metric as a fundamental dynamical variable. LQG and causal dynamical triangulation are approaches that attempt to make sense of quantum general relativity without employing perturbation theory. At this time it would be fair to say that there is not yet any fully successful quantum theory of gravity.

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But being unable to do everything is not at all the same as lacking the ability to do anything. Perturbative quantum general relativity (with or without matter) can still be used in the standard sense of low-energy effective field theory,43,44 in the same way that Fermi theory was for years employed to understand the weak interactions, even including loop effects.45,46 The most interesting effects tend to occur in curved spacetime backgrounds, on which one can also consider quantizing only matter, without dynamical gravity.47 When working on curved backgrounds one often finds the formalism of flat space scattering theory to be inappropriate because it is based on incorrect assumptions about free vacuum at very early times (which actually began with an initial singularity) and at very late times (which may be filled with ensembles of particles created by the curved geometry). Solving the effective field equations under these conditions is especially troublesome because the in–out effective field equations at µ a spacetime point xµ contain contributions from points x
far in its future, and because even the in–out matrix elements of Hermitian operators such as the metric and the Maxwell field strength tensor can develop imaginary parts. For these reasons it is often more appropriate to employ the Schwinger–Keldysh formalism. This is a diagrammatic technique that is almost as simple to use as the standard Feynman rules, which gives true expectation values instead of in–out matrix elements. It was devised for quantum mechanics in 1960 by Schwinger.48 Over the next few years it was generalized to quantum field theory by Mahanthappa and Bakshi,49–51 and to statistical field theory by Keldysh.52 Although Schwinger–Keldysh effective µ field equations are nonlocal, they are causal — in the sense that only points x
in the past of xµ contribute — and the solutions for Hermitian fields are real.53,54 They are the natural way to study quantum effects in cosmology55 and nonequilibrium quantum field theory.56 Weinberg recently devised a variant of the formalism which is especially adapted to computing the correlation functions of primordial inflation.57,58 For a recent review of perturbative quantum gravity, see Ref. 59. 3. String Theory On the other hand, the search for a completely consistent theory of quantum gravity has never stopped. Although it does not embrace general relativity at the foundational level and it is disputable whether it can be genuinely formulated in a background-independent, nonperturbative fashion, string theory is arguably considered by many as the most promising candidate for a consistent theory of quantum gravity on the grounds that its low-energy limit surprisingly gives rise to (the supersymmetric extension of) the theory of (modified) gravity plus other force and matter fields. In the framework of string theory, the point-like particles of particle physics are replaced by (the different quantum excitations of) minuscule one-dimensional objects called strings. In addition to being a quantum theory of gravity, string theory also aims to unify all fundamental forces and all forms of matter, striving

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for the ultimate goal of being a “theory of everything”. (For textbooks on string theory, see Refs. 60–62; for a detailed account of its history and development, see Ref. 63.) String theory was born in the late 1960s as a never completely successful theory of the strong nuclear force and was later recognized as a suitable framework for a quantum theory of gravity (see Ref. 64 for its “prehistory”). The idea of identifying the string action as the area of the worldsheet of the string traveling in spacetime was introduced independently by Nambu,65 Goto66 and Hara67 in the early 1970s. The modern treatment of string theory based on the worldsheet path integral of the Polyakov action was initiated by Polyakov in 1981 (see Refs. 68 and 69) and has led to an intimate link with the conformal field theory (CFT). The worldsheet CFT in the presence of spacetime background fields was studied by Callan et al.70 ; the conformal invariance demands beta functions of the worldsheet field theory to vanish identically and consequently yields the equations of motion of the background fields that bear remarkable resemblance to the Einstein field equation and (non-Abelian) Maxwell’s equations with higher-order corrections. Quantization of the bosonic string requires the number of spacetime dimensions to be 26, of which 22 extra spatial dimensions are thought to be compactified in the deep microscopic scale and thus undetectable at low energies. The bosonic string theory is unsatisfactory in two aspects: First, it entails existence of negative-normed tachyon fields and consequently is nonunitary and inconsistent; second, it does not contain fermions and thus cannot account for the quarks and leptons in the standard model. In the 1980s, bosonic string theory was extended into superstring theory, which incorporates both bosonic and fermionic degrees of freedom via supersymmetry and requires six extra spatial dimensions to be compactified on a Calabi–Yau manifold. Superstring theory not only gets rid of negative-normed fields but also includes fermions as desired. There are basically two (equivalent) approaches to embody supersymmetry in string theory: (i) The Ramond–Neveu–Schwarz (RNS) formalism with manifest supersymmetry on the string worldsheet71,72 ; (ii) the Green–Schwarz (GS) formalism with manifest supersymmetry on the background spacetime.73–75 The first superstring revolution began in 1984 with the discovery by Green and Schwarz76 that the cancellation of gauge and gravitational anomalies demands a very strong constraint on the gauge symmetry: The gauge group must be either SO(32) or E8 × E8 . Eventually, five consistent but distinct superstring theories were found: Type I, type IIA, type IIB, SO(32) heterotic and E8 × E8 heterotic. In the late 1980s, it was realized that the two type II theories and the two heterotic theories are related by T-duality, which, simply speaking, maps a string winding around a compactified dimension of radius R to that of radius
2s /R, where
s is the string length scale. (For reviews on T-duality, see Refs. 77 and 78.) A few years later, S-duality was discovered as another kind of duality that maps the string coupling constant gs to 1/gs . The two basic examples are the duality that maps the type I theory to the SO(32) heterotic theory79 and the duality that maps the type

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IIB theory to itself.80 As S-duality relates a strongly coupled theory to a weakly coupled theory, it provides a powerful tool to explore nonperturbative behaviors of a theory with gs > 1, given the knowledge of the dual theory perturbatively obtained with gs < 1. In the mid-1990s, understanding of the nonperturbative physics of superstring theory progressed significantly, revealing that superstring theory contains various dynamical objects with p spatial dimensions called p-branes, in addition to the fundamental strings. Particularly, type I and II theories contain a class of p-branes, known as D-branes (or Dp-branes, more specifically), upon which open strings can end in Dirichlet boundary conditions. In 1995, Polchinski identified D-branes as solitonic solutions of supergravity that are understood as (generalized) black holes,81 a landmark discovery that heralded the prominence of D-brane dynamics. Furthermore, studies on the S-duality of type II82,83 and E8 × E8 heterotic84,85 theories sprang a big surprise: These two theories grow an eleventh dimension of size gs
s at strong coupling. These discoveries together with the aforementioned relations between different superstring theories via T- and S-dualities brought about the second superstring revolution that took place around 1994–1997, suggesting the existence of a more fundamental theory in 11 dimensions. The existence of such a theory was first conjectured and named as M-theory by Witten at a string theory conference at the University of Southern California in 1995. M-theory is supposed to be the fundamental theory of everything, of which the five distinct superstring theories along with the eleven-dimensional theory of supergravity are regarded as different limits. Although the complete formulation of M-theory remains elusive (and whether “M” should stand for “mother”, “magic”, “mystery”, “membrane”, “matrix”, etc. remains a matter of taste), a great deal about it can be learned by virtue of the fact that the theory should describe two- and five-dimensional objects known as M-branes (M2- and M5-branes, respectively) and reduce to elevendimensional supergravity theory at the low-energy limit. Investigations of M-theory have inspired a great number of important theoretical results in both physics and mathematics. A collection of coincident D-branes in string theory or M-branes in M-theory produces a warped spacetime with flux of gauge fields akin to a charged black hole, as the branes are sources of gauge flux and gravitational curvature. The low-energy limit of the gauge theory on the branes’ worldvolume (referred to as “boundary”) is found to describe the same physics of string theory or M-theory in the near-horizon geometry (referred to as “bulk”). In this way, one is led to conjecture a remarkable duality that relates conventional (nongravitational) quantum field theory on the boundary to string theories or M-theory on the bulk. This gauge theory/string theory duality is often referred to as anti-de Sitter (Ads)/CFT correspondence, which was first spelled out in a seminal paper by Maldacena in 1997.86 (A detailed review on AdS/CFT correspondence was given in Ref. 87.) By considering N coincident D3-branes in the type IIB theory, one obtains the celebrated example: The

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type IIB theory in AdS5 × S 5 (the product space of five-dimensional anti-de Sitter space and five-dimensional sphere) is equivalent to the SU (N ) super Yang–Mills theory with N = 4 supersymmetry on the four-dimensional boundary (which is a superconformal field theory). Many other examples have also been substantiated to various degrees of rigor. In the attempt to construct the complete formulation of M-theory, the AdS/CFT correspondence sets out a new strategy based on the holographic principle, which posits that the physics of a bulk region is completely encoded on its lower-dimensional boundary, as originally propounded by ’t Hooft in 199388 and elaborated by Susskind in 1994.89 In string theory, the shape and size of the compactified manifold and consequently the fundamental physical constants (lepton masses, coupling constants, cosmological constant, etc.) are dynamically determined by vacuum expectation values of scalar (moduli) fields. The crucial question arises known as the modulispace problem or moduli-stabilization problem: What mechanism stabilizes the compactified manifold and uniquely determines the physical constants? Answers to this problem have been proposed in the approach of flux compactifications as a generalization of conventional Calabi–Yau compactifications, which were first introduced by Strominger in 198690 and by de Wit et al. in 198791 and now have become a rapidly developing area of research. The idea is to generate a potential that stabilizes the moduli fields by compactifying string theory or M-theory on a warped geometry. In a warped geometry, the fluxes associated with certain tensor fields thread cycles of the compactified manifold; as the magnetic flux of an n-form field strength through an n-cycle depends only on the homology of the n-cycle, the flux stabilizes due to the flux quantization condition and gives rise to a nonvanishing potential. Flux compactifications were first studied in the context of M-theory by Becker and Becker in 1996.92 (A comprehensive review on flux compactifications was given in Ref. 93.) In 1999, Gukov et al. made it evident that flux compactfications generate nonvanishing potential for moduli fields, leading to a solution to the moduli-space problem.94 However, since the fluxes can take many different discrete values over different homology cycles of different (generalized) Calabi–Yau manifolds, flux compactifications typically yield a vast multitude (commonly estimated to be of the order 10500 (see Ref. 95)) of possible vacuum expectation values, referred to as the string theory landscape, and therefore we have to abandon the long-cherished hope that the fundamental physical constants are supposed to be uniquely fixed by string theory or M-theory. In response to this problem, Susskind in 200396 proposed the anthropic argument of the string theory landscape as a concrete implementation of the “anthropic principle”, which suggests that physical constants take their values not because fundamental laws of physics dictate so but rather because such values are at the loci of the landscape that are suitable to the existence of (intelligent) life (in order to measure these constants). The properties of the string theory landscape was shortly analyzed in a statistical approach by Douglas.95 The scientific relevance of the anthropic landscape has sparked fierce debate and remained highly

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controversial, yet it has gradually gained popularity (especially in the context of cosmology). The application of flux compactifications as well as the string theory landscape to the study of cosmology has spawned a whole new growing research area known as string cosmology. A notable subfield of it is brane cosmology, which posits that our four-dimensional universe is restricted to a brane inside a higher-dimensional space, as first proposed by Randall and Sundrum in 1999.97 The primary goal of string cosmology is to place field-theoretic models of cosmological inflation on a firmer logical ground from the perspective of string theory or, more boldly, to open entirely new realms of pre-big bang scenarios that cannot be described by any fieldtheoretic models. In 2003, Kachru et al. outlined the construction of metastable de Sitter vacua98 and shortly investigated the application to inflation.99 String cosmology has made some concrete predictions that will soon be confronted with the near-future astrophysical observations. (For reviews on string cosmology, see Refs. 100 and 101 and the textbook Ref. 102.)

4. Loop Quantum Gravity The primary competitor of string theory for a complete quantum theory is LQG. In sharp contrast to the ambitious goal of string theory, LQG does not intend to pursue a theory of everything that unifies all force and matter fields but deliberately adopts the “minimalist” approach in the sense that it focuses solely on the search for a consistent quantum theory of gravity without entailing any extraordinary ingredients such as extra dimensions, supersymmetry, etc. (although many of these can be incorporated compatibly). The beauty of LQG lies in its faithful attempt to establish a conceptual framework whereby the apparently conflicting tenets of quantum theory and Einstein’s theory of general relativity conjoin harmoniously, essentially in a nonperturbative, background-independent fashion. (For textbooks on LQG, see Refs. 103–105; for a detailed account of its history and development, see Ref. 106; for a recent review of LQG, see Ref. 107.) The research of LQG originated from the works by Ashtekar in 1986–1987,108,109 which reformulated Einstein’s general relativity into a new canonical formalism in terms of a self-dual spinorial connection and its conjugate momentum, now known as Ashtekar variables, casting general relativity in a language closer to that of the Yang–Mills gauge theory. The shift from metric to Ashtekar variables provided the possibility of employing nonperturbative techniques in gauge theories and opened a new avenue eventually leading to LQG. Shortly thereafter, Jacobson and Smolin discovered that Wilson loops of the Ashtekar connection are solutions of the Wheeler–DeWitt equation (the formal equation of quantum gravity).110 This remarkable discovery led to the “loop representation of quantum general relativity” introduced by Rovelli and Smolin in 1988–1990.111,112 The early developments of LQG were reviewed in Ref. 113.

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In the 1990s, by the works of Br¨ ugmann et al.,114,115 it became clear that intersections of loops are essential for the consistency of the theory and quantum states of gravity should be formulated in terms of intersecting loops, i.e. graphs with links and nodes. Later on, inspired by Penrose’s speculation on the combinatorial structure of space,116 Rovelli and Smolin117 obtained an explicit basis of states of quantum geometry known as spin networks, which are oriented graphs labeled by numbers associated with spin representations of SU (2) on each link and node. The mathematical construction of spin networks was systematized by Baez.118,119 The idea that LQG could predict discrete quantum geometry was first suggested in the study of “weave states”.120 In 1995, the area and volume operators were defined upon spin networks and their eigenvalue spectra were found to be discrete,121 showing that quantum geometry is indeed quantized in the Planck scale as had long been speculated. Rigorous and systematic studies of geometry operators and their eigenvalues were further elaborated (e.g. see Refs. 122–124). The physical interpretation of spin networks (or more precisely, diffeomorphisminvariant knot classes of spin networks, known as s-knots) is extremely appealing: They represent different quantized three-dimensional geometries, which are not quantum excitations in space but of space. This picture manifests the paradigm of background independence of general relativity, as any reference to the localization of spin networks is dismissed. Furthermore, inclusion of matter fields into LQG does not require a major revamp of the underlying framework. (For a systematic account of inclusion of matter, see Ref. 104.) The quantum states of space plus matter naturally extend the notion of spin networks with additional degrees of freedom. In the presence of matter fields, the background independence becomes even more prominent, as geometry and matter fields are genuinely on the equal footing and reside on top of one another via their contiguous relations in the spin network without any reference to a given background. While kinematics of LQG is well understood in terms of spin networks, its dynamics (i.e. evolution of spin networks) and low-energy (semiclassical) physics is much more difficult and unclear. An anomaly-free formulation of the quantum dynamics was first obtained by Thiemann in 1996,125 and shortly the formulation was fully developed in his remarkable “QSD” series of papers.126–128 Since then, a great number of variant approaches have been pursued, but the quantum dynamics remains very obscure, largely because implementation of the quantum Hamiltonian constraint is very intricate. The Master constraint program 129 initiated by Thiemannn proposed an elegant solution to the difficulties and has evolved into a fully combinatorial theory known as algebraic quantum gravity (AQG).130–133 Meanwhile, various techniques, notably by the idea of holomorphic coherent states,134–139 have been devised to investigate the low-energy physics. Up to now, LQG has achieved considerable progress in understanding the quantum dynamics and providing contact with the low-energy physics.

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Applying principles of LQG to cosmological settings leads to the symmetryreduced theory known as loop quantum cosmology (LQC), which was originally proposed by Bojowald in 1999 and reached a rigorous formulation around 2006. Providing a “bottom–up” approach to the full theory of LQG, LQC has become the most well-developed subfield of LQG and led to many significant successes. Most notably, it suggests a new cosmological scenario where the big bang is replaced by the quantum bounce, which bridges the present expanding universe with a preexistent contracting counterpart. Absence of cosmological singularities has been shown to be robust for a great variety of LQC models, therefore affirming the long-held conviction that singularities in classical general relativity should be resolved by the effects of quantum gravity. (For textbooks and reviews on LQC, see Refs. 140–143.) The standard formalism of LQG adopts the canonical (Hamiltonian) approach but is closely related to the covariant (sun-over-histories) approach of quantum gravity known as the spin foam theory. The concept of a spin foam, which can be viewed as a (discrete) “worldsurface” swept out by a spin network traveling and transmuting in time, was introduced in 1993 as inspired by the work of Ponzano– Regge model144 and later developed into a systematic framework of quantum gravity by Perez et al. in the 2000s. A spin foam represents a quantized spacetime in the same sense that a spin network represents a quantized space; the transition amplitude from one spin network to another is given as the discrete sum (with appropriate weights) over all possible spin foams that connect the initial and final spin networks. (For reviews on the spin foam theory, see Refs. 145–148.) LQG has grown into a very active research field pursued in many directions, both in the traditional canonical approach and in the covariant (i.e. spin foam) approach. Recently, the precise connection between the canonical and covariant approaches has become one of the central topics of LQG. The merger of different approaches has yielded profound insight and suggested a new theoretical framework referred to as covariant LQG, which provides new conceptual principles and could pave a royal road to a complete theory of quantum gravity as advocated by Rovelli149–151 (also see Ref. 105 for a comprehensive account). 5. Black Hole Thermodynamics Perhaps the most notable achievement in the study of quantum gravity so far was the discovery that black holes are not really black, but emit thermal Hawking radiation. The surprising fact that black holes behave as thermodynamic objects has radically affected our understanding of general relativity and given valuable hints about the nature of quantum gravity. The study of black hole thermodynamics can be traced back to 1972 when Hawking proved that an area of an event horizon can never decrease152 — a property reminiscent of the second law of thermodynamics. The resemblance between black hole mechanics and thermodynamics was further enhanced by the discovery of analogs of other thermodynamical laws and formally elaborated in the paper by

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Bardeen et al.153 in terms of the “four laws of black hole mechanics”, which are in clear parallel with the four usual laws of thermodynamics as the surface gravity (κ) plays the role of temperature and the area of horizon (Ahor ) of entropy. Around the same time, considering a series of thought experiments initiated by his advisor, Wheeler,154 Bekenstein argued that the black hole entropy has to be of the form SBH = ηkB Ahor /(
G), where kB is Boltzmann’s constant and η is a proportional constant of order one, and correspondingly obtained the “generalized second law of thermodynamics”, which asserts that the total of ordinary entropy of matter plus the black hole entropy never decreases.155–157 In 1974, using the newly developed techniques of quantum field theory in curved spacetime, Hawking158,159 showed that all black holes radiate as black bodies with the Hawking temperature TH =
κ/(2π). The first law of black hole mechanics then determines the entropy as SBH = 14 kB Ahor /(
G), which confirms Bekenstein’s expression with η = 1/4 and is often referred to as the Bekenstein–Hawking formula. As the equivalence principle implies that the gravitational field near a black hole horizon is locally equivalent to uniform acceleration in a flat spacetime, it is expected that an accelerated observer should perceive an effect similar to the Hawking radiation. In 1976, Unruh160 demonstrated that an observer moving with a constant proper acceleration a in Minkowski spacetime indeed see a thermal flux of particles with the Unruh temperature TU =
a/(2π), which is in almost exact analogy with the Hawking temperature. At almost the same time, Bisognano and Wichmann gave an independent and mathematically rigorous proof of the Unruh effect in quantum field theory.161 To resolve the worrying issue whether the flux of particles formally obtained via the standard quantum field-theoretical definitions of vacuum states and particle numbers can be physically interpreted as “particles” as seen by detectors, Unruh160 and DeWitt162 devised simple models of particle detectors and showed that the answer is affirmative for these detectors. The Hawking temperature and the Bekenstein–Hawking entropy are inherently quantum gravitational, in the sense that they depend explicitly on both Planck’s constant
and Newton’s constant G. They are also surprisingly universal. The entropy, for example, depends only on the horizon area, and takes the same simple form regardless of the black hole’s charges, angular momentum, horizon topology and even the number of spacetime dimensions. The thermal properties of black holes have not been directly observed, but by now they have been derived in so many different ways — from Hawking’s original calculation of quantum field theory in curved spacetime and Unruh’s approach of an accelerated observer to Euclidean partition functions,161,163–165 canonical quantization in Hamiltonian formulation,166 quantum tunneling,167,168 anomaly techniques,169–174 the computation of pair production amplitudes175–178 and many others developed since early on to very recently — that their existence seems very nearly certain. The natural question is then whether black hole thermodynamics, like the ordinary thermodynamics of matter, has a microscopic “statistical mechanical” explanation. In 1996, using methods based on D-branes and string duality in string

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theory, Strominger and Vafa suggested a strategy of “turning down” the strength of the gravitational interaction until a black hole becomes a weakly coupled system of strings and D-branes so that at weak coupling one can count the number of states; this strategy led to the same Bekenstein–Hawking entropy for extremal (i.e. maximally charged) supersymmetric (BPS) black holes in five dimensions.179 This remarkable result was quickly extended to a large number of extremal, nearextremal and some particular nonextremal black holes.180,181 However, it becomes harder and uncertain to obtain the right factor η = 1/4 for far-from-extremal black holes (the Schwarzschild black hole is a typical case). This issue is currently under intense investigation in various approaches but has yet to be fully elucidated. In 2005, Mathur proposed running the analysis backwards: Starting at weak coupling with a particular collection of strings and D-branes, one then turns the gravitational coupling up and see what geometry appears at strong coupling.182 The result is typically not a black hole but instead a “fuzzball” — a configuration with no horizon and no singularity but with a geometry akin to that of a black hole outside a would-be horizon.183 In a few special cases, one can count the number of such classical fuzzball geometries and reproduce the Bekenstein–Hawking entropy. In general, however, many of the relevant states may not have classical geometric descriptions. In the context of string theory, the notable AdS/CFT correspondence also provides a feasible scheme to derive the black hole entropy. For asymptotically Ads black holes, one can in principle compute the entropy by counting states in the (nongravitational) dual CFT. The most straightforward application of this correspondence is for the (2+1)-dimensional BTZ black hole.184 In 1997–1998, Strominger185 and Birmingham et al.186 independently computed the BTZ black hole entropy, which precisely reproduces the Bekenstein–Hawking expression. Since many higherdimensional near-extremal black holes have a near-horizon geometry of the form BTZ × trivial, the BTZ results can be used to obtain the entropy of a large class of string theoretical black holes.187 The second major research program to derive the black hole entropy from a microscopic picture is LQG. The suggestion that counting the quanta of area in LQG could offer the statistical description of black hole thermodynamics was first proposed by Krasnov in 1997.188 The idea essentially is that the statistical ensemble is composed of the microstates of the horizon geometry that give rise to a specified total area. By introducing the concept of “isolated horizons”, Ashtekar et al. rigorously developed the framework of the idea and derived the right Bekenstein– Hawking formula for the Schwarzschild black hole.189–191 These results were soon extended to other black holes with rotation, distortion, etc.192–197 In 2004, the combinatorial problem of counting the microscopic states in LQG was rephrased in a more manageable way by Domagala and Lewandowski198 and by Meissner.199 These works corrected a flawed assumption considered true for several years and enabled one to compute the formula for Ahor
G
to the subleading

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−1 1 A hor order as kB S = γγM A 4G
− 2 ln G
+· · ·, by which the Barbero–Immirzi parameter γ is fixed as γ = γM ≈ 0.23653. For small areas, the precise number counting was first suggested in 2006200,201 and later thoroughly investigated by employing combinatorial methods (see Ref. 202 for a detailed account and Ref. 203 for a review). The key discovery is that, for microscopic black holes, the so-called black hole degeneracy spectrum when plotted as a function of the area exhibits a striking “staircase” structure, which makes contact in a nontrivial way with the evenly-spaced black hole horizon area spectrum predicted by Bekenstein and Mukhanov.204 Recently, an explicit SU (2) formulation for the black hole entropy has been developed by Engle et al. based on covariant Hamiltonian methods,205–207 giving rigorous support for the earlier proposal that the quantum black hole degrees of freedom could be described by an SU (2) Chern–Simons theory. (Also see the review Ref. 203 and references therein for other recent advances.) In addition to string theory and LQG, we now have a number of statistical mechanical approaches to black hole thermodynamics. These include entanglement entropy,208–212 induced gravity213–215 and many others. While they may differ on the subleading corrections,216 they all seem to give the correct Bekenstein–Hawking −1 A hor S=A entropy. (In general, one expects kB 4G
− α ln G
+ · · ·, where the coefficient α depends on the quantum theory.) A new puzzle is why such a diverse set of quantum gravitational approaches should all agree. (There are some indications that even the subleading corrections might be universal, up to differences on the treatment of angular momentum and conserved charges.216) Even more puzzling is the “information loss problem”: If a black hole is formed by collapse of matter in a pure quantum state, how can the seemingly thermal final state be compatible with unitary evolution? This question is currently the subject of intense research, and may yield surprising new insights into the nature of space and time. (For a detailed account of the information paradox, see Ref. 217.) For a recent review of black hole thermodynamics, see Ref. 218.

6. Quantum Gravity Phenomenology Quantum gravity is often denounced as merely a theoretical enterprise that has no direct contact with experimental or observational realms, as quantum gravitational effects are appreciable only at the Planck scale, which is thought to be completely out of current reach. Today, as idea flourishes and technology advances drastically, the situation could be changed soon and quantum gravity might become reachable in the foreseeable future. Many quantum gravity models suggest departures from the equivalence principle, CPT symmetry and/or local Lorentz invariance in the deep Planck regime; among them the specific examples are string theory,219 braneworld scenarios,220 LQG and spin foam theory,221,222 noncommutative geometry,223–225 emergent gravity,226 etc. Deviations from the symmetries at the Planck scale in turn modify the

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pure metric spacetime structure and the local Lorentzian energy–momentum dispersion relation at lower-energy scale. Modifications of the pure metric spacetime structure implies deviations from the Einstein Equivalence Principle (EEP). The most stringent test of EEP comes from cosmic observations. The null result of birefringence in cosmic propagation of polarized photons and polarized gamma rays so far has ascertained the light-cone structure and the core metric to ultrahigh precision of 10−38 ; this high precision already probes the second-order in the ratio of W -boson mass or proton mass to the Planck mass.227 To this precision, the only freedoms left over regarding the metric structure are a scalar degree of freedom (dilaton) and a pseudo-scalar degree of freedom (axion). The dilaton alters the amplitude while the axion rotates the linear polarization of the cosmic propagation. Based on the cosmic propagation since the last scattering surface of the cosmic microwave background (CMB), the fractional variation of the dilaton degree of freedom is constrained to 8×10−4 by the precision agreement of the CMB spectrum to the black-body spectrum.228 The cosmic axion degree of freedom is constrained by comic polarization rotation (CPR): Uniform CPR to less than 0.02 (rad);229 CPR fluctuations to 0.02 (rad).229,230 Observations on ultrahigh energy cosmic Rays (UHECR)231 — cosmic rays with kinetic energy greater than 1018 eV, the most energetic particles ever observed — provide a chance to see possible modifications to the local Lorentzian energy– momentum dispersion relation of ultrarelativistic (v ∼ c) particles. The UHECR spectrum measured at the Pierre Auger Observatory thus far has put very strong constraints on deviations from the Lorentzian dispersion relation.232 Future investigation with improved precision will either impose even stronger constraints or pick up signals of new physics including quantum gravitational effects (see Refs. 233 and 234 for detailed surveys). Similarly, observations on propagation of ultrahigh energy cosmic electromagnetic waves can be used to test any momentum dependence of the speed of photons, as an indication of a breakdown of the Lorentz symmetry. Particularly, the detailed analysis of the relationship between the arrival time, photon momentum and redshift of gamma-ray bursts is approaching very closely to the desired sensitivity to the Planck-scale physics.235 On the other hand, the ultraprecise measurements of “atom-recoil frequency” in the cold-atom experiments allow us to probe the energy–momentum dispersion relation of nonrelativistic (v  c) particles with Planck-scale sensitivity, thus providing revealing insight into various types of modifications to the dispersion relation considered in the quantum gravity literature.236,237 Combining studies of the two complementary regimes of the nonrelativistic and ultrarelativistic dispersion relations would tell us a great deal about the nature of quantum spacetime.237 In the context of astrophysics, many theories — such as attractor theory,238 the DGP model,239 massive gravity,240 etc. — predict the effects of quantum gravity in the solar system dynamics. None of these effects have been observed yet, but

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they might be testable in the proposed space missions designed to test relativistic gravity241,242 and/or to detect and measure gravitational waves by using laser interferometry (see Ref. 243 for a detailed survey on space-based gravitationalwave detectors). Sensitive tests of the equivalence principle, local Lorentz invariance, modifications to Einstein’s field equations, etc. might also be possible in the coming space missions.244 In the context of cosmology, quantum effects in the pre-inflationary era may give rise to sufficient deviations from the standard inflationary scenario and leave footprints on the CMB. In string theory, various observable imprints on anisotropies and polarization of the CMB have been suggested by different models of string cosmology and by the existence of cosmic strings (see Ref. 102 for a review). In LQG, the standard inflationary scenario is extended to the epoch from the big bounce in the Planck era to the onset of slow-roll inflation.245 Some research works have attempted to reveal possible observable imprints of loop quantum effects on the CMB.246–251 There are a few experiments currently in operation or in ongoing development, aiming to make accurate measurements of anisotropies and polarization of the CMB, and their results will soon lead to great excitement. Very recently, the joint analysis of BICEP2/Keck Array and Planck data found no statistically significant evidence for tensor modes in the B-mode polarization, yielding an upper limit r < 0.12 at 95% confidence on the tensor-to-scalar ratio r.252 This upper limit already ruled out many “large-r scenarios” of pre-inflationary models, including those in string cosmology. The conclusive result of B-mode polarization expected to arrive in the near future will further discriminate between different quantum gravity theories. There are many other cosmological and astrophysical observations as well as laboratory experiments that could reveal possible quantum gravitational effects. At the turn of the centennial of the birth of Einstein’s general relativity, quantum gravity phenomenology will in time become an important area of relevant research. (For a comprehensive survey on quantum gravity phenomenology, see Ref. 253; also see Ref. 254 for a survey from the perspective of LQG.) Acknowledgements S.C. was partially supported by US Department of Energy grant DE-FG0291ER40674; D.W.C. by the Ministry of Science and Technology of Taiwan under the Grant No. 101-2112-M-003-002-MY3; and R.W. by NSF Grant PHY-1506513 and by the Institute for Fundamental Theory at the University of Florida. References 1. 2. 3. 4.

A. Einstein, Sitz.ber. Preuss. Akad. Wiss. Berl. Math. Phys. 1916 (1916) 688. O. Klein, Z. Phys. 46 (1927) 688. O. Klein, Kosmos (Sweden) 32 (1954) 33. O. Klein, Generalization of Einsteins theory of gravitation considered from the point of view of quantum field theory, in F¨ uenfzig Jahre Relativit¨ atstheorie: Proc. of the

May 4, 2017 9:53 b2167-ch20

II-340

5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

20. 21. 22.

23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46.

S. Carlip et al.

Jubilee of Relativity Theory, Bern, 11–16 July, 1955, eds. A. Mercier and M. Kervaire, Helvetica Physica Acta Supplementum, Vol. 4 (Birkh¨ auser, Basel, 1956). S. Deser, Rev. Mod. Phys. 29 (1957) 417. J. Stachel, Early history of quantum gravity (1916–1940), in Black Holes, Gravitational Radiation and the Universe, eds. B. R. Iyer and B. Bhawal (Kluwer, 1999). M. Fierz and W. Pauli, Proc. Roy. Soc. Lond. A 173 211 (1939). R. Arnowitt and S. Deser, Phys. Rev. 113 745 (1959). R. L. Arnowitt, S. Deser and C. W. Misner, Phys. Rev. 116 (1959) 1322. R. L. Arnowitt, S. Deser and C. W. Misner, Phys. Rev. 117 (1960) 1595. S. Deser, R. Arnowitt and C. W. Misner, Il Nuovo Cimento 15 (1960) 487. R. Arnowitt, S. Deser and C. W. Misner, Phys. Rev. 118 (1960) 1100. S. Deser, R. Arnowitt and C. W. Misner, J. Math. Phys. 1 (1960) 434. R. L. Arnowitt, S. Deser and C. W. Misner, Phys. Rev. 120 (1960) 313. S. Deser, R. Arnowitt and C. W. Misner, Ann. Phys. 11 (1960) 116. S. Deser, R. Arnowitt and C. W. Misner, Il Nuovo Cimento 19 (1961) 668. R. L. Arnowitt, S. Deser and C. W. Misner, Phys. Rev. 121 (1961) 1556. R. L. Arnowitt, S. Deser and C. W. Misner, Phys. Rev. 122 (1961) 997. R. L. Arnowitt, S. Deser and C. W. Misner, Gen. Rel. Gravit. 40 (2008) 1997, arXiv:gr-qc/0405109: Originally appeared as Chap. 7, pp. 227–264, in Gravitation: An Introduction to Current Research, ed. L. Witten (Wiley, New York, 1962). R. Schoen and S. T. Yau, Commun. Math. Phys. 65 (1979) 45. R. Schoen and S. T. Yau, Commun. Math. Phys. 79 (1981) 231. C. Rovelli, Strings, loops and others: A Critical survey of the present approaches to quantum gravity, in Gravitation and Relativity: At the Turn of the Millennium (Inter-University Centre for Astronomy and Astrophysics, Pune, 1998), pp. 281–331. S. Carlip, Rep. Prog. Phys. 64 (2001) 885, arXiv:gr-qc/0108040. R. P. Woodard, Rep. Prog. Phys. 72 (2009) 126002, arXiv:0907.4238 [gr-qc]. C. Kiefer, Quantum Gravity (Oxford University Press, Oxford, 2012). C. Rovelli, arXiv:gr-qc/0006061. L. Smolin, Three Roads to Quantum Gravity (Weidenfeld and Nicolson, London, 2000). B. S. DeWitt, Phys. Rev. 160 (1967) 1113. B. S. DeWitt, Phys. Rev. 162 (1967) 1195. B. S. DeWitt, Phys. Rev. 162 (1967) 1239. L. F. Abbott, Acta Phys. Pol. B 13 (1982) 33. L. D. Faddeev and V. N. Popov, Phys. Lett. B 25 (1967) 29. G. ’t Hooft and M. J. G. Veltman, Nucl. Phys. B 44 (1972) 189. C. G. Bollini and J. J. Giambiagi, Nuovo Cimento B 12 (1972) 20. G. ’t Hooft and M. J. G. Veltman, Ann. Poincar´e Phys. Theor. A 20 (1974) 69. S. Deser and P. van Nieuwenhuizen, Phys. Rev. Lett. 32 (1974) 245. S. Deser and P. van Nieuwenhuizen, Phys. Rev. D 10 (1974) 401. S. Deser and P. van Nieuwenhuizen, Phys. Rev. D 10 (1974) 411. S. Deser, H. S. Tsao and P. van Nieuwenhuizen, Phys. Rev. D 10 (1974) 3337. M. H. Goroff and A. Sagnotti, Phys. Lett. B 160 (1985) 81. M. H. Goroff and A. Sagnotti, Nucl. Phys. B 266 (1986) 709. A. E. M. van de Ven, Nucl. Phys. B 378 (1992) 309. J. F. Donoghue, Phys. Rev. Lett. 72 (1994) 2996, arXiv:gr-qc/9310024. J. F. Donoghue, Phys. Rev. D 50 (1994) 3874, arXiv:gr-qc/9405057. G. Feinberg and J. Sucher, Phys. Rev. 166 (1968) 1638. S. D. H. Hsu and P. Sikivie, Phys. Rev. D 49 (1994) 4951, arXiv:hep-ph/9211301.

May 4, 2017 9:53 b2167-ch20

Quantum gravity: A brief history of ideas and some prospects

47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60.

61.

62. 63. 64. 65.

66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81.

II-341

B. S. DeWitt, Phys. Rep. 19 (1975) 295. J. S. Schwinger, J. Math. Phys. 2 (1961) 407. K. T. Mahanthappa, Phys. Rev. 126 (1962) 329. P. M. Bakshi and K. T. Mahanthappa, J. Math. Phys. 4 (1963) 1. P. M. Bakshi and K. T. Mahanthappa, J. Math. Phys. 4 (1963) 12. L. V. Keldysh, Zh. Eksp. Teor. Fiz. 47 (1964) 1515 [Sov. Phys.-JETP 20 (1965) 1018]. K. C. Chou, Z. B. Su, B. L. Hao and L. Yu, Phys. Rep. 118 (1985) 1. R. D. Jordan, Phys. Rev. D 33 (1986) 444. E. Calzetta and B. L. Hu, Phys. Rev. D 35 (1987) 495. E. Calzetta and B. L. Hu, Phys. Rev. D 37 (1988) 2878. S. Weinberg, Phys. Rev. D 72 (2005) 043514, arXiv:hep-th/0506236. S. Weinberg, Phys. Rev. D 74 (2006) 023508, arXiv:hep-th/0605244. R. P. Woodard, Chapter 21 in this book; Int. J. Mod. Phys. D 23 (2014) 1430020, arXiv:1407.4748 [gr-qc]. M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory: Introduction, Vol. 1 (Cambridge University Press, Cambridge, 1987); Superstring Theory: Loop Amplitudes, Anomalies and Phenomenology, Vol. 2 (Cambridge University Press, Cambridge, 1987). J. Polchinski, String Theory: An Introduction to the Bosonic String, Vol. 1 (Cambridge University Press, Cambridge, 1998); String Theory: Superstring Theory and Beyond, Vol. 2 (Cambridge University Press, Cambridge, 1998). K. Becker, M. Becker and J. H. Schwarz, String Theory and M-Theory: A Modern Introduction (Cambridge University Press, Cambridge, 2007). D. Rickles, A Brief History of String Theory: From Dual Models to M-Theory (Springer-Verlag, Berlin, 2014). J. H. Schwarz, arXiv:0708.1917 [hep-th]. Y. Nambu, Duality and Hadrodynamics, Notes prepared for the Copenhagen High Energy Symposium (1970); Reprinted in Broken Symmetry: Selected Papers of Y. Nambu, eds. T. Eguchi and K. Nishijima (World Scientific, Singapore, 1995). T. Goto, Prog. Theor. Phys. 46 (1971) 1560. O. Hara, Prog. Theor. Phys. 46 (1971) 1549. A. M. Polyakov, Phys. Lett. B 103 (1981) 207. A. M. Polyakov, Phys. Lett. B 103 (1981) 211. C. G. Callan, Jr., E. J. Martinec, M. J. Perry and D. Friedan, Nucl. Phys. B 262 (1985) 593. P. Ramond, Phys. Rev. D 3 (1971) 2415. A. Neveu and J. H. Schwarz, Nucl. Phys. B 31 86 (1971). M. B. Green and J. H. Schwarz, Nucl. Phys. B 181 (1981) 502. M. B. Green and J. H. Schwarz, Nucl. Phys. B 198 (1982) 252. M. B. Green and J. H. Schwarz, Phys. Lett. B 109 (1982) 444. M. B. Green and J. H. Schwarz, Phys. Lett. B 149 (1984) 117. A. Giveon, M. Porrati and E. Rabinovici, Phys. Rep. 244 (1994) 77, arXiv:hep-th/ 9401139. E. Alvarez, L. Alvarez-Gaume and Y. Lozano, Nucl. Phys. Proc. Suppl. 41 (1995) 1, arXiv:hep-th/9410237. J. Polchinski and E. Witten, Nucl. Phys. B 460 (1996) 525, arXiv:hep-th/9510169. C. M. Hull and P. K. Townsend, Nucl. Phys. B 438 (1995) 109, arXiv:hep-th/ 9410167. J. Polchinski, Phys. Rev. Lett. 75 (1995) 4724, arXiv:hep-th/9510017.

May 4, 2017 9:53 b2167-ch20

II-342

S. Carlip et al.

82. 83. 84. 85. 86.

P. K. Townsend, Phys. Lett. B 350 (1995) 184, arXiv:hep-th/9501068. E. Witten, Nucl. Phys. B 443 (1995) 85, arXiv:hep-th/9503124. P. Hoˇrava and E. Witten, Nucl. Phys. B 460 (1996) 506, arXiv:hep-th/9510209. P. Hoˇrava and E. Witten, Nucl. Phys. B 475 (1996) 94, arXiv:hep-th/9603142. J. M. Maldacena, Int. J. Theor. Phys. 38 (1999) 1113; Adv. Theor. Math. Phys. 2 (1998) 231, arXiv:hep-th/9711200. O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz, Phys. Rep. 323 (2000) 183, arXiv:hep-th/9905111. G. ’t Hooft, Dimensional reduction in quantum gravity, in Salamfestschrift: A Collection of Talks from the Conf. on Highlights of Particle and Condensed Matter Physics, Trieste, Italy, 8–12 March, 1993 (World Scientific, 1993), arXiv:gr-qc/9310026. L. Susskind, J. Math. Phys. 36 (1995) 6377, arXiv: hep-th/9409089. A. Strominger, Nucl. Phys. B 274 (1986) 253. B. de Wit, D. J. Smit and N. D. H. Dass, Nucl. Phys. B 283 (1987) 165. K. Becker and M. Becker, Nucl. Phys. B 477 (1996) 155, arXiv:hep-th/9605053. M. Gra˜ na, Phys. Rep. 423 (2006) 91, arXiv:hep-th/0509003. S. Gukov, C. Vafa and E. Witten, Nucl. Phys. B 584 (2000) 69, Erratum 608 (2001) 477, arXiv:hep-th/9906070. M. R. Douglas, J. High Energy Phys. 0305 (2003) 046, arXiv:hep-th/0303194. L. Susskind, The Anthropic landscape of string theory, in Universe or Multiverse? ed. C. Bernard (Cambridge University Press, 2009), pp. 247–266. L. Randall and R. Sundrum, Phys. Rev. Lett. 83 (1999) 4690, arXiv:hep-th/9906064. S. Kachru, R. Kallosh, A. D. Linde and S. P. Trivedi, Phys. Rev. D 68 (2003) 046005, arXiv:hep-th/0301240. S. Kachru, R. Kallosh, A. D. Linde, J. M. Maldacena, L. P. McAllister and S. P. Trivedi, J. Cosmol. Astropart. Phys. 0310 (2003) 013, arXiv:hep-th/0308055. F. Quevedo, Class. Quantum Grav. 19 (2002) 5721, arXiv:hep-th/0210292. U. H. Danielsson, Class. Quantum Grav. 22 (2005) S1, arXiv:hep-th/0409274. D. Baumann and L. McAllister, Inflation and String Theory (Cambridge University Press, Cambridge, 2015). C. Rovelli, Quantum Gravity (Cambridge University Press, Cambridge, 2004). T. Thiemann, Modern Canonical Quantum General Relativity (Cambridge University Press, Cambridge, 2007). C. Rovelli and F. Vidotto, Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory (Cambridge University Press, Cambridge, 2014). C. Rovelli, Class. Quantum Grav. 28 (2011) 153002, arXiv:1012.4707 [gr-qc]. D. W. Chiou, Chapter 23 in this book; Int. J. Mod. Phys. D 24 (2014) 1530005, arXiv:1412.4362 [gr-qc]. A. Ashtekar, Phys. Rev. Lett. 57 (1986) 2244. A. Ashtekar, Phys. Rev. D 36 (1987) 1587. T. Jacobson and L. Smolin, Nucl. Phys. B 299 (1988) 295. C. Rovelli and L. Smolin, Phys. Rev. Lett. 61 (1988) 1155. C. Rovelli and L. Smolin, Nucl. Phys. B 331 (1990) 80. C. Rovelli, Class. Quantum Grav. 8 (1991) 1613. B. Br¨ ugmann, R. Gambini and J. Pullin, Phys. Rev. Lett. 68 (1992) 431. B. Br¨ ugmann, R. Gambini and J. Pullin, Nucl. Phys. B 385 (1992) 587, arXiv:hepth/9202018. R. Penrose, Angular momentum: An approach to combinatorial spacetime, in Quantum Theory and Beyond, ed. T. Bastin (Cambridge University Press, Cambridge 1971).

87. 88.

89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105.

106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116.

May 4, 2017 9:53 b2167-ch20

Quantum gravity: A brief history of ideas and some prospects

II-343

117. C. Rovelli and L. Smolin, Phys. Rev. D 52 (1995) 5743, arXiv:gr-qc/9505006. 118. J. C. Baez, Adv. Math. 117 (1996) 253, arXiv:gr-qc/9411007. 119. J. C. Baez, Spin networks in nonperturbative quantum gravity, in The Interface of Knots and Physics, ed. L. H. Kauffman (American Mathematical Soc. San Francisco, 1995), pp. 167–203. 120. A. Ashtekar, C. Rovelli and L. Smolin, Phys. Rev. Lett. 69 (1992) 237, arXiv:hepth/9203079. 121. C. Rovelli and L. Smolin, Nucl. Phys. B 442 (1995) 593, Erratum Nucl. Phys. B 456 (1995) 753, arXiv:gr-qc/9411005. 122. R. De Pietri and C. Rovelli, Phys. Rev. D 54 (1996) 2664, arXiv:gr-qc/9602023. 123. T. Thiemann, J. Math. Phys. 39 (1998) 3347, arXiv:gr-qc/9606091. 124. J. Lewandowski, Class. Quantum Grav. 14 (1997) 71, arXiv:gr-qc/9602035. 125. T. Thiemann, Phys. Lett. B 380 (1996) 257, arXiv:gr-qc/9606088. 126. T. Thiemann, Class. Quantum Grav. 15 (1998) 839, arXiv:gr-qc/9606089. 127. T. Thiemann, Class. Quantum Grav. 15 (1998) 875, arXiv:gr-qc/9606090. 128. T. Thiemann, Class. Quantum Grav. 15 (1998) 1207, arXiv:gr-qc/9705017. 129. T. Thiemann, Class. Quantum Grav. 23 (2006) 2211, arXiv:gr-qc/0305080. 130. K. Giesel and T. Thiemann, Class. Quantum Grav. 24 (2007) 2465, arXiv:gr-qc/ 0607099. 131. K. Giesel and T. Thiemann, Class. Quantum Grav. 24 (2007) 2499, arXiv:gr-qc/ 0607100. 132. K. Giesel and T. Thiemann, Class. Quantum Grav. 24 (2007) 2565, arXiv:gr-qc/ 0607101. 133. K. Giesel and T. Thiemann, Class. Quantum Grav. 27 (2010) 175009, arXiv: 0711.0119 [gr-qc]. 134. A. Ashtekar, J. Lewandowski, D. Marolf, J. Mourao and T. Thiemann, J. Funct. Anal. 135 (1996) 519, arXiv:gr-qc/9412014. 135. T. Thiemann, Class. Quantum Grav. 23 (2006) 2063, arXiv:gr-qc/0206037. 136. B. Bahr and T. Thiemann, Class. Quantum Grav. 26 (2009) 045011, arXiv:0709.4619 [gr-qc]. 137. B. Bahr and T. Thiemann, Class. Quantum Grav. 26 (2009) 045012, arXiv:0709.4636 [gr-qc]. 138. C. Flori, arXiv:0904.1303 [gr-qc]. 139. E. Bianchi, E. Magliaro and C. Perini, Phys. Rev. D 82 (2010) 024012, arXiv:0912.4054 [gr-qc]. 140. M. Bojowald, Living Rev. Rel. 11 (2008) 4, http://www.livingreviews.org/ Articles/lrr-2008-4. 141. A. Ashtekar and P. Singh, Class. Quantum Grav. 28 (2011) 213001, arXiv:1108.0893 [gr-qc]. 142. M. Bojowald, Lect. Notes Phys. 835 (2011) 1. 143. M. Bojowald, Canonical Gravity and Applications: Cosmology, Black Holes, and Quantum Gravity (Cambridge University Press, Cambridge, 2011). 144. N. J. LaFave, arXiv:gr-qc/9310036. 145. A. Perez, Class. Quantum Grav. 20 (2003) R43, arXiv:gr-qc/0301113. 146. D. Oriti, arXiv:gr-qc/0311066. 147. A. Perez, arXiv:gr-qc/0409061. 148. A. Perez, Living Rev. Rel. 16 (2013) 3, http://www.livingreviews.org/Articles/ lrr2013-3, arXiv:1205.2019 [gr-qc]. 149. C. Rovelli, Class. Quantum Grav. 28 (2011) 114005, arXiv:1004.1780 [gr-qc]. 150. C. Rovelli, PoS (QGQGS) 2011 (2011) 003, arXiv:1102.3660 [gr-qc].

May 4, 2017 9:53 b2167-ch20

II-344

151. 152. 153. 154. 155. 156. 157. 158. 159. 160. 161. 162.

163. 164. 165. 166. 167. 168. 169. 170. 171. 172. 173. 174. 175. 176. 177. 178. 179. 180.

181. 182. 183.

S. Carlip et al.

C. Rovelli, Lect. Notes Phys. 863 (2013) 57. S. W. Hawking, Commun. Math. Phys. 25 (1972) 152. J. M. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys. 31 (1973) 161. J. A. Wheeler and K. Ford, Geons, Black Holes, and Quantum Foam (W. W. Norton, New York, 1998), Chap. 4. J. D. Bekenstein, Lett. Nuovo Cimento 4 (1972) 737. J. D. Bekenstein, Phys. Rev. D 7 (1973) 2333. J. D. Bekenstein, Phys. Rev. D 9 (1974) 3292. S. W. Hawking, Nature 248 (1974) 30. S. W. Hawking, Commun. Math. Phys. 43 (1975) 199, Erratum 46 (1976) 206. W. G. Unruh, Phys. Rev. D 14 (1976) 870. J. J. Bisognano and E. H. Wichmann, J. Math. Phys. 17 (1976) 303. B. S. DeWitt, Quantum gravity: The new synthesis, in General Relativity: An Einstein Centenary Survey, eds. S. W. Hawking and W. Israel (Cambridge University Press, Cambridge, 1979). G. W. Gibbons and M. J. Perry, Phys. Rev. Lett. 36 (1976) 985. G. W. Gibbons and M. J. Perry, Proc. R. Soc. Lond. A 358 (1978) 467494. G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15 (1977) 2752. S. Carlip and C. Teitelboim, Class. Quantum Grav. 12 (1995) 1699, arXiv:gr-qc/ 9312002. M. K. Parikh and F. Wilczek, Phys. Rev. Lett. 85 (2000) 5042, arXiv:hep-th/9907001. L. Vanzo, G. Acquaviva and R. Di Criscienzo, Class. Quantum Grav. 28 (2011) 183001, arXiv:1106.4153 [gr-qc]. S. P. Robinson and F. Wilczek, Phys. Rev. Lett. 95 (2005) 011303, arXiv:gr-qc/ 0502074. S. Iso, H. Umetsu and F. Wilczek, Phys. Rev. Lett. 96 (2006) 151302, arXiv:hep-th/ 0602146. S. Iso, H. Umetsu and F. Wilczek, Phys. Rev. D 74 (2006) 044017, arXiv:hep-th/ 0606018. S. Iso, T. Morita and H. Umetsu, Phys. Rev. D 77 (2008) 045007, arXiv:0710.0456 [hep-th]. L. Bonora, M. Cvitan, S. Pallua and I. Smoli´c, J. High Energy Phys. 0812 (2008) 021, arXiv:0808.2360 [hep-th]. L. Bonora, M. Cvitan, S. Pallua and I. Smoli´c, Phys. Rev. D 80 (2009) 084034, arXiv:0907.3722 [hep-th]. D. Garfinkle, S. B. Giddings and A. Strominger, Phys. Rev. D 49 (1994) 958, arXiv:gr-qc/9306023. J. D. Brown, Phys. Rev. D 51 (1995) 5725, arXiv:gr-qc/9412018. F. Dowker, J. P. Gauntlett, D. A. Kastor and J. H. Traschen, Phys. Rev. D 49 (1994) 2909, arXiv:hep-th/9309075. R. B. Mann and S. F. Ross, Phys. Rev. D 52 (1995) 2254, arXiv:gr-qc/9504015. A. Strominger and C. Vafa, Phys. Lett. B 379 (1996) 99, arXiv:hep-th/9601029. A. W. Peet, TASI lectures on black holes in string theory, in TASI 99 : Strings, Branes and Gravity, eds. J. Harvey, S. Kachru and E. Silverstein (World Scientific, Singapore, 2001). S. R. Das and S. D. Mathur, Annu. Rev. Nucl. Part. Sci. 50 (2000) 153, arXiv:grqc/0105063. S. D. Mathur, Fortschr. Phys. 53 (2005) 793, arXiv:hep-th/0502050. S. D. Mathur, Class. Quantum Grav. 23 (2006) R115, arXiv:hep-th/0510180.

May 4, 2017 9:53 b2167-ch20

Quantum gravity: A brief history of ideas and some prospects

II-345

184. M. Banados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. 69 (1992) 1849, arXiv:hep-th/9204099. 185. A. Strominger, J. High Energy Phys. 9802 (1998) 009, arXiv:hep-th/9712251. 186. D. Birmingham, I. Sachs and S. Sen, Phys. Lett. B 424 (1998) 275, arXiv:hep-th/ 9801019. 187. K. Skenderis, Lect. Notes Phys. 541 (2000) 325, arXiv:hep-th/9901050. 188. K. V. Krasnov, Phys. Rev. D 55 (1997) 3505, arXiv:gr-qc/9603025. 189. A. Ashtekar, J. Baez, A. Corichi and K. Krasnov, Phys. Rev. Lett. 80 (1998) 904, arXiv:gr-qc/9710007. 190. A. Ashtekar, A. Corichi and K. Krasnov, Adv. Theor. Math. Phys. 3 (1999) 419, arXiv:gr-qc/9905089. 191. A. Ashtekar, J. C. Baez and K. Krasnov, Adv. Theor. Math. Phys. 4 (2000) 1, arXiv:gr-qc/0005126. 192. A. Ashtekar, S. Fairhurst and B. Krishnan, Phys. Rev. D 62 (2000) 104025, arXiv:grqc/0005083. 193. A. Ashtekar, C. Beetle and J. Lewandowski, Phys. Rev. D 64 (2001) 044016, arXiv:grqc/0103026. 194. A. Ashtekar, C. Beetle and J. Lewandowski, Class. Quantum Grav. 19 (2002) 1195, arXiv:gr-qc/0111067. 195. A. Ashtekar, A. Corichi and D. Sudarsky, Class. Quantum Grav. 20 (2003) 3413, arXiv:gr-qc/0305044. 196. A. Ashtekar and A. Corichi, Class. Quantum Grav. 20 (2003) 4473, arXiv:grqc/0305082. 197. A. Ashtekar, J. Engle and C. Van Den Broeck, Class. Quantum Grav. 22 (2005) L27, arXiv:gr-qc/0412003. 198. M. Domagala and J. Lewandowski, Class. Quantum Grav. 21 (2004) 5233, arXiv:grqc/0407051. 199. K. A. Meissner, Class. Quantum Grav. 21 (2004) 5245, arXiv:gr-qc/0407052. 200. A. Corichi, J. Diaz-Polo and E. Fernandez-Borja, Class. Quantum Grav. 24 (2007) 243, arXiv:gr-qc/0605014. 201. A. Corichi, J. Diaz-Polo and E. Fernandez-Borja, Phys. Rev. Lett. 98 (2007) 181301, arXiv:gr-qc/0609122. 202. I. Agullo, J. Fernando Barbero, E. F. Borja, J. Diaz-Polo and E. J. S. Villasenor, Phys. Rev. D 82 (2010) 084029, arXiv:1101.3660 [gr-qc]. 203. J. F. G. Barbero, J. Lewandowski and E. J. S. Villasenor, PoS (QGQGS) 2011 (2011) 023, arXiv:1203.0174 [gr-qc]. 204. J. D. Bekenstein and V. F. Mukhanov, Phys. Lett. B 360 (1995) 7, arXiv:grqc/9505012. 205. J. Engle, A. Perez and K. Noui, Phys. Rev. Lett. 105 (2010) 031302, arXiv:0905.3168 [gr-qc]. 206. J. Engle, K. Noui, A. Perez and D. Pranzetti, Phys. Rev. D 82 (2010) 044050, arXiv:1006.0634 [gr-qc]. 207. J. Engle, K. Noui, A. Perez and D. Pranzetti, J. High Energy Phys. 1105 (2011) 016, arXiv:1103.2723 [gr-qc]. 208. R. D. Sorkin, On the entropy of the vacuum outside a horizon, in Proc. of the Tenth Int. Conf. General Relativity and Gravitation, Padova, 4–9 July, 1983, Vol. II (Consiglio Nazionale della Ricerche, Roma, 1983), pp. 734–736. 209. L. Bombelli, R. K. Koul, J. Lee and R. D. Sorkin, Phys. Rev. D 34 (1986) 373. 210. M. Srednicki, Phys. Rev. Lett. 71 (1993) 666, arXiv:hep-th/9303048. 211. S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96 (2006) 181602, arXiv:hep-th/0603001.

May 4, 2017 9:53 b2167-ch20

II-346

S. Carlip et al.

212. V. E. Hubeny, M. Rangamani and T. Takayanagi, J. High Energy Phys. 0707 (2007) 062, arXiv:0705.0016 [hep-th]. 213. A. D. Sakharov, Sov. Phys.-Dokl. 12 (1968) 1040, Dokl. Akad. Nauk Ser. Fiz. 177 (1967) 70; Sov. Phys.-Usp. 34 (1991) 394; Gen. Relat. Gravit. 32 (2000) 365. 214. V. P. Frolov, D. V. Fursaev and A. I. Zelnikov, Nucl. Phys. B 486 (1997) 339, arXiv:hep-th/9607104. 215. V. P. Frolov and D. V. Fursaev, Phys. Rev. D 56 (1997) 2212, arXiv:hep-th/9703178. 216. S. Carlip, Class. Quantum Grav. 17 (2000) 4175, arXiv:gr-qc/0005017. 217. S. D. Mathur, Lect. Notes Phys. 769 (2009) 3, arXiv:0803.2030 [hep-th]. 218. S. Carlip, Chapter 22 in this book; Int. J. Mod. Phys. D 23 (2014) 1430023, arXiv:1410.1486 [gr-qc]. 219. V. A. Kostelecky and S. Samuel, Phys. Rev. D 39 (1989) 683. 220. C. P. Burgess, J. M. Cline, E. Filotas, J. Matias and G. D. Moore, J. High Energy Phys. 0203 (2002) 043, arXiv:hep-ph/0201082. 221. R. Gambini and J. Pullin, Phys. Rev. D 59 (1999) 124021, arXiv:gr-qc/9809038. 222. C. Rovelli and S. Speziale, Phys. Rev. D 83 (2011) 104029, arXiv:1012.1739 [gr-qc]. 223. J. Lukierski, H. Ruegg and W. J. Zakrzewski, Ann. Phys. 243 (1995) 90, arXiv:hepth/9312153. 224. G. Amelino-Camelia and S. Majid, Int. J. Mod. Phys. A 15 (2000) 4301, arXiv:hepth/9907110. 225. S. M. Carroll, J. A. Harvey, V. A. Kostelecky, C. D. Lane and T. Okamoto, Phys. Rev. Lett. 87 (2001) 141601, arXiv:hep-th/0105082. 226. C. Barcelo, S. Liberati and M. Visser, Living Rev. Rel. 8 (2005) 12; Living Rev. Rel. 14 (2011) 3, http://relativity.livingreviews.org/Articles/lrr-2011-3/, arXiv:grqc/0505065. 227. W. T. Ni, Phys. Lett. A 379 (2015) 1297, arXiv:1411.0460 [gr-qc]. 228. W. T. Ni, Phys. Lett. A 378 (2014) 3413, arXiv:1410.0126 [gr-qc]. 229. S. di Serego Alighieri, Int. J. Mod. Phys. D 24 (2015) 1530016, arXiv:1501.06460 [astro-ph.CO]. 230. H. H. Mei, W. T. Ni, W. P. Pan, L. Xu and S. di Serego Alighieri, Astrophys. J. 805 (2015) 107, arXiv:1412.8569 [astro-ph.CO]. 231. K. Kotera and A. V. Olinto, Annu. Rev. Astron. Astrophys. 49 (2011) 119, arXiv:1101.4256 [astro-ph.HE]. 232. Pierre Auger Collab. (T. Yamamoto), arXiv:0707.2638 [astro-ph]. 233. F. W. Stecker and S. T. Scully, New J. Phys. 11 (2009) 085003, arXiv:0906.1735 [astro-ph.HE]. 234. S. Liberati and L. Maccione, J. Phys., Conf. Ser. 314 (2011) 012007, arXiv:1105.6234 [astro-ph.HE]. 235. G. Amelino-Camelia, A. Marciano, M. Matassa and G. Rosati, arXiv:1006.0007 [astro-ph.HE]. 236. G. Amelino-Camelia, C. L¨ ammerzahl, F. Mercati and G. M. Tino, Phys. Rev. Lett. 103 (2009) 171302, arXiv:0911.1020 [gr-qc]. 237. F. Mercati, D. Mazon, G. Amelino-Camelia, J. M. Carmona, J. L. Cortes, J. Indurain, C. L¨ ammerzahl and G. M. Tino, Class. Quantum Grav. 27 (2010) 215003, arXiv:1004.0847 [gr-qc]. 238. T. Damour and K. Nordtvedt, Phys. Rev. Lett. 70 (1993) 2217. 239. G. R. Dvali, G. Gabadadze and M. Porrati, Phys. Lett. B 485 (2000) 208, arXiv:hepth/0005016. 240. C. de Rham, Living Rev. Rel. 17 (2014) 7, http://relativity.livingreviews.org/ Articles/lrr-2014-7/, arXiv:1401.4173 [hep-th].

May 4, 2017 9:53 b2167-ch20

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II-347

241. W. T. Ni, Class. Quantum Grav. 26 (2009) 075021, arXiv:0812.0887 [astro-ph]. 242. ASTROD Collab. (C. Braxmaier et al.), Exp. Astron. 34 (2012) 181, arXiv:1104.0060 [gr-qc]. 243. J. R. Gair, M. Vallisneri, S. L. Larson and J. G. Baker, Living Rev. Rel. 16 (2013) 7, http://relativity.livingreviews.org/Articles/lrr-2013-7/, arXiv:1212.5575 [gr-qc]. 244. C. L¨ ammerzahl, arXiv:gr-qc/0402122. 245. A. Ashtekar, arXiv:1303.4989 [gr-qc]. 246. J. Grain and A. Barrau, Phys. Rev. Lett. 102 (2009) 081301, arXiv:0902.0145 [gr-qc]. 247. A. Barrau, arXiv:0911.3745 [gr-qc]. 248. J. Mielczarek, T. Cailleteau, J. Grain and A. Barrau, Phys. Rev. D 81 (2010) 104049, arXiv:1003.4660 [gr-qc]. 249. J. Grain, A. Barrau, T. Cailleteau and J. Mielczarek, Phys. Rev. D 82 (2010) 123520, arXiv:1011.1811 [astro-ph.CO]. 250. A. Barrau, PoS (ICHEP) 2010 (2010) 461, arXiv:1011.5516 [gr-qc]. 251. I. Agullo, A. Ashtekar and W. Nelson, Class. Quantum Grav. 30 (2013) 085014, arXiv:1302.0254 [gr-qc]. 252. BICEP2 and Planck Collab. (P. A. R. Ade et al.), Phys. Rev. Lett. 114(10) (2015) 101301, arXiv:1502.00612 [astro-ph.CO]. 253. G. Amelino-Camelia, Living Rev. Rel. 16 (2013) 5, http://relativity.livingreviews. org/Articles/lrr-2013-5/, arXiv:0806.0339 [gr-qc]. 254. L. Smolin, Lect. Notes Phys. 669 (2005) 363.

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Chapter 21 Perturbative quantum gravity comes of age

R. P. Woodard Department of Physics, University of Florida, Gainesville, FL 32611, USA [email protected]

I argue that cosmological data from the epoch of primordial inflation is catalyzing the maturation of quantum gravity from speculation into a hard science. I explain why quantum gravitational effects from primordial inflation are observable. I then review what has been done, both theoretically and observationally, and what the future holds. I also discuss what this tells us about quantum gravity. Keywords: Inflation; quantum gravity; cosmology. PACS Number(s): 04.60.−m, 04.62.+v, 98.80.−k

1. Introduction Gravity was the first of the fundamental forces to impress its existence upon our ancestors, because it is universally attractive and long range. These same features ensure its precedence in cosmology. Gravity also couples to stress-energy, which is why quantum general relativity is not perturbatively renormalizable,1–13 and why identifiable effects are unobservably weak at low energies.14 These problems have hindered the study of quantum gravity until recently. This paper is about how interlocking developments in the theory and observation of inflationary cosmology have changed that situation, and what the future holds. The experiences of two Harvard graduate students serve to illustrate the situation before inflation. The first is Leonard Parker who took his degree in 1967, based on his justly famous work quantifying particle production in an expanding universe.15–17 Back then people believed that the expansion of the universe had been constantly slowing down or “decelerating.” Parker’s work was greeted with indifference on account of the small particle production associated with the current expansion, and on the inability of a decelerating universe to preserve memories of early-times when the expansion rate was much higher. The ruling dogma of the 1960s was S-matrix theory, whose more extreme proponents believed they could II-349

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guess the strong interaction S-matrix based on a very few properties such as analyticity and unitarity. Through a curious process this later morphed into string theory. Quantum field theory was regarded as a failed formalism whose success for quantum electrodynamics was an accident. Confirmation of the Standard Model had changed opinions about quantum field theory by my own time at Harvard (1977–1983). However, the perturbative nonrenormalizability of quantum general relativity led to dismissive statements such as, “only old men should work on quantum gravity.” The formalism of quantum field theory had also become completely tied to asymptotic scattering experiments. For example, no one worried about correcting free vacuum because infinite time evolution from “in” states to “out” states was supposed to do this automatically. Little attention was paid to making observations at finite times because the S-matrix was deemed the only valid observable, the knowledge of which completely defined a quantum field theory. My thesis on developing an invariant extension of local Green’s functions for quantum gravity was only accepted because Brandeis Professor Stanley Deser vouched for it. I left it unpublished for eight years.18 The situation was no better during the early stages of my career. As a postdoc, I worked with a very bright graduate student who dismissed the quantum gravity community as “la-la land” and made no secret of his plan to change fields. And there is no denying that any number of crank ideas were treated with perfect seriousness in those days, which validated our critics. I recall knowledgeable people questioning why anyone bothered trying to quantize gravity in view of the classical theory’s success. That opinion was never viable in view of the fact that the lowest divergences of quantum gravity4–9 derive from the gravitational response to matter theories which are certainly quantum, whether or not gravitons exist.14 The difference between then and now is that I can point to data — and quite a lot of it — from the same gravitational response to quantum matter. Today cosmological particle production is recognized as the source of the primordial perturbations which seeded structure formation. There is a growing realization that these perturbations are quantum gravitational phenomena,14,19,20 and that they cannot be described by any sort of S-matrix or by the use of in–out quantum field theory.21,22 This poses a challenge for fundamental theory and an opportunity for its practitioners, which dismays some physicists and delights others. All of the problems that had to be solved for flat space scattering theory in the mid 20th century are being re-examined, in particular, defining observables which are infrared (IR) finite, renormalizable (at least in the sense of low energy effective field theory) and in rough agreement with the way things are measured.23,24 People are also thinking seriously about how to perturbatively correct the initial state.25 This revolutionary change of attitude did not result from any outbreak of sobriety within the quantum gravity community, or of toleration from our colleagues. The transformation was forced upon us by the overwhelming data in support of inflationary cosmology. In the coming sections of this paper I review the theory

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behind that data, in particular: • • • • •

Why quantum gravitational effects from inflation are observable; Why the tree order power spectra are quantum gravitational effects; Loop corrections to the primordial power spectra; Other potentially observable effects; and What the future holds.

2. Why Quantum Gravitational Effects from Primordial Inflation are Observable Three things are responsible for the remarkable fact that quantum gravitational effects from the epoch of primordial inflation can be observed today: • The inflationary Hubble parameter is large enough that quantum gravitational effects are small, but not negligible; • Long wavelength gravitons and massless, minimally coupled scalars experience explosive particle production during inflation; and • The process of first horizon crossing results in long wavelength gravitons and massless, minimally coupled scalars becoming fossilized so that they can survive down to the current epoch. I will make the first point at the beginning, in the subsection on the inflationary background. Then the subsection on perturbations discusses the second and third points. 2.1. The background geometry On scales larger than about 100 Mpc the observable universe is approximately homogeneous, isotropic and spatially flat. The invariant element for such a geometry can be put in the form, ds2 = −c2 dt2 + a2 (t)dx · dx.

(1)

Two derivatives of the scale factor a(t) have great significance, the Hubble parameter H(t) and the first slow roll parameter
(t), H(t) ≡

a˙ , a


(t) ≡ −

H˙ . H2

(2)

Inflation is defined as H(t) > 0 with
(t) < 1. One can see that it is possible from the current values of the cosmological parameters (denoted by a subscript zero),26 H0 ≈ 2.2 × 10−18 Hz,


0 ≈ 0.47.

(3)

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However, the important phase of inflation for my purposes is Primordial Inflation, which is conjectured to have occurred during the first 10−32 s of existence. If the BICEP2 detection of primordial B-mode polarization is accepted, then we finally know the values of H(t) and
(t) near the end of primordial inflation,27 Hi ≈ 1.8 × 10+38 Hz,


i ≈ 0.013.

(4)

I will comment later on the significance of Hi . Let us here note that
i is very near the de Sitter limit of
= 0 at which the Hubble parameter becomes constant. This is a very common background to use when estimating quantum effects during primordial inflation. We have direct observational evidence that both the scale factor and its logarithmic time derivative H(t) have changed over many orders of magnitude during cosmic history. In contrast, the deceleration parameter only varies over the small range 0 ≤
(t) ≤ 2. Figure 1 shows what we think we know about
(t) as a function of the number of e-foldings since the end of primordial inflation at t = te ,
 a(t) N (t) ≡ ln . (5) a(te ) It is a tribute to decades of observational work that only a small portion of this figure is really unknown, corresponding to the phase of re-heating at the end of inflation.

2 as a function of the ˙ Fig. 1. The red curve shows the first slow roll parameter
(t) ≡ −H/H number of e-foldings N since the end of primordial inflation. A question mark stands for the phase of reheating between the epochs of primordial inflation and radiation domination, because there are many models for this period. Significant events marked on the graph are Big Bang Nucleosynthesis (BBN) when the seven lightest isotopes were produced, matter-radiation equality (eq) when the energy density was composed of equal amounts of relativistic and nonrelativistic matter, and recombination (rec) when neutral Hydrogen formed and the cosmic microwave radiation began free-streaming. Observable cosmological perturbations experience first horizon crossing near the lower left hand corner of the graph.

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Primordial inflation was advanced in the late 1970s and early 1980s to explain the absence of observed relics (primordial black holes, magnetic monopoles, cosmic strings) and the initial conditions (homogeneous, isotropic and spatially flat) for the long epoch of radiation domination which is visible on Fig. 1. After some notable precursors,28–31 the paper of Guth32 focused attention on the advantages of a early epoch of inflation and, incidentally, coined the name. Important additional work concerned finding an acceptable way to commence inflation and to make it end.33,34 The first completely successful model was Linde’s “Chaotic Inflation.”35 One of the most powerful motivations for primordial inflation is that it explains the Horizon Problem of why events far back in our past light-cone seem so uniform. I will review the argument here because the same analysis is useful for the next subsection. From the cosmological geometry (1), we can easily compute the coordinate distance R(t2 , t1 ) traversed by a light ray whose trajectory obeys ds2 = 0,  t1 cdt R(t2 , t1 ) ≡ . (6) a(t) t2 Now note the relation,



 d 1 1
˙ = 1− . dt (
− 1)Ha a (
− 1)2 H

(7)

One can see from Fig. 1 that
(t) was nearly constant over long periods of cosmic evolution, in particular during the epoch of radiation domination, which would extend back to the beginning if it were not for primordial inflation. So we can drop the second term of (7) to conclude, R(t2 , t1 ) ≈

c c − . (
1 − 1)H1 a1 (
2 − 1)H2 a2

(8)

One additional exact relation brings the horizon problem to focus, d [H(t)a(t)] = −[
(t) − 1]H 2 (t)a(t). dt

(9)

Combining Eq. (9) with (8) reveals a crucial distinction between inflation (
(t) < 1) and deceleration (
(t) > 1): During deceleration the radius of the light-cone is dominated by its upper limit, whereas the lower limit dominates during inflation. The horizon problem derives from assuming that there was no phase of primordial inflation so that the epoch of radiation domination extends back to the beginning of the universe. Suppose that the universe began at t = t2 and we view some early event such as recombination (rec on Fig. 1) or big bang nucleosynthesis (BBN on Fig. 1). At time t = t1 we can see things out to the radius of our past light-cone R(t1 , t0 ) which is vastly larger than the radius of the forward light-cone R(t2 , t1 ) ≈ c/[(
1 − 1)H1 a1 ] that anything can have traveled from the beginning of time. For example, the cosmic microwave radiation is uniform to one part in 105 , which is far better thermal equilibrium than the air of the room in which you are sitting. Without a phase of primordial inflation we are seeing about 2200 different

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patches of the sky which have not even had time to exchange a single photon, much less achieve a high degree of thermal equilibrium.14 Of course the problem just gets worse the further back we look. At the time of big bang nucleosynthesis BBN we are seeing about 1015 causally disconnected regions, which are nonetheless in rough thermal equilibrium.14 Without inflation the radius R(t2 , t1 ) of the forward light-cone is almost independent of the beginning of time t2 . No matter how early we make t2 , it is not possible to increase R(t2 , t1 ) more than about c/[(
1 − 1)H1 a1 ]. Hence, the high degree of uniformity we observe in the early universe would have to be a spectacularly unlikely accident. Primordial inflation solves the problem neatly by making the lower limit of the forward light-cone dominate, R(t2 , t1 ) ≈ c/[(1 −
2 )H2 a2 ]. We can make the radius of the forward light-cone much larger than the radius of the past light-cone, so that causal processes would have had plenty of time to achieve the high degree of equilibrium that is observed. Before closing this subsection, I want to return to the numerical values quoted for H0 and Hi in relations (3)–(4). The loop counting parameter of quantum gravity can 2 ≡
G/c5 ≈ 2.9×10−87 s2 . be expressed in terms of the square of the Planck time, TPl Quantum gravitational effects from a process whose characteristic frequency is ω 2 . For inflationary particle production the characteristic are typically of order ω 2 TPl frequency is of course the Hubble parameter, so we can easily compare the strengths of quantum gravitational effects during the current phase of inflation and from the epoch of primordial inflation,
GH02 ≈ 1.4 × 10−122 , c5


GHi2 ≈ 9.4 × 10−11 . c5

(10)

The minuscule first number is why we will never detect quantum gravitational effects from the current phase of inflation. Although the second number is tiny, it is not so small as to preclude detection, if only the signal can persist until the present day. In the next subsection, I will explain how that can happen. The loop counting parameter
GH 2 /c5 is the quantum gravitational analog of the quantum electrodynamics fine structure constant α ≡ e2 /4π
0
c ≈ 7.3 × 10−3 . Both parameters control the strength of perturbative corrections. Recall that a result in quantum electrodynamics — for example, the invariant amplitude of Compton scattering — typically consists of a lowest, tree order contribution of strength α, then each additional loop brings an extra factor of α. In the same way, the lowest, tree order quantum gravity effects from inflationary particle production have strength
GH 2 /c5 , and each addition loop brings an extra factor of
GH 2 /c5 . Because the quantum gravitational loop counting parameter from primordial inflation is so much smaller than its quantum electrodynamics cousin, we expect that quantum gravitational perturbation theory should be wonderfully accurate. In fact, all that can be resolved with current data is the tree order effect, although I will argue in Sec. 4.4 that the one loop correction may eventually be resolved. Beyond that there is no hope.

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2.2. Inflationary particle production The phenomenon of polarization in a medium is covered in undergraduate electrodynamics. The medium contains a vast number of bound charges. The application of an electric field makes positive charges move with the field and the negative changes move opposite. That charge separation polarizes the medium and tends to reduce the electric field strength. One of the amazing predictions of quantum field theory is that virtual particles are continually emerging from the vacuum, existing for a brief period, and then disappearing. How long these virtual particles can exist is controlled by the energy– time uncertainty principle, which gives the minimum time ∆t needed to resolve and energy difference ∆E, ∆t∆E

.

(11)

If one imagines the emergence of a pair of positive and negatively charged particles of mass m and wave vector ±k then the energy went from zero to E = 1 2[m2 c4 +
2 c2 k 2 ] 2 . To not resolve a violation of energy conservation, the energy– time uncertainty principle requires the pair to disappear after a time ∆t given by,


∆t ∼ √ . m 2 c4 +
2 c2 k 2

(12)

The rest is an exercise in classical (that is, nonquantum) physics. If we ignore the change in the particles’ momentum then their positions obey, d2  2 2 ±
2 eE 2 k 2 ∆x ) = ±eE ⇒ ∆x (∆t) = ( m c +
(13) ± ± 3 . cdt2 2c[m2 c2 +
2 k 2 ] 2 Hence the polarization induced by wave vector k is, p = +e∆x+ (∆t) − e∆x− (∆t) =


2 e 2 E 3

c[m2 c2 +
2 k 2 ] 2

.

(14)

The full vacuum polarization density comes from integrating d3 k/(2π)3 . The simple analysis I have just sketched gives pretty nearly the prediction from one loop quantum electrodynamics, which is in quantitative agreement with experiment. It allows us to understand two features of vacuum polarization which would be otherwise obscure: • That the largest effect derives from the lightest charged particles because they have the longest persistence times ∆t and therefore induce the greatest polarization; and • That the electrodynamic interaction becomes stronger at short distances because the longest wavelength (hence smallest k) virtual particles could induce more polarization than is allowed by the travel time between two very close sources. Cosmological expansion can strengthen quantum effects because it causes the virtual particles which drive them to persist longer. This is easy to see from the

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geometry (1). Because spatial translation invariance is unbroken, particles still have conserved wave numbers k. However, because the physical distance is the coordinate distance scaled by a(t), the physical energy of a particle with mass m and wave number k = 2π/λ becomes time-dependent, 
2 c2 k 2 E(t, k) = m2 c4 + 2 . (15) a (t) Hence, the relation for the persistence time ∆t of a virtual pair which emerges at time t changes from (12) to,  t+∆t dt
E(t
, k) ∼
. (16) t

Massless particles persist the longest, just as they do in flat space. However, for inflation it is the lower limit of (16) which dominates, so that even taking ∆t to infinity does not cause the integral to grow past a certain point. One can see this from the de Sitter limit,  t+∆t
ck
ck dt

= [1 − e−Hi ∆t ]. (17) a(t ) H i a(t) t A particle with ck < H(t)a(t) is said to be super-horizon, and we have just shown that any massless virtual particle which emerges from the vacuum with a superhorizon wave number during inflation will persist forever. It turns out that almost all massless particles possess a symmetry known as conformal invariance which suppresses the rate at which they emerge from the vacuum. This keeps the density of virtual particles small, even though any that do emerge can persist forever. One can see the problem by specializing the Lagrangian of a massless, conformally coupled scalar ψ(t, x) to the cosmological geometry (1),   R 2√ a3 ψ˙ 2 1 ∂i ψ∂i ψ (H˙ + 2H 2 )ψ 2 µν √ −g − ψ −g → L = − ∂µ ψ∂ν ψg − − . 2 12 2 c2 a2 c2 (18) The equation for a canonically normalized, spatial plane wave of the form ψ(t, x) = v(t, k)eik · x can be solved for a general scale factor a(t),
  t dt

2 2  exp −ick
c k
ti a(t ) ˙ + 2H 2 v = 0 ⇒ v(t, k) = v¨ + 3H v˙ + + H . (19) 2 a 2ck a(t) The factor of 1/a(t) in (19) suppresses the emergence rate, even though destructive interference from the phase die off, just as the energy–time uncertainty principle (17) predicts. The stress-energy contributed by this field is,
 
1 1 1 ρ σ ρσ Tµν = δµ δν − gµν g ∂ρ ψ∂σ ψ + Rµν − gµν R + gµν − Dµ Dν ψ 2 , (20) 2 6 2

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is the covariant d’Alembertian. We where Dµ is the covariant derivative and can get the zero-point energy of a single wave vector k by specializing T00 to the cosmological geometry (1) and multiplying by a factor of a3 (t), 2 2


a3 c k
ck 2 2 2 ∗ ∗ E(t, k) = |v| ˙ + + H |v| + H(v v˙ + vv ˙ ) = . (21) 2 a2 2a(t) This is just the usual 12
ω term which is not strengthened but rather weakened by the cosmological expansion. Only gravitons and massless, minimally coupled scalars are both massless and not conformally invariant so that they can engender significant quantum effects during inflation. Because they obey the same mode equation36,37 it will suffice to specialize the scalar Lagrangian to the cosmological geometry (1),   1 1 3 φ˙ 2 1 µν √ L = − ∂µ φ∂ν φg −g → a − 2 ∂i φ∂i φ . (22) 2 2 c2 a The equation for a canonically normalized, spatial plane wave of the form φ(t, x) = u(t, k)eik · x is simpler than that of its conformally coupled cousin (19) but more difficult to solve, so I will specialize the solution to de Sitter,
  t
dt 
 exp −ick
c2 k 2
iHi a(t) ti a(t ) u ¨ + 3H u˙ + 2 u = 0 ⇒ u(t, k) = 1+ . a 2ck ck a(t) (23) The minimally coupled mode function u(t, k) has the same phase factor as the conformal mode function (19), and they both fall off like 1/a(t) in the far subhorizon regime of ck  Hi a(t). However, they disagree strongly in the super-horizon regime during  which v(t, k) continues to fall off whereas u(t, k) approaches a phase times Hi
/2c3 k 3 . One can see from the equation on the left of (23) that u(t, k) approaches a constant for any inflating geometry. The zero-point energy in wave vector k is, 


2  c2 k 2 2 Hi a(t) 1 3
ck 1 2 E(t, k) = a |u| ˙ + 2 |u| = + . (24) 2 a a(t) 2 2ck Because each wave vector is an independent harmonic oscillator with mass proportional to a3 (t) and frequency k/a(t) we can read off the occupation number from expression (24),
2 Hi a(t) N (t, k) = . (25) 2ck As one might expect, this number is small in the sub-horizon regime. It becomes of order one at the time tk of horizon crossing, ck = H(tk )a(tk ), and N (t, k) grows explosively afterward. This is crucial because it means that inflationary particle production is an IR effect. That means we can study it reliably using quantum general relativity, even though that theory is not perturbatively renormalizable.

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The final point I wish to make is that the mode function u(t, k) becomes constant after first horizon crossing. For de Sitter this constant is calculable, 

ick u(t, k)|dS → iHi exp − . (26) 2c3 k 3 Hi a i However, one can see from the mode equation on the left-hand side of (23) that the approach to a constant happens for any inflating geometry. Recall from equation (9) that the inverse horizon length c−1 H(t)a(t) grows during inflation and decreases during the later phase of deceleration which encompasses so much of the cosmological history depicted in Fig. 1. Hence, we can give the following rough summary of the “life cycle of a mode” of wave number k: • At the onset of primordial inflation the mode has ck  H(t)a(t) so the mode function oscillates and fall off like 1/a(t); • If inflation lasts long enough the mode will eventually experience first horizon crossing ck = H(tk )a(tk ), after which mode function becomes approximately constant; and • During the phase of deceleration which follows primordial inflation, modes which experienced first horizon crossing near the end of inflation re-enter the horizon ck = H(Tk )a(Tk ), after which they begin participating in dynamical processes with amplitude larger by a factor of a(Tk )/a(tk ) than they would have had without first horizon crossing. This is how quantum gravitational effects from the epoch of primordial inflation become fossilized so that they can be detected now. 3. Tree Order Power Spectra Although the evidence for primordial inflation is overwhelming, there is not yet any compelling mechanism for causing it. The simplest class of successful models is based on general relativity plus a scalar inflaton ϕ(x) whose potential V (ϕ) is regarded as a free function,35
4  √ c R 1 L= − ϕ,µ ϕ,ν gµν − V (ϕ) −g. (27) 16πG 2 Here gµν (x) is the D-dimensional, spacelike metric with Ricci curvature R. (I will work in D spacetime dimensions to facilitate the use of dimensional regularization, even though the D = 4 limit must eventually be taken for physical results). The purpose of this section is to show how this simple model can not only drive primordial inflation but also the quantum gravitational fluctuations whose imprint on the cosmic background radiation has been imaged with stunning accuracy.26,38–40 I first demonstrate that the potential V (ϕ) can be chosen to support the cosmological geometry (1) with any scale factor a(t) for which the Hubble parameter is monotonically decreasing. I also comment on the many problems of plausibility which seem to point to the need for a better model. I then decompose perturbations

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about the background (1) into a scalar ζ(x) and a transverse-traceless tensor hij (x). Owing to the particle production mechanism adumbrated in Sec. 2.2, certain modes of ζ and hij become hugely excited during primordial inflation, and then freeze-in so that they can survive to much later times. The strength of this effect is quantified by the primordial scalar and tensor power spectra, which I define and compute at tree order, along with associated observables. I then discuss the controversy which has arisen concerning an alternate definition of the tree order power spectra. The section closes with an explanation of why the tree order power spectra are the first quantum gravitational effects ever to have been resolved. I will adopt the notation employed in recent studies by Maldacena41 and by Weinberg,21 however, the original work for tensors was done in 1979 by Starobinsky,42 and for scalars in 1981 by Mukhanov and Chibisov.43 Important subsequent work was done over the course of the next several years by Hawking,44 Guth and Pi,45 Starobinsky,46 Bardeen et al.47 and by Mukhanov.48 Some classic review articles on the subject are Refs. 49–51. 3.1. The background for single-scalar inflation There is no question that a minimally coupled scalar potential model of the form (27) can support inflation because there is a constructive procedure for finding the potential V (ϕ) given the expansion history a(t).52–55 For the geometry (1) the scalar depends just on time ϕ0 (t) and only two of Einstein’s equations are nontrivial,
 1 8πG ϕ˙ 20 (D − 2)(D − 1)H 2 = 2 + V (ϕ ) (28) 0 , 2 c c2
 1 8πG ϕ˙ 20 2 ˙ −(D − 2)H − (D − 2)(D − 1)H = 2 − V (ϕ0 ) . (29) 2 c c2 By adding (28) and (29) one obtains the relation, 8πG −(D − 2)H˙ = 4 ϕ˙ 20 . (30) c Hence, one can reconstruct the scalar’s evolution provided the Hubble parameter is monotonically decreasing, and that relation can be inverted (numerically if need be) to solve for time as a function of   t ˙
) −(D − 2)c4 H(t ϕ0 (t) = ϕ0 (t2 ) ± dt
⇔ t = T (ϕ0 ). (31) 8πG t2 One then determines the potential by subtracting (29) from (28) and evaluating the resulting function of time at t = T (ϕ0 ), (D − 2)c2 ˙ [H(T (ϕ0 )) + 3H 2 (T (ϕ0 ))]. (32) 16πG Just because scalar potential models (27) can be adjusted to work does not mean they are particularly plausible. They suffer from six sorts of sometimes contradictory V (ϕ0 ) =

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fine-tuning problems: (1) Initial Conditions — Inflation must begin with the inflaton approximately homogeneous, and potential-dominated, over more than a Hubble volume56 ; (2) Duration — The inflation potential must be flat enough to make inflation last for at least 50 e-foldings33,34 ; (3) Scalar Perturbations — Getting the right magnitude for the scalar power spec2 trum requires
G3 /c11 × V 3 /V
∼ 10−11 (Ref. 49); (4) Tensor Perturbations — Getting the right magnitude for the tensor power spectrum requires c4 /G × (V
/V )2 ∼ 1 (Ref. 49); (5) Reheating — The inflation must couple to ordinary matter (its gravitational couplings do not suffice) so that its post-inflationary kinetic energy produces a hot, radiation-dominated universe57 ; (6) Cosmological Constant — The minimum of the scalar potential must have the right value
G2 Vmin /c7 ≈ 10−123 to contribute the minuscule vacuum energy we observe today.58–61 Note that adding the matter couplings required to produce reheating puts 2–4 at risk because matter loop effects induce Coleman–Weinberg corrections to the inflation effective potential. Nor does fundamental theory provide any explanation for why the cosmological constant is so small.62,63 The degree of fine-tuning needed to enforce these six conditions strains credulity, and disturbs even those who devised the early models.64–67 Opinions differ, but I feel it is a mistake to make too much of the defects of single-scalar inflation. The evidence for an early phase of accelerated expansion is overwhelming and really incontrovertible, irrespective of what caused it. Further, all that we know about low energy effective field theory confirms that the general relativity portion of Lagrangian (27) must be valid, even at the scales of primordial inflation. That suffices to establish the quantum gravitational character of primordial perturbations, even without a compelling model for what caused inflation. So I will go forward with the analysis on the basis of the single-scalar model (27), firm in the belief that whatever eventually supplants it must exhibit many of the same features. 3.2. Gauge-fixed, constrained action We decompose gµν into lapse, shift and spatial metric according to Arnowitt, Deser and Misner (ADM),68–70 gµν dxµ dxν = −N 2 c2 dt2 + gij (dxi − N i cdt)(dxj − N j cdt).

(33)

ADM long ago showed that the Lagrangian has a very simple dependence upon the lapse,68–70  √
c4 g B L = (Surface Terms) − N ·A+ . (34) 16πG N

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The quantity A is a potential energy, A = −R +


 16πG 1 ij V (ϕ) + ϕ ϕ g , ,i ,j c4 2

(35)

where R is the (D − 1)-dimensional Ricci scalar formed from gij . The quantity B in (34) is a sort of kinetic energy.

2 8πG ϕ˙ i 2 ij i B = (E i ) − E Eij − 4 − ϕ,i N , (36) c c where Eij /N is the extrinsic curvature, 1 [Ni;j + Nj;i − c−1 g˙ ij ] (37) 2 and a semicolon denotes spatial covariant differentiation using the connection compatible with gij . ADM fix the gauge by specifying N (t, x) and N i (t, x), however, Maldacena41 and Weinberg21 instead impose the conditions, Eij ≡

G0 (t, x) ≡ ϕ(t, x) − ϕ0 (t) = 0,

(38)

Gi (t, x) ≡ ∂j hij (t, x) = 0.

(39)

The transverse-traceless graviton field hij (t, x) is defined by decomposing the spatial metric into a conformal part and a unimodular part gij , √ gij (t, x) ⇒ g = aD−1 e(D−1)ζ . gij = a2 (t)e2ζ(t,x)  (40) The unimodular part is obtained by exponentiating the transverse-traceless graviton field hij (t, x), 1 gij ≡ [eh ]ij = δij + hij + hik hjk + O(h3 ), 2

hii = 0.

(41)

The Faddeev–Popov determinant associated with (38), (39) depends only on hij , 2 ˙ . and becomes singular for
= 0. Recall that H(t) = a/a ˙ and
(t) = −H/H Of course, no gauge can eliminate the physical scalar degree of freedom which is evident in (27). With condition (38) the inflation degree of freedom resides in ζ(t, x) and linearized gravitons are carried by hij (t, x). In this gauge the lapse and shift are constrained variables which mediate important interactions between the dynamical fields but contribute no independent degrees of freedom. Varying (34) with respect to N produces an algebraic equation for N ,  B B A− 2 =0⇒N = . (42) N A This gives the constrained Lagrangian a “virial” form,71 √ c4 g √ Lconst. = (Surface Terms) − AB. 8πG

(43)

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From relations (30–32) one can see that the background values of the potential and kinetic terms are equal, A0 = B0 = (D − 2)c−2 [H˙ + (D − 1)H 2 ]. Hence, the background value of the lapse is unity. There is unfortunately no nonperturbative solution for the shift N i (t, x) in terms of ζ and hij , so its constraint equation must be solved perturbatively. One first employs (40) to exhibit how the potential (35) depends on ζ and hij , A = A0 − R ≡ A0 (1 + α).

(44)

Here the spatial Ricci scalar is, R=

e−2ζ   2 ζ − (D − 2)(D − 3)ζ ,k  [R − 2(D − 2)∇ g k ζ, ], a2

(45)

 2 ≡ ∂i gij ∂j is the  = O(h2 ) is the Ricci scalar formed from gij and ∇ where R conformal scalar Laplacian. The full scalar Laplacian is, 1 √ ∇2 ≡ √ ∂i [ gg ij ∂j ]. g

(46)

At this stage one can recognize that the dimensionless three-curvature perturbation R is just ζ, in D = 4 dimensions and to linearized order,50

D−2 1 R(t, x) ≡ − 2 R = (47) ζ(t, x) + O(ζ 2 , ζh, h2 ). 4∇ 2 The kinetic energy (36) can be expressed as,  k) − N k  k,k ] − E  k E B ≡ A0 (1 + β) = A0 + 2(D − 2)c−1 H[(D − 1)(c−1 ζ˙ − ζ,k N

2

˙ ζ˙ ζ k  k + (N k N  k )2 . + (D − 2)(D − 1) − ζ,k N − 2(D − 2) − ζ,k N ,k ,k c c (48)  j and E i ≡ gij N ij ≡ 1 [N  i ≡ N i, N i;j + N j;i − c−1 h˙ ij ]. Here we define N 2 The next step is to expand the volume part of the constrained Lagrangian in powers of α and β, √ c4 g √ c4 aD−1 e(D−1)ζ  − AB = − A0 (1 + α)(1 + β) (49) 8πG 8πG   c4 aD−1 e(D−1)ζ (α + β) (α − β)2 =− A0 1 + − + ··· . (50) 8πG 2 8 As Weinberg noted, the terms involving no derivatives of ζ or hij sum up to a total derivative,21 aD−1 A0 =

∂ [(D − 2)HaD−1 ]. ∂t

(51)

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 i and ζ can be elimiAnother important fact is that quadratic mixing between N 71 nated with the covariant field redefinition,
−2ζ   k + gk ∂ 1 ce  2 ζ −
(c−1 ζ˙ − ζ,i N  i) . Sk ≡ N ∇ (52)  2 Ha2 ∇ After much work the quadratic Lagrangians emerge,  

D−3+
c4 aD−1 LS 2 = ∂ Sk ∂ Sk + ∂ Sk ∂k S , 32πG D−1−
  (D − 2)c4
aD−1 ζ˙2 ∂k ζ∂k ζ Lζ 2 = − , 16πG c2 a2 Lh2

c4 aD−1 = 64πG



h˙ ij h˙ ij ∂k hij ∂k hij − c2 a2

(53)

(54)

 .

(55)

These results suffice for the analysis of this section. To consider loop corrections (or non-Gaussiantity) one must solve the constraint equation for Si ,

D−3+
i  ∂j ∂j S + (56) ∂i ∂j Sj = O(ζ 2 , ζh, h2 , Sζ). D−1−
That is a tedious business which has only been carried out to a few orders. I will review what is known in Sec. 4.1.

3.3. Tree order power spectra As we will see in Sec. 4.3, there is not yet general agreement on how to define the primordial power spectra when loop corrections are included.23,24 At tree order we can dispense with dimensional regularization, and also forget about the distinction between ζ and the dimensionless three-curvature perturbation (47). The following definitions suffice:  k3 (57) d3 x e−ik·x Ω|ζ(t, x)ζ(t, 0)|Ω , ∆2R (k, t) ≡ 2π 2  k3 ∆2h (k, t) ≡ (58) d3 x e−ik·x Ω|hij (t, x)hij (t, 0)|Ω . 2π 2 Although it is useful to retain the time dependence in expressions (57), (58), the actual predictions of primordial inflation are obtained by evaluating the timedependent power spectra safely between the first and second horizon crossing times tk and Tk described in Sec. 2.2, ∆2R (k) ≡ ∆2R (k, t)|tk tTk ,

∆2h (k) ≡ ∆2h (k, t)|tk tTk .

(59)

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The state | Ω in expressions (57), (58) is annihilated by α(k) and β(k, λ) in the free field expansions of ζ and hij ,  d3 k ζ(t, x) = {v(t, k)eik·x α(k) + v ∗ (t, k)e−ik·x α† (k)}, (60) (2π)3  d3 k  hij (t, x) = {u(t, k)eik·x
ij (k, λ)β(k, λ) + Conjugate}. (61) (2π)3 λ=±

The polarization tensors
ij (k, λ) are the same as those of flat space. If we adopt the usual normalizations for the creation and annihilation operators, [α(k), α† (k
)] = (2π)3 δ 3 (k − k
),

[β(k, λ), α† (k
, λ
)] = δλλ
(2π)3 δ 3 (k − k
) (62)

then canonical quantization of the free Lagrangians (54), (55) implies that the mode functions obey,

˙ c2 k 2 i4π
G v¨ + 3H + v˙ + 2 v = 0, v v˙ ∗ − vv ˙ ∗= 2 3 , (63)
a c
a u ¨ + 3H u˙ +

c2 k 2 u = 0, a2

uu˙ ∗ − uu ˙ ∗=

i32π
G . c2 a 3

(64)

It has long been known that the graviton mode function u(t, k) obeys the same Eq. (23) as that of a massless, minimally coupled scalar.36,37 Only their normalizations differ by the square root of 32πG/c2 . By substituting the free field expansions (60), (61) into the definitions (57), (58) of the power spectra, and then making use of the canonical commutation relations (62), one can express the tree order power spectra in terms of scalar and tensor mode functions, ∆2R (k, t) =

k 3 |v(t, k)| , 2π 2

∆2h (k, t) =

k 3 |u(t, k)|2  k 3 |u(k, t)|2
ij
∗ij = . 2 2π π2

(65) (66)

λ=±

One of the frustrating things about primordial inflation is that we do not know what a(t) is, so we need results which are valid for any reasonable expansion history. This means that even tree order expressions such as (65), (66) can only be evaluated approximately because there are no simple expressions for the mode functions for a general scale factor a(t).72–74 One common approximation is setting
(t) to a constant, the reliability of which can be gauged by studying the region (at N ≈ −40) of Fig. 1 at which currently observable perturbations freeze-in. (The necessity of nonconstant
(t) later is not relevant for the validity of assuming constant
(t) to estimate the amplitude at freeze-in.) For constant
(t) both mode functions are proportional to a Hankel

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function of the first kind, u(t, k)
˙ = 0 ⇒ v(t, k) = √ , 8


8π 2
G ck (1) u(t, k) = H , ν (1 −
)c2 H(t)a3 (t) (1 −
)H(t)a(t)

(67) ν=

1 2



3−
. 1−


(68)

Between first and second horizon crossing (tk t Tk ) we can take the small argument limit of the Hankel function,
(t) ˙ = 0,

tk t T k 

⇒ u(t, k) ≈  =


ν 8π 2
G −iΓ(ν) 2(1 −
)H(t)a(t) × (1 −
)c2 H(t)a3 (t) π ck


 1 16π 2
G −iΓ(ν) 2(1 −
)H(t)a (t) 1−
× . c5 k 3 π (ck)

(69)

(70)

Constant
(t) also implies H(t)a (t) is a constant, which we may as well evaluate at the time of first horizon crossing, H(t)a (t) = H 1− (tk )(ck) . With the doubling √ formula (Γ(2x) = 22x−1 / π × Γ(x)Γ(x + 12 )) we at length obtain, 
 1 2 1 −
1−
Γ( 1− ) 16π
GH 2 (tk )
(t) ˙ = 0, tk t Tk ⇒ u(t, k) ≈ −i × . 1  2 c5 k 3 Γ( 1− ) (71) The factor multiplying the square root has nearly unit modulus for small
— and the BICEP2 result is
i = 0.013,27 while previous data sets give the even smaller bound of
i < 0.007 at 95% confidence.38–40 Hence it should be reliable to drop this factor, resulting in the approximate forms, ∆2R (k) ≈


GH 2 (tk ) , πc5
(tk )

∆2h (k) ≈

16
GH 2 (tk ) . πc5

(72)

The Wentzel–Kramers–Brillouin (WKB) approximation is another common technique for estimating the freeze-in amplitudes of v(t, k) and u(t, k) which appear in expressions (65), (66) for the tree order power spectra. Recall that the method applies to differential equations of the form f¨ + ω 2 (t)f = 0. From expression (64) one can see that reaching this form for the tensor mode function requires, the rescal3 ing f (t, k) = a 2 (t) × u(t, k). It is then simple to recognize the correctly normalized WKB solution and its associated frequency as, 
 t  16π
G

exp −i u(t, k) ≈ dt ω(t , k) , (73) c2 a3 (t)|ω(t, k)| t2
 9 3 c2 k 2 ω (t, k) ≡ 2 − −
(t) H 2 (t). a (t) 4 2 2

(74)

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In the sub-horizon regime of ck  H(t)a(t) the frequency is real ω(t, k) ≈ ck/a(t) and the solution (73) both oscillates and fall off like 1/a(t). Freeze-in occurs in the super-horizon regime of ck H(t)a(t) during which the frequency is imaginary ω(t, k) ≈ iH(t)[ 32 − 12
(t)]. We can estimate the freeze-in amplitude by computing the real part of the exponent,     3  t  t 1
˙
) 2 (t)H 2 (t) 3 a(t ˙ ) 1 H(t a dt
ω(t
, k) ≈ dt
−i + . (75) = ln 1 3 2 a(t
) 2 H(t
) a 2 (tk )H 2 (tk ) tk tk Substituting (75) in (73) and using H(tk )a(tk ) = ck gives the following result for the freeze-in modulus,   16π
GH 2 (t )  k  . tk t Tk ⇒ |u(t, k)|WKB ≈  (76)   5 3  3 1 c k  −
(t) 2 2 If we drop the order one factor of | 32 − 12
(t)| the result is (72). From expression (63) we see that reaching the WKB form for the scalar mode 1 3 function requires the rescaling f (t, k) =
2 (t) × a 2 (t) × v(t, k). Its frequency is simpler to express if I first introduce the (Hubble form of the) second slow roll parameter η(t),
(t) ˙ . (77) 2H(t)
(t) The correctly normalized WKB approximation for the scalar mode function and its frequency is, 
 t  2π
G

v(t, k) ≈ exp −i dt ν(t , k) , (78) c2
(t)a3 (t)|ν(t, k)| t2




 3 3 c2 k 2
˙ − η˙ ν 2 (t, k) ≡ 2 − +
(t) − η(t) − η(t) + (79) H 2 (t). a (t) 2 2 H(t) η(t) ≡
(t) −

Freeze-in occurs as one evolves from the sub-horizon regime of ν(t, k) ≈ ck/a(t) to the super-horizon regime of ν(t, k) ≈ iH(t)[ 32 + 12
(t) − η(t)]. The real part of the exponent of (78) is,    t  t

˙
) 1
(t 3 a(t ˙ ) 1 H(t ˙ ) dt
ν(t
, k) ≈ dt
−i + + 2 a(t
) 2 H(t
) 2
(t
) tk tk   1 1 3 a 2 (t)H 2 (t)
2 (t) = ln . (80) 1 1 3 a 2 (tk )H 2 (tk )
2 (tk ) Substituting in (78) gives a freeze-in modulus which is again roughly consistent with (72),   2π
GH 2 (tk )   . tk t Tk ⇒ |v(t, k)|WKB ≈  (81) 3 1   5 3   c k
(tk )  +
(t) − η(t) 2 2

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It should be obvious from the discordant factors of order one in expressions (71), (76) and (81), that the results (72) for the tree order power spectra are only approximate. This is confirmed by numerical integration of explicit models.75,76 In addition to order one factors, depending on the geometry at t = tk there is also a nonlocal “memory factor” which depends on the precise manner in which the mode evolves up to first horizon crossing.74 Most of the ambiguity derives from not having a definitive model for what caused inflation. Once the expansion history is known it is possible to derive wonderfully accurate results by numerically integrating the mode functions.77 It is even more efficient to numerically evolve the time-dependent power spectra directly, without the irrelevant phase information.78 The tree order power spectra (59) give the tensor-to-scalar ratio r(k), the scalar spectral index ns (k), and the tensor spectral index nt (k), r(k) ≡

∆2h (k) ≈ 16
(tk ), ∆2R (k)

ns (k) ≡ 1 + nt (k) ≡

∂ ln(∆2R (k)) ≈ 1 − 4
(tk ) + 2η(tk ), ∂ ln(k)

∂ ln(∆2h (k)) ≈ −2
(tk ). ∂ ln(∆2h (k))

(82) (83) (84)

In each case the definition is exact, and the approximate result derives from expressions (72) with an additional approximation to relate dk to dtk , cdk = [1 −
(tk )]H 2 (tk )a(tk )dtk ≈ ck × H(tk )dtk .

(85)

Comparison of (82) and (84) implies an important test on single-scalar inflation which is violated in more general models,79–81 r ≈ −8nt .

(86)

Certain general trends are also evident from the approximate results (72): • r < 1 because
1; • nt < 0 because H(t) decreases; and • ns − 1 < nt because
(t) tends to increase. To anyone who works in quantum gravity it is breath-taking that we have any data, so it seems petulant to complain that the some of the parameters, and particularly their dependences upon k, are still poorly constrained. The scalar power spectrum can be inferred from the measurements of the intensity and the E-mode of polarization in cosmic microwave radiation which originates at the time of recombination (rec on Fig. 1) and then propagates through the fossilized metric perturbations left over from the epoch of primordial inflation. The latest full-sky results come from the Planck satellite and are fit to the form,26 s ln( k ) ns −1+ ddn k0 ln(k) k 2 ∆R (k) ≈ As . (87) k0

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The fiducial wave number is k0 = 0.050 Mpc−1 , and the quantities “ns ” and “dns /d ln(k)” are the scalar spectral index and its derivative evaluated at k = k0 . When combined with the polarization data from the WMAP satellite38 the Planck team reports,26 +0.051 109 × As = 2.196−0.060 , ns = 0.9603 ± 0.0073, dns = −0.013 ± 0.018. d ln(k)

(88)

The tensor-to-scalar ratio is reported at a much smaller wave number of k = 0.002 Mpc−1 . Bounds on r0.002 can be derived from analyzing how the intensity and the E-mode of polarization of the cosmic microwave background radiation depend upon k. Because this sort of k dependence might also indicate “running” of the scalar spectral index (dns /d ln(k) = 0), the limits on r0.002 become significantly weaker if one allows for running. Combining Planck and previous data sets gives the following bounds at 95% confidence,26 r0.002 < 0.11,

dns = 0, d ln(k)

(89)

r0.002 < 0.26,

dns = 0. d ln(k)

(90)

Direct detection requires a measurement of the B-mode of polarization. The BICEP2 team have done this and they report a result consistent with, r0.002 ≈ 0.20.

(91)

Although the tensor power spectrum is still poorly known and controversial,82,83 resolving it (and ruling out alternative interpretations84 ) is terrifically important because it tests single-scalar inflation through relation (86) and incidentally fixes the scale of primordial inflation. If only ∆2R (k) is resolved one can always construct a single-scalar potential V (ϕ) which will explain it. To see this, suppose we have measured the scalar power spectrum for some range of wave numbers k2 < k < k1 and use the approximate formula (72), along with the small
relation (85) between dtk and dk, to reconstruct the inflationary Hubble parameter,  1 1 2
G k dk
− = . (92) H 2 (tk ) H22 πc5 k2 k
∆2R (k
) The Hubble parameter H2 ≡ H(t2 ) is an integration constant which we can choose to make the tensor power spectrum smaller than any bound. Now use (92), with (85), to reconstruct the relation between t and k,  k
 
dk 2
GH22 k dk

1 + H2 (t − t2 ) = . (93)


2

πc5 k2 k k2 k ∆R (k ) This expression can always be inverted numerically, and the rest of the construction is the same as that given in Sec. 3.1.

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3.4. The controversy over adiabatic regularization It is obvious from their free Lagrangians (60)–(55) that the two-point correlators of ζ(t, x) and hij (t, x) diverge quadratically when the two fields are evaluated at the same spacetime point. This is not enough to induce any tree order divergence in my definitions (57), (58). However, it is problematic for the more common definition which is based on a spectral resolution of the coincident two-point function,  ∞ dk 2 Ω|ζ(t, x)ζ(t, x)|Ω = ∆ (k, t), (94) k R 0  ∞ dk 2 Ω|hij (t, x)hij (t, x)|Ω = ∆ (k, t). (95) k h 0 In 2007 Parker,85 pointed out that removing this divergence with the standard technique of adiabatic regularization16,86–89 can change the power spectra by several orders of magnitude. Subsequent work by Parker and his collaborators showed that adiabatic regularization of the scalar and tensor power spectra would alter the single-scalar consistency relation (86) and would also reconcile the conflict between even WMAP data38 and a quartic inflation potential V (ϕ) = λϕ4 .90,91 Such profound changes in the labor of three decades provoked the natural objection that no technique for addressing ultraviolet divergences ought to affect the IR regime in which inflationary particle production takes place.92,93 Parker and his collaborators replied that consistency of renormalization theory requires adiabatic subtractions which affect all modes, including those in the IR.94 I find this debate fascinating because it is an example of how inflationary cosmology is challenging the way we think about, hitherto abstract issues in quantum gravity and viceversa. I do not know the answer, but I have encountered the same problem when trying to work out the pulse of gravitons which would be produced by a very peculiar model in which H(t) oscillates from positive to negative at the end of inflation.95,96 The resolution may not lie with any change in the way we renormalize but rather with greater care in how we connect theory to observation, for example, defining the tree-order power spectra from expressions (57), (58) rather than from spectral resolutions of the coincident two-point functions (94), (95).97 Whatever we find, it is worthwhile to reflect on the wonder of what is taking place. These are the same problems which the men of genius who founded flat space quantum field theory had to puzzle out when they settled on noncoincident one-particle-irreducible (1PI) functions as the basis for renormalization and computation of the S-matrix. It is a privilege to reprise their roles. 3.5. Why these are quantum gravitational effects The factors of
G in expressions (72) ought to establish that both power spectra are legitimate quantum gravitational, the first ever detected. Unfortunately, three

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objections seem to be delaying general recognition of this simple but revolutionary fact: • Expressions (72) are tree order results; • There is still debate over whether or not the graviton signal ∆2h (k) has been resolved27,82,83; and • There is not yet a compelling model for what caused primordial inflation. I will argue below that all three objections result from imposing unreasonably high standards on what qualifies as a quantum gravitational effect. The first objection might be re-stated as, “it’s not quantum gravity if it’s only tree order.” This is applied to no other force. For example, both the photo-electric effect and beta decay occur at tree order, yet no one disputes that they are quantum manifestations of the electroweak interaction. The same thing could be said of Planck’s blackbody spectrum, and any number of other tree order effects such as Bhabha scattering. The second objection might be restated as, “it’s not quantum gravity if it doesn’t involve gravitons.” This is also silly because ∆2R (k) has certainly been resolved and it is just as certainly a quantum gravitational effect in view of the factor of
G evident in expression (72). We saw in Secs. 3.2 and 3.3 that the scalar power spectrum derives from the gravitational response to quantum matter, the same way that all the solar system tests of general relativity derive from the gravitational response to classical matter. Were we to insist that only gravitons can test quantum gravity then logical consistency would imply that only gravitational radiation tests classical gravity, at which point we are left with only the binary pulsar data! Indeed, a little reflection on the problem of perturbative quantum gravity14 reveals that the lowest order problem is not from gravitons — which cause no uncontrollable divergences until two loop order12,13 — but rather from exactly the same gravitational response to quantum matter which the scalar power spectrum tests. All experimentally confirmed matter theories engender quantum gravitational divergences at just one loop order.4–9 If a sensible quantum gravity expert was told he could only know one of the two power spectra and then asked to choose which one, he ought to pick ∆2R (k) because it tells him about the lowest order problem. Fortunately, we will know both power spectra, and probably sooner rather than later. It is even possible we will eventually resolve one loop corrections. The final objection could be re-stated as, “it isn’t quantum gravity if we can’t make a unique prediction for it.” This seems as ridiculous as trying to argue that galactic rotation curves don’t necessarily derive from gravity just because we are not yet certain whether their shapes are explained by Newtonian gravity with dark matter or by some modification of gravity. Which is not to deny how wonderfully improved the situation would be with a compelling model for inflation. If we had one then the two power spectra would provide a definitive test of quantum gravity, the same way that the photo-electric effect and Bhabha scattering test quantum electrodynamics.

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Sceptics are free to accuse me of unwarranted optimism, but I believe that working out what drove primordial inflation is just a matter of time in the datarich environment which is developing. Measurements of ns with increasingly tight upper bounds on r0.002 have already ruled out some potentials such as V (ϕ) ∼ ϕ4 ,38 and all models with constant
(t).39,40 This process is bound to continue, and even accelerate, as the data gets better. There are plans to reduce the errors on ns by a factor of five using galaxy surveys.98 (This will begin filling in the question mark region of reheating on Fig. 1.) If the BICEP2 detection really means r0.002 ≈ 0.20, it would rule out a host of models with small r0.002 .27 We will know within the next five years by checking if the BICEP2 signal possesses the key frequency and angular dependences needed to distinguish it as primordial gravitons. If so then it should be possible to reduce the errors on r0.002 to the percent level within the next decade. As higher resolution polarization measurements are made over the course of the next 15 years, it should be possible to remove the gravitational lensing signal (known as “de-lensing”) to reach the sensitivity needed to measure nt . It is inconceivable to me that theorists will remain idle while these events transpire. Past experience shows that theory and experiment develop synergistically. The data are not going to run out any time soon, and I believe fundamental theorists will eventually receive enough guidance to develop a truly compelling model for primordial inflation. Let me close this section by pointing out that just the fact of observing scalar perturbations from primordial inflation tells us two significant things about quantum gravity14: • It is no longer viable to avoid quantizing gravity; and • The problem of ultraviolet divergences cannot be explained by making spacetime discrete. The first point is obvious from the fact that the scalar power spectrum represents the gravitational response to quantum fluctuations of matter, which would be absent if the source of classical gravity were taken to be the expectation value of the matter stress tensor in some state. To see the second point, note that although discretization at any scale makes quantum gravitational loop integrals  finite, it will not keep them small unless the discretization length is larger than
G/c3 ∼ 1.6 × 10−34 m. But primordial inflation posits that the universe has expanded by the staggering factor of about e100 ∼ 1043 from a time when quantum gravitational effects were small. Hence, the current comoving scale of discretization would correspond to about a million kilometers! 4. Loop Corrections to the Power Spectra From (88) one can see that the scalar power spectrum is currently measured with an accuracy of more than two significant figures. However, resolving the one loop correction would require about 10 significant figures because the loop counting parameter of inflationary quantum gravity is no larger than
GHi2 /c5 ∼ 10−10 .

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Although there is no hope of achieving this precision within the next two decades, the data is potentially recoverable and theorists have begun thinking about how to predict the results when (and if) one loop corrections are resolved in the far future. This section describes the basic formalism and the significant issues. I close by adumbrating a process through which the missing eight significant figures might be made up. 4.1. How to make computations I will return later in this section to the issue of precisely what theoretical quantities correspond to the observed scalar and tensor power spectra. For now let me assume that the tree order definitions (57), (58) remain valid. One striking fact about these expressions is that neither of them is an S-matrix element. Nor is either the matrix element of some product of noncoincident local operators (because both are at the same time) between an “in” state which is free vacuum at asymptotically early-times and an “out” state which is free vacuum at asymptotically late-times. One can define a formal S-matrix for the simplest cosmologies99 but it calls for measurements which are precluded by causality. More generally, the entire formalism of in–out matrix elements — which is all most of us were taught to calculate — is inappropriate for cosmology because the universe began with an initial singularity100 and no one knows how it will end. Persisting with in–out quantum field theory would make loop corrections possess two highly undesirable features: • They would be dominated by assumptions about the “out” vacuum owing to vast expansion of spacetime; and • The matrix elements of even Hermitian operators would be complex numbers because the “in” and “out” vacua must differ due to inflationary particle production. The more appropriate quantity to study in cosmology is the expectation value of some operator in the presence of a prepared state which is released at a finite time. Of course one could always employ the canonical formalism to make such computations, but particle physicists yearn for a technique that is as simple as the Feynman rules are for in–out matrix elements. Schwinger devised such a formalism for quantum mechanics in 1961.101 Over the next two years, it was generalized to quantum field by Mahanthappa102 and by Bakshi and Mahanthappa.103,104 Keldysh applied it to statistical field theory in 1964 (Ref. 105) where the technique has become routine. Until very recently, its use in quantum field theory was limited to a handful of people working on phase transitions and gravity.106–109 Most particle theorists were majestically ignorant of the technique and so attached to the in–out formalism that they dismissed as mistakes what are significant and deliberate deviations of the Schwinger–Keldysh formalism, such as the absence of an imaginary part. The stifling atmosphere which prevailed is well-conveyed by the lofty disdain in the words of a referee I had for a 2003 grant renewal proposal to the Department of

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Energy: In his work with Tsamis, Woodard has focused on what they interpret as an instability of de Sitter space due to a two loop IR divergence associated with long wavelength virtual gravitons. They describe this as the accumulation of gravitational attraction of “large-wavelength virtual gravitons.” That is a puzzling statement in itself — the accumulation that they describe would build up only if gravitons were really being produced. In fact, they think these virtual gravitons are rendered real as they are “pulled apart by rapid expansion of spacetime.” I believe that there is absolutely no evidence for this. Real particle production should show up as an imaginary contribution to the graviton vacuum polarization tensor, at least if unitarity in de Sitter space resembles flat space. The thinking of particle theorists underwent a radical transformation in 2005 when Nobel laureate Weinberg undertook a study of loop corrections to the power spectra.21 He quickly realized that the in–out formalism was inappropriate and, because he did not then know of the Schwinger–Keldysh formalism, he independently discovered a version of it which is better suited to this problem than the usual one. (His student Bua Chaicherdsakul told Weinberg of the older technique, and he gave full credit to Schwinger in his paper). I well recall the day Weinberg’s paper appeared on the arXiv. I chanced to be visiting the University of Utrecht then and a very knowledgeable and not unsympathetic colleague commented, “I guess I will finally have to learn the Schwinger–Keldysh formalism.” Weinberg’s words on the general problem of computing loop effects in primordial inflation are also worth quoting in defence of the intellectual curiosity which is sometimes lacking in particle theory: This paper will discuss how calculations of cosmological correlations can be carried to arbitrary orders of perturbation theory, including the quantum effects represented by loop graphs. So far, loop corrections to correlation functions appear to be much too small ever to be observed. The present work is motivated by the opinion that we ought to understand what our theories entail, even where in practice its predictions cannot be verified experimentally, just as field theorists in the 1940s and 1950s took pains to understand quantum electrodynamics to all orders of perturbation theory, even though it was only possible to verify results in the first few orders. The best way to understand the Schwinger–Keldysh formalism is by relating its functional integral representation to the canonical formalism. Recall how this goes for the in–out formalism in the context of a real scalar field φ(t, x) whose Lagrangian is the spatial integral of its Lagrangian density,  (96) L[φ(t)] ≡ dD−1 x L[φ(t, x)].

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The in–out formalism gives matrix elements of T ∗ -ordered products of operators, which means that any derivatives are taken outside the time-ordering symbol. The usual relation is adapted to asymptotic scattering problems but, for our purposes, it is better to consider the matrix element between a state |Ψ whose wave functional at time t = t2 is Ψ[φ(t2 )] and a state Φ| with wave functional Φ[φ(t1 )]. The well-known functional integral expression for the matrix element of the T ∗ -ordered product of some operator Oa [φ] is,   i

Φ|T ∗ (Oa [φ])|Ψ = [dφ] Oa [φ] Φ∗ [φ(t1 )] e


R t1 t2

dtL[φ(t)]

Ψ[φ(t2 )].

(97)

We can use (97) to obtain a similar expression for the matrix element of the antiT ∗ -ordered product of some operator Ob [φ] in the presence of the conjugate states, ∗

Ψ|T (Ob [φ])|Φ = Φ|T ∗ (Ob† [φ])|Ψ ∗  

(98) i

= [dφ] Ob [φ] Φ[φ(t1 )] e−


R t1 t2

dtL[φ(t)]

Ψ∗ [φ(t2 )].

(99)

Summing over a complete set of wave functionals Φ[φ(t1 )] gives a delta functional,  Φ[φ− (t1 )]Φ∗ [φ+ (t1 )] = δ[φ− (t1 ) − φ+ (t1 )]. (100) Φ

Multiplying (97) by (99), and using (100), gives a functional integral expression for the expectation value of any anti-T ∗ -ordered operator Ob multiplied by any T ∗ -ordered operator Oa , ∗

Ψ|T (Ob [φ])T ∗ (Oa [φ])|Ψ   = [dφ+ ][dφ− ] δ[φ− () − φ+ ()] i

× Ob [φ− ]Oa [φ+ ]Ψ∗ [φ− (t2 )]e


R s

dt{L[φ+ (t)]−L[φ− (t)]}

Ψ[φ+ (t2 )].

(101)

This is the fundamental Schwinger–Keldysh relation between the canonical operator formalism and the functional integral formalism. What we might term the “Feynman rules” of the Schwinger–Keldysh formalism follow from (101) in close analogy to those for in–out matrix elements. Because the same field operator is represented by two different dummy fields, φ± (x), the endpoints of lines carry a ± polarity. External lines associated with the operator Ob [φ] have the − polarity while those associated with the operator Oa [φ] have the + polarity. Interaction vertices are either all + or all −. The same is true for counterterms, which means that mixed-polarity diagrams cannot harbor primitive divergences. Vertices with + polarity are the same as in the usual Feynman rules whereas vertices with the − polarity have an additional minus sign. Propagators can be ++, −+, +− and −−.

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The four propagators can be read off from the fundamental relation (101) when the free Lagrangian is substituted for the full one. I denote canonical expectation values in the free theory with a subscript 0. With this convention one sees that the ++ propagator is the ordinary Feynman result, i∆++ (x; x
) = Ω|T (φ(x)φ(x
))|Ω 0 = i∆(x; x
).

(102)

The other cases are simple to read off and to relate to (102), i∆−+ (x; x
) = Ω|φ(x)φ(x
)|Ω 0 = θ(t − t
)i∆(x; x
) + θ(t
− t)[i∆(x; x
)]∗ ,

(103)

i∆+− (x; x
) = Ω|φ(x
)φ(x)|Ω 0 = θ(t − t
)[i∆(x; x
)]∗ + θ(t
− t)i∆(x; x
),

(104)

i∆−− (x; x
) = Ω|T (φ(x)φ(x
))|Ω 0 = [i∆(x; x
)]∗ .

(105)

The close relations between the various propagators and the minus signs from − vertices combine to enforce causality and reality in the Schwinger–Keldysh formalism. For example, in a diagram with the topology depicted in Fig. 2, suppose the µ vertex at xµ is connected to an amputated external + line. If the vertex at x
is internal then we must sum over + and − variations and integrate to give a result proportional to,   t  dD x
{[i∆++(x; x
)]3 − [i∆+− (x; x
)]3 } = 2i dct
dD−1 x
Im([i∆(x; x
)]3 ). t2

(106) Although expression (97) is simple to derive from the canonical formalism, few particle theorists would have recognized it before Weinberg’s paper for two reasons: • The action integral runs between the finite times t2 ≤ t ≤ t1 ; and • It contains state wave functionals Ψ[φ(t2 )] and Φ∗ [φ(t1 )]. The over-specialization of quantum field theory to asymptotic scattering problems led to generations of particle theorists being inculcated with the dogma that it is irrelevant to consider any state but “the” vacuum (often defined as “the unique, normalizable energy eigenstate”), and that this state is automatically selected by

Fig. 2. A typical Schwinger–Keldysh loop. The vertex at xµ connects to an amputated + leg. The vertex at x
µ is internal and must be summed over + and − polarities. The cancellation between polarities makes the integral (106) pure imaginary and restricts the integration to points x
µ on or within the past light-cone of xµ .

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extending the temporal integration to −∞ < t < +∞. This was always nonsense, but it sufficed for asymptotic scattering theory as long as IR problems were treated using the Bloch–Nordsieck technique,110 the universal applicability of which also became dogma in spite of simple counterexamples.111 Inflationary cosmology has forced us to consider releasing the universe in a prepared state at a finite time. When this is done one realizes that the state wave functional Ψ[φ(t2 )] can be broken up into a free part, whose logarithm is quadratic in the perturbation field, and a series of perturbative corrections involving higher powers of the field, Ψ[φ(t2 )] = Ψ0 [φ(t2 )]{1 + O(φ3 (t2 ))}.

(107)

For example, the free vacuum state wave functional of a massive scalar in flat space is,     2 c2 1 m Ω0 [φ(t2 )] ∝ exp − dD−1 x φ(t2 , x) −∇2 + 2 φ(t2 , x) . (108) 2c

It can be shown that the free part of the vacuum wave functional combines with the quadratic surface variations of the action to enforce Feynman boundary conditions.112 Rather than the usual hand-waving, that is how inverting the kinetic operator gives a unique solution for the propagator. The perturbative correction terms (107), which must be present even to recover the flat space limit, correspond to nonlocal interactions on the initial value surface.25 4.2.
-Suppression and late-time growth Making exact computations requires the ζ(t, x) and hij (t, x) propagators and their interaction vertices. From the free Lagrangian (54), and the appropriate D-dimensional generalization of the scalar mode function (63), one can give a formal expression for the ζ Feynman propagator,  dD−1 k {θ(t − t
)v(t, k)v ∗ (t
, k) + θ(t
− t)v ∗ (t, k)v(t
, k)}eik·x . i∆ζ (x; x
) = (2π)D−1 (109) Expressions (55) and (64) give a similar result for the graviton Feynman propagator,
  1 dD−1 k Πij Πk × i[ij ∆k ](x; x
) = Πi(k Π)j − D−2 (2π)D−1 × {θ(t − t
)u(t, k)u∗ (t
, k) + θ(t
− t)u∗ (t, k)u(t
, k)}eik·x ,

(110)

where the transverse projection operator is Πij ≡ δij − ∂i ∂j /∂k ∂k . Unfortunately, we do not possess simple expressions for either the scalar or tensor mode functions for a general expansion history a(t), nor are all the gauge-fixed and constrained interactions yet known to the order required for a full one loop computation, and

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nothing has been done about renormalization. I will therefore concentrate on characterizing how loop corrections behave with respect to the two most important issues which control their strength: • Enhancement by inverse factors of the slow roll parameter
(t); and • Enhancement by secular growth from IR effects. To understand the issue of
-enhancement, let us first note from the free Lagrangians (54), (55) that the scalar and tensor propagators have the following dependences upon
(t) and the various fundamental constants, i∆ζ ∼


G × Frequency2 , c5


i∆h ∼


G × Frequency2 . c5

(111)

For the effects of inflationary particle production the relevant frequency is the Hubble parameter H(t). (Of course it could be at any time in the past, as could the factor of 1/
(t) in the ζ propagator). This offers a very simple explanation for the approximate forms (72) I derived for the tree order power spectra in Sec. 3.3. The relevant diagrams are given in Fig. 3. To find the gauge-fixed and constrained interactions one must solve the constraint equation (56) for S i [ζ, h], then substitute back into (49). There are many terms, even at the lowest orders, and they generally combine (sometimes after partial temporal integrations) so that the final result is suppressed by crucial powers of
. Each term has two net derivatives, however, this counting must include −1 derivatives from factors of 1/H, and −2 derivatives from factors of 1/∂k ∂k which arise in solving the constraint equation (56). The ζ 3 interaction was derived in 2002 and Maldacena,41 and simple results were obtained in 2006 for the ζ 4 terms by Seery et al.113 At the level of detail I require these two interactions take the form, 1 c4
2 aD−1 Lζ 3 ∼ ζ∂ζ∂ζ,
16π
G

1 c4
2 aD−1 2 Lζ 4 ∼ ζ ∂ζ∂ζ.
16π
G

(112)

In 2007 Jarhus and Sloth discussed the next two interactions,114 1 c4
3 aD−1 3 Lζ 5 ∼ ζ ∂ζ∂ζ,
16π
G

Lζ 6 ∼

c4
3 aD−1 4 ζ ∂ζ∂ζ. 16π
G

(113)

Results for the lowest ζ-hij interactions were reported in 2012 by Xue et al.115 Making no distinction between which fields are differentiated, these interactions take the general form, 1 c4
aD−1 [Lζh2 + Lζ 2 h + Lζ 2 h2 ] ∼ [ζ∂h∂h + h∂ζ∂ζ + h2 ∂ζ∂ζ].
16π
G

(114)

Fig. 3. Diagrammatic representation for the tree order power spectra. A straight line represents the ζ propagator while the graviton propagator is wavy.

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Fig. 4. One loop corrections to the scalar power spectrum. Straight lines represent the ζ propagator while the graviton propagator is wavy.

And because they persist even in the de Sitter limit of
= 0 it is obvious that the purely graviton interactions are not
-suppressed, 1 c4 aD−1 [Lh3 + Lh4 + · · ·] ∼ [h∂h∂h + h2 ∂h∂h + · · ·].
16π
G

(115)

The various diagrams which contribute to the one loop correction to ∆2R (k) are depicted in Fig. 4. In each case, the leftmost point is fixed at xµ = (t, x) and the µ rightmost point is fixed at x
= (t, 0). Interior points are integrated. For example, the leftmost diagram on the first line has the general form,   (116) dD y i∆ζ (x; y)Vζ 3 (y) dD y
i∆ζ (x
; y
)Vζ 3 (y
)[i∆ζ (y; y
)]2 , where Vζ 3 (y) and Vζ 3 (y
) denote the vertex operators one can read off from the ζ 3 interaction. To recover the ordering in (57) the xµ line must have − polarity µ µ and the x
must be +, while the y µ and y
vertices would be summed over all ± variations. I will return to the possibility that vertex integrations lead to temporal growth, but for now let me assume that the two net derivatives in each vertex combine with the associated integral to produce a factor of c/H 2 . Under this assumption one can estimate the strength of any diagram by combining: • A factor of
GH 2 /c5
for each ζ propagator; • A factor of
GH 2 /c5 for each hij propagator; and • A factor of c5
N /
GH 2 for each vertex with either 2N − 1 or 2N ζ fields and any number of hij fields. For example, the estimated result for the leftmost diagram on the first line of Fig. 4 is,

0 5 2 2

4

2


GH 2
GH 2 c

GH 2
GH 2
GH 2
× × = = . × c5
c5
GH 2 c5
c5
c5 (117) In the final expression of (117) I have extracted the tree order result (72), so one sees that this one loop correction is down by the factor of
GH 2 /c5 (which was

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Fig. 5. One loop corrections to the tensor power spectrum. Straight lines represent the ζ propagator while the graviton propagator is wavy.

inevitable on dimensional grounds) times an extra factor of
. Neither of the two diagrams on the bottom line of Fig. 4 has this extra suppression,

3

0 5 2 1




GH 2
GH 2 c

GH 2
GH 2 × × = × , (118) c5
c5
GH 2 c5
c5

2

1 5 1




GH 2
GH 2 c

GH 2
GH 2 × × = × . (119) c5
c5
GH 2 c5
c5 The one loop corrections to ∆2h (k) are depicted in Fig. 5. The same rules suffice to estimate the strengths of these corrections, although one must recall that the tree order result (72) has no
-enhancement. For example, the leftmost diagram on the first line of Fig. 5 contributes,

2

2 5 2




GH 2
GH 2 c

GH 2
GH 2 × × = × . (120) c5
c5
GH 2 c5 c5 None of the one loop corrections to ∆2h (k) is any stronger than (120); the central diagram on the first line is actually suppressed by an additional factor of
. We therefore conclude that one loop corrections to each of the power spectra are generically suppressed from the tree results (72) by a factor of
GH 2 /c5 ∼ 10−10 . In these estimates it will be noted that I have not specified when the various factors of H(t) and
(t) are evaluated. Both quantities are thought to be nearly constant during much of primordial inflation — in which case, it does not matter much when they are evaluated. However, it is well to recall that the actual loop corrections are integrals of sometimes differentiated propagators, like expression (116). Weinberg noted the possibility for these integrations to grow with the co-moving time.21 That this can happen is associated with IR divergences of the ζ and hij propagators which are evident from the small k limiting form (69) of both mode functions for constant
.116 This leads to two sources of possible secular growth: • A coincident propagator — such as the four diagrams on the bottom lines of Figs. 4 and 5 — grows like
GH 2 /c5 × Ht;46,117,118 and • When the analogous in–out expression would be IR divergent, a Schwinger– Keldysh integration such as (106) is finite but grows with time.119

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The physical origin of both effects is that even the long wavelength parts of the ζ and hij effective actions are affected by the on-going process of inflationary particle production. Weinberg proved an important theorem which limits the growth of loop corrections to the primordial power spectra for the single-scalar model (27) plus an arbitrary number of free scalars which are minimally coupled to gravity.22 His result is that the largest possible secular enhancement to the depressingly small estimates (119) and (120) consists of powers of the number of inflationary e-foldings. His student Bua Chaicherdsakul extended the result to cover fermions and gauge particles.120 However, the situation changes radically if one allows matter couplings to the inflation because the resulting Coleman–Weinberg corrections to its effective potential can induce important changes in the expansion history. For example, if an m2 ϕ2 inflation were coupled to a massless fermion the resulting negative energy Coleman–Weinberg correction would cause the universe to end in a Big Rip singularity.121 Because the gauge (38) forces the inflation to agree with its classical trajectory, changes in the physical expansion history manifest in secular growth of ζ(t, x) correlators. It should also be noted that Weinberg’s theorem is limited to the inflationary power spectra. Explicit computations show that loop corrections to other correlators such as the vacuum polarization122 and the fermion self-energy123 can grow like powers of the inflationary scale factor. 4.3. Nonlinear extensions No one disputes Weinberg’s bound, but some cosmologists disagree that there can be any secular corrections. Weinberg gave two examples,21 which other authors confirmed.124 However, Senatore and Zaldarriaga identified a problem with the use of dimensional regularization in one of these examples, and went on to argue that no secular enhancements are possible under any circumstances.125 It seems very clear that models can be devised for which quantum corrections to the naive correlators (57), (58) grow with timelike powers of the number of e-foldings, just as Weinberg stated.71 Close examination of claims to the contrary126–128 reveals that the authors are not actually disputing this, but rather arguing that the naive correlators (57), (58) should be replaced with other theoretical quantities which fail to show secular growth. That brings up the fascinating and crucial issue of what operators represent the measured power spectra. The problem with trying to overcome the loop suppression through secular enhancements is that the growth begins at first horizon crossing and terminates with the end of inflation. But observable modes experienced first horizon crossing at most 50 e-foldings before the end of inflation, which means the enhancement can be at most some small power of 50. The issue which focused people’s attention on modifying the naive observables was not secular growth but rather the closely associated problem of sensitivity to the IR cutoff. Ford and Parker showed in 1977 that the propagator of a massless, minimally coupled scalar has an IR divergence

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for any constant
(t) cosmology in the range 0 ≤
≤ 32 .116 In view of relations (67), (68) this same problem afflicts both the ζ and hij propagators. Like all IR divergences, this one derives from posing an unphysical question. The problem in this case is arranging large correlations for super-horizon modes which no local observer can control. There are two fixes which have been suggested: • Either arrange for the initially super-horizon modes to be in some less highly correlated state129 ; or else • Work on a spatially compact manifold such as T D−1 whose coordinate radius is such that there are no initially super-horizon modes.130 In practice, each fix amounts to cutting off the Fourier mode sum at some minimum value k = L−1 . If IR divergences could be shown to afflict loop corrections to the power spectra, and if the cutoff L were large enough, then loop corrections might be significant. I recommend the review article by Seery on IR loop corrections to inflationary correlators.131 Important work was done by a number of authors.132–145 In 2010 Giddings and Sloth were able to give a convincing argument that graviton loop corrections to ∆2R (k) are indeed sensitive to the IR cutoff L,146,147 and hence able to make significant corrections. This disturbed people who think about gauge invariance in gravity because the actual IR divergence — as opposed to the closely associated secular growth factor — is a constant in space and time, and a constant field configuration hij (t, x) ought to be gauge equivalent to zero. Even before the work of Giddings and Sloth, the possibility of such corrections had prompted Urakawa and Tanaka to argue for modifying the original definition (57) so that the spatial argument of the first field ζ(t, x) is replaced by the metric-dependent geodesic X[g](x) which is a constant invariant length x from the other point 0 in the direction x,148,149  k3 2 ∆R (k, t) → (121) d3 x e−ik·x Ω|ζ(t, X[g](x)ζ(t, 0)|Ω . 2π 2 After the paper by Giddings and Sloth, it was quickly established that these sorts of partially invariant observables are free of the IR divergence.150–156 In subsequent work, Giddings and Sloth have sought to identify invariant observables which still show the enhancement.157,158 Tanaka and Urakawa have also continued their work on the problem,159–162 The discussion of IR effects attracted me because I had for years been working on these in de Sitter background. I was also fascinated by the struggle to identify physical observables in quantum gravity because my long-neglected doctoral thesis dealt with that very subject.18 In fact had I considered corrections involving precisely the same sort of geodesics as in (121)! One thing I discovered is that they introduce new ultraviolet divergences associated with integrating graviton fields over the one-dimensional background geodesic.18 These new divergences change the power spectrum into the expectation value of a nonlocal composite operator

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which no one currently understands how to renormalize.23 Shun-Pei Miao and I also demonstrated that (121) disturbs the careful pattern of
-suppression which we saw in Sec. 4.2; one loop corrections to (121) go like the tree order result (72) times
GH 2 /c5
.23 For the very same reason, non-Gaussianity would also be unsuppressed.23 So changing what theoretical quantity, we identify with the scalar power spectrum from (57) to (121) in order to avoid sensitivity on the IR cutoff would come at the high price of introducing uncontrollable ultraviolet divergences and observable non-Gaussianity. It seems a bad bargain, and I mean no disrespect to colleagues who are struggling to puzzle out the truth, as am I, when I say we must do better. It seems clear to me that we need new ideas. One radical and thought-provoking proposal is the suggestion by Miao and Park to abandon correlators altogether and instead quantum-correct the mode function relations (65), (66).24 Among other things, this would avoid the new ultraviolet divergence which Fr¨ ob et al. have found in one loop corrections to the tensor power spectrum because the two times coincide.163 It also seems to me that too few physicists appreciate the wondrous opportunity which has befallen us to shape a new discipline by defining its observables. The debate on this vital subject is sometimes confused, and too often degenerates into shouting matches. In an effort to clarify matters Shun-Pei Miao and I laid out 10 principles which are worth repeating here23 : (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

IR divergence differs from IR growth; The leading IR logs might be gauge independent; Not all gauge dependent quantities are unphysical; Not all gauge invariant quantities are physical; Nonlocal “observables” can null real effects; Renormalization is crucial and unresolved; Extensions involving ζ must be
-suppressed; It is important to acknowledge approximations; Sub-horizon modes cannot have large IR logs; and Spatially constant quantities are observable.

4.4. The promise of 21 cm radiation Particle physicists are familiar with the saying, “yesterday’s discovery is tomorrow’s background.” Cosmologists are today witnessing the final stages of this process in the context of observations of the cosmic microwave background, as interlocking developments in technology and understanding of astrophysical processes have permitted fundamental theory to be probed more and more deeply. A brief survey of the history is instructive: • 1964 — discovery of the monopole, for which Penzias and Wilson received the 1978 Nobel Prize;

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• 1970s — discovery of the dipole, which gives the Earth’s motion relative to the CMB; • 1992 — discovery of lowest higher multipoles in the temperature–temperature correlator by COBE, for which Mather and Smoot received the 2006 Nobel Prize; • 1999 — detection of the first Doppler peak by BOOMERanG and MAXIMA, supporting inflation and not cosmic strings as the primary source of structure formation; • early 2000s — detection of E-mode polarization by DASI and CBI, and demonstration by WMAP of the T -E anti-correlation predicted by inflation; • 2003–2010 — full-sky maps of temperature and E-mode correlators by WMAP, and their use for precision determinations of cosmological parameters; • 2013 — full-sky map of Planck resolves seven Doppler peaks and give tighter bounds on ΛCDM parameters; • 2013 — First detection of B-mode polarization from gravitational lensing by the South Pole Telescope; • 2014 — Detection of primordial B-mode polarization claimed by BICEP2, confirming another key prediction of primordial inflation, fixing the inflationary energy scale to be ∼ 2 × 1016 GeV, and incidentally establishing the existence and quantization of gravitons; and • 2014 — Resolution of six acoustic peaks of E-mode polarization by the Atacama Cosmology Telescope Polarimeter, which provides an independent determination of ΛCDM parameters. The first few steps are even now being taken in what could be an equally fruitful evolution, whose full realization will consume decades as it yields a steady series of discoveries. I refer to the project of surveying large volumes of the universe using the 21 cm line.164 The discovery potential is obvious from the comparison between an X-ray and a CT scan: all that has been learned from the cosmic microwave background derives from the surface of last scattering, whereas 21 cm radiation allows us to make a tomograph of the universe. Current and planned projects probe two regimes of cosmic redshift: • 0  z  4 — in which the radiation from unresolved galaxies is observed to probe baryon acoustic oscillations.165–170 • 6  z  10 — in which intergalactic Hydrogen is observed to probe the epoch of reionization.171–175 The first of these provides important information for understanding the mysterious physics which is causing the current universe to accelerate, the discovery of which earned Perlmutter, Schmidt and Riess the 2011 Nobel Prize. The second is crucial to understanding the first generation of stars, and will eventually be an important foreground in future observations. As technology and engineering improve, and as astrophysical effects are better understood, it is possible to foresee a time (decades from now) when redshifts as

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high as z ∼ 50 are observed to measure the matter power spectrum with staggering accuracy. There is enough potentially recoverable data in the 21 cm radiation to resolve one loop corrections.176 Current measurements of ∆2R (k) do not test fundamental theory because we lack a compelling mechanism for driving primordial inflation, but that is bound to change over the decades required for the full maturity of 21 cm cosmology. And when we do understand the driving mechanism, it will be possible to untangle the one loop correction from the tree order effect, which will test quantum gravity. This could be for quantum gravity what the measurement of g − 2 was for quantum electrodynamics. The data is there, and people will be working for decades to harvest it. 5. Other Quantum Gravitational Effects As I have explained, the driving force for quantum gravitational effects during inflation is the production of nearly massless, minimally coupled scalars (if there are any) and gravitons. The presence of these particles is quantified by the scalar and tensor power spectra. Because Einstein + anything is an interacting quantum field theory, the newly created particles must interact, at some level, both with themselves and with other particles. This section describes how to study those interactions. I first list the various linearized effective field equations, then I describe the propagators and how to represent the tensor structure of the associated 1PI two-point functions. The section closes with a review of results and open problems. However, the issue of back-reaction is so convoulted and contentious that it merits its own subsection. 5.1. Linearized effective field equations We want to study how the propagation of a single particle is affected by the vast sea of IR gravitons and scalars produced by inflation. That can be done by computing the 1PI two-point function of the particle in question and then using it to quantum-correct the linearized effective field equation. Recall from (10) that quantum gravitational loop corrections from inflationary particle production are suppressed by
GHi2 /c5 ∼ 10−10 . Because this number is so small it is seldom necessary include nonlinear effects or to go beyond one loop order. The usual unit conventions of relativistic quantum field theory apply in which time is measured so that c = 1, and mass is measured so that
= 1. The loop counting parameter of quantum gravity is κ2 ≡ 16πG(×
/c3 ) ≈ 1.3 × 10−68 m2 . Because
i ≈ 0.013 is so small, most work is done on de Sitter background, for which
(t) = 0 and H(t) is a constant. Computations are done on a portion of the full de Sitter manifold which is termed “the cosmological patch” in the recent literature, and sometimes “open conformal coordinates” in the older literature, ds2 = a2 (η)[−dη 2 + dx · dx],

a(η) ≡ −

1 = eHt . Hη

(122)

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The D − 1 spatial coordinates exist in the same range −∞ < xi < +∞ as Minkowski space, but the conformal time η is limited to the range −∞ < η ≤ 0. The de Sitter metric is gµν = a2 ηµν , where ηµν is the Minkowski metric. In contrast to Sec. 4, metric fluctuations are characterized by the conformally rescaled and canonically normalized graviton field hµν , full gµν (x) ≡ a2 [ηµν + κhµν (x)].

(123)

Graviton indices are raised and lowered with the Minkowski metric, hµν ≡ η µρ hρν , hµν ≡ η µρ η νσ hρσ , h ≡ η µν hµν . Fermion fields are also conformally rescaled, ψifull (x) ≡ a

D−1 2

× ψi (x).

(124)

The various 1PI two-point functions are evaluated using dimensional regularization, then fully renormalized with the appropriate counterterms in the sense of Bogoliubov and Parasiuk,177 Hepp178 and Zimmermann179,180 (BPHZ). After this the unregulated limit of D = 4 is taken. As explained in Sec. 4.1, quantum corrections to the in–out effective field equations at spacetime point xµ are dominated µ by contributions from points x
in the infinite future when the three-volume has been expanded to infinity. Quantum corrections to the in–out matrix elements of field operators are also generally complex, even for real fields. These results are correct for in–out scattering theory, but they have no physical relevance for cosmology where the appropriate question is, what happens to the expectation value of the field operator in the presence of a prepared state which is released at some finite time. One solves that sort of problem using the Schwinger–Keldysh formalism.106–109 In this technique each of the 1PI N -point functions of the in–out formalism gives rise to 2N Schwinger–Keldysh N -point functions. It is the sum of the ++ and +− 1PI two-point functions which appears in the linearized Schwinger–Keldysh effective field equation. This combination is both real and causal. The 1PI two-point function for a scalar is known as its “self-mass-squared,” −iM 2 (x; x
). The quantum-corrected, linearized field equation for a minimally coupled scalar with mass m is,  a2 (−∂02 − 2Ha∂0 + ∇2 )ϕ(x) − m2 a4 ϕ(x) − d4 x
M 2 (x; x
)ϕ(x
) = 0. (125) The (conformally rescaled) fermion’s 1PI two-point function is called its “selfµ stands for the usual gamma matrices then the energy,” −i[i Σj ](x; x
). If γij quantum-corrected, linearized field equation for a fermion with mass m is,  µ (126) iγij ∂µ ψj (x) − maψi (x) − d4 x
[i Σj ](x; x
)ψj (x
) = 0. The 1PI two-point function for a photon has the evocative name “vacuum polarization,” +i[µ Πν ](x; x
). If F νµ ≡ η νρ η µσ (∂ρ Aσ − ∂σ Aρ ) is the usual field strength tensor then the quantum-corrected Maxwell equation can be written,  ∂ν F νµ (x) + d4 x
[µ Πν ](x; x
)Aν (x
) = J µ (x), (127)

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where J µ (x) is the current density. Finally, the 1PI two-point function for a (conformally rescaled and canonically normalized) graviton is termed the “graviton self-energy,” −i[µν Σρσ ](x; x
). It is used to quantum-correct the linearized Einstein equation as,  κa2 µρ νσ ∂α [a2 Lµναβρσ ∂β hρσ (x)] − d4 x
[µν Σρσ ](x; x
)hρσ (x
) = − η η Tρσ (x), 2 (128) where Tρσ is the linearized stress tensor and I define, Lµνρσαβ ≡

1 αβ µ(ρ σ)ν 1 η [η η − η µν η ρσ ] + η µν η ρ(α η β)σ 2 2 1 + η ρσ η µ(α η β)ν − η α)(ρ η σ)(µ η ν)(β) . 2

(129)

The scalar equation (125) and its spinor counterpart (126) of course describe the propagation of scalars and spinors, respectively. The vector and tensor equations (127), (128) can be similarly used to study the propagation of dynamical photons and gravitons, but they also describe modifications of the electrodynamic and gravitational forces. Dynamical quanta show no modification in flat space quantum field theory (enforcing that is what typically fixes the field strength renormalization), so it is at least possible that nothing happens as well during inflation. However, the force laws are guaranteed to show an effect during inflation because they do so in flat space background.181–183 5.2. Propagators and tensor 1PI functions The symmetries of the general cosmological geometry (1) are homogeneity and isotropy. However, the de Sitter limit of
(t) = 0 results in the appearance of two additional symmetries. Although it obvious to cosmologists that these extra symmetries can be at best approximate for inflationary cosmology, they have exerted a powerful influence on mathematical physicists owing to the expectation that the full de Sitter group should play the same role in organizing and simplifying quantum field theory on de Sitter that Poincar´e invariance has played for flat space. That expectation has remained unfulfilled owing to the time dependence intrinsic to inflationary particle production. In our D-dimensional conformal coordinate system (122), the 12 D(D + 1) de Sitter transformations can be decomposed as follows: • (D − 1) spatial translations, η
= η, •

1 2 (D

x
i = xi +
i .

(130)

x
i = Rij xj .

(131)

− 1)(D − 2) spatial rotations, η
= η,

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• One dilatation, η
= kη,

x
i = kxj .

(132)

• (D − 1) spatial special conformal transformations, η
=

η , 1 − 2θ · x + θ2 x · x

x
=

xi − θi x · x . 1 − 2θ · x + θ2 x · x

(133)

Homogeneity is (130) and isotropy is (131). The two additional symmetries which appear in the de Sitter limit of
(t) = 0 are (132), (133). Although IR divergences induce de Sitter breaking, they do so in a limited way that leaves the largest part of the result de Sitter invariant. For dimensional regularization computations the best way to express this de Sitter invariant part in terms of the length function y(x; z), y(x; x
) ≡ aa
H 2 [x − x
2 − (|η − η
| − iε)2 ].

(134)

Except for the factor of iε (whose purpose is to enforce Feynman boundary conditions) the function y(x; x) is related to the invariant length (x; x
) from xµ to µ x
,

1 y(x; x
) = 4 sin2 H(x; x
) . (135) 2 The four Schwinger–Keldysh polarity variations (102)–(105) never affect the de Sitter breaking terms. They can all be obtained by making simple changes of the iε term in (134), y++ (x; x
) = aa
H 2 [x − x
2 − (|η − η
| − iε)2 ],

(136)

y−+ (x; x
) = aa
H 2 [x − x
2 − (η − η
− iε)2 ],

(137)

y+− (x; x
) = aa
H 2 [x − x
2 − (η − η
+ iε)2 ],

(138)

y−− (x; x
) = aa
H 2 [x − x
2 − (|η − η
| + iε)2 ].

(139)

The best way of expressing higher spin propagators on de Sitter, as on flat space, is by acting differential operators on scalar propagators. I work with a general scalar propagator i∆b (x; x
) which obeys the equation, [

+ (b2 − b2A )H 2 ]i∆b (x; x
) =

Here and henceforth, the index bA is bA ≡ scalar d’Alembertian,

D−1 2

iδ D (x − x
) √ . −g and

(140)

stands for the covariant

√ 1 1 ≡ √ ∂µ ( −gg µν ∂ν ) = 2 (−∂02 − (D − 2)Ha∂0 + ∇2 ). −g a

(141)

For the case of b < bA , the propagator has a positive mass-squared m2 = (b2A − b2 )H 2 and its propagator is a de Sitter invariant hypergeometric function of y(x; x
).

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Its expansion for b = ν is,

 D  D 

D2 −1 Γ( 2 )Γ 1 − ∞  D 4 H D−2  2




Γ − − 1 (x; x ) = i∆dS D ν 1 1 2 y (4π) 2    Γ +ν Γ − ν n=0 2 2



 3 3 Γ 2 + ν + n Γ 2 − ν + n y !n− D2 +2

×  D 4 Γ 3− + n (n + 1)! 2     ! Γ(bA + ν + n)Γ(bA − ν + n) y n   .

− (142) D 4    Γ + n n! 2 When b ≥ bA the naive mode sum is IR divergent for the same reason as the problem discovered by Ford and Parker116 which I described in Sec. 4.3. One sometimes encounters contrary statements in the mathematical physics literature,184 but close examination reveals that the authors admit they are constructing a formal solution to the propagator equation which is not a true propagator by the illegitimate technique of adding negative norm states to the theory.185 The proper technique129,130 of cutting off the mode sum amounts to adding to (142) a de Sitter breaking, IR correction,186,187
∆IR ν (x; x ) =

H D−2 (4π)

D 2

Γ(ν)Γ(2ν)

× θ(ν − bA ) 1 Γ(bA )Γ ν + 2

n [  N 2 ]  (aa
)ν−bA −N  a a
× + CN nm (y − 2)N −n−2m , ν − bA − N n=0 a
a m=0 N −n

[ν−bA ]

N =0

(143) where the coefficients CN nm are,

N 1 − Γ(bA + N + n − ν) 4 CN nm = × m!n!(N − n − 2m)! Γ(bA + N − ν) ×

Γ(bA ) Γ(1 − ν) Γ(1 − ν) × × . Γ(bA + N − 2m) Γ(1 − ν + n + 2m) Γ(1 − ν + m)

(144)

The full propagator is therefore,
IR
i∆b (x; x
) = lim [i∆dS ν (x; x ) + ∆ν (x; x )]. ν→b

(145)

It is often useful to discuss integrated propagators which obey, [

+ (b2 − b2A )H 2 ]i∆bc (x; z) = i∆c (x; z).

(146)

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The solution is easily seen to be,187,188 1 i∆bc (x; z) = 2 [i∆c (x; z) − i∆b (x; z)] = i∆cb (x; z). (b − c2 )H 2

II-389

(147)

I also employ a doubly integrated propagator which obeys, [

+ (b2 − b2A )H 2 ]i∆bcd (x; z) = i∆cd (x; z).

(148)

The solution can be written in a form which is manifestly symmetric under any interchange of the three indices a, b and c, i∆bcd (x; z) = =

i∆bd (x; z) − i∆bc (x; z) (c2 − d2 )H 2

(149)

(d2 − c2 )i∆b (x; z) + (b2 − d2 )i∆c (x; z) + (c2 − b2 )i∆d (x; z) . (150) (b2 − c2 )(c2 − d2 )(d2 − b2 )H 4

The propagator for a (conformally rescaled) fermion of mass m was constructed m by Candelas and Raine189 in terms of the scalar propagator (142) with ν = − 12 +i H , D

i[i Sj ](x; x
) = [iγ µ ∂µ + am](aa
) 2 −1 



I − γ0 I + γ0
dS
× (x; x ) + i∆dS (x; x ) . i∆ ∗ ν ν 2 2

(151)

Except for the conformal rescaling, expression (151) is de Sitter invariant as long as the mass is real. Tachyonic fermions with m2 ≤ −DH 2 break de Sitter invariance the same way that tachyonic scalars do. Vector and tensor fields raise the issue of gauge fixing. Normally we would accomplish this by adding to the action gauge fixing terms which respect the coordinate isometries of the background, however, there is an obstacle to doing this on any background such as de Sitter which possesses a linearization instability.188 This leaves two alternatives: • One either add a de Sitter breaking gauge fixing term which cannot be extended to the full de Sitter manifold; or • One can impose some gauge condition on the field operators. After some discussion of the problem, I will give results for both alternatives. The obstacle to adding invariant gauge fixing terms came as a surprise because one can actually derive them in flat space quantum field theory by starting from an exact gauge and making functional changes of variables.190 In 2009 Miao, Tsamis and I identified precisely where this procedure breaks down when a linearization instability is present188 but one can see the physics problem quite simply by considering flat space electrodynamics on the manifold T 3 × R. Because the spatial sections are compact, both sides of the spatially averaged, µ = 0 Maxwell equation must vanish separately,   d3 x ∂i F i0 (t, x) = d3 x J 0 (t, x) = 0. (152) ∂ν F νµ = J µ ⇒ T3

T3

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This zero charge constraint follows from the invariant field equations so it must be true in any valid gauge. However, adding a Poincar´e invariant gauge fixing term results in a very different theory. The field equations of Feynman gauge are, [−∂t2 + ∇2 ]Aµ (t, x) = J µ (t, x).

(153)

These equations can be solved for any total charge so there cannot be any question that the theory has been changed. The use of covariant gauge fixing terms is so ingrained that some people’s first reaction to the obstacle on de Sitter is, “let’s just go ahead and do it anyway!” As it happens, Emre Kahya and I had stumbled upon what goes wrong if that is done in 2005. We used two different gauges to compute and fully renormalize the one loop self-mass-squared −iM 2 (x; x
) for a charged, massless, minimally coupled scalar on de Sitter.191,192 There was no problem with a de Sitter-breaking gauge fixing term193 — nor is there any problem194 when Lorentz gauge is enforced as a strong operator condition195 — but we found on-shell singularities when using the de Sitter–Feynman gauge propagator.196 The origin of these singularities seems to be that integrating the self-mass-squared against the scalar wave function measures the A0 J 0 interaction of the particle with its own field, and one can see from Eq. (153) that the solution for A0 (t, x) must grow like Qt2 /2 in Feynman gauge. My favorite de Sitter-breaking gauge fixing term for electromagnetism is,193 1 LGF = − aD−4 (η µν Aµ,ν − (D − 4)HaA0 )2 . 2

(154)

Because space and time components are treated differently it is useful to have an expression for the purely spatial part of the Minkowski metric, η µν ≡ ηµν + δµ0 δν0 .

(155)

In this gauge, the photon propagator takes the form of a sum of constant tensor factors times scalar propagators, i[µ ∆ν ](x; x
) = η µν aa
i∆B (x; x
) − δµ0 δν0 aa
i∆C (x; x
).

(156)

The B-type and C-type propagators are special cases of (142) with ν = (D − 3)/2 and ν = (D − 5)/2, respectively,    

D2 −1 D−2  D 4 H
Γ i∆B (x; x ) = − 1 D  2 y 2 (4π)     +

0

1

D A 2 ( y )n+2− D2

∞  Γ@n+    Γ(n+2)  n=0

4

     ( y )n  Γ(n+D−2) , 1 0  − 4  D   A @ Γ n+  2

(157)

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i∆C (x; x
) =

H D−2 D

(4π) 2

II-391

  

D2 −1  D 4 Γ −1  2 y  



 D D ∞   n− 2 +3 Γ n+ 2 −1 y !n+2− D2  −  Γ(n + 2) 4 n=0

−

(n + 1)Γ(n + D − 3) Γ(n + D 2)

    y !n   .   4  

(158)

Note that the infinite sums in (157) and (158) vanish for D = 4, which means they only need to be included when multiplied by a divergence, and even then only the lowest terms of the sums are required. In fact the B-type and C-type propagators agree in D = 4, and the photon propagator in this gauge is the same for D = 4 as it is in flat space! Despite the noncovariant gauge, this propagator shows no physical breaking of de Sitter invariance. My favorite de Sitter-breaking gauge fixing term for gravity is,193,197 1 LGF = − aD−2 η µν Fµ Fν , Fµ 2

1 ρσ 0 ψµρ,σ − ψρσ,µ + (D − 2)Haψµρ δσ . ≡η 2

(159)

Note that it breaks (133) but preserves (130)–(132), just like its electromagnetic cousin (154). In this gauge the graviton propagator takes the form of a sum of three constant index factors times a scalar propagator,  I [µν Tρσ ]i∆I (x; x
). (160) i[µν ∆ρσ ](x; x
) = I=A,B,C

The constant index factors are, A [µν Tρσ ] = 2 ηµ(ρ η σ)ν −

2 η η , D − 3 µν ρσ

B 0 0 ] = −4δ(µ [µν Tρσ η ν)(ρ δσ) , C ]= [µν Tρσ

2 [(D − 3)δµ0 δν0 + η µν ][(D − 3)δρ0 δσ0 + η ρσ ]. (D − 2)(D − 3)

(161) (162) (163)

The de Sitter invariant B-type and C-type propagators appear as well in the photon propagator (156). The A-type propagator corresponds to (145) with b = (D − 1)/2, and it is better known as the propagator of a massless, minimally coupled

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scalar,198,199 i∆A (x; x
)





D D  D 

D2 −2 Γ Γ + 1 −1 H D−2  4 2 4 Γ(D − 1) 2 2



= + + ln(aa
) D  D D D y y (4π) 2   −1 −2 Γ 2 2 2

  D  + 1 Γ n + ∞ D  !n+2− 2 !n   y Γ(n + D − 1) y 2   ,



+ A1 − −  D D 4 4   n=1  n− + 2 Γ(n + 2) nΓ n + 2 2 (164)

where the constant A1 is, 



 Γ(D − 1) D−1 D −ψ 1 − A1 = +ψ + ψ(D − 1) + ψ(1) . D 2 2 Γ 2

(165)

The de Sitter breaking factor of ln(aa
) on the first line of (164) is intensely disturbing to mathematical physicists who insist that it must be an artefact of the de Sitter breaking gauge (159).200 However, there are several ways of seeing that gravitons show real de Sitter breaking.201 The simplest and most physical is to note that linearized gravitons in transverse-traceless gauge obey the same equation as the massless, minimally coupled scalar,36,37 whose propagator is admitted to break de Sitter invariance.186 In fact, the breaking of de Sitter invariance is a consequence of the time dependence of inflationary particle production and is embedded in the primordial power spectra.185 One should also note that the gauge fixing term (159) preserves dilatation invariance (132) while the propagator (164) breaks this symmetry. Finally, one can show that the propagator is not invariant even when naive de Sitter transformations are augmented to include the compensating gauge transformation needed to restore (159).202 It is also possible to enforce gauge conditions — even de Sitter invariant ones — as strong operator equations. When this is done, the resulting propagators can no longer be expressed using constant tensor factors. The most economical representation is to write them as differential projection operators acting on scalar structure functions. The differential operators are constructed using the covariant derivative operator Dµ based on the de Sitter connection, Γρµν = aH(δ ρµ δ 0ν + δ ρν δ 0µ − η 0ρ ηµν ).

(166)

We raise and lower indices using the de Sitter metric, Dµ ≡ g µν Dν . The Lorentz gauge condition Dµ Aµ = 0 implies that the photon propagator obeys, Dµ i[µ ∆ν ](x; x
) = 0 = D
i[µ ∆ν ](x; x
). ν

(167)

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If the photon has mass MV (which can happen due to spontaneous symmetry breaking) then its propagator obeys the equation, [

− (D − 1)H 2 − MV2 ]i[µ ∆ρ ](x; x
) =

igµρ δ D (x − x
) √ + Dµ D
ρ i∆A (x; x
). −g

(168)

The term involving i∆A on the right-hand side of (168) is required for consistency with the gauge condition of Eq. (167) of Ref. 195 and was missing from an earlier µ by writing the electromagnetic analysis.196 We can find the transverse projector Pαβ µ field strength as Fαβ ≡ Pαβ Aµ . The photon propagator takes the form,203 i[µ ∆ρ ](x; x
) = P αβµ × P


κλ ρ

×



Dα Dκ

Dβ Dλ y × S (x; x ) . 1 2H 2 2H 2

(169)

The vector structure function can be expressed using a singly integrated propagator (147),203 

2 MV2 2H 2 2H 2 D−3 S1 = + 2 i∆BB + [i∆ − i∆ ], where c = − . (170) B c MV MV4 2 H2 It is de Sitter invariant so long as MV2 > −(D − 2)H 2 , which includes the usual massless photon. Recall from expression (123) that we define the graviton field hµν by conformal rescaling. Although this is simplest for computations, enforcing de Sitter covariant gauge conditions is better done with the field χµν ≡ a2 hµν whose indices are raised and lowered with the de Sitter metric. The most general de Sitter covariant gauge condition can be parametrized with a real number β = 2, Dµ χµν −

β Dν χµµ = 0. 2

(171)

In any gauge of this form the χµν propagator is the sum of a spin zero part and a spin two part,203,204 i[αβ ∆ρσ ](x; x
) = i[αβ ∆0ρσ ](x; x
) + i[αβ ∆2ρσ ](x; x
).

(172)

The spin zero part is diagonal on the primed and unprimed index groups, i[µν ∆0ρσ ](x; x
) = Pµν × P
ρσ [S0 (x; x
)].

(173)

The spin zero projector Pµν is,
Pµν ≡ Dµ Dν + gµν

2−β Dβ − 2





+2

D−1 Dβ − 2



H

2

.

(174)

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And the spin zero structure function can be expressed in terms of a doubly integrated propagator (150),203,204 S0 (x; x
) = −

2(Dβ − 2)2 i∆WNN (x; x
), (2 − β)2 (D − 2)(D − 1)

(175)

D−1 2 where bW ≡ (D + 1)/2 and b2N ≡ ( D−1 2 ) + 2( 2−β ). Although BW corresponds to a tachyonic mass of m2 = −DH 2 , it turns out that this source of de Sitter breaking drops out when acted upon by the projectors.205 The index bN corresponds to a 2 scalar of mass m2 = −2( D−1 2−β )H , and it will give rise to physical de Sitter breaking for β < 2. Constructing the spin two part of the propagator is roughly analogous to what was done for the photon. I first define the projector by expanding the Weyl tensor, C αβγδ ≡ Pµναβγδ κχµν + O(κ2 χ2 ). This operator can be used to build a manifestly transverse and traceless quantity,

Dα Dγ D
κ D
θ 4H 4



Dβ D λ y Dδ D φ y × S2 (x; x ) . 4H 4

i[µν ∆2ρσ ](x; x
) = Pµναβγδ × P
ρσ

κλθφ

×

(176)

The (gauge independent) spin two structure function involves doubly integrated propagators (150),203 32 S2 (x; z) = [i∆AAA (x; z) − 2i∆AAB + i∆ABB (x; z)]. (177) (D − 3)2 Although the B-type propagator is de Sitter invariant, we saw from expression (164) that the A-type propagator is not. An explicit computation shows that this de Sitter breaking does not drop out when all the various derivatives in (176) are acted.205 I have already noted that mathematical physicists are very loath to accept de Sitter breaking in the graviton propagator, so it is perhaps not surprising that a final attempt was made to avoid it.206 However, the net result was simply to clarify the illegitimate analytic continuations which must be employed to derive formal de Sitter invariant solutions that are not true propagators.185 There are still some who believe that the de Sitter breaking evident in the all correct solutions (160)– (163) and (172)–(177) will drop out when gauge invariant operators are studied, as it does from the linearized Weyl–Weyl correlator.207,208 I doubt this, and I will point out in Sec. 5.4 that it amounts to re-fighting the same controversy over fields versus potentials which was decided for electromagnetism by the Aharomov–Bohm effect. However, the important thing to note here is that everyone now agrees on the propagators which must be used to make such computations. Just as for their propagators, so it is best to represent the 1PI two-point functions of vector and tensor fields in terms of differential operators acting on structure functions. When the particular loop under study shows no physical de Sitter breaking one could employ a de Sitter invariant representation. However, long experience

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shows that it is usually superior to a employ simple, de Sitter breaking representation, even when there is no physical breaking of de Sitter invariance.209–211 For the vacuum polarization that form is, i[µ Πν ] = (η µν η ρσ − η µσ η νρ )∂ρ ∂σ
F (x; x
) + (η µν η ρσ − η µσ η νσ )∂ρ ∂σ
G(x; x
).

(178)

For the graviton self-energy based on the conformally rescaled field hµν defined in expression (123) it is, − i[µν Σρσ ](x; x
) = F µν × F
[F0 (x; x
)] + G µν × G
ρσ

ρσ

[G0 (x; x
)]

+ F µνρσ [F2 (x; x
)] + G µνρσ [G2 (x; x
)].

(179)

The scalar differential operators are, (µ

F µν = ∂ µ ∂ ν + 2f1 aHδ 0 ∂ ν) + f2 a2 H 2 δ µ0 δ ν0 − η µν [∂ 2 + f3 aH∂0 + f4 a2 H 2 ], (180) µ ν

(µ ν)

G µν = ∂ ∂ + 2g1 aHδ 0 ∂

+ g2 a2 H 2 δ µ0 δ ν0 − η µν [∇2 + g3 aH∂0 + g4 a2 H 2 ], (181)

where ordinary derivative indices are raised using the Minkowski metric and I remind the reader that a bar denotes the suppression of temporal indices, η µν ≡ ηµν + δ0µ δ0ν . The various constants in (180), (181) are, f1 = f3 = f4 = (D − 1),

f2 = (D − 2)(D − 1),

(182)

g1 = g3 = g4 = (D − 2),

g2 = (D − 2)(D − 1).

(183)

The spin-2 operators are constructed using a transverse-traceless projector obtained by expanding the Weyl tensor, Cαβγδ ≡ a2 Cαβγδµν × κhµν + O(κ2 h2 ), F µνρσ ≡ Cαβγδµν × C
κλθφ

× η ακ η βλ η γθ η δφ ,

(184)

G µνρσ ≡ Cαβγδµν × C
κλθφ

× η ακ η βλ η γθ η δφ .

(185)

ρσ ρσ

5.3. Results and open problems As explained in Sec. 2, inflationary particle production is significant for gravitons and for massless, minimally coupled scalars. Any effect due to gravitons is, by definition, quantum gravitational, however, scalar effects may or may not be. An important example of scalar quantum gravitational effects is correlators of the field ζ(t, x) which represents the gravitational response to a scalar inflation. On the other hand, it is possible to imagine a massless, minimally coupled scalar which is a spectator to de Sitter inflation, with gravity considered to be a nondynamical background. The effects from such scalars are not strictly quantum gravitational, but I will mention them as well because their physics is so similar, and because

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they are so much simpler to study. For four such spectator scalar models there are complete, dimensionally regulated and fully renormalized results: • For a real scalar with a quartic self-interaction both the expectation value of the stress tensor198,199 and the self-mass-squared212,213 have been computed at one and two loop orders. These calculations show that inflationary particle production pushes the scalar up its potential, which increases the vacuum energy and leads to a violation of the weak energy condition on cosmological scales without any instability. • For a massless fermion which is Yukawa-coupled to a real scalar, one loop computations have been made of the fermion self-energy,123,214 the scalar selfmass-squared,215 and the effective potential.216 There is also a two loop computation of the coincident vertex function.216 These calculations show that the inflationary production of scalars affects super-horizon fermions like a mass, while the scalar mass remains small. The fermion mass decreases the vacuum energy without bound in such a way that the universe eventually undergoes a Big Rip singularity.216 • For scalar quantum electrodynamics, one loop computations have been made of the vacuum polarization from massless scalars122,217 — and also slightly massive scalars218 — of the scalar self-mass-squared,191,192,194 and of the effective potential.219,220 Much more difficult two loop computations have been made for the square of the scalar field strength and for its kinetic energy,194 for two coincident field strength tensors,221 and for the expectation value of the stress tensor.221 These calculations show that the inflationary production of charged scalars causes super-horizon photons behave as if they were massive, while the scalar remains light and the vacuum energy decreases slightly.220,222 Comoving observers, at an exponentially increasing distance from the sources, perceive screening of electric charges and magnetic dipoles, while observers at a fixed invariant distance perceive the same sources to be enhanced.223 • The nonlinear sigma model has been exploited to better understand the derivative interactions of quantum gravity,224 and explicit two loop results have been obtained for the expectation value of the stress tensor.225,226 These computations show that while the leading secular corrections to the stress tensor cancel, there are sub-leading corrections. Spectator scalar effects are simpler than those of gravitons, and generally stronger because they can avoid derivative interactions. Although scalar effects avoid the gauge issue, they are less universal because they depend upon the existence of light, minimally coupled scalars at inflationary scales. In five models with gravitons there are complete, dimensionally regulated and fully renormalized results: • For pure quantum gravity, the graviton one-point function has been computed at one loop order.227 This shows that the effect of inflationary gravitons at one loop order is a slight increase in the cosmological constant.

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• For quantum gravity plus a massless fermion, the fermion self-energy has been computed at one loop order.121,228 This shows that spin–spin interactions with inflationary gravitons drive the fermion field strength up by an amount that increases without bound.229,230 • For quantum gravity plus a slightly massive fermion, the fermion self-energy has been computed to lowest order in the mass.231 However, the dynamical consequences of this have not yet been explored. • For quantum gravity plus a massless, minimally coupled scalar there are one loop computations of the scalar self-mass-squared232 and the graviton self-energy.233 The effective field equations reveal that the scalar kinetic energy redshifts too rapidly for it to experience a significant effect from inflationary gravitons,234 or for a graviton to experience significant effects from inflationary scalars.211,235 The effects of inflationary scalars on the force of gravity, are still under study. • For quantum gravity plus electromagnetism, there is a one loop computation of the vacuum polarization.236 Even though gravitons are uncharged, they do carry momentum which can be added to virtual photons to make comoving observers perceive an enhanced force, while static observers experience no secular change.237 The effect for magnetic dipoles is opposite.237 The effect on dynamical photons is still being studied. It is worth noting that all these computations were made with the simple graviton propagator (160) in the noncovariant gauge (159). Dolgov did a very early computation of the small cosmological contribution to the vacuum polarization from fermionic quantum electrodynamics.238 Another early result is Ford’s approximate evaluation of the one loop graviton one-point function in pure quantum gravity.239 A more recent computation was made using adiabatic regularization.240 Momentum cutoff computations have also been performed of the one loop graviton self-energy241 and the two loop graviton one-point function.242 For gravity plus various sorts of scalars, the one loop scalar contribution to the noncoincident graviton self-energy has been computed.243 More recently, Kitamoto and Kitazawa have been exploring IR graviton corrections to various couplings in the off-shell effective field equations.244–248 Given a Lagrangian, it is quite simple to work out the interaction vertices in any background. The obstacle to making computations is finding the propagators. In Sec. 5.2 I have given all of them, in all known gauges, so anyone wishing to make computations should be able to do so. In spite of all the work that has been done, many interesting calculations remain which can be done within the existing formalism: • Use the previously derived one loop graviton contribution to the vacuum polarization to work out the effects on dynamical photons.236 • Use the previously derived one loop scalar contribution to the graviton selfenergy211,233 to work out corrections to the force of gravity.

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• Perform a fully dimensionally regulated and renormalized computation of the one loop graviton contribution to the graviton self-energy in the noncovariant gauge (159) and use it to work out, what happens to dynamical gravitons249 and to the force of gravity. • Re-compute the one loop graviton contributions to the scalar, fermion, vector and graviton 1PI two-point functions in a general de Sitter invariant gauge (171). This is especially important to check the conjecture23 that the leading IR logarithms might be gauge independent. • Compute the two loop expectation value of C αβγδ (x)Cαβγδ (x) to see if the secular corrections which are certainly present in individual diagrams cancel out all the diagrams are added to produce a scalar. There are also two improvements of the basic formalism which need to be developed: • Work out corrections to the various initial states; and • Develop reasonable and consistent approximations for including the effects of
(t) = 0 in realistic models of inflation. The state corrections would allow us to solve the effective field equations at all times, rather than only at asymptotically late-times. This is important to check for corrections to field strengths which develop for a while and then approach a constant at late-times when the particle under study has redshifted too much to interact further with the sea of IR quanta produced by inflation. It has been suggested that this could produce an observable k-dependent tilt to the scalar power spectrum.234 The goal of the second improvement would be to finally perform complete and fully renormalized computations of the one loop corrections to the power spectra. Note from Sec. 5.2 that the technology for representing vector and tensor propagators and 1PI functions can be applied to any geometry, so the real problem is just generalizing the scalar propagator to arbitrary
(t).250 Finally, there are a number of less focused but very important issues which require study: • Develop gauge independent and physically reasonable ways of quantifying changes in particle dynamics due to inflationary particle production; and • Work out observable consequences of the interactions other particles have with inflationary gravitons and scalars. The first point is closely related to the important issue I discussed in Sec. 4.3 of how we correspond the observed power spectra to quantities in fundamental theory. Of course the second issue is crucial if the calculations described in this section are ever to be tested. One obvious point of phenomenological contact is the origin of cosmic magnetic fields.251,252 It has been suggested that the vacuum polarization induced during inflation by a light, charged scalar — possibly part of the Higgs doublet — might produce the super-horizon correlations needed to seed cosmic

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magnetic fields.253–257 More needs to be done in connecting the undoubted inflationary effect to post-inflationary cosmology. Another point of phenomenological contact might be inflationary baryogenesis.258 More generally, we need to understand what happens after inflation to the ubiquitous factors of “H” which occur in one loop corrections to scalar effective potentials from fermions,216 from gauge particles220 and from other scalars.259 5.4. Back-Reaction I commented in Sec. 3.1 on the fine-tuning issues of single-scalar inflation. The worst of these is the problem of the cosmological constant.62,63 Many people have suspected that quantum effects associated with a positive cosmological constant might lead to the cosmological constant screening itself. As far as I know, the first person to suggest this was Sasha Polyakov back in 1982.260 Variations of the same idea were put forward in the 1980s by Myhrvold,261 Mottola,262 Ford,239 Antoniadis et al.263 Mazur and Mottola264 and Antoniadis and Mottola.265,266 In 1992 Tsamis and I proposed a model of inflation based on three contentions267,268 : • The bare cosmological constant is not unreasonably small; • This triggered primordial inflation; and • Inflation was brought to an end by the gradual accumulation of self-gravitation between IR gravitons which are ripped out of the vacuum by the accelerated expansion and whose gravitational attraction is still holding the universe together. This is the idea that provoked the hostile referee report from which I quoted at the beginning of Sec. 4.1. It still amazes me that our model generates so much opposition because it is really a natural way to start inflation, and to make it last a long time without fine-tuning. Our proposal is also wonderfully economical, making a virtue out of the absence of any reason for the bare cosmological constant to be small, and using gravity to solve a gravitational problem with no new particles. I will devote the remainder of this section to discussing our model but one should take note of significant work on related proposals by Polyakov,269–272 by Mottola and collaborators,273–285 by Brandenberger and collaborators,286–292 by Boyanovsky, de Vega and Sanchez,293–295 by Boyanovsky and Holman,296 by Marolf and Morrison,297–299 and by Akhmedov and collaborators.300–309 Of course scepticism about my proposal with Tsamis is centered on the crucial third bullet point,310 which we have so far been unable to prove but is at least not obviously wrong.311 IR gravitons are certainly produced during inflation and it is difficult to understand why they would not attract one another, at least a little. It is also difficult to understand how that effect, which starts from zero, can avoid growing as more and more of the newly created gravitons come into contact with one another. Indeed, it is easy to show that as few as 10 e-foldings of inflation

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produces enough IR gravitons to make the universe collapse if they were all in causal contact.312,313 One reason we have not been able to demonstrate the third point is that it requires the finite part of a two loop computation in quantum gravity on de Sitter background. This might seem confusing after my statement in Sec. 3.3 that the primordial power spectra are tree order effects but there is not any contradiction. The diagram which represents how the tree order power spectra of Fig. 3 change the background geometry is obtained by attaching the two end points to a threepoint vertex and connecting this to an external line, as in the leftmost diagram of Fig. 6. That one loop diagram gives the average gravitational response to the lowest order stress-energy of inflationary gravitons. It is constant and must actually be absorbed into a renormalization of the cosmological constant if the universe is really to begin inflating at the stated rate.227 (This is just like the renormalization condition of flat space quantum electrodynamics which makes the mass that appears in the electron propagator agree with the physical electron mass). The effect of selfgravitation between inflationary gravitons begins at the next order, for example, in the right-hand side diagram of Fig. 6. This diagram can show secular growth in the Schwinger–Keldysh formalism because the internal vertex points are integrated over the past light-cone of the external point, which grows as one waits to later and later times. Another reason we have not yet been able to prove the third point is that it requires an invariant quantification of the expansion rate. We actually performed a two loop computation of the graviton one-point function in a fixed gauge (but in D = 4, with a three-momentum cutoff that I do not trust) which does show slowing if we infer the expansion rate from the expectation value of the metric the same way one does from a classical metric.241 Although the two loop computation consumed a year’s time (with a computer!), using it to get the expansion rate was simple because the expectation value of the metric in a homogeneous and isotropic state is itself homogeneous and isotropic, so the procedure is identical to the passage from expression (1) to (2). However, Unruh noted correctly that one cannot treat the expectation value of the metric as a metric.314 The correct procedure is to instead define an invariant operator which quantifies the expansion rate, and then compute its expectation value, just as I did for Green’s functions in my long-forgotten thesis work.18 So my career has come full circle!

Fig. 6. One loop and two loop contributions to the graviton one-point function in pure quantum gravity.

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The importance of Unruh’s observation was demonstrated by another proposal for significant back-reaction in scalar-driven inflation. Although the lowest order effect in pure gravity is at two loops, Mukhanov et al. realized in 1996 that mixing with dynamical scalars affords scalar-driven inflation the opportunity of showing back-reaction at one loop order. Their initial work seemed to show such an effect.315,316 In 1998 Raul Abramo and I confirmed that their result appears as well in different gauges.317,318 However, the effect was absent when we formulated an invariant operator to quantify the expansion rate and computed its expectation value in 2001.319–321 A better expansion invariant was proposed and computed in 2002 by Geshnizjani and Brandenberger, with the same result.322 Subsequent refinements and extensions have been made by Morozzi and Vacca.323–326 Pure quantum gravity lacks the scalar upon which the expansion observable of Geshnizjani and Brandenberger is based. However, Tsamis and I were eventually able to devise what seems to be a reasonable substitute based on a nonlocal functional of the metric.327 With Shun-Pei Miao we are currently engaged in computing its expectation value at one loop order. There should be no effect at one loop, but demonstrating that in a dimensionally regulated and fully renormalized computation is an important test of the observable. If it passes then we can begin the year-long labor involved in computing its expectation value at two loop order. Reaching that order will require a determination of the one loop corrections to the initial state. Although computing the expectation value of the expansion observable is a worthy goal, it will not end the controversy. The predicted time dependence takes the form,220,224  


2 
GH22
GH22 a(t) 2 3 H(t) = H2 1 + ×0+ × K ln + O(G ln (a)) , (186) c5 c5 a(t2 ) where H2 is the expansion rate of an initially empty universe released at time t = t2 . Determining the value of K is expected to require a year. Assuming it is negative, we would have proven that the production of inflationary gravitons slows inflation by an amount which eventually becomes nonperturbatively strong. However, this only means that perturbation theory breaks down, not that inflation eventually stops. The unknown higher loop contributions might push the universe into deceleration, or they might sum up to produce only a small fractional decrease in the expansion rate. Both cases occur in scalar models.216,220 Working out what happens requires some sort of nonperturbative resummation technique. For scalar potential models (with nondynamical gravity) Starobinsky and Yokoyama showed how to sum the series of leading logarithms of the scale factor a(t).224,328,329 This technique was successfully extended to a Yukawacoupled scalar216 and to scalar quantum electrodynamics,220 but there is still no

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generalization to quantum gravity.230 One important piece of recent progress is the demonstration that a classical configuration of gravitational radiation can indeed hold the universe together for a least a little while.330 This is crucial because particles created during inflation decohere so that their physics is essentially classical at late-times. If Tsamis and I are right, it must be a classical configuration of gravitational radiation which is holding the universe together today. The existence of such a configuration also shows that inflationary particle production can actually stop inflation, if only it can produce the necessary configuration. There is absolutely no doubt that individual two loop diagrams make secular contributions to the expansion observable of the form (186). Sceptics believe that all such contributions will cancel when the many, many diagrams are summed to produce a gauge invariant result. This certainly does not happen in scalar models which show the same sort of secular growth.216,220 Sceptics are convinced that gravity will be different because the growth factors derive from long wavelength gravitons which are nearly constant, and exactly constant gravitons are gauge equivalent to zero. Of course the key distinction is between approximately constant and exactly constant. Sceptics insist that there can be no gravitational effects unless there is curvature, and effects must be small if the curvature is small. I take Lagrangians more seriously than pre-conceived opinions, and there is no doubt that matter (and the metric itself) couples to the metric, not to the curvature. Hence it must be possible, under certain circumstances, for these couplings to produce effects even in places and at times when the curvature of the original source has redshifted to nearly zero. The computation will decide who is right, but it is fitting to end this section by commenting on the similarity of this controversy to a classic dispute in the history of quantum electrodynamics. Electromagnetic Lagrangians show unambiguously that quantum matter couples to the electromagnetic vector potential rather than to the field strength, hence it must be possible for these couplings to produce effects even at times and in places where the field strength is small. However, physicists are as prone to prejudice as other humans and generations of theorists ignored what the theory was telling them and insisted that nothing can happen unless the field strength is nonzero. In 1949 Ehrenberg and Siday predicted331 what is better known today as the Aharonov–Bohm effect.332 We all know how the experiment turned out. I am confident the quantum gravitational reprise will have a similar outcome, but the important point for this paper is that posing such classic questions, and assembling the technology to answer them, is one more example of how quantum gravity is becoming a mature subject. 6. Conclusions Another man described another revolutionary era in the following terms: It was the best of times, it was the worst of times. Fortunately, no one is likely to get his head chopped off as quantum gravity makes the difficult passage to maturity! It is

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nonetheless a time of great change, and that affects different physicists in different ways. Some are delighted by the opportunity to participate in founding a new field; others want the changes to stop with the recognition of their own work; and some pretend that nothing is changing. The majority of those who call themselves quantum gravity researchers are in the last camp. To them the key problem of quantum gravity is resolving what happens to black holes in the final stages of evaporation, a process for which we have no data and are not likely to acquire data any time soon. Many of these people are my friends and I wish them well, but the last three decades of unrequited toil in string theory have proven that humans are not good at guessing fundamental theory without the guidance of data. Inflationary cosmology is already changing the way we think about quantum gravity. For example, neither discretization nor simply refusing to quantize the metric are any longer viable solutions to the divergence problem of quantum gravity.14 Cosmology has required us to abandon in–out quantum field theory because the universe began with an initial singularity and no one knows how it will end, or even if it will end. And releasing the universe in a prepared state at a finite time implies we must stop dodging the issue of perturbative corrections to the initial state. Inflationary cosmology is also forcing us to re-think the issue of observables. For example, it used to be pronounced, with haughty dogmatism, that the S-matrix defines all that is observable about a quantum field theory. That statement was always dubious, but we now have counterexamples — which cannot be dismissed because they are being observed — in the form of the primordial power spectra discussed in Sec. 3. The recent controversy over the use of adiabatic regularization (see Sec. 3.4) makes it clear that we need to think more carefully about how to connect theory to observation even at tree order. The one loop corrections discussed in Sec. 4 — which may well be observable in a few decades — pose the further problem of how to define loop corrections to the power spectrum so that they are IR finite, ultraviolet renormalizable, and so that they show the expected pattern of
-suppression. And we are just beginning to understand what other quantum gravitational effects (see Sec. 5) might be engendered by inflationary gravitons and scalars. Some people dismiss the impact of inflationary cosmology on quantum gravity because the detection of primordial gravitons is still tentative27,82,83 and because fundamental theory has yet to provide a compelling model of primordial inflation. I believe these objections are purblind and transient. Within no more than a few years we will know whether or not BICEP2 has detected primordial gravitons.98 It is not so easy to forecast when fundamental theorists will bring forth a compelling model of inflation. The six fine-tuning problems I mentioned in Sec. 3.1 indicate that there is something very wrong with our current thinking. I suspect only a painful collision with data is going to straighten us out. Fortunately, the continuing flow of data makes that collision inevitable. When our observational colleagues have finally uncovered enough of the truth for us to guess the rest it will be possible to

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replace the rough estimates (72) of the tree order power spectra with precise results, and also to compute loop corrections reliably, both for the power spectra and for other quantum gravitational effects. In the fullness of time all of these predictions will be tested by the data present in 21 cm radiation. When that process has gone to completion perturbative quantum gravity will assume its place as a possession for the ages. Acknowledgments I have profited from conversation and collaboration on this subject with L. R. Abramo, B. Allen, R. H. Brandenberger, S. Deser, L. H. Ford, J. Garriga, G. Geshnizjani, A. H. Guth, S. Hanany, A. Higuchi, B. L. Hu, E. O. Kahya, H. Kitamoto, K. E. Leonard, A. D. Linde, D. Marolf, S. P. Miao, P. J. Mora, I. A. Morrison, V. F. Mukhanov, W. T. Ni, V. K. Onemli, S. Park, L. Parker, U. L. Pen, T. Prokopec, A. Roura, A. A. Starobinsky, T. Tanaka, N. C. Tsamis, W. G. Unruh, Y. Urakawa, A. Vilenkin, C. Wang and S. Weinberg. This work was partially supported by NSF Grant PHY-1205591, and by the Institute for Fundamental Theory at the University of Florida. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

B. S. DeWitt, Phys. Rev. 160 (1967) 1113. B. S. DeWitt, Phys. Rev. 162 (1967) 1195. B. S. DeWitt, Phys. Rev. 162 (1967) 1239. G. ’t Hooft and M. J. G. Veltman, Ann. Poincare Phys. Theor. A 20 (1974) 69. S. Deser and P. van Nieuwenhuizen, Phys. Rev. Lett. 32 (1974) 245. S. Deser and P. van Nieuwenhuizen, Phys. Rev. D 10 (1974) 401. S. Deser and P. van Nieuwenhuizen, Phys. Rev. D 10 (1974) 411. S. Deser, H.-S. Tsao and P. van Nieuwenhuizen, Phys. Lett. B 50 (1974) 491. S. Deser, H.-S. Tsao and P. van Nieuwenhuizen, Phys. Rev. D 10 (1974) 3337. K. S. Stelle, Phys. Rev. D 16 (1977) 953. M. H. Goroff and A. Sagnotti, Phys. Lett. B 160 (1985) 81. M. H. Goroff and A. Sagnotti, Nucl. Phys. B 266 (1986) 709. A. E. M. van de Ven, Nucl. Phys. B 378 (1992) 309. R. P. Woodard, Rep. Prog. Phys. 72 (2009) 126002, arXiv:0907.4238 [gr-qc]. L. Parker, Phys. Rev. Lett. 21 (1968) 562. L. Parker, Phys. Rev. 183 (1969) 1057. L. Parker, Phys. Rev. D 3 (1971) 346, Erratum 3 (1971) 2546. N. C. Tsamis and R. P. Woodard, Ann. Phys. 215 (1992) 96. A. Ashoorioon, P. S. B. Dev and A. Mazumdar, Implication of purely classical gravity for inflationary tensor modes, arXiv:1211.4678 [hep-th]. L. M. Krauss and F. Wilczek, Phys. Rev. D 89 (2014) 047501, arXiv:1309.5343 [hep-th]. S. Weinberg, Phys. Rev. D 72 (2005) 043514, arXiv:hep-th/0506236. S. Weinberg, Phys. Rev. D 74 (2006) 023508, arXiv:hep-th/0605244. S. P. Miao and R. P. Woodard, J. Cosmol. Astropart. Phys. 1207 (2012) 008, arXiv:1204.1784 [astro-ph.CO]. S. P. Miao and S. Park, Phys. Rev. D 89 (2014) 064053, arXiv:1306.4126 [hep-th].

May 4, 2017 9:53 b2167-ch21

Perturbative quantum gravity comes of age

II-405

25. E. O. Kahya, V. K. Onemli and R. P. Woodard, Phys. Rev. D 81 (2010) 023508, arXiv:0904.4811 [gr-qc]. 26. Planck Collab. (P. A. R. Ade et al.), arXiv:1303.5076 [astro-ph.CO]. 27. BICEP2 Collab. (P. A. R. Ade et al.), Phys. Rev. Lett. 112 (2014) 241101, arXiv:1403.3985 [astro-ph.CO]. 28. R. Brout, F. Englert and E. Gunzig, Ann. Phys. 115 (1978) 78. 29. A. A. Starobinsky, Phys. Lett. B 91 (1980) 99. 30. D. Kazanas, Astrophys. J. 241 (1980) L59. 31. K. Sato, Mon. Not. R. Astron. Soc. 195 (1981) 467. 32. A. H. Guth, Phys. Rev. D 23 (1981) 347. 33. A. D. Linde, Phys. Lett. B 108 (1982) 389. 34. A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. 48 (1982) 1220. 35. A. D. Linde, Phys. Lett. B 129 (1983) 177. 36. E. Lifshitz, J. Phys. (USSR) 10 (1946) 116. 37. L. P. Grishchuk, Sov. Phys. JETP 40 (1975) 409; Zh. Eksp. Teor. Fiz. 67 (1974) 825. 38. WMAP Collab. (G. Hinshaw et al.), Astrophys. J. Suppl. 208 (2013) 19, arXiv:1212.5226 [astro-ph.CO]. 39. Z. Hou et al., Astrophys. J. 782 (2014) 74, arXiv:1212.6267 [astro-ph.CO]. 40. Atacama Cosmology Telescope Collab. (J. L. Sievers et al.), J. Cosmol. Astropart. Phys. 1310 (2013) 060, arXiv:1301.0824 [astro-ph.CO]. 41. J. M. Maldacena, J. High Energy Phys. 0305 (2003) 013, arXiv:astro-ph/0210603. 42. A. A. Starobinsky, JETP Lett. 30 (1979) 682; Pis’ma Zh. Eksp. Teor. Fiz. 30 (1979) 719. 43. V. F. Mukhanov and G. V. Chibisov, JETP Lett. 33 (1981) 532; Pis’ma Zh. Eksp. Teor. Fiz. 33 (1981) 549. 44. S. W. Hawking, Phys. Lett. B 115 (1982) 295. 45. A. H. Guth and S. Y. Pi, Phys. Rev. Lett. 49 (1982) 1110. 46. A. A. Starobinsky, Phys. Lett. B 117 (1982) 175. 47. J. M. Bardeen, P. J. Steinhardt and M. S. Turner, Phys. Rev. D 28 (1983) 679. 48. V. F. Mukhanov, JETP Lett. 41 (1985) 493; Pisma Zh. Eksp. Teor. Fiz. 41 (1985) 402. 49. V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger, Phys. Rep. 215 (1992) 203. 50. A. R. Liddle and D. H. Lyth, Phys. Rep. 231 (1993) 1, arXiv:astro-ph/9303019. 51. J. E. Lidsey, A. R. Liddle, E. W. Kolb, E. J. Copeland, T. Barreiro and M. Abney, Rev. Mod. Phys. 69 (1997) 373, arXiv:astro-ph/9508078. 52. N. C. Tsamis and R. P. Woodard, Ann. Phys. 267 (1998) 145, arXiv:hep-ph/9712331. 53. T. D. Saini, S. Raychaudhury, V. Sahni and A. A. Starobinsky, Phys. Rev. Lett. 85 (2000) 1162, arXiv:astro-ph/9910231. 54. S. Capozziello, S. Nojiri and S. D. Odintsov, Phys. Lett. B 634 (2006) 93, arXiv:hepth/0512118. 55. Z.-K. Guo, N. Ohta and Y.-Z. Zhang, Mod. Phys. Lett. A 22 (2007) 883, arXiv:astroph/0603109. 56. T. Vachaspati and M. Trodden, Phys. Rev. D 61 (1999) 023502, arXiv:gr-qc/9811037. 57. R. Allahverdi, R. Brandenberger, F.-Y. Cyr-Racine and A. Mazumdar, Annu. Rev. Nucl. Part. Sci. 60 (2010) 27, arXiv:1001.2600 [hep-th]. 58. Supernova Search Team Collab. (A. G. Riess et al.), Astron. J. 116 (1998) 1009, arXiv:astro-ph/9805201.

May 4, 2017 9:53 b2167-ch21

II-406

R. P. Woodard

59. Supernova Cosmology Project Collab. (S. Perlmutter et al.), Astrophys. J. 517 (1999) 565, arXiv:astro-ph/9812133. 60. Y. Wang and P. Mukherjee, Astrophys. J. 650 (2006) 1, arXiv:astro-ph/0604051. 61. U. Alam, V. Sahni and A. A. Starobinsky, J. Cosmol. Astropart. Phys. 0702 (2007) 011, arXiv:astro-ph/0612381. 62. S. Weinberg, Rev. Mod. Phys. 61 (1989) 1. 63. S. M. Carroll, Living Rev. Relativ. 4 (2001) 1, arXiv:astro-ph/0004075. 64. A. Ijjas, P. J. Steinhardt and A. Loeb, Phys. Lett. B 723 (2013) 261, arXiv:1304.2785 [astro-ph.CO]. 65. A. H. Guth, D. I. Kaiser and Y. Nomura, Phys. Lett. B 733 (2014) 112, arXiv:1312.7619 [astro-ph.CO]. 66. A. Linde, arXiv:1402.0526 [hep-th]. 67. A. Ijjas, P. J. Steinhardt and A. Loeb, arXiv:1402.6980 [astro-ph.CO]. 68. R. Arnowitt and S. Deser, Phys. Rev. 113 (1959) 745. 69. R. L. Arnowitt, S. Deser and C. W. Misner, Phys. Rev. 117 (1960) 1595. 70. R. L. Arnowitt, S. Deser and C. W. Misner, Gen. Relativ. Gravit. 40 (2008) 1997, arXiv:gr-qc/0405109. 71. E. O. Kahya, V. K. Onemli and R. P. Woodard, Phys. Lett. B 694 (2010) 101, arXiv:1006.3999 [astro-ph.CO]. 72. N. C. Tsamis and R. P. Woodard, Class. Quantum Grav. 20 (2003) 5205, arXiv:astroph/0206010. 73. N. C. Tsamis and R. P. Woodard, Class. Quantum. Grav. 21 (2003) 93, arXiv:astroph/0306602. 74. N. C. Tsamis and R. P. Woodard, Phys. Rev. D 69 (2004) 084005, arXiv:astroph/0307463. 75. L.-M. Wang, V. F. Mukhanov and P. J. Steinhardt, Phys. Lett. B 414 (1997) 18, arXiv:astro-ph/9709032. 76. J. Martin and D. J. Schwarz, Phys. Rev. D 62 (2000) 103520, arXiv:astroph/9911225. 77. R. Easther, H. Finkel and N. Roth, J. Cosmol. Astropart. Phys. 1010 (2010) 025, arXiv:1005.1921 [astro-ph.CO]. 78. M. G. Romania, N. C. Tsamis and R. P. Woodard, J. Cosmol. Astropart. Phys. 1208 (2012) 029, arXiv:1207.3227 [astro-ph.CO]. 79. D. Polarski and A. A. Starobinsky, Phys. Lett. B 356 (1995) 196, arXiv:astroph/9505125. 80. J. Garcia-Bellido and D. Wands, Phys. Rev. D 52 (1995) 6739, arXiv:gr-qc/9506050. 81. M. Sasaki and E. D. Stewart, Prog. Theor. Phys. 95 (1996) 71, arXiv:astroph/9507001. 82. R. Flauger, J. C. Hill and D. N. Spergel, arXiv:1405.7351 [astro-ph.CO]. 83. P. J. Steinhardt, Nature 510 (2014) 281. 84. R. H. Brandenberger, Probing particle physics from top down with cosmic strings, 1 (2013) 4, 6, arXiv:1401.4619 [astro-ph.co]. 85. L. Parker, arXiv:hep-th/0702216 [Hep-th]. 86. L. Parker and S. A. Fulling, Phys. Rev. D 9 (1974) 341. 87. S. A. Fulling, L. Parker and B. L. Hu, Phys. Rev. D 10 (1974) 3905. 88. T. S. Bunch, J. Phys. A 13 (1980) 1297. 89. P. R. Anderson and L. Parker, Phys. Rev. D 36 (1987) 2963. 90. I. Agullo, J. Navarro-Salas, G. J. Olmo and L. Parker, Phys. Rev. Lett. 101 (2008) 171301, arXiv:0806.0034 [gr-qc].

May 4, 2017 9:53 b2167-ch21

Perturbative quantum gravity comes of age

II-407

91. I. Agullo, J. Navarro-Salas, G. J. Olmo and L. Parker, Phys. Rev. Lett. 103 (2009) 061301, arXiv:0901.0439 [astro-ph.CO]. 92. R. Durrer, G. Marozzi and M. Rinaldi, Phys. Rev. D 80 (2009) 065024, arXiv:0906.4772 [astro-ph.CO]. 93. G. Marozzi, M. Rinaldi and R. Durrer, Phys. Rev. D 83 (2011) 105017, arXiv:1102.2206 [astro-ph.CO]. 94. I. Agullo, J. Navarro-Salas, G. J. Olmo and L. Parker, Phys. Rev. D 84 (2011) 107304, arXiv:1108.0949 [gr-qc]. 95. M. G. Romania, N. C. Tsamis and R. P. Woodard, Class. Quantum Grav. 28 (2011) 075013, arXiv:1006.5150 [gr-qc]. 96. M. G. Romania, N. C. Tsamis and R. P. Woodard, Class. Quantum Grav. 30 (2013) 025004, arXiv:1108.1696 [gr-qc]. 97. M. Bastero-Gil, A. Berera, N. Mahajan and R. Rangarajan, Phys. Rev. D 87 (2013) 087302, arXiv:1302.2995 [astro-ph.CO]. 98. Inflation Probe Science Interest Group, CMB Polarization Measurements: 2015 and Onward. 99. D. Marolf, I. A. Morrison and M. Srednicki, Class. Quantum Grav. 30 (2013) 155023, arXiv:1209.6039 [hep-th]. 100. A. Borde, A. H. Guth and A. Vilenkin, Phys. Rev. Lett. 90 (2003) 151301, arXiv:grqc/0110012. 101. J. S. Schwinger, J. Math. Phys. 2 (1961) 407. 102. K. T. Mahanthappa, Phys. Rev. 126 (1962) 329. 103. P. M. Bakshi and K. T. Mahanthappa, J. Math. Phys. 4 (1963) 1. 104. P. M. Bakshi and K. T. Mahanthappa, J. Math. Phys. 4 (1963) 12. 105. L. V. Keldysh, Zh. Eksp. Teor. Fiz. 47 (1964) 1515; Sov. Phys. JETP 20 (1965) 1018. 106. K.-C. Chou, Z.-B. Su, B.-L. Hao and L. Yu, Phys. Rep. 118 (1985) 1. 107. R. D. Jordan, Phys. Rev. D 33 (1986) 444. 108. E. Calzetta and B. L. Hu, Phys. Rev. D 35 (1987) 495. 109. E. Calzetta and B. L. Hu, Phys. Rev. D 37 (1988) 2878. 110. F. Bloch and A. Nordsieck, Phys. Rev. 52 (1937) 54. 111. G. Veneziano, Nucl. Phys. B 44 (1972) 142. 112. N. C. Tsamis and R. P. Woodard, Class. Quantum Grav. 2 (1985) 841. 113. D. Seery, J. E. Lidsey and M. S. Sloth, J. Cosmol. Astropart. Phys. 0701 (2007) 027, arXiv:astro-ph/0610210. 114. P. R. Jarnhus and M. S. Sloth, J. Cosmol. Astropart. Phys. 0802 (2008) 013, arXiv:0709.2708 [hep-th]. 115. W. Xue, X. Gao and R. Brandenberger, J. Cosmol. Astropart. Phys. 1206 (2012) 035, arXiv:1201.0768 [hep-th]. 116. L. H. Ford and L. Parker, Phys. Rev. D 16 (1977) 245. 117. A. Vilenkin and L. H. Ford, Phys. Rev. D 26 (1982) 1231. 118. A. D. Linde, Phys. Lett. B 116 (1982) 335. 119. N. C. Tsamis and R. P. Woodard, Ann. Phys. 238 (1995) 1. 120. K. Chaicherdsakul, Phys. Rev. D 75 (2007) 063522, arXiv:hep-th/0611352. 121. S.-P. Miao and R. P. Woodard, Class. Quantum Grav. 23 (2006) 1721, arXiv:grqc/0511140. 122. T. Prokopec, O. Tornkvist and R. P. Woodard, Ann. Phys. 303 (2003) 251, arXiv:grqc/0205130. 123. T. Prokopec and R. P. Woodard, J. High Energy Phys. 0310 (2003) 059, arXiv:astroph/0309593.

May 4, 2017 9:53 b2167-ch21

II-408

R. P. Woodard

124. P. Adshead, R. Easther and E. A. Lim, Phys. Rev. D 79 (2009) 063504, arXiv:0809.4008 [hep-th]. 125. L. Senatore and M. Zaldarriaga, J. High Energy Phys. 1012 (2010) 008, arXiv:0912.2734 [hep-th]. 126. L. Senatore and M. Zaldarriaga, J. High Energy Phys. 1301 (2013) 109, arXiv:1203.6354 [hep-th]. 127. L. Senatore and M. Zaldarriaga, J. Cosmol. Astropart. Phys. 1208 (2012) 001, arXiv:1203.6884 [astro-ph.CO]. 128. G. L. Pimentel, L. Senatore and M. Zaldarriaga, J. High Energy Phys. 1207 (2012) 166, arXiv:1203.6651 [hep-th]. 129. A. Vilenkin, Nucl. Phys. B 226 (1983) 527. 130. N. C. Tsamis and R. P. Woodard, Class. Quantum Grav. 11 (1994) 2969. 131. D. Seery, Class. Quantum Grav. 27 (2010) 124005, arXiv:1005.1649 [astro-ph.CO]. 132. M. S. Sloth, Nucl. Phys. B 748 (2006) 149, arXiv:astro-ph/0604488. 133. M. S. Sloth, Nucl. Phys. B 775 (2007) 78, arXiv:hep-th/0612138. 134. A. Bilandzic and T. Prokopec, Phys. Rev. D 76 (2007) 103507, arXiv:0704.1905 [astro-ph]. 135. M. van der Meulen and J. Smit, J. Cosmol. Astropart. Phys. 0711 (2007) 023, arXiv:0707.0842 [hep-th]. 136. D. H. Lyth, J. Cosmol. Astropart. Phys. 0712 (2007) 016, arXiv:0707.0361 [astro-ph]. 137. D. Seery, J. Cosmol. Astropart. Phys. 0711 (2007) 025, arXiv:0707.3377 [astro-ph]. 138. D. Seery, J. Cosmol. Astropart. Phys. 0802 (2008) 006, arXiv:0707.3378 [astro-ph]. 139. N. Bartolo, S. Matarrese, M. Pietroni, A. Riotto and D. Seery, J. Cosmol. Astropart. Phys. 0801 (2008) 015, arXiv:0711.4263 [astro-ph]. 140. Y. Urakawa and K.-I. Maeda, Phys. Rev. D 78 (2008) 064004, arXiv:0801.0126 [hep-th]. 141. A. Riotto and M. S. Sloth, J. Cosmol. Astropart. Phys. 0804 (2008) 030, arXiv:0801.1845 [hep-ph]. 142. K. Enqvist, S. Nurmi, D. Podolsky and G. I. Rigopoulos, J. Cosmol. Astropart. Phys. 0804 (2008) 025, arXiv:0802.0395 [astro-ph]. 143. D. Seery, J. Cosmol. Astropart. Phys. 0905 (2009) 021, arXiv:0903.2788 [astroph.CO]. 144. J. Kumar, L. Leblond and A. Rajaraman, J. Cosmol. Astropart. Phys. 1004 (2010) 024, arXiv:0909.2040 [astro-ph.CO]. 145. C. P. Burgess, L. Leblond, R. Holman and S. Shandera, J. Cosmol. Astropart. Phys. 1003 (2010) 033, arXiv:0912.1608 [hep-th]. 146. S. B. Giddings and M. S. Sloth, J. Cosmol. Astropart. Phys. 1101 (2011) 023, arXiv:1005.1056 [hep-th]. 147. S. B. Giddings and M. S. Sloth, J. Cosmol. Astropart. Phys. 1007 (2010) 015, arXiv:1005.3287 [hep-th]. 148. Y. Urakawa and T. Tanaka, Prog. Theor. Phys. 122 (2009) 779, arXiv:0902.3209 [hep-th]. 149. Y. Urakawa and T. Tanaka, Prog. Theor. Phys. 122 (2010) 1207, arXiv:0904.4415 [hep-th]. 150. C. T. Byrnes, M. Gerstenlauer, A. Hebecker, S. Nurmi and G. Tasinato, J. Cosmol. Astropart. Phys. 1008 (2010) 006, arXiv:1005.3307 [hep-th]. 151. Y. Urakawa and T. Tanaka, Phys. Rev. D 82 (2010) 121301, arXiv:1007.0468 [hep-th]. 152. Y. Urakawa and T. Tanaka, Prog. Theor. Phys. 125 (2011) 1067, arXiv:1009.2947 [hep-th].

May 4, 2017 9:53 b2167-ch21

Perturbative quantum gravity comes of age

II-409

153. M. Gerstenlauer, A. Hebecker and G. Tasinato, J. Cosmol. Astropart. Phys. 1106 (2011) 021, arXiv:1102.0560 [astro-ph.CO]. 154. T. Tanaka and Y. Urakawa, J. Cosmol. Astropart. Phys. 1105 (2011) 014, arXiv:1103.1251 [astro-ph.CO]. 155. D. Chialva and A. Mazumdar, arXiv:1103.1312 [hep-th]. 156. Y. Urakawa, Prog. Theor. Phys. 126 (2011) 961, arXiv:1105.1078 [hep-th]. 157. S. B. Giddings and M. S. Sloth, Phys. Rev. D 84 (2011) 063528, arXiv:1104.0002 [hep-th]. 158. S. B. Giddings and M. S. Sloth, Phys. Rev. D 86 (2012) 083538, arXiv:1109.1000 [hep-th]. 159. T. Tanaka and Y. Urakawa, Prog. Theor. Exp. Phys. 2013 (2013) 083E01, arXiv:1209.1914 [hep-th]. 160. T. Tanaka and Y. Urakawa, Prog. Theor. Exp. Phys. 2013 (2013) 063E02, arXiv:1301.3088 [hep-th]. 161. T. Tanaka and Y. Urakawa, Class. Quantum Grav. 30 (2013) 233001, arXiv:1306.4461 [hep-th]. 162. T. Tanaka and Y. Urakawa, Prog. Theor. Exp. Phys. 2014, 073E01, arXiv:1402.2076 [hep-th]. 163. M. B. Fr¨ ob, A. Roura and E. Verdaguer, J. Cosmol. Astropart. Phys. 1208 (2012) 009, arXiv:1205.3097 [gr-qc]. 164. S. Furlanetto, S. P. Oh and F. Briggs, Phys. Rep. 433 (2006) 181, arXiv:astroph/0608032. 165. J. C. Pober et al., arXiv:1210.2413 [astro-ph.CO]. 166. R. Ansari, J.-E. Campagne, P. Colom, C. Magneville, J.-M. Martin, M. Moniez, J. Rich and C. Yeche, C. R. Acad. Sci. Ser. IV Phys. Astrophys. 13 (2012) 46, arXiv:1209.3266 [astro-ph.CO]. 167. R. A. Battye, I. W. A. Browne, C. Dickinson, G. Heron, B. Maffei and A. Pourtsidou, Mon. Not. R. Astron. Soc. 434 (2013) 1239, arXiv:1209.0343 [astro-ph.CO]. 168. http://chime.phas.ubc.ca/VanderLinde 2013 cifar.pdf. 169. S. S. Ali and S. Bharadwaj, arXiv:1310.1707 [astro-ph.CO]. 170. X. Chen, Int. J. Mod. Phys. Conf. Ser. 12 (2012) 256, arXiv:1212.6278 [astro-ph.IM]. 171. J. C. Pober et al., arXiv:1310.7031 [astro-ph.CO]. 172. M. P. van Haarlem et al., arXiv:1305.3550 [astro-ph.IM]. 173. C. J. Lonsdale et al., arXiv:0903.1828 [astro-ph.IM]. 174. H. Zheng et al., arXiv:1309.2639 [astro-ph.IM]. 175. A. R. Parsons et al., arXiv:0904.2334 [astro-ph.CO]. 176. K. W. Masui and U.-L. Pen, Phys. Rev. Lett. 105 (2010) 161302, arXiv:1006.4181 [astro-ph.CO]. 177. N. N. Bogoliubov and O. S. Parasiuk, Acta Math. 97 (1957) 227. 178. K. Hepp, Commun. Math. Phys. 2 (1966) 301. 179. W. Zimmermann, Commun. Math. Phys. 11 (1968) 1. 180. W. Zimmermann, Commun. Math. Phys. 15 (1969) 208, Lect. Notes Phys. 558 (2000) 217. 181. N. E. J. Bjerrum-Bohr, Phys. Rev. D 66 (2002) 084023, arXiv:hep-th/0206236. 182. J. F. Donoghue, Phys. Rev. Lett. 72 (1994) 2996, arXiv:gr-qc/9310024. 183. J. F. Donoghue, Phys. Rev. D 50 (1994) 3874, arXiv:gr-qc/9405057. 184. M. Faizal and A. Higuchi, Phys. Rev. D 85 (2012) 124021, arXiv:1107.0395 [gr-qc]. 185. S. P. Miao, P. J. Mora, N. C. Tsamis and R. P. Woodard, Phys. Rev. D 89 (2014) 104004, arXiv:1306.5410 [gr-qc]. 186. B. Allen and A. Folacci, Phys. Rev. D 35 (1987) 3771.

May 4, 2017 9:53 b2167-ch21

II-410

R. P. Woodard

187. S. P. Miao, N. C. Tsamis and R. P. Woodard, J. Math. Phys. 51 (2010) 072503, arXiv:1002.4037 [gr-qc]. 188. S. P. Miao, N. C. Tsamis and R. P. Woodard, J. Math. Phys. 50 (2009) 122502, arXiv:0907.4930 [gr-qc]. 189. P. Candelas and D. J. Raine, Phys. Rev. D 12 (1975) 965. 190. S. R. Coleman, Subnucl. Ser. 11 (1975) 139. 191. E. O. Kahya and R. P. Woodard, Phys. Rev. D 72 (2005) 104001, arXiv:grqc/0508015. 192. E. O. Kahya and R. P. Woodard, Phys. Rev. D 74 (2006) 084012, arXiv:grqc/0608049. 193. R. P. Woodard, arXiv:gr-qc/0408002. 194. T. Prokopec, N. C. Tsamis and R. P. Woodard, Class. Quantum Grav. 24 (2007) 201, arXiv:gr-qc/0607094. 195. N. C. Tsamis and R. P. Woodard, J. Math. Phys. 48 (2007) 052306, arXiv:grqc/0608069. 196. B. Allen and T. Jacobson, Commun. Math. Phys. 103 (1986) 669. 197. N. C. Tsamis and R. P. Woodard, Commun. Math. Phys. 162 (1994) 217. 198. V. K. Onemli and R. P. Woodard, Class. Quantum Grav. 19 (2002) 4607, arXiv:grqc/0204065. 199. V. K. Onemli and R. P. Woodard, Phys. Rev. D 70 (2004) 107301, arXiv:grqc/0406098. 200. A. Higuchi, D. Marolf and I. A. Morrison, Class. Quantum Grav. 28 (2011) 245012, arXiv:1107.2712 [hep-th]. 201. S. P. Miao, N. C. Tsamis and R. P. Woodard, Class. Quantum Grav. 28 (2011) 245013, arXiv:1107.4733 [gr-qc]. 202. G. Kleppe, Phys. Lett. B 317 (1993) 305. 203. S. P. Miao, N. C. Tsamis and R. P. Woodard, J. Math. Phys. 52 (2011) 122301, arXiv:1106.0925 [gr-qc]. 204. P. J. Mora, N. C. Tsamis and R. P. Woodard, J. Math. Phys. 53 (2012) 122502, arXiv:1205.4468 [gr-qc]. 205. E. O. Kahya, S. P. Miao and R. P. Woodard, J. Math. Phys. 53 (2012) 022304, arXiv:1112.4420 [gr-qc]. 206. I. A. Morrison, arXiv:1302.1860 [gr-qc]. 207. P. J. Mora and R. P. Woodard, Phys. Rev. D 85 (2012) 124048, arXiv:1202.0999 [gr-qc]. 208. P. J. Mora, N. C. Tsamis and R. P. Woodard, Phys. Rev. D 86 (2012) 084016, arXiv:1205.4466 [gr-qc]. 209. K. E. Leonard, T. Prokopec and R. P. Woodard, Phys. Rev. D 87 (2013) 044030, arXiv:1210.6968 [gr-qc]. 210. K. E. Leonard, T. Prokopec and R. P. Woodard, J. Math. Phys. 54 (2013) 032301, arXiv:1211.1342 [gr-qc]. 211. K. E. Leonard, S. Park, T. Prokopec and R. P. Woodard, arXiv:1403.0896 [gr-qc]. 212. T. Brunier, V. K. Onemli and R. P. Woodard, Class. Quantum Grav. 22 (2005) 59, arXiv:gr-qc/0408080. 213. E. O. Kahya and V. K. Onemli, Phys. Rev. D 76 (2007) 043512, arXiv:gr-qc/0612026. 214. B. Garbrecht and T. Prokopec, Phys. Rev. D 73 (2006) 064036, arXiv:gr-qc/0602011. 215. L. D. Duffy and R. P. Woodard, Phys. Rev. D 72 (2005) 024023, arXiv:hepph/0505156. 216. S.-P. Miao and R. P. Woodard, Phys. Rev. D 74 (2006) 044019, arXiv:gr-qc/0602110.

May 4, 2017 9:53 b2167-ch21

Perturbative quantum gravity comes of age

II-411

217. T. Prokopec, O. Tornkvist and R. P. Woodard, Phys. Rev. Lett. 89 (2002) 101301, arXiv:astro-ph/0205331. 218. T. Prokopec and E. Puchwein, J. Cosmol. Astropart. Phys. 0404 (2004) 007, arXiv:astro-ph/0312274. 219. B. Allen, Nucl. Phys. B 226 (1983) 228. 220. T. Prokopec, N. C. Tsamis and R. P. Woodard, Ann. Phys. 323 (2008) 1324, arXiv:0707.0847 [gr-qc]. 221. T. Prokopec, N. C. Tsamis and R. P. Woodard, Phys. Rev. D 78 (2008) 043523, arXiv:0802.3673 [gr-qc]. 222. T. Prokopec and R. P. Woodard, Ann. Phys. 312 (2004) 1, arXiv:gr-qc/0310056. 223. H. Degueldre and R. P. Woodard, Eur. Phys. J. C 73 (2013) 2457, arXiv:1303.3042 [gr-qc]. 224. N. C. Tsamis and R. P. Woodard, Nucl. Phys. B 724 (2005) 295, arXiv:grqc/0505115. 225. H. Kitamoto and Y. Kitazawa, Phys. Rev. D 83 (2011) 104043, arXiv:1012.5930 [hep-th]. 226. H. Kitamoto and Y. Kitazawa, Phys. Rev. D 85 (2012) 044062, arXiv:1109.4892 [hep-th]. 227. N. C. Tsamis and R. P. Woodard, Ann. Phys. 321 (2006) 875, arXiv:gr-qc/0506056. 228. S. P. Miao and R. P. Woodard, Phys. Rev. D 74 (2006) 024021, arXiv:gr-qc/0603135. 229. S.-P. Miao, arXiv:0705.0767 [hep-th]. 230. S.-P. Miao and R. P. Woodard, Class. Quantum Grav. 25 (2008) 145009, arXiv:0803.2377 [gr-qc]. 231. S. P. Miao, Phys. Rev. D 86 (2012) 104051, arXiv:1207.5241 [gr-qc]. 232. E. O. Kahya and R. P. Woodard, Phys. Rev. D 76 (2007) 124005, arXiv:0709.0536 [gr-qc]. 233. S. Park and R. P. Woodard, Phys. Rev. D 83 (2011) 084049, arXiv:1101.5804 [gr-qc]. 234. E. O. Kahya and R. P. Woodard, Phys. Rev. D 77 (2008) 084012, arXiv:0710.5282 [gr-qc]. 235. S. Park and R. P. Woodard, Phys. Rev. D 84 (2011) 124058, arXiv:1109.4187 [gr-qc]. 236. K. E. Leonard and R. P. Woodard, Class. Quantum Grav. 31 (2014) 015010, arXiv:1304.7265 [gr-qc]. 237. D. Glavan, S. P. Miao, T. Prokopec and R. P. Woodard, arXiv:1308.3453 [gr-qc]. 238. A. D. Dolgov, Sov. Phys. JETP 54 (1981) 223; Zh. Eksp. Teor. Fiz. 81 (1981) 417. 239. L. H. Ford, Phys. Rev. D 31 (1985) 710. 240. F. Finelli, G. Marozzi, G. P. Vacca and G. Venturi, Phys. Rev. D 71 (2005) 023522, arXiv:gr-qc/0407101. 241. N. C. Tsamis and R. P. Woodard, Ann. Phys. 253 (1997) 1, arXiv:hep-ph/9602316. 242. N. C. Tsamis and R. P. Woodard, Phys. Rev. D 54 (1996) 2621, arXiv:hepph/9602317. 243. G. Perez-Nadal, A. Roura and E. Verdaguer, J. Cosmol. Astropart. Phys. 1005 (2010) 036, arXiv:0911.4870 [gr-qc]. 244. H. Kitamoto and Y. Kitazawa, Phys. Rev. D 87 (2013) 124007, arXiv:1203.0391 [hep-th]. 245. H. Kitamoto and Y. Kitazawa, Phys. Rev. D 87 (2013) 124004, arXiv:1204.2876 [hep-th]. 246. H. Kitamoto and Y. Kitazawa, Nucl. Phys. B 873 (2013) 325, arXiv:1211.3878 [hep-th]. 247. H. Kitamoto and Y. Kitazawa, J. High Energy Phys. 1310 (2013) 145, arXiv:1305.2029 [hep-th].

May 4, 2017 9:53 b2167-ch21

II-412

R. P. Woodard

248. H. Kitamoto and Y. Kitazawa, Int. J. Mod. Phys. A 29 (2014) 1430016, arXiv:1402.2443 [hep-th]. 249. P. J. Mora, N. C. Tsamis and R. P. Woodard, J. Cosmol. Astropart. Phys. 1310 (2013) 018, arXiv:1307.1422. 250. T. M. Janssen, S. P. Miao, T. Prokopec and R. P. Woodard, Class. Quantum Grav. 25 (2008) 245013, arXiv:0808.2449 [gr-qc]. 251. P. P. Kronberg, Rep. Prog. Phys. 57 (1994) 325. 252. D. Grasso and H. R. Rubinstein, Phys. Rep. 348 (2001) 163, arXiv:astro-ph/0009061. 253. A.-C. Davis, K. Dimopoulos, T. Prokopec and O. Tornkvist, Phys. Lett. B 501 (2001) 165; Phys. Rev. Focus 10 (2002) STORY9, arXiv:astro-ph/0007214. 254. O. Tornkvist, A.-C. Davis, K. Dimopoulos and T. Prokopec, arXiv:astro-ph/0011278. 255. K. Dimopoulos, T. Prokopec, O. Tornkvist and A. C. Davis, Phys. Rev. D 65 (2002) 063505, arXiv:astro-ph/0108093. 256. T. Prokopec and R. P. Woodard, Am. J. Phys. 72 (2004) 60, arXiv:astro-ph/0303358. 257. T. Prokopec and E. Puchwein, Phys. Rev. D 70 (2004) 043004, arXiv:astroph/0403335. 258. B. Garbrecht and T. Prokopec, Phys. Rev. D 78 (2008) 123501, arXiv:0706.2594 [astro-ph]. 259. T. M. Janssen, S. P. Miao, T. Prokopec and R. P. Woodard, J. Cosmol. Astropart. Phys. 0905 (2009) 003, arXiv:0904.1151 [gr-qc]. 260. A. M. Polyakov, Sov. Phys. Usp. 25 (1982) 187; Usp. Fiz. Nauk. 136 (1982) 538. 261. N. P. Myhrvold, Phys. Rev. D 28 (1983) 2439. 262. E. Mottola, Phys. Rev. D 31 (1985) 754. 263. I. Antoniadis, J. Iliopoulos and T. N. Tomaras, Phys. Rev. Lett. 56 (1986) 1319. 264. P. Mazur and E. Mottola, Nucl. Phys. B 278 (1986) 694. 265. I. Antoniadis and E. Mottola, J. Math. Phys. 32 (1991) 1037. 266. I. Antoniadis and E. Mottola, Phys. Rev. D 45 (1992) 2013. 267. N. C. Tsamis and R. P. Woodard, Phys. Lett. B 301 (1993) 351. 268. N. C. Tsamis and R. P. Woodard, Nucl. Phys. B 474 (1996) 235, arXiv:hepph/9602315. 269. A. M. Polyakov, Nucl. Phys. B 797 (2008) 199, arXiv:0709.2899 [hep-th]. 270. A. M. Polyakov, Nucl. Phys. B 834 (2010) 316, arXiv:0912.5503 [hep-th]. 271. D. Krotov and A. M. Polyakov, Nucl. Phys. B 849 (2011) 410, arXiv:1012.2107 [hep-th]. 272. A. M. Polyakov, arXiv:1209.4135 [hep-th]. 273. I. Antoniadis, P. O. Mazur and E. Mottola, Phys. Rev. Lett. 79 (1997) 14, arXiv:astro-ph/9611208. 274. I. Antoniadis, P. O. Mazur and E. Mottola, arXiv:astro-ph/9705200. 275. S. Habib, C. Molina-Paris and E. Mottola, Phys. Rev. D 61 (2000) 024010, arXiv:grqc/9906120. 276. P. R. Anderson, W. Eaker, S. Habib, C. Molina-Paris and E. Mottola, Phys. Rev. D 62 (2000) 124019, arXiv:gr-qc/0005102. 277. P. R. Anderson, W. Eaker, S. Habib, C. Molina-Paris and E. Mottola, Int. J. Theor. Phys. 40 (2001) 2217. 278. P. R. Anderson, C. Molina-Paris and E. Mottola, Phys. Rev. D 72 (2005) 043515, arXiv:hep-th/0504134. 279. E. Mottola and R. Vaulin, Phys. Rev. D 74 (2006) 064004, arXiv:gr-qc/0604051. 280. I. Antoniadis, P. O. Mazur and E. Mottola, New J. Phys. 9 (2007) 11, arXiv:grqc/0612068.

May 4, 2017 9:53 b2167-ch21

Perturbative quantum gravity comes of age

II-413

281. P. R. Anderson, C. Molina-Paris and E. Mottola, Phys. Rev. D 80 (2009) 084005, arXiv:0907.0823 [gr-qc]. 282. E. Mottola, Int. J. Mod. Phys. A 25 (2010) 2391, arXiv:1006.3567 [gr-qc]. 283. I. Antoniadis, P. O. Mazur and E. Mottola, J. Cosmol. Astropart. Phys. 1209 (2012) 024, arXiv:1103.4164 [gr-qc]. 284. P. R. Anderson and E. Mottola, Phys. Rev. D 89 (2014) 104038, arXiv:1310.0030 [gr-qc]. 285. P. R. Anderson and E. Mottola, Phys. Rev. D 89 (2014) 104039, arXiv:1310.1963 [gr-qc]. 286. J. P. Zibin, R. H. Brandenberger and D. Scott, Phys. Rev. D 63 (2001) 043511, arXiv:hep-ph/0007219. 287. R. H. Brandenberger, arXiv:hep-th/0210165. 288. G. Geshnizjani and R. Brandenberger, J. Cosmol. Astropart. Phys. 0504 (2005) 006, arXiv:hep-th/0310265. 289. R. H. Brandenberger and J. Martin, Phys. Rev. D 71 (2005) 023504, arXiv:hepth/0410223. 290. P. Martineau and R. H. Brandenberger, Phys. Rev. D 72 (2005) 023507, arXiv:astroph/0505236. 291. P. Martineau and R. Brandenberger, arXiv:astro-ph/0510523. 292. P. Martineau and R. Brandenberger, Mod. Phys. Lett. A 23 (2008) 727, arXiv:0709.2671 [astro-ph]. 293. D. Boyanovsky, H. J. de Vega and N. G. Sanchez, Phys. Rev. D 71 (2005) 023509, arXiv:astro-ph/0409406. 294. D. Boyanovsky, H. J. de Vega and N. G. Sanchez, Nucl. Phys. B 747 (2006) 25, arXiv:astro-ph/0503669. 295. D. Boyanovsky, H. J. de Vega and N. G. Sanchez, Phys. Rev. D 72 (2005) 103006, arXiv:astro-ph/0507596. 296. D. Boyanovsky and R. Holman, J. High Energy Phys. 1105 (2011) 047, arXiv:1103.4648 [astro-ph.CO]. 297. D. Marolf and I. A. Morrison, Phys. Rev. D 82 (2010) 105032, arXiv:1006.0035 [gr-qc]. 298. D. Marolf and I. A. Morrison, Phys. Rev. D 84 (2011) 044040, arXiv:1010.5327 [gr-qc]. 299. D. Marolf and I. A. Morrison, Gen. Relativ. Gravit. 43 (2011) 3497, arXiv:1104.4343 [gr-qc]. 300. E. T. Akhmedov and P. V. Buividovich, Phys. Rev. D 78 (2008) 104005, arXiv:0808.4106 [hep-th]. 301. E. T. Akhmedov, P. V. Buividovich and D. A. Singleton, Phys. Atom. Nucl. 75 (2012) 525, arXiv:0905.2742 [gr-qc]. 302. E. T. Akhmedov, Mod. Phys. Lett. A 25 (2010) 2815, arXiv:0909.3722 [hep-th]. 303. E. T. Akhmedov, A. Roura and A. Sadofyev, Phys. Rev. D 82 (2010) 044035, arXiv:1006.3274 [gr-qc]. 304. E. T. Akhmedov, J. High Energy Phys. 1201 (2012) 066, arXiv:1110.2257 [hep-th]. 305. E. T. Akhmedov and A. V. Sadofyev, Phys. Lett. B 712 (2012) 138, arXiv:1201.3471 [hep-th]. 306. E. T. Akhmedov and P. Burda, Phys. Rev. D 86 (2012) 044031, arXiv:1202.1202 [hep-th]. 307. E. T. Akhmedov, Phys. Rev. D 87 (2013) 044049, arXiv:1209.4448 [hep-th]. 308. E. T. Akhmedov, F. K. Popov and V. M. Slepukhin, Phys. Rev. D 88 (2013) 024021, arXiv:1303.1068 [hep-th].

May 4, 2017 9:53 b2167-ch21

II-414

R. P. Woodard

309. E. T. Akhmedov, Int. J. Mod. Phys. D 23 (2014) 1430001, arXiv:1309.2557 [hep-th]. 310. J. Garriga and T. Tanaka, Phys. Rev. D 77 (2008) 024021, arXiv:0706.0295 [hep-th]. 311. N. C. Tsamis and R. P. Woodard, Phys. Rev. D 78 (2008) 028501, arXiv:0708.2004 [hep-th]. 312. N. C. Tsamis and R. P. Woodard, Int. J. Mod. Phys. D 20 (2011) 2847, arXiv:1103.5134 [gr-qc]. 313. R. P. Woodard, arXiv:1403.6763 [astro-ph.CO]. 314. W. Unruh, arXiv:astro-ph/9802323. 315. V. F. Mukhanov, L. R. W. Abramo and R. H. Brandenberger, Phys. Rev. Lett. 78 (1997) 1624, arXiv:gr-qc/9609026. 316. L. R. W. Abramo, R. H. Brandenberger and V. F. Mukhanov, Phys. Rev. D 56 (1997) 3248, arXiv:gr-qc/9704037. 317. L. R. W. Abramo and R. P. Woodard, Phys. Rev. D 60 (1999) 044010, arXiv:astroph/9811430. 318. L. R. W. Abramo and R. P. Woodard, Phys. Rev. D 60 (1999) 044011, arXiv:astroph/9811431. 319. L. R. Abramo and R. P. Woodard, Phys. Rev. D 65 (2002) 043507, arXiv:astroph/0109271. 320. L. R. Abramo and R. P. Woodard, Phys. Rev. D 65 (2002) 063515, arXiv:astroph/0109272. 321. L. R. Abramo and R. P. Woodard, Phys. Rev. D 65 (2002) 063516, arXiv:astroph/0109273. 322. G. Geshnizjani and R. Brandenberger, Phys. Rev. D 66 (2002) 123507, arXiv:grqc/0204074. 323. G. Marozzi and G. P. Vacca, Class. Quantum Grav. 29 (2012) 115007, arXiv:1108.1363 [gr-qc]. 324. G. Marozzi, G. P. Vacca and R. H. Brandenberger, J. Cosmol. Astropart. Phys. 1302 (2013) 027, arXiv:1212.6029 [hep-th]. 325. G. Marozzi and G. P. Vacca, Phys. Rev. D 88 (2013) 027302, arXiv:1304.2291 [gr-qc]. 326. G. Marozzi and G. P. Vacca, arXiv:1405.3933 [gr-qc]. 327. N. C. Tsamis and R. P. Woodard, Phys. Rev. D 88 (2013) 044040, arXiv:1306.6441 [gr-qc]. 328. A. A. Starobinsky and J. Yokoyama, Phys. Rev. D 50 (1994) 6357, arXiv:astroph/9407016. 329. R. P. Woodard, Nucl. Phys. Proc. Suppl. 148 (2005) 108, arXiv:astro-ph/0502556. 330. N. C. Tsamis and R. P. Woodard, arXiv:1405.6281 [gr-qc]. 331. W. Ehrenberg and R. E. Siday, Proc. Phys. Soc. B 62 (1949) 8. 332. Y. Aharonov and D. Bohm, Phys. Rev. 115 (1959) 485.

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Chapter 22 Black hole thermodynamics

S. Carlip Physics Department, University of California at Davis, Davis, CA 95616, USA [email protected]

The discovery in the early 1970s that black holes radiate as black bodies has radically affected our understanding of general relativity, and offered us some early hints about the nature of quantum gravity. In this paper, I will review the discovery of black hole thermodynamics and summarize the many independent ways of obtaining the thermodynamic and (perhaps) statistical mechanical properties of black holes. I will then describe some of the remaining puzzles, including the nature of the quantum microstates, the problem of universality, and the information loss paradox. Keywords: Black hole thermodynamics; Hawking radiation; Bekenstein-Hawking entropy. PACS Number(s): 04.70.Dy, 04.60.−m

1. Introduction The surprising discovery that black holes behave as thermodynamic objects has radically affected our understanding of general relativity and its relationship to quantum field theory. In the early 1970s, Bekenstein1,2 and Hawking3,4 showed that black holes radiate as black bodies, with characteristic temperatures and entropies kTH =


κ , 2π

SBH =

Ahor , 4
G

(1)

where κ is the surface gravity and Ahor is the area of the horizon. These quantities appear to be inherently quantum gravitational, in the sense that they depend on both Planck’s constant
and Newton’s constant G. The resulting black body radiation, Hawking radiation, has not yet been directly observed: The temperature of an astrophysical black hole is on the order of a microkelvin, far lower than the cosmic microwave background temperature. But the Hawking temperature and the Bekenstein–Hawking entropy have been derived in so many independent ways, in different settings and with different assumptions, that it seems extraordinarily unlikely that they are not real. In ordinary thermodynamic systems, thermal properties reflect the statistical mechanics of underlying microstates. The temperature of a cup of tea is a measure II-415

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of the average energy of its molecules. Its entropy is a measure of the number of possible microscopic arrangements of those molecules. It seems natural to suppose the same to be true for black holes, in which case the temperature and entropy (1) might tell us something about the underlying quantum gravitational states. This idea has been used as a consistency check for a number of proposed models of quantum gravity, and has suggested important new directions for research. In one key aspect, though, black hole entropy is atypical. For an ordinary nongravitational system, entropy is extensive, scaling as volume. Black hole entropy, on the contrary, is “holographic,” scaling as area. If the Bekenstein–Hawking entropy really counts black hole microstates, this holographic scaling suggests that a black hole has far fewer degrees of freedom than we might expect. But it has also been argued that a black hole provides an upper bound on the number of degrees of freedom in a given volume: If one tries to pack too many degrees of freedom into a region of space, they may inevitably collapse to form a black hole. This has led to the conjecture that all of nature may be fundamentally holographic, and that our usual counting of local degrees of freedom vastly overestimates their number. Black hole thermodynamics leads to a number of other interesting puzzles as well. One is the “problem of universality.” If the Bekenstein–Hawing entropy SBH counts the black hole degrees of freedom in an underlying quantum theory of gravity, one might expect that the result would depend on that theory. Oddly, though, the expression (1) can be obtained from a number of very different approaches to quantum gravity, from string theory to loop quantum gravity to induced gravity, in which the microscopic states appear to be quite different. This suggests the existence of an underlying structure shared by all of these approaches, but although we have some promising ideas, we do not know what that structure is. A second puzzle is the “information loss paradox.” Consider a black hole initially formed by the collapse of matter in a pure quantum state, which then evaporates completely into Hawking radiation. If the radiation is genuinely thermal, this would represent a transition from a pure state to a mixed state, a process that violates unitarity of evolution and is forbidden in ordinary quantum mechanics. If, on the other hand, the Hawking radiation is secretly a pure state, this would appear to require correlations between “early” and “late” Hawking particles that have never been in causal contact. The problem was recently sharpened by Almheiri et al.,5 who argue that one must sacrifice some cherished principle: the equivalence principle, low energy effective field theory, or the nonexistence of high-entropy “remnants” at the end of black hole evaporation. These issues are currently areas of active research, and the matter remains in flux.

2. Prehistory: Black Hole Mechanics and Wheeler’s Cup of Tea The prehistory of black hole thermodynamics might be traced back to Hawking’s 1972 proof that the area of an event horizon can never decrease.6 This property is reminiscent of the second law of thermodynamics, and the correspondence

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was strengthened with the discovery of analogs of other laws of thermodynamics. This work culminated in the publication by Bardeen et al.7 of the “four laws of black hole mechanics”: For a stationary asymptotically flat black hole in four dimensions, uniquely characterized by a mass M , an angular momentum J, and a charge Q,a

(0) The surface gravity κ is constant over the event horizon. (1) For two stationary black holes differing only by small variations in the parameters M , J and Q κ δM = δAhor + ΩH δJ + ΦH δQ, (2) 8πG where ΩH is the angular velocity and ΦH is the electric potential at the horizon. (2) The area of the event horizon of a black hole never decreases, δAhor ≥ 0.

(3)

(3) It is impossible by any procedure to reduce the surface gravity κ to zero in a finite number of steps.

These laws are numbered in parallel with the four usual laws of thermodynamics, and they are clearly formally analogous, with κ playing the role of temperature and Ahor of entropy. As in ordinary thermodynamics, there are a number of formulations of the third law, not all strictly equivalent. For a summary of the current status, see Ref. 8. While the four laws were originally formulated for four-dimensional “electrovac” spacetimes, they can be extended to more dimensions, more charges and angular momenta, and to other “black” objects such as black strings, rings, and branes. The first law, in particular, holds for arbitrary isolated horizons,9,10 and for much more general gravitational actions, for which the entropy can be understood as a Noether charge.11 But despite the formal analogy with the laws of thermodynamics, Bardeen, Carter and Hawking argued that κ cannot be a true temperature and Ahor cannot be a true entropy. The defining characteristic of temperature, after all, is that heat flows from hot to cold. But if one places a classical black hole in contact with a heat reservoir of any temperature, energy will flow into the black hole, but never out. A classical black hole thus has a temperature of absolute zero. A second thread in the development of black hole thermodynamics began in 1972, when John Wheeler asked his student, Jacob Bekenstein, what would happen if he dropped his cup of tea into a passing black hole.12 The initial state would be a See

the Appendix for a brief summary of the relevant classical black hole physics.

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a cup of tea (with a nonzero entropy) plus a back hole. The final state would be no tea, but a slightly larger black hole. Where did the entropy go? Bekenstein’s ultimate answer was that the black hole must have an entropy, which, he contended, must be proportional to its horizon area. Using a series of thought experiments,1,2 he argued further that the entropy ought to be of the form S=η

Ahor ,
G

(4)

where η is a constant of order one. For example, a spherical particle with a proper radius equal to its Compton wavelength will increase the entropy (4) by an amount of order one bit,2 a result that is, remarkably, independent of the spin and charge of the black hole. A small harmonic oscillator dropped into a black hole will increase (4) by an amount at least as great as its entropy,2 as will a beam of black body radiation.1 Bekenstein’s “generalized second law of thermodynamics” — the claim that the total of ordinary entropy of matter plus black hole entropy SBH never decreases — has now been tested, and confirmed, over a very wide range of settings.8,13,14 3. Hawking Radiation Despite the intriguing parallels between area and entropy, the fact that a classical black hole had temperature T = 0 seemed to present an insurmountable obstacle to identifying the two. Zel’dovich had suggested in 1970 that quantum effects might be relevant, causing spinning black holes to radiate in particular modes,15 but the arguments were not worked out in detail. In 1974, however, Hawking3,4 used newly developed techniques for treating quantum field theory in curved spacetime17 to show that all black holes radiate, with a black body temperature TH . With this identification of temperature, the first law of black hole mechanics (2) determines the entropy, and Bekenstein’s expression (4) is confirmed, with η = 1/4.b There are several ways to describe Hawking’s results. Perhaps the most intuitive is to say that quantum mechanics allows particles to tunnel out through the event horizon. But while Hawking himself used this as a heuristic picture,4 the full mathematical description of such a tunneling process was not worked out until much later.18 Hawking’s calculation was based, rather, on a key feature of quantum field theory in curved spacetime: the fact that the vacuum is not unique, but depends on a choice of time. In particular, Hawking showed that the Minkowski vacuum of a past observer watching the collapse of a star differed from the vacuum of a future observer looking at the resulting black hole.

b Strictly speaking, the first law determines black hole entropy only up to an additive constant. But Pretoruis et al. have designed an (idealized) process that reversibly forms a black hole with entropy SBH , strongly hinting that no additive constant appears (Ref. 16).

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3.1. Quantum field theory in curved spacetime To be more explicit, consider a free massless scalar field ϕ. In Minkowski space, the standard approach to quantization19 is to first expand ϕ in its Fourier modes
ϕ= (ak fk (t, x) + a†k fk∗ (t, x)) with fk = ck eik·x−iωk t , ωk = |k|. (5) k

The fk are positive frequency, positive energy modes, which may be completely characterized by the conditions
fk (t, x) = 0,

∂t fk (t, x) = −iωk fk (t, x) (with ωk > 0).

(6)

fk∗

Similarly, the are negative frequency, negative energy modes. The theory has a natural scalar product  ↔ √ na (ϕ1 ∂ a ϕ∗2 ) gΣ d3 x, (ϕ1 , ϕ2 ) = −i (7) Σ

where Σ is an arbitrary Cauchy surface and na is its unit normal. The modes (6) are then orthonormal, provided that we choose the normalization constants ck = 

1 (2π)3 2ω

.

(8)

To quantize the theory, we now interpret the coefficients of the fk as annihilation operators, and the coefficients of the fk∗ as creation operators. These obey the standard commutation relations, and can be used to construct the usual Fock space of free particle states. In particular, the vacuum is the state annihilated by the ak ak |0 = 0.

(9)

In a more general spacetime, or even a noninertial frame in Minkowski space, the standard Fourier modes no longer exist. Suppose, though, that we can still choose a preferred time coordinate t. We can then find a complete set of orthonormal modes satisfying (6), perform an expansion of the form (5) with respect to these modes, and once again define creation and annihilation operators and a vacuum state. What is new, though, is that there may now be more than one preferred time. Given two choices, say t and t , two expansions exist

† ∗ (ai fi + a†i fi∗ ) = (ai fi + ai fi ). (10) ϕ= i

i

Furthermore, since the {fi , fi∗ } are a complete set of functions, we can write
(αji fi + βji fi∗ ) (11) fj = i

for some coefficients αji and βji . This relation is known as a Bogoliubov transformation, and the coefficients αji and βji are Bogoliubov coefficients.20 The coefficients may be read off from (11) by using the inner product (7). In particular βji = −(fi∗ , fj ).

(12)

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We now have two vacua, a state |0 annihilated by the ai and a state |0  † annihilated by the ai , and two number operators, Ni = a†i ai and Ni = ai ai . Using (11) and the orthonormality of the mode functions, it may be easily checked that
0 |Ni |0  = |βji |2 . (13) j

Thus, if the coefficients βji are not all zero, the “primed” vacuum will have a nonvanishing “unprimed” particle content. 3.2. Hawking’s calculation In his seminal work on black hole radiation and evaporation,3,4 Hawking computed the Bogoliubov coefficients between an initial vacuum outside a collapsing star and a final vacuum after the formation of a black hole. He showed that a “primed” observer in the distant future would see a thermal distribution of particles at the Hawking temperature (1). While Hawking’s full calculation is too technical to include in this review — see, for example, Traschen’s very nice paper21 for details — it is possible to sketch out the basic argument. Hawking’s starting point is the collapse of a star to form a black hole, as shown in the Carter–Penrose diagram of Fig. 1(a). The shaded area is the collapsing star. The horizon H forms at some time after the collapse has begun. In the distant past, the spacetime far from the star is nearly flat, and we can define a standard Minkowski vacuum at past null infinity I − . In the distant future, the black hole has settled down to a (nearly) stationary configuration, and we can define a vacuum at future null infinity I + . From the previous discussion, we need to find the orthonormal modes at I ± , propagate them to a common Cauchy surface — Σ in Fig. 1(b) — and take the inner product (12) to determine the Bogoliubov coefficients.

(a)

(b)

Fig. 1. The setup for Hawking’s calculation. (a) Carter–Penrose diagram for a star collapsing to form a black hole. (b) Coordinates, ray-tracing, and mode functions.

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For the first step, let us choose null coordinates u (“retarded time”) and v (“advanced time”) in the exterior of the black hole, chosen so that u is an affine parameter along I + and v is an affine parameter along I − , as shown in Fig. 1(b). It is easy to check that the affine condition implies that near I ± , the metric takes the form ds2 = dudv + angular terms. As a result, the Klein–Gordon equation greatly simplifies, and the asymptotic modes become 1 Ym (θ, φ)e−iωv r

near I − ,

1 ∼ Ym (θ, φ)e−iωu r

near I .

fωm ∼  fωm

(14) +

 To calculate the inner product (12), Hawking evolved the mode fωm from I + backwards to I − . In principle, this requires solving the Klein–Gordon equation with boundary conditions at I + . For the relevant frequencies, though, the geometric optics approximation, in which massless particles move along null geodesics, is good enough. The dotted line labeled γ in Fig. 1(b) shows a null geodesic propagating backwards from I + , “reflecting” off r = 0, and continuing to I − . (One may visualize this as an ingoing spherical wave shrinking to a point at the origin, passing through itself, and expanding outward.) This geodesic maps a point u on I + to a point p(u)  in turn map to functions of the form on I − , and the mode functions fωm −1 1 Ym (θ, φ)e−iωp (v) . r

(15)

The meat of Hawking’s calculation is the determination of this function p(u), which he finds by tracking the position of the geodesic γ relative to the horizon H. The result is that p(u) ∼ A − Be−κu ,

(16)

where κ is the surface gravity and A and B are constants. The exponential dependence on u reflects the extreme blue shift of waves passing near the horizon: Two successive crests at I + , labeled by u and u + δu, will be mapped to a separation p (u)δu ∼ κBe−κu δu at I − . Viewed in the opposite direction, the function p(u) describes the “peeling” of null geodesics away from the horizon.22 The computation of the Bogoliubov coefficients is now straightforward. The inner product (12) on a Cauchy surface Σ near I − now involves integrals of the form 
ω dv eiω v e−i κ ln v .

These yield gamma functions of complex arguments, whose absolute squares give the exponential behavior of a thermal distribution with the Hawking temperature. I have left out many steps, of course. In particular, future null infinity I + is not  in (14) are not a complete set. They must be a Cauchy surface, so the modes fωm

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supplemented by modes on the horizon H. These account for backscattering into the black hole. This scattering is energy-dependent, and distorts the black body spectrum, yielding instead a “grey body spectrum” Γω N  = 2πω , (17) e κ −1 where the reflection coefficient Γω can be calculated explicitly.23,24 Generalizations to particles other than massless scalars are straightforward, and give a consistent thermodynamic picture. 4. Back-of-the-Envelope Estimates The description I have given of Hawking’s derivation of black hole radiation is not, for most people, terribly intuitive. It is useful to understand some back-ofthe-envelope estimates, which can give a better feel for the physics. The following examples, illustrated in Figs. 2 and 3, are taken from a set of my earlier lectures,25 slightly updated. 4.1. Entropy We start with an estimate of black hole entropy, using the generalized second law.26 Consider a cubic box of gas with volume 3 , mass m, and temperature T , dropped into a Schwarzschild black hole of mass M (and therefore having a horizon area of A = 16πG2 M 2 ). Let us assume that the box is at least as large as the thermal wavelength of the gas, ∼
/T . Then the descent of the gas into the black hole will lead to a decrease of entropy m m ∆Sgas ∼ − ∼ − . (18) T
The box of gas will coalesce with the black hole when its proper distance ρ from the horizon is of order . For a Schwarzschild black hole, this proper distance is  2GM+δr √ dr  ρ= ∼ GM δr, 1 − 2GM 2GM r

Fig. 2.

A thought experiment for Bekenstein–Hawking entropy.

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A thought experiment for Hawking temperature.

so ρ ∼ when δr ∼ 2 /GM . The gas initially has mass m, but its energy as seen from infinity is redshifted as the box falls toward the black hole. When the box reaches r = 2GM + δr, the black hole will gain a mass  2GM m ∆M ∼ m 1 − ∼ . 2GM + δr GM The change in the horizon area is thus ∆A ∼ G2 M ∆M ∼ Gm .

(19)

Comparing (18) and (19), we see that the black hole must gain an entropy ∆SBH ∼ −∆Sgas ∼

∆A .
G

(20)

4.2. Temperature We next estimate the Hawking temperature, at a similar level of hand-waving.27 As in Sec. 3, we consider inequivalent vacua, but following a heuristic approach suggested by Hawking,4 we now compare vacua inside and outside the horizon. As usual in quantum field theory, let us view the vacuum as a sea of virtual pairs of particles, each pair consisting of a particle with positive energy E relative to an observer at infinity and a particle with negative energy −E. Normally, by the uncertainty principle, a negative energy particle can exist only for a time t ∼
/|E|. Recall, however, that the distinction between “positive” and “negative” energy can depend on the choice of time. For a Schwarzschild black hole, with a metric    −1 2M 2M 2 2 dr2 − r2 dΩ2 , (21) dt − 1 − ds = 1 − r r the coordinate t is a time coordinate in the exterior, but a spatial coordinate in the interior, where the coefficient of dt2 changes sign.c Similarly, the coordinate c Strictly speaking, the coordinates labeled r and t outside the horizon are not the same as those inside, since the two coordinate patches do not overlap. The argument can be rephrased in terms of proper times of infalling observers, though, in a way that avoids this issue.

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r is a spatial coordinate in the exterior, but a time coordinate in the interior, where “forward in time” means “toward the center of the black hole.” A particle with negative energy relative to an exterior observer may thus have positive energy relative to an interior observer, and if it crosses the horizon quickly enough, this will allow us to evade the uncertainty principle. Now consider a virtual pair of massless particles momentarily at rest at a coordinate distance δr from the horizon. As in Sec. 4.1, the proper distance — and therefore the proper time for the negative energy partner to reach the horizon — will be √ τ ∼ GM δr. Setting this equal to the lifetime
/E of the pair, we find that
|E| ∼ √ , GM δr which should match the energy of the escaping positive-energy partner. This is the energy at 2GM + δr, though. As in the preceding section, the energy at infinity will be redshifted, becoming 
2GM
E∞ ∼ √ 1− ∼ , (22) 2GM + δr GM GM δr independent of the initial position δr. We thus expect a black hole to radiate with a characteristic temperature kT ∼
/GM , matching the Hawking temperature (1). 5. The Many Derivations of Black Hole Thermodynamics Hawking’s derivation of the black hole temperature seems to depend only on simple properties of quantum field theory in curved spacetime. Still, one may worry that these properties have been pushed beyond their realm of validity. In particular, it is evident from Sec. 3.2 that the calculation of the Bogoliubov coefficients requires that we follow outgoing modes backwards in time to a region in which they are very highly blueshifted, with energies far above the Planck scale.28–30 This “trans-Planckian problem” has been addressed in a number of ways: by imposing high-energy cut-offs,29 by restricting fields to discrete lattices,31 by altering dispersion relations to break the connection between high frequencies and high energies,32,33 and by considering analog systems such as Unruh’s “dumb holes,” fluid flows that generate sonic event horizons.28 It turns out to be very hard to “break” Hawking’s results: Standard thermal Hawking radiation, with the usual Hawking temperature, appears even in models that never involve Planck scale physics.34 Perhaps the strongest evidence for the reality of black hole thermodynamics, though, comes from the number of independent derivations, each relying on different assumptions and approximations. In this section, I will briefly summarize some of these results.

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5.1. Other settings Hawking’s original derivation involved a particular physical setting, a star collapsing to form a black hole. This determined the choice of vacuum state, and one could worry that the result might be too narrow. It has been subsequently shown, though, that one can generalize the derivation considerably, for instance by replacing the initial Minkowski vacuum in the distant past with the vacuum for a freely falling observer near the horizon.35 Indeed, there is now fairly strong evidence that only a few fundamental features are needed to obtain thermal radiation: a vacuum near the horizon as seen by a freely falling observer, vacuum fluctuations that start in the ground state, and subsequent adiabatic evolution.33,36 In particular, the Einstein field equations are not required.37 We may also ask which ingredients in the derivation of Sec. 3.2 were essential to the result.22 Consider any setting in which, as in Fig. 1(b), one can trace null geodesics (such as γ in the figure) from part of I + to part of I − , defining a map p(u). One can then define a time-dependent “surface gravity” κ(u) by κ(u) = −

p (u) . p (u)

(23)

Now pick a point u∗ on I + . Barcel´o et al. show22 that an observer near u∗ will observe nearly thermal Hawking radiation, with a time-dependent temperature
κ(u∗ )/2π, provided an adiabatic condition κ (u∗ )  κ(u∗ )2

(24)

holds. In particular, it is not even necessary for an event horizon to form. The exponential “peeling” (16) is sufficient. One can also analyze an eternal black hole at equilibrium with its Hawking radiation,38 obtaining consistent results. Moreover, the computation may be extended beyond the number operator (13) to find an expression for the full final state in terms of the initial vacuum. It is exactly thermal, at least in the approximation that one ignores back-reaction of Hawking radiation on the black hole.39,40 5.2. Unruh radiation By the principle of equivalence, the gravitational field near a black hole horizon should be locally equivalent to uniform acceleration in a flat spacetime. Hawking radiation is not entirely local, so one might not expect an exact equivalence, but one should be able to test Hawking’s results by comparing the vacua of an inertial observer and a uniformly accelerating observer in Minkowski space. This was first done by Unruh.35 The Bogoliubov coefficients can be derived fairly easily, and the conclusion is that the accelerated observer with a constant proper acceleration a sees a thermal flux of particles with a temperature
a , (25) TU = 2π in almost exact analogy with the Hawking temperature (1).

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Coincidentally, at almost the same time as Unruh’s work, Bisognano and Wichmann gave an independent, more difficult but mathematically rigorous proof of the Unruh effect in quantum field theory.41 Their results are quite general. While they require flat spacetime, the quantum field theory need not be a free theory. 5.3. Particle detectors Hawking’s derivation and similar approaches to black hole thermodynamics depend heavily on the standard quantum field theoretical definitions of vacuum states, Fock space and particle number. But since one of the main conclusions is that states in curved spacetime are rather different from those we are used to in Minkowski space — for instance, the vacuum is no longer unique — we might worry that we are overinterpreting the formalism. True physical observables are notoriously difficult to find in quantum gravity, so this is not a trivial concern. To investigate this question, Unruh35 and DeWitt42 developed simple models of particle detectors in black hole backgrounds. They showed that such detectors do, in fact, see thermal radiation at the Hawking temperature. Similarly, Yu and Zhou43 have shown that a two-level atom outside a black hole will spontaneously excite as if it were in a thermal bath at the Hawking temperature. 5.4. Tunneling Hawking radiation is a quantum process, and we may try to apply our intuition about quantum mechanics to seek a deeper understanding. As noted in Sec. 3, an obvious guess is that while escape from a black hole is classically forbidden, quantum particles might tunnel out. For instance, instead of visualizing virtual pairs forming outside the horizon as in Sec. 4.2, we might reverse the picture and imagine pairs forming inside the horizon, with one member then tunneling out. While the idea of a tunneling description can be traced to Damour and Ruffini44 and Srinivasan and Padmanabhan,45 its modern incarnation owes itself to a key insight of Parikh and Wilczek,18 who realized that rather than thinking of a particle as tunneling through the horizon, one could think of the horizon as tunneling past the particle. Consider an initial Schwarzschild black hole of mass M that emits a particle in an s-wave state, represented as a spherical shell of mass ω  M . In the WKB approximation, the tunneling rate is Γ = e−2 Im I/
,

(26)

where I is the action for the outgoing shell, moving in a background metric of a black hole as it crosses the horizon from rin to rout . The exponent, in turn, can be written as  rout  rout  pr  M−ω  rout dr  dH, (27) Im I = Im pr dr = Im dpr dr = Im r˙ M rin rin 0 rin

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where I have used Hamilton’s equations of motion to write dpr = dH/r˙ and noted that the horizon moves inward from GM to G(M − ω) as the particle is emitted. To evaluate this integral, Parikh and Wilczek use the Painlev´e–Gullstrand form of the black hole metric    2G(M − ω) 2G(M − ω) 2 2 ds = 1 − dt − 2 dt dr − dr2 − r2 dΩ2 , (28) r r which avoids coordinate singularities at the horizon. Outgoing radial null geodesics then satisfy  2G(M − ω) r˙ = 1 − , r and the integral (27) can be evaluated by a standard contour deformation, yielding a tunneling rate ω

Γ = e−8πωG(M− 2 )/
= e∆SBH ,

(29)

where ∆SBH is the change in the Bekenstein–Hawking entropy (1) due to the emission of the shell. By the first law of thermodynamics, ∆SBH = −
ω/TH, and we recover the expected emission rate for thermal Hawking radiation. The tunneling approach was initially developed for the simplest black holes, but has by now been vastly extended, to apply to virtually every known black hole solution, including even nonstationary configurations.46 The method has also been generalized. In particular, one can replace the assumption that outgoing particles follow null geodesics by the more general approximation that the action I satisfies a Hamilton–Jacobi equation.47 5.5. Hawking radiation from anomalies From early on, it was clear that black hole thermodynamics should be visible in the behavior of the stress–energy tensor near a horizon.42,48 The computation of the expectation value Tab  is a complicated and technical subject, beyond the scope of this paper (see for instance, the book by Birrell and Davies19 for a pedagogical introduction). The results, however, are consistent with Hawking’s discovery. In particular, the heuristic approach of Sec. 4.2 is supported: An ingoing flux of negative energy near the horizon balances the outgoing flux of positive energy at infinity. Energy is conserved, and as in the tunneling derivation of Sec. 5.4, the back-reaction of Hawking radiation reduces the mass of the black hole. In general, the computation of the expectation value Tab  requires a mixture of analytic approximations and numerical methods. In one special case, though, the situation becomes drastically simpler. Consider a conformally invariant field (for instance, a massless scalar) in two spacetime dimensions. Classically, conformal invariance forces the trace T a a of the stress–energy tensor to vanish, and its

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quantum correction, the “trace anomaly,” is completely determined as c T a a  = R, 24π where c, the central charge, is fully determined by the properties of the field in a flat background.49 Christensen and Fulling50 realized in 1977 that in such a case, the full stress–energy tensor is completely determined from T a a  by conservation laws and boundary conditions. The result exactly matched expectations, with fluxes of positive energy at infinity, describing thermal radiation, balanced by fluxes of negative energy through the horizon. Christensen and Fulling’s original analysis relied crucially on the restriction to two dimensions. More recently, though Wilczek and collaborators51–53 have shown how to generalize the argument to a much wider setting. The central idea is that one can impose an effective description of a vacuum state near the horizon, the equivalent of the freely falling observer’s vacuum, by excluding matter fields that correspond to outgoing (“horizon-skimming”) modes. This restriction makes the field theory chiral, though, and a chiral theory contains a new anomaly, a diffeomorphism anomaly. By analyzing the resulting theory one partial wave at a time, one can again use this anomaly to reconstruct the stress–energy tensor, recovering the characteristic thermal radiation. This method is actually closely related to that of Christensen and Fulling — in two dimensions, one can always trade a conformal anomaly for a diffeomorphism anomaly by adding appropriate counterterms — but the newer approach seems more suitable for treating separate partial waves, and thus for application to more than two dimensions. In a remarkable piece of work, Iso et al.54 and Bonora et al.55,56 have extended this approach to obtain a much more detailed picture of Hawking radiation. They show that if one considers not just the stress–energy tensor, but higher spin currents as well, one can recover the full thermal spectrum. Hawking radiation can thus be obtained, with only fairly minimal assumptions, from the symmetries of the nearhorizon region of a black hole. 5.6. Periodic Greens functions In ordinary quantum field theory, thermal Greens functions have a hidden periodicity. Consider, for instance, the Greens function of a scalar field ϕ in a thermal ensemble at temperature T = 1/β Gβ (x, 0; x , t) = T r(e−βH ϕ(x, 0)ϕ(x , t)) = T r(e−βH ϕ(x, 0)e−βH eβH ϕ(x , t)) = T r(ϕ(x, 0)e−βH eβH ϕ(x , t)e−βH ) = T r(ϕ(x, 0)e−βH ϕ(x , t + iβ)) = Gβ (x , t + iβ; x, 0),

(30)

where I have used cyclicity of the trace and the fact that the Hamiltonian generates time translations, so eβH ϕ(x , t)e−βH = ϕ(x , t+iβ). In particular, a thermal Greens function that is symmetric in its arguments must be periodic in imaginary time,

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with period iβ. Conversely, in axiomatic quantum field theory such periodicity, formalized as the KMS condition,57–59 serves to define a thermal state. For a uniformly accelerating observer in Minkowski space, it was shown in 1976 that the Greens function has just such a periodicity,41 with a temperature equal to the Unruh temperature (25). By the arguments of Sec. 5.2, one should expect an analogous result for a stationary observer near a black hole horizon. This is indeed the case,60,61 and β is just the inverse of the Hawking temperature (1). Moreover, the response of a particle detector of the sort discussed in Sec. 5.3 is determined by a closely related Greens function, and the periodicity ensures quite generally that such a detector will see a thermal bath at temperature TH . 5.7. Gravitational partition function One may ask whether the periodicity properties of the Greens functions may be extended to the gravitational field itself. For ordinary quantum mechanical systems, it is well known that a thermal partition function may be obtained from the usual path integral by analytically continuing to periodic imaginary time with period β.62 The meaning of such a continuation is not so clear in general relativity, though: There is usually no preferred time coordinate to make imaginary, and there is no simple relationship between solutions of the field equations with “real time” (Lorentzian signature) and “imaginary time” (Riemannian signature). In one setting, though, there is a natural choice. If a spacetime is stationary — that is, if it admits a timelike Killing vector — the Killing vector defines a preferred time coordinate. In particular, for a nonextremal stationary black hole, the generic metric near enough to the horizon takes the form ds2 = 2κ(r − r+ )dt2 −

1 2 dr2 − r+ dΩ2 , 2κ(r − r+ )

where r+ is the location of the horizon and κ is again the surface gravity. Setting t = iτ and replacing r by the proper distance to the horizon 1 ρ= 2κ(r − r+ ), κ we obtain the “Euclidean black hole” metric 2 ds2 = dρ2 + κ2 ρ2 dτ 2 + r+ dΩ2 .

(31)

We can immediately recognize the first two terms of (31) as the metric of a flat plane in polar coordinates. The horizon r = r+ has collapsed to a point ρ = 0: The “Euclidean section” includes only the black hole exterior. But the ρ–τ plane is flat only if κτ has a period 2π. Otherwise, the origin is a conical singularity, and the metric will not extremize the action at that point. This periodicity, in turn, requires that τ have a period 2π/κ = 1/TH, exactly as in the preceding section. We now see that the periodicity of Greens functions did not depend on details of quantum fields, but arose directly from the geometry. But we can do more: as

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Gibbons and Hawking argued,64 we can use this result to obtain a saddle point approximation to the gravitational path integral. By analogy with ordinary quantum field theory, the Euclidean path integral for the gravitational partition function can be formally written as  (32) Z[β] = [dg]eIEuc , where IEuc is the “Euclidean” Einstein–Hilbert action — that is, the Einstein– Hilbert action for metrics of Riemannian signature — and the integral is over all metrics that are, in some sense, periodic in “Euclidean” time with period β. This is, of course, an ill-defined expression: General relativity is nonrenormalizable, so even if we knew exactly what “periodic” meant here, we would not know how to make sense of the functional integral. Still, though, there ought to be some sense in which (32) has a meaning in effective field theory,63 and even if the full expression is ill-defined, a saddle point approximation could give a physically reasonable result. Naively, the saddle point contributions (31) gives a vanishing contribution to IEuc : the “bulk” Einstein–Hilbert action   1 d4 x |g|R 16πG is zero for any classical solution of the vacuum field equations. The key observation of Gibbons and Hawking64 was that on a manifold with boundary, the “bulk” action must be supplemented by a boundary term, without which the action will typically have no true extrema.65 The boundary was originally taken to be at infinity, but subsequent work has shown that it may also be placed at the horizon.66–68 The resulting boundary term may be evaluated in a number of ways: For instance, if one dimensionally reduces the action to the ρ–τ plane, the problem becomes purely topological.66 The final result is that IEuc =

Ahor − β(M + ΩJ + ΦQ). 4
G

(33)

The saddle point contribution eIEuc to the partition function (32) may then be recognized as the grand canonical partition function for a system with inverse temperature β = 2π/κ and entropy SBH = Ahor /4
G, just as expected. The same analysis applies to much more general stationary geometries.69 Just as above, Killing horizons in the Lorentzian configurations translate into zeros of the Killing vector in the Riemannian continuation, and boundary terms resembling (33) again appear. Recently Neiman has argued that even in Lorentzian signature, the usual Gibbons–Hawking boundary term acquires an imaginary piece whenever the boundary involves “signature flips,” points at which the signature of the boundary changes from spacelike to timelike.70 For black hole spacetimes, the resulting boundary term looks very much like the one in the Euclidean action. Alternatively, one may try to evade the ambiguities in “imaginary time” by starting with the Hamiltonian formulation of general relativity.67 Perhaps

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unsurprisingly, the same boundary term (33) appears. In canonical quantization, this boundary term gives rise to a new term in the Wheeler–DeWitt equation, from which one can again recover the Bekenstein–Hawking entropy.71 5.8. Pair production of black holes As Heisenberg and Euler noticed in 1936,72 the quantum vacuum is unstable in the presence of a strong electric field. Virtual pairs of charged particles can be “pulled out of the vacuum” by the field, becoming physical particle–antiparticle pairs. The pair production rate per unit volume for a fermion of mass m and charge e in a constant field E was worked out in detail by Schwinger,73 and takes the form α2 E 2 −πm2 /|eE| e . π2 For N identical species of fermions, this expression would be multiplied by N . Now consider pair production of charged black holes in an electric field. Without a full quantum theory of gravity we cannot compute an exact production rate, but as in the preceding section, we can use the Euclidean path integral to obtain a semiclassical approximation. If the Bekenstein entropy is a statistical mechanical entropy, we would expect a black hole to have exp{SBH } microscopic states, with a corresponding enhancement of the pair production rate by N = exp{SBH }. This is exactly what is found. The first computation, by Garfinkle et al.74 considered magnetically charged black holes, and compared the pair production rate to the rate for magnetic monopoles of the same mass. Subsequent work extended these results to a much wider variety of black holes.75–77 In every known case, pair production is enhanced just as one would expect if the Bekenstein–Hawking entropy represented an actual counting of states. W ∼

5.9. Quantum field theory and the eternal black hole Hawking’s derivation of black hole thermodynamics was based on a collapsing star, with a Minkowski-like vacuum in the distant past. But we can also write down solutions of the Einstein field equations that represent eternal black holes. Recall that the maximally extended eternal black hole spacetime has a Carter– Penrose diagram with four regions, as shown in Fig. 4. Now consider a quantum state Ψ defined on a Cauchy surface Σ passing through the black hole bifurcation sphere, as shown in the figure. No information can pass from region II to region I, so even if Ψ is a pure state, an observer in region I will see only a density matrix, obtained by tracing over region II. This at least opens up the possibility that physics for an observer in region I will be thermal. Indeed, for a free quantum field there is at most one quantum state, the HartleHawking vacuum state, that is regular everywhere on the horizon.78,79 For a scalar field, a direct computation shows that the density matrix obtained by tracing over region II is thermal, with a temperature TH .80 For more general fields, the same can

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Fig. 4.

A Carter–Penrose diagram for an eternal black hole.

be shown by means of fairly sophisticated quantum field theoretical arguments78,79 or by an analysis of the path integral.81 Thus, even if matter is secretly in a pure state, the eternal black hole is effectively also a thermal system. 5.10. Quantized gravity and classical matter Although they differ in detail, the derivations of Hawking radiation so far have all analyzed quantum matter in a classical black hole background. This is a reasonable approximation, except perhaps for the “trans-Planckian problem” discussed above. In one special case, though, the direction can be reversed: Hawking radiation can be derived from classical matter in the presence of a quantum black hole. This special case is that of the BTZ black hole,82 a black hole in (2 + 1)dimensional asymptotically anti-de Sitter space. This is a rather peculiar solution: The vacuum Einstein field equations in 2 + 1 dimensions imply that spacetime has constant negative curvature, and, indeed, the BTZ black hole can be described as a region of anti-de Sitter space with certain identifications.83 Nevertheless, it is a genuine black hole, the final state of collapsing matter, with a normal event horizon and (in the rotating case) a normal inner Cauchy horizon.84 The BTZ black hole presents a puzzle for any statistical mechanical explanation of black hole entropy. As a corollary to the constant curvature required by the field equations, (2+1)-dimensional gravity contains no propagating degrees of freedom that could serve as candidates for microscopic states. As first suggested by Carlip85 and confirmed in detail by Strominger87 and, independently, Birmingham et al.,88 the natural candidate for the microscopic degrees of freedom are “edge states,” additional degrees of freedom at infinity that appear because the boundary conditions break diffeomorphism invariance. I will discuss these states further in Sec. 9. For now, the key feature is that they allow an “effective” description of the quantum BTZ black hole in terms of a particular two-dimensional conformal field theory at infinity.89 Emparan and Sachs have shown how to couple this quantum theory to classical matter,90 and to use

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the coupling to calculate transition amplitudes between quantum black hole states induced by interactions with this “background” matter. Detailed balance arguments then allow them to determine the emission rate of the matter fields, obtaining the correct Hawking radiation spectrum, even including the correct greybody factors. 5.11. Other approaches I have tried to give an overview of the most common approaches to black hole thermodynamics, but I have necessarily omitted a few. For example • Instead of considering only the region outside the horizon, one may choose a time-slicing that crosses the horizon and continues to the singularity. A black hole interior is not static — the Killing vector that gives time translation invariance in the exterior becomes spacelike in the interior — and the Hamiltonian in horizoncrossing coordinates is therefore time-dependent. Nevertheless, one can perform a Heisenberg picture quantization to obtain time-dependent states which, although pure, behave “thermally” in the sense that they excite radiation detectors and have a net flux of radiation at infinity that matches Hawking’s results.91 • One may repeat the computation of Unruh radiation in the functional Schr¨ odinger picture, in which the quantum state is a wave functional of the field configuration. The results are again what would be expected92 : The ground states for the stationary and accelerated observers are different, and the difference appears as a thermal distribution of “Minkowski” particles in the accelerated observer’s state. • York has proposed a model in which fluctuations of the event horizon lead to a “quantum ergosphere” through which particles can tunnel.93 The picture is too complicated to compute an exact Hawking temperature, but by restricting the horizon fluctuations to those corresponding to the lowest quasinormal mode, he obtains a value for the temperature that is accurate to to within about 2%. The derivations in this section are not entirely independent. For instance, the response function of a particle detector (Sec. 5.3) depends on a Greens function, and it is the periodicity of this function (Sec. 5.6) that determines the thermal response. This periodicity, in turn, may be traced the periodicity of the black hole metric in imaginary time (Sec. 5.7). The tunneling approach (Sec. 5.4) and the original analysis by Hawking both lead to relations of the form P (emis) = e−βE P (absor) between emission and absorption probabilities, and both results come from the behavior of complex functions near the horizon. Still, the derivations are sufficiently independent, with different enough assumptions and approximations, that together they provide very powerful evidence for the reality of Hawking radiation.

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6. Thermodynamic Properties of Black Holes There is more to thermodynamics than temperature and entropy. If black holes are thermal objects, we should be able to apply the rest of thermodynamic theory, from Carnot cycles to phase changes, to systems containing black holes. In this section, after a short discussion of black hole evaporation, I will review four selected aspects: heat capacity, phase transitions, thermodynamic volume, and an argument that black hole thermodynamics prohibits certain violations of Lorentz invariance. 6.1. Black hole evaporation We have seen that a black hole radiates as a grey body, at a temperature TH . By the Stefan–Boltzmann law, it will therefore lose mass at a rate dM (34) = −εσTH4 Ahor , dt where ε is some averaged emissivity, summed over the species of particles that can be radiated. For the Schwarzschild black hole, TH ∼ 1/M and Ahor ∼ M 2 , so the lifetime goes as M 3 . To determine the exact coefficients one must treat the greybody factors carefully.23 The result is  3 M τ ∼ 1071 , M where M is the mass of the Sun. For charged or rotating black holes, evaporation is more complicated94 : Black holes can “discharge” via Schwinger pair production, but the Hawking temperature decreases with increasing charge and angular momentum. 6.2. Heat capacity A Schwarzschild black hole has a temperature T =
/8πM . As an isolated black hole evaporates, it radiates energy, its mass decreases, and its temperature consequently increases. That is, a Schwarzschild black hole has a negative heat capacity C=T

1 ∂S =− . ∂T 8πT 2

(35)

For a rotating black hole, the situation is more complicated: As Davies has shown,95 the heat capacity at fixed angular velocity is always negative, but the heat capacity at fixed angular momentum is negative for low J, diverges as a critical value, and then becomes positive, a behavior indicative of a phase transition. While negative heat capacity is unusual, it is not a peculiar feature of black holes, but holds even for Newtonian self-gravitating systems. As a satellite in low Earth orbit loses energy to atmospheric friction, its orbit drops to a lower altitude, increasing its kinetic energy. For gravitationally bound stellar systems, LyndenBell has analyzed this behavior, which he calls it the “gravothermal catastrophe,” in detail.96 In Newtonian gravity, the runaway behavior is a consequence of the purely

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attractive, long range nature of the gravitational interaction, and it stops when gravity is no longer the principle interaction governing the system. For a relativistic black hole, on the other hand, there seems to be no competing interaction, at least until the evaporating black hole has shrunk to the Planck scale. There are two standard (theoretical) ways to stabilize a black hole. The first is to place it in a reflecting cavity, allowing it to come into equilibrium with its 2 , the heat capacity for Hawking radiation. For a spherical cavity of area A = 4πrB 97 a Schwarzschild black hole becomes   −1 3M 2M 2 CA = 8πM 1 − −1 , (36) rB rB which is positive for a small enough cavity, rB < 3M . The second is to consider a black hole in asymptotically anti-de Sitter space,98 in which the geometry serves as a sort of “cavity.” For a given temperature, one finds two possible anti-de Sitter black hole configurations, a “small” black hole with negative heat capacity and a “large” black hole with positive heat capacity. 6.3. Phase transitions As already indicated, black holes can have a complicated phase structure, including phase transitions of various types. The most famous of these is the Hawking–Page transition for a black hole in asymptotically anti-de Sitter space,98 a first order phase transition between thermal radiation at low temperatures and a black hole at higher temperatures. In the AdS/CFT correspondence of string theory, this transition has an interesting “dual” description as a confinement–deconfinement transition.99 We have come to understand that this kind of behavior is not at all exceptional. As one example among many,100 consider a black hole with charge Q in asymptotically flat space, enclosed in a cavity whose walls are maintained at a fixed temperature 1/β. One finds a phase diagram in the Q–β plane containing a line of first-order phase transitions, terminating at a critical point which is the location of a second-order phase transition with calculable critical exponents. 6.4. Thermodynamic volume The first law of black hole mechanics (2) closely resembles the standard form of the first law of thermodynamics, but it is missing a term, the pressure term −pdV . There is, in fact, a candidate for “pressure”: The cosmological constant Λ has the correct dimension, and acts as a pressure in cosmological settings. But Λ is not normally considered a state function. It is a coupling constant that is fixed by the theory, not by the state of the thermodynamic system. There are, however, situations in which Λ is a dynamical field, forced to be constant only by virtue of its field equations.101 In that case, it may make sense to

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treat it as a state function,102 and to generalize the first law to the form103,104 δM = T δS + ΩδJ + ΦδQ + Vtherm δp,

(37)

where p = −Λ/8πG. Note that in this formulation, the mass M should be interpreted as enthalpy (that is, U + pV , where U is the internal energy).104 The “thermodynamic volume” Vtherm is defined by (37), and exists even in the limit Λ → 0. Its physical meaning is not very well understood, though. For a Schwarzschild black hole, it is easy to check that Vtherm =

4 πr+ 3 , 3

(38)

where r+ is the location of the horizon in Schwarzschild coordinates. This relation generalizes to higher dimensional static black holes, including those with charge: The thermodynamic volume is the volume of a flat round ball whose surface area is equal to the area of the event horizon. But the generalization to rotating black holes is complicated, and the corresponding thermodynamic volume does not yet have a simple physical interpretation.105 The inclusion of a volume term in the first law (37) enriches the phase structure of the theory, with a number of interesting consequences.106,107 For example, the equation of state for a charged, rotating anti-de Sitter black hole becomes very close to that of a van der Waals gas. Rotating anti-de Sitter black holes in more than six dimensions exhibit reentrant phase transitions; and anti-de Sitter black holes in six dimensions with multiple spins have a triple point. 6.5. Lorentz violation and perpetual motion machines Lorentz invariance guarantees that there is a single “maximum speed” c, the same for all particles. I have implicitly used this assumption throughout this review, choosing units c = 1. Suppose, though, that Lorentz invariance is broken, in such a way that two species of particles — say A and B — have different maximum speeds cB > cA . Then Dubovsky and Sibiryakov have shown108 that a black hole can be used to build a “perpetual motion machine of the second kind,” a device that transfers heat from a cold reservoir to a hot reservoir without any net use of energy. In some Lorentz-violating theories, there is even a classical mechanism, analogous to the Penrose process, for “mining” a black hole.109 To understand the argument, first note that the event horizon for the more slowly moving species A will lie outside the event horizon for the faster species B. This is physically intuitive, but also follows directly if we restore the factors of c in the definition of the Schwarzschild event horizon, r+ = 2GM/c2 . Correspondingly, the Hawking temperature TA,H of species A should be lower than the Hawking temperature TB,H of species B. Dubovsky and Sibiryakov’s “construction manual” is then the following. As heat reservoirs, construct a shell A at temperature TA,shell that interacts only with

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species A, and a second shell B at temperature TB,shell that interacts only with species B. Arrange the temperatures so that TB,H > TB,shell > TA,shell > TA,H .

(39)

This will result in a net energy flux from shell A into the black hole, and from the black hole out to shell B. It is not hard to check that one can adjust the shell temperatures so that the flux into the black hole balances the flux out. The net result, then, is a flux of energy from the “cold” shell A to the “hot” shell B, violating the second law of thermodynamics. Arguments like this should be treated cautiously. Historically, the generalized second law of thermodynamics has shown remarkably resilience, often involving subtle and unexpected effects. There may be natural Lorentz-violating theories in which all matter shares a single universal horizon, for instance, or theories in which the black hole interior carries hidden entropy. But the apparent paradox shows, at least, that simple thermodynamic arguments about black holes may place surprising restrictions on nongravitational physics. 7. Approaches to Black Hole Statistical Mechanics In ordinary thermodynamic systems, thermodynamic properties reflect the statistical mechanics of underlying microscopic states. Entropy, in particular, is a measure of the number of accessible states. One might hope for the same to be true for black holes. Then, since the Bekenstein–Hawking entropy depends on both Newton’s constant G and Planck’s constant
, it is plausible that black hole thermodynamics is telling us something about quantum gravitational states. Black hole thermodynamics may therefore present us with a rare opportunity. There is very little that we actually know about quantum gravity, so any insight is likely to be valuable. Like many opportunities, though, this one is accompanied by a serious difficulty: In the absence of a quantum theory of gravity, how do we start to understand the quantum states of a black hole? Fortunately, while we do not yet have a complete quantum theory of gravity, we have a number of research programs, at various levels of maturity, attempting to develop such a theory. Some of these are advanced enough to be able to make predictions about at least certain classes of black holes. Indeed, as I shall discuss in Sec. 9, the real problem may be not that we have no microscopic explanation of black hole entropy, but that we have too many. 7.1. “Phenomenology” From the earliest days of black hole thermodynamics, there was an obvious simple “phenomenological” model for Bekenstein’s area law2 : One had to merely assume that black hole horizon area was quantized, with discrete “plaquettes” of area on the order of the Planck area. The simplest choice — partially justified by the observation

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that the horizon area is an adiabatic invariant110 — would a discrete, evenly spaced area spectrum. Suppose, for instance, that the Bekenstein–Hawking entropy were exactly the logarithm of the number of states. This number is, of course, an integer, so successive values of the area would have to differ by an amount ∆A = 4
G ln k,

(40)

for some integer k. As Hod first observed,111 one could then appeal to the Bohr correspondence principle to determine the area spacing. While black holes have no stable excitations, they do have damped “ringing modes,” called quasinormal modes, with complex frequencies.112 The most strongly damped quasinormal modes of the Schwarzschild black hole have frequencies Re ω ∼

ln 3 . 8πGM

(41)

The Bohr correspondence principle would then suggest that these frequencies should correspond to transitions between adjacent area eigenstates, ∆M =
ω ⇒

∆A
ln 3 = , 2 32πG M 8πGM

(42)

which exactly matches (40) with k = 3. Arguments of this sort are suggestive, but must also be treated with care. For instance, within the same framework, Maggiore113 has argued — based in part on an analogy with a damped harmonic oscillator — that relevant frequency one should use for the Bohr correspondence principle is not the real part Re ω of the quasinormal frequency, but the modulus |ω|. For highly damped modes   1 1 n+ |ω| ∼ , (43) 4GM 2 and the Bohr correspondence principle then gives ∆A = 8π
G.

(44)

A wide range of simple models have been explored — Medved’s review114 gives a partial list — and they do not agree about the area spectrum. The lesson seems to be, not surprisingly, that thought experiments and analogies can only go so far. One needs a genuine quantum theory of gravity to obtain a real answer. In the remainder of this section, I will briefly summarize some of the more specific approaches to black hole statistical mechanics from the point of view of particular models of quantum gravity. 7.2. Entanglement entropy Consider a space that is divided into two disjoint pieces A and B. Suppose a quantum system on that space is characterized by a pure state |Ψ, with a corresponding

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density matrix ρAB = |ΨΨ|. The von Neumann entropy of such a pure state SvN = Tr ρ ln ρ

(45)

is, of course, zero. Now, however, imagine an observer who is restricted to the region A, and cannot observe anything in B. All observations of such an observer can be described by a reduced density matrix, ρA = T rB ρAB , whose entropy will in general be nonzero. For a local theory with a finite-dimensional Hilbert space, for instance, we can write the state |Ψ in a Schmidt decomposition
(46) |Ψ = ci |φi |ψi , where {|φi } and {|ψi } are orthonormal bases of states in regions A and B. Then
ρA = |ci |2 |φi φi |, (47) and the von Neumann entropy of region A is simply
|ci |2 ln |ci |2 , SA = Tr ρA ln ρA =

(48)

the Shannon entropy for the probabilities pi = |ci |2 . This “entanglement entropy” is now commonplace in quantum theory, but it was first introduced by Sorkin in the context of black hole thermodynamics.115 The idea is that an observer outside the horizon (“region A”) knows nothing about the state inside the horizon (“region B”), and therefore sees a mixed state with nonvanishing entropy. This picture arises naturally from the eternal black hole of Sec. 5.9: The density matrix that gives the entropy (48) is the same as the thermal density matrix that gives the Hawking spectrum. For many states, including the vacuum state, the entanglement entropy for a black hole is proportional to the horizon area,115–117 since the main contribution comes from correlations among degrees of freedom very close to the horizon, for which the “bulk” state is irrelevant. The coefficient of this area term, on the other hand, diverges, for essentially the same reason: The more closely degrees of freedom are localized to the horizon, the higher their energies. The simplest cut-off, a “brick wall” approximately one Planck length in proper distance from the horizon,118 gives a coefficient for the Bekenstein–Hawking entropy of the right order of magnitude. In general, though, the exact value depends sensitively on the number and type of entangled fields — that is, on the particle content of the Universe. A possible solution to this “species problem” comes from the observation by Susskind and Uglum that the same modes responsible for the entanglement entropy also renormalize Newton’s constant, which also appears in the Bekenstein–Hawking entropy formula.119 It is plausible that these contributions cancel. In particular, Cooperman and Luty120 have recently argued that for a certain class of states defined by a path integral, the leading renormalized term in the entanglement entropy is given by the corresponding renormalized Bekenstein–Hawking formula, independent of the particle content of the theory.

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Inspired by the AdS/CFT correspondence of string theory, Ryu and Takayanagi have proposed an alternative “holographic” approach to defining and regulating entanglement entropy.121,122 In the description I have given above, the regions A and B lie on a (d − 1)-dimensional spacelike slice of a d-dimensional spacetime. Ryu and Takayanagi propose embedding this spacetime as the asymptotic boundary of (d+1)-dimensional anti-de Sitter space. The boundary between regions A and B can then be extended into the “bulk” anti-de Sitter space as a d-dimensional minimal surface; Ryu and Takayanagi propose that the entanglement entropy should be proportional to the area of this surface. While this approach is obviously inspired by black hole thermodynamics, can be proven without any assumptions about black holes, at least for the static case.123 The holographic entanglement entropy can then be used to determine the entropy of a black hole, and once again obtains the Bekenstein–Hawking expression, with no divergences or species problem.124 7.3. String theory The leading research program in quantum gravity today is string theory. I will not attempt to summarize the theory here, but will instead focus on three rather different approaches to black hole statistical mechanics within string theory. 7.3.1. Weakly coupled strings and branes The first successful calculation of black hole entropy in string theory, by Strominger and Vafa,125 looked at a class of multiply charged, extremal (that is, maximally charged), supersymmetric (BPS) black holes in five dimensions. As classical objects, such black holes are completely determined by their charges: The mass, horizon area, and all other macroscopic quantities can be expressed as functions of the charges. As string theoretical objects, on the other hand, they are strongly bound states of the fundamental constituents of string theory, strings and D-branes. Strominger and Vafa suggested “turning down” the strength of the gravitational interaction, until a black hole became a weakly coupled system of strings and D-branes. These constituents would still have the prescribed charges, but at weak coupling one could also count the number of states with such charges. The result, translated back into the black hole parameters, exactly matched the Bekenstein–Hawking entropy. These results were quickly extended to a large number of extremal and nearextremal black holes, and, through dualities, to some particular nonextremal black holes as well.126,127 A straightforward extension of the argument gives Hawking radiation, coming from decay of excited states of the constituent strings and D-branes, complete with the correct greybody factors.128 One might worry about the consistency of the procedure: The process of turning down the gravitational coupling need not preserve the number of states. For supersymmetric black holes, though, the number of states is protected by nonrenormalization theorems. For nearly supersymmetric black holes, the procedure continues to work well, perhaps

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even better than one might expect. Far from extremality, it becomes harder to obtain the right factor of 1/4 in the entropy. There is one peculiar feature of these calculations, which I will return to in Sec. 9. While the final result is an expression for entropy in terms of horizon area, the connection is indirect. On the strong coupling side, one may start with the horizon area, but one must reexpress it in terms of charges. On the weak coupling side, one may start with the number of states, but one must again reexpress it in terms of charges. It is only when the two processes are compared that one obtains the Bekenstein–Hawking entropy. This means that each new type of black hole requires a new calculation. Some crucial aspect of universality is missing.

7.3.2. Fuzzballs Mathur has proposed running the analysis of the preceding section backwards.129 Suppose one starts on the weak coupling side, with a particular collection of strings and D-branes whose charges match those of a desired black hole. One may then turn the gravitational coupling up, and try to see what geometry appears at strong coupling. The result is typically not a black hole, but rather a “fuzzball,” a configuration with no horizon and no singularity, but with a geometry that looks very much like that of a black hole outside a would-be horizon.130,131 This phenomenon seems to require the extra dimensions available in string theory, but it provides an intriguing new picture of black holes. For example, Hawking radiation can appear as a result of a classical instability in a fuzzball geometry.132 In a few special cases, one can count the number of such classical “fuzzball” geometries and reproduce the Bekenstein–Hawking entropy. In general, though, it is likely that many of the states relevant for determining the Bekenstein–Hawking entropy will not have classical geometric descriptions.

7.3.3. The AdS/CFT correspondence A third approach to black holes in string theory exploits Maldacena’s celebrated AdS/CFT correspondence.133,134 This extremely well-supported conjecture states that string theory in a d-dimensional asymptotically anti-de Sitter spacetime is dual to a conformal field theory in a flat (d − 1)-dimensional space that can, in a sense, be viewed as the boundary of the AdS spacetime. For asymptotically anti-de Sitter black holes, this means that one can, in principle, compute the entropy by counting states in a (nongravitational) dual conformal field theory. The most straightforward application of this duality occurs for the (2 + 1)dimensional BTZ black hole discussed in Sec. 5.10. Here, the dual description is given by a two-dimensional conformal field theory. As I shall describe in more detail in Sec. 9, two-dimensional conformal field theories have an exceptionally large symmetry group that controls many of their properties. In particular, the density of states is determined by a single parameter c, the central charge, which characterizes

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the conformal anomaly. As we saw in Sec. 5.5, anomalies can provide very useful tools; here, they are powerful enough to determine the density of states. For asymptotically anti-de Sitter gravity in 2 + 1 dimensions, the central charge is dominated by a classical contribution, which was discovered some time ago by Brown and Henneaux.86 Strominger87 and Birmingham et al.88 independently realized that this result could be used to compute the BTZ entropy, precisely reproducing the Bekenstein–Hawking expression. As noted in Sec. 5.10, the same methods can be used to compute Hawking radiation for the BTZ black hole. While this result appears to be rather specialized, it has an important extension. Many higher dimensional near-extremal black holes, including black holes that are not themselves asymptotically anti-de Sitter, have a near-horizon geometry of the form BTZ × trivial , where the “trivial” part merely renormalizes constants in the calculation of entropy. As a result, the BTZ results can be used to obtain the entropy of a large class of string theoretical black holes, including most of the black holes whose states could be counted in the weak coupling approach of Sec. 7.3.1.135 7.4. Loop quantum gravity The second major research program in quantum gravity is loop quantum gravity, or “quantum geometry.” I will again focus only on those elements relevant to black hole thermodynamics. The following are the key features relevant to this question: (1) A basis for the kinematical states of the theory is given by spin networks, graphs with edges labeled by spins and vertices labeled by SU(2) intertwiners. (2) Given any surface Σ, each spin network is an eigenstate of the area operator AˆΣ , with eigenvalue
 AΣ = 8πγG j(j + 1), (49) j

where the sum is over the spins j of edges of the spin network that cross Σ. Note that this area is quantized, but in a rather complicated way, with a spectrum that can be shown to have an elaborate substructure.137,138 (3) Operators such as the area and volume depend on a parameter γ, the Barbero– Immirzi parameter. The significance of this parameter is quite poorly understood. Theories with different values of γ are probably inequivalent, but it has been suggested that γ may not appear in properly renormalized observables139 or in a somewhat different approach to quantization.140 (4) The physical states are obtained from the spin networks by imposing the Hamiltonian constraint, a procedure that is not yet completely under control. 7.4.1. Microcanonical approach Given this structure, a natural first step is to choose the surface Σ to be a black hole horizon and count the number of spin network states that give a prescribed

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area.141,142 This is a bit tricky, since the true event horizon is defined by the global properties of the spacetime. On the other hand, we know that Hawking radiation depends only on local properties, so a local characterization should be adequate. The relevant definition is probably that of an “isolated horizon,” a null surface with vanishing expansion that obeys the laws of black hole mechanics (see the Appendix for details).9,10,136 As we shall see in Sec. 9, the specification of such a horizon is a sort of boundary condition, which requires the introduction of boundary terms in the Einstein–Hilbert action. For loop quantum gravity, these boundary terms induce a three-dimensional Chern–Simons action on the horizon. The careful version of the state-counting143,144 can then be translated into a counting of Chern–Simons states, a well-understood procedure,145 although with slightly subtle combinatorics. The ultimate result is a microcanonical black hole entropy146,147 S=

γM Ahor , γ 4
G

(50)

where γ is the Barbero-Immirzi parameter and γM ≈ 0.23753

(51)

is a constant determined as the solution of a particular combinatoric problem. If one chooses γ = γM , one recovers the standard Bekenstein–Hawking entropy. The significance of this peculiar value γM is not understood, and it may reflect an inadequacy in the quantization procedure or the definition of the area operator.140 Note, though, that the Barbero–Immirzi parameter appears only in the combination Gγ, where G is the “bare” Newton’s constant, i.e. the parameter appearing in the action. The relevant constant in the Bekenstein–Hawking entropy, however, is the renormalized value,139 and since it is not known how Newton’s constant is renormalized in loop quantum gravity, the “physical” value of γ remains uncertain. Ideally, one could address this question by computing the attraction between two test masses in the Newtonian limit, but this seemingly straightforward problem is not yet solved — loop quantum gravity is defined at the Planck scale, and the extrapolation to physically realistic distances remains extremely difficult. In any case, though, once γ is fixed for one type of black hole — the Schwarzschild solution, for instance — the loop quantum gravity computations give the correct entropy for a wide variety of others, including charged black holes, rotating black holes, black holes with dilaton couplings, black holes with higher genus horizons, and black holes with arbitrarily distorted horizons.148,149 In particular, there is no need to restrict oneself to near-extremal black holes. As an interesting variation within this framework,150 one may again look at the horizon area (49), but instead of counting states of the boundary Chern–Simons theory one may count the number of ways the boundary spins can be joined to a single interior vertex. This means, in essence, that one is completely coursegraining the interior state of the black hole, much as one does in an entanglement

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entropy computation. There are, in fact, interesting relationships to entanglement entropy. The end result is again an entropy proportional to the horizon area, but now requiring a different value of the Barbero–Immirzi parameter to match the Bekenstein–Hawking result. 7.4.2. Other ensembles The microcanonical approach described above involves a hidden assumption that all spin networks that give the same horizon area occur with equal probability. This is a standard assumption in ordinary thermodynamics, but gravitating systems are not quite ordinary. Moreover, the microcanonical approach uses only properties of the “kinematical Hilbert space,” that is, the space of states before the Hamiltonian constraint has been imposed. But we know in other contexts that although Hawking radiation is basically kinematical, the Bekenstein–Hawking entropy depends on the dynamics,37 including the Hamiltonian constraint. In the past several years, interesting progress has been made in moving away from these assumptions. In particular (i) The quantum Hamiltonian that generates translations along the horizon has been identified,151 and yields a local temperature and energy, as measured by an idealized particle detector, that agree with semiclassical expectations.152 This is not yet an enumeration of microscopic states, but it is a version of the thermodynamic computation of Sec. 5.3 in a fully quantum gravitational setting. (ii) One can define a sort of grand canonical ensemble, in which the number of punctures — that is, of edges of the spin network that cross the horizon — appears with a corresponding chemical potential.153 The Barbero–Immirzi parameter still appears, but in a different form: the entropy becomes S=

Ahor + N σ(γ), 4
G

(52)

where σ(γ) is a Lagrange multiplier that vanishes when γ takes the value (51). It has been argued that if one considers additional couplings to matter, and makes the “holographic” assumption that the matter entropy near the horizon scales with the area, then the Lagrange multiplier term vanishes in the classical limit, giving back the standard Bekenstein–Hawking entropy.154 (iii) The Barbero–Immirzi parameter can in principle take any value, but there is one “natural” value, γ = i. At this value, the Ashtekar–Sen connection of loop quantum gravity is self-dual, and the Hamiltonian constraint becomes much simpler. Loop quantum gravity is unfortunately much harder to work with when γ = i, because of the need for added constraints to ensure that the metric is real, but several recent computations155–157 indicate that this choice also gives the correct Bekenstein–Hawking entropy, perhaps even more simply.

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7.5. Induced gravity Induced gravity, as first proposed by Sakharov,158 is a model in which the Einstein– Hilbert action is not fundamental, but appears a consequence of the presence of other fields. If one starts with a theory of quantum fields propagating in an initially nondynamical curved spacetime, counterterms from renormalization will automatically induce a gravitational action, which almost always includes an Einstein– Hilbert term at lowest order.159 In Sakharov’s language, the resulting spacetime dynamics is a kind of “metric elasticity” induced by quantum fluctuations of matter. One can write down an explicit set of “heavy” fields that can be integrated out in the path integral to induce the Einstein–Hilbert action. The gravitational counterterms normally have divergent coefficients that must be renormalized, but by including an appropriately chosen collection of nonminimally coupled scalar fields, one can obtain finite values for Newton’s constant and the cosmological constant.160 Given such a model, one can then count quantum states of the “heavy” fields in a black hole background.160,161 Some subtleties occur in the definition of entropy in the presence of nonminimally coupled fields, but in the end the computation reproduces the standard Bekenstein–Hawking expression, with the correct coefficient. Furthermore, if one dimensionally reduces to a two-dimensional conformally invariant system near the horizon, one can count states by standard methods of conformal field theory,162 as described below in Sec. 9. This offers a new, and apparently completely different, picture of black hole microstates as those of the ordinary quantum fields responsible for inducing the gravitational action. 7.6. Other approaches Just as in black hole thermodynamics, there are a number of other approaches to black hole statistical mechanics that also show promise. For example • Classical black holes do not have excited states, but they have characteristic damped oscillations, called quasinormal modes.112 York has argued that these modes should be quantized,93 and as noted in Sec. 5.11, obtains a value close to the Bekenstein–Hawking entropy with an approximate quantization. • The “causal set” program is an attempt to quantize gravity that starts by replacing a continuous spacetime by a discrete set of points with specified causal relations.163 While it is not yet known how to perform an exact calculation of black hole entropy, there are indications that one can obtain a reasonable value by counting the number of points in the future domain of dependence of a crosssection of the horizon164 or the number of causal links crossing the horizon.165 • Zurek and Thorne have shown that there is sense in which the entropy of a black hole counts the number of distinct ways in which a black hole with prescribed macroscopic properties such as mass and angular momentum can be formed from the collapse of quantum matter.166

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7.7. Logarithmic corrections While many approaches to quantum gravity give the correct Bekenstein–Hawking entropy, they may differ on the next order corrections. In general, one expects   Ahor Ahor SBH = + α ln + ···, (53) 4
G 4
G where the coefficient α may depend on the quantum theory. There are some arguments for a “universal” answer that, as in Sec. 9, depends on conformal field theory,167 but these are not conclusive, and particular models seem to give varying results.168–171 The problem is further complicated by the fact that the logarithmic terms in (53) may differ depending on one’s choice of thermodynamic ensemble.172 8. The Holographic Conjecture In ordinary thermodynamic systems, entropy is an extensive quantity, scaling with volume. For black holes, in contrast, the entropy scales with area. It is not so surprising that entropy is not quite extensive for a gravitating system: Gravity cannot be shielded, so when one increases the size of a system one is necessarily changing the internal interactions. Nor is it surprising that entropy cannot be simply defined in terms of a density that can be integrated over a volume: There is a general result that, as a consequence of diffeomorphism invariance, observables in general relativity cannot be defined in terms of local densities.173 Still, an entropy proportional to area is a rather dramatic departure. ’t Hooft174 and Susskind175 have proposed that this feature is not unique to black holes, but is a general property of any gravitating system. They suggest that the entropy inside any region, whether or not it is a black hole, should scale as the area rather than the volume. This conjectured “holographic principle” would imply that our usual view of local physics drastically overcounts degrees of freedom. As a first argument for the plausibility of this holographic viewpoint,176 consider an approximately spherical surface S with a surface area AS = 4πR2 . Let us try to fill the interior of this surface with quantum excitations. To be contained in the region, each excitation should have a wavelength no larger than 2R, and thus an energy E 
/2R. N such excitations will have an energy E 
N/2R, and to avoid forming a black hole, we must have R  2GE. Hence R2 . (54)
G If we interpret N as the number of degrees of freedom in the system, we see that (54) implies a holographic bound. As a second argument, suppose the surface S now contains almost enough matter to form a black hole. Now surround the region by a collapsing shell of matter, with enough additional mass to form a black hole. The initial state has an entropy N

Sinit = Sinterior + Sshell ,

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while the final state has an entropy Sfinal =

AS . 4
G

By the generalized second law of thermodynamics, Sfinal ≥ Sinit . Hence, the matter initially inside S must have had no more entropy than that of a black hole. Neither of these arguments is conclusive. The first implicitly assumed that the region inside S was flat. One may be able to fit more degrees of freedom into a curved space. There are, in fact, classical configurations — so-called “monsters” — in which the entropy within a region is greater than that of the corresponding black hole,177 although all known examples form from initial singularities (they cannot be “made in the laboratory”) and collapse quickly to form black holes. The second argument fails for a more subtle reason: the area of S is not a gauge invariant quantity.178,179 We are really imagining a timelike “world tube” traced out by S and asking its cross-sectional area at a fixed time. But this area depends on the choice of time-slicing, and by choosing a slice that “wiggles” enough in a timelike direction — and is therefore nearly null over much of its intersection with the world tube — we can make the area as small as we like. One might worry that the same problem occurs for black holes, and that the “horizon area” might also not be well-defined. In fact, black holes evade the problem, essentially because the horizon is not timelike, but null (and, for stationary black holes, nonexpanding). Bousso180 has proposed a covariant entropy bound that exploits a similar idea. He starts with a closed surface S, no longer necessarily spherical, and extends it not as a timelike world tube, but as a “collapsing” light sheet, a null hypersurface of decreasing area extending to the past and future of S, as illustrated in Fig. 5. Bousso’s conjecture is that the total entropy flux through either light sheet — ∆+ or ∆− in the figure — is bounded by AS /4
G. For the special case of a black hole, ∆− may be interpreted as the event horizon, and the Bousso entropy bound is basically the generalized second law of thermodynamics. Classically, Bousso’s conjecture can be proven correct, given suitable energy conditions and bounds on the entropy flux and its gradient.178,181 Quantum mechanically, the situation is less clear, though there is strong evidence for the case of free fields with negligible gravitational backreaction.182 Although the holographic principle in its larger sense remains a conjecture, there is one case in which it has been dramatically successful. The AdS/CFT correspondence of string theory133,134 describes the “bulk” physics of an asymptotically anti-de Sitter space in terms of a lower-dimensional “surface” conformal field theory. If this holographic viewpoint can be extended beyond the anti-de Sitter case, black hole thermodynamics will have had a fundamental impact on a far wider field, profoundly altering our approach to fundamental physics.

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Fig. 5. Future-directed null geodesics orthogonal to S form two light sheets, usually one expanding outward (not shown) and one collapsing inward (∆+ ). The same holds for past-directed null geodesics. In this figure, symmetry causes the light sheets to collapse to single points. In general, they will end at caustics where neighboring geodesics cross and the expansion becomes positive.

9. The Problem of Universality For a physicist working on quantum gravity, black hole thermodynamics is, at first sight, a huge source of hope. The Hawking temperature and Bekenstein–Hawking entropy are “quantum gravitational,” depending on both Planck’s and Newton’s constants. Moreover, as Wheeler’s famous aphorism states, “a black hole has no hair” — a classical black hole has no distinguishing characteristics beyond its mass, charges, and spins. Hence, if the entropy of a black hole is an ordinary statistical mechanical entropy, it seems that the only states it can be counting are quantum gravitational states. We may thus have a window into quantum gravity. Upon closer examination, though, the situation is not so simple. As we saw in Sec. 7, we suffer an embarrassment of riches: There seem to be many different quantum descriptions of black hole statistical mechanics, with very different quantum states, all giving the same entropy. Moreover, while the simple Bekenstein area law seems natural in some of these approaches, in others it seems almost miraculous. The “problem of universality” is really two problems, which are logically distinct, although their solutions may be related. The first is that black hole entropy has such a simple, universal value, one-fourth of the horizon area in Planck units, regardless of the mass, spin, charges, or the number of dimensions. Even the horizon topology is irrelevant: The same area law holds for black holes, black strings, black rings, and black branes. The second is that so many different approaches to quantum gravity give the same result, regardless of any of the details of the black hole states.

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It is tempting to address the first problem with a claim that the horizon area must be quantized, as in Sec. 7.1. But such an answer is at best incomplete: It does not explain the universal coefficient of 1/4, nor does it tell us why no other features matter. There is an interesting explanation of the factor of 1/4 in terms of the topology of the Euclidean black hole,66 but it is not clear why this feature would be reflected in the counting of states. A cynical answer to the second problem is that only quantum theories that give the “right” answer are published. But this, too, is at best incomplete: It does not explain why any such models work at all. Consider, for example, the weakly coupled string approach of Sec. 7.3.1. As we saw there, the theory gives the correct entropy for a large class of near-extremal black holes. But each new type of black hole requires a separate computation, and the relationship between entropy and area only appears at the very last step. Such miracles cry out for explanation. In the remainder of this section, I will describe one attempt to explain these miracles. This is still very much a case of work in progress, but there have been some hopeful signs. 9.1. State-counting in conformal field theory Statistical mechanical entropy is basically a measure of the number of quantum states, so we are looking for a universal mechanism to explain the density of states of a black hole. There is one context in which such a mechanism is known: that of two-dimensional conformal field theory. The metric for a two-dimensional manifold can always be written locally as z, ds2 = 2gzz¯dzd¯

(55)

in terms of complex coordinates z and z¯. Holomorphic and antiholomorphic coordi¯ z ) merely rescale the metric, and provide the nate changes z → z + ξ(z), z¯ → z¯ + ξ(¯ basic symmetries of a conformal field theory. Unlike the conformal symmetry group in higher dimensions, the two-dimensional group has infinitely many generators, ¯ These satisfy a Virasoro algebra49 ¯ ξ]. denoted L[ξ] and L[  c   [L[ξ], L[η]] = L[ηξ − ξη ] + dz(η  ξ  − ξ  η  ), 48π  (56) ¯ L[¯ ¯ ξ], ¯ η ]] = L[¯ ¯ η ξ¯ − ξ¯η¯ ] + c¯ [L[ d¯ z (¯ η  ξ¯ − ξ¯ η¯ ), 48π ¯ η ]] = 0, [L[ξ], L[¯ uniquely determined by the values of the two constants c and c¯, the central charges. ¯ 0 of the symmetry generators As in ordinary field theory, the zero modes L0 and L are conserved quantities, the “conformal weights,” which can be seen as linear combinations of mass and angular momentum. Two-dimensional conformal symmetry is remarkably powerful. In particular, with a few mild restrictions, Cardy has shown that the asymptotic density of states

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of any two-dimensional conformal field theory is given by183,184   ¯0 cL0 c¯L ¯ ln ρ(L0 ) ∼ 2π , ln ρ¯(L0 ) ∼ 2π (microcanonical) 6 6 π2 ln ρ(T ) ∼ cT, 3

π2 ln ρ¯(T ) ∼ c¯T (canonical) 3

(57)

where T is the temperature. The entropy is thus uniquely determined by a few parameters, regardless of the details of the conformal field theory — just the kind of universal behavior we would like for black holes. 9.2. Application to black holes Black holes are neither conformally invariant nor two-dimensional, so one might wonder whether the preceding results are relevant. There is a sense, however, in which the near-horizon region is nearly conformally invariant: The extreme redshift washes out dimensionful quantities such as masses for an observer who remains outside the black hole.185 The same redshift also makes transverse excitations negligible relative to those in the r − t plane, effectively reducing the problem to two dimensions. This is the same dimensional reduction that allowed many of the anomaly-based calculations described in Sec. 5.5. Conformal techniques of this type were first applied to the (2 + 1)-dimensional BTZ black hole.87,88 Here, the connection to two-dimensional conformal field theory is clear: The asymptotic boundary at infinity is a two-dimensional cylinder, and the canonical generators of diffeomorphisms obey a Virasoro algebra. Moreover, as Brown and Henneaux showed long ago,86 this Virasoro algebra has a classical central charge, whose presence can be traced back to the need to add boundary terms ¯ 0 for the BTZ black to the canonical generators. The conformal weights L0 and L hole have simple expressions in terms of charge and mass, and the microcanonical form of the Cardy formula (57) then gives the standard Bekenstein–Hawking entropy. The generalization to higher dimensional black holes186–188 is more subtle, and not so firmly established, but there has been some significant progress.d The key trick is to treat the horizon as a sort of boundary — not in the sense that matter cannot pass through it, but in the sense that it is a place where one must impose “boundary conditions,” namely the condition that it is a horizon. As in the BTZ case, the canonical generators of diffeomorphisms acquire boundary terms, and for a number of sensible “stretched horizon” boundary conditions these pick out a Virasoro subalgebra with a calculable central charge. Similar techniques have been used for the near-horizon geometry of the near-extremal Kerr black hole,190 again giving the expected Bekenstein–Hawking entropy. d See

Ref. 189 for a more thorough treatment.

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9.3. Effective descriptions While conformal methods may allow a “universal” computation of Bekenstein– Hawking entropy, they do not tell us a lot about the quantum states of the black hole. In one sense, this is a good thing: The point, after all, is to explain why many different descriptions of the states yield the same entropy. Still, one may be able to learn something useful about “effective” descriptions. It is well known that the presence of a boundary, either at infinity or at a finite location, alters the canonical generators of diffeomorphisms by requiring the addition of boundary terms.65 The exact form of these boundary terms depend on the choice of boundary conditions, and the resulting generators necessarily respect these boundary conditions. As a result, some of the “gauge transformations” of the theory without boundary — those that change the boundary conditions — are no longer invariances of the theory. This boundary symmetry-breaking increases the number of states in the theory: States that would previously have been considered gauge-equivalent are now distinct.191 Moreover, these new states appear only at the boundary, since the gauge symmetry remains unbroken elsewhere. This phenomenon is not quite the same as the Goldstone mechanism,192 but it is similar in spirit. By analogy, it suggests a possible effective description of black hole horizon states in terms of the parameters that label the boundary-conditionbreaking “would-be diffeomorphisms.” For the case of the BTZ black hole, such a description can be made explicitly.89,193 The effective field theory is a particular conformal field theory, Liouville theory, with a central charge that matches the Brown–Henneaux value. The counting of states in this theory is poorly understood: Technically, the relevant states are in the “nonnormalizable sector,” which is not under good control. But as discussed in Sec. 5.10, one can couple the Liouville theory to external matter and recover a correct description of Hawking radiation. The effective description described here is holographic, and is reminiscent of the “membrane paradigm” for black holes,194,195 in which a black hole is also described by a collection of degrees of freedom at the horizon. It is possible that a better understanding of these degrees of freedom could help us understand the information loss problem, the next topic of this review. 10. The Information Loss Problem Let us turn finally to one of the most puzzling aspects of black hole thermodynamics,196 the “information loss paradox.” This is a topic that is very much in flux, with nothing near a consensus concerning its resolution. The purpose of this section is therefore simply to introduce some of the central ideas. Consider a process in which a shell of quantum matter in a pure state is allowed to collapse to form a black hole, which then evaporates completely into Hawking radiation. It appears that this process has allowed an initial pure state to evolve into a final highly mixed (thermal) state. But this is not possible in quantum mechanics: Unitarity requires that a pure state evolve to a pure state. If one thinks of the von

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Neumann entropy as a measure of lack of information, this process has led to an unallowable loss of information. This argument contains several hidden assumptions, of course, any one of which could point to a resolution. We have assumed that a pure state can collapse to form a black hole; that the final state consists only of Hawking radiation, with no long-lived “remnants” of the black hole; that the final spacetime is simply nearly flat space, with no residue of the singularity and no “baby universes” that could swallow up lost correlations; and that the Hawking radiation is genuinely thermal, with no hidden correlations. Let us consider each of these in turn. 10.1. Nonunitary evolution One simple answer to the information loss problem is simply that quantum mechanics is not quite correct, and the evolution is not unitary. Figure 6 shows a proposed Carter–Penrose diagram for an evaporating black hole. In this figure, it is clear where the “missing” information has gone: It has simply vanished into the singularity. Of course, one might hope that quantum gravity would eliminate the singularity, but that need not guarantee that the lost correlations would have to reappear for a future observer. Restricted to black holes, such a loss of unitarity might seem innocuous. But there are presumably quantum amplitudes for any physical process that include virtual black holes, so the effect, though perhaps tiny, would be pervasive. It has been argued that such violations of unitarity lead to either violations of causality or energy nonconservation at an unacceptable level,197 but there are known counterexamples,198 and the possibility remains open. 10.2. No black holes We really understand the formation of a black hole only as a classical process, in which the collapsing matter is typically in a complicated thermal state. It is possible

Fig. 6.

A possible Carter–Penrose diagram for an evaporating black hole.

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that matter in a pure state simply does not collapse to form a black hole. In the fuzzball picture of Sec. 7.3.2, for instance, the microscopic states are not black holes: Properties such as horizons appear only statistically as features of ensembles.131 One can also write down “phenomenological” metrics for which the black hole singularity is replaced by a nonsingular de Sitter-like region.199–202 In such models, a true horizon never forms, but there can be a “quasi-horizon” that looks much like an event horizon for a long period of time. Recall from Sec. 5.1 that a horizon is not required for Hawking radiation — it is enough to have exponential blue shift and a slowly varying effective surface gravity — so for the most part, the thermodynamics will not change. But there is no longer a singularity at which correlations can be lost, and when the quasi-horizon eventually disappears, the information in the interior becomes accessible. Such models share some similarities with the “remnant” scenarios discussed below. In particular, there are unsettled issues concerning how fast the information can “escape” after the disappearance of a quasi-horizon. Nevertheless, they capture a feature that must be shared by almost any model of unitary evolution: If the black hole eventually disappears without any loss of information, then almost by definition there cannot have been a true event horizon.

10.3. Remnants and baby universes Calculations of Hawking radiation seem likely to be reliable down to the length scales at which quantum gravitational effects become important. Beyond that scale, though, we do not know what to expect. In particular, it is possible that black hole evaporation might stop,203 leaving a black hole “remnant.” If remnants exist, they offer a new resolution to the information loss problem. A remnant could be correlated with the Hawking radiation, allowing the combined state to remain pure. Of course, this requires that remnants have a huge entropy, that is, a huge number of possible states. This is generally understood to be incompatible with the AdS/CFT correspondence, and there is a danger of remnants being overproduced by pair production in the vacuum.204 On the other hand, particular exact models of black hole evaporation do avoid the information loss problem through a remnant mechanism,205 providing at least an existence proof. Remnants need not have infinite lifetimes. They, too, may eventually decay into something like Hawking radiation. There are, however, constraints on such a process, if unitarity is to be preserved.203,206 As an estimate of the lifetime,207 suppose the remnant has mass mR , and decays by emitting N photons. Each photon carries roughly one bit of information, and if the process is to preserve unitarity, their total entropy must be on the order of that of the original black hole. We thus require N ∼ GM 2 /
, so each photon should have an energy and wavelength E∼


mR , GM 2

λ∼

GM 2 . mR

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For the photons to be uncorrelated, they should be emitted far enough apart that their wave packets do not significantly overlap. The total time for emission is thus  4 M
τ ∼ Nλ ∼ , (58) mP l mR typically far longer than the age of the universe. By carefully analyzing the entanglement entropy at future null infinity, Bianchi has found a similar time scale, although with a bit more flexibility depending on the details of the evaporation.208 A related approach to the problem is the “baby universe” scenario.210 While we might expect quantum gravity to eliminate the singularity in Fig. 6, the result could plausibly be a new region of the Universe that is causally disconnected from I + — something one could perhaps think of as a peculiar, large remnant. In this case, unitarity is technically maintained, but it cannot be tested by a single observer. 10.4. Hawking radiation as a pure state The most widely held expectation among physicists in this field is that the information loss problem will be resolved by a demonstration that Hawking radiation has subtle hidden correlations and is actually in a pure state. This belief gains support from the AdS/CFT correspondence: In the setting in which we think we best understand quantum gravity, the boundary conformal field theory is unitary, and there does not seem to be room in its Hilbert space for remnants. At first sight, this idea does not seem unreasonable. The density matrix of a pure state can be very close to a thermal density matrix,209 differing only by terms of order e−S . But while the difference between a pure state and a thermal state may be very small, it can still be large enough to cause trouble. Consider two Hawking quanta, one “early” and one “late,” emitted by a black hole, as shown in Fig. 7. As in Sec. 4.2, each escaping particle is accompanied by a “partner” inside the horizon. In order for a freely falling observer to see the standard vacuum near H, particle B must initially be entangled with its “partner” A. This correlation is part of the basic structure of the quantum vacuum. It can be violated only at the cost of introducing large vacuum expectation values of such quantities as the stress–energy tensor.

Fig. 7.

Early and late Hawking quanta emitted from near the horizon H.

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On the other hand, if the Hawking radiation is to be in a pure state in the far future, particle A must be entangled with other particles in the radiation — for simplicity, say particle C. But this violates a fundamental result of quantum mechanics, “monogamy of entanglement,” which states that a state cannot be simultaneously maximally entangled with two others.211 More technically, the problem can be restated as a violation of the strong subadditivity of entropy.5 Of course, two quantum states that are initially entangled need not remain so, and it might be possible to transfer the entanglement of particle B from A to C. But A and C are not causally connected — they are separated by a horizon — so such a process would have to be nonlocal.212 One might respond that quantum gravity is always nonlocal,213,214 but it remains unclear whether “enough” nonlocality can be made compatible with local effective field theory without doing violence to our notions of the scales at which quantum gravitational effects ought to be important. For a slightly different perspective,215 consider the “membrane” description of effective degrees of freedom of Sec. 9.3, or more generally a holographic picture in which the black hole is described by horizon degrees of freedom. As particle A enters the black hole, its state should be captured by these horizon degrees of freedom, which could then be correlated with the later emission of particle C. An infalling observer, on the other hand, should be able to see particle A inside the horizon. This would seem to violate the “no cloning” theorem of quantum theory,216 which prohibits the duplication of a state. The notion of “black hole complementarity” is that this need not be a problem, because no single observer can measure both states. But in a much-cited paper, Almheiri et al. have disputed this,5 arguing instead that the entanglement across the horizon is likely to be broken; see also Braunstein217 for a similar prediction. The resulting large expectation values of the stress–energy tensor would then form a “firewall” for any infalling observer. As Almheiri et al. point out, this implies a breakdown of the principle of equivalence near a horizon, a rather dramatic claim. At this writing, the controversy is far from being settled. 11. Conclusion Classically, black holes are very nearly the simplest structures in general relativity. With the advent of black hole thermodynamics, however, we have come to see them as highly complex thermal systems, rarely at equilibrium, with a truly remarkable number of internal states. Perhaps most surprisingly, we have learned that ordinary methods of general relativity and quantum theory — quantum field theory in curved spaces, WKB approximations, semiclassical path integrals, and the like — allow us to probe properties of quantum gravitational states. Such a claim should be greeted with skepticism, of course. But over the past few decades we have gathered such a weight of self-consistent results, based on enough different and independent approximations, that it seems almost certain we are seeing something real.

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At the same time, black hole thermodynamics is having a profound impact on the rest of physics. The holographic conjecture suggests that our fundamental notions of local physics, and perhaps of space and time, are only low energy approximations. Attempts to understand black hole universality, while less sweeping, also point to the key role of “boundary” states. The information loss problem has led us to question basic aspects of quantum mechanics, and has exhibited remarkable connections among very different aspects of physics. And perhaps in the future, black hole thermodynamics will tell us something profound about quantum gravity. We have interesting times ahead. Acknowledgment This work was supported in part by U.S. Department of Energy Grant No. DE– FG02-91ER40674. Appendix A. Classical Black Holes The Schwarzschild and Kerr solutions, discussed elsewhere in this book, are the archetypes of black holes. For black hole thermodynamics, though, we need more general configurations: “dirty” black holes whose horizons are distorted by nearby matter, dynamical black holes formed from collapsing matter, black holes in other spacetime dimensions. For this, we need a generalization of the notion of a horizon. A black hole is defined as a “region of no return,” an area of spacetime from which light can never escape. This idea can be made more precise — see Hawking and Ellis218 for details — by considering a spacetime whose Carter-Penrose diagram looks roughly like that of Fig. 1, in the sense that it contains a future null infinity I + that describes the “end points at infinity” of light rays. Points in the causal past of I + are ordinary events from which light can escape to infinity. Points that do not lie in the past of I + are cut off from infinity, and form a black hole region. The dividing line — the boundary of the past of I + — is the event horizon; in Fig. 1(a), it is the dashed line labeled H. The event horizon defined in this way has many interesting global properties. For instance, it cannot bifurcate or decrease in area.218 But it might be argued that this definition does not quite capture the right physics. The problem can be traced to the word “never” in the phrase “light can never escape” — to truly make such a statement, one must know about the entire future. The event horizon is teleological in nature: it is determined by “final causes,” events in the indefinite future. Imagine, for instance, that the Earth is sitting at the center of a highly energetic incoming spherical shell of light, one so energetic that it has a Schwarzschild radius of one light year. To be sure, this is not likely, but we also cannot rule out the possibility observationally: no signal can propagate inward faster than such a shell, so we would not know of its existence until it reached us. Suppose now that the shell is presently two light years away, collapsing inward toward us at the speed of light. Step outside and point a flashlight up into the sky. One year from now, the

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light from your flashlight will have traveled one light year, where it will meet the collapsing shell just as the shell reaches its Schwarzschild radius. At that point, the light will be trapped at the horizon of the newly formed Schwarzschild black hole, and will be unable to travel any farther outward. In other words, in this scenario we are already at the event horizon of a black hole, even though we will see no evidence for this fact until we are suddenly vaporized two years from now. This seems odd; surely Hawking radiation “now” should not need to know about the infinite future. A number of attempts have been made to find more local versions of the event horizon,219 leading to a variety of more or less useful definitions. The most useful for our purposes is the “isolated horizon,”9 a locally defined surface with properties suitable for describing equilibrium black holes. An isolated horizon is a null surface — a surface traced out by light rays — whose area remains constant in time, as the horizon of a stationary black hole does. A thought experiment may again be helpful. Picture a spherical lattice studded with flashbulbs, set to all flash simultaneously in the lattice rest frame. The bulbs will produce two spherical shells of light, one traveling inward and one traveling outward. As we know from our experience in nearly flat spacetime, the area of the ingoing shell will normally decrease with time, while the area of the outgoing shell will increase. If the lattice is placed at the horizon of a Schwarzschild black hole, though, it is not hard to show that the area of the outgoing shell will remain constant. Similarly, if the lattice is placed inside the horizon, both shells will decrease in area.e To express this idea mathematically, let us first define a nonexpanding horizon H in a d-dimensional spacetime to be a (d− 1)-dimensional submanifold such that9,136 (1) H is null: that is, its normal vector a is a null vector; (2) the expansion of H vanishes: ϑ() = q ab ∇a b = 0, where qab is the induced metric on H; (3) the field equations hold on H, and −T ab b is future-directed and causal. The first condition expresses the fact that the horizon is traced out by light rays. The expansion ϑ() is the logarithmic derivative of the area of a cross-section of H,220 so the second condition implies time independence of the horizon area. The third condition is a relatively weak prohibition of negative energy at the horizon. These three conditions imply that ∇a b = ωa b

on H

for some one-form ωa . The surface gravity κ() for the normal a — the quantity that appears in the Hawking temperature (1) — is then defined as κ() = a ωa . e The

(A.1)

outgoing shell must still be outgoing with respect to the lattice, of course; as the lattice itself collapses inward, its area will decrease even faster than that of the “outgoing” shell of light. This is one way of understanding why any object inside a black hole must necessarily collapse inward.

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The normal a is not quite unique, though: if a is a null normal to H and λ is an arbitrary function, then eλ a is also a null normal to H. We can reduce this ambiguity by demanding further time independence: a weakly isolated horizon is one for which, as an added condition, we require that (4) L ω = 0 on H , where L is the Lie derivative. This constraint implies the zeroth law of black hole mechanics, that the surface gravity is constant on the horizon. Even with this fourth condition, the null normal a may be rescaled by an arbitrary constant. This also rescales the surface gravity, so the numerical value of κ() is not uniquely determined. At first sight this seems to be a terrible feature, but it reflects a genuine physical ambiguity: even at a horizon, the choice of time coordinate is not completely fixed. In fact, the first law of black hole mechanics (2) requires such an ambiguity: the scale of the mass M is also fixed only after one has normalized the scale of time at infinity. To clarify this issue, it will help to take a small detour. For a stationary black hole, another type of a horizon can be defined. A Killing horizon is a (d − 1)dimensional submanifold HK such that (1) some Killing vector χa is null, that is, χa χa = 0; and (2) HK is itself a null surface, that is, its normal vector is null. If both conditions hold, it follows that χa is itself a null normal to HK , and that χa ∇a χb = κ(χ) χb

on HK .

(A.2)

Note also that the expansion q ab ∇a χb is automatically zero on HK , simply by virtue of the Killing equation ∇a χb + ∇b χa = 0. In this case, the null vector a that defines an isolated horizon may be chosen to equal χa at HK , and the two horizons coincide. This does not quite solve the problem of normalization: a Killing vector also requires normalization, since if χa is a Killing vector, so is cχa . But in the stationary case, the normalization of χa can be fixed at infinity, for example by requiring that χt ∼ 1. In other words, for stationary black holes one can use global properties of the spacetime to adjust clocks at the horizon by comparing them with clocks at infinity. If, on the other hand, one is only concerned with physics at or near the horizon, the normalization of κ() becomes more problematic. One can use known properties of exact solutions to express the surface gravity in terms of other quantities at the horizon,221 thereby fixing a , but so far the procedure seems rather artificial. Alternatively, one can simply accept the ambiguity, and note that other quantities such as quasilocal masses defined near H require a similar choice of normalization. As noted in Sec. 2, weakly isolated horizons obey the four laws of black hole mechanics, the second law in the equilibrium form that the horizon area remains constant. Generalizations to dynamical, evolving horizons are also possible,221 and

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may be used to prove the inequality version (3) of the second law. These generalizations also provide a potential setting for nonequilibrium black hole thermodynamics, allowing us, for instance, to describe flows of energy and angular momentum that include the contribution of the gravitational field. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

16. 17. 18. 19. 20. 21.

22. 23. 24. 25. 26.

27.

J. D. Bekenstein, Lett. Nuovo Cim. 4 (1972) 737–740. J. D. Bekenstein, Phys. Rev. D 7 (1973) 2333–2346. S. W. Hawking, Nature 248 (1974) 30–31. S. W. Hawking, Commun. Math. Phys. 43 (1975) 199–220. A. Almheiri, D. Marolf, J. Polchinski and J. Sully, J. High Energy Phys. 1302 (2013) 062, arXiv:1207.3123. S. W. Hawking, Commun. Math. Phys. 25 (1972) 152–166. J. M. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys. 31 (1973) 161. A. C. Wall, J. High Energy Phys. 0906 (2009) 021, arXiv:0901.3865. A. Ashtekar, C. Beetle and S. Fairhurst, Class. Quantum Gravit. 16 (1999) L1–L7, arXiv:gr-qc/9812065. A. Ashtekar, S. Fairhurst and B. Krishnan, Phys. Rev. D 62 (2000) 104025, arXiv: gr-qc/0005083. R. M. Wald, Phys. Rev. D 48 (1993) 3427–3431, arXiv:gr-qc/9307038. J. A. Wheeler and K. Ford, Geons, Black Holes, and Quantum Foam, Chap. 4 (W. W. Norton & Company, New York, 1998). R. M. Wald, Living Rev. Rel. 4 (2001) 6, arXiv:gr-qc/9912119. A. C. Wall, Phys. Rev. D 85 (2012) 104049, arXiv:1105.3445; Erratum-ibid. D 87 (2013) 069904. Y. B. Zel’dovich and A. A. Starobinsky, Zh. Eksp. Teor. Fiz. 61 (1971) 2161–2175; Y. B. Zel’dovich and A. A. Starobinsky, Sov. Phys. J. Exp. Theor. Phys. 34 (1972) 1159–1166. F. Pretorius, D. Vollick and W. Israel, Phys. Rev. D 57 (1998) 6311–6316, arXiv:grqc/9712085. L. Parker, Phys. Rev. 183 (1969) 1057–1068. M. K. Parikh and F. Wilczek, Phys. Rev. Lett. 85 (2000) 5042–5045, arXiv:hepth/9907001. N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space (Cambridge University Press, Cambridge, 1982). N. N. Bogoliubov, Sov. Phys. J. Exp. Theor. Phys. 7 (1958) 51. J. H. Traschen, An introduction to black hole evaporation, in Mathematical Methods in Physics, Proceedings of the 1999 Londrona Winter School, eds. A. A. Bytsenko and F. L. Williams (World Scientific, Singapore, 2000), arXiv:gr-qc/0010055. C. Barcelo, S. Liberati, S. Sonego and M. Visser, Phys. Rev. D 83 (2011) 041501, arXiv:1011.5593. D. N. Page, Phys. Rev. D 13 (1976) 198–206. D. N. Page, New J. Phys. 7 (2005) 203, arXiv:hep-th/0409024. S. Carlip, Lect. Notes Phys. 769 (2009) 89–123, arXiv:0807.4520. C. Kiefer, Thermodynamics of black holes and Hawking radiation, in Classical and Quantum Black Holes, eds. P. Fre, V. Gorini, G. Magli and U. Moschella (IOP Publishing, Bristol, 1999). B. Schutz, A First Course in General Relativity, Chap. 11.5, 2nd edn. (Cambridge University Press, Cambridge, 2009).

May 4, 2017 9:53 b2167-ch22

II-460

S. Carlip

28. 29. 30. 31. 32. 33.

W. G. Unruh, Phys. Rev. Lett. 46 (1981) 1351–1353. T. Jacobson, Phys. Rev. D 44 (1991) 1731–1739. A. D. Helfer, Rept. Prog. Phys. 66 (2003) 943–1008, arXiv:gr-qc/0304042. S. Corley, Phys. Rev. D 57 (1998) 6280–6291, arXiv:hep-th/9710075. T. Jacobson, Phys. Rev. D 53 (1996) 7082–7088, arXiv:hep-th/9601064. W. G. Unruh and R. Schutzhold, Phys. Rev. D 71 (2005) 024028, arXiv:grqc/0408009. C. Barcelo, S. Liberati and M. Visser, Living Rev. Rel. 14 (2011) 3, arXiv:grqc/0505065. W. G. Unruh, Phys. Rev. D 14 (1976) 870–892. M. Visser, Phys. Rev. Lett. 80 (1998) 3436–3439, arXiv:gr-qc/9712016. M. Visser, Int. J. Mod. Phys. D 12 (2003) 649–661, arXiv:hep-th/0106111. J. B. Hartle and S. W. Hawking, Phys. Rev. D 13 (1976) 2188–2203. R. M. Wald, Commun. Math. Phys. 45 (1975) 9–34. L. Parker, Phys. Rev. D 12 (1975) 1519–1525. J. J. Bisognano and E. H. Wichmann, J. Math. Phys. 17 (1976) 303–321. B. S. DeWitt, Quantum gravity: The new synthesis, in General Relativity: An Einstein Centenary Survey, eds. S. W. Hawking and W. Israel (Cambridge University Press, Cambridge, 1979). H. Yu and W. Zhou, Phys. Rev. D 76 (2007) 044023, arXiv:0707.2613. T. Damour and R. Ruffini, Phys. Rev. D 14 (1976) 332–334. K. Srinivasan and T. Padmanabhan, Phys. Rev. D 60 (1999) 024007, arXiv:grqc/9812028. L. Vanzo, G. Acquaviva and R. Di Criscienzo, Class. Quantum Gravit. 28 (2011) 183001, arXiv:1106.4153. R. Kerner and R. B. Mann, Phys. Rev. D 73 (2006) 104010, arXiv:gr-qc/0603019. P. C. W. Davies, S. A. Fulling and W. G. Unruh, Phys. Rev. D 13 (1976) 2720–2723. P. Di Francesco, P. Mathieu and D. S´en´echal, Conformal Field Theory (Springer, New York, 1997). S. M. Christensen and S. A. Fulling, Phys. Rev. D 15 (1977) 2088–2104. S. P. Robinson and F. Wilczek, Phys. Rev. Lett. 95 (2005) 011303, arXiv:grqc/0502074. S. Iso, H. Umetsu and F. Wilczek, Phys. Rev. Lett. 96 (2006) 151302, arXiv:hepth/0602146. S. Iso, H. Umetsu and F. Wilczek, Phys. Rev. D 74 (2006) 044017, arXiv:hepth/0606018. S. Iso, T. Morita and H. Umetsu, Phys. Rev. D 77 (2008) 045007, arXiv:0710.0456. L. Bonora, M. Cvitan, S. Pallua and I. Smoli´c, J. High Energy Phys. 0812 (2008) 021, arXiv:0808.2360. L. Bonora, M. Cvitan, S. Pallua and I. Smoli´c, Phys. Rev. D 80 (2009) 084034, arXiv:0907.3722. R. Kubo, J. Phys. Soc. Jpn. 12 (1957) 570–586. P. C. Martin and J. S. Schwinger, Phys. Rev. 115 (1959) 1342–1373. R. Haag, N. M. Hugenholtz and M. Winnink, Commun. Math. Phys. 5 (1967) 215– 236. G. W. Gibbons and M J. Perry, Phys. Rev. Lett. 36 (1976) 985–987. G. W. Gibbons and M. J. Perry, Proc. Roy. Soc. Lond. A 358 (1978) 467–494. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGrawHill, New York, 1965). C. P. Burgess, Living Rev. Rel. 7 (2004) 5–56, arXiv:gr-qc/0311082.

34. 35. 36. 37. 38. 39. 40. 41. 42.

43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63.

May 4, 2017 9:53 b2167-ch22

Black hole thermodynamics

II-461

64. G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15 (1977) 2752–2756. 65. C. Regge and C. Teitelboim, Annals Phys. 88 (1974) 286–318. 66. M. Ba˜ nados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. 72 (1994) 957–960, arXiv:gr-qc/9309026. 67. C. Teitelboim, Phys. Rev. D 51 (1995) 4315–4318, arXiv:hep-th/9410103. 68. S. W. Hawking and G. T. Horowitz, Class. Quantum Gravit. 13 (1996) 1487–1498, arXiv:gr-qc/9501014. 69. S. W. Hawking and C. J. Hunter, Phys. Rev. D 59 (1999) 044025, arXiv:hepth/9808085. 70. Y. Neiman, J. High Energy Phys. 1304 (2013) 071, arXiv:1301.7041. 71. S. Carlip and C. Teitelboim, Class. Quantum Gravit. 12 (1995) 1699–1704, arXiv:grqc/9312002. 72. W. Heisenberg and H. Euler, Z. Phys. 98 (1936) 714–732. 73. J. S. Schwinger, Phys. Rev. 82 (1951) 664–679. 74. D. Garfinkle, S. B. Giddings and A. Strominger, Phys. Rev. D 49 (1994) 958–965, arXiv:gr-qc/9306023. 75. J. D. Brown, Phys. Rev. D 51 (1995) 5725–5731, arXiv:gr-qc/9412018. 76. F. Dowker, J. P. Gauntlett, D. A. Kastor and J. H. Traschen, Phys. Rev. D 49 (1994) 2909–2917, arXiv:hep-th/9309075. 77. R. B. Mann and S. F. Ross, Phys. Rev. D 52 (1995) 2254–2265, arXiv:gr-qc/9504015. 78. R. M. Wald, Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics (University of Chicago Press, Chicago, 1994). 79. B. S. Kay and R. M. Wald, Phys. Rept. 207 (1991) 41–136. 80. W. Israel, Phys. Lett. A 57 (1976) 107–110. 81. T. Jacobson, Phys. Rev. D 50 (1994) 6031–6032, arXiv:gr-qc/9407022. 82. M. Banados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. 69 (1992) 1849–1851, arXiv:hep-th/9204099. 83. M. Banados, M. Henneaux, C. Teitelboim and J. Zanelli, Phys. Rev. D 48 (1993) 1506–1525, arXiv:gr-qc/9302012. 84. S. Carlip, Class. Quantum Gravit. 12 (1995) 2853–2880, arXiv:gr-qc/9506079. 85. S. Carlip, Field Theory, Integrable Systems and Symmetries, eds. F. Khanna and L. Vinet (Les Publications CRM, Montreal, 1997), arXiv:gr-qc/9509024. 86. J. D. Brown and M. Henneaux, Commun. Math. Phys. 104 (1986) 207–226. 87. A. Strominger, J. High Energy Phys. 9802 (1998) 009, arXiv:hep-th/9712251. 88. D. Birmingham, I. Sachs and S. Sen, Phys. Lett. B 424 (1998) 27–2805, arXiv:hepth/9801019. 89. S. Carlip, Class. Quantum Gravit. 22 (2005) R85–R124, arXiv:gr-qc/0503022. 90. R. Emparan and I. Sachs, Phys. Rev. Lett. 81 (1998) 2408–2411, arXiv:hepth/9806122. 91. K. Melnikov and M. Weinstein, Int. J. Mod. Phys. D 13 (2004) 1595–1636, arXiv:hepth/0205223. 92. K. Freese and C. T. Hill, Nucl. Phys. B 255 (1985) 693–716. 93. J. W. York, Phys. Rev. D 28 (1983) 2929–2945. 94. W. A. Hiscock and L. D. Weems, Phys. Rev. D 41 (1990) 1142–1151. 95. P. C. W. Davies, Proc. R. Soc. Lond. A 353 (1977) 499–521. 96. D. Lynden-Bell and R. Wood, Mon. Not. R. Astron Soc. 138 (1968) 495–525. 97. J. W. York, Phys. Rev. D 33 (1986) 2092–2099. 98. S. W. Hawking and D. N. Page, Commun. Math. Phys. 87 (1983) 577–588. 99. E. Witten, Adv Theor. Math. Phys. 2 (1998) 253–291, arXiv:hep-th/9802150.

May 4, 2017 9:53 b2167-ch22

II-462

S. Carlip

100. S. Carlip and S. Vaidya, Class. Quantum Gravit. 20 (2003) 3827–3838, arXiv:grqc/0306054. 101. M. Henneaux and C. Teitelboim, Phys. Lett. B 143 (1984) 415–420. 102. M. M. Caldarelli, G. Cognola and D. Klemm, Class. Quantum Gravit. 17 (2000) 399–420, arXiv:hep-th/9908022. 103. Y. Sekiwa, Phys. Rev. D 73 (2006) 084009, arXiv:hep-th/0602269. 104. D. Kastor, S. Ray and J. Traschen, Class. Quantum Gravit. 26 (2009) 195011, arXiv:0904.2765. 105. M. Cvetic, G. W. Gibbons, D. Kubiznak and C. N. Pope, Phys. Rev. D 84 (2011) 024037, arXiv:1012.2888. 106. B. P. Dolan, Class. Quantum Gravit. 28 (2011) 125020, arXiv:1008.5023. 107. D. Kubiznak and R. B. Mann, arXiv:1404.2126. 108. S. L. Dubovsky and S. M. Sibiryakov, Phys. Lett. B 638 (2006) 509–514. 109. C. Eling, B. Z. Foster, T. Jacobson and A. C. Wall, Phys. Rev. D 75 (2007) 101502(R). 110. A. Barvinsky, S. Das and G. Kunstatter, Phys. Lett. B 517 (2001) 415–420, arXiv:hep-th/0102061. 111. S. Hod, Phys. Rev. Lett. 81 (1998) 4293–4296, arXiv:gr-qc/9812002. 112. G. Siopsis, Lect. Notes Phys. 769 (2009) 471–508, arXiv:0804.2713. 113. M. Maggiore, Phys. Rev. Lett. 100 (2008) 141301, arXiv:0711.3145. 114. A. J. M. Medved, Mod. Phys. Lett. A 24 (2009) 2601–2609, arXiv:0906.2641. 115. R. D. Sorkin, On the entropy of the vacuum outside a horizon, in Tenth Int. Conf. General Relativity and Gravitation, Vol. 2 (1983), pp. 734–736, arXiv:1402.3589. 116. L. Bombelli, R. K. Koul, J. Lee and R. D. Sorkin, Phys. Rev. D 34 (1986) 373–383. 117. M. Srednicki, Phys. Rev. Lett. 71 (1993) 666–669, arXiv:hep-th/9303048. 118. G.’t Hooft, Nucl. Phys. B 256 (1985) 727–745. 119. L. Susskind and J. Uglum, Phys. Rev. D 50 (1994) 2700–2711, arXiv:hep-th/9401070. 120. J. H. Cooperman and M. A. Luty, arXiv:1302.1878. 121. S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96 (2006) 181602, arXiv:hep-th/0603001. 122. V. E. Hubeny, M. Rangamani and T. Takayanagi, J. High Energy Phys. 0707 (2007) 062, arXiv:0705.0016. 123. A. Lewkowycz and J. Maldacena, J. High Energy Phys. 1308 (2013) 090, arXiv:1304.4926. 124. R. Emparan, J. High Energy Phys. 012 (2006) 0606, arXiv:hep-th/0603081. 125. A. Strominger and C. Vafa, Phys. Lett. B 379 (1996) 99–104, arXiv:hep-th/9601029. 126. A. W. Peet, TASI lectures on black holes in string theory, in TASI 99 : Strings, Branes and Gravity, eds. J. Harvey, S. Kachru and E. Silverstein (World Scientific, Singapore, 2001), arXiv:hep-th/0008241. 127. S. R. Das and S. D. Mathur, Ann. Rev. Nucl. Part. Sci. 50 (2000) 153–206, arXiv:grqc/0105063. 128. J. Maldacena and A. Strominger, Phys. Rev. D 55 (1997) 861–870, arXiv:hepth/9609026. 129. S. D. Mathur, Fortsch. Phys. 53 (2005) 793–827, arXiv:hep-th/0502050. 130. S. D. Mathur, Class. Quantum Gravit. 23 (2006) R115–R168, arXiv:hep-th/0510180. 131. S. D. Mathur, Lect. Notes Phys. 769 (2009) 3–48, arXiv:0803.2030. 132. B. D. Chowdhury and S. D. Mathur, Class. Quantum Gravit. 25 (2008) 135005, arXiv:0711.4817. 133. J. M. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231–252, arXiv:hep-th/9711200. 134. O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz, Phys. Rept. 323 (2000) 183–386, arXiv:hep-th/9905111.

May 4, 2017 9:53 b2167-ch22

Black hole thermodynamics

II-463

135. K. Skenderis, Lect. Notes Phys. 541 (2000) 325–364, arXiv:hep-th/9901050. 136. A. Ashtekar, C. Beetle and S. Fairhurst, Class. Quantum Gravit. 17 (2000) 253–298, arXiv:gr-qc/9907068. 137. A. Corichi, J. Diaz-Polo and E. Fernandez-Borja, Phys. Rev. Lett. 98 (2007) 181301, arXiv:gr-qc/0609122. 138. I. Agullo, J. Fernando Barbero, E. F. Borja, J. Diaz-Polo and E. J. S. Villase˜ nor, Phys. Rev. D 82 (2010) 084029, arXiv:1101.3660. 139. T. Jacobson, Class. Quantum Gravit. 24 (2007) 4875–4879, arXiv:0707.4026. 140. S. Alexandrov and E. R. Livine, Phys. Rev. D 67 (2003) 044009, arXiv:grqc/0209105. 141. K. V. Krasnov, Phys. Rev. D 55 (1997) 3505–3513, arXiv:gr-qc/9603025. 142. C. Rovelli, Phys. Rev. Lett. 77 (1996) 3288–3261, arXiv:gr-qc/9603063. 143. A. Ashtekar, J. C. Baez, A. Corichi and K. Krasnov, Phys. Rev. Lett. 80 (1998) 904–907, arXiv:gr-qc/9710007. 144. A. Ashtekar, J. C. Baez and K. Krasnov, Adv. Theor. Math. Phys. 4 (2000) 1–94, arXiv:gr-qc/0005126. 145. E. Witten, Commun. Math. Phys. 121 (1989) 351–399. 146. M. Domagala and J. Lewandowski, Class. Quantum Gravit. 21 (2004) 5233–5244, arXiv:gr-qc/0407051. 147. K. A. Meissner, Class. Quantum Gravit. 21 (2004) 5245–5252, arXiv:gr-qc/0407052. 148. A. Ashtekar and J. Lewandowski, Class. Quantum Gravit. 21 (2004) R53–R152, arXiv:gr-qc/0404018. 149. A. Ashtekar, J. Engle and C. Van Den Broeck, Class. Quantum Gravit. 22 (2005) L27–L34, arXiv:gr-qc/0412003. 150. E R. Livine and D. R. Terno, Nucl. Phys. B 741 (2006) 131–161, arXiv:grqc/0508085. 151. E. Bianchi, arXiv:1204.5122. 152. E. Frodden, A. Ghosh and A. Perez, Phys. Rev. D 87 (2013) 121503, arXiv:1110.4055. 153. A. Ghosh and A. Perez, Phys. Rev. Lett. 27 (2011) 241301, arXiv:1107.1320. 154. A. Ghosh, K. Noui and A. Perez, arXiv:1309.4563. 155. E. Frodden, M. Geiller, K. Noui and A. Perez, arXiv:1212.4060. 156. J. B. Achour, A. Mouchet and K. Noui, arXiv:1406.6021. 157. S. Carlip, arXiv:1503.02981. 158. A. D. Sakharov, Sov. Phys. Dokl. 12 (1968) 1040–1041; reprinted in Gen. Relativ. Gravit. 32 (2000) 365–367. 159. S. L. Adler, Rev. Mod. Phys. 54 (1982) 729–766; Erratum ibid. 55 (1983) 837. 160. V. P. Frolov, D. V. Fursaev and A. I. Zelnikov, Nucl. Phys. B 486 (1997) 339–352, arXiv:hep-th/9607104. 161. V. P. Frolov and D. V. Fursaev, Phys. Rev. D 56 (1997) 2212–225, arXiv:hepth/9703178. 162. V. P. Frolov, D. Fursaev and A. Zelnikov, J. High Energy Phys. 0303 (2003) 038, arXiv:hep-th/0302207. 163. L. Bombelli, J. Lee, D. Meyer and R. Sorkin, Phys. Rev. Lett. 59 (1987) 521–524. 164. D. Rideout and S. Zohren, Class. Quantum Gravit. 23 (2006) 6195–6213, arXiv:grqc/0606065. 165. D. Dou and R. D. Sorkin, Found. Phys. 33 (2003) 279–296, arXiv:gr-qc/0302009. 166. W. H. Zurek and K. S. Thorne, Phys. Rev. Lett. 54 (1985) 2171–2175. 167. S. Carlip, Class. Quantum Gravit. 17 (2000) 4175–4186, arXiv:gr-qc/0005017. 168. A. Ghosh and P. Mitra, Phys. Rev. D 71 (2005) 027502, arXiv:gr-qc/0401070.

May 4, 2017 9:53 b2167-ch22

II-464

S. Carlip

169. A. J. M. Medved and E. C. Vagenas, Phys. Rev. D 70 (2004) 124021, arXiv:hepth/0411022. 170. A. Sen, J. High Energy Phys. 04 (2013) 156, arXiv:1205.0971. 171. J. Engle, K. Noui, A. Perez and D. Pranzetti, J. High Energy Phys. 1105 (2011) 016, arXiv:1103.2723. 172. G. Gour and A. J. M. Medved, Class. Quantum Gravit. 20 (2003) 3307–3326, arXiv:gr-qc/0305018. 173. C. G. Torre and I. M. Anderson, Phys. Rev. Lett. 70 (1993) 3525–3529, arXiv:grqc/9302033. 174. G. ’t Hooft, Dimensional reduction in quantum gravity, in Salamfestschrift: A Collection of Talks, eds. A. Ali, J. Ellis and S. Randjbar-Daemi (World Scientific, Singapore, 1993), arXiv:gr-qc/9310026. 175. L. Susskind, J. Math. Phys. 36 (1995) 6377–6396, arXiv:hep-th/9409089. 176. U. Yurtsever, Phys. Rev. Lett. 91 (2003) 041302, arXiv:gr-qc/0303023. 177. S. D. H. Hsu and D. Reeb, Phys. Lett. B 658 (2008) 244–248, arXiv:0706.3239. ´ E. ´ Flanagan, D. Marolf and R. M. Wald, Phys. Rev. D 62 (2000) 084035, 178. E. arXiv:hep-th/9908070. 179. R. Bousso, J. High Energy Phys. 9907 (1999) 004, arXiv:hep-th/9905177. 180. R. Bousso, Rev. Mod. Phys. 74 (2002) 825–874, arXiv:hep-th/0203101. ´ E. ´ Flanagan and D. Marolf, Phys. Rev. D 68 (2003) 064001, arXiv:hep181. R. Bousso, E. th/0305149. 182. R. Bousso, H. Casini, Z. Fisher and J. Maldacena, arXiv:1404.5635. 183. J. L. Cardy, Nucl. Phys. B 270 (1986) 186–204. 184. H. W. J. Bl¨ ote, J. L. Cardy and M. P. Nightingale, Phys. Rev. Lett. 56 (1986) 742–745. 185. H. E. Camblong and C. R. Ord´ on ˜ez, Phys. Rev. D 71 (2005) 104029, arXiv:hepth/0411008. 186. S. Carlip, Phys. Rev. Lett. 82 (1999) 2828–2831, arXiv:hep-th/9812013. 187. S. Carlip, Class. Quantum Gravit. 16 (1999) 3327–3348, arXiv:gr-qc/9906126. 188. S. Carlip, J. High Energy Phys. 1104 (2011) 076, arXiv:1101.5136; Erratum-ibid. 1201 (2012) 008. 189. S. Carlip, Entropy 13 (2011) 1355–1379, arXiv:1107.2678. 190. M. Guica, T. Hartman, W. Song and A. Strominger, Phys. Rev. D 80 (2009) 124008, arXiv:0809.4266. 191. S. Carlip, Nucl. Phys. Proc. Suppl. 57 (1997) 8–12, arXiv:gr-qc/9702017. 192. J. Goldstone, A. Salam and S. Weinberg, Phys. Rev. 127 (1962) 965–970. 193. S. Carlip, Class. Quantum Gravit. 22 (2005) 3055–3060, arXiv:gr-qc/0501033. 194. K. S. Thorne, R H. Price and D. A. Macdonald, Black Holes: The Membrane Paradigm (Yale University Press, New Haven, 1986). 195. M. Parikh and F. Wilczek, Phys. Rev. D 58 (1998) 064011, arXiv:gr-qc/9712077. 196. S. W. Hawking, Phys. Rev. D 14 (1976) 2460–2473. 197. T. Banks, L. Susskind and M. E. Peskin, Nucl. Phys. B 244 (1984) 125–134. 198. W. G. Unruh and R. M. Wald, Phys. Rev. D 52 (1995) 2176–2182, arXiv:hepth/9503024. 199. V. P. Frolov and G. A. Vilkovisky, Phys. Lett. B 106 (1981) 307–313. 200. S. A. Hayward, Phys. Rev. Lett. 96 (2006) 031103, arXiv:gr-qc/0506126. 201. V. P. Frolov, J. High Energy Phys. 1405 (2014) 049, arXiv:1402.5446. 202. J. M. Bardeen, arXiv:1406.4098. 203. Y. Aharonov, A. Casher and S. Nussinov, Phys. Lett. B 191 (1987) 51–55. 204. S. B. Giddings, Phys. Rev. D 49 (1994) 947–957.

May 4, 2017 9:53 b2167-ch22

Black hole thermodynamics

II-465

205. A. Almheiri and J. Sully, J. High Energy Phys. 1402 (2014) 108, arXiv:1307.8149. 206. R. D. Carlitz and R. S. Willey, Phys. Rev. D 36 (1987) 2336–2341. 207. S. B. Giddings, Quantum mechanics of black holes, in Summer School in High Energy Physics and Cosmology: Proceedings, eds. E. Gava, A. Masiero, K. S. Narain, S. Randjbar-Daemi and Q. Shafi (World Scientific, Singapore, 1995), arXiv:hepth/9412138. 208. E. Bianchi, Talk at Peyresq Physics 11, June 2014. 209. K. Papadodimas and S. Raju, Int. J. Mod. Phys. D 22 (2013) 1342030. 210. S. W. Hawking, Mod. Phys. Lett. A 5 (1990) 453–466. 211. D. Yang, Phys. Lett. A 360 (2006) 249–250, arXiv:quant-ph/0604168. 212. S. B. Giddings, Class. Quantum Gravit. 28 (2011) 025002, arXiv:0911.3395. 213. T. Jacobson, Int. J. Mod. Phys. D 22 (2013) 1342002, arXiv:1212.6944. 214. K. Papadodimas and S. Raju, Phys. Rev. Lett. 112 (2014) 051301, arXiv:1310.6334. 215. L. Susskind, L. Thorlacius and J. Uglum, Phys. Rev. D 48 (1993) 3743–3761, arXiv:hep-th/9306069. 216. W. K. Wootters and W. H. Zurek, Nature 299 (1982) 802–803. 217. S. L. Braunstein, S. Pirandola and K. Zyczkowski, Phys. Rev. Lett. 110 (2013), 101301, arXiv: 0907.1190. 218. S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time (Cambridge University Press, Cambridge, 1973). 219. I. Booth, Can. J. Phys. 83 (2005) 1073–1099, arXiv:gr-qc/0508107. 220. E. Poisson, A Relativist’s Toolkit (Cambridge University Press, Cambridge, 2004). 221. A. Ashtekar and B. Krishnan, Living Rev. Rel. 7 (2004) 10, arXiv:gr-qc/0407042.

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Chapter 23 Loop quantum gravity

Dah-Wei Chiou Department of Physics, National Taiwan Normal University, Taipei 11677, Taiwan Department of Physics and Center for Condensed Matter Sciences, National Taiwan University, Taipei 10617, Taiwan

This paper presents an “in-a-nutshell” yet self-contained introductory review on loop quantum gravity (LQG) — a background-independent, nonperturbative approach to a consistent quantum theory of gravity. Instead of rigorous and systematic derivations, it aims to provide a general picture of LQG, placing emphasis on the fundamental ideas and their significance. The canonical formulation of LQG, as the central topic of the paper, is presented in a logically orderly fashion with moderate details, while the spin foam theory, black hole thermodynamics, and loop quantum cosmology are covered briefly. Current directions and open issues are also summarized. Keywords: Loop quantum gravity; spin foams; black hole thermodynamics; loop quantum cosmology. PACS Number: 04.60.Pp

1. Introduction Quantum gravity (QG) is the research in theoretical physics that seeks a consistent quantum theory of gravity. It is considered by many as the open problem of paramount importance in fundamental physics, as its task is to unify quantum mechanics (more specifically, quantum field theory, QFT) and general relativity (GR), which are the two greatest theories discovered in the twentieth century and have become the cornerstones of modern physics. Ever since it was recognized that the gravitational field needs to be quantized, the quest for a satisfactory quantum description of spacetime has never stopped, and gradually QG has grown into a vast area of research along many different paths with various doctrines. The two most developed approaches are string theory and loop quantum gravity (LQG). Other directions include causal sets, dynamical triangulation, emergent gravity, H space theory, noncommutative geometry, supergravity, thermogravity, twistor theory, and many more. (See Refs. 1–4 for surveys and more references on different theories of QG.) Among other approaches, the strength of LQG is that it provides a compelling description of quantum spacetime in a nonperturbative, background-independent fashion. The beauty lies in its faithful attempt to establish a conceptual framework II-467

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that consolidates the apparently conflicting tenets of QFT and GR. LQG deliberately adopts the “minimalist” approach in the sense that it focuses solely on the search for a consistent quantum theory of gravity without requiring any extraordinary ingredients such as extra dimensions, supersymmetry, and so on (although many of these can be incorporated compatibly). Unlike string theory, the aim of which is much more ambitious, LQG does not intend to address the unification problem of finding a “theory of everything” in which all forces, including gravity, are unified. The central problem confronted by LQG simply is: What is the consistent quantum field theory of which the low-energy limit is classical GR, if it exists after all? Having been developed about 25 years (see Ref. 5 for its history and current status), LQG has faced many challenges and achieved encouraging progress. Many long-standing open problems within the approach have been solved, and LQG now provides a rigorous mathematical foundation for QG. One of the main results is the discovery that the spectra of area and volume are discrete and the corresponding quantum states, called spin networks, reveal the microscopic structure of space that is granular in Planck scale. The discreteness of space is not postulated ad hoc but a direct consequence of quantization in the same nature as the discrete levels of energy in an atom. Because of the discreteness, LQG enables one to derive the Bekenstein–Hawking entropy of black holes from first principles, offering a microscopic explanation of its proportionality to the area of the horizon. Furthermore, the construction of LQG relies heavily on diffeomorphism invariance. Once the gauge degrees of diffeomorphism are factored away, the resulting quantum states, called s-knots, are not to be interpreted as quantum excitations on a space but, rather, of the space itself, because any external reference to localization of quantum states becomes irrelevant and only their contiguous relation remains significant. In terms of this relational structure, LQG profoundly manifests background independence — the key principle of classical GR — in the context of QFT. With many successes achieved, the major weakness of LQG is in our limited understanding about its quantum dynamics and low-energy physics, but these two aspects have been intensively investigated and some promising solutions have emerged. LQG is closely related to the spin foam theory, an alternative approach of QG. While LQG is based on the canonical (Hamiltonian) formalism, the spin foam theory can be viewed as the covariant (sun-over-histories) approach of LQG. The history of a spin network (more precisely, an s-knot) evolving over time is called a spin foam, which represents a quantized spacetime in the same sense that a spin network represents a quantized space. The transition amplitude is given by a discrete version of path integral, which is the sum, with proper weight factors, over all possible spin foams sending the initial spin network to the final one. Extensive research on the spin foam theory has made rapid advances, suggesting that a specific class of spin foam models might provide a covariant definition of the LQG dynamics. Applying principles of LQG to cosmological settings, one obtains loop quantum cosmology (LQC). LQC is a symmetry-reduced model of LQG with only a finite

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number of degrees of freedom. The simplification of symmetry reduction makes possible a detailed investigation on the ramifications of loop quantum effects, many of which remain obscure in the full theory of LQG. One of the striking results of LQC is that the big bang singularity is resolved by the loop quantum effects and replaced by a quantum bounce, which bridges the present universe with a preexistent one. The bouncing scenario opens up a new paradigm of cosmology. Research of LQC has led to many successes and represents a very active research field in recent years. There are many books and review articles available on LQG,6–13 the spin foam theory,14–17 and LQC.18–21 This paper serves as an “in-a-nutshell” yet self-contained introductory review on LQG. It is centered on the canonical formulation of LQG with moderate details but covers the spin foam theory, black hole thermodynamics, and LQC only very briefly. The paper is organized as follows. After a quick review on the problems of QG in Sec. 2, the main topic of canonical approach of LQG is presented in Secs. 3–8 in a logically orderly fashion. Additionally, the spin foam theory, black hole thermodynamics, and LQC are briefly described in Secs. 9–11, respectively. Finally, Sec. 12 outlines the current directions and open issues. A large part of this paper is heavily based on Refs. 8 and 9 without delving into all technicalities. Conventions are given as follows. The signature of the spacetime metric gαβ is given by (−, +, +, +). The Greek letters in lower case are used for the coordinates of the four-dimensional spacetime manifold, while the Latin letters in lower case starting a, b, . . . are for the coordinates of the three-dimensional space manifold. The Latin letters in upper case starting I, J, K, . . . = 0, 1, 2, 3 are used for the four-dimensional “internal space” (the vector space on which the Lorentz group SO(3, 1) is represented), while the Latin letters in lower case starting i, j, k, . . . = 1, 2, 3 are for the three-dimensional internal space (the vector space on which SU (2), the covering group of SO(3), is represented). In case the Euclidean GR, in place of the Lorentzain GR, the metric signature is given by (+, +, +, +) and the Lorentz group SO(3, 1) is replaced by the Euclidean rotational group SO(4). The speed of light is set to c = 1, but we keep the (reduced) Planck constant
and the gravitational constant G explicit. 2. Motivations Despite the fact that enormous efforts in various approaches have been devoted for QG, it remains the Holy Grail of theocratical physics. In the section, we briefly discuss the conceptual problems in QG. (See Refs. 3, 4 and 22 for full elaboration.)

2.1. Why quantum gravity ? The very first question about QG is: Why do we even bother to quantize gravity at all? Apart from many aesthetic considerations for an elegant unification of known

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fundamental physics, the logical necessity of a quantum description of gravity follows from the conflicts between the two fundamental theories of GR and QFT. The fundamental theories collide in the classical Einstein field equation 1 Rµν − R gµν = 8πGTµν , (1) 2 which relates the dynamics of nongravitational matter (in terms of Tµν ) described by the theory of QFT to the dynamics of geometry (in terms of gµν ) described by the theory of GR. Note that the metric gµν enters the definition of the energy– momentum tensor via Tµν ≡ Tµν (g) = δS/δg µν (S is the action) and that QFT is defined only on a given (flat or curved) background. As matter fields are quantized in QFT, the only way to make sense of Eq. (1) without quantizing gµν is to start with a given background g0 and replace the righthand side with the expectation value  Tˆµν (g0 ). Once we solve the classical field gµν by this prescription, we treat the solution as the new background and iterate the procedure. However, the iteration is not guaranteed to yield convergent solutions and thus leads to inconsistency. In fact, one cannot consistently couple a classical system to a quantum one,3,23 even though QFT in curved spacetime provides an adequate effective theory in low-energy regime. The conflicts force one to quantize both geometry and matter fields simultaneously or to invoke other more radical modifications. 2.2. Difficulties of quantum gravity What after all are the difficulties that prevent one from quantizing both geometry and matter easily at the same time? It turns out the difficulties lie at the very contradictory difference between the fundamental principles of GR and QFT. In GR, spacetime is dynamical as well as matter fields. Matter and geometry are essentially on the equal footing in this regard. In QFT, by contrast, all dynamical variables are quantized except spacetime. Spacetime is treated as a given fixed background, which parametrizes all dynamical fields and provides the a priori causal structure needed for field quantization. Therefore, one encounters the stark paradox when joining GR and QFT together. There are various approaches trying to circumvent the paradox, differing on which features of GR and QFT are viewed as fundamental and unchangeable and which features as inessential and modifiable, as surveyed in Refs. 1–4. 2.3. Background-independent approach In conventional QFT, dynamics of dynamical fields takes place on the “stage” of a fixed spacetime with given metric. By analogy, dynamical fields are animals that roam around and chase one another upon the stage of an island of spacetime. GR, on the other hand, has taught us a very different paradigm of background independence, according to which spacetime is itself a dynamical entity

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(gravitational field) in many respects the same as other nongravitational entities. That said, physical entities, both gravitational and nongravitational, are not residing and moving in spacetime but, rather, they reside and move on top of one another. In the words of Rovelli’s metaphor9 : “It is as if we had observed in the ocean many animals living on an island: animals on the island. Then we discover that the island itself is in fact a great whale. So the animals are no longer on the island, just animals on animals.” Among various theories of QG, LQG is special in the sense that it reveres the paradigm of background independence earnestly and aims to formulate a nonconventional QFT of GR in genuine conformity with the paradigm, essentially in a nonperturbative fashion, so as to reconcile the aforementioned paradox. This of course is a very challenging task and the first step is to begin with the “proper” mathematical framework, which turns out to be the canonical description of connection dynamics of GR in terms of Ashtekar connection. The shift from metric to Ashtekar connection opens the possibility of employing nonperturbative techniques in gauge theories (especially, lattice gauge theories) to QG.

3. Connection Theories of General Relativity Classical GR is usually presented as geometrodynamics, i.e. a dynamical theory of metrics, but it can also be recast as connection dynamics, i.e. a dynamical theory of connections. The merit of this reformulation is to cast GR in a language closer to that of gauge theories, for which quantization is better understood. In this section, we present a brief introduction to connection theories of GR, largely excerpted from Sec. 2 of Ref. 8, and address some remarks in Sec. 3.3. For introductory purpose, we will focus only on the gravitational part of the action and phase space. For inclusion of matter and the cosmological constant, see e.g. Ref. 24 (also see Sec. 7); for extension to supergravity and other spacetime dimensions, see e.g. Refs. 25–27 (also see Sec. 12.14).

3.1. Connection dynamics The most widely known formulation of classical GR is the dynamical theory of metric gµν , which is based on the Riemannian geometry and governed by the Einstein– Hilbert action
 1 S[g] = (2) d4 x |g| R, 16πG where R is the Ricci scalar and g is the determinant of gµν . Classical GR also admits many other formulations, which are equivalent to one another to various extent. In the Palatini formulation28,29 based on the Riemann–Cartan geometry, instead of gµν , the basic gravitational variables are taken to be the coframe field eµ I , which gives the orthonormal cotetrad (or vierbein), and the so(1, 3)-valued connection

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1-form field ωµ I J , which corresponds to the gauge group SO(3, 1) of local Lorentz transformation. The Palatini action is given by
1 S[e, ω] = IJKL eI ∧ eJ ∧ ΩKL , (3) 32πG where Ω := dω + ω ∧ ω

(4)

is the curvature of the connection ωµ I J . The coframe eµ I determines the spacetime metric via gµν = ηIJ eµ I eν J .

(5)

Meanwhile, the connection ωµ I J is completely determined by the coframe via de + ω ∧ e = 0,

(6)

which is the equation of motion obtained by varying the Palatini action with respect to ωµ . As far as equations of motion are concerned, the Palatini action Eq. (3), imposed by Eq. (6), reduces to the familiar Einstein–Hilbert action Eq. (2). Thus, these two actions lead to the same theory at least at the classical level. It is tedious yet straightforward to perform the Legendre transform on the Palatini action to obtain the Hamiltonian theory,30 but the resulting theory has certain second-class constraints. When the second-class constraints are solved, one is led to the standard Hamiltonian description of geometrodynamics (in terms of cotriads) and thus loses all reference to connection dynamics; meanwhile, the form of geometrodynamics is rather complicated and seems insurmountable for quantum theory. This problem can be avoided by modifying the Palatini action with Holst’s augmentation31 as

1 1 I J KL S[e, ω] = − (7) IJKL e ∧ e ∧ Ω eI ∧ eJ ∧ ΩIJ , 32πG 16πGγ where γ is an arbitrary but fixed number, called the Barbero–Immirzi parameter. The celebrated feature is that this augmentation does not change the equations of motion of Eq. (3). Inclusion of Holst’s modification renders it more elegant to perform the Legendre transform in the sense that all constraints are first class (as we will see shortly) and consequently makes the quantum theory founded on the Hamiltonian description more manageable as significance of connection dynamics is retained. (While different numerical values of γ yield equivalent theories at the classical level, it should be noted that, in the quantum theory, they give rise to different sectors that are not unitarily equivalent to one another as will be shown in Sec. 5.) In the non-abelian gauge theory, the Yang–Mills action can also be augmented with a topological term called the θ-term as

θ 1  F ∧ F + Tr Tr F ∧ F, (8) S= 2α2 8π 2

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where F is the field strength (curvature) of the Yang–Mills connection, is the Hodge dual, and α and θ are known as the coupling constant and the θ-angle respectively.a It is noticeable that Eq. (7) bears close resemblance to Eq. (8). Whether this intriguing resemblance carries any profound meaning arouses a lot of curiosity. LQG, in particular, exploits the resemblance to the extent that we can adopt the nonperturbative techniques (such as the quantization scheme in terms of Wilson loops) well developed in the context of gauge theories. 3.2. Canonical (Hamiltonian) formulation We have reformulated the classical theory of GR in a fashion very close to nonabelian gauge theories. Prior to quantizing the theory, we have to take a further step to translate the covariant description into the canonical (or Hamiltonian) description in order to apply the standard scheme of equal-time quantization used in QFT. First, we apply the ADM foliation, which foliates the four-dimensional spacetime manifold M into a family of spacelike surfaces (called “leaves”) Σt , labeled by a time coordinate t and with spatial coordinates on each slice given by xi . Under the ADM foliation, variables on the spacetime manifold are split into 3+1 decomposition. The coframe (cotetrad) eµ I is split into the cotriad eia plus the lapse function N and the shift vector field N a . The lapse and shift describe how each of the leaves Σt are welded together in the foliation.b Correspondingly, the four-dimensional metric gµν of M is split into the three-dimensional metric qab of Σ plus N and N a . Similarly, apart from (ω i · t) := − 21 ijk ωjk · t, the so(3, 1)-valued connection 1-form ωµIJ on M is naturally decomposed into two su(2)-valued 1-forms on Σ: Γ = Γia τi dxa and K = Kai τi dxa , where τi = τ i = σi /(2i) are the generators of SU (2) with σi being the Pauli matrices. As a consequence of Eq. (6), Γia is the su(2) ∼ = so(3) spin connection on Σt associated with the cotriad eia , i.e. dei + i jk Γi ∧ ek = 0,

(9)

and Kai is the extrinsic curvature of Σt imbedded in M. The 3+1 splitting is summarized in Table 1. Performing the Legendre transformation on the Palatini–Holst action Eq. (7), we obtain

 a  1 ˜ Lt Ai − h(E˜ a , Ai , N, N a , (ω i · t)) , dt d3 x E (10) S= a a i i 8πGγ Σt where L is the Lie derivative, and the Hamiltonian density h is given by h = (ω i · t)Gi + N a Ca + N C.

(11)

to the Barbero–Immirze parameter γ, different numerical values of θ yield equivalent theories at the classical level but give rise to inequivalent θ-sectors in the quantum theory. b More precisely, let nα be the unit vector normal to the slice Σ , we have tα = N nα + N α with t N α nα = 0. a Analogous

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II-474 Table 1.

Decomposition under the ADM foliation.

M µ, ν, . . . I, J, . . . = (0, 1, 2, 3) so(3, 1)

→ → → →

Σt a, b, . . . i, j, . . . = (1, 2, 3) so(3) ∼ = su(2)

eµ I gµν ωµ I J deI + ω I J ∧ eJ = 0

→ → → →

eia , N, N a qab , N, N a Γia , Kai , (ω i · t) dei + i jk Γi ∧ ek = 0

b

˜ j ) is given by the Ashtekar connection (1-form) The canonical pair (Aia , E Aia (x) := Γia (x) + γKai (x)

(12)

and the densitized inverse triad (vector density of weight 1) a 1 E˜ i := ejb ekc abc ijk , 2

(13)

which gives the three-metric qab via ˜ a E˜ b δ ij q q ab = E i j

(14)

˜ a. q ≡ |det qab | = det E i

(15)

with

The Hamiltonian density h is given by three (local) constraints Gi = 0 (Gauss constraint), Ca = 0 (diffeomorphism constraint), and C = 0 (scalar constraint) associated with the Lagrange multipliers (ω i · t), N a , and N , respectively. The three constraints are given by a

a

a

˜ := ∂a E ˜ + ij k Aj E ˜ Gi = Da E a k, i i 2 ˜ b F i − σ − γ K i Gi , Ca = E i ab a σγ   γ a b ij ˜  k F k + (σ − γ 2 )2K i K j C =  E˜ i E ab j [a b] 2 |q|  a  ˜ E 2 + 8πG(γ − σ)∂a  i Gi , |q|

(16a) (16b)

(16c)

k where Fab is the curvature of the connection Aia and σ = −1.c

we considered the Euclidean GR instead of the Lorentzain GR, the Lorentz group SO(3, 1) would have been replaced by the Euclidean rotational group SO(4). Correspondingly, σ = +1 and Aia := Γia − σγKai = Γia − γKai for the Euclidean case. c Had

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˜ a , which satisfy the canonical relation The fundamental variables are Aia and E i b {Aia (x), E˜ j (y)} = 8πGγ δji δab δ 3 (x − y),

and all three constraints can be expressed in terms of Aia and su(2)-valued field λ = λi τi on Σ, the functional
1 CG [λ] := d3 xλi Gi 8πGγ Σ

(17) ˜ ai .d E

For any smooth

generates precisely the infinitesimal internal SU (2) gauge rotations by λ i

a

˜a. Aa , CG [λ] = −Da λi and E˜ i , CG [λ] = ij k λj E k

(18)

(19)

Consequently, the Gauss constraint Gi = 0 is the “Gauss law” that ensures the gauge invariance under the internal SU (2) rotations. Furthermore, removing from C a and C a suitable multiple of the Gauss constraint, for any smooth vector field N a and scalar field N , we can show that the functional
 1 ˜ i F i − (N a Ai )Gi CDiff [N] := (20) d3 x N a E a b ab 8πGγ Σ generates the infinitesimal 3d diffeomorphism (Diff) on Σ along the direction N i

a

˜ i , CDiff [N] = LN E ˜ ai , Aa , CDiff [N] = LN Aia and E (21) and the functional 1 C[N ] := 16πG




˜ aE ˜b E i j ij j k i d xN  + (σ − γ 2 )2K[a Kb]  k Fab |q| Σ 3

generates the infinitesimal translation in time along the direct N nα : i

a

˜ i , C[N ] = LN nα E ˜ ai , Aa , C[N ] = LN nα Aia and E

(22)

(23)

where L is the Lie derivative.e Altogether, the three constraints ensure invariance under the SU (2) gauge rotation, the diffeomorphism on Σ, and the translation in time. The three constraints are first class in Dirac’s terminology; that is, the Poisson bracket of any two of the constraints vanishes in the restricted phase space imposed by the three constraints. Explicitly, the Poisson brackets between the constraints, which give the so-called “constraint algebra,” read asf

CG [λ], CG [λ
] = CG [[λ, λ
]], (24a) ˜a. that Kai = (Γia − Aia )/(σγ) and Γia is a non-polynomial function of E i a i ˜ should be noted that Eqs. (21) and (23) are valid only if Aa and E i already satisfy the Gauss constraint, because we have removed a multiple of the Gaussian constraint from C a and C. f Note that the constraint algebra Eq. (24) is not closed in the usual sense, as the arguments of ˜a. CDiff and CG on the right-hand side of Eq. (24f) involve the phase space variables Aa and E d Note e It

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CG [λ], CDiff [N] =

CG [λ], C[N ] =

CDiff [N], CDiff [N
] =

CDiff [N], C[M ] =

C[N ], C[M ] =

−CG [LN λ],

(24b)

0,

(24c)

CDiff [[N, N
]] ≡ C[LN N
] ≡ −C[LN
N],

(24d)

C[LN M ],  σ CDiff [S] + CG [S a Aa ]

(24e)

+

 σ − γ2 a ˜ b ∂b M ] , CG |q|−1 [E˜ ∂a N, E 2 (8πGγ)

(24f)

where in the last equation the vector field S a is given by b

S a = (N ∂b M − M ∂b N )

a

˜ E ˜ δ ij E i j ≡ σ q ab (N ∂b M − M ∂b N ). |q|

(25)

While the three constraints have to be satisfied, the evolution of the canonical pair is dictated by the Hamilton’s equations ∂Aia = {Aia , H} and ∂t

a

˜i ∂E ˜ a , H}, = {E (26) i ∂t  where the (total) Hamiltonian is H := (8πGγ)−1 Σ d3 x h.g Because all three con˜ b ), straints are first class, once all of them are satisfied for an initial state of (Ai , E a

j

they will continue to be satisfied under the evolution of Eq. (26). The two equations of motion in Eq. (26) together with the three constraints Gi = 0, Ca = 0, C = 0 are completely equivalent to the Einstein field equation. 3.3. Remarks on connection theories Historically, the Ashtekar connection was not obtained by starting from the Palatini–Holst action. Rather, the original idea32 is to replace the Lorentz connection ωµ IJ in the Palatini theory with the complex Lorentz connection Aµ IJ defined as   1 i Aµ IJ := ωµ IJ − IJ KL ωµ KL , (27) 2 2 which satisfies the self-dual condition: 12 IJ KL Aµ IJ = iAµ IJ . Correspondingly, ΩKL in Eq. (3) is replaced by F IJ , which is the curvature of the connection AIJ and can also be viewed as the self-dual part of ΩIJ , i.e. F IJ := 12 ΩIJ − 14 IJ KL ΩKL . It turns out, in terms of the complex connection, all equations in the classical theory are simplified dramatically,33 as the new variable manifests the fact that so(3, 1)C is isomorphic to su(2)C ⊕ su(2)C (where gC denotes complexification of g). g This Hamiltonian is peculiar and sometimes referred to as “super-Hamiltonian” in the sense that it vanishes identically when the three constraints are imposed. The fact that the Hamiltonian vanishes is a characteristic of reparametrization-invariant theories.

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The essence of using the new variable is closely related to the constructions of the twistor theory34 and the H space theory.35 However, the complex connection takes values in the Lie algebra of a noncompact group, posing an obstacle to constructing the corresponding quantum theory. It is Holst31 who first realized that the imaginary number i in Eq. (27) can be replaced by any (real or complex) parameter γ and this prescription corresponds to adding the second term in Eq. (7). Since then, most progress toward the quantum theory has been made with real γ. Taking γ = ±i in the Lorentzian GR (σ = −1) and γ = ±1 in the Euclidean GR (σ = +1), one encounters the special cases, in which the (classical) theory has a richer geometrical structure — particularly, the three constraints in Eq. (16) are drastically simplified (note σ − γ 2 = 0) and the Poisson brackets in Eq. (24) form a closed Lie algebra. Rigorously speaking, the formulation we adopted is not a theory of “connection” dynamics but of “coframe-connection” dynamics, as we also include the coframe eµ I in addition to the connection ωµ IJ as the fundamental variables. Connection and coframe are the gauge variables in the Poincar´e gauge theory of gravity (see Ref. 36 for a review). If one postulates a Lagrangian quadratic in the gauge variables for the Poincar´e gauge theory, the most general Lagrangian is given in Eq. (5.13) of Ref. 36
1 S= (a0 R + b0 X − 2Λ)η 16πG + 12 more terms quadratic in curvature, torsion, and X,

(28)

1 where R := Rαβ βα is the Ricci scalar, X := 4! ηαβγδ Rαβγδ , Λ is the cosmological constant, and η is the volume four-form. The Palatini–Holst action Eq. (7) only takes the first line of Eq. (28) with a0 = 1, b0 = γ −1 (and usually Λ = 0).h From the standpoint of the Poincar´e gauge theory, it seems rather ad hoc to include Holst’s term, but neglect the second line completely, since Holst’s term is equivalent to a certain torsion square term via Nieh–Yan identity. Indeed, it would be theoretically more compelling if one could construct the quantum theory based on the generic Lagrangian Eq. (28), but inclusion of any terms in the second line unfortunately makes the constraints in the canonical description very complicated and no longer first class, rendering the Hamiltonian formulation inadequate for quantization. The fact that the Palatini–Holst action per se is privileged seems to suggest that, even with real γ, the su(2)C ⊕ su(2)C structure of the complex Lorentz algebra still plays

Lagrangian with the first two terms ∼a0 R + b0 X was first discussed by Hojman et al.,37 but the second term ∼b0 X is referred to as Holst’s modification in the literature of LQG, mainly because Holst was the first to relate it to the Ashtekar variable. Likewise, the coframe-connection formulation of the first term ∼a0 R is known as the Einstein–Cartan or the Einstein–Cartan– Sciama–Kibble theory in the literature of connection theories (see Ref. 36 for a historical account) but often referred to as the Palatini formulation in LQG, partly because Palatini’s particular idea was underscored in Ashtekar’s seminal paper Ref. 32. Although the eponyms adopted in LQG are not historically faithful, we adhere to them in conformity with the LQG convention. h The

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a profound role in QG (as advocated by the twistor theory and the H space theory), although its direct relevance remains obscure.i Finally, it should be noted that in the case of vacuum the Barbero–Immirzi parameter γ does not appear in the equations of motion and thus have no physical effect at the classical level, but this is not true in general. In the presence of minimally coupled fermions, the parameter appears in the equations of motion, giving rise to a four-fermion interaction.38 LQG nevertheless takes the Palatini–Holst action (with or without matter) as the starting point, adopting the attitude that the four-fermion interaction is very likely to be real on account of the fact that the effect of gravity on fermions is very difficult to measure. 4. Quantum Kinematics Departing from the canonical description of GR described in Sec. 3, it is time to construct the quantum theory. 4.1. Quantization scheme Following the standard strategy for quantization in gauge theories, as A = Aia τi dxa ˜ = E ˜ ai τ i ∂a is analogous to the is considered as the connection potential and E SU (2) electric field (note τ i = δ ij τj = τi ), one would take functionals of A as the kinematical quantum states to start with. However, this straight approach does not lead us far enough. In LQG, instead of the connection field A(x), we consider the holonomy (i.e. Wilson line) hγ defined as the path-ordered integral
hγ := P exp Aia τi dxa (29) γ

over an oriented one-dimensional curve γ in Σ. Correspondingly, instead of the ˜ we consider the surface integral electric field E,
c abc E˜ i f i dxa ∧ dxb (30) ES,f := S

over a two-dimensional surface S in Σ and with a smooth smearing su(2)-valued function f (x) = f (x)i τi . The Poisson bracket {hγ , ES,f } can be straightforwardly computed by Eq. (17). The bracket vanishes if γ and S do not intersect or γ lies within S. Furthermore, {hγ } and {ES,f } are closed under the Poisson bracket and form the loop algebra. The reason why we should take hγ instead of A(x) as the fundamental variable is essentially because hγ transforms more “nicely” than A does under both the SU (2) and the 3d diffeomorphism transformations, and consequently it is much more manageable to remove the gauge overcounting in the quantum theory. i Accordingly, LQG is most naturally constructed for four-dimensional spacetime. Nevertheless, generalization to other dimensions is still possible, depending on what principles of LQG are to be upheld. (See Sec. 12.14.)

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Under the SU (2) gauge transformation, A(x) is transformed via A(x) → A
(x) = Λ(x)A(x)Λ† (x) + Λ(x)d Λ(x)† ,

(31)

while hγ is transformed via hγ → h
γ = Λ(xγf )hγ Λ(xγi )† , xγi

(32)

xγf

where and are the initial and final endpoints of γ. While Eq. (31) involves every point x for A, Eq. (32) involves only the endpoints of γ for hγ . On the other hand, consider a 3d diffeomorphism map ϕ : Σ → Σ that is smooth and invertible everywhere. Under ϕ, the connection A transforms as a 1-form A → A
= ϕ∗ A,

(33)

∗

where ϕ is the pullback of ϕ. It follows that hγ (A) → hγ (A
) = hγ (ϕ∗ A) ≡ hϕγ (A),

(34)

where ϕγ ≡ ϕ(γ) is the image of γ by ϕ. Compared with Eq. (33) for A, the transformation Eq. (34) for hγ is more manageable. We will first construct the kinematical Hilbert space H using the functions of holonomies, and then implement the three constraints, one after another, to obtain the SU (2)-invariant Hilbert space HG , the SU (2)- and Diff-invariant Hilbert space Hinv , and finally the physical Hilbert space Hphys . The scheme can be summarized as H −→ HG −→ Hinv −→ Hphys . SU(2)

Diff

C

(35)

The states of H will be called cylindrical functions; the states of HG called spin networks; and the states of Hinv called s-knots. Finally, the physical Hilbert space Hphys is supposed to reveal the quantum dynamics. 4.2. Cylindrical functions Let Γ be an oriented graph, which is defined as a collection of a finite number of oriented and (piecewise) smooth edges γl with l = 1, . . . , L embedded in Σ. Consider a smooth function f of L SU (2) elements. The couple (Γ, f ) defines a functional of A as ΨΓ,f [A] ≡ A | ΨΓ,f  := f (hγ1 (A), . . . , hγL (A)).

(36)

These functionals (of A) are called cylindrical functions (of holonomies). Let Cyl denote the linear space of cylindrical functions for all Γ and f . With a suitable topology (the detail is not important here), Cyl is dense in the space of all continuous functionals of A; in this sense, cylindrical functions grasp all information of continuous functionals of A. If two cylindrical functions are defined for the same graph Γ, we can define the inner product between them as
∗ dµ1 · · · dµL f (hγ1 , . . . , hγL ) g(hγ1 , . . . , hγL ), (37) ΨΓ,f | ΨΓ,g  := SU(2)L

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where dµl is the Haar measure on SU (2). If two cylindrical functions are defined by two different couples (Γ
, f
) and (Γ

, g

), let Γ = {γ1 , . . . , γL } = {γ1
, . . . , γL

} ∪ {γ1

, . . . , γL



} be the union of the two graphs Γ
= {γ1
, . . . , γL

} and Γ

= {γ1

, . . . , γL



}. We can extend the functions f
and g

to be defined in Γ in the obvious way as f (hγ1 , . . . , hγL ) := f
(hγ1
, . . . , hγL

) and g(hγ1 , . . . , hγL ) := g

(hγ1

, . . . , hγL



). The inner product Eq. (37) can then be extended to any two given cylindrical functions as ΨΓ
, f
| ΨΓ

, g

 := ΨΓ,f | ΨΓ,g .

(38)

This implements the inner product measure to the space Cyl. We can then define the kinematical Hilbert space H as the Cauchy completion of Cyl with respect to the norm of the inner product Eq. (38), and the dual space Cyl∗ as the completion of Cyl with respect to the weak topology defined by Eq. (38).j This complete the Gelfand triple Cyl ⊂ H ⊂ Cyl∗ . (See Refs. 39–41 for the rigorous construction.) The cylindrical functions with support on a given graph Γ form a finite˜ Γ = L2 (SU (2)L , dµL ) with L being ˜ Γ of H. Obviously, H dimensional subspace H ˜ Γ is a subspace of the the number of edges in Γ. If Γ ⊂ Γ
, the Hilbert space H ˜ Hilbert space HΓ
. This nested structure is called a projective family of Hilbert spaces and H is (and can be defined as) the projective limit of this family.40,41 It turns out H can be viewed as the space of square integrable functionals in the Ashtekar–Lewandowski measure; i.e. H = L2 [A, dµAL ], where A is an extension of the space of smooth connections.39 The Peter–Weyl theorem states that a basis for the Hilbert space of L2 functions on SU (2) is given by the matrix elements of the irreducible representations of the group. The irreducible representation of SU (2) is labeled by an half-integer number j and the matrix elements of the j representation R(j) are denoted as R(j)α β (U ) ∈ C ˜ Γ is composed of the states for U ∈ SU (2). Then a basis of H |Γ, jl , αl , βl  ≡ |Γ, j1 , . . . , jL , αl , . . . , αL , βl , . . . , βL ,

(39)

which are defined via A | Γ, jl , αl , βl  = R(j1 )α1 β1 (hγ1 (A)) · · · R(jL )αL βL (hγL (A)).

(40)

˜Γ For each graph Γ, the proper subspace HΓ associated with Γ is the subspace of H spanned by |Γ, jl , αl , βl  with jl = 0. The proper subspaces HΓ are orthogonal to one another and span the whole H; i.e.  HΓ . (41) H∼ = Γ

The “null” graph Γ = ∅ is included in the sum. It corresponds to the onedimensional Hilbert space spanned by the trivial state |∅ defined as A | ∅ = 1. precisely, H is the space of the Cauchy sequences |Ψn  in Cyl (i.e. Ψm − Ψn  converges to zero); Cyl∗ is the space of the sequences |Ψn  such that Ψn | Ψ converges for all |Ψ ∈ Cyl.

j More

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An orthonormal basis of H is simply composed of the states |Γ, jl , αl , βl  with all graphs Γ (including ∅) and all spin labels jl = 1/2, 1, 3/2, 2, . . . (but jl = 0). Among the basis states, an important special case is that the graph is given by a closed curve (or a “loop”) α. As α has only one edge and no endpoints, the corresponding state |Γ, jl , αl , βl  becomes |α, j, which is defined as    (42) A | α, j = Tr R(j) P exp A . α

These states |α, j are called loop states. For j = 1/2 particularly, A | α, j = Tr hα (A) is the Wilson loop. Similarly, for a “multiloop” [αl ] = (α1 , . . . , αn ) given by a collection of a finite number of loops (without intersection), a multiloop state |[αl ], jl  is defined as    n  Tr R(jl ) P exp A . (43) A | [αl ], jl  = l=1

αl

The kinematical Hilbert space H carries a natural representation for both SU (2) and Diff groups. Under the SU (2) gauge transformation, the holonomy hγ transforms by Eq. (32) and it follows that the basis states |Γ, jl , αj , βl  transform by ˆΛ |Γ, jl , αj , βl  |Γ, jl , αj , βl  → U  L   γl † γl (jl )αl
(jl )βl


= R βl (Λ(xi )) |Γ, jl , αl , βl . αl (Λ(xf ) )R α
l ,βl


l=1

(44) On the other hand, under the Diff transformation, Eq. (34) leads to ˆϕ |ΨΓ,f  = |ΨϕΓ,f . |ΨΓ,f  → U

(45)

That is, a cylindrical function ΨΓ,f [A] with support on Γ is sent to a new cylindrical function ΨϕΓ,f [A] with support on the relocated graph ϕΓ. A moment of reflection tells that the inner product is Diff-invariant ˆϕ ΦΓ
,g  = ΨϕΓ,f | ΦϕΓ
,g  = ΨΓ,f | ΦΓ
,g . ˆϕ ΨΓ,f | U U

(46)

In fact, the uniqueness theorem tells that the Ashtekar–Lewandowski measure is the only unique measure that gives rise to the (∗ -algebra) representation of the kinematical algebra invariant under spatial diffeomorphisms.42 4.3. Spin networks We are now ready to implement the Gauss constraint upon H to obtain the SU (2)invariant space HG . In the space of functionals Ψ[A], we promote Aia (x) to Aˆia (x) as the multiplicative operator: (Aˆia (x)Ψ)[A] := Aia (x)Ψ[A],

(47)

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˜ a as the differential operator and E i ˜ a Ψ)[A] := −i8πG
γ (Eˆ i

δΨ[A] , δAia (x)

(48)

in accordance with Eq. (17). The Gauss constraint CG [λ] = 0 in Eq. (18) is then promoted to the operator CˆG [λ]: (CˆG [λ]Ψ)[A] = Ψ[A − Dλ],

(49)

according to Eq. (19). The SU (2)-invariant states are those lying in the kernel of CˆG [λ] for arbitrary λ. In the quantum theory based on cylindrical functions of connections, in place of functionals of connections, it is the finite gauge transformation that is of primary importance. Accordingly, the space HG as the kernel of CˆG [λ] is obtained as the ˆΛ , which act on |Ψ as invariant subspace of H under finite transformations U ˆΛ Ψ)[A] = Ψ[ΛA Λ† + ΛdΛ† ] (U

(50)

in accordance with Eq. (31). For a given graph Γ, call the endpoints of the edges “nodes” and assume that, without loss of generality, the edges overlap, if they do at all, only at nodes. The oriented edges are then also referred to as “links.” Under a finite SU (2) ˆΛ , the basis states transform according to Eq. (44), which is transformation U invariant in the subspace HΓ and acts only on the nodes of Γ. At each node vn , assume there are nin “ingoing” links and nout “outgoing” links. The task is to find the SU (2)-invariant (i.e. j = 0) spin states within the tensor product j1 ⊗ · · · jnin ⊗ j¯
1 ⊗ · · · j¯
nout = 0 ⊕ 0 ⊕ · · · at each node vn . This can be achieved by finding appropriate linear superpositions  

in m1 ··· mnout m1 ··· mnin |j1 , m1  ⊗ · · · ⊗ |jnin , mnin  |J = 0, M = 0 = mi ,m
j

⊗ (|j1
, −m
1  ⊗ · · · ⊗ |jn
in , −m
nout ),

(51)

···

where in ··· are called intertwiners and regarded as generalized Clebsch–Gordan coefficients, which can be obtained by the standard method of the recoupling theory43 (also see Appendix A of Ref. 9). With every node specified with an intertwiner denoted as in , an SU (2)-invariant state |Γ, jl , in  is given by  N    (n) (n) β1 ···βn out (n) in (52) |Γ, jl , αl , βl , |Γ, jl , in  = (n) α ···α αl ,βl

1

n=1 (n)

nin

where N is the number of nodes, αi (i = 1, . . . , nout ) is one of αl that is “ingoing” (n) to the node vn , and βi (i = 1, . . . , nin ) is one of βl that is “outgoing” from the node vn . See Fig. 1 for a simple example. The states |Γ, jl , in  are called spin networks, and choice of jl and in is called the “coloring” of the links and the nodes. Note that

May 10, 2017 11:10

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Fig. 1. The left diagram represents the state |Γ, jl , αl , βl  while the right one represents the spin networks |Γ, jl , in , both for the same simple graph Γ composed of three oriented links and two nodes. The state on the right is a linear superposition of states on the left via the intertwiners.

the loop states |α, j and multiloop states |[αl ], jl  are trivial spin networks without nodes. It is clear that the SU (2)-invariant states of HΓ form a proper subspace of HΓ , which inherits the same inner product structure of HΓ . Consequently, we have the SU (2)-invariant Hilbert space HG as the SU (2)-invariant subspace of H and the corresponding Gelfand triple CylG ⊂ HG ⊂ Cyl∗G in the obvious manner. See Ref. 44 for the original idea of spin networks and Refs. 45 and 46 for the systematic implementation. 4.4. S-knots The next step to implement the second constraint CDiff (N) = 0 for the far more crucial invariance: the 3d diffeomorphism invariance. (We will follow the lines of Sec. 6.2 in Ref. 8.) To solve the constraint, we adopt the “group averaging procedure”: the invariant states are obtained by averaging over the elements of Cyl that are transitive to one another by the invariance group.47,48 It turns out the Diff-invariant states are not in H but in the extended space Cyl∗ . Given a graph Γ, denote by Diff Γ the subgroup of Diff that maps Γ to itself, and by TDiff Γ the subgroup of Diff Γ that preserves every edge of Γ and its orientation. It is easy to see that the induced action of TDiff Γ on CylΓ is trivial. The quotient GSΓ =

Diff Γ TDiff Γ

(53)

is the group of graph symmetries of Γ, which permutates the ordering and/or flips the orientations of the edges of Γ. GSΓ is a finite and discrete group and it induces  Γ on CylΓ in the obvious way. a nontrivial action GS We construct the general solutions to the diffeomorphism constraint in two steps. First, given a state |ΨΓ  ∈ HΓ , we average it over the group of GS and obtain a Γ projection map PˆDiff,Γ from HΓ to its subspace that is invariant under GS 1  ˆ PˆDiff,Γ |ΨΓ  := Uϕ |ΨΓ , NΓ ϕ∈GSΓ

(54)

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ˆϕ is defined in Eq. (45). The state PˆDiff,Γ |ΨΓ  where NΓ is the number of GSΓ and U
is GSΓ -invariant, as for any ϕ ∈ GSΓ we have   ˆϕ
PˆDiff,Γ |ΨΓ  = 1 ˆϕ U ˆϕ |ΨΓ  = ˆϕ
◦ϕ |ΨΓ  U U U NΓ ϕ∈GSΓ

=



ϕ∈GSΓ

ˆϕ

|ΨΓ  = PˆDiff,Γ |ΨΓ . U

(55)

ϕ
−1 ◦ϕ

∈GSΓ

The map PˆDiff,Γ naturally extends to the projection PˆDiff , which projects H to its  Γ for all Γ. subspace that is invariant under GS Next, we average over the remaining diffeomorphisms that move the graph Γ. This is a very huge group and the resulting group-averaged states are genuinely “distributions” and belong to Cyl∗ instead of H. For any given state |ΨΓ  ∈ HΓ , we can define the group-averaged state (η(ΨΓ )| ∈ Cyl∗ by its linear action on arbitrary cylindrical functions |ΦΓ
 ∈ Cyl as  ˆϕ PˆDiff,Γ ΨΓ | ΦΓ
, (η(ΨΓ )|ΦΓ
 := U (56) ϕ∈Diff/Diff Γ

where · | · is the inner product of H. This action is well defined because only a finite number of terms in the sum are nonzero when Γ
= ϕΓ for some ϕ. It follows from Eq. (46) that the state (η(Ψγ )| is invariant under the action of Diff: ˆϕ ΦΓ
 = (η(ΨΓ )|ΦΓ
 (η(ΨΓ )|U

(57)

for all ϕ ∈ Diff. The space of these solutions to the diffeomorphism constraint is denoted by Cyl∗Diff , and we have constructed a map η : Cyl → Cyl∗Diff ,

(58)

which sends every element of Cyl, upon group averaging, to a Diff-invariant state of Cyl∗Diff . It should be noted that η is not a projection as it maps Cyl onto a different space Cyl∗Diff . Nevertheless, the group averaging procedure naturally endows Cyl∗Diff with the inner product (η(Ψ) | η(Φ)) := (η(Ψ)|Φ,

(59)

which is well defined as the right-hand side is independent of the specific choice of Ψ and Φ for the same states (η(Ψ)| and (η(Φ)|. With respect to this inner product, the Hilbert space HDiff of Diff-invariant states is the Cauchy completion of Cyl∗Diff . Finally, we can obtain the general solutions to both the Gauss and the diffeomorphism constraints by simply restricting the starting state |Ψ ∈ Cyl to be SU (2) invariant, i.e. we start with |Ψ ∈ Cyl ∩ HG . The space Cyl∗inv of solutions to both the Gauss and diffeomorphism constraints is then given by Cyl∗inv = η(Cyl ∩ HG ),

(60)

and the Hilbert space Hinv of SU (2)- and Diff-invariant states is the Cauchy completion of Cyl∗inv .

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What happens to operators? For a given (SU (2) and) diffeomorphism invariˆ acting on H (i.e. O ˆ commutes with U ˆϕ ; we will see examples of ant operator O diffeomorphism invariant operators in Sec. 5.4), one can define the corresponding ˆ ∗ acting on Cyl∗ as operator O Diff ˆ ∗ η(Ψ)|Φ := (η(Ψ)|O ˆ † Φ, (O ˆ∗

(61) Cyl∗Diff

Cyl∗Diff ,

which is well defined in the sense that O maps from into since ˆϕ Φ = (O ˆ ∗ η(Ψ)|Φ provided that O ˆ† ˆ ∗ η(Ψ)|U one can show that, by Eq. (57), (O ∗ ˆ ˆ commutes with Uϕ . Furthermore, the operator O is hermitian in HDiff with respect ˆ is hermitian in H. to the inner product Eq. (59) if and only if O The fact that the (SU (2) and) diffeomorphism invariant Hilbert space can be rigorously constructed is very significant, in contrast to the quantization program of geometrodynamics using metrics as fundamental variables, in which the precise construction remains elusive. To understand the structure of Hinv , we consider the action of η on the basis states of spin networks |S ≡ |Γ, jl , in  of HG . By Eq. (56), we have (η(S
) | η(S)) ≡ (η(S
)|S  0     = ˆg |S S
|U k   gk ∈GSΓ

if ϕΓ = Γ
for all ϕ ∈ Diff, if ϕΓ = Γ
for some ϕ ∈ Diff,

(62)

 Γ , and the state U ˆg |S is obtained from |S by changing the orderˆg ∈ GS where U k k ing and/or the orientations of the edges of Γ. As equivalence classes of unoriented graphs under diffeomorphisms are called “knots” and classified by their knotting structures of edges, the first line of Eq. (62) tells that two spin networks |S and |S
 in HG give rise to two orthogonal states (η(S)| and (η(S
)| in Hinv , unless Γ and Γ
are knotted in the same way. The states in Hinv are then distinguished by the knots, denoted as K, as well the colorings of links and nodes of K. The states with the same K but different colorings, however, are not necessarily orthonormal to one another, due to the nontrivial action of the discrete symmetry group GSΓ in the second line of Eq. (62). To obtain an orthonormal basis of Hinv , we have to further diagonalize the quadratic form defined by the second line of Eq. (62). Denote (K, c| the resulting states, where the discrete label c corresponds to the coloring of links and nodes of Γ up to the complications caused by the discrete graph symmetry group GSΓ . We call the states (K, c| s-knots and c the coloring of the knot K. It should be noted that diagonalizing the quadratic form in Eq. (62) may yield degenerate eigenstates, as the projection map PˆDiff,Γ defined in Eq. (54) might have a nontrivial kernel. As a result, the coloring c for s-knots (K, c| in general has less choice than the coloring jl , in for spin networks |Γ, jl , in . Knots without nodes have been widely studied in the knot theory, while knots with nodes have also been studied but to a lesser extent.49,50 One peculiarity of knots with nodes is that there are graphs with four-valent or higher valent nodes

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that cannot be mapped to one another by smooth diffeomorphisms.k Consequently, the knot classes are labeled by continuous parameters (moduli) and therefore the space Hinv remains nonseparable as the kinematical Hilbert space H.44,51 This problem can be resolved by extending the group Diff of smooth diffeomorphisms to the group Diff ∗ of extended diffeomorphisms.52 Extended diffeomorphisms are maps from Σ to Σ that are smooth and invertible everywhere in Σ except at a finite number of isolated points. With Diff replaced by Diff ∗ , the knot classes are discretely classified and consequently the Hilbert space Hinv is separable. The “excessive size” of H in terms of nonseparability turns out to be just a gauge artifact. It is argued in Ref. 52 (also see Sec. 6.7 of Ref. 9) that Diff ∗ in place of Diff is in fact more natural as the quantum theory is concerned, for diffeomorphisms in Diff are too “rigid” in the sense that they leave invariant the linear structures of tangent spaces at nodes, which however have no direct physical significance. The physical interpretation of the s-knots states will be clear after we define the operators. 5. Operators and Quantum Geometry The two fundamental variables in the canonical theory is the connection Aia and ˜ a . In the space of functionals of A, the corresponding its conjugate momentum E i ˜ a are given by Eqs. (47) and (48), respectively. These two operators Aˆia and Eˆ i

operators are however not good operators in the space Cyl of cylindrical functions. We will follow the lines of Sec. 6.6 of Ref. 9 to construct the appropriate quantum operators. 5.1. Holonomy operator In Cyl, the operators of connections have to be replaced by the operators of holonomies defined in Eq. (29). Let (hγ )A B be the matrix elements of the holonˆ γ )A acting on |Ψ ∈ Cyl is simply omy hγ . Then the corresponding operator (h B the multiplicative operator defined as ˆ γ )A Ψ)[A] := (hγ )A (A)Ψ[A]. ((h B B

(63)

The right-hand side is clearly in Cyl. In fact, any cylindrical function is immediately well defined as a multiplicative operator in Cyl. For example, the operator Sˆ associated with the spin network |S acts on another spin network |S
 as ˆ
 = |S ∪ S
, S|S

(64)

3d diffeomorphism ϕ induces a linear map Jp (pushforward) from the tangent space of a given point p to the tangent space of ϕ(p). In order for ϕ to send two nodes into each other, we must have Jp vi = vi , where vi , i = 1, . . . , n, are the n tangent vectors of the links at p and vi , i = 1, . . . , n, are the n tangent vectors of the links at ϕ(p). In general, however, there is no pushforward map sending n given vectors into n other given directions when n ≥ 4, as the rank of Jp is only 3. k Any

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where |S ∪ S
 is the spin network formed by gluing S and S
together in the obvious way ˆ
 = A | SA | S
. A|S|S

(65)

ˆΛ ) and Furthermore, it is clear that Sˆ is SU (2) invariant (i.e. Sˆ commutes with U leaves the Hilbert space HG invariant. Spin networks |S can be constructed as ˆ the action of Sˆ acting on the trivial state |∅; i.e. S|∅ = |S. In this sense, |∅ is analogous to the Fock vacuum. 5.2. Area operator On the other hand, when acting on cylindrical functions, the operator defined in Eq. (48) leads to
i δ ˆ a ˜ E (x) hγ = hγ = ds γ˙ a (s) δ 3 (γ(s), x) (hγ1 τi hγ2 ) , (66) 8πG
γ i δAia (x) γ where s is an arbitrary parametrization of the curve γ, γ˙ a (s) ≡ dγ a (s)/ds is the tangent to the curve at the point γ(s), and γ1 and γ2 are the two segments into which γ is separated by the point γ(s). The right-hand side is a two-dimensional Dirac distribution (δ 3 (· · ·) is integrated over one dimension) and thus does not ˆ˜ a is not a well-defined operator in H, we instead seek the belong to Cyl. As E i ˜ over a two-dimensional surface S. In accordance desired operator by smearing E with Eq. (30) (with f = 1), we define the operator
ˆi (S) := −8πiG
γ E dσ 1 dσ 2 na (σ) S

δ δAia (x(σ))

,

(67)

where σ = (σ 1 , σ 2 ) are the parametrizations of the surface S, and na (σ) = abc

∂xb (σ) ∂xc (σ) ∂σ 1 ∂σ a

(68)

is the one-form normal to the surface S at the point x(σ). When acting on ˆi (S) yields holonomies, by Eq. (66), E  ˆi (S)hγ = −8πiG
γ E ±hγ1p τi hγ2p , (69) p∈S∩γ

where p are intersection points (one, many, or none) between the curve γ and the surface S, γ1p and γ2p are the two segments of γ separated by the point p, and the sign ± is given by + if the curve γ pierces S at p “upwards” to the orientation of S and − if “downwards” to the orientation of S. The generalization to an arbitrary representation of the holonomy is obvious  ±hγ1p R(j) (τi )hγ2p . (70) Eˆi (S)R(j) (hγ ) = −8πiG
γ p∈S∩γ

ˆi (S) is a well-defined operator on H. Therefore, E

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ˆi (S) is not SU (2) invariant, and we cannot obtain an SU (2)The operator E invariant operator by simply contracting the index i as 2 (S) := δ ij E ˆi (S)Eˆj (S), E

(71)

ˆi (S) is complicated by the integral over S. because the SU (2) transformation of E Instead, we first partition S into N small surfaces SI , which become smaller and  smaller as N → ∞ (for each N , N I=1 SI = S), and then define the area operator associated with S as  2 (SI ), ˆ A(S) := lim (72) E N →∞

I

which corresponds to the classical area of S,
 ˜ a nb E ˜ b δ ij d2 σ. A(S) = na E i j

(73)

S

ˆ Let us study the action of A(S) on a spin network |Γ, jl , in , assuming that no spin network nodes lies on S. For sufficiently large N , none of SI will contain more than one intersection point with Γ and the sum over I becomes a sum over the intersection points p between S and Γ. Consequently, by Eq. (70), we have   ˆ A(S)|Γ, jl , in  = 8πG
γ jp (jp + 1) |Γ, jl , in , (74) p∈S∩Γ

where p are the intersection points between S and Γ, and jp is the color of the link ˆ that pierces S at p. Since the operator A(S) is diagonal on spin network states and ˆ its eigenvalues are real, the operator A(S) for an arbitrary S is well defined in HG and is hermitian. For a complete and rigorous construction of the area operator, see Refs. 10, 53 and 54. The general result (with the possibility that the spin network nodes lie on S) is given by   ˆ 2jpu (jpu + 1) + 2jpd (jpd + 1) − jpt (jpt + 1) |Γ, . . ., (75) A(S)|Γ, . . . = 4πG
γ p∈S∩Γ

jpu , jpd ,

jpt

and are the colors of the links that emerge upwards (u), downwards where (d), and tangentially (t) to the surface S, respectively. It is the key result of LQG that the spectrum of area is discrete. The smallest nonzero area eigenvalue is given by (with j = 1/2) √ √ (76) ∆ = 2 3 πG
γ ≡ 2 3 πγ2Pl , which is of the order of the Planck area 2Pl ≡ G
(assuming γ is of the order of unity). Intrinsic discreteness of space at the Planck scale has long been expected in QG. In the context of LQG, this discreteness is not postulated or imposed by hand but rather arises as a direct consequence of the quantization in the same sense that the energy spectrum of an harmonic oscillator or of an atom is quantized. Also note that different choices of the numerical value of γ give rise to nonequivalent quantum theories as the difference is reflected in the spectrum of the area operator.

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5.3. Volume operator ˆ The two operators Sˆ in Eq. (64) and A(S) in Eq. (72) are in principle sufficient to define the quantum theory. To better understand quantum states, we also define the volume operator, which plays a key role in the physical interpretation of quantum states. Consider a three-dimensional region R. The classical volume of R is given by 
1 3 ˜ bj E ˜ ck |. ˜ ai E V (R) = d x abc ijk |E (77) 3! R To construct the quantum counterpart, we first have to regularize the quantity b c 1 ijk ˜ a E i (x)E˜ j (x)E˜ k (x) by the “three-hand holonomy” defined as 3!  T abc (x; s, t, r) :=

1 ˜ a (r) ijk R(1) (hγxr )il E l 3! b

c

˜ m (s)R(1) (hγxt )kn E ˜ n (t), × R(1) (hγxs )jm E

(78)

where s, t and r are three points (close to but different from the point x), and γxx
are paths from x to x
. When s, t, and r are very close to x, T abc (x; s, t, r) approx2 (S), for a ˜ ai (x)E˜ bj (x)E˜ ck (x). As the “three-hand generalization” of E imates 1 ijk E 3!

given closed surface, we define


3 (S) := d2 σ d2 σ
d2 σ

na (σ)nb (σ
)nc (σ

) T abc (x; σ, σ
, σ

), E S

S

(79)

S

where x is a point in the interior of S (the exact position of x is irrelevant as we will always consider the limit of small S).  Partition the region R into small cubes RI (for each N , N n=I RI = R) such 3 that the coordinate volume of each cube is smaller than  as N → ∞. In the same spirit of Eq. (72), we can now define the volume operator associated with R as  1 3 (∂RI )| , ˆ V (R) := √ lim |E (80) 3! (N→0 →∞) I where ∂RI is the boundary surface of the cube RI . When the operator Vˆ (R) acts on a spin network state |Γ, . . ., the three surface integrals over ∂RI in Eq. (79) give three intersection points, as in the case of the area operator. For  small enough, the only cubes whose surfaces have at least three intersections with the spin network are those containing a node of the spin network. Therefore, the sum over cubes I reduces to the sum over the nodes n ∈ Γ ∩ R, and we have   1 ˆ In | |Γ, . . ., lim Vˆ (R)|Γ, . . . = √ |W (81) 3! (N→0 →∞) n∈Γ∩R where 3 (∂RI )|Γ, . . . = ˆ In |Γ, . . . ≡ E W n

 r,s,t∈Γ∩∂RI n r=s=t=r

T (r, s, t)|Γ, . . .

(82)

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3 (∂RI ) on |Γ, . . ., which is the sum over the triplets (r, s, t) of is the action of E n distinct intersections between the spin network and the boundary of the cube RIn containing the node n. For each triplet (r, s, t), the result of the action is denoted ˆ In does as T (s, t, r)|Γ, . . .. The key point is that, in the limit  → 0, the operator W not change the graph of the spin network, nor the coloring of the links. Its only possible action is on the intertwiners. Consequently, we have 
Vin in |Γ, jl , i1 · · · i
n · · · iN , (83) Vˆ (R)|Γ, jl , i1 · · · iN  = (16πG
γ)3/2 n∈Γ∩R i
n

where the coefficients Vin can be calculated by the recoupling theory. A complete and rigorous construction of the volume operator can be found in
Refs. 55 and 56. The detailed calculation of Vin in is presented in Refs. 57 and 58, where a list of eigenvalues is also given. It turns out that the node must be at least four-valent in order to have a nonvanishing volume.59 The volume operator is well defined and hermitian with a discrete spectrum of non-negative eigenvalues. Furthermore, we can choose a basis of intertwiners in
that diagonalize the matrices Vin in in Eq. (83) so that the resulting spin networks are eigenstates of the area and the volume operators simultaneously. We denote Vin the corresponding eigenvalues. In addition to area and volume operators, the length operator can also be defined and also yields a discrete spectrum,60 but it is far more complicated and less understood than the area and volume operators. 5.4. Quantum geometry The volume operator essentially has contributions only from the nodes of a spin network, while the area operator has contributions from the links. Therefore, each node represents a quantum of volume and each link represents a quantum of area. That a spin network |Γ, j1 · · · jL , i1 . . . , iN  can be interpreted as an ensemble of N quanta of volume, or N “chunks” of space, which are separated from one another by the adjacent surfaces of L quanta of area. Each chunk of space is located “around” the note n with a quantized volume Vin . Two chunks are regarded as adjacent to each other if the two corresponding notes are connected by  a link jl , which corresponds to the adjacent surface with a quantized area 8πG
γ jl (jl + 1). The graph Γ dictates the adjacency relation among the chunks of space. The physical picture is compelling that a spin network state determines a quanˆ tized 3d metric. However, it should be noted that both the area operator A(S) ˆ and the volume operator V (R) are not diffeomorphism invariant (i.e. they do ˆϕ ), as the specification of S or R relies on spatial coordinot commute with U l nates. Nevertheless, we can specify the surface and region intrinsically on the knot of the spin network itself: A “region” is simply as a collection of nodes l The

only exception is the total volume operator Vˆ ≡ Vˆ (Σ), which is diffeomorphism invariant.

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{n1 , n2 , . . .}K := {n1 , . . . |ni ∈ notes of K} of the knot K associated with a graph Γ; a “surface,” as the boundary of a region, is a collection of knot links each of which connects to only one of {n1 , n2 , . . .}K . In this way, the volume operator Vˆ ({n1 , . . .}K ) defined as   0 if [Γ] = K,    ˆ V ({n1 , . . .}K )|Γ, . . . = (84) Vinj if [Γ] = K,    {n1 ,...}

where [Γ] denotes the knot class represented by Γ, and the area operator ˆ A(∂{n 1 , . . .}K ) defined as   0 if [Γ] = K,     ˆ A(∂{n (85) 1 , . . .}K )|Γ, . . . = 8πG
γ jl (jl + 1) if [Γ] = K,    jl ∈∂{n1 ,...}K

are both diffeomorphism invariant. Thanks to diffeomorphism invariance, we can define the corresponding operators Vˆ ∗ ({n1 , . . .}K ) and Aˆ∗ (∂{n1 , . . .}K ) acting on Cyl∗Diff via Eq. (61). The resulting operators are well defined on s-knot states.m Furthermore, in principle, the eigenvalues of the area and the volume operators represent possible outcomes of the corresponding physical measurements.61 The situation is precisely the same as that in the classical theory. In classical GR, we distinguish between a three-metric qab and a three-geometry [q]; the latter is an equivalence class of the former modulo diffeomorphisms. The notion of geometry is diffeomorphism invariant while the notion of metric is not. Given a metric qab with coordinates xa , we can compute the area of S or the volume of R, if we define S by a map (σ 1 , σ 2 ) → xa (σ) and R by a map (σ 1 , σ 2 , σ 3 ) → xa (σ). However, it makes no sense to ask what the area of S or the volume of R is in a given geometry [q], because the coordinates have no significance in the geometry. Instead, given a geometry [q], we should specify surfaces or regions intrinsically on the geometry itself. For example, given the three-geometry of the solar system, the region and the surface of the earth are well defined, without any reference to coordinate localization, and it is meaningful to ask what the volume and area of the earth are. In this sense, a spin network can be regarded as a quantum (discretized) 3d metric, and an s-knot as a quantum 3d geometry. The physical interpretation of the s-knots is extremely appealing: They represent different quantized 3d geometries, each of which is an abstract aggregate of chunks of discrete space separated by discrete adjacent surfaces. S-knots are not quantum excitations in space; rather, they are excitations on top of one another, as any reference to localization of the chunks and surfaces is dismissed. Also note that the “empty” s-knot (∅| associated with the trivial spin network |∅ describes a space with no volume and no area at all. We therefore have brought off the m Here,

we have disregard the technicalities due to the discrete group GSΓ of graph symmetries.

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paradigm of background independence as advocated in Sec. 2.3. The manifestation of background independence will become even more prominent if we also include nongravitational matter fields as will be seen in Sec. 7.

6. Scalar Constraint and Quantum Dynamics We have constructed the SU (2) and diffeomorphism invariant kinematical Hilbert space Hinv by imposing the Gauss and diffeomorphism constraints. While the quantum kinematics is well understood, the crux of the problem in LQG lies in the scalar constraint, implementation of which is supposed to reveal the quantum dynamics. One might attempt to repeat the group averaging procedure for the scalar constraint as we did for the diffeomorphism constraint, but this strategy turns out very difficult because the finite transformations generated by the scalar constraint are poorly understood even at the classical level. Instead, we adopt the strategy: First, we regularize the classical expression of the scalar constraint; and second, we promote the regulated classical constraint to a quantum operator and then remove the regulator. (We will follow closely the lines of Sec. 6.3 in Ref. 8.) Since the scalar constraint is very intricate, its implementation is far less clean and complete than that of the other two constraints. Consequently, the quantum dynamics remains a challenging open problem in LQG. Readers are referred to Refs. 62–65 for more details.

6.1. Regulated classical scalar constraint The classical scalar constraint is given by Eq. (22). Had we considered the Euclidean GR and chosen γ = 1, we would have γ 2 = σ = 1 and the second term in Eq. (22) would vanish. Therefore, the first term has the interpretation of the scalar constraint for the Euclidean GR with γ = 1. Accordingly, we rewrite the full Lorentzian constraint, i.e. Eq. (22) with σ = −1, as C[N ] =

√ Eucl γC [N ] − 2(1 + γ 2 )T [N ],

(86)

where C Eucl[N ] :=

a b √
˜ E˜ i E γ j k d3 x  ij k Fab 16πGγ Σ ˜ det E

(87)

and T [N ] :=

1 16πG




˜ bj ˜ ai E E j i d3 x  K[a Kb] . ˜ Σ det E

(88)

One of the difficulties  to deal with C[N ] is that it involves nonpolynomial func˜ a through det E˜ and K i . Fortunately, the nonpolynomiality can be tions of E i a

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circumvented by Thiemann’s trick.62,63 First, by the canonical relation Eq. (17), we can rewrite the combination for the cotriad eia eia

˜bE ˜c E 1 j k ijk = abc   2 ˜ det E

(89)

as a manageable Poisson bracket eia (x) =

1 {Ai (x), V }, 4πGγ a

(90)

where V = V (Σ) is the total volume of Σ given by Eq. (77) with R = Σ. This allows us to express C Eucl [N ] as
 1 C Eucl[N ] = − d3 x N (x) abc Tr Fab (x){Ac (x), V } (91) 2 2 3/2 32π G γ Σ in a form more suitable for loop quantization. Second, we have 1 ¯ Kai = (92) {Ac (x), K}, 8πγ ¯ is defined as where K
a ¯ K := d3 x Kai E˜ i , (93) Σ

which can also be expressed as a Poisson bracket ¯ = 1 {C Eucl[1], V }. K γ 3/2

(94)

Therefore, T [N ] can be recast as
2 ¯ ¯ T [N ] = − d3 x N (x) abc Tr({Aa (x), K}{A b (x), K}{Ac (x), V }). (8πG)4 γ 3 Σ (95) We have expressed C Eucl[N ] and T [N ] in terms of A, F and V . The next step is to replace A and F with holonomies, which are the “right” variables to be used for loop quantization. For a small path γx,u of coordinate length  starting at x and tangent to u, the holonomy hγx,u takes the expansion hγx,u = 1 +  ua Aa (x) + O(2 ),

(96a)

a 2 −1 = 1 −  u Aa (x) + O( ). h−1 γx,u ≡ hγx,u

(96b)

Similarly, for a rectangular loop αx,uv with one vertex at x and two sides tangent to u and v, each of coordinate length , the holonomy hαx,uv takes the expansion hαx,uv = 1 + 2 ua v b Fab (x) + O(3 ),

(97a)

= 1 − 2 ua v b Fab (x) + O(3 ). h−1 αx,uv ≡ hα−1 x,uv

(97b)

Now, partition Σ into small cubic cells, edges of which are of coordinate length . Denote by s1 , s2 , and s3 the edges of an elementary cell
based at a vertex v
,

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and by β1 , β2 , and β3 the three oriented loops that are the boundaries of the three rectangular faces based at v
and orthogonal to s1 , s2 , and s3 , respectively (see Fig. 2 in Ref. 8). By Eqs. (96) and (97), we can then regulate Eq. (91) as  Eucl (N ), where the contribution from each cell is
C
C
Eucl(N ) =

N (v
)  Tr((hβI − hβ −1 )hs−1 {hsI , V }), I I 32π 2 G2 γ 3/2 I

and similarly regulate Eq. (95) as T
(N ) =






(98)

T
(N ), where

2N (v
)  IJK ¯ ¯  Tr(hs−1 {hsI , K}h {hsJ , K}h {hsK , V }). s−1 s−1 I J K (8πG)4 γ 3

(99)

IJK

−1 Note that we regulate {A, V } by h−1 γ {hγ , V } ≡ V − hγ V hγ instead of simply −1 ¯ ¯ by h−1 ¯ ¯ ¯ {hγ , V } and {A, K} γ {hγ , K} ≡ K − hγ Khγ instead of simply {hγ , K}, because we want to keep the scalar constraint to be SU (2) invariant after the regularization. The regularization can be more generic than the above prescription. Instead of the simple cubic partition, we can partition Σ into cells
of arbitrary shape (particularly, the partition can be chosen to be a triangulation of Σ), and in every cell
we define edges sJ , J = 1, . . . , ns and loops βi , i = 1, . . . , nβ all based at a vertex v
inside
, where ns , nβ may be different for different cells (see Fig. 3 in Ref. 8); furthermore, we can choose an arbitrary representation R(j) of the SU (2) group other than R(1/2) . The entire prescription is denoted by R and called a permissible classical regulator if both the following conditions hold Eucl = C Eucl[N ], lim CR 

(100a)

→0

lim TR = T [N ],

(100b)

→0

where Eucl [N ] = CR 

Eucl CR (N ) = 





Eucl CR (N ), 


(101a)

N (v
)  iJ C Tr((R(j) (hβi ) − R(j) (hβ −1 )) i 32π 2 G2 γ 3/2 i,J × R(j) (hs−1 ){R(j) (hsJ ), V }),

(101b)

J

TR [N ] =



TR
(N ),

(102a)




TR
(N ) =

2N (v
)  IJK ¯ T Tr(R(j) (hs−1 ){R(j) (hsI ), K} I (8πG)4 γ 3 IJK

¯ (j) (h −1 ){R(j) (hsK ), V }), × R(j) (hs−1 ){R(j) (hsJ ), K}R s J

K

(102b)

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and C iJ and T IJK are fixed constants, independent of the scale parameter . There exists a great variety of permissible classical regulators, and this nonuniqueness is the source of quantization ambiguities. 6.2. Quantum scalar constraint Eucl into a quantum operator acting on the It is fairly straightforward to promote CR  space H of cylindrical functions. Recall that the total volume operator Vˆ ≡ Vˆ (Σ) and the holonomy operator ˆ hγ are both well defined in H. Therefore, simply by ˆ ˆ replacing V with V , hγ with hγ , and Poisson brackets with commutators, we obtain Eucl on H: the operator CˆR   Eucl Eucl CˆR CˆR [N ] = (N ), (103a)  



Eucl CˆR (N ) = − 


 iN (v
) ˆ β ) − R(j) (h ˆ −1 )) C iJ Tr((R(j) (h i βi 2 2 3/2 32π G γ
i,J

ˆ −1 )[R(j) (h ˆ s ), Vˆ ]). × R(j) (h J s J

(103b)

Eucl is regularized in a way preserving SU (2) invariance, Furthermore, because CR  Eucl CˆR is SU (2) invariant and leaves the space HG of spin networks invariant. When 

 ˆ s ), Vˆ will yield acting on a given spin network state |Γ, . . ., the part R(j) (h J zero unless the segment sJ intersects a node of Γ, because the volume operator has contributions only from nodes of the spin network. Also note that while nodes must Eucl can have be at least four-valent to have nonvanishing volume, the operator CˆR  contributions from trivalent nodes, as ˆ hsJ adds one more edge to the intersection node. Eucl is well defined in H, here comes the critical issue: We have to ensure As CˆR  that the resulting quantum operator is covariant under diffeomorphisms. For this purpose, we have to restrict our regularization scheme to a diffeomorphism covariant quantum regulator, which is a family of permissible classical regulators that are not fixed but transform covariantly as a graph Γ is moved under diffeomorphisms. The precise definition of a diffeomorphism covariant quantum regulator is not important here, but the essential point is that such regulators exist and lead to the regulated Eucl that is densely defined in the full Hilbert space H with domain Cyl operator CˆR  and diffeomorphism covariant.63,65 The simplest and most convenient case of such regulators can be summarized as follows. For a given graph Γ, make the partition refined enough such that each cell
contains at most one node of Γ. For a cell
containing a node n, the edges sI are assigned to the proper segments of the links of Γ incident at n; orientations of sI are all assigned to be outgoing at n. The loops βi ≡ β[IJ] ≡ βIJ are chosen to be the triangular loops spanned by sI and sJ ; the loop βIJ contains no other points of Γ except for the edges sI , sJ , and its orientation is defined as the same as the plane by the ordered pair of sI and sJ . See Fig. 2

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eJ eK

sJ

eI

C IJK = ±κ1 , 0, T IJK = ±κ2 , 0,

sK n sI

if ( eI × eJ) · eK > 0, < 0, = 0

Fig. 2. The edges sI , sJ , sK and the triangular loop βIJ at a trivalent node n. The pattern is similar at other multivalent nodes. C IJ K and T IJ K are given by constants in accordance with the orientation of eI , eJ , eK tangent to sI , sJ , sK .

for the illustration. Finally, the constants C iK ≡ C IJK are given by ±κ1 , 0, and the constants T IJK given by ±κ2 , 0, depending on the orientation of a triple vector tangent to sI , sJ , sK relative to the background orientation of Σ, where κ1 , κ2 are  fixed constants. With this regulator, the sum
in Eq. (103a) effectively becomes  the sum n∈Γ over the nodes of Γ, and the lapse function N (v
) and the volume operator Vˆ in Eq. (103b) become Nn and Vˆ ({n}), respectively, for each node. Eucl is well defined in H, we can define the corresponding operator acting As CˆR  ∗ on Cyl via its action on the states of Cyl by Eucl Eucl Ψ|Φ := (Ψ|CˆR Φ (CˆR  

(104)

∗

for every (Ψ| ∈ Cyl and |Φ ∈ Cyl.n The final step is to remove the regulator and obtain the operator CˆEucl acting on Cyl∗ . It does not work if one attempts to take the limit directly ? Eucl Φ, (CˆEucl Ψ|Φ = (Ψ|lim CˆR  →0

(105)

Eucl since CˆR becomes ill-defined in Cyl in the limit  → 0. Instead, we should define  the operator CˆEucl acting on Cyl∗ by Eucl Φ, (CˆEuclΨ|Φ := lim (Ψ| CˆR  →0

(106)

where the limit is now a limit of a sequence of numbers, instead a sequence of operators. The limit exists if (Ψ| is a diffeomorphism invariant state, namely (Ψ| ∈ Cyl∗Diff , as we can see in the following crucial observation. Given |Φ a spin network Eucl on the right-hand side of Eq. (106) modifies |Γ, . . . in two |Γ, . . ., the operator CˆR  ways by changing its graph Γ and its coloring. The graph Γ is changed by the two hβIJ . The former superimposes an edge of coordinate classes of operators: ˆ hsK and ˆ length  to a link of Γ, and the latter adds the triangular loop as depicted in Fig. 2. When  is sufficiently small, changing  in the operator changes the resulting state, but the resulting state remains in the same diffeomorphism equivalence class. Eucl Φ for (Ψ| ∈ Cyl∗Diff becomes independent of  Consequently, the value of (Ψ| CˆR  Eucl∗† be precise, we should call the operator on the left-hand side CˆR in the same spirit of  Eq. (61), but we neglect ∗ and † to keep the notation simple.

n To

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once  is sufficiently small, and thus the  → 0 limit in Eq. (106) is finite. Therefore, we can well define the quantum operator CˆEucl acting on the domain Cyl∗Diff , as the regulator is removed trivially. The Hamiltonian constraint operator, albeit defined upon spin networks with reference to coordinates in the first place, turns out to be well defined and independent of the regularization scale  on the domain of diffeomorphism invariant states, i.e. s-knots. To sum up, the small  of coordinate length loses its physical significance at the diffeomorphism invariant level and the  → 0 limit becomes finite because making the regulator smaller cannot modify anything below the Planck scale as there is really nothing below the short-scale discreteness.66 This striking feature as a consequence of the intimate interplay between diffeomorphism invariance and short-scale discreteness profoundly cures the ultraviolet pathology that has long plagued quantum field theory of gravity. The term T [N ] in the scalar constraint can be dealt with in a completely parallel manner. Specifically, we promote Eq. (102) to the operator  TˆR
(N ), (107a) TˆR [N ] =


TˆR
(N ) =

2iN (v
)  IJK ˆ −1 )[R(j) (h ˆ s ), K ¯ˆ R ] T Tr(R(j) (h I sI (8πG)4 γ 3
3 IJK

ˆ¯ ]R(j) (h ˆ −1 )[R(j) (h ˆ s ), K ˆ −1 )[R(j) (h ˆ s ), Vˆ ]), × R(j) (h R J K s s J

(107b)

K

ˆ¯ is defined as where K R ˆ¯ := K R

i Eucl [Vˆ , CˆR ], 
γ 3/2

(108)

and in the end define the operator Tˆ on the domain Cyl∗Diff by (Tˆ Ψ|Φ := lim (Ψ|TˆR Φ. →0

The total quantum scalar operator ˆ ] = √γ CˆEucl[N ] − 2(1 + γ 2 )Tˆ [N ] C[N is well defined on the domain

Cyl∗Diff ,

(109)

(110)

namely

ˆ ] : Cyl∗ → Cyl∗ . C[N Diff

(111)

Up to diffeomorphisms, the operator CˆR [N ] with the diffeomorphism covariant quantum regulator is independent of . That is, for any sufficiently small  and 
, given any |Φ ∈ Cyl, there is a diffeomorphism ϕ such that ˆϕ CˆR [N ] U ˆ † |Φ. CˆR
[ϕ∗ N ]|Φ = U ϕ

(112)

ˆ ] does not leave Cyl∗ It should be noted, however, that the operator C[N Diff invariant, because C[N ] does not commute with CDiff [N] even at the classical level. (Also note that C[N ] depends on coordinates via N (x).) More precisely, let

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(η(ΨΓ )| ∈ Cyl∗Diff be a group-averaged state as defined in Eq. (56). As defined ˆ ] on the diffeomorphism invariant state (η(ΨΓ )| is in Eq. (106), the action of C[N given by ˆ ]η(ΨΓ )|Φ := lim (η(ΨΓ )|CˆR [N ]Φ (C[N →0

≡ lim

→0

= lim

→0

= lim

→0

= lim

→0



ˆϕ PˆDiff,Γ ΨΓ | CˆR [N ]Φ U

ϕ∈Diff/Diff Γ



ˆ † CˆR [N ] Φ PˆDiff,Γ ΨΓ | U ϕ

ϕ∈Diff/Diff Γ



ˆϕ† Φ PˆDiff,Γ ΨΓ | CˆR
[ϕ−1∗ N ] U

ϕ∈Diff/Diff Γ



ˆϕ CˆR
[ϕ−1∗ N ]† PˆDiff,Γ ΨΓ | Φ, U 

(113)

ϕ∈Diff/Diff Γ

where we have used Eqs. (56) and (112), and the regulator R
= R
(, ϕ) nontrivially depends on  and ϕ. It follows from Eq. (113) that  ˆϕ CˆR
ˆ ]η(ΨΓ )| ∼ lim U [ϕ−1∗ N ]† PˆDiff,Γ ΨΓ |. (114) (C[N  (,ϕ) →0

ϕ∈Diff/Diff Γ

The resulting state on the right-hand side is a distribution belonging to Cyl∗ as the sum is over the huge group Diff/Diff Γ , but it is not in Cyl∗Diff as the summands are not arranged in the style of group averaging. The quantum theory defined by the ˆ ]Ψ| = 0 nevertheless is diffeomorphism invariant, quantum scalar constraint (C[N ˆ ], which is a proper subspace of Cyl∗ because what matters is the kernel of C[N Diff and independent of N (x). ˆ ] is manifestly SU (2) invariant, as the regularization On the other hand, C[N in Sec. 6.1 has been adopted to be SU (2) invariant. Taking into account both the Gauss constraint and the diffeomorphism constraint, we have the quantum operator ˆ ] well defined on the domain Cyl∗inv , namely C[N ˆ ] : Cyl∗inv → Cyl∗G . C[N

(115)

ˆ ], which is a proper subspace of Cyl∗inv , The Cauchy completion of the kernel of C[N is the sought-after physical Hilbert space Hphys in Eq. (35). 6.3. Solutions to the scalar constraint ˆ ] has been rigorously defined on the domain The scalar constraint operator C[N Cyl∗inv . The next step is to construct the physical Hilbert space Hphys by solving ˆ ]Ψ| = 0. the solutions to the scalar constraint (C[N The actions of the resulting operators CˆEucl and Tˆ on a given SU (2) and diffeomorphism invariant state (η(ΨΓ,j,i )| are rather simple. As indicated in Eq. (114) with Eqs. (103) and (107), it turns out that, if |ΨΓ,j,i  ≡ |Γ, jl , in  contains any

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extraordinary loops (labeled by R(j) ), CˆEucl will remove one such loop and Tˆ will remove two in (η(ΨΓ,j,i )|, in addition to possible changes in the intertwiner at the node.63 The extraordinary loop is of the type introduced by the regulator at any of the nodes as depicted in Fig. 2. The details of the actions of CˆEucl and Tˆ are discussed in Ref. 67. (Also see Ref. 68 for the general structure of the scalar constraint.) The action of adding or removing an extraordinary loop at a node is schematically illustrated in Fig. 3. An s-knot state (η(ΨΓ0 ,j0 )| is said to be simple if any spin networks belonging to the same diffeomorphism class of (η(ΨΓ0 ,j0 )| do not contain an extraordinary loop labeled by R(j) . Obviously, simple s-knots are annihilated by both CˆEucl and Tˆ and thus are solutions to both the Euclidean constraint and the full (Lorentzian) scalar constraint. In a sense, these simple solutions are analogues of time-symmetric solutions to the classical Hamiltonian constraint. Particularly, s-knot states of loops or multiloops are trivial examples of simple solutions. More solutions can be obtained by starting from the simple solutions. Let (n) (0) (η(ΨΓ0 ,j0 )| be an s-knot obtained from (η(ΨΓ0 ,j0 )| ≡ (η(ΨΓ0 ,j0 )| by attachment of n extraordinary loops labeled by R(j) (note that there are many ways of the (n) attachment), and denote by DΓ0 ,j0 the subspace of Cyl∗Diff spanned by those s-knot (n)

states with n extraordinary loops. The resulting spaces DΓ0 ,j0 are finite-dimensional and have trivial intersection with one another, i.e. (n
)

if (Γ0 , j0 ), n = (Γ
0 , j0
), n
.

(116)

Every (Ψ| ∈ Cyl∗Diff can be uniquely decomposed as  (n) (n) (n) (Ψ| = (η(ΨΓ,j )|, where (η(ΨΓ,j )| ∈ DΓ,j .

(117)

(n)

DΓ0 ,j0 ∩ DΓ
,j
= ∅, 0

0

(Γ,j),n

This unique decomposition enables us to find the solutions to the scalar constraint in a systematic way. First, for the solutions to the Euclidean constraint, we have (n)

(Ψ|CˆEucl [N ] = 0 ⇔ (η(ΨΓ,j )|CˆEucl[N ] = 0,

for every (Γ, j) and n.

(118)

That is, (Ψ| is a solution to the Euclidean constraint if and only if its components with respect to the decomposition Eq. (117) are all solutions. This property reduces

Fig. 3. When an extraordinary loop is added or removed at a node, the change is between the left diagram and the middle one, which can be understood as the right diagram. Here, we show the case that the extraordinary loop is labeled by R(j=1/2) .

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the problem of finding the solutions to the Euclidean scalar constraint to that of finding solutions in finite-dimensional subspaces, which is equivalent to the tractable task (by a computer) to find the kernel of finite matrices. Now, consider the full (Lorentzian) scalar constraint. In the scheme with respect to the decomposition Eq. (117), the problem of finding solutions to the full scalar constraint is reduced to a hierarchy of steps. That is, the scalar constraint equation ˆ ] = √γ (Ψ|CˆEucl[N ] − 2(1 + γ 2 )(Ψ|Tˆ [N ] = 0 (119) (Ψ|C[N is equivalent to the hierarchy of equations (1) (η(ΨΓ,j )|Tˆ [N ] = 0,

√ (2) (1) 2(1 + γ 2 )(η(ΨΓ,j )|Tˆ [N ] = γ (η(ΨΓ,j )|CˆEucl [N ], .. . (n+1)

2(1 + γ 2 )(η(ΨΓ,j

)|Tˆ [N ] =

(120) √

(n)

γ (η(ΨΓ,j )|CˆEucl[N ],

.. . since CˆEucl removes one extraordinary loop and Tˆ removes two. This hierarchical procedure gives a good control on the solutions and suggests a sound postulation that the series should terminate at a finite number of steps. Even though we have a good understanding of solutions to the scalar constraint, our knowledge about them is far less clear and complete than that about s-knot states. Additionally, we encounter various quantization ambiguities, notably the choice of R(j) for the SU (2) representation, which further complicate the solutions. ˆ ] is evidently not It should also be remarked that the scalar constraint operator C[N hermitian in the space HDiff , as it removes extraordinary loops but does not add them. One can simply obtain the hermitian scalar constraint operator by replacing it with (Cˆ + Cˆ† )/2, and the hermitian operator is expected to be better behaved for some technical issues as well as for some aspects of the classical limit. However, as a nonhermitian scalar constraint operator does not lead to any inconsistency, there is no logical necessity of being hermitian. Meanwhile, a great number of variant approaches of the quantum scalar constraint have been considered in the literature (see Sec. 7.4.1 of Ref. 9). 6.4. Quantum dynamics ˆ ], which The space of solutions to the quantum scalar constraint is the kernel of C[N ∗ is a proper subspace of Cylinv . Consequently, the physical inner product between any two solutions are naturally inherited from the inner product of the space Cyl∗inv ; ˆ the physical inner product is given by i.e. for (Ψ|, (Φ| ∈ Kernel of C, (Ψ | Φ)phys := (Ψ | Φ),

(121)

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where the right-hand side is defined in Eq. (59). With respect to the physical inner product (· | ·)phys , the Cauchy completion of the kernel of Cˆ is the physical Hilbert space Hphys . Even though Hphys can been rigorously constructed (at least formally), the quantum dynamics in terms of evolution remains very obscure. Given physical state (Ψ| ∈ Hphys , to read out the “evolution” (with respect to the arbitrary coordinate time t used in the ADM foliation), one might attempt to designate the quantum counterpart of the classical Hamilton’s equation Eq. (26) as

 d
ˆ ˆ ˆ |Ψ)phys , (Ψ| O(t)|Ψ) (Ψ| O(t), H (122) phys = dt i  ˆ ] , and ˆ := (8πGγ)−1 CˆG [(ω · t)] + CˆDiff [N] + C[N where the quantum Hamiltonian H ˆ is the operator of an observable O corresponding to a measurement performed O(t) at coordinate time t. However, this strategy does not work, because the right-hand side simply vanishes as physical states are annihilated by all three constraints. In a sense, the evolution, in the conventional notion, is completely “frozen.” This difficulty is closely related to the renowned “problem of time” in QG69,70 and has entailed the timeless description of quantum mechanics (see Chap. 5 of Ref. 9 for an in-depth elaboration). In the literature, there are mainly two strategies to unveil the quantum dynamics: the “complete observable approach” versus the “partial observable approach.”o Complete observables, also referred to as “Dirac observables” or “physical observables,” are those that commute with all three constraints; by contrast, partial observables are those that do not commute with all three constraints (the scalar constraint in particular). In other words, complete observables are free of any gauge ambiguities, while partial observables still have a remnant of gauge dependence in the strict sense. (For more on the distinction, see Refs. 71 and 72.) ˆ In the complete observable approach, instead of O(t), we construct a famΦ is nontrivially  ily of Dirac operators O|Φ parametrized by Φ. The operator O| constructed so that it is well defined in Hphys and commutes with all constraint  operators for any value of Φ. We interpret the operator O| Φ0 as representing a measurement O performed at the instance when the observable Φ takes the value Φ0 .p For a given physical state (Ψ| ∈ Hphys , the quantum dynamics is portrayed in terms of the expectation values Φ |Ψ)phys , (123) (Ψ| O| which do not describe the evolution of observables with respect to a preferred time variable but, instead, the correlation between observables (O and Φ). In this regard, the variable Φ is said to serve as the “internal time” (also known as “internal o It

should be remarked that there are many various approaches in the literature that are more or less in one form or the other of the two main strategies but different in finer detail. The aim here is neither to give decisive definitions of these two strategies nor to exhaust all possible approaches but to give general ideas about how the problem of quantum dynamics could be tackled. p In the timeless language, a measurement is said to be conducted at some “instance,” rather than at some “instant.”

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D.-W. Chiou

clock”). Unfortunately, it is extremely difficult to construct Dirac operators that are nontrivial and physically sound. Usually, one also has to include nongravitational matter into the system in order to provide a privileged dynamical reference as the internal time.q In some symmetry-reduced theories, such as LQC, the Dirac operators can be explicitly formulated and consequently the quantum dynamics can be clearly deciphered, as will be shown in Sec. 11. The partial observable approach, on the other hand, adopts a very different philosophy. While it is asserted in the complete observable approach that only complete observables are physical, the partial observable approach contends that it would be too restrictive to dismiss all partial observables, because one can never grasp “totality” of the whole system and therefore some gauge degrees of freedom, especially those for the reparametrization invariance in time, nevertheless bear physical significance as far as one concerns real measurements, which are conducted without the knowledge of totality. Unattainability of totality is even more crucial in quantum mechanics, as one can never disregard the observer, who is not viewed as a part of the system to be observed. By the standard Copenhagen interpretation of quantum mechanics, it is impossible to know and even invalid to ask what the physical state (Ψ| ∈ Hphys is for the whole world. In this sense, the quantum dynamics described in the style of Eq. (123) seems inadequate and heuristic at best. In accord with the Copenhagen interpretation, the partial observable approach describes the quantum dynamics in terms of the “if-then-what” prediction. Provided that a measurement OA performed at some instance yields the outcome a, the quantum dynamics is to predict what the probability is for the measurement OB performed at another instance to yield the outcome b. The only difference from the conventional quantum mechanics is that we do not need to specify the time separation between the two measurements, because any notion of time separation is in principle encoded in the measurement outcomes a and b themselves in the timeless description. (In fact, we do not even need to specify the time-ordering of the two measurements. See Sec. 3.5 of Ref. 73 for more discussions.) For LQG, to make sense of the quantum dynamics of spacetime, we naturally choose the relevant measurements to be those for the intrinsic areas and volumes as studied in Sec. 5.4. Note that the corresponding operators defined in Eqs. (84) and (85) correspond to partial observables as opposed to complete observables, as they ˆ ]. Consequently, after a complete do not commute with the scalar constraint C[N measurement of the quantum geometry, the kinematical state is collapsed into an eigenstate of the quantum geometry, which is an s-knot state (K, c| ∈ Hinv . The quantum dynamics of spacetime is then posed as a predictive question: What then is the probability for another complete measurement of the quantum geometry to yield the outcome associated with the s-knot (K
, c
|? It is natural to postulate q The

strategy of the complete observable approach is closely related to the idea of the reduced phase space quantization, in which one finds a one-parameter family of complete observables O|Φ at the classical level before quantization. See Sec. 12.3.

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that the probability is given by the |W (K, c; K
, c
)| , where W (K, c; K
, c
) is the transition amplitude 2

W (K, c; K
, c
) := (K, c|PˆC |K
, c
)

(124)

and PˆC is the “projector” that projects an s-knot state in Hinv into the subspace Hphys . More precisely, the projector PˆC : Hinv → Hphys ⊂ Hinv is formally given by PˆC :=



|Ψ)(Ψ|,

(125)

(Ψ|∈Hphys

where {(Ψ| ∈ Hphys } forms an orthonormal basis of Hphys . Therefore, the quantum dynamics in principle can be inferred from the complete knowledge about the physical Hilbert space Hphys . The transition amplitude given by Eq. (124) bears close similarity to that in the spin foam formalism as will be discussed in Sec. 9. Both the complete observable approach and the partial observable approach are far from fully developed and remain tentative, and our understanding about the quantum dynamics of spacetime is still very limited. The obstacles not only lie in the mathematical difficulties but are also deeply rooted from the conceptual (and even philosophical) riddles of interpreting the fundamental notions of space, time, quantum measurements, etc.

7. Inclusion of Matter Fields So far, to bring out the main ideas of LQG, we have ignored nongravitational matter fields. Inclusion of matter fields does not require a major revamp of the underlying framework. The quantum states of space plus matter naturally extend the notion of s-knots with additional degrees of freedom. We will follow the lines of Sec. 7.2 of Ref. 9. For more details, see Chap. 12 of Ref. 10 and references therein (and also see Chap. 9 of Ref. 6 for the corresponding classical theories).

7.1. Yang–Mills fields The easiest extension is to incorporate Yang–Mills fields. Let GYM be the Yang– Mills group. The two 3d connections, Ashtekar connection A and Yang–Mills connection AYM , can be considered together as a single connection A = (A, AYM ) associated with the group SU (2) × GYM . As the holonomies of AYM and surface integrals of the Yang–Mills electric field can be defined exactly in the same ways as ˜ the construction of H, HG , and Hinv extends straightforwardly those of A and E, to the total connection A without much difficulty. The gauge (G) and diffeomorphism (Diff) invariant quantum states of space plus Yang–Mills fields are given by the s-knot states that are classified by knotted graphs and carry irreducible representations of the group G on the links and the

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corresponding intertwiners at the nodes. Because G is the direct product of SU (2) and GYM , its irreducible representations are simply the products of those of SU (2) and GYM . Consequently, the coloring for each link is extended as (jl , kl ) and the coloring for each nodes is extended as (in , wn ), where kl is the irreducible representation of GYM representing the Yang–Mills electric flux passing through the surface l, and wn is the intertwiner of GYM representing the Yang–Mills field strength at the node n. 7.2. Fermions The next is to include fermions. Let η(x) be a Grassmann-valued fermionic field. It transforms as an irreducible representation k under the gauge transformation of the Yang–Mills group GYM and as a fundamental (j = 1/2) representation under the SU (2)  transformation. It is more convenient in LQG to take the densitized field ˜ η(x) as the fundamental field variable.74 ψ(x) := det E As an extension of (36), the cylindrical function of the connection A and the fermion field ψ is a smooth (Grassmann-valued) function of L group elements of G = SU (2) × GYM and 2N |k| Grassmann variables (|k| is the dimension of k) defined as  ¯ = f hγ (A), . . . , hγ (A), ΨΓ,f [A, ψ, ψ] 1 L mi ¯i (xn ), . . . , (126) . . . , ψ (xn ), . . . , ψ¯m where Γ is the graph comprised of L edges γl and N nodes located at xn=1,...,N , ¯ i=1,...,|k| ∈ mi=1,....|k| ∈ w(k) are indices (i.e. weights) of the representation k, and m ¯ are indices of the conjugate representation k. ¯ Since Grassmann variables antiw(k) commute with one another, cylindrical functions are at most linear in ψnm ≡ ψ m (xn ) ¯ ¯ ≡ ψ¯m (xn ) for each pair (n, m). If two cylindrical functions are defined for and ψ¯nm the same graph Γ, as an extension of (37), the inner product between them is defined as ΨΓ,f | ΨΓ,g 
 L := dµl GL l=1



 

N   ¯ ¯m∗ dψ¯nm dψn¯ Iψnm ,ψnm∗ + dψnm dψnm∗ Iψ¯nm¯ ,ψ¯nm∗ ¯ +

m∈w(k) n=1

∗ m m ¯ ¯1 , . . . , ψ¯N |k| ) × f (hγ1 , . . . , hγL , ψ1m1 , . . . , ψN |k| , ψ¯1m m m ¯ ¯1 × g(hγ1 , . . . , hγL , ψ1m1 , . . . , ψN |k| , ψ¯1m , . . . , ψ¯N |k| ),

(127)

where dµ is the Haar measure on G, Iψ,ψ∗ is a linear transformation that maps the integrand into a new (Grassmann-valued) function by assigning ψ = 0 and  ψ ∗ = 0, dψdψ ∗ is the Grassmann integral over the superspace spanned by ψ and ¯ are treated as independent Grassmann variables. The extension ψ ∗ , and ψnm , ψ¯nm of this inner product to any two cylindrical functions is completely analogous to

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that in the case of pure gravity. The construction of H, HG , and Hinv can then be readily repeated. The gauge and diffeomorphism invariant quantum states are again given by the s-knots with extended colorings. For each node n, in addition to the intertwiner (in , wn ), we also have to specify an integer Fn as the degree of the monomial in ψ m (xn ), which determines the fermion number in the region of the node, and ¯ (xn ), which determines the an integer F¯n as the degree of the monomial in ψ¯m antifermion number in the region of the node. In the presence of fermions, at each node n, the intertwiner (in , wn ) is not only intertwined with the coloring (jl , kl ) of the links attached to n but also with the (anti)fermion numbers Fn , F¯n . This is because fermions carry the j = 1/2 representation indices of SU (2) and the k representation indices of GYM , which have to be properly contracted with the matrix elements of R(jl ) (hγl (A)) and R(kl ) (hγl (AYM )) in a way such that the generalized Clebsh–Gordon conditions are satisfied for both SU (2) and GYM . In other words, fermions are quanta of “charged” particles, with both SU (2) and GYM charges, and the quanta of electric flux labeled by jl for SU (2) and kl for GYM can emerge from and end at the charged particles. (The fact that fermions are SU (2) charged implies that they contribute to the spectrum of the area operator. See Ref. 75 for more discussions.) In the same regard, both the gravitational and Yang–Mills fields are self-sourcing, as they yield colorings for both links and nodes. 7.3. Scalar fields Scalar fields can also be included in LQG, but in a less natural manner than those for Yang–Mills fields and fermions. Let φ(x) be a scalar field, which transforms as a trivial (j = 0) representation under the SU (2) transformation and as an irreducible representation k
under the gauge transformation of the Yang–Mills group GYM . Thus, φ takes the value in the vector space of the k
representation. The problem is that the k
representation vector space is noncompact with respect to natural GYM -invariant measures, thus rendering the definition of the inner product between cylindrical functions difficult. (Fermions do not suffer from this problem, since the Grassmann integral naturally gives the appropriate measure.) One way to get around this problem is to assume k
to be the adjoint representation of GYM and replace φ(x) with the associated “point holonomy” U (x) := exp (φ(x)), which takes the value of GYM and admits the natural Haar measure. The cylindrical function of A, ψ, and φ is a smooth (Grassmann-valued) function of L group elements of G = SU (2) × GYM , 2N |k| Grassmann variables, and N group elements of GYM defined as  ¯ φ] = f hγ (A), . . . , hγ (A), ψ m1 (x1 ), . . . , ψ m|k| (xN ), ΨΓ,f [A, ψ, ψ, 1 L ¯1 ¯ |k| (x1 ), . . . ψ¯m (xN ), U (x1 ), . . . , U (xN ) . (128) ψ¯m

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The inner product between two cylindrical functions of the same graph is then naturally defined as

L N  dµl dµn ΨΓ,f | ΨΓ,g  := GL l=1



GN YM n=1

 

N   ¯ ¯m∗ + dψ¯nm dψn¯ Iψnm ,ψnm∗ + dψnm dψnm∗ Iψ¯nm¯ ,ψ¯nm∗ ¯

m∈w(k) n=1 ∗ m m ¯ ¯1 , . . . ψ¯N |k| , U1 , . . . , UN ) × f (hγ1 , . . . , hγL , ψ1m1 , . . . , ψN |k| , ψ¯1m m m ¯ ¯1 × g(hγ1 , . . . , hγL , ψ1m1 , . . . , ψN |k| , ψ¯1m , . . . ψ¯N |k| , U1 , . . . , UN ),

(129) where dµl is the Haar measure on G, and dµn the Haar measure on GYM . Extension to any two cylindrical functions is straightforward as before. In the presence of scalar fields, of the resulting s-knot state, the coloring for each node n is augmented with an additional integer number Sn , which represents the total adjoint charge, or equivalently the total number, of the scalar particles in the region of the node. Like fermions, scalar fields carry GYM (adjoint) charge and accordingly Sn is coupled with wn , kl , Fn in the generalized Clebsh–Gordon condition for GYM . Unlike fermions, on the other hand, scalar fields are trivial for SU (2) and Sn is decoupled from in , jl in the generalized Clebsh–Gordon condition for SU (2). 7.4. S-knots of geometry and matter In summary, with inclusion of all kinds of matter (Yang–Mills fields, fermions, scalar fields), the gauge and diffeomorphism quantum states are s-knots (K, c| ≡ (Γ, jl , in , kl , wn , Fn , F¯n , Sn | labeled by the following quantum numbers: • • • • • • •

Γ: an abstract knotted graph with oriented links l and nodes n. jl : an irreducible j representation of SU (2) associated with each link l. in : an SU (2) intertwiner associated with each node n. kl : an irreducible k representation of GYM associated with each link l. wn : a GYM intertwiner associated with each node n. Fn , F¯n : two integers associated with each node n. Sn : an integer associated with each node n.

These quantum numbers correspond to physical quantities as listed in Table 2. The physical interpretation of s-knots as discussed in Sec. 5.4 can be directly generalized to the s-knots of both geometry and matter in the obvious way. The paradigm of background independence is even more remarkable with inclusion of matter fields, as geometry and matter fields are truly on the equal footing and live on top of one another via their contiguous relations without any reference to a given background.

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Loop quantum gravity Table 2.

Physical quantities for the quantum numbers of s-knots.

Quantum number Γ jl in kl wn Fn , F¯n Sn

II-507

Physical quantity Adjacency relation of regions n and surfaces l Area of the surface l Volume of the region n Yang–Mills electric flux through the surface l Yang–Mills field strength at the region n Numbers of fermions and antifermions at the region n Number (total adjoint charge) of scalars at the region n

Furthermore, in the presence of nongravitational matter fields, there is no difficulty to perform the ADM foliation and obtain the three constraints. Particularly, the scalar constraint of the classical theory can be regulated in the same fashion as that in Sec. 6.1 and then promoted to the quantum operator as in Sec. 6.2. Therefore, the quantum theory of dynamics of spacetime plus matter in principle can be constructed in the same manner as in Secs. 6.3 and 6.4. 8. Low-Energy Physics The essential premise of LQG is that quantum states are background-independent excitations out of nothingness — a concept in direct conflict with the foundation of conventional QFTs. This poses the big challenge to LQG to show how the lowenergy description of conventional QFTs arises from the underlying Planckian world in an appropriate sense (of coarse graining). The final resolution is still far out of reach, but the research is being undertaken step by step. The low-energy physics remains one of the most wanted missing pieces in LQG — not until we figure out the low-energy physics of LQG and show its agreement with Einstein’s theory of classical GR and compatibility with the QFT of the Standard Model can we declare LQG to be an adequate quantum theory of GR, because both classical GR and the Standard Model have been intensively tested in the low-energy regime. In this section, we present the basic ideas of recovering low-energy physics and refer readers to Chap. 11 of Ref. 10 for more systematic treatments and more details. 8.1. Weave states It is impressive that LQG yields discrete spectra of geometry as a natural consequence of quantization. The eigenstates of geometry, i.e. s-knots, however do not look akin at all to the smooth macroscopic spaces (curved or flat) we are familiar with. What then is the connection between the discrete quantum geometry and the smooth classical geometry? Consider a very huge spin network with a very large number of nodes and links, each of which is of area or volume in the Planck scale. This is a big lattice of Planck-scale lattice size, but it appears as a smooth 3d geometry when probed at a macroscopic scale much larger than the Planck scale. This is analogous to the

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fabric of a piece of cloth, which is composed of thousands of granular threads but appear smooth at a distance. More precisely, given a fixed classical 3d metric qab (x) = δij eia (x)ejb (x), it is possible to construct a spin network state |S that approximates the metric at scale l  Pl , much larger than the Planck length Pl . That is, for any region R and surface S, we have ˆ A(S)|S = (A(S)|e + O(Pl /l))|S,

(130a)

Vˆ (R)|S = (V (R)|e + O(Pl /l))|S,

(130b)

where A(S)|e is the classical area of S defined in Eq. (73) and V (R)|e is the classical volume of R defined in Eq. (77) with the given cotriad eia , and O(Pl /l) denotes small corrections in Pl /l. Such a spin network |S is called a weave state of the metric qab . This definition is given for spin networks but can be easily carried over to the diffeomorphism invariant level for s-knots. Several weave states have been constructed for flat space, Schwarzschild space, space with gravitational waves, and more.76 It should be noted that Eq. (130) does not determine a unique weave state. When all macroscopic physics is taken into account, it seems more likely that the proper quantum state for a macroscopic geometry is a coherent superposition of the weave state solutions.

8.2. Loop states versus Fock states The basic variables of LQG are holonomies (Wilson loops) of the connection A ˜ across along one-dimensional curves and the fluxs of the conjugate momentum E two-dimensional surfaces. These variables however fail to be well defined in the Fock space of excitations of A in the perturbative QFTs. A further inquiry into the low-energy physics of LQG then is to understand how the perturbative description of quantum excitations in terms of Fock states in conventional QFTs arise as a low-energy limit of the nonperturbative theory in terms of loop states. These issues have been studied for simple examples. The main effort so far is to construct mathematical and conceptual tools that will facilitate a systematic analysis of quantum fields on semiclassical states of quantum geometry.77,78 Particularly, the connection between loop states and Fock states has been studied in detail for the Maxwell field79–82 and linearized gravity83 in Minkowski spacetime.

8.3. Holomorphic coherent states The relation between quantum states and the classical theory is most likely to be understood by the techniques of coherent states. Various constructions of coherent states have been suggested. Here we briefly describe the formulation of holomorphic coherent states 84–89 by following the lines of Ref. 89 for the Euclidean theory.

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Consider the heat kernel Kt (h, h0 ) on SU (2), which is given by the Peter–Weyl expansion:  (2j + 1) e−j(j+1)t Tr R(j) (hh−1 (131) Kt (h, h0 ) = 0 ), j

where the positive real number t is the “heat kernel time.” For a given graph Γ of N nodes, a holomorphic coherent state associated with Γ is defined as the SU (2)invariant projection of a product over links of heat kernels
  dga Ktab (hab , ga Hab gb−1 ), (132) ΨΓ,Hab ,tab (hab ) = SU(2)N

ab

where, to simplify the notation, we use a, b, . . . = 1, . . . , N to denote the nodes of Γ and the (ordered) couples ab to denote the (oriented) links.r The label Hab for each link ab is an element of SL(2, C), which is diffeomorphic to the SU (2) cotangent bundle T ∗ SU (2) ∼ = SU (2) × su(2)∗ ∼ = SU (2) × su(2). The state given in Eq. (132) is analogous to the standard wave packet of nonrelativistic quantum theory (x−x )2 i 1 − 4σ20 (133) e
p0 (x−x0 ) , ψx0 ,p0 ,σ = e (2πσ 2 )1/4 which is peaked in the position x0 as well as in the momentum p0 and can be written (up to a constant phase) as a Gaussian function ψX0 ,σ =

(x−X0 )2 1 e− 4σ2 2 1/4 (2πσ )

(134)

peaked on a complex position X0 := x0 + 2iσ 2
−1 p0 . As Kt (h, H0 ) is the analytic continuation to SL(2, C) of the heat kernel Kt (h, h0 ) on SU (2) and t = 0 reduces Kt (h, h0 ) to a delta function, Hab is analogous to X0 (while hab analogous to x0 ) and tab is analogous to σ. As SL(2, C) is diffeomorphic to SU (2)× su(2), we can decompose each SL(2, C) element as Hab = hab e2itLab ,

(135)

where hab ∈ SU (2) and Lab ∈ su(2). Alternatively, Hab can be written in the form Hab = nab e−i(ξab +iηab )

σ3 2

n−1 ba ,

(136)

where nab , nba ∈ SU (2) are two unrelated group elements of SU (2), ξ ∈ [0, 2π) is an angle, and ηab ∈ R+ is a positive real numbers. Any element n of SU (2) can be associated with a unit vector n ∈ R3 via n := R(j=1) (n) · z,

(137)

where z = (0, 0, 1) ∈ R3 and R(j=1) (n) acts as a rotation matrix. Consequently, for each link l ≡ ab, we have four labels of two unit vectors, one angle, and one r This notation scheme is well-suited for complete graphs, but generalization to arbitrary (noncomplete) graphs is obvious.

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real number: (ni(l) , nf (l) , ξl , ηl ) ≡ (nab , nba , ξab , ηab ), where i(l) and f (l) denote the initial and final endpoints (nodes) of the link l. Therefore, apart from the labels tab (its meaning will be clarified shortly), the state given in Eq. (132) associated with a spin network graph Γ is labeled by a number ηl and an angle ξl for each link l, and for each node a set of unit vectors n, one for each link at that node. These variables together admit a geometric interpretation of a simplicial three-complex as follows. The graph Γ is assumed to be dual to a simplicial decomposition of the spatial manifold, as each node is dual to a 3-simplex (chunk of space) and each link is dual to a face of two adjacent simplices. The vectors n’s at a node are outgoing unit vectors normal to the faces of the simplex dual to the node,s and the positive parameter ηl for a link l is related to a spin jl0 , which is the average of the area of the face dual to the link l. Furthermore, the simplicial extrinsic curvature is specified by an angle for each face of two adjacent simplices and is identified by ξl . To sum up, the state in Eq. (132) is specified by both an intrinsic geometry (labeled by n and jl0 ) and an extrinsic geometry (labeled by ξl ) for a simplicial three-complex due to Γ. To see that the holomorphic coherent states represent semiclassical states at some appropriate large-scale limit, we study their asymptotics for large ηl . Using the asymptotic formula R(jab ) (e−i(ξab +iηab )σ3 /2 ) ≈ e−iξab jab eηab jab |jab , +jab jab , +jab |

(138)

for ηab → ∞, it was shown in Ref. 89 that the holomorphic coherent state in Eq. (132) is given by the superposition     0 ,ξ ,σ 0 (jab ) (2jab + 1) cjab ΨΓ,jab ,Φa (nab ) (hab ), (139) ΨΓ,Hab ,tab (hab ) ≈ ab ab jab

ab

where cj 0 ,ξ,σ0 (j) is a Gaussian function multiplied by a phase:   (j − j 0 )2 0 0 (140) cj ,ξ,σ (j) := exp − e−iξj 2σ 0 with ηab 1 0 0 (2jab + 1) := , and σab := . (141) tab 2tab Here, Φa (nab ) is the coherent intertwiner introduced in Ref. 90, which is a linear superposition of an orthonormal intertwiner basis {ia }, given by 
m
··· Φia (nab )via 1 m ··· (142) Φa (nab )m1 ··· m1 ··· = 1

ia

with





Φia (nab ) = via ·

|jab , nab  ,

(143)

b

that in general nab = −nba . The difference between nab and −nba encodes connections of the intrinsic geometry.

s Note

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where |jab , nab  := R(jab ) (nab )|jab , +jab . The states ΨΓ,jab ,Φa (nab ) are spin networks with coherent intertwiners ΨΓ,jab ,Φa (nab ) (hab ) :=

   ia

 Φia (nab ) ΨΓ,jab ,ia (hab ).

(144)

a

The asymptotic states given by Eq. (139) are exactly the boundary semiclassical states used in Ref. 91 and the coefficients cj 0 ,ξ,σ0 (j) in Eq. (140) are the same as proposed in Ref. 92, which demand that the states are peaked both on the area (j 0 ) and on the extrinsic angle (ξ) with appropriate spread widths (specified by σ 0 ≡ 1/(2t) ≈ (j 0 )k with k < 2). These states represent the boundaries of Euclidean 4-geometries and are used to compute transition amplitudes for given boundary states in the Euclidean theory (see Sec. 12.6). The analysis of graviton propagators provides a methodology for studying the low-energy physics, and particularly the comparison to the classical theory of Euclidean GR confirms that the value ξ is fixed by ξ = γ cos−1 (−1/4). The construction of holomorphic coherent states for the Lorentzian theory however is much less established and demands more works of further research.

9. Spin Foam Theory Conventional QFTs admit two different formulations: the canonical (Hamiltonian) formalism and the sum-over-histories (path integral) formalism. So far we have concentrated on the canonical formulation of LQG. It is time to discuss the “spin foam” theory, which can be viewed as the covariant approach of LQG, alternative to the canonical approach. While the general structure of spin foam models matches nicely with the canonical theory of LQG, the precise relation between these two formalisms however is still not entirely clear and remains a key topic in current research (see Sec. 12.5). Up to now, the canonical theory of LQG and the theory of spin foams should be regarded as two closely related but independent approaches of QG. In this section, we present the basic ideas of the spin foam theory and refer readers to Refs. 13–17 for detailed construction of spin foam models. Also see Chap. 9 of Ref. 9 and Chap. 14 of Ref. 10 for shorter accounts.

9.1. From s-knots to spin foams Spin foams can be regarded as the “worldsurfaces” swept out by s-knots traveling and transmuting in time. A spin foam represents a quantized spacetime, in the same sense that an s-knot represents a quantized space. Consider the transition amplitude from one s-knot (k
| to another (k| W (k, k
) := (k|PˆC |k
)

(145)

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ˆ ]. as given in Eq. (124). The operator PˆC projects s-knots into the kernel of C[N Heuristically, we can write the projector as   ˆ |E) e−Et (E| = |E)(E| (146) PˆC = lim e−C[N ]t = lim t→∞

t→∞

E≥0

E=0

ˆ ] has a non-negative spectrum of E.t Consequently, assuming the operator C[N ˆ

W (k, k
) = lim (k|e−C[N ]t |k
). t→∞

(147)

In the same fashion of the path integral in quantum mechanics, inserting resolutions  of the identity 1 = k |k)(k|, we can expand the above expression as a product of small time-step evolutions  ˆ ˆ W (k, k
) = lim lim (k|e−C[N ]∆t |kM )(kM |e−C[N ]∆t |kM−1 ) t→∞ M→∞

k1 ,...,kM

ˆ

ˆ

· · · (k2 |e−C[N ]∆t |k1 )(k1 |e−C[N ]∆t |k
),

(148)

where ∆t ≡ t/M . For a fixed t, we can always make M big enough such that ∆t smaller than any given number. When ∆t is sufficiently small, we have ˆ

ˆ (kn+1 |e−C[N ]∆t |kn ) ≈ δkn+1 ,kn − ∆t N (v)(kn+1 | C[1]|k n ),

(149)

ˆ ]|kn ) ≡ N (v)(kn+1 | C[1]|k ˆ where (kn+1 | C[N n ) is nonvanishing only between two sknots kn , kn+1 that differ by the action of Cˆ at a node v. As discussed in Sec. 6.3, the action of Cˆ is to add or remove one or two extraordinary loops and also modify the colorings. Therefore, while δkn+1 ,kn keeps kn+1 to be the same as kn , the action ˆ ]|kn ) is on the nodes of s-knots as schematically illustrated in Fig. 3. of (kn+1 | C[N Consequently, for k = k
, Eq. (148) leads to W (k, k
) = lim

∆t→0

∞ 

(−∆t)N +1

N =0



ˆ ˆ (k|C[1]|k N )(kN |C[1]|kN −1 )

k1 ,...,kN kn+1 =kn


ˆ ˆ · · · (k2 |C[1]|k 1 )(k1 |C[1]|k ),

(150)

where we have absorbed the factor N (v) into ∆t and also identify k
≡ k0 and k ≡ kN +1 for shorthand. Now, recall that in Sec. 6.2, as a consequence of the intimate interplay between 3d diffeomorphism invariance and 3d short-scale discreteness, the value Eucl Φ in Eq. (106) turns out to be independent of  once  is sufficiently of (Ψ| CˆR  t As commented in Sec. 6.2, the projector P ˆC is independent of the choice of N (x), but we have ˆ ] in Eq. (146). Hermiticity can be easily to assume hermiticity and positive definiteness of C[N prescribed as discussed in the last paragraph in Sec. 6.3. Positive definiteness, however, is not p guaranteed, but heuristically we can replace the local scalar constraint C(x) with C(x)2 before regularization and quantization to yield positive definiteness. This heuristic tweak in fact makes good sense from the standpoint of the Master constraint program, which will be outlined in Sec. 12.1.

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small and consequently the regulator for  → 0 can be removed. The regulator ∆t on the right-hand side of Eq. (150) is analogous to  in the sense that reparametrization invariance in time now plays the role of 3d diffeomorphism invariance. That is, as long as ∆t is small enough, making it smaller will not change anything further. In addition to discreteness of space at the Planck length scale, let us postulate also discreteness of time at the Planck time scale. It is then suggested, somehow by the interplay between reparametrization invariance and short-scale discreteness in time, that the dependence on ∆t in Eq. (150) should go away as ∆t becomes sufficiently small and consequently the regulator for ∆t → 0 should be removed.u By removing the regulator ∆t, Eq. (150) then leads to


W (k, k ) =

∞  N =0

w(N )



A(σ (N ) ) ≡



w(N (σ))A(σ),

(151)

σ=(k,...,k
) kn+1 =kn

σ(N )

where w(N ) is a weight factor for different N arising from (−∆t)N +1 , and σ (N ) := (k ≡ kN +1 , kN , kN −1 , . . . , k1 , k
≡ k0 ) with kn+1 = kn is a discrete sequence of s-knots, which represents a “history” from k
to k with N + 1 times of intermediate state change. The amplitude associated with a particular history σ = (k, · · · , kv+1 , kv , · · · , k
) is given by  Av (σ), (152) A(σ) = v

where v labels the steps of the history, and Av (σ) is determined by ˆ Av (σ) ∼ (kv+1 | C[1]|k v)

(153)

up to a constant proportional factor that can be absorbed into rescaling of ∆t. In Eq. (151), the transition amplitude is cast in a sum-over-histories formulation, which is not a functional integral over continuous histories of fields but a sum over discrete histories of s-knots. A history σ = (k, . . . , kv+1 , kv , . . . , k
), kn+1 = kn , is called a spin foam. More precisely, imagine that a graph of an s-knot in an abstract 4d space moves upward along the “time” direction and the graph is changed by branching of its edges ˆ the worldsheet swept out by the moving and at each step under the action of C, changing graph is the spin foam. Call “faces” and denote by f the worldsurfaces of the links of the graph; call “edges” and denote by e the worldlines of the nodes of the graph; and call “vertices” and denote by v the points at which the edges branch. Figure 4 depicts a vertex of a spin foam corresponding to the action in Fig. 3, and Fig. 5 shows a simple examples of spin foams. u The short-scale discreteness in time is only postulated. Whether the canonical theory of LQG gives rise to temporal discreteness is unknown. The suggestion is only heuristic. Rigorously, to make sense of it, we might have to reformulate the regulator R in Sec. 6.2 such that it is intricately matched with the lapse function N in a 4d diffeomorphism covariant manner.

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Fig. 4. Left: A typical spin foam vertex (colorings of adjacent edges and faces are not indicated). Right: Time-slicing (foliation) of the left diagram gives the evolution of s-knots (time is chosen to be vertically upward). The trivalent node on the bottom is branched into three trivalent nodes on the top.

Fig. 5. Left: A simple spin foam composed of 2 vertices, 6 edges, and 6 faces (3 faces inside the cylinder are shaded; the other 3 on the surface of the cylinder are not). Its top and bottom boundaries, (k| and (k  |, are given by two θ-shaped s-knots. Middle: Time-slicing of the left spin foam shows the evolution from (k  | to (k| through an intermediate s-knot state. Right: The left and right diagrams overlapped for better visualization.

The combinatorial object defined by the collection and the adjacency relation of faces f , edges e, and vertices v is called a “two-complex” and denoted by Γ. As an sknot is identified not only by its graph but also the colorings of its links (irreducible representations) and nodes (intertwiners), accordingly a spin form denoted by σ = (Γ, jf , ie )

(154)

is determined by a two-complex Γ, the coloring of irreducible representations jf associated with faces f , and the coloring of intertwiners ie associated with edges e. 9.2. Spin foam formalism The discussion in the previous subsection motivates a sum-over-histories formulation of QG. For a given spin foam σ = (Γ, jf , ie ), it is natural to implement Av (σ) in Eq. (153) as a function Av (jf , ie ), called the vertex amplitude associated with the vertex v, where jf and ie are the colorings for the faces and edges adjacent to

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v. Consequently, Eqs. (151) and (152) lead to the formal spin foam expression of the amplitude transition   W (k, k
) = w(Γ(σ)) Av (jf , ie ), (155) ∂σ=k∪k


jf ,ie v

where ∂σ = k ∪ k
indicates that the boundary (i.e. initial and final s-knots) of σ is given by the union of k
and k, and w(Γ) is a weight factor that depends only on the two-complex. The weight factor w(Γ) is a generalized form of w(N ), where N = N (Γ) is the number of vertices of Γ. It is often more convenient to recast Eq. (155) in the extended form     w(Γ(σ)) Af (jf ) Ae (jf , ie ) Av (jf , ie ), (156) W (k, k
) = ∂σ=k∪k


e

jf ,ie f

v

where Af and Ae are the amplitudes associated with faces and edges, which in principle can be absorbed into a redefinition of Av . For most models in the literature, Af (jf ) is simply given by the dimension dim(jf ) of the irreducible representation jf . Consequently the transition amplitude is given in the form     W (k, k
) = w(Γ(σ)) dim(jf ) Ae (jf , ie ) Av (jf , ie ), (157) ∂σ=k∪k


e

jf ,ie f

v

and accordingly the partition function is given by     Z= w(Γ(σ)) dim(jf ) Ae (jf , ie ) Av (jf , ie ). σ

jf ,ie f

e

(158)

v

A spin foam model is then formally defined by the specification of (i) A category of two-complexes Γ and the weight factor w(Γ). (ii) A Lie (or deformed Lie) group and associated irreducible representations jf and intertwiners ie . (iii) Functions of the vertex amplitude Av (jf , ie ) and the edge amplitude Ae (je , ie ). There are various spin foam models that have been elaborated.14–17 They provide tentative quantum theories of 3d GR, 4d BF theories, 4d Euclidean GR, etc. A remarkable feature is that many very different models all coincide with the form of Eq. (158), suggesting that this expression can be viewed as a general backgroundindependent formalism of covariant QFTs (also see Sec. 12.6).

10. Black Hole Thermodynamics From theorems proved by Hawking et al.,93,94 which reveal a remarkable resemblance between a set of laws obeyed by black holes and the principles of thermodynamics, Bekenstein suggested that a Schwarzschild black hole should carry an

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entropy proportional to the area of its event horizon divided by the Planck area95–97 SBH = a kB

A , G


(159)

where kB is Boltzmann’s constant, a is a proportional constant, and A = 4π(2GM )2 is the area of the event horizon with M being the mass of the black hole. By the standard thermodynamical relation T −1 = dS/dE ≡ dS/dM , this implies that the black hole has a finite temperature T =


. a32πkB GM

(160)

Shortly after Bekenstein’s conjecture, by studying QFT in a gravitational background, Hawking showed that black holes emit thermal Hawking radiation98,99 corresponding to the Hawking temperature T =


, 8πkB GM

(161)

which tightly confirms Bekenstein’s conjecture and fixes the constant a = 1/4. Equation (159) with a = 1/4 is often referred to as the Bekenstein–Hawking formula A , (162) 4G
where the subscript “BH” coincidentally stands for both “black hole” and “Bekenstein–Hawking”. In the modern interpretation of entropy in statistical mechanics, entropy (divided by kB ) is defined as the logarithm of the number of admissible microstates for a given macroscopic state. What then are the microscopic degrees of freedom responsible for the black hole entropy? Can we derive Eq. (162) from first principles? As
enters in Eq. (162), answers to these questions require a quantum theory of gravity. In LQG, a detailed description of black hole thermodynamics has become an active direction of research. One of the major achievements of LQG is to derive the Bekenstein–Hawking formula from the first principles for the Schwarzschild and other black holes.100–108 In Secs. 10.1 and 10.2, we present the basic ideas, following Refs. 109 and 110 (also see Sec. 8.2 of Ref. 9). Readers are referred to Chap. 15 of Ref. 10 for a rigorous treatment and Sec. 8 of Ref. 8 for a shorter account. More recent advances in the black hole entropy can be found in the excellent review paper Ref. 111. We mention some of the important results in Sec. 10.3. SBH = kB

10.1. Statistical ensemble The microscopic degrees of freedom responsible for the black hole entropy SBH are those that can participate in the energy exchanges with the exterior. Because, by definition of black holes, the states of gravity and matter inside a black hole have no effect on the exterior, the microstates of the interior of the black hole are irrelevant

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for SBH measured from the exterior.v Observed from the exterior, therefore, the entropy of the black hole (for its thermal interaction with the surrounding exterior) is completely determined by the geometry of the event horizon. More precisely, the microcanonical ensemble is composed of the microstates of the horizon geometry that give rise to an exactly specified total area. Denote by N (A) the number of the microcanonical ensemble of area A, the quantity S(A) = kB ln N (A) is then the entropy for the black hole as far as its thermal interaction with the exterior is concerned. Given any arbitrary surface (say, not a horizon of any kind), we can of course ask how many microstates of the surface correspond to a specific area, but the number of the microstates usually has no thermodynamical significance. As heat or information can flow across a surface without changing its geometry, the geometry of a surface and thus the number of its microstates in general have nothing to do with the heat exchange or the notion of entropy. As originally argued by Jacobson,112 it is for any causal horizon that the heat flow always accompanies a stress–energy tensor that distorts the horizon geometry and consequently the entropy measured by the observer who is separated from the system by the causal barrier is given by the geometry of the causal horizon (see also Refs. 113–115, in the context of LQG). The black hole entropy given by the geometry of the event horizon is a special case of Jacobson’s argument. 10.2. Bekenstein–Hawking entropy Let us now compute the number N (A) from the quantum geometry of LQG. First, consider that the quantum state of the geometry of a 3d ADM leaf Σt is given by an spin network state |s. The horizon is a 2d surface S imbedded in Σt and its geometry is determined by its intersections with the spin network |s. As we are interested in the geometry as probed from the exterior, the surface S we consider is, more rigorously, the one that is outside but infinitesimally close to the horizon. Therefore, we exclude the possibility that the spin network nodes lie on S and, instead of Eq. (75), the area of S is simply given by Eq. (74)  ji (ji + 1), (163) A(S) = 8πG
γ i

where j1 , . . . , jn are the colorings of the links intersecting the surface S. For a given A, how many admissible microstates of the horizon are there then? From the perspective of an external observer, an admissible microstate corresponds to a possible choice of the sequence j1 , . . . , jn that gives rise to A via Eq. (163) and a possible way of “ending” the links to the exterior. A possible “end” of a link with coloring j is simply a vector in the vector space of the j representation, which has a (2j + 1)-fold multiplicity. To sum up, the admissible microstates are obtained v For instance, in the Kruskal extension of a Schwarzschild black hole, the region III may contain billions of galaxies, but these do not bear any detectable consequences for us in the region I.

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by considering all sequences j1 , . . . , jn that give the area A, and for each sequence ! there is a multiplicity of i (2ji + 1) possible states. For a large A much greater than the Planck area 2P 1 ≡ G
, we assume that the number of admissible microstates is dominated by the case of ji = 1/2 for all intersecting links (we will come back to this assumption shortly). In this case, the area of a single link is given by √ Aj=1/2 = 4π 3 G
γ (164) and thus the number of intersections is n=

A Aj=1/2

=

A √ . 4π 3 G
γ

(165)

Each j = 1/2 link has a multiplicity of (2j+1) = 2, so the total number of admissible microstates for a given large A is given by N (A) = 2n = 2 4π

√A 3 G
γ

.

(166)

Consequently, we obtain the entropy of the black hole SBH = kB ln N (A) =

1 ln 2 A √ kB , γ 4π 3 G


(167)

which agrees perfectly with the Bekenstein–Hawking entropy Eq. (162) provided that the Barbero–Immirzi parameter is fixed at the value γ=

ln 2 √ = 0.127384 . . . . π 3

(168)

The Bekenstein–Hawking entropy can be calculated for different kinds of black holes by LQG. The striking feature is that they all lead to the same value of γ and thus ensure the consistency of the black hole thermodynamics under the framework of LQG. We have made the assumption that the leading contribution to the entropy comes entirely from the lowest states (ji = 1/2), as had been suggested in Ref. 102 and considered true for several years. The detailed computation of Refs. 116 and 117 based on a practical rephrasing of the combinatorial problem pointed out that in fact all quantum states (not only the lowest ones) must be taken into account. This leads to a revision from Eq. (168) for the value of γ, which can be numerically calculated at arbitrary accuracy117 as γ = γM ≡ 0.23753295796592 . . ..

(169)

The simple calculation presented above on the flawed assumption nevertheless grasps the conceptual essence of relating the number of microstates to the black hole entropy.

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10.3. More on black hole entropy Section 10.2 gives a simple account for the black hole entropy from first principles of LQG. A more rigorous approach is the modeling of black holes in LQG by using isolated horizons as inner boundaries.100–102 Quantization of the degrees of freedom for the isolated horizon can be understood as a Chern–Simons theory. Obtained by tensoring the Chern–Simons boundary Hilbert space and the LQG bulk Hilbert space, the Hilbert space of the resulting model provides the groundwork to address the statistical problems of black holes. The combinatorial problem associated with the computation of the black hole entropy can be rephrased in a more manageable way,116,117 allowing one to compute the asymptotic behavior of the black hole entropy in the limit of large horizon area. For A  2Pl , the computation gives an exact formula to the subleading order as    0 γ A 1 A A M −1 kB S = ln N (A) = − ln +O , (170) 4γ G
2 G
2Pl where γM is a constant, whose numerical value is given by Eq. (169). This formula agrees with Eq. (159) on the proportionality to A in the leading order and also obtains the precise subleading order correction as a logarithmic function of A. Comparison to the Bekenstein–Hawking formula in Eq. (162) fixes the Barbero– Immirzi parameter γ to be γ = γM . For small areas, the precise number counting was first suggested in Refs. 118 and 119 and later has been thoroughly investigated by employing numbertheoretical and combinatorial methods (see Ref. 120 for a detailed account and Ref. 111 for a review). The key discovery is that, for microscopic black holes, the so-called black hole degeneracy spectrum when plotted as a function of the area exhibits a persistent “periodicity” (more precisely, a modulation with a regular period of growing magnitude). This produces an effectively evenly-spaced area spectrum, despite the fact that the area spectrum in LQG is not evenly spaced, and makes contact in a nontrivial way with the evenly-spaced black hole horizon area spectrum predicted by Bekenstein and Mukhanov under general conditions.121 The periodicity in the degeneracy spectrum leads to a striking “staircase” structure with regular steps when the black hole entropy is plotted as a function of the area. The staircase behavior eventually disappears in the large area limit. The framework of precise number counting also predicts the subleading correction to the Bekenstein–Hawking law. For various model settings, the subleading correction generically takes the form a1 kB ln A/2Pl , where the coefficient a1 is independent of the value of γ but differs for different models. The logarithmic correction is qualitatively in agreement with those obtained by different approaches (including those based on asymptotic symmetries, horizon symmetries, and certain string theories), despite very different physical assumptions. There are some indications that even the coefficient a1 might be universal, up to differences on the treatment of angular momentum and conserved charges.122

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Recently, an explicit SU (2) formulation for the black hole entropy has been developed based on covariant Hamiltonian methods.123–125 This formulation avoids the partial gauge fixing of the standard approach and gives rigorous support for the earlier proposal that the quantum black hole degrees of freedom could be described by an SU (2) Chern–Simons theory. 11. Loop Quantum Cosmology Loop quantum cosmology (LQC) is a finite, symmetry-reduced model of LQG that applies principles of the full theory to cosmological settings. Thanks to the mathematical simplifications, many obscure aspects (e.g. quantum dynamics and semiclassical physics) in the full theory become transparent in LQC. The framework of LQC provides a “bottom-up” approach to the full theory, in which many ideas of LQG can be explicitly implemented and tested. The distinguishing feature of LQC is that the quantum geometry of LQG gives rise to a brand new quantum force that is inappreciable at low spacetime curvature but rises very rapidly and opposes the classical gravitational force in the Planck regime. As a consequence, for a variety of models of LQC, the cosmological singularity (big bang, big crunch, big rip, etc.) is avoided by the opposing force, therefore affirming the long-held conviction that singularities in GR signal a breakdown of the classical theory and should be resolved by the quantum effects of gravity. Particularly, the big bang singularity is replaced by a quantum bounce, which bridges the present expanding universe with a preexistent contracting universe. The new cosmological scenario suggests a change of the paradigm in the standard big-bang cosmology. In what follows, to illustrate the basic ideas of LQC, we consider the simplest setting — LQC in the k = 0 Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) model and particularly follow the lines of Ref. 126. See Refs. 18–21 for more detailed construction and more models of LQC.

11.1. Symmetry reduction The first step is to impose symmetries of cosmology — spatial homogeneity and oftentimes also isotropy — on the phase space variables before quantization. Symmetry reduction deliberately ignores infinitely many degrees of freedom and thus brings up the question whether the results from the full theory of LQG, if can be obtained after all, should resemble any predictions by LQC. The arguments in Sec. 1.2 of Ref. 19 suggest that the answer is likely to be affirmative, provided that the framework of LQC captures the essential features of the full theory. In the k = 0 FLRW model, both homogeneity and isotropy are assumed and the spacetime metric in the comoving coordinates is given by dτ 2 ≡ gµν xµ dxν = −N (t)dt2 + a(t)2 dx2 .

(171)

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With the imposition of homogeneity and isotropy, the fundamental fields Aia (x) and ˜ ai (x) are reduced to two variables c and p E Aia (x) → c,

˜ ai (x) → p, E

(172)

which are independent of x because of homogeneity and satisfy the canonical relation 8 (173) {c, p} = πGγ, 3 a which is the symmetry-reduced counterpart of Eq. (17). As E˜ i is related to “area” in the sense of Eq. (73), the symmetry-reduced variable p represents the area of the surfaces of a finite-sized  3 cubic cell V (which is chosen to make sense of the spatial 3 integral Σ d x → V d x). More precisely

p = a2 L 2 ,

(174)

where a is the scalar factor in Eq. (171) and L is the coordinate length of the edges of V. Under the symmetry reduction, the Gauss constraint is trivially satisfied, and the diffeomorphism constraint is no longer an issue as the diffeomorphism invariance is gauge fixed in accord with the homogeneity. Only the scalar constraint remains significant. In the k = 0 FLRW setting, we have F = dA + A ∧ A = A ∧ A and K = (A − Γ)/γ = A/γ, which lead to F = A ∧ A = γ 2 K ∧ K, or equivalently j k i = 2Ai[a Ajb] = 2γ 2 K[a Kb] . Correspondingly, Eq. (22) becomes ij k Fab


a b
˜ a E˜ b ˜ E E˜ i E N i j ij j k 3  d x  k Fab , ≡ − d x 2Ai[a Ajb] , (175) 2 8πGγ |q| |q| V V  where N = N (t) is independent of x, and the spatial integral Σ d3 x is restricted to L L L  3 d x ≡ 0 dx1 0 dx2 0 dx3 to make C[N ] finite. By Eq. (172), it follows from V Eq. (175) that the gravitational part of the scalar constraint in terms of c and p is given by

N C[N ] = − 8πGγ 2

3

Cgrav (N ) = −

6N p2  c2 , 2 8πGγ p3

(176)

where the lapse function N (x, t) = N (t) is chosen to be homogeneous. ˜ In the full theory, instead of A(x) and E(x), the holonomy hγ and the electric flux over a surface ES,f are used as the fundamental variables for loop quantization. In the symmetry-reduced theory, analogous to Eq. (29), the variable c is replaced by the “holonomy” of c, defined as Nµ := eiµc/2

(177)

for an arbitrary µ ∈ R. On the other hand, corresponding to Eq. (30), p remains the good variable, as it is redundant to specify S and f (x) in the homogeneous and

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Symmetry reduction in the k = 0 FLRW model.

In the full theory

In the k = 0 FLRW model

Aia (x) ˜a E i (x)

p

c

˜ b (y)} = 8πGγ δi δb δ3 (x − y) {Aia (x), E j j a

{c, p} =

8 πGγ 3

CG [λ]

—

CDiff [N]

—

C[N ] in Eq. (22)

Cgrav (N ) in Eq. (176)

holonomy: hγ in Eq. (29)

Nµ := eiµc/2 , µ ∈ R

electric flux: ES,f in Eq. (30)

p

loop algebra: {hγ , ES,f } = . . .

µNµ {Nµ , p} = i 4πGγ 3

isotropic setting. The variables Nµ and p form the closed “loop algebra” {Nµ , p} = i

4πGγ µ Nµ 3

and {Nµ , Nν } = 0.

(178)

The symmetry reduction into the k = 0 FLRW model is summarized in Table 3. 11.2. Quantum kinematics Analogous to the linear space Cyl of cylindrical functions of A, i.e. ΨΓ,f [A], defined in Eq. (36) in the full theory, we begin with the linear space CylS of almost periodic functions of c defined as   ξk Nµk (c) ≡ ξk eiµk c/2 , (179) Ψ(c) = k

k

where k runs over a finite number of integers, µk ∈ R, and ξk ∈ C. Analogous to the inner product of Cyl defined in Eqs. (37) and (38), the inner product of CylS is defined via Nµ1 | Nµ2  ≡ µ1 | µ2  = δµ1 µ2 ,

(180)

where the right-hand side is the Kronecker delta, but µ1 , µ2 ∈ R take continuous values. This inner product is very different from that defined via µ1 | µ2  = δ(µ1 − µ2 ) with the Dirac delta function. To highlight this nuance, CylS is said to be in the “polymer representation” of µ. S is the Cauchy The (gravitational part) of the kinematical Hilbert space Hkin S S completion of Cyl with respect to the inner product Eq. (180). That is, Hkin = 2 L (RBohr , dµBohr ), where RBohr is the Bohr compactification of R and dµBohr is the S is given by {Nµ | µ ∈ R}. By corresponding measure. An orthonormal basis of Hkin iµc/2 is denoted by |µ with Dirac’s bra-ket notation, the map c → e ei

µc 2

= c | µ,

(181)

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and consequently we have |Ψ =



|µµ | Ψ ≡



µ

Ψ(µ)|µ.

(182)

µ

Analogous to the holonomy, area, and volume operators defined in Sec. 5, we S have corresponding operators acting on Hkin i µc i µc 2 Ψ(c) := e 2 Ψ(c), Nˆµ Ψ(c) ≡ e

pˆ Ψ(c) := −i

8πγ 2 ∂Ψ(c)  , 3 Pl ∂c

3/2 Vˆ Ψ(c) := |ˆ p| Ψ(c),

(183) (184) (185)

√ where Pl ≡ G
is the Planck length. When acting on the basis state |µ, these operator behave as
i µc ˆµ
|µ ≡ e 2 |µ = |µ + µ , N

(186)

8πγ 2  µ|µ, 6 Pl  3/2 8πγ ˆ V |µ = Vµ |µ ≡ 3Pl |µ. |µ| 6 pˆ |µ =

(187) (188)

The correspondence for quantum kinematics between the full theory and the symmetry-reduced theory is listed in Table 4.

Table 4.

Quantum kinematics of LQG and LQC in the k = 0 FLRW model.

LQG

LQC in the k = 0 FLRW model

Cyl

CylS

cylindrical functions:

almost periodic functions: P P Ψ(c) = k ξk Nµk ≡ k ξk eiµk c/2

ΨΓ,f [A] in Eq. (36) inner product: ΨΓ,f | ΨΓ
,g
 in Eqs. (37), (38) H = L2 [A, dµAL ]

inner product: Nµ1 | Nµ2  ≡ µ1 | µ2  = δµ1 µ2 2 HS kin = L (RBohr , dµBohr )

|Γ, jl , αl , βl  in Eqs. (39), (40)

|µ

spin networks |Γ, jl , in 

|µ (Gauss constraint trivial)

s-knots (K, c|

|µ (no diffeomorphism constraint)

ˆ γ in Eqs. (63), (64) holonomy operator: h

iµ
c/2 |µ = |µ + µ  ˆµ
|µ ≡ e
N

ˆ area operator: E(S) in Eq. (75)

pˆ |µ =

volume operator: Vˆ (R) in Eq. (83)

Vˆ |µ =

8πγ 6

“

2Pl µ|µ ”3/2 8πγ |µ| 3Pl |µ ≡ Vµ |µ 6

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11.3. Quantum constraint operator Just as in the full theory, we have to regularize the scalar constraint before quantization. In Eq. (175), the nonpolynomial factor 1/ |q| is dealt with Thiemann’s trick and Fab is replaced by a holonomy long a square loop
. Corresponding to Eqs. (100) and (101) (with R(j) = R(1/2) , C iJ = 1), it turns out Cgrav with N = 1 can be written as (λ) , Cgrav = lim Cgrav

(189a)

λ→0

(λ) =− Cgrav

4 sgn(p)  ijk  (λ) (λ)  (λ) −1  (λ) −1 (λ)  (λ) −1 hj hk  Tr hi hj hi hk ,V 8πγ 3 λ3 G "

ijk

% # $ 4 sgn(p)  (λ)  (λ) −1 = sin λc − hk Tr τk hk ,V sin λc, 8πγ 3 λ3 G

(189b)

k

where (λ)

hi

= cos

λc λc + 2τi sin 2 2

(190)

is the holonomy along the edge of
, which is in the xi direction and of coordinate length λL, and λ plays the role of  in Eq. (100) (more precisely, λL = ). (λ) (λ) It is straightforward to obtain the quantum operator Cˆgrav associated with Cgrav , but the limit λ → 0 is ill-defined. Unlike the full theory of LQG, the dependence (λ) of Cˆgrav on the regulator λ does not go away because the diffeomorphism has been gauge fixed. In order to inherit the spatial discreteness from LQG appropriately, we take the prescription by hand to shrink the square loop
to a finite area equal to the area gap ∆, i.e. the smallest nonzero eigenvalue of area, given by Eq. (76).w Consequently, we are led to choose for λ a specific function λ = µ ¯(p) given by Area of
= a2 (λL)2 = µ ¯2 |p| = ∆, or equivalently

& µ ¯=

∆ = |p|

 √ 3 3 1 √  . 2 |µ|

(191)

(192)

Equation (186) suggests that, in a heuristical sense, d µc ¯  ei 2 Ψ(µ) = eµ¯ dµ Ψ(µ),

even though the differential operator define the affine parameter v := K sgn(µ)|µ| w As

d dµ

3/2

,

(193)

is ill-defined in L2 (RBohr , dµBohr ). Let us √ 2 2 with K =  √ , 3 3 3

(194)

this is only a phenomenological prescription, different numerical values for ∆ of the same order of magnitude are also used in the literature.

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such that formally µ ¯

d d = . dµ dv

(195)

µc ¯  In the v-representation, the action of ei 2 can then be defined as

µc ¯  ei 2 Ψ(v) = Ψ(v + 1),

and correspondingly Eq. (188) leads to  3/2 4πγ |v| 3 Vˆ |v =  |v. 3 K Pl

(196)

(197)

µc ¯  (λ) As ei 2 and Vˆ are now well defined, Cgrav in Eq. (189b) can be promoted to the quantum operator ' ( 3i sgn(µ)  µ ¯c ˆ µ ¯c µ ¯c ˆ µ ¯c  ˆ Cgrav = sin(¯ µc) sin V cos − cos V sin sin(¯ µc), (198) πγ 3 µ ¯3 2Pl 2 2 2 2

where we have chosen the symmetric ordering to make Cˆgrav hermitian. Its action on Ψ(v) is given by Cˆgrav Ψ(v) = f+ (v)Ψ(v + 4) + f0 (v)Ψ(v) + f− (v)Ψ(v − 4), where 27 f+ (v) = 16



4π KPl |v + 2|||v + 1| − |v + 3||, 3 γ 3/2

(199)

(200a)

f− (v) = f+ (v − 4),

(200b)

f0 (v) = −f+ (v) − f− (v).

(200c)

Meanwhile, to make the quantum dynamics more transparent, we add a free massless scalar φ(x, t) = φ(t) to serve as the internal time. At the classical level, the matter part of the scalar constraint is given by −3/2 2 pφ ,

Cmatt = 8πG|p|

(201)

where pφ = a3 L3 φ˙

(202)

is the momentum of φ, and φ˙ is the derivative with respect to the proper time. The matter part of the kinematical Hilbert space is given by the ordinary construction L2 (R, dφ) without any “holonomization.” However, the inverse triad factor  −3/2 still has to be regularized by Thiemann’s trick as for 1/ |q| in Cgrav . Up |p| to quantization ambiguities, the quantum operator for |p|−3/2 turns out to be  3/2 3  − 32 |p| Ψ(v) = B(v)Ψ(v), (203) 4πγ2Pl

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where B(v) =

 3 3 1/3 1/3 3 K|v| ||v + 1| − |v − 1| | . 2

(204)

Consequently, the total quantum constraint equation is given by ˆ CΨ(v, φ) = (Cˆgrav + Cˆmatt )Ψ(v, φ) = 0,

(205)

which is explicitly expressed as ∂φ2 Ψ(v, φ) = B(v)−1 (C+ (v)Ψ(v + 4, φ) + C0 (v)Ψ(v, φ) + C− (v)Ψ(v − 4, φ)) =: −ΘΨ(v, φ),

(206)

where C+ (v) =

3πKG |v + 2|||v + 1| − |v + 3||, 8

(207a)

C− (v) = C+ (v − 4),

(207b)

C0 (v) = −C+ (v) − C− (v).

(207c)

11.4. Physical Hilbert space The form of the constraint equation (206) suggests that it is natural to regard φ as the internal time, with respect to which Ψ(v) evolves. To implement this idea, an appropriate kinematical Hilbert space for both geometry and the scalar field is chosen to be Hkin = L2 (RBohr , B(v)dvBohr ) ⊗ L2 (R, dφ). It is straightforward to show that both the operators pˆφ = −i
∂φ and Θ are hermitian in Hkin , and furthermore Θ is positive definite. The operator Θ is a second-order difference (cf. differential) operator. Consequently, the space of physical states (i.e. the space of solutions to the constraint equation) is naturally divided into sectors each of which is preserved by the “evolution.” Denote by L the “lattice” of points {|| + 4n; n ∈ Z} ∪ {−|| + 4n; n ∈ Z} in the v-axis and denote by H grav the subspace of L2 (RBohr , B(v)dvBohr ) with states whose support is restricted to lattice L . Each of H grav is mapped to itself by Θ. Let ek (v) in H grav be the eigenstate of Θ Θek (v) = ω(k)2 ek (v),

(208)

where ω(k)2 is the corresponding eigenvalue. The general solution to the quantum constraint Eq. (206) with initial data in H grav is then given by
∞  ˜ + (k)ek (v)eiωφ + Ψ ˜ − (k)ek (v)∗ e−iωφ , Ψ(v, φ) = (209) dk Ψ −∞

˜ ± (k) are in L (R, dk). The positive/negative-frequency solutions satisfy a where Ψ Schr¨ odinger type first order differential equation in φ: ∂Ψ± √ = Θ Ψ± . ∓i (210) ∂φ 2

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Therefore, the positive/negative-frequency solutions with the initial state Ψ(v, φ0 ) are given by √ Θ (φ−φ0 )

Ψ± (v, φ) = e±i

Ψ(v, φ0 ).

(211)

The positive-frequency and negative-frequency solutions can be discussed separately, and we focus on the former ones in what follows. labeled by  ∈ [0, 2] consists of The sector of the physical Hilbert space Hphys positive-frequency solutions Ψ(v, φ) to Eq. (210) with initial data Ψ(v, φ0 ) in the symmetric sector of H grav .x The physical inner product is defined as  Ψ1 | Ψ2 phys := B(v)Ψ1 (v, φ0 )∗ Ψ2 (v, φ0 ). (212) v∈{±| |+4n;n∈Z}

It is easy to show that this definition is independent of the choice of φ0 as long as Ψ1,2 (v, φ) are solutions to Eq. (210).  Dirac operators acting on Hphys are pˆφ and a family of operators |v| φ parametrized by φ, defined as pˆφ Ψ(v, φ) := −i
√

 Ψ(v, φ) := ei |v| φ

∂Ψ(v, φ) , ∂φ

Θ (φ−φ0 )

|v|Ψ(v, φ0 ).

(213)

(214)

 preserve the sector H Both pˆφ and |v| phys and are hermitian with respect to φ · | ·phys . The expectation value Ψ| pˆφ |Ψphys gives the momentum of φ, which is a constant of motion, i.e. independent of φ. On the other hand, the expectation value  |Ψphys tells how big the volume of V is when the internal time takes the Ψ||v| φ  in value φ, in the style of Eq. (123). It should be noted that the definition of |v| φ

Eq. (214) is independent of the choice of φ0 . 11.5. Quantum dynamics

The eigenvalue problem Eq. (208) can be numerically solved (by a computer), and the eigenfunctions ek (v) can be used to build coherent states. Particularly, ˜ − (k) = 0 and in Eq. (209), a natural choice is to set Ψ ˜ + (k) = e− Ψ

k−k 2σ2

e−iωφ , 

(215)

with a suitable small spread σ, to obtain a coherent physical state Ψpφ ,vφ (v, φ). 0 √ The resulting state is peaked at Ψpφ ,vφ | pˆφ |Ψpφ ,vφ phys = pφ ≡ − 12πG
k  and 0 0  |Ψp ,v phys = v  , where v  is determined by φ . at Ψp ,v | |v| φ

φ0

φ0

φ

φ0

φ0

φ0

To give a coherent state that is semiclassical at late times, we have to choose large values for pφ (i.e. k   −1) and for vφ 0 (i.e. vφ 0  1). Given such a coherent sign flip v → −v corresponds to reverse of the triad orientation. Because of no parity violating processes, we are led to choose the symmetric sector in which Ψ(−v) = Ψ(v).

x The

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physical state, the evolution with respect to the internal time φ can be read out from the expectation value v(φ) :=

 |Ψp ,v phys Ψpφ ,vφ | |v| φ φ φ 0

0

Ψpφ ,vφ | Ψpφ ,vφ  0

0

.

(216)

phys

The numerical results show that the trajectory of v(φ) follows the classical solution at large scale but at the Planck regime undergoes the quantum bounce, which bridges the expanding classical solution with the contracting one (see Fig. 4 in Ref. 126). The quantum bounce takes place when the energy density √ ρφ of the scalar field approaches the critical density ρcrit ≈ 3/(8πGγ 2 ∆) = 3/(16π 2 γ 3 G2
) of the Planck scale. More precisely, the energy density operator is defined as  3   p2φ p2φ K2 3 | := ρ ≡ (217) φ φ 2 |p3 |φ 2 4πG
γ |v 3 |φ and its expectation value ρφ (φ) :=

 phys Ψpφ ,vφ | ρ φ |φ |Ψp φ ,vφ 0

Ψpφ ,vφ | Ψpφ ,vφ  0

≈

K2 2



3 4πG
γ

0

3

0

phys

(pφ )2 v(φ)3

(218)

is bounded above by ρcrit (see Fig. 6 in Ref. 126). If the internal time φ is expressed in terms of the proper time t via a3 L3 dφ = ˆ pφ dt = pφ dt in accord with Eq. (202),y the trajectory of v(φ) in relation to ρφ (φ) can be described rather accurately by the effective equation for the scalar factor a(t)  2   8πG a˙ ρφ = ρφ 1 − , (219) a 3 ρcrit which is the Friedmann equation modified by the opposing quantum force. 11.6. Other models A slight simplification in the model discussed above leads to the “solvable LQC” (sLQC).127 In sLQC, analytic solutions are available and it is shown that the quantum bounce is generic not only for coherent states, but for any arbitrary physical states and the matter density has an absolute upper bound given by ρcrit =

3 . 8πGγ 2 ∆

(220)

y The change of variables from φ to t makes sense only for sharply peaked coherent states. In the quantum theory, the internal time is more fundamental than the proper time, which is a derived notion only meaningful for sharply peaked coherent states.

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The construction of LQC has been extended to a variety of models with various degrees of rigor, including k = +1 (Ref. 128) and k = −1 (Ref. 129) FLRW models, possibly with a nonzero cosmological constant,130–132 Bianchi I,133–137 Bianchi II138 and Bianchi IX139 models, Kantowski–Sachs model,140,141 unimodular model,142 higher-order holonomy extension,143,144 and much more. The bouncing scenario has been shown to be robust in different models. Particularly, the formalism of LQC in the Kantowski–Sachs model can be used to study the loop quantum geometry of the interior of a Schwarzschild black hole.145–149 The study suggests that, just like resolution of the cosmological singularity, the classical black hole singularity is also resolved by the loop quantum effects and replaced by the quantum bounce, which either bridges the black hole interior to a white hole interior or gives birth to a baby black hole.149,z Furthermore, inhomogeneity has also been taken into consideration in the Gowdy model (the simplest of the inhomogeneous models)150–155 as well as in the framework of “lattice LQC”.156,157 12. Current Directions and Open Issues In this review article, we gave a self-contained introduction to the canonical formulation of LQG and briefly covered the ideas of the spin foam theory, black hole thermodynamics, and LQC. The core formalism of LQG provides a coherent framework in which the fundamental principles of GR and QFT consort with each other in harmony without invoking additional hypotheses. The key result of LQG is a compelling physical picture of quantum space, which is made up of quantized areas and volumes of the Planck scale in a profoundly background-independent fashion. Despite many remarkable achievements, the theory of LQG is still far from complete, especially for the aspects of quantum dynamics and low-energy limit. A wide range of research has been developed for the open problems. We conclude this review by outlining a nonexhaustive list of recent advances that are not covered in the main text and also addressing some open issues along the list. One can easily appreciate that LQG has grown into a vast and active area of research along many various directions. 12.1. The master constraint program The scalar constraint remains the major unsolved problem in LQG as noted in Sec. 6. The master constraint program158 (also see Sec. 10.6 of Ref. 10 for a systematic account) proposed an elegant solution to the difficulties concerning the algebra z Validity

of this approach however might be questionable, since the Schwarzschild interior is extensible beyond its boundary (event horizon) and thus cannot be considered as a self-contained universe (i.e. it is not a genuine Kantowski–Sachs spacetime). The more rigorous approach is to consider the black hole interior and exterior as a whole in the framework of spherically symmetric midisuperspaces (see Sec. 12.10).

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of commutators among smeared scalar constraints in terms of the so-called master constraint, which combines the smeared scalar constraints C[N ] := Σ d3 xN (x)C(x) for all smearing functions N (x) into a single constraint
C(x)2 d3 x  , M= (221) |q| Σ  where the factor 1 |q| has been incorporated to make the integrand a scalar density of weight 1 so that M is diffeomorphism invariant, i.e. {CDiff [N], M} = 0. It is clear that M = 0 implies the infinitely many constraints C(x) = 0 for ∀x ∈ Σ and vice versa. The single constraint M in place of C[N ] greatly reduces the number of constraints and drastically simplifies the Poisson bracket structures among constraints: Eq. (24e) is replaced by {CDiff [N], M} = 0, and more notably the complicated structure of Eq. (24f) is now reduced to the trivial algebra {M, M} = 0. As the master constraint renders the constraint algebra as a genuine Lie algebra, the program opens a new avenue for understanding the quantum dynamics and establishing the semiclassical limit. The master constraint program has various advantages in an attempt to define the physical inner product and formulate rigorous path integrals,159,160 and it seems to be more directly related to the spin foam formalism than the standard formulation of LQG. 12.2. Algebraic quantum gravity The master constraint program has evolved into a fully combinatorial theory known as algebraic quantum gravity (AQG)161–164 (also see Sec. 10.6.5 of Ref. 10). AQG is closely related to yet very different from LQG by the fact that no topology or differential structure of space is assumed a priori. In this sense, background independence of AQG is even more compelling, and it is possible to talk about topology change. A viable semiclassical machinery that involves only nongraphchanging operators has been suggested, and considerable progress has been made in providing contact with the low-energy physics. 12.3. Reduced phase space quantization There are two major approaches to the canonical quantization of a field theory with gauge symmetries: the so-called “Dirac approach” and the “reduced phase space approach.” In the Dirac approach, one first construct the Hilbert space that represents the partial (gauge-variant) observables and then impose the constraints at the quantum level to select the physical (gauge-invariant) states. In the reduced phase space approach, one first constructs the physical (gauge-invariant) observables at the classical level and then directly construct the Hilbert space that represents the physical observables. (Of course, it is possible to take a “mixed” approach by implementing different strategies for different gauge symmetries.)

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The advantage of the Dirac approach is that the algebra of partial observables is usually simple enough and thus the (kinematical) Hilbert space is easy to construct, but one has to deal with spurious degrees of freedom, which are the possible source of quantization ambiguities and quantum anomalies. On the other hand, the advantage of the reduced phase space approach is that one never has to care about the kinematical Hilbert space at all, but the induced algebra of physical observables is typically so complicated that the corresponding (physical) Hilbert space is extremely difficult to find. The standard formulation of LQG adopts the Dirac approach, as constructing the reduced phase space of GR with standard matter is uncontrollably nontrivial. However, there is a hope of obtaining the reduced phase space with a manageable induced algebra, if one adds pressure-free dust165 or a massless scalar field166 to the theory. This is essentially because, at the price of introducing additional matter, the constraints can be deparametrized with respect to the dust or the scalar field, which serves as a material reference system. Schematically (consider only the scalar constraint), when the system is deparametrized, one can find a partial variable T serving as an “internal clock” such that the local scalar constraint (at least locally) reads as C[q a , pa ] = P + h[q a , pa ], where P is the momentum conjugate to T and both T and P are independent of other conjugate pairs (q a (x), pa (x)). One can identify a one-parameter family of the physical observables OT associated with the partial operator O such that the map Fτ : O → OT =τ for any τ is a homomorphism between the Poisson bracket of partial observables and a certain Dirac bracket (uniquely determined by the constraints and the choice of T ) of physical observables. This implies that the reduced phase space approach is achievable, as finding the Hilbert space representation of OT is as easy as that of O. In the resulting Hilbert space, the physical observable OT =τ is interpreted as the observable O measured when the internal clock T takes the value τ , and (as heuristically P acts as −i∂/∂T ) the dynamics is generated by the evolution equation 

d  ˆ O τ , Oτ = −i H, dτ where

(222)


ˆ := H

ˆ a (x), pa (x)] d3 x h[q

(223)

is called the physical Hamiltonian in contrast to the Hamiltonian constraint as it corresponds to a nonvanishing physical observable that generates the genuine “physical” evolution. Based on the model of Ref. 165 with pressure-free dust, the reduced phase space approach to a quantum theory of GR has been formulated in the styles of LQG167,168 as well as of the Master constraint program and AQG.164 Based on the model of Ref. 166 with a massless scalar field, the reduced phase space approach

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has also been constructed in the style of LQG.169 These studies have led to a new appealing direction of QG alongside the standard LQG. 12.4. Off-shell closure of quantum constraints In the Dirac approach of canonical quantization, quantizing a constrained system faithfully, one would expect that the corresponding quantum operators represent the same structure of the classical constraint algebra. That is, in the case of gravity, in accordance with the constraint algebra given by Eq. (24), there exists a vector space V of spin networks upon which, for all |Ψ ∈ V, the quantum operators act as ˆϕ2 |Ψ = U ˆϕ1 ◦ϕ2 |Ψ, ˆ ϕ1 U U ˆϕ† C[N ˆ ]U ˆϕ |Ψ = C[ϕ ˆ ∗ N ]|Ψ, U 

ˆ ], C[M ˆ ] |Ψ = CˆDiff [ q ab (N ∂b M − M ∂b N )]|Ψ, C[N

(224a) (224b) (224c)

where we have disregarded the Gauss constraint CG = 0, which is trivially satisfied via intertwiners. This, however, is not the case in the standard formulation of LQG. In the diffeomorphism covariant regularization scheme as addressed in Sec. 6.2, Eq. (224a) is trivially satisfied and Eq. (224b) corresponds to Eq. (112), but Eq. (224c) on the other hand is replaced by [see Eq. (113)]

 

ˆ ], C[M ˆ ] η(Ψ)|Φ := lim (η(Ψ)| CˆR [N ], CˆR [M ] Φ = 0, (225) ( C[N →0

for all Φ ∈ V, where η is the linear map in Eq. (58). In this sense, the standard LQG is said to be anomaly-free only at the on-shell level, but not off-shell. Although there is no obvious self-inconsistency, the fact that the quantum algebra is not off-shell closed arouses various worries. It was argued in Refs. 170 and 171 that quantum anomaly is hidden on-shell simply because the local scalar constraint C in Eq. (16c) is specifically of density weight 1. To see this heuristically, consider (k−2)/2 ˜E ˜ +· · · of density weight k associated FE the local scalar constraint C (k) ∼ |q| with the smearing function N (1−k) of density weight 1 − k. The scalar constraint in d spatial dimensions (d ≥ 2) is given by C[N ] = Σ dd xN (1−k) (x)C (k) (x) and scaling analysis tells that the regulated classical scalar constraint CR [N ] scales as ∝ d −2d(k−2)/2 −2 −(d−1) −(d−1) = d(1−k) . In the special case of k = 1, there is no explicit dependence on  and consequently the right-hand side of Eq. (225) is trivialˆ ] ized (as long as  becomes sufficiently small). In a sense, the quantum operator C[N implemented in the standard approach fails to grasp the geometrical significance dictated by Eq. (224c) and hence the quantum theory might not faithfully correspond to the starting classical theory (however, see Ref. 172 for a counterargument). If one choose to construct the scalar constraint operator by starting with C (k) of higher density weight, one would has enough scaling factors of −1 to match the scaling of CDiff [S] on the right-hand side of Eq. (224c) (as CDiff [S] involves spatial derivatives via N ∂b M − M ∂b N , which gives rises to at least one factor of

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−1 ) and be able to obtain an off-shell quantum representation for the constraint algebra. This “off-shell closure” approach indeed can be rigorously formulated for a generally covariant U (1)3 gauge theory, which can be understood as the G → 0 limit of Euclidean gravity.173–175 For Euclidean LQG, the new study of Ref. 176 presented some evidence that, working with k = 4/3, there exists such a scalar constraint operator well defined on a vector space V, which turns out to be a suitable generalization of the Lewandowski–Marolf habitat.171 In the off-shell closure approach, the geometrical interpretation of the action of the scalar constraint operator becomes more transparent, which should help to elucidate the quantum dynamics and the low-energy physics. Furthermore, as the requirement of being anomaly-free at the off-shell level is supposed to impose more restrictions on permissible regulators, the infinite multitude of quantization ambiguities could be enormously reduced or even uniquely fixed. 12.5. Loop quantum gravity versus spin foam theory The canonical (Hamiltonian) formalism of LQG and the covariant (sum-overhistories) formalism of the spin foam theory are closely related to each other, yet up to now spin foam models have not been systematically derived from the standard canonical theory of LQG. Over the past years, the spin foam theory in relation to the kinematics of LQG have been clearly established.90,177–181 For the dynamics, on the other hand, the two formalisms bear close resemblance to each other but it remains a challenging open problem to clearly derive the relation in the fourdimensional theory. (In the three-dimensional theory, a clear-cut relation between the canonical quantization of three-dimensional gravity and spin foam models has been nicely established.182 ) 12.6. Covariant loop quantum gravity Even though the precise connection between LQG and the spin foam theory is still not completely clear, the merger of the canonical and covariant approaches has suggested a rather well-established theory formulated in a covariant formalism known as “covariant loop quantum gravity,” which provides us a brand new perspective on LQG183–185 (also see Ref. 13 for a comprehensive account). Since various quantization procedures (canonical, spin foam, etc.) all lead to the overlapped results, perhaps it is the time to “leave the ladder behind” (as advocated in Ref. 183) and take the merged theory of covariant LQG seriously as the defining quantum theory of gravity instead of “deriving” it by quantizing the classical GR. The kinematics of covariant LQG is well defined and background independent, and the dynamics of it is given in terms of a simple vertex function, largely determined by locality, diffeomorphism invariance, and local Lorentz invariance. In the framework of covariant LQG, transition amplitudes for given boundary states can be computed explicitly and then compared with the classical theory by the techniques of holomorphic coherent states (see Sec. 8.3 and references therein).

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Particularly, the two-point function of the Euclidean theory over a flat spacetime has been computed to the first order in the vertex expansion, and has been shown to converge to the free graviton propagator of QG in the large distance limit.91 The n-point functions of the Lorentzian theory have yet to be computed and compared with the vertex amplitude of conventional perturbative QG on Minkowski space. 12.7. Spin foam cosmology In the framework of covariant LQG, by choosing the holomorphic coherent states of geometry peaked on homogeneous isotropic metrics, it is possible to compute the transition amplitude in the vertex and graph expansions for the evolution of a homogenous isotropic universe. The calculation has been performed and the resulting amplitude in the classical limit appears to be consistent with the FLRW evolution.186 Furthermore, if a cosmological constant term is added, the result matches the de Sitter solution in the classical limit.187 This approach starting from the full theory of covariant LQG and then taking an approximation of the vertex and graph expansions is called “spin foam cosmology.” By comparison, in LQC one studies the exact solutions in a symmetry-reduced quantum theory, while in spin foam cosmology one studies the approximate solutions in the full quantum theory. Casting LQC in an spin foam-like expansion has also been considered.188–191 Spin foam cosmology and the spin foam-like expansion of LQC can be regarded as two converging approaches to the sum-over-histories formalism of quantum cosmology. 12.8. Quantum reduced loop gravity Quantum reduced loop gravity (QRLG) is a framework introduced for the quantum theory of a symmetry-reduced sector of GR, based on a projection from the kinematical Hilbert space of the full theory of LQG down to a subspace representing the proper arena for the symmetry-reduced sector. It was first proposed in Refs. 192 and 193 for an inhomogeneous extension of the Bianchi I cosmological model. This approach provides a direct link between the full theory and its cosmological sector and also sheds light on the relation between LQC and LQG. It was later shown in Ref. 194 that the QRLG technique can be applied to broader cases beyond the cosmological context whenever the spatial metric can be gauge-fixed to a diagonal form. The technique projects the Hilbert space to the states based on the reduced graphs with Livine–Speziale coherent intertwiners.90 The framework of QRLG could simplify the analysis of the dynamics in the full theory and other issues, such as the coupling between quantum geometry and matter and the relation between canonical and covariant approaches. 12.9. Cosmological perturbations in the Planck era Cosmological inflation is a popular paradigm in modern cosmology. The theory of inflation has successfully explained how quantum fluctuations sow the primordial

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seeds that grow into the large-scale structure at late times of our universe and made a number of predictions that have been confirmed by a range of observations. In the standard inflationary scenario, cosmological perturbations are described by QFT on classical cosmological spacetimes, and thus the domain of validity excludes the Planck era. In a series of papers Refs. 195–197 (also see Ref. 198 for a nice summary), by studying the dynamics of quantum fields representing scalar and tensor perturbations on quantum cosmological spacetimes using techniques from LQG, the inflationary paradigm is extended to a self-consistent theory covering the epoch from the big bounce in the Planck era to the onset of slow-roll inflation. This pre-inflationary dynamics could yield deviations from the standard inflationary scenario and give rise to novel effects, such as a source for non-Gaussianity. These novel effects might catch a glimpse into the deep Planck regime of the early universe in future cosmological observations. 12.10. Spherically symmetric loop gravity LQC is the loop quantum theory of minisuperspaces in the sense that degrees of freedom are truncated to a finite number by the symmetry of homogeneity (and possibly also isotropy). The next simplest symmetry-reduced framework is the spherically symmetric midisuperspace, in which the spherical symmetry is imposed but inhomogeneity along the radial direction is retained, thus still giving rise to a fieldtheoretical system of infinite degrees of freedom. Ashtekar’s formalism for spherically symmetric midisuperspaces and its loop quantization have been studied and developed with various degrees of rigor.199–203 The theory of spherically symmetric loop gravity provides an arena for testing important issues that are too difficult in the full theory of LQG and trivialized in minisuperspace (LQC) models. Particularly, the SU (2) internal gauge is reduced to U (1) and the three-dimensional diffeomorphism is reduced to one-dimensional diffeomorphism (in the radial direction), making the constraint algebra much simpler yet nontrivial. The framework of spherically symmetric loop gravity can be used to study loop quantum geometry of spherically symmetric black holes (Schwarzschild black hole, spherical gravitational collapse, etc.). It is indicated that in the resulting quantum spacetime the black hole singularity is avoided,202–206 suggesting that the information loss problem could be resolved accordingly. 12.11. Planck stars and black hole fireworks One of the key insights obtained in LQC is that the quantum geometry of LQG gives rise to opposing quantum gravitational force, which becomes appreciable when the matter density comes close to the Planck scale density ∼1/(G2
). Based on the insight, recently, a new possible scenario of collapsing black holes has been suggested207 : The gravitational collapse of a star does not lead to a singularity but to

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the quantum gravitational phase called a “Planck star,” where the very huge gravitational attraction is counterbalanced by the opposing quantum force. Accordingly, a black hole hides a core of a Planck star, which is of the Planck scale density but can be much larger than the Planck scale size (depending on the initial mass of the collapsing star). As the black hole evaporates, the core remembers the initial mass and the final explosion can occur at macroscopic scale, thus providing a possible mechanism for recovery of information loss. More interestingly, the objects of Planck stars could produce detectable signals of quantum gravitational origin.207,208 Furthermore, the macroscopic remnant can develop into a white hole. As shown in a recent paper Ref. 209, there is indeed a classical metric that satisfies the Einstein field equation outside a finite spacetime region where matter collapses into a black hole and then emerges from a white hole. Therefore, a black hole can quantumtunnel into a white hole in the similar fashion that the wave packet representing a collapsing universe tunnels into a wave packet representing an expanding universe in LQC. A distant observer thus sees a dimming star, after a very long time, reemerge and burst out matter — a phenomenon called “black hole fireworks.” Under the scenario of black hole fireworks, the arguments over the information paradox have to be drastically revised. 12.12. Information loss problem It is widely expected that QG, once fully developed, should resolve several important problems in QG, particularly the singularity problem and the black hole information paradox. As both the cosmological and black hole singularities are indicated to be resolved by the LQG effects, it is strongly suggested that the information lost in the process of black hole evaporation should be recovered in a singularity-free scenario. In Ref. 210, it was analyzed in detail for two-dimensional black holes that quantum geometry effects indeed recover the information loss, primarily because the black hole singularity is resolved and consequently the quantum spacetime is sufficiently larger than the classical counterpart. For four-dimensional collapsing black holes, the aforementioned scenarios of Planck stars and black hole fireworks in particular suggest a similar mechanism for information recovery. The model of Ref. 210 provides a concrete example of avoiding the information loss problem under the “remnant scenario,” in which black hole evaporation stops at some point, leaving a black hole remnant that is correlated with the Hawking radiation and allowing the combined state to remain pure. The remnant scenario generally implies the existence of a very long-lived remnant, which, unfortunately, may cause other problems (e.g. the problem of infinite remnant production211 ). Meanwhile, a recent paper212 investigated an example from LQG in which the singularity problem is solved but the information loss problem is made worse. It was argued that the aggravated information loss problem is likely to be a generic feature of LQG, therefore putting considerable conceptual pressure on the theory of LQG.

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The information loss problem has been around for a long time. There is still no consensus over how the problem is to be resolved and it remains one of the focal points in the research of QG. 12.13. Quantum gravity phenomenology A legitimate quantum theory of gravity must make unambiguous predictions that in principle can be empirically tested by experiments or observations. Contrary to popular opinion, LQG does make definite predictions. For example, any measured physical area (such as the total cross-section of a scattering process) must be given by the discrete spectrum of area. (Also see Ref. 213 for the possible effects of angle quantization on scattering.) The true problem is that these predictions demand experiments or observations probing the Planck scale, which is believed to be far out of current reach. The impasse could be overcome soon, as the Planckian physics may not be completely unreachable after all by present and near-future technology. As mentioned earlier, loop quantum effects in the pre-inflationary era could give rise to sufficient deviations from the standard inflationary scenario. Some research works have attempted to reveal possible observable footprints of these effects on the cosmic microwave background along the lines of Refs. 214–218. Also as noted previously, the existence of Planck stars could produce detectable signals. The possible phenomenological consequences of Planck stars in gamma-ray bursts were studied in Ref. 208. Additionally, while the Lorentz invariance can be made manifest both in LQG and in the spin foam theory (see Ref. 219 and references therein), many other models suggest violation of Lorentz invariance in the deep Planck regime, which will modify the Lorentzian energy–momentum dispersion relation at lower-energy scale. Observations of ultra-high-energy cosmic rays — the most energetic particles ever observed — have thus far put strong constraints on deviations from the Lorentzian dispersion relation. Future investigation with improved precision will either impose even stronger constraints or unveil breakdown of the Lorentz invariance, thus enabling theorists to discriminate between different models of LQG and the spin foam theory. See Refs. 220 and 221 for more discussions. 12.14. Supersymmetry and other dimensions It is a celebrated virtue that LQG does not require any hypothetical ingredients, yet it has always been desirable to incorporate supersymmetry and extra dimensions at one’s disposal for the sake of both theoretical interest and comparison with other QG approaches (string theory in particular). Early attempts can be found e.g. in Refs. 25–27. Recently, a series of papers222–229 have been devoted to rigorous formulation of the loop quantum theory for supergravity and for all dimensions. Some progress has been made toward calculating black hole entropy from loop quantum theory in higher dimensions.230,231 It would be very instructive to compare these results to those obtained by string theory inspired approaches.

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Acknowledgments The author would like to thank Wei-Tou Ni for warmly inviting and encouraging him to write this paper, and is deeply grateful to Steven Carlip, Friedrich Hehl, and Chun-Yen Lin for their detailed comments and suggestions on the manuscript, which have been extremely helpful and have led to many improvements. Deep gratitude goes to Chih-Wei Chang for having inspiring discussions with the author on quantum gravity and providing him with a comfortable office at CCMS. This paper was written over the time when the author was supported in part by the Ministry of Science and Technology of Taiwan under the Grant Nos. 101-2112-M-002-027-MY3 and 101-2112-M-003-002-MY3.

References 1. C. Rovelli, Strings, loops and others: A Critical survey of the present approaches to quantum gravity, in Gravitation and Relativity: At the Turn of the Millenium, (Inter-University Centre for Astronomy and Astrophysics, Pune, 1998), pp. 281–331, arXiv:gr-qc/9803024. 2. S. Carlip, Rep. Prog. Phys. 64 (2001) 885, arXiv:gr-qc/0108040. 3. R. P. Woodard, Rep. Prog. Phys. 72 (2009) 126002, arXiv:0907.4238 [gr-qc]. 4. C. Kiefer, Quantum Gravity (Oxford University Press, Oxford, 2012). 5. C. Rovelli, Class. Quantum Grav. 28 (2011) 153002, arXiv:1012.4707 [gr-qc]. 6. A. Ashtekar, Adv. Ser. Astrophys. Cosmol. 6 (1991) 1. 7. T. Thiemann, Lect. Notes Phys. 631 (2003) 41, arXiv:gr-qc/0210094. 8. A. Ashtekar and J. Lewandowski, Class. Quantum Grav. 21 (2004) R53, arXiv:grqc/0404018. 9. C. Rovelli, Quantum Gravity (Cambridge University Press, Cambridge, 2004). 10. T. Thiemann, Modern Canonical Quantum General Relativity (Cambridge University Press, Cambridge, 2007), arXiv:gr-qc/0110034. 11. C. Rovelli, Living Rev. Relativ. 11 (2008) 5, http://www.livingreviews.org/lrr2008-5. 12. R. Gambini and J. Pullin, A First Course in Loop Quantum Gravity (Oxford University Press, Oxford, 2011). 13. C. Rovelli and F. Vidotto, Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory (Cambridge University Press, Cambridge, 2014.) 14. A. Perez, Class. Quantum Grav. 20 (2003) R43, arXiv:gr-qc/0301113. 15. D. Oriti, arXiv:gr-qc/0311066. 16. A. Perez, arXiv:gr-qc/0409061. 17. A. Perez, Living Rev. Relativ. 16 (2013) 3, http://www.livingreviews.org/lrr-2013-3, arXiv:1205.2019 [gr-qc]. 18. M. Bojowald, Living Rev. Relativ. 11 (2008) 4, http://www.livingreviews.org/lrr2008-4. 19. A. Ashtekar and P. Singh, Class. Quantum Grav. 28 (2011) 213001, arXiv:1108.0893 [gr-qc]. 20. M. Bojowald, Lect. Notes Phys. 835 (2011) 1. 21. M. Bojowald, Canonical Gravity and Applications: Cosmology, Black Holes and Quantum Gravity (Cambridge University Press, Cambridge, 2011).

May 4, 2017 9:52 b2167-ch23

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II-539

22. C. J. Isham, Conceptual and geometrical problems in quantum gravity, in Recent Aspects of Quantum Fields, eds. H. Mitter and H. Gausterer (Springer-Verlag, Berlin, 1991), pp. 123–229. 23. R. M. Wald, General Relativity (University of Chicago Press, Chicago, 1984), Sec. 14.1. 24. A. Ashtekar, J. D. Romano and R. S. Tate, Phys. Rev. D 40 (1989) 2572. 25. T. Jacobson, Class. Quantum Grav. 5 (1988) 923. 26. H. J. Matschull and H. Nicolai, Nucl. Phys. B 411 (1994) 609, arXiv:gr-qc/9306018. 27. L. Freidel, K. Krasnov and R. Puzio, Adv. Theor. Math. Phys. 3 (1999) 1289, arXiv:hep-th/9901069. 28. R. Capovilla, T. Jacobson and J. Dell, Phys. Rev. Lett. 63 (1989) 2325. 29. R. Capovilla, T. Jacobson, J. Dell and L. J. Mason, Class. Quantum Grav. 8 (1991) 41. 30. A. Ashtekar, A. P. Balachandran and S. Jo, Int. J. Mod. Phys. A 4 (1989) 1493. 31. S. Holst, Phys. Rev. D 53 (1996) 5966, arXiv:gr-qc/9511026. 32. A. Ashtekar, Phys. Rev. Lett. 57 (1986) 2244. 33. A. Ashtekar, Phys. Rev. D 36 (1987) 1587. 34. R. Penrose, Chaos Solitons Fractals 10 (1999) 581. 35. M. Ko, M. Ludvigsen, E. T. Newman and K. P. Tod, Phys. Rep. 71 (1981) 51. 36. M. Blagojevi´c and F. W. Hehl, Gauge Theories of Gravitation: A Reader with Commentaries (Imperial College Press, London, 2013). 37. R. Hojman, C. Mukku and W. A. Sayed, Phys. Rev. D 22 (1980) 1915. 38. A. Perez and C. Rovelli, Phys. Rev. D 73 (2006) 044013, arXiv:gr-qc/0505081. 39. A. Ashtekar and J. Lewandowski, Representation theory of analytic holonomy algebras, in Knots and Quantum Gravity, ed. J. C. Baez (Oxford University Press, Oxford, 2014), arXiv:gr-qc/9311010. 40. D. Marolf and J. M. Mourao, Commun. Math. Phys. 170 (1995) 583, arXiv:hepth/9403112. 41. A. Ashtekar and J. Lewandowski, J. Math. Phys. 36 (1995) 2170, arXiv:grqc/9411046. 42. J. Lewandowski, A. Okolow, H. Sahlmann and T. Thiemann, Commun. Math. Phys. 267 (2006) 703, arXiv:gr-qc/0504147. 43. L. H. Kauffman and S. L. Lins, Temperle-Lieb Recoupling Theory and Invariants of 3-Manifolds (Princeton University Press, Princeton, 1994). 44. C. Rovelli and L. Smolin, Phys. Rev. D 52 (1995) 5743, arXiv:gr-qc/9505006. 45. J. C. Baez, Adv. Math. 117 (1996) 253, arXiv:gr-qc/9411007. 46. J. C. Baez, Spin networks in nonperturbative quantum gravity, in Interface of Knots and Physics, ed. L. Kauffman (American Mathematical Society, Providence, Rhode Island, 1996), pp. 167–203, arXiv:gr-qc/9504036. 47. D. Marolf, arXiv:gr-qc/9508015. 48. A. Ashtekar, J. Lewandowski, D. Marolf, J. Mourao and T. Thiemann, J. Math. Phys. 36 (1995) 6456, arXiv:gr-qc/9504018. 49. B. Br¨ uegmann, R. Gambini and J. Pullin, Phys. Rev. Lett. 68 (1992) 431. 50. B. Br¨ uegmann, R. Gambini and J. Pullin, Nucl. Phys. B 385 (1992) 587. 51. N. Grot and C. Rovelli, J. Math. Phys. 37 (1996) 3014, arXiv:gr-qc/9604010. 52. W. Fairbairn and C. Rovelli, J. Math. Phys. 45 (2004) 2802, arXiv:gr-qc/0403047. 53. A. Ashtekar and J. Lewandowski, Class. Quantum Grav. 14 (1997) A55, arXiv:grqc/9602046. 54. S. Frittelli, L. Lehner and C. Rovelli, Class. Quantum Grav. 13 (1996) 2921, arXiv:grqc/9608043.

May 4, 2017 9:52 b2167-ch23

II-540

D.-W. Chiou

55. A. Ashtekar and J. Lewandowski, Adv. Theor. Math. Phys. 1 (1998) 388, arXiv:grqc/9711031. 56. J. Lewandowski, Class. Quantum Grav. 14 (1997) 71, arXiv:gr-qc/9602035. 57. R. De Pietri and C. Rovelli, Phys. Rev. D 54 (1996) 2664, arXiv:gr-qc/9602023. 58. T. Thiemann, J. Math. Phys. 39 (1998) 3347, arXiv:gr-qc/9606091. 59. R. Loll, Nucl. Phys. B 460 (1996) 143, arXiv:gr-qc/9511030. 60. T. Thiemann, J. Math. Phys. 39 (1998) 3372, arXiv:gr-qc/9606092. 61. C. Rovelli, Nucl. Phys. B 405 (1993) 797. 62. T. Thiemann, Phys. Lett. B 380 (1996) 257, arXiv:gr-qc/9606088. 63. T. Thiemann, Class. Quantum Grav. 15 (1998) 839, arXiv:gr-qc/9606089. 64. T. Thiemann, Class. Quantum Grav. 15 (1998) 875, arXiv:gr-qc/9606090. 65. T. Thiemann, Class. Quantum Grav. 15 (1998) 1207, arXiv:gr-qc/9705017. 66. C. Rovelli and L. Smolin, Phys. Rev. Lett. 72 (1994) 446, arXiv:gr-qc/9308002. 67. R. Borissov, R. De Pietri and C. Rovelli, Class. Quantum Grav. 14 (1997) 2793, arXiv:gr-qc/9703090. 68. C. Rovelli, J. Math. Phys. 36 (1995) 6529, arXiv:gr-qc/9503067. 69. C. J. Isham, Canonical quantum gravity and the problem of time, in Integrable Systems, Quantum Groups, and Quantum Field Theories, eds. L. A. Ibort and M. A. Rodr´ıguez (Springer, 1993), pp. 157–287, arXiv:gr-qc/9210011. 70. K. V. Kuchaˇr, Int. J. Mod. Phys. Proc. Suppl. D 20 (2011) 3. 71. B. Dittrich, Gen. Relativ. Gravit. 39 (2007) 1891, arXiv:gr-qc/0411013. 72. B. Dittrich, Class. Quantum Grav. 23 (2006) 6155, arXiv:gr-qc/0507106. 73. D.-W. Chiou, Class. Quantum Grav. 30 (2013) 125004, arXiv:1009.5436 [gr-qc]. 74. T. Thiemann, Class. Quantum Grav. 15 (1998) 1487, arXiv:gr-qc/9705021. 75. M. Montesinos and C. Rovelli, Class. Quantum Grav. 15 (1998) 3795, arXiv:grqc/9806120. 76. A. Ashtekar, C. Rovelli and L. Smolin, Phys. Rev. Lett. 69 (1992) 237, arXiv:hepth/9203079. 77. H. Sahlmann and T. Thiemann, Class. Quantum Grav. 23 (2006) 867, arXiv:grqc/0207030. 78. H. Sahlmann and T. Thiemann, Class. Quantum Grav. 23 (2006) 909, arXiv:grqc/0207031. 79. M. Varadarajan, Phys. Rev. D 61 (2000) 104001, arXiv:gr-qc/0001050. 80. M. Varadarajan, Phys. Rev. D 64 (2001) 104003, arXiv:gr-qc/0104051. 81. A. Ashtekar and J. Lewandowski, Class. Quantum Grav. 18 (2001) L117, arXiv:grqc/0107043. 82. J. M. Velhinho, Commun. Math. Phys. 227 (2002) 541, arXiv:math-ph/0107002. 83. M. Varadarajan, Phys. Rev. D 66 (2002) 024017, arXiv:gr-qc/0204067. 84. A. Ashtekar, J. Lewandowski, D. Marolf, J. Mourao and T. Thiemann, J. Funct. Anal. 135 (1996) 519, arXiv:gr-qc/9412014. 85. T. Thiemann, Class. Quantum Grav. 23 (2006) 2063, arXiv:gr-qc/0206037. 86. B. Bahr and T. Thiemann, Class. Quantum Grav. 26 (2009) 045011, arXiv:0709.4619 [gr-qc]. 87. B. Bahr and T. Thiemann, Class. Quantum Grav. 26 (2009) 045012, arXiv:0709.4636 [gr-qc]. 88. C. Flori, arXiv:0904.1303 [gr-qc]. 89. E. Bianchi, E. Magliaro and C. Perini, Phys. Rev. D 82 (2010) 024012, arXiv:0912.4054 [gr-qc]. 90. E. R. Livine and S. Speziale, Phys. Rev. D 76 (2007) 084028, arXiv:0705.0674 [gr-qc].

May 4, 2017 9:52 b2167-ch23

Loop quantum gravity

II-541

91. E. Bianchi, E. Magliaro and C. Perini, Nucl. Phys. B 822 (2009) 245, arXiv:0905.4082 [gr-qc]. 92. C. Rovelli, Phys. Rev. Lett. 97 (2006) 151301, arXiv:gr-qc/0508124. 93. S. W. Hawking, Commun. Math. Phys. 25 (1972) 152. 94. J. M. Bardeen, B. Carter and S. W. Hawking, Commun. Math. Phys. 31 (1973) 161. 95. J. D. Bekenstein, Lett. Nuovo Cim. 4 (1972) 737. 96. J. D. Bekenstein, Phys. Rev. D 7 (1973) 2333. 97. J. D. Bekenstein, Phys. Rev. D 9 (1974) 3292. 98. S. W. Hawking, Nature 248 (1974) 30. 99. S. W. Hawking, Commun. Math. Phys. 43 (1975) 199, Erratum 46 (1976) 206. 100. A. Ashtekar, J. Baez, A. Corichi and K. Krasnov, Phys. Rev. Lett. 80 (1998) 904, arXiv:gr-qc/9710007. 101. A. Ashtekar, A. Corichi and K. Krasnov, Adv. Theor. Math. Phys. 3 (1999) 419, arXiv:gr-qc/9905089. 102. A. Ashtekar, J. C. Baez and K. Krasnov, Adv. Theor. Math. Phys. 4 (2000) 1, arXiv:gr-qc/0005126. 103. A. Ashtekar, S. Fairhurst and B. Krishnan, Phys. Rev. D 62 (2000) 104025, arXiv:grqc/0005083. 104. A. Ashtekar, C. Beetle and J. Lewandowski, Phys. Rev. D 64 (2001) 044016, arXiv:grqc/0103026. 105. A. Ashtekar, C. Beetle and J. Lewandowski, Class. Quantum Grav. 19 (2002) 1195, arXiv:gr-qc/0111067. 106. A. Ashtekar, A. Corichi and D. Sudarsky, Class. Quantum Grav. 20 (2003) 3413, arXiv:gr-qc/0305044. 107. A. Ashtekar and A. Corichi, Class. Quantum Grav. 20 (2003) 4473, arXiv:grqc/0305082. 108. A. Ashtekar, J. Engle and C. Van Den Broeck, Class. Quantum Grav. 22 (2005) L27, arXiv:gr-qc/0412003. 109. C. Rovelli, Phys. Rev. Lett. 77 (1996) 3288. 110. C. Rovelli, Helv. Phys. Acta 69 (1996) 582, arXiv:gr-qc/9608032. 111. J. F. Barbero, J. Lewandowski and E. J. S. Villasenor, PoS QGQGS 2011 (2011) 023, arXiv:1203.0174 [gr-qc]. 112. T. Jacobson, Phys. Rev. Lett. 75 (1995) 1260, arXiv:gr-qc/9504004. 113. A. C. Wall, Phys. Rev. D 82 (2010) 124019, arXiv:1007.1493 [gr-qc]. 114. E. Bianchi and A. Satz, Phys. Rev. D 87 (2013) 124031, arXiv:1305.4986 [gr-qc]. 115. E. Bianchi, arXiv:1204.5122 [gr-qc]. 116. M. Domagala and J. Lewandowski, Class. Quantum Grav. 21 (2004) 5233, arXiv:grqc/0407051. 117. K. A. Meissner, Class. Quantum Grav. 21 (2004) 5245, arXiv:gr-qc/0407052. 118. A. Corichi, J. Diaz-Polo and E. Fernandez-Borja, Class. Quantum Grav. 24 (2007) 243, arXiv:gr-qc/0605014. 119. A. Corichi, J. Diaz-Polo and E. Fernandez-Borja, Phys. Rev. Lett. 98 (2007) 181301, arXiv:gr-qc/0609122. 120. I. Agullo, J. Fernando Barbero, E. F. Borja, J. Diaz-Polo and E. J. S. Villasenor, Phys. Rev. D 82 (2010) 084029, arXiv:1101.3660 [gr-qc]. 121. J. D. Bekenstein and V. F. Mukhanov, Phys. Lett. B 360 (1995) 7, arXiv:grqc/9505012. 122. S. Carlip, Class. Quantum Grav. 17 (2000) 4175, arXiv:gr-qc/0005017. 123. J. Engle, A. Perez and K. Noui, Phys. Rev. Lett. 105 (2010) 031302, arXiv:0905.3168 [gr-qc].

May 4, 2017 9:52 b2167-ch23

II-542

D.-W. Chiou

124. J. Engle, K. Noui, A. Perez and D. Pranzetti, Phys. Rev. D 82 (2010) 044050, arXiv:1006.0634 [gr-qc]. 125. J. Engle, K. Noui, A. Perez and D. Pranzetti, J. High Energy Phys. 1105 (2011) 016, arXiv:1103.2723 [gr-qc]. 126. A. Ashtekar, T. Pawlowski and P. Singh, Phys. Rev. D 74 (2006) 084003, arXiv:grqc/0607039. 127. A. Ashtekar, A. Corichi and P. Singh, Phys. Rev. D 77 (2008) 024046, arXiv:0710.3565 [gr-qc]. 128. A. Ashtekar, T. Pawlowski, P. Singh and K. Vandersloot, Phys. Rev. D 75 (2007) 024035, arXiv:gr-qc/0612104. 129. K. Vandersloot, Phys. Rev. D 75 (2007) 023523, arXiv:gr-qc/0612070. 130. E. Bentivegna and T. Pawlowski, Phys. Rev. D 77 (2008) 124025, arXiv:0803.4446 [gr-qc]. 131. W. Kami´ nski and T. Pawlowski, Phys. Rev. D 81 (2010) 024014, arXiv:0912.0162 [gr-qc]. 132. T. Pawlowski and A. Ashtekar, Phys. Rev. D 85 (2012) 064001, arXiv:1112.0360 [gr-qc]. 133. D.-W. Chiou, Phys. Rev. D 75 (2007) 024029, arXiv:gr-qc/0609029. 134. D.-W. Chiou and K. Vandersloot, Phys. Rev. D 76 (2007) 084015, arXiv:0707.2548 [gr-qc]. 135. D.-W. Chiou, Phys. Rev. D 76 (2007) 124037, arXiv:0710.0416 [gr-qc]. 136. A. Ashtekar and E. Wilson-Ewing, Phys. Rev. D 79 (2009) 083535, arXiv:0903.3397 [gr-qc]. 137. M. Martin-Benito, G. A. M. Marugan and T. Pawlowski, Phys. Rev. D 80 (2009) 084038, arXiv:0906.3751 [gr-qc]. 138. A. Ashtekar and E. Wilson-Ewing, Phys. Rev. D 80 (2009) 123532, arXiv:0910.1278 [gr-qc]. 139. E. Wilson-Ewing, Phys. Rev. D 82 (2010) 043508, arXiv:1005.5565 [gr-qc]. 140. L. Modesto, Int. J. Theor. Phys. 45 (2006) 2235, arXiv:gr-qc/0411032. 141. D.-W. Chiou, Phys. Rev. D 78 (2008) 044019, arXiv:0803.3659 [gr-qc]. 142. D.-W. Chiou and M. Geiller, Phys. Rev. D 82 (2010) 064012, arXiv:1007.0735 [gr-qc]. 143. D. W. Chiou and L. F. Li, Phys. Rev. D 79 (2009) 063510, arXiv:0901.1757 [gr-qc]. 144. D.-W. Chiou and L.-F. Li, Phys. Rev. D 80 (2009) 043512, arXiv:0907.0640 [gr-qc]. 145. A. Ashtekar and M. Bojowald, Class. Quantum Grav. 23 (2006) 391, arXiv:grqc/0509075. 146. L. Modesto, Class. Quantum Grav. 23 (2006) 5587, arXiv:gr-qc/0509078. 147. L. Modesto, Adv. High Energy Phys. 2008 (2008) 459290, arXiv:gr-qc/0611043. 148. C. G. Boehmer and K. Vandersloot, Phys. Rev. D 76 (2007) 104030, arXiv:0709.2129 [gr-qc]. 149. D.-W. Chiou, Phys. Rev. D 78 (2008) 064040, arXiv:0807.0665 [gr-qc]. 150. M. Martin-Benito, L. J. Garay and G. A. M. Marugan, Phys. Rev. D 78 (2008) 083516, arXiv:0804.1098 [gr-qc]. 151. L. J. Garay, M. Martin-Benito and G. A. M. Marugan, Phys. Rev. D 82 (2010) 044048, arXiv:1005.5654 [gr-qc]. 152. D. Brizuela, G. A. M. Marugan and T. Pawlowski, Class. Quantum Grav. 27 (2010) 052001, arXiv:0902.0697 [gr-qc]. 153. M. Martin-Benito, G. A. M. Marugan and E. Wilson-Ewing, Phys. Rev. D 82 (2010) 084012, arXiv:1006.2369 [gr-qc]. 154. M. Martin-Benito, D. Martin-de Blas and G. A. M. Marugan, Phys. Rev. D 83 (2011) 084050, arXiv:1012.2324 [gr-qc].

May 4, 2017 9:52 b2167-ch23

Loop quantum gravity

II-543

155. D. Brizuela, G. A. M. Marugan and T. Pawlowski, Phys. Rev. D 84 (2011) 124017, arXiv:1106.3793 [gr-qc]. 156. M. Bojowald, Gen. Relativ. Gravit. 38 (2006) 1771, arXiv:gr-qc/0609034. 157. M. Bojowald, D. Cartin and G. Khanna, Phys. Rev. D 76 (2007) 064018, arXiv:0704.1137 [gr-qc]. 158. T. Thiemann, Class. Quantum Grav. 23 (2006) 2211, arXiv:gr-qc/0305080. 159. M. Han and T. Thiemann, Class. Quantum Grav. 27 (2010) 225019, arXiv:0911.3428 [gr-qc]. 160. M. Han and T. Thiemann, J. Math. Phys. 51 (2010) 092501, arXiv:0911.3431 [gr-qc]. 161. K. Giesel and T. Thiemann, Class. Quantum Grav. 24 (2007) 2465, arXiv:grqc/0607099. 162. K. Giesel and T. Thiemann, Class. Quantum Grav. 24 (2007) 2499, arXiv:grqc/0607100. 163. K. Giesel and T. Thiemann, Class. Quantum Grav. 24 (2007) 2565, arXiv:grqc/0607101. 164. K. Giesel and T. Thiemann, Class. Quantum Grav. 27 (2010) 175009, arXiv:0711.0119 [gr-qc]. 165. J. D. Brown and K. V. Kuchaˇr, Phys. Rev. D 51 (1995) 5600, arXiv:gr-qc/9409001. 166. K. V. Kuchaˇr and J. D. Romano, Phys. Rev. D 51 (1995) 5579, arXiv:gr-qc/9501005. 167. K. Giesel, S. Hofmann, T. Thiemann and O. Winkler, Class. Quantum Grav. 27 (2010) 055005, arXiv:0711.0115 [gr-qc]. 168. K. Giesel, S. Hofmann, T. Thiemann and O. Winkler, Class. Quantum Grav. 27 (2010) 055006, arXiv:0711.0117 [gr-qc]. 169. M. Domagala, K. Giesel, W. Kami´ nski and J. Lewandowski, Phys. Rev. D 82 (2010) 104038, arXiv:1009.2445 [gr-qc]. 170. R. Gambini, J. Lewandowski, D. Marolf and J. Pullin, Int. J. Mod. Phys. D 7 (1998) 97, arXiv:gr-qc/9710018. 171. J. Lewandowski and D. Marolf, Int. J. Mod. Phys. D 7 (1998) 299, arXiv:grqc/9710016. 172. T. Thiemann, arXiv:1010.2426 [gr-qc]. 173. C. Tomlin and M. Varadarajan, Phys. Rev. D 87(4) (2013) 044039, arXiv:1210.6869 [gr-qc]. 174. A. Henderson, A. Laddha and C. Tomlin, Phys. Rev. D 88(4) (2013) 044028, arXiv:1204.0211 [gr-qc]. 175. A. Henderson, A. Laddha and C. Tomlin, Phys. Rev. D 88(4) (2013) 044029, arXiv:1210.3960 [gr-qc]. 176. A. Laddha, arXiv:1401.0931 [gr-qc]. 177. J. Engle, R. Pereira and C. Rovelli, Phys. Rev. Lett. 99 (2007) 161301, arXiv:0705.2388 [gr-qc]. 178. J. Engle, R. Pereira and C. Rovelli, Nucl. Phys. B 798 (2008) 251, arXiv:0708.1236 [gr-qc]. 179. L. Freidel and K. Krasnov, Class. Quantum Grav. 25 (2008) 125018, arXiv:0708.1595 [gr-qc]. 180. J. Engle, E. Livine, R. Pereira and C. Rovelli, Nucl. Phys. B 799 (2008) 136, arXiv:0711.0146 [gr-qc]. 181. W. Kami´ nski, M. Kisielowski and J. Lewandowski, Class. Quantum Grav. 27 (2010) 095006, Erratum 29 (2012) 049502, arXiv:0909.0939 [gr-qc]. 182. K. Noui and A. Perez, Class. Quantum Grav. 22 (2005) 1739, arXiv:gr-qc/0402110. 183. C. Rovelli, Class. Quantum Grav. 28 (2011) 114005, arXiv:1004.1780 [gr-qc]. 184. C. Rovelli, PoS QGQGS 2011 (2011) 003, arXiv:1102.3660 [gr-qc].

May 4, 2017 9:52 b2167-ch23

II-544

D.-W. Chiou

185. C. Rovelli, Lect. Notes Phys. 863 (2013) 57. 186. E. Bianchi, C. Rovelli and F. Vidotto, Phys. Rev. D 82 (2010) 084035, arXiv:1003.3483 [gr-qc]. 187. E. Bianchi, T. Krajewski, C. Rovelli and F. Vidotto, Phys. Rev. D 83 (2011) 104015, arXiv:1101.4049 [gr-qc]. 188. A. Ashtekar, M. Campiglia and A. Henderson, Phys. Lett. B 681 (2009) 347, arXiv:0909.4221 [gr-qc]. 189. A. Ashtekar, M. Campiglia and A. Henderson, Class. Quantum Grav. 27 (2010) 135020, arXiv:1001.5147 [gr-qc]. 190. M. Campiglia, A. Henderson and W. Nelson, Phys. Rev. D 82 (2010) 064036, arXiv:1007.3723 [gr-qc]. 191. A. Henderson, C. Rovelli, F. Vidotto and E. Wilson-Ewing, Class. Quantum Grav. 28 (2011) 025003, arXiv:1010.0502 [gr-qc]. 192. E. Alesci and F. Cianfrani, Europhys. Lett. 104 (2013) 10001, arXiv:1210.4504 [grqc]. 193. E. Alesci and F. Cianfrani, Phys. Rev. D 87 (2013) 083521, arXiv:1301.2245 [gr-qc]. 194. E. Alesci, F. Cianfrani and C. Rovelli, Phys. Rev. D 88 (2013) 104001, arXiv:1309.6304 [gr-qc]. 195. I. Agullo, A. Ashtekar and W. Nelson, Phys. Rev. Lett. 109 (2012) 251301, arXiv:1209.1609 [gr-qc]. 196. I. Agullo, A. Ashtekar and W. Nelson, Phys. Rev. D 87(4) (2013) 043507, arXiv:1211.1354 [gr-qc]. 197. I. Agullo, A. Ashtekar and W. Nelson, Class. Quantum Grav. 30 (2013) 085014, arXiv:1302.0254 [gr-qc]. 198. A. Ashtekar, arXiv:1303.4989 [gr-qc]. 199. M. Bojowald, Class. Quantum Grav. 21 (2004) 3733, arXiv:gr-qc/0407017. 200. M. Bojowald and R. Swiderski, Class. Quantum Grav. 21 (2004) 4881, arXiv:grqc/0407018. 201. M. Bojowald and R. Swiderski, Class. Quantum Grav. 23 (2006) 2129, arXiv:grqc/0511108. 202. M. Campiglia, R. Gambini and J. Pullin, Class. Quantum Grav. 24 (2007) 3649, arXiv:gr-qc/0703135. 203. D.-W. Chiou, W.-T. Ni and A. Tang, arXiv:1212.1265. 204. R. Gambini and J. Pullin, Phys. Rev. Lett. 101 (2008) 161301, arXiv:0805.1187 [gr-qc]. 205. R. Gambini and J. Pullin, Phys. Rev. Lett. 110(21) (2013) 211301, arXiv:1302.5265 [gr-qc]. 206. R. Gambini, J. Olmedo and J. Pullin, Class. Quantum Grav. 31 (2014) 095009, arXiv:1310.5996 [gr-qc]. 207. C. Rovelli and F. Vidotto, arXiv:1401.6562 [gr-qc]. 208. A. Barrau and C. Rovelli, arXiv:1404.5821 [gr-qc]. 209. H. M. Haggard and C. Rovelli, arXiv:1407.0989 [gr-qc]. 210. A. Ashtekar, V. Taveras and M. Varadarajan, Phys. Rev. Lett. 100 (2008) 211302, arXiv:0801.1811 [gr-qc]. 211. S. B. Giddings, Phys. Rev. D 49 (1994) 947, arXiv:hep-th/9304027. 212. M. Bojowald, arXiv:1409.3157 [gr-qc]. 213. S. A. Major, Class. Quantum Grav. 27 (2010) 225012, arXiv:1005.5460 [gr-qc]. 214. J. Grain and A. Barrau, Phys. Rev. Lett. 102 (2009) 081301, arXiv:0902.0145 [gr-qc]. 215. A. Barrau, arXiv:0911.3745 [gr-qc].

May 4, 2017 9:52 b2167-ch23

Loop quantum gravity

II-545

216. J. Mielczarek, T. Cailleteau, J. Grain and A. Barrau, Phys. Rev. D 81 (2010) 104049, arXiv:1003.4660 [gr-qc]. 217. J. Grain, A. Barrau, T. Cailleteau and J. Mielczarek, Phys. Rev. D 82 (2010) 123520, arXiv:1011.1811 [astro-ph.CO]. 218. A. Barrau, PoS ICHEP 2010 (2010) 461, arXiv:1011.5516 [gr-qc]. 219. C. Rovelli and S. Speziale, Phys. Rev. D 83 (2011) 104029, arXiv:1012.1739 [gr-qc]. 220. L. Smolin, Lect. Notes Phys. 669 (2005) 363. 221. S. Liberati and L. Maccione, J. Phys. Conf. Ser. 314 (2011) 012007, arXiv:1105.6234 [astro-ph.HE]. 222. N. Bodendorfer, T. Thiemann and A. Thurn, Class. Quantum Grav. 30 (2013) 045001, arXiv:1105.3703 [gr-qc]. 223. N. Bodendorfer, T. Thiemann and A. Thurn, Class. Quantum Grav. 30 (2013) 045002, arXiv:1105.3704 [gr-qc]. 224. N. Bodendorfer, T. Thiemann and A. Thurn, Class. Quantum Grav. 30 (2013) 045003, arXiv:1105.3705 [gr-qc]. 225. N. Bodendorfer, T. Thiemann and A. Thurn, Class. Quantum Grav. 30 (2013) 045004, arXiv:1105.3706 [gr-qc]. 226. N. Bodendorfer, T. Thiemann and A. Thurn, Class. Quantum Grav. 30 (2013) 045005, arXiv:1105.3708 [gr-qc]. 227. N. Bodendorfer, T. Thiemann and A. Thurn, Class. Quantum Grav. 30 (2013) 045006, arXiv:1105.3709 [gr-qc]. 228. N. Bodendorfer, T. Thiemann and A. Thurn, Class. Quantum Grav. 30 (2013) 045007, arXiv:1105.3710 [gr-qc]. 229. N. Bodendorfer, T. Thiemann and A. Thurn, Phys. Lett. B 711 (2012) 205, arXiv:1106.1103 [gr-qc]. 230. N. Bodendorfer, T. Thiemann and A. Thurn, Class. Quantum Grav. 31 (2014) 055002, arXiv:1304.2679 [gr-qc]. 231. N. Bodendorfer, Phys. Lett. B 726 (2013) 887, arXiv:1307.5029 [gr-qc].

May 4, 2017 11:2

Subject Index

3+1 spacetime split,

I-214

II-282, II-283, II-288–II-293, II-295, II-314, II-318, II-338

A Abelian axion, I-265, I-270, I-278, I-279, I-288, I-292, I-298, I-301, I-302 accelerated observer, II-335, II-425, II-433 accelerated reference system, I-274 active gravitational mass, I-92, I-110, I-111 ADM, I-200, I-203, I-213, I-215, I-248, II-326, II-360, II-361, II-473, II-474, II-501, II-507, II-517 advanced LIGO, I-480, I-492, I-497, I-506, I-543, I-546–I-549, I-551–I–553, I–555, I-561, I-567, II-298 advanced Virgo, I-497, I-506, I-552, I-555 affine connection, I-230, I-275 age of universe, II-314 angular resolution, I-319, I-494, I-495, I-579, I-595, I-602–I-605, I-607, I-611, I-612, I-625, II-44, II-48, II-49, II-54, II-94, II-97, II-100, II-105, II-112 anthropic principle, II-331 apparent horizon, II-59, II-69 ASTROD, I-361, I-363, I-364, I-384, I-388, I-394, I-396, I-481, I-579, I-582, I-592, I-593, I-596, I-599, I-601, I-603, I-607, I-612, I-613, I-623, I-625 ASTROD-GW, I-363, I-384, I-481–I-485, I-494, I-495, I-582, I-584, I-585, I-589, I-593, I-595–I-599, I-600, I-602–I-605, I-607, I-611–I-614, I-616, I-619, I-620, I-621, I-622, I-623, I-625 ASTROD I, I-361, I-363, I-364, I-384, I-394–I-396, I-481, I-579, I-581, I-588, I-589, I-591, I-619 axion, I-265, I-270, I-278–I-281, I-288–I-293, I-296, I-298–I-304, I-306, I-308, I-309, I-326, II-56, II-90, II-275,

B B modes, I-322, II-43, II-85, II-86, II-88–II-90, II-94–II-96, II-106, II-117, II-134, II-136, II-204, II-205, bar detector, I-478, I-520, I-521 Bepi-Colombo, I-394 Big Bang, I-309, I-310, I-479, I-481–I-483, I-489, I-493, I-495, I-579, I-582, I-584, I-594, I-596, I-604, I-612, II-4–II-6, II-14, II-24, II-44, II-54, II-56–II-58, II-62, II-66, II-134, II-135, II-225, II-226, II-227, II-229, II-231, II-236, II-239, II-273, II-274, II-278, II-285, II-293, II-298, II-332, II-334, II-352–II-354, II-469, II-520 Big Bang Observer, I-481–I-483, I-495, I-579, I-582, I-584, I-594, I-596, I-604, I-612 binary black holes, I-492, I-508, I-612 binary neutron stars, I-491, I-492, I-508, I-509, I-519 binary pulsar, I-276, I-378, I-409, I-412, I-413, I-415, I-421–I-426, I-431, I-462, I-508, II-370 black hole binaries, I-481, I-506, I-509, I-583 black hole entropy, II-335–II-337, II-416, II-418, II-422, II-432, II-437, II-440, II-443, II-445, II-448, II-516–II-520, II-537 black hole mechanics, II-334, II-335, II-416–II-418, II-435, II-443, II-458 black holes, I-109, I-127, I-130, I-154, I-174, I-248, I-407, I-432, I-435, I-437, I-441–I-443, I-453, I-492–I-494, I-498, I-508, I-509, I-579, I-601, I-612, I-624, II-160, II-274, II-330, II-334–II-337, II-353, II-403, II-415–II-418, II-427, II-431, II-434, II-435–II-438, II-440, I

May 4, 2017 11:2

II

Subject Index

II-441–II-443, II-445, II-446, II-447, II-449, II-450–II-453, II-455, II-457, II-458, II-468, II-515, II-516, II-518, II-519, II-535, II-536 black hole statistical mechanics, II-437, II-438, II-440, II-445, II-448 black hole thermodynamics, I-155–I-157, II-325, II-334, II-335–II-337, II-415–II-417, II-419, II-424, II-426, II-427, II-431, II-433, II-434, II-435, II-437, II-439, II-440, II-442, II-445, II-447, II-448, II-451, II-455, II-456, II-459, II-467, II-469, II-515, II-516, II-518, II-529 blackbody radiation, 292, 495, II-135 blackbody spectrum, I-292, I-295, II-5, II-44, II-62, II-63, II-71, II-72, II-99, II-105, II-115, II-134, II-370 C causal sets, II-326, II-467 causal structure, I-134, I-228, II-91, II-470 CGC ephemeris framework, I-613 Chandrasekhar limit, II-158 Christoffel connection, I-96 Christoffel symbols, I-118, I-130 clocks, I-166, I-167, I-62, I-73, I-307, I-331, I-332, I-335–I-341, I-348, I-349, I-353–I-355, I-357, I-361, I-662, I-364, I-365, I-407, I-409, I-452, I-591, I-623, I-625, II-458 CMB, see cosmic microwave background CMB anisotropy, I-296, I-489, II-9, II-44–II-46, II-50, II-66, II-67, II-77, II-78, II-100, II-106, II-134, II-261 CMB polarization, I-292, I-321, I-322, I-325, I-327, I-490, II-43, II-68, II-89, II-94, II-96 CMB temperature anisotropy, II-49, II-52, II-69, II-78, II-81, II-98, II-100, II-135, II-136 COBE satellite, II-31, II-45 compact sources, II-130, II-131 connection, I-4, I-28, I-33, I-90, I-102, I-118, I-172, I-175, I-187, I-189, I-196, I-197, I-200, I-203–I-205, I-209, I-212, I-223, I-229, I-230, I-231, I-235, I-241, I-244–I-246, I-248, I-252, I-267, I-269, I-275, I-318, I-335, I-396, I-543, II-3,

II-5, II-69, II-134, II-227, II-229, II-302, II-327, II-332, II-334, II-361, II-392, II-424, II-441, II-444, II-450, II-471–II-474, II-476–II-480, II-486, II-503, II-504, II-507, II-508, II-533 connection theories, II-471, II-476, II-477 core metric, I-267, I-269, I-284, I-297, I-298, I-303, I-305, I-306, I-309, II-338 cosmic background radiation, I-317, II-24, II-358 cosmic discovery, II-19 cosmic microwave background, I-317, I-423, I-424, I-426, I-463, I-580, II-5, II-12, II-26, II-28, II-43–II-48, II-50–II-54, II-56, II-58, II-60–II-73, II-75–II-78, II-80–II-87, II-89–II-91, II-93–II-95, II-97–II-100, II-102–II-108, II-110, II-112, II-114, II-117, II-130–II-134, II-192, II-225, II-273, II-275, II-338, II-363, II-368, II-382, II-383, II-415, II-537 cosmic polarization rotation (CPR), I-292, I-317, I-490 cosmic rays, I-624, II-105, II-106, II-338, II-537 cosmological constant, I-85, I-101, I-109, I-119, I-122, I-136, I-168, I-169, I-172, I-173, I-464, I-603, II-5, II-11, II-12, II-14, II-24, II-28, II-57, II-76–II-78, II-163, II-164, II-166, II-167, II-175, II-195, II-196, II-209, II-230, II-249, II-261, II-282, II-284, II-331, II-360, II-396, II-399, II-400, II-435, II-445, II-471, II-477, II-529, II-534 cosmological perturbation theory, II-58 cosmological perturbations, II-12, II-26, II-59–II-61, II-64, II-69, II-81, II-89, II-91, II-92, II-94, II-352, II-534, II-535 cosmological principle, I-101, II-3 cosmology, I-85, I-101, I-102, I-173, I-265, I-304, I-322, I-371, I-383, I-396, I-421, I-437, I-462, I-489, I-495, I-498, I-505, I-603, I-613, II-3, II-4, II-5, II-12, II-14, II-19, II-25, II-43, II-49, II-54, II-57, II-60, II-66, II-67, II-69, II-76, II-77, II-83, II-86, II-134–II-136, II-152, II-157, II-158, II-160, II-162–II-168, II-173, II-195, II-213, II-219, II-220,

May 4, 2017 11:2

Subject Index

II-225, II-226, II-229, II-230, II-231, II-239, II-246, II-260, II-261, II-267, II-273–II-275, II-280, II-281, II-295, II-296, II-302–II-304, II-315, II-328, II-332, II-334, II-339, II-349, II-350, II-369, II-372, II-376, II-381 covariance, I-93, I-99, I-102, I-203, I-215, I-265, I-267, I-412, I-446, II-22–II-24, II-86, II-89, II-91, II-111, II-219, II-326 covariant derivative, I-96, I-130, I-142, I-228, I-230, I-237, I-245, II-88, II-89, II-247, II-357, II-392 CPR, see cosmic polarization rotation cryogenic bar detector, I-520, I-521 curvature, I-96, I-98, I-100, I-114, I-118, I-119, I-123, I-143, I-161, I-165, I-172, I-175, I-187, I-188, I-190, I-195–I-198, I-203–I-205, I-210, I-228, I-229–I-232, I-240, I-252, I-274, I-275, I-292, I-340, I-412, I-423, I-464, I-494, I-496, II-26, II-47, II-59, II-61, II-76, II-83, II-173, II-176, II-177, II-178, II-226, II-229, II-230, II-231, II-234, II-237, II-241, II-244–II-248, II-250, II-251, II-259, II-261–II-267, II-278, II-327, II-330, II-361–II-363, II-402, II-432, II-472–II-474, II-476, II-477, II-510, II-520 curvature perturbations, II-61, II-246, II-248, II-259, II-262, II-263 curvature scalar, I-98, I-119 curvature tensor, I-96, I-118, I-119, I-143, I-161, I-165, I-190, I-197, I-275, I-464 D dark energy, I-389, I-392, I-396, I-497, I-579, I-599, I-603, I-604, I-624, II-6, II-11, II-12, II-14, II-28, II-57, II-79, II-79, II-136, II-151, II-152, II-163, II-166–II-168, II-175, II-192, II-196, II-197, II-214, II-220 dark matter, I-195, I-396, I-431, I-437, I-447, II-6, II-8–II-12, II-14, II-19–II-23, II-26, II-27, II-29–II-31, II-34–II-38, II-40, II-56, II-57, II-63, II-69, II-78, II-173, II-174, II-191, II-196, II-211–II-213, II-217, II-261, II-273,

III

II-281, II-293, II-295, II-300, II-301, II-305–II-307, II-310, II-315, II-370 DE ephemeris, I-387 de Sitter inflationary solution, I-102 de Sitter universe, II-78 deceleration parameter, II-164, II-352 DECIGO, I-481, I-482, I-485, I-495, I-579, I-582, I-594, I-596, I-600, I-601, I-604, I-612, I-625, II-268 density of the universe, I-436, II-22, II-24, II-32, II-57, II-78, II-164, II-166, II-188, II-238 deflection of light, I-19, I-96, I-97, I-362, I-373, I-374, I-376, I-379, II-5, II-174 deployment, I-365, I-579, I-586, I-619, I-625 diffeomorphism, I-190, I-191, I-197, I-202, I-219, I-220, I-232, I-237, I-248, II-333, II-428, II-432, II-446, II-450, II-451, II-468, II-474, II-475, II-478, II-479, II-481, II-483, II-484, II-485, II-486, II-490–II-492, II-495, II-496, II-497, II-498, II-499, II-503, II-505, II-506, II-508, II-512, II-513, II-521, II-524, II-530, II-532, II-533, II-535 dilaton, I-265, I-269, I-270, I-279, I-280, I-288–I-293, I-295, I-296, I-302, I-303, I-306, I-309, II-90, II-338, II-443 Doppler effect, I-42, I-99 Doppler tracking, I-307, I-371, I-393, I-472, I-480, I-481, I-585, I-586, I-587, I-589, I-590, I-591, I-592 dragging of inertial frames, I-393 dust, I-85, I-97, I-143, I-171, I-490, I-561, II-39, II-40, II-43, II-95, II-114–II-122, II-126–II-133, II-135, II-136, II-152, II-154, II-264, II-276, II-280, II-292, II-310, II-531 dust polarization, II-116 dwarf galaxies, II-19, II-30, II-31, II-34, II-36, II-37, II-39 E Earth orbits, I-594 Earthlike solar orbit, I-340, I-363, I-395, I-579, I-581, I-586, I-588, I-594, I-596, I-600, I-615, I-619, I-622, I-625 Eddington–Finkelstein coordinates, I-130, I-132, I-133

May 4, 2017 11:2

IV

Subject Index

eikonal approximation, I-273, I-277, I-279, I-280, I-288, I-289, II-90 Einstein equation, I-85, I-101, I-133, I-141, I-142, I-145, I-171, I-190, I-196, I-198, I-270, I-374, I-375, I-464, II-135, II-231, II-234, II-237, II-386 Einstein equivalence principle, I-85, I-88, I-265, I-274, I-275, I-303, I-310, I-317, I-372, I-490, II-338 Einstein’s field equation, I-109, I-118–I-120, I-124, I-142, I-144, I-147, I-160, II-4, II-339 Einstein tensor, I-96, I-119, I-145, I-146, I-175, I-195, I-198, I-251, I-252 electromagnetic energy tensor, I-99 energy–momentum tensor, I-119, I-136, I-137, I-139, I-142, I-196, I-198, I-226, I-229, I-233, I-470 energy tensor, I-85, I-91, I-95, I-97–I-101, I-270, I-374, I-464, II-6, II-55, II-59, II-427, II-428, II-454, II-455, II-517 E¨ otv¨ os experiment, I-85, I-88, I-91, I-92, I-265, I-270, I-273, I-277, I-293, I-372 ephemeris, I-87, I-375, I-382–I-385, I-387–I-389, I-393, I-397, I-410, I-595, I-607, I-611, I-613, I-614, I-616, I-617, II-6 EPM ephemerides, I-384, I-387 equivalence principle (EP), I-85, I-88, I-82, I-93, I-120, I-122, I-164, I-182, I-252, I-265, I-269–I-277, I-288, I-291, I-303, I-304, I-306, I-309, I-310, I-371–I-374, I-379, I-393, I-396, I-421, I-431, I-490, I-580, II-14, II-416 equivalence principles for photons, I-273 Einstein Telescope (ET), I-480, I-507, I-564–I-566 ether, I-3–I-5, I-8–I-11, I-13, I-15–I-18, I-20, I-22, I-24–I-27, I-29, I-30, I-32, I-35, I-36, I-37, I-38, I-39, I-41–I-46, I-50, I-51, I-52, I-53, I-54, I-55, I-56, I-57, I-59, I-66, I-73, I-75–I-80, I-82, I-83, I-87, I-88 event horizon, I-109, I-126, I-128–I-131, I-151, I-152, I-155–I-158, I-169, I-508, II-334, II-416–II-418, II-424, II-425, II-432, II-433, II-436, II-443, II-447, II-453, II-456, II-457, II-516, II-517, II-529

event timer, I-331, I-343–I-345, I-347, I-350, I-351, I-355, I-357, I-361, I-362, I-364–I-366, I-395, I-591 expansion of the universe, II-5, II-12, II-60, II-64, II-156, II-163, II-164, II-166, II-168, II-277, II-349 F Fabry–Perot interferometer, I-555, I-558, I-571 fiber bundle, I-200 Fizeau’s experiments, I-55, I-59, I-77 four-dimensional excitation (density), I-94, I-95 four-dimensional field strength, I-93–I-95 frame dragging, I-102, I-392, I-393 free–free emission, II-113, II-114, II-130, II-131 frequency sensitivity spectrum, I-482, I-596, I-622 Friedman–Lemaˆıtre–Robertson–Walker (FLRW) cosmology, I-603, II-4, II-54, II-176, II-277, II-520 G G-inflation, II-225, II-231, II-240, II-250–II-255, II-259, II-260, II-266, II-268 GAIA, I-392, I-394, I-420, I-623, II-309, II-315 galactic synchrotron emission, II-112–II-114 galaxy correlation function, II-22 Galilean transformation, I-87, I-93 Galileo equivalence principle, I-88, I-265, I-271–I-273, I-288, I-291, I-303, I-306, I-310, I-580 gauge, I-68, I-97, I-172, I-173, I-187, I-188, I-189, I-192, I-195, I-196, I-197, I-201–I-206, I-209–I-212, I-214, I-215, I-217, I-219, I-223, I-225–I-227, I-229–I-242, I-244, I-245, I-247, I-252, I-253, I-275, I-280, I-281, I-288, I-289, I-304, I-308, I-382, I-464, I-465, I-466, I-467, I-471, I-510, I-511, I-515, I-516, I-589, I-613, II-65, II-66, II-227, II-230, II-245, II-246, II-247, II-248, II-249, II-255, II-274, II-286, II-292, II-295, II-327, II-329, II-330, II-332, II-360, II-361, II-364, II-376, II-377,

May 4, 2017 11:2

Subject Index

II-380–II-382, II-389–II-394, II-396–II-402, II-447, II-451, II-468, II-471–II-473, II-475, II-477–II-479, II-481, II-482, II-486, II-501–II-506, II-520, II-521, II-524, II-530, II-533, II-534, II-535 general covariance, I-190 general relativity (GR), I-86, I-118, I-187, I-188, I-317, I-461, II-467 GEO, I-478, I-479, I-482, I-491, I-505, I-506, I-531, I-545, I-552–I-555, I-583 geodesic, I-118, I-119, I-125, I-132, I-134, I-142, I-152, I-164, I-228, I-304, I-380, I-381, I-383, I-510, I-622, I-696, II-65, II-84, II-90, II-176, II-177, II-381, II-421, II-425, II-427, II-448 geodesic deviation, I-119, I-164 geodetic precession, I-142, I-143, I-415, I-416, I-418, I-419, I-421, I-426 GPS, I-337, I-355–I-357 grain alignment, II-116, II-126, II-128 gravitational lenses, II-38, II-40, II-194 gravitational lensing, I-490, II-5, II-9, II-10, II-12, II-13, II-20, II-38, II-40, II-85–II-87, II-94, II-107, II-112, II-134, II-135, II-167, II-173–II-176, II-183, II-184, II-190, II-191, II-194, II-196, II-197, II-213, II-219, II-220, II-371, II-383 gravitational mass, I-88, I-92, I-110, I-111, I-274, I-424 gravitational redshift, I-85, I-96, I-102, I-103, I-277, I-338, I-373, I-374, I-413, II-66 gravitational wave, I-85, I-92, I-101, I-307, I-371, I-384, I-394, I-395, I-407, I-410, I-413, I-427, I-428, I-432, I-435, I-441, I-443, I-459, I-461, I-482, I-505–I-519, I-525–I-528, I-533, I-534, I-538, I-539, I-541, I-545–I-547, I-551, I-555, I-556, I-562, I-564–I-567, I-579, I-581, I-583, I-603, I-604, II-3, II-12, II-81, II-94, II-95, II-245, II-249, II-273, II-276, II-279, II-295, II-296, II-297, II-298, II-299, II-300, II-301, II-303, II-311, II-312, II-313, II-315, II-326, II-339, II-508 gravitational wave detection, I-307, I-432, I-505, I-506, I-579

V

gravitational wave sources, I-435, I-506–I-508, I-510, I-517, I-565 gravitational waves, classification of, I-461, I-499 graviton, I-603, I-229, I-249, I-274, I-281, I-284, I-350, I-351, I-357, I-361, I-364, I-369, I-370, I-371, I-373, I-376–I-379, I-381, I-383–I-386, I-391–I-403, I-511, I-534 Gravity Probe B, I-142, I-164, I-307, I-393 GW sensitivities, I-461, I-476 gyrogravitational ratio, I-165, I-265, I-307, I-308, I-310 H harmonic gauge, I-375, I-382, I-464–I-466 Hawking radiation, II-334, II-335, II-415, II-416, II-418, II-424–II-428, II-432, II-433, II-435, II-440–II-444, II-451–II-455, II-457, II-516, II-536 helicity, I-164, I-470 holographic conjecture, II-446, II-456 Hubble constant, I-436, I-468, I-483, I603, II-11, II-12, II-80, II-163, II-193, II-299, II-305 I indefinite metric, I-94, I-95, I-103, I-267 inertial frame, I-269, I-275, I-280, I-288, I-291, I-299, I-304, I-307, I-348, I-392, II-117, II-419 inertial mass, I-88, I-92, I-112, I-274, I-277, I-424 inflation, I-439, I-489, I-496, I-579, II-6, II-19, II-22, II-26, II-30, II-59–II-61, II-69–II-71, II-75–II-77, II-81, II-92, II-94, II-95, II-97, II-134, II-225–240, II-242, II-243, II-244, II-246, II-248–II-256, II-258–II-261, II-263–II-268, II-273, II-275, II-276, II-278, II-280, II-282, II-283, II-285–II-289, II-291–II-296, II-299, II-314, II-328, II-332, II-339, II-349, II-351–II-354, II-356–II-360, II-362–II-364, II-367–II-371, II-373, II-379, II-380, II-383, II-384, II-386, II-398, II-399, II-401–II-403, II-534, II-535

May 4, 2017 11:2

VI

Subject Index

information loss problem, II-337, II-451–II-454, II-456, II-535–II-537 INPOP, I-384, I-385, I-387, I-388, I-390, I-391, I-401–I-403, I-613 interferometer, I-88, I-321, I-335, I-362, I-366, I-392, I-407, I-432, I-440, I-441, I-452, I-453, I-470, I-471, I-476, I-477, I-478, I-480–I-482, I-485, I-488, I-491, I-492, I-496, I-497, I-505–I-508, I-510, I-515–I-520, I-527–I-540, I-543–I-549, I-551–I-553, I-555, I-556, I-558–I-567, I-569–I-573, I-579, I-580, I-582, I-583, I-591, I-592, I-597, I-600, I-601, I-611, II-268 intermediate range, I-265, I-270, I-305, I-308, I-379, I-392 K Kerr metric, I-141, I-144, I-149–I-152, I-155, I-157, I-158, I-160, I-161, I-165–I-174, I-250, I-252 Killing horizon, I-130, I-131, I-155, I-156, I-157, I-170, II-430, II-458 Kruskal–Szekeres coordinates, I-132–I-134 Kruskal–Szekeres diagram, I-135 L LAGEOS, I-307, I-393 LAGEOS2, I-393 LARES, I-307, I-393 large r scenarios, II-288, II-339 large scale flows, II-11, II-31, II-34 Lense–Thirring effect, I-85, I-142, I-307, I-393 Levi–Civita connection, I-118, I-172, I-196 LCGT, I-480, I-497, I-555, I-559, I-560, I-561 LIGO, I-432, I-433, I-446, I-448, I-453, I-472, I-479, I-480, I-485, I-492, I-493, I-497, I-505–I-507, I-519, I-535, I-536, I-538, I-543, I-545–I-553, I-555–I-559, I-561, I-567, I-579, I-624, II-298, II-302, II-311 LISA, I-384, I-394, I-395, I-441, I-443, I-453, I-481, I-482, I-484, I-485, I-494, I-495, I-500, I-579–I-583, I-585, I-589,

I-592–I-595, I-597–I-608, I-611, I-612, I-614, I-615, I-619–I-624, II-13, II-302, II-311–II-313 LISA Pathfinder, I-394, I-395, I-485, I-581, I-592, I-622–I-624 local inertial frame, I-269, I-280, I-288, I-291, I-299, I-304, I-392 local Lorentz invariance, I-424, II-337, II-339, II-533 long range, I-224, I-265, I-270, I-305, I-308, I-310, II-349, II-435 loop corrections, II-327, II-351, II-363, II-370–II-373, II-377–II-382, II-384, II-398, II-399, II-401, II-403, II-404 loop quantum cosmology, II-334, II-467, II-468, II-520 loop quantum gravity, II-325, II-326, II-332, II-416, II-442–II-444, II-467, II-533 Lorentz invariance, I-92, I-276, I-318, I-321, I-325, I-421, I-424, II-14, II-337, II-339, II-434, II-436, II-533, II-537 Lorentz transformation, I-67–I-71, I-73, I-90, I-93, I-228, I-230, I-232, I-251, I-252, I-267, I-282, I-348, II-472 low-energy physics, II-33, II-468, II-507, II-508, II-511, II-530, II-533 low tension cosmic strings, II-273, II-275, II-293, II-315 LRO Spacecraft, I-357–I-359 lunar laser ranging, I-143, I-276, I-342, I-343, I-371, I-386, I-392, I-424, I-425, I-429, I-430, I-580, I-591 M M-theory, II-330, II-331 mass-energy equivalence, I-90, I-372 matter fields, I-170, I-235, I-237, I-246, I-269, I-495, II-328, II-332, II-333, II-428, II-433, II-470, II-492, II-503, II-506, II-507 Maxwell equations, I-3, I-43, I-67–I-69, I-71, I-93, I-94, I-132, I-163, I-169, I-225, I-266–I-269, I-490 Mercury perihelion advance, I-85, I-88, I-89, I-98, I-99, I-100, I-103, I-371, I-372, I-373, I-386 MESSENGER, I-334, I-387, I-388, I-589 metric, I-85, I-91, I-93–I-99, I-101, I-103, I-109, I-114, I-115, I-116, I-118,

May 4, 2017 11:2

Subject Index

I-120, I-121, I-122, I-123, I-124, I-125, I-126, I-132–I-135, I-139, I-141–I-146, I-148–I-150, I-152, I-155, I-157, I-158, I-160, I-161, I-165–I-175, I-177, I-187, I-197, I-200, I-203–I-205, I-215, I-216, I-221–I-223, I-228–I-230, I-248–I-250, I-252, I-253, I-265–I-268, I-289, I-270, I-274–I-276, I-278–I-280, I-284, I-287, I-288, I-291–I-293, I-295–I-298, I-300, I-303–I-306, I-309, I-310, I-317, I-318, I-374, I-375, I-377–I-383, I-429, I-464, I-466–I-468, I-470, I-471, I-510, I-511, I-515, I-589, I-590, I-613, II-14, II-54, II-57, II-59, II-65, II-69, II-176, II-229, II-233, II-246, II-249, II-251, II-255, II-258, II-277, II-278, II-326, II-327, II-328, II-332, II-338, II-360, II-361, II-367, II-381, II-385, II-390, II-392, II-393, II-395, II-400–II-403, II-421, II-423, II-426, II-427, II-429, II-430, II-433, II-444, II-445, II-449, II-453, II-457, II-469–II-475, II-490, II-491, II-499, II-508, II-520, II-534 Michelson–Morley experiment, I-85, I-87, I-88, I-89, I-372 microlensing, II-175, II-273, II-276, II-295, II-302, II-307, II-308, II-309, II-310, II-311, II-313, II-315 Microscope (MICROSCOPE), I-88, I-104, I-396, I-406, I-623, I-630 Minkowski metric, I-93, I-94, I-96, I-115, I-200, I-223, I-248, I-280, I-288, I-464, II-385, II-390, II-395 Minkowski space, I-109, I-114, I-117, I-128, I-132, I-36, I-141, I-149, I-190, I-195, I-205, I-225, I-230, I-248, I-252, I-253, II-335, II-385, II-419, II-425, II-426, II-429, II-508, II-534 Minkowski spacetime, I-115, I-116, I-135, I-155, I-194, I-203, I-208, I-229, I-214, I-231, I-249, I-335, I-508 multi-formation configuration, I-605 N Newton’s theory, I-16, I-19, I-20, I-86, I-88, I-92, I-109, I-142, I-167, I-371, I-373, II-7 Newtonian gravity, I-89, I-93, I-109, I-120, I-123, I-128, I-372, I-373, I-421,

VII

I-528, I-539, I-540, I-543, I-588, II-7, II-8, II-173, II-370, II-434 Newtonian noise, I-573 Noether energy–momentum current, I-194 Noether theorems, I-189, I-208 non-Gaussianity, II-86, II-92–II-94, II-112, II-134, II-267, II-285, II-287, II-292, II-382, II-535 nonmetric theory, I-277, I-304, I-312, I-329 numerical relativity, I-469 O Odyssey, I-395, I-396 OMEGA, I-481, I-482, I-579, I-583, I-593–I-596, I-599, I-600, I-611 one-way lunar laser link, I-357 optical clock, I-307, I-335, I-336, I-363, I-365, I-396, I-481, I-590, I-591, I-622, I-623, I-625 orbit configuration, I-482, I-579–I-583, I-588, I-594, I-595, I-597, I-605–I-607, I-610, I-611, I-614, I-616, I-619 orbit design, I-382–I-384, I-579, I-585, I-586, I-612–I-614 orbit observations, I-585, I-586 orbit optimization, I-579, I-586, I-605, I-612–I-614, I-616 OSS, I-363, I-395, I-396 P parametrized post-Newtonian framework, I-379 passive gravitational mass, I-92 payload concept, I-579, I-622 Penrose–Carter diagram, I-150, I-160 Penrose diagram, I-115, I-117, I-120, I-135, I-160, I-161, II-420, II-431, II-432, II-452, II-456 Penrose–Kruskal diagram, I-129, I-135, I-136 perfect fluid, I-137, I-376, I-378, I-379, II-56, II-59 perihelion advance, I-85–I-89, I-98–I-100, I-103, I-371–I-373, I-386, I-395, I-413, I-588, II-7 perturbative quantum gravity, II-325, II-327, II-328, II-349, II-370, II-404

May 4, 2017 11:2

VIII

Subject Index

Planck, I-90–I-92, I-269, I-274, I-278, I-292, I-295, I-296, I-305, I-306, I-310, I-317, I-324, I-372, I-468, I-478, I-488, I-489, I-490, I-527, I-528, II-28, II-49–II-54, II-61, II-67, II-83, II-85, II-91, II-92, II-94–II-98, II-100, II-101, II-104–II-106, II-109, II-110, II-112, II-115, II-116, II-128, II-132–II-137, II-234–II-236, II-239, II-261, II-264, II-267, II-268, II-273, II-275, II-280, II-296, II-333, II-335, II-337, II-339, II-354, II-367, II-368, II-370, II-415, II-424, II-432, II-435, II-437, II-439, II-443, II-448, II-468, II-488, II-507, II-508, II-513, II-516, II-518, II-520, II-523, II-528, II-529, II-534, II-536, II-537 PLANCK, II-279, II-286, II-289, II-297 Planck length, II-439, II-508, II-513, II-523 Polarization, I-21, I-23, I-29, I-32, I-33, I-39–I-41, I-46, I-53, I-55, I-265, I-273, I-279, I-284, I-286–I-292, I-300, I-304, I-307, I-308, I-310, I-317–I-327, I-416, I-423, I-433, I-443, I-447, I-467–I-470, I-476, I-477, I-489, I-490, I-495, I-512, I-514, I-515, I-517, I-518, I-580, I-605, II-5, II-43, II-48, II-50, II-51, II-54, II-62, II-64, II-67–II-69, II-71–II-74, II-80–II-84, II-87–II-90, II-93–II-98, II-100, II-106–II-110, II-112, II-115, II-116, II-120–II-122, II-125, II-130, II-135, II-136, II-140, II-230, II-250, II-257, II-261, II-263, II-264, II-267, II-268, II-275, II-276, II-280, II-289, II-327, II-338, II-339, II-352, II-355, II-364, II-367, II-368, II-371, II-373, II-380, II-383, II-385, II-395–II-398 PPN parameters, I-377, I-382, I-386, I-396, I-423–I-425, I-427, I-613 primordial CMB anisotropies, II-134 primordial power spectrum, II-48, II-61–II-63, II-70, II-94, II-256 principle of relativity, I-3, I-11, I-61, I-65–I-67, I-70, I-72, I-73, I-75, I-76, I-89, I-90, I-92, I-274, I-348 progenitors, I-604, II-151, II-157, II-158, II-161 (pseudo-)Riemannian geometry, I-96

pseudoscalar–photon interaction, I-277, I-490 pseudotensors, I-188, I-189, I-196, I-198, I-200, I-201, I-247, I-252 PTA, I-307, I-407, I-434–I-436, I-439–I-445, I-447, I-448, I-450–I-453, I-471, I-481, I-486, I-487, I-493–I-495, I-497, I-580, I-591, I-599, I-601, I-602, I-624 pulsars, I-137, I-286, I-325, I-326, I-378, I-379, I-396, I-407–I-410, I-412, I-415, I-421, I-423–I-427, I-429, I-431–I-435, I-440, I-442–I-446, I-448–I-453, I-471, I-479, I-486–I-488, I-492, I-508, I-550, I-551 Q quadrupole moment, I-383, I-386, I-387, I-492, I-509, I-613, II-70, II-197, II-200, II-201, II-207 quantum dynamics, II-333, II-468, II-479, II-492, II-500, II-501–II-503, II-520, II-525, II-529, II-530, II-533 quantum geometry, II-333, II-442, II-486, II-490, II-502, II-507, II-508, II-517, II-520, II-529, II-534, II-535, II-536 quantum gravity, I-85, I-371, I-395, I-421, I-497, I-505, I-579, II-3, II-14, II-274, II-314, II-323, II-325–II-328, II-332, II-333, II-334, II-337, II-338, II-339, II-349, II-350, II-354, II-367, II-369–II-371, II-381, II-384, II-396, II-397, II-400–II-404, II-415, II-416, II-426, II-437, II-438, II-440, II-442, II-443, II-444, II-446, II-448, II-452, II-454, II-455, II-456, II-467, II-469, II-470, II-530, II-533, II-537, II-538 quantum gravity phenomenology, II-325, II-337, II-339, II-537 quantum fluctuations, I-489, I-496, II-6, II-62, II-227, II-229, II-230, II-236, II-239, II-242, II-243, II-244, II-278, II-371, II-445, II-534 quantum kinematics, II-478, II-492, II-522, II-523 quasars, I-319, I-326, I-327, I-488, I-493, II-38, II-39, II-80, II-81, II-133, II-192, II-195, II-196

May 4, 2017 11:2

Subject Index

quasi-local energy, I-187–I-189, I-203, I-248, I-249, I-251, I-252, I-253 R radiation reaction (pressure) noise, I-462 Radio polarization, I-319, I-326, I-327 recombination, I-533, II-26, II-27, II-43, II-63–II-65, II-68, II-79, II-80, II-82, II-84, II-87, II-114, II-352, II-353, II-367 redshift, I-85, I-93, I-96, I-102, I-103, I-131, I-155, I-265, I-270, I-274, I-277, I-287, I-288, I-292, I-293, I-296, I-319–I-321, I-326, I-338, I-340, I-373, I-374, I-413, I-432, I-433, I-436, I-438, I-489, I-583, I-585–I-587, I-602–I-604, II-12–II-14, II-23, II-25, II-26–II-30, II-38, II-57, II-58, II-66, II-67, II-78, II-79, II-80, II-81, II-85, II-132, II-133, II-135, II-151, II-152, II-157, II-158, II-160–II-164, II-168, II-174, II-194, I-195, II-196, II-197, II-206, II-209, II-211, II-212, II-213, II-216, II-217, II-220, II-294, II-296, II-302, II-303, II-338, II-383, II-397, II-398, II-402, II-423, II-424, II-450 reheating, II-61, II-63, II-231, II-232, II-236, II-238, II-239, II-242, II-253, II-268, II-296, II-352, II-360, II-371 relativistic gravity, I-87, I-96, I-271, I-371, I-374, I-375, I-377, I-379, I-382, I-385, I-387, I-388, I-392, I-394, I-395, I-396, I-397, I-407, I-410, I-412, I-421, I-453, I-462, I-481, I-581, I-582, I-588, I-603, I-625, II-339 renormalization, II-369, II-377, II-382, II-386, II-400, II-445 resonant antenna, I-492, I-505, I-506, I-510, I-512, I-519–I-527, I-546, I-566, I-567 Ricci (curvature) tensor, I-85, I-99, I-119, I-123, I-136, II-249 Riemannian (curvature) tensor, I-96 Riemannian geometry, I-91, I-172, I-202, I-204, I-230, I-246, II-471 S Sachs–Wolfe formula, II-43, II-65–II-67, II-72, II-94 SAGAS, I-361, I-363, I-394–I-396

IX

Sagnac effect, I-348, I-351, I-593 scalar constraint, II-474, II-492, II-494, II-495, II-497–II-502, II-507, II-512, II-521, II-524, II-525, II-529–II-533 Schwarzschild metric, I-120–I-122, I-124, I-126, I-132, I-133, I-135, I-142, I-151, I-157, I-379 Schwarzschild radius, I-119, I-127, I-128, I-469, II-456, II-457 SEP, see strong equivalence principle Shapiro time delay, I-374, I-376, I-379, I-380, I-381, I-389, I-588 shot noise, I-482, I-484, I-485, I-528–I-531, I-533–I-535, I-537–I-539, I-544, I-545, I-553, I-554, I-556, I-558, I-562, I-570, I-596–I-598, II-28 single-field inflation, II-93, II-225, II-231, II-246, II-250, II-261 slow-roll inflation, II-230, II-233–II-236, II-239, II-248, II-250, II-251, II-254, II-260, II-266, II-267, II-277, II-278, II-339, II-535 small r scenarios, II-283 SNe Ia, II-12, II-151–II-163, II-165, II-167–II-169 solar oblateness, I-384, I-386, I-387 solar quadrupole, I-383, I-386, I-387, I-613 solar system ephemeris, I-375, I-382, I-410, I-595 solar system test, I-371, I-374, I-379, I-385, I-390, I-391, I-392, I-396, I-423, I-424, I-431, II-370 spacetime, I-85, I-93, I-95, I-96, I-98, I-102, I-103, I-114, I-115–I-119, I-120, I-122, I-128, I-129–I-136, I-141–I-143, I-149, I-151, I-152, I-155, I-157, I-158, I-160–I-162, I-166–I-170, I-172, I-174, I-177–I-181, I-183, I-187–I-189, I-194, I-198, I-200, I-202, I-203–I-209, I-212–I-217, I-219, I-221, I-222, I-223, I-227–I-235, I-242, I-243, I-245, I-249, I-250, I-252, I-253, I-256, I-261, I-265, I-267, I-269, I-270–I-275, I-278, I-279, I-284, I-287, I-288, I-291, I-292, I-293, I-295, I-296, I-297, I-300, I-303, I-305, I-306, I-309, I-310, I-372, I-393, I-396, I-414, I-432, I-462, I-466, I-489, I-505, I-510, I-589, I-591, I-603, II-14, II-70, II-72, II-73, II-91, II-136, II-173, II-229,

May 4, 2017 11:2

X

Subject Index

II-234, II-238, II-244–II-246, II-251, II-261, II-267, II-274, II-282, II-292, II-297, II-303, II-307, II-325, II-328–II-330, II-334, II-335, II-338, II-358, II-369, II-371, II-372, II-373, II-385, II-417–II-420, II-424–II-427, II-429, II-430–II-432, II-440, II-441, II-443, II-445, II-452, II-456–II-458, II-467–II-473, II-478, II-502, II-503, II-507, II-508, II-511, II-529, II-534–II-537 spacetime structure, I-160, I-265, I-270, I-279, I-295, I-300, I-310, I-396, II-338 special relativity, I-3, I-4, I-72, I-76, I-85, I-90, I-91, I-93, I-94–I-96, I-102, I-109, I-114, I-128, I-132, I-229, I-273, I-274, I-275, I-279, I-372, II-275 spin foam, II-334, II-337, II-467–II-469, II-503, II-511, II-513, II-514, II-515, II-529, II-530, II-533, II-534, II-537 standardizable distance candle, II-152 static spacetime, I-143 stationary black holes, II-417, II-429, II-447, II-457, II-458 stress-energy tensor, I-85, I-91, I-95, I-97, I-98, I-100–I-102, I-270, I-374, I-464, II-6, II-56, II-60, II-427, II-428, II-454, II-455, II-517 stress-momentum-energy tensor, I-95 string theory, I-174, II-6, II-273–II-277, II-280–II-284, II-286, II-288, II-289, II-292, II-293, II-294–II-297, II-302, II-304–II-306, II-310, II-314, II-315, II-325–II-332, II-336, II-337, II-339, II-350, II-403, II-416, II-435, II-440, II-441, II-447, II-467, II-468, II-537 strong equivalence principle, I-273, I-276, I-278, I-379, I-388, I-391, I-421, I-423, I-424, I-430, I-431, I-624, II-14 strong lensing, II-20, II-173–II-175, II-184, II-190, II-191, II-193, II-195, II-196, II-209, II-211 Sunyaev–Zeldovich effects, II-44, II-99 Super-ASTROD, I-394–I-396, I-481, I-483, I-579, I-582, I-584, I-596, I-601, I-612, I-625 supergravity, I-174, II-282, II-289, II-291, II-292, II-326, II-330, II-467, II-471, II-537

supernova, I-325, I-408, I-415, I-492, I-493, I-506, I-509, I-521, I-604, II-12, II-13, II-36, II-58, II-113, II-151, II-152–II-154, II-158–II-160, II-164, II-165, II-167, II-168, II-192, II-220, II-227, II-306 supernova luminosity distance, I-436, I-446, I-449, I-603, I-604, II-12, II-163, II-164 supersymmetry, II-283, II-287, II-329, II-331, II-332, II-468, II-537 surface gravity of black holes, I-131, I-157, I-335, I-415, I-417, II-421, II-425, II-429, II-453, II-457, II-458 surface of last scattering, II-383 synchronization of clocks, I-331, I-333, I-334, I-341, I-348, I-364, I-365 T T2L2, I-331, I-333, I-334, I-336–I-348, I-350, I-352–I-357, I-359, I-361, I-364–I-366, I-591, I-625 TAMA, I-478, I-506, I-536, I-538, I-541, I-542, I-555–I-558, I-562 tensor perturbation, I-489, II-229, II-244, II-245, II-250, II-255–II-259, II-263, II-264, II-266, II-267, II-276, II-360, II-535 thermal dust emission, II-44, II-96, II-114–II-116, II-121, II-126, II-130, II-131, II-132 thermal history of the universe, II-43, II-56 thermal noise, I-505, I-519, I-520, I-528, I-531, I-535, I-536, I-539–I-543, I-548, I-552, I-559, I-567, I-568, I-569 tidal force, I-112–I-114, I-164, I-491, I-508, I-510, II-205 time delay interferometry (TDI), I-384, I-579, I-586, I-592, I-593, I-616, I-619–I-623 trapped surface, I-129 U uniformly accelerated frame, I-93, I-307 universal metrology, I-270, I-305, I-306 universality, I-88, I-265, I-271–I-273, I-304, I-305, I-580, II-415, II-416, II-441, II-448, II-456

May 4, 2017 11:2

Subject Index

universe, I-8, I-9, I-12, I-52, I-64, I-134, I-136, I-195, I-275, I-276, I-296, I-309, I-310, I-317, I-319, I-372, I-407, I-410, I-432, I-436, I-439, I-442, I-446, I-448, I-468, I-482, I-488, I-489, I-491, I-493, I-497, I-498, I-505, I-506, I-508, I-517, I-581, I-582, I-603, II-3–II-6, II-11, II-12, II-14, II-19, II-20, II-22, II-23, II-24, II-26, II-27, II-29–II-34, II-36, II-40, II-43, II-45, II-56–II-65, II-68, II-70, II-76–II-81, II-84, II-85, II-91, II-133, II-135, II-136, II-151, II-156, II-163–II-168, II-173, II-176, II-188, II-196, II-209, II-214, II-216, II-219, II-220, II-225–II-231, II-233, II-234, II-236, II-238, II-239–II-241, II-250, II-253, II-257, II-259, II-261, II-267, II-273, II-274, II-276–II-278, II-281–II-286, II-292–II-295, II-299, II-300, II-302, II-305, II-311, II-313, II-314, II-325, II-327, II-332, II-334, II-349, II-351, II-353, II-360, II-372, II-376, II-380, II-383, II-396, II-399–II-403, II-439, II-452–II-454, II-469, II-520, II-534, II-535, II-536 UV polarization, I-292, I-319, I-320, I-321, I-327 V vacuum, I-7, I-9, I-10, I-13, I-17, I-27, I-56, I-59, I-95, I-109, I-113, I-119, I-120, I-122, I-132, I-136, I-138, I-141–I-146, I-148, I-149, I-160, I-165, I-168, I-170–I-172, I-200, I-206, I-214, I-222, I-225, I-227, I-246, I-248, I-265, I-280, I-305, I-306, I-309, I-318, I-520, I-528, I-544, I-545, I-548, I-553, I-554, I-556, I-558–I-560, I-580, I-592, II-30,

XI

II-61, II-62, II-91, II-226, II-227, II-229, II-237, II-238, II-242, II-249, II-266, II-282, II-284, II-285, II-288, II-328, II-331, II-335, II-350, II-355, II-356, II-360, II-372, II-373, II-375, II-376, II-380, II-385, II-395–II-399, II-418, II-419, II-420, II-423, II-425, II-426, II-428, II-430–II-432, II-439, II-453, II-454, II-478, II-487 Virgo, I-443, I-448, I-449, I-472, I-478–I-480, I-493, I-497, I-505–I-507, I-528, I-532, I-536, I-538, I-543, I-546, I-548–I-552, I-555, I-556, I-559, I-561, II-31 W weak equivalence principle, I-88, I-92, I-270, I-273, I-276, I-304, I-307, I-309, I-318, I-393, I-396, I-421 weak lensing, I-625, II-85, II-135, II-173–II-175, II-182, II-184, II-193, II-196–II-199, II-202, II-203, II-205–II-211, II-213, II-214, II-219 WEP, see weak equivalence principle WEP I, I-270, I-271, I-273, I-276–I-278, I-284, I-288, I-291, I-304 WEP I for photon, I-273, I-288 WEP II, I-273, I-276, I-278, I-291, I-304, I-307 WEP II for photon, I-273 Wheeler–DeWitt equation, II-332, II-431 Wheeler’s cup of tea, II-416 WMAP, I-322, I-489, II-46, II-49–II-51, II-83, II-86, II-94, II-97, II-98, II-101, II-105, II-113, II-135, II-167, II-192, II-196, II-261, II-279, II-280, II-289, II-297, II-368, II-369, II-383

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May 4, 2017 10:56

Author Index Authors of chapters and the first 3 authors of references

A Aasi, J., I-500, I-577, II-320 Abadie, J., I-500, I-503, I-574 Abazajian, K. N., II-137 Abbott, B., I-108, I-454, I-575, I-625, II-320 Abbott, L. F., II-137, II-270, II-340 Abgrall, M., I-367, I-368 Ables, J. G., I-313 Abraham, M., I-82, I-106 Abramo, L. R., II-414 Abramovici, A., I-573, I-575 Accadia, T., I-577 Acef, O., I-369 Acernese, F., I-500, I-573, I-574, I-576 Achour, J. B., II-463 Acquaviva, G., II-344, II-460 Acquaviva, V., II-137 Adam, R., I-503, II-144, II-149 Adams, F. C., II-316 Ade, P. A. R., I-313, I-327, I-499, I-503, II-16, II-137, II-143–145, II-149, II-269, II-271, II-316, II-321, II-347, II-405 Adelberger, E. G., I-315 Adhikari, R. X., I-575 Adler, R., I-176 Adler, S. L., II-463 Adshead, P., II-408 Affeldt, C., I-577 Agatsuma, K., I-577 Aghanim, N., I-329, II-145 Aguiar, O. D., I-500, I-501, I-574, I-627 Agullo, I., II-345, II-347, II-406, II-407, II-463, II-541, II-544 Aharonov, Y., II-414, II-464 Aharony, O., II-342, II-462 Ahmed, Z., I-327 Akeley, E. S., I-176 Akhlaghi, M., II-222 Akhmedov, E. T., II-413, II-414

Akutsu, T., I-500 Alam, U., II-406 Albanese, D., I-369 Albin, E., II-319 Albrecht, A., I-502, II-137, II-268, II-270, II-316, II-405 Alesci, E., II-544 Alexandrov, S., II-463 Alexandrow, W., I-176 Ali, S. S., II-409 Ali-Ha¨ımoud, Y., II-137 Alishahiha, M., II-271 II-317 Allahverdi, R., II-405 Allahverdizadeh, M., I-176 Allen, B., I-454, I-502, II-320, II-409, II-410, II-411 Allen, G., I-629 Allmendinger, F., I-314 Almheiri, A., II-459, II-465 Alpher, R. A., II-137, II-268 Altamimi, Z., I-366 Altschul, B., I-369 Alvarez, E., II-341 Alvarez-Gaume, L., II-341 Amanullah, R., II-170 Amaro-Seoane, P., I-454, I-503, I-629 Amelino-Camelia, G., II-346, II-347 Ames, A., II-40 Amp`ere, A.-M., I-81 Anderson, A. J., I-628 Anderson, I. M., II-464 Anderson, J. D., I-403, I-628 Anderson, L., II-41 Anderson, P. R., II-406, II-412, II-413 Anderson, S. B., I-503 Ando, M., I-501, I-573, I-575, I-576, I-629 Ando, S., I-503 Andr´e, P., II-145 Andreas, B., I-314 XIII

May 4, 2017 10:56

XIV

Andress, W. R., I-177 Ang´elil, R., I-177 Angulo, R. E., II-223 Anholm, M., I-454 Ansari, R., II-409 Ansorg, M., I-182 Antoniadis, I., II-316, II-412, II-413 Antoniadis, J., I-454 Antonucci, R., I-327 Aplin, K., I-630 Appleby, G., I-368 Applegate, D. E., II-223 Appouchaux, T., I-405, I-627 Arago, F., I-81 Arai, K., I-576 Araujo, H., I-630 Arends, M., II-318 Arenou, F., I-404 Aristotle, I-80 Arkani-Hamed, N., II-271 Armano, M., I-405, I-626 Armendariz-Picon, C., II-269 Armitage, P. J., I-454 Armstrong, J. W., I-500, I-628 Arnett, D., II-170 Arnold, K., II-137 Arnowitt, R. L., I-254, II-340, II-406 Arun, K. G., I-501, I-629 Arzoumanian, Z., I-454, I-502, I-629 Ashby, N., I-574 Ashoorioon, A., II-404 Ashtekar, A., II-342, II-343, II-345, II-347, II-459, II-463, II-465, II-538, II-539, II-540–542, II-544 Asmar, S. W., I-456 Aso, Y., I-574 Astier, P., II-170, II-172 Aston, S. M., I-629 Astone, P., I-500, I-574, I-575 Aubourg, E., II-41 Aubry, J. P., I-367 Aumont, J., I-327 Auriol, A., I-367 Avgoustidis, A., II-318, II-321 B Babak, S., I-502 Babich, D., II-137, II-271 Babul, A., II-223 Bach, R., I-177

Author Index

Bachlechner, T. C., II-318 B¨ ackdahl, T., I-177 Backer, D. C., I-313, I-454, I-455 Bacon, D. J., II-221, II-224 Bae, Y.-B., I-503, I-574 Baekler, P., I-177 Baez, J. C., II-343, II-345, II-463, II-539, II-541 Bahr, B., II-343, II-540 Bailes, M., I-454, I-455 Bailey, S., II-170, II-41 Bakshi, P. M., II-341, II-407 Balachandran, A. P., II-539 Ballmer, S., I-454 Banados, M., II-345, II-461 Banday, A. J., II-141, II-318 Banks, T., II-464 Barausse, E., I-458 Barbero, J. F., II-345, II-463, II-541 Barcelo, C., II-346, II-459, II-460 Bardeen, J. M., I-502, II-137, II-270, II-344, II-405, II-459, II-464, II-541 Barkana, R., II-137 Barker, B. M., I-454 Barnwell, N., I-367 Barrau, A., II-347, II-544, II-545 Barreiro, R. B., II-146 Bartelmann, M., II-221–223 Bartlett, D. F., I-400 Bartolo, N., II-137, II-271, II-408 Barton, E. J., I-457 Barvinsky, A. O., II-270, II-462 Bastero-Gil, M., II-407 Bateman, H., I-105, I-310 Battistellie, E., II-137 Battye, R. A., I-328, I-457, I-504, II-319, II-321, II-409 Bauer, H., I-254 Baugh, C. M., II-223 Baumann, D., II-316, II-317, II-342 Bautz, M. W., II-223 Bazin, G., II-172 Bazin, M., I-176 Beacom, J. F., I-503 Bean, R., II-317, II-318 Bec-Borsenberger, A., I-500 Becker, G., II-41 Becker, K., II-341, II-342 Becker, M., II-341, II-342 Becker, P., I-314

May 4, 2017 10:56

Author Index

Becker, R. H., II-137 Beetle, C., II-345, II-459, II-463, II-541 Begelman, M. C., I-328 Beig, R., I-177, I-254 Bekenstein, J. D., I-177, I-454, II-16, II-40, II-344, II-345, II-459, II-541 Beker, M. G., I-575 Bell, J. F., I-454 Ben-Dayan, I., II-318 Bender, P. L., I-498, I-501, I-626, I-627, I-629, I-630 Benjamin, J., II-224 Bennett, C. L., I-502, II-141, II-147, II-148, II-271, II-316, II-318 Bennett, D. P., II-320 Bentivegna, E., II-542 Bercy, A., I-369 Berczik, P., I-455 Berentzen, I., I-455 Berera, A., II-407 Berg, M., II-317, II-318 Bergamin, L., I-313 Bergamini, R., I-177 Bergmann, P. G., I-254 Bernardeau, F., II-319 Bernardi, M., II-222 Bernstein, G., II-224 Bernstein, R. A., I-328 Berry, C. P. L., I-456, I-499 Berti, E., I-503 Bertin, E., II-222 Bertotti, B., I-404, I-454 Bertschinger, E., II-137, II-142 Besso, M., I-106, I-398 Bethe, H., II-137, II-268 Beutler, F., II-41 Bevis, N., II-320, II-321 Bezrukov, F. L., II-270 Bharadwaj, S., II-409 Bhat, N. D. R., I-454 Bhattacharya, A., II-223 Bhattacharya, D., I-454 Bianchi, E., II-343, II-463, II-465, II-540, II-541, II-544 Biˇca ´k, J., I-257 Bilandzic, A., II-408 Bildsten, L., II-170 Billing, H., I-573 Bin´etruy, P., I-313 Bini, D., I-177

XV

Binney, J., II-322 Binns, D. A., I-626 Biot, J.-B., I-81 Birch, P., I-328 Birkhoff, G. D., I-177 Birkinshaw, M., I-502, II-137 Birmingham, D., II-345, II-461 Birrell, N. D., II-459 Biskupek, L., I-404, I-455 Bisnovatyi-Kogan, G. S., I-504 Bisognano, J. J., II-344, II-460 Bjerrum-Bohr, N. E. J., II-409 Blagojevi´c, M., I-177, I-254, II-539 Blair, D. G., I-575 Blanchard, A., II-149 Blanco-Pillado, J. J., II-321 Blandford, R. D., I-328, II-41, II-137, II-221, II-222 Blas, D., I-458 Blau, S. K., II-269 Bloch, F., II-407 Bloom, B. J., I-367 Bloomfield, J. K., II-322 Bl¨ ote, H. W. J., II-464 Blumenthal, G. R., II-41 Bock, J., II-139 Bode, P., II-41 Bodendorfer, N., II-545 Boehmer, C. G., II-542 Boggess, N. W., II-137 Boggs, D. H., I-401, I-403, I-404, I-455, I-458, I-630 Boggs, J. G., I-402 Bogoliubov, N. N., II-409, II-459 Bohm, D., II-414 B¨ ohringer, H., II-223 Bojowald, M., II-343, II-538, II-542–II-544 Bollini, C. G., II-340 Bolte, M., II-41 Bolton, A. S., II-41 Bolton, J., II-41 Bombelli, L., II-345, II-462, II-463 Bond, J. R., II-41, II-137, II-316 Bonnefond, P., I-367, I-369, I-628 Bonnor, W. B., I-177 Bonora, L., II-344, II-460 Boos, J., I-177 Booth, I., I-254, II-465 Borde, A., II-407

May 4, 2017 10:56

XVI

Borissov, R., II-540 Borja, E. F., II-345, 463, II-541 Born, M., II-137 Bosi, F., I-314 Bosworth, J. M., I-368 Boubekeur, L., II-269 Bouchet, F. R., II-137, II-320 Bouchet, F., II-138 Boughn, S., II-138 Boughn, S. P., I-575 Bousso, R., II-316, II-464 Bouwens, R. J., II-221 Bovy, J., I-457 Boyanovsky, D., II-413 Boyd, M. M., I-314, I-630 Boyer, R. H., I-177 Boylan-Kolchin, M., I-454, II-41 Boyle, L., I-454, II-316 Br¨ uegmann, B., II-539, II-342 Braccini, S., I-577 Bradaˇc, M., II-221 Brading, K., I-254, I-255 Bradley, J., I-80, I-178 Braginskii, V. B., I-573, I-576, I-577, I-628 Braibant, L., I-328 Brainerd, T. G., II-221, II-222 Branch, D., II-169, II-170 Brandenberger, R. H., II-143, II-270, II-405–II-407, II-413, II-414 Brandt, T. D., II-171 Braunstein, S. L., II-465 Braxmaier, C., I-369, I-401, I-500, I-627, II-347 Breton, R. P., I-454 Bridle, A. H., I-328 Bridle, S., II-142, II-223 Briggs, F., II-409 Brihaye, Y., I-177 Brill, D., I-177 Brizuela, D., II-542, II-543 Broadhurst, T., II-221, II-222, II-224, II-318 Brodin, G., I-504 Brooks, A., I-577, II-41 Brout, R., II-405 Brown, E. F., II-171 Brown, H. R., I-254, I-255 Brown, J. D., I-255, II-344, II-461, II-543

Author Index

Brown, J. M., I-315 Brown, M. L., I-328 Browne, I. W. A., I-328, II-409 Brumberg, V. A., I-401, I-402, I-630 Brun, A. S., I-403 Brunet, M., I-369 Brunier, T., II-410 Bruno, G., I-80 Buchan, S., II-317 Buchdahl, H. A., I-177 Bucher, M., I-108, I-626, II-15, II-3, II-43, II-138, II-139 Buchman, S., I-501, I-627 I-629 Buisson, H., I-399 Buividovich, P. V., II-413 Bulatowicz, M., I-314 Bullock, J., II-41 Bullock, J. S., I-457, II-41 Bunch, T. S., II-270, II-406 Bunn, E. F., I-328 Buonanno, A., II-321 Burda, P., II-413 Burgay, M., I-454 Burgess, C. P., II-316, II-317, II-346, II-408, II-460 Burgett, W., II-221 Burigana, C., II-138 Burinskii, A., I-177 Burke-Spolaor, S., I-454 Bush, R. I., I-403 Byrnes, C. T., II-408 C Caballero, R. N., I-400, I-457 Cabanac, R., I-328 Cacciapuoti, L., I-367, I-626 Cacciato, M., II-224 Cailleteau, T., II-347, II-545 Cain, B., II-223 Caldarelli, M. M., II-462 Caldwell, R. R., I-454, I-504, II-319, II-320 Callahan, P. S., I-628 Callan, Jr., C. G., II-341 Callen, H. B., I-576 Calzetta, E., II-341, II-407 Camblong, H. E., II-464 Cameron, P. B., I-455 Campagne, J.-E., II-409

May 4, 2017 10:56

Author Index

Campiglia, M., II-544 Candelas, P., II-410 Candelier, V., I-368 Cao, S., II-222 Capaccioli, M., II-319 Capelo, P. R., II-222 Capitaine, N., I-402 Capovilla, R., II-539 Capozziello, S., I-178, II-405 Caprini, C., I-504 Cardoso, J.-F., II-138 Cardy, J. L., II-464 Carilli, C., I-454, I-502 Carlip, S., I-107, I-177, II-325, II-340, II-344, II-346, II-415, II-459, II-461–464, II-538, II-541 Carlitz, R. D., II-465 Carlstrom, J. E., II-138 Carnot, S., I-83 Caro, L., I-367 Carroll, S. M., I-177, I-328, II-171, II-346, II-406 Carswell, R. F., II-42, II-220 ´ Cartan, E., I-311, I-312 Carter, B., I-178, II-344, II-459, II-541 Carter, W. E., I-404 Cartin, D., II-543 Casher, A., II-464 Casini, H., II-464 Cattani, C., I-255 Caux, E., II-140 Cavanaugh, J. F., I-367 Cavendish, H., I-398 Caves, C. M., I-574 Celotti, A., I-329 Cerdonio, M., I-575 Cernicharo, J., II-143 Cervantes-Cota, J. L., II-270 Chadburn, S., II-318, II-322 Chae, K.-H., II-222 Chaicherdsakul, K., II-407 Challinor, A., II-142 Chamberlin, S. J., I-454 Chambers, K., II-221 Champion, D. J., I-454, I-456 Chandrasekhar, S., I-400 Chang, C.-C., I-255 Chang, P., II-170 Chapon, D., I-454

XVII

Chen, C.-M., I-107, I-187, I-255, I-257, I-258, I-260, I-261 Chen, H., I-260 Chen, P.-N., I-255 Chen, S.-J., I-328 Chen, X., II-170, II-171, II-317, II-409 Cheng, K.-S., II-321 Cheng, S.-L., I-314 Chernoff, D. F., I-108, I-504, II-16, II-273, II-321, II-322 Cherubini, C., I-178 Chialva, D., II-409 Chiba, M., II-221 Chibisov, G. V., I-502, II-143, II-269, II-405 Chieppa, F., I-178 Childress, M., II-171 Chinnapared, K., I-182 Chiodo, N., I-369 Chiou, D.-W., I-107, I-401, I-630, II-325, II-342, II-467, II-540, II-542, II-544 Chiueh, T., II-318 Chluba, J., II-138, II-146 Choi, K., II-317 Chou, K.-C., II-341, II-407 Chowdhury, B. D., II-462 Christensen, S. M., II-460 Christiansen, J. L., II-319 Christoffel, E. B., I-105 Christophe, B., I-369, I-405, I-406 Chru´sciel, P. T., I-178 Chu, M., II-141 Chu, P.-H., I-314 Chu, S., I-312 Chua, S. S. Y., I-576 Chwolson, O., II-220 Cianfrani, F., II-544 Cicoli, M., II-317 Cimatti, A., I-328 Cinquegrana, C., I-574 Cirelli, M., II-16 Ciufolini, I., I-178, I-314, I-405 Clairon, A., I-369 Clarke, C. J. S., I-178, I-179 Clarke, J. N., I-328 Clarkson, C., I-504 Clifton, T., I-454 Cline, J. M., II-346

May 4, 2017 10:56

XVIII

Author Index

Clohessy, W. H., I-630 Clowe, D., II-222, II-224 Coe, D., II-221, II-222 II-224 Cognard, I., I-454, I-455 Cognola, G., II-462 Cohen, J. M., I-178 Cohen, M. H., I-329 Colacino, C. N., I-457, I-503, I-629 Cole, R. H., I-456, I-499 Coleman, S. R., II-410 Collilieux, X., I-366 Colom, P., II-409 Conklin, J. W., I-367, I-501, I-627 Conley, A., II-169, II-171 Conlon, J. P., II-317 Contaldi, C. R., II-138 Contardo, G., II-169 Cooperman, J. H., II-462 Cooperstock, F. I., I-255 Copeland, E. J., I-504, I-629, II-318, II-321 Copi, C. J., I-502 Corbitt, T., I-576 Cordes, J. M., I-454, I-457 Corey, B. E., II-41 Corichi, A., II-345, II-463, II-541, II-542 Corley, S., II-460 Cornish, N. J., I-313, I-501, I-627 Cornu, A., I-83 Corry, L., I-255 Costa, E., I-328 Cote, P. J., I-576 Couch, E., I-182 Coulson, D., II-137 Courde, C., I-367, I-368 Covino, S., I-313, I-328 Cramer, C. E., I-315 Creighton, J. D. E., I-254, I-454, I-457, II-320 Creminelli, P., II-137, II-270, II-271 Crill, B. P., II-138 Crittenden, R., II-138 Croft, R., II-41 Croom, S. M., II-222 Cropper, M., II-224 Crowder, J., I-501, I-627 Cruise, A. M., I-499, I-500 Cruz, M., II-146 Cudell, J. R., I-329 Curie P., I-83

Curtis, H., II-15 Cusano, N., I-80 Cutler, C., I-499, I-629 Cvetic, M., II-462 Cvitan, M., II-344, II-460 Cypriano, E. S., II-222 Cyr-Racine, F.-Y., II-405 D Dadhich, N., I-178 Dado, S., II-169 Dahl, M. F., I-313 Dahle, H., II-222 Dalal, N., II-42 Dame, T. M., II-138 Damour, T., I-405, I-454, I-455, II-269, II-319, II-320, II-346, II-460 Danielsson, U. H., II-342 Danzmann, K., I-575, II-322 Dar, A., II-169 Das, Sauryo, II-462 Das, Sudeep, II-138 Das, S. R., II-344, II-462 Dasgupta, K., II-317 da Silva Alves, M. E., I-628 Dass, N. D. H., II-342 Dautcourt, G., I-178 Dave, R., II-41 Davidson, C., I-106, I-399 Davies, P. C. W., II-270, II-459, II-460, II-461 Davis, A.-C., II-412 Davis, J. L., I-404 Davis, L., II-138 Davis, M. M., I-313 Davis, M., I-108, II-15, II-19, II-41 Davis, R. L., I-502, II-138 Davis, R. W., I-628 Day, P., II-138 de Andrade, V. C., I-255 de Araujo, J. C. N., I-501, I-627 de Bernardis, P., I-329, II-137, II-138 De Felice, A., II-271 de Felice, F., I-178, I-179 De Lucia, G., II-223 de Laix, A. A., II-319 De Maria, M., I-255 de Oliveira-Costa, A., II-138 De Pietri, R., II-343, II-540

May 4, 2017 10:56

Author Index

de Sitter, W., I-107, I-178 de Rham, C., II-346 de Vega, H. J., II-413 de Vine, G., I-630 de Waard, A., I-574 de Wit, B., II-342 De Yoreo, M., I-456 de Zeeuw, T., I-328 DePies, M. R., II-322 DePoy, D. L., II-224 DeWitt, B., I-178 DeWitt, B. S., I-178, II-340, II-341, II-344, II-404, II-460 DeWitt, C., I-178 Dean, A. J., I-328 Debaisieux, A., I-367 Debra, D., I-501, I-627 Deffayet, C., II-270 Defraigne, P., I-367 Degnan, J. J., I-366, I-368 Degueldre, H., II-411 Dehnen, H., II-270 Delabrouille, J., II-138 Dell, J., II-539 Deller, A. T., I-455 Delporte, J., I-369 Demarque, P., II-16 Demia´ nski, M., I-183 Demorest, P. B., I-455, I-629, II-320 Depies, M. R., II-320 Descartes, R., I-80 Deser, S., I-254, II-340, II-404, II-406 Desvignes, G., I-455 Detweiler, S., I-455 Dev, P. S. B., II-404 Dhurandhar, S. V., I-401, I-628, I-629 Di Casola, E., I-312 Di Criscienzo, R., II-344, II-460 di Serego Alighieri, S., I-313, I-317, I-328, I-503, II-346 Dias, M., II-318 Diaz-Polo, J., II-345, II-463, II-541 Dick, G. J., I-500, I-628, I-628 Dicke, R. H., I-312, I-399, I-402, II-15, II-139, II-268 Dickey, J. O., I-404 Dickinson, C., II-137, II-409 Diddams, S. A., I-367 Diemand, J., II-223

XIX

Dillinger, W. H., I-404 Dimmelmeier, H., I-503, I-574 Dimopoulos, K., II-412 Dimopoulos, S., I-501 Dimopoulos, S., I-629 Dimopoulos, S., II-317 Dine, Fischler, M., I-313 Dini, D., I-178 Dirac, P. A. M., I-255 Dittrich, B., II-540 Dittus, H., I-627 Djerroud, K., I-369 Dodds, S. J., II-224 Dodelson, S., II-141, II-271 Dolan, B. P., II-462 Dolan, L., II-269 Dolch, T., I-502 Dolgov, A. D., II-411 Domagala, M., II-345, II-463, II-541, II-543 Dominik, M., I-503, I-574 Donoghue, J. F., II-340, II-409 Dor´e, O., II-146, II-318 Doran, C., I-178 Doroshenko, O., I-458 Doroshkevich, A. G., II-40 Dotti, M., I-457 Dou, D., II-463 Douglas, M. R., II-316, II-342 Douglass, D. H., I-573 Dowker, F., II-344, II-461 Downs, G. S., I-455, I-499 Doyle, S., II-139 Draine, B. T., II-139, II-147, II-149 Dreitlein, J., I-574 Drever, R. W. P., I-573, I-575 Driggers, J. C., I-577 Droste, J., I-178 Dubovsky, S. L., II-462 Duffy, L. D., II-410 Duhem, P., I-80 Duley, W. W., II-139 Dultzin-Hacyan, D., I-458 Dunkley, J., I-502, II-139, II-148, II-321 Durrer, R., I-504, II-407 Duvall, Jr., T. L., I-402 Dvali, G., II-316, II-317, II-346 Dvorkin, C., II-316 Dwek, E., II-139

May 4, 2017 10:56

XX

Author Index

Dyer, C. C., I-329 Dymnikova, I. G., II-269 Dyson, F., I-106, I-399 E Eaker, W., II-412 Eardley, D. M., I-499 Earman, J., I-106, I-255, I-399 Easther, R., I-504, II-317, II-406, II-408 Eatough, R. P., I-456 Ebeling, H., II-223 Economou, A., II-319 Eddington, A. S., I-106, I-107, I-311, I-313, I-399, I-401, II-15 Efstathiou, G., II-137, II-41 Ehlers, J., I-178, II-221 Ehrenberg, W., II-414 Eichler, D., II-137 Einhorn, M. B., II-268 Einstein, A., I-83, I-104–I-107, I-178, I-255, I-256, I-311, I-313, I-397–I-399, I-455, I-498, II-15, II-139, II-171, II-220, II-339 Eisenberg, M., II-139 Eisenstaedt, J., I-178 Eisenstein, D. J., I-328, II-41 Eling, C., II-462 Ellis, G. F. R., I-180, I-181, II-465 Ellis, J. A., I-457, I-504 Ellis, R. S., II-40, II-220, II-221 Emilio, M., I-403 Emparan, R., II-461, II-462 Engle, J., II-345, II-463, II-464, II-541, II-542, II-543 Englert, F., II-405 Enqvist, K., II-408 E¨ otv¨ os, R. V., I-104, I-311, I-397 Erben, T., II-222 Eriksen, H. K., II-139 Ernst, F. J., I-179, I-180 Esposito-Far`ese, G., I-454, I-498 Estabrook, F. B., I-500, I-628 Eubanks, T. M., I-502 Euler, H., II-461 Evans, M., I-577 Everitt, C. W. F., I-179, I-181, I-314, I-405 Evershed, J., I-399 Exertier, P., I-367, I-368, I-403, I-628

F Fabbri, R., II-149 Fabry, C., I-399 Faddeev, L. D., II-340 Fahlman, G., II-223 Fairbairn, W., II-539 Fairhurst, S., II-345, II-459, II-463, II-541 Faizal, M., II-409 Falcke, H., I-179 Falco, E. E., II-221 Faller, J. E., I-626 Fang, X., I-404 Fang, Z., I-500 Faraday, M., I-81 Farhi, E., II-270 Farmer, A. J., I-503, I-629, II-322 Favaro, A., I-313 Favata, M., I-256, I-455 Feinberg, G., II-340 Fejer, M. M., I-576 Feldman, H. A., II-143, II-270, II-405 Ferdman, R. D., I-455, I-503, II-320 Fern´ andez-Soto, A., I-313 Fernandez-Borja, E., II-345, II-463, II-541 Ferreira, P. G., I-179, II-137, II-139 Fert´e, A., II-139 Feynman, R. P., II-460 Field, G. B., I-328 Fienga, A., I-401–I-403, I-630 Fierz, M., II-340 Figueroa, D. G., I-504 Filippenko, A. V., II-169 Filotas, E., II-346 Findlay, J. R., I-458 Finelli, F., I-328, II-411 Finkel, H., II-406 Finkelstein, D., I-179 Finn, L. S., I-455, I-499 Firouzjahi, H., II-317–I-319 Fisher, P., II-222 Fisher, Z., II-464 FitzGerald, G., I-83 Fivian, M. D., I-403 Fixsen, D. J., I-313, II-139, II-143 Fizeau, H., I-81, I-82 Flambaum, V. V., I-315 Flaminio, R., I-575, I-576 Flamm, L., I-179

May 4, 2017 10:56

Author Index

´ E., ´ Flanagan, E. I-329, I-499, II-464 Flauger, R., II-139, II-316, II-406 Fleischer, J., I-179 Fletcher, T., II-319 Flori, C., II-343, II-540 Fokker, A. D., I-106 Folacci, A., II-409 Foley, R. J., II-170 Folkner, W. M., I-401–I-403, I-455, I-456, I-629, I-630 Fomalont, E. B., I-404 Fonseca, E., I-455, I-503 Font, J. A., I-503 Ford Jr., W. K., II-16 Ford, K., II-344, II-459 Ford, L. H., II-270, II-407, II-411 Ford, W. K., II-40 Foreman, S. M., I-369 Fort, B., II-220 Forward, R. I., I-574 Fosbury, R. A. E., I-329, I-328 Foster, B. Z., II-462 Foulon, B., I-627 Fowler, L. A., I-457 Fowler, W. A., II-170 Fraisse, A. A., II-320 Frankel, T., I-256 Franx, M., II-222, II-224 Franzen, A., I-179 Frasconi, F., I-577 Frauendiener, J., I-256 Frazer, J., II-318 Freese, K., II-316, II-317, II-461 Freidel, L., II-539, II-543 Freire, P. C. C., I-400–I-456, I-498, II-17 Frenk, C. S., II-41, II-223, II-224 Fresnel, A., I-81 Freud, Ph., I-256 Fridelance, P., I-366, I-367, I-369 Friedmann, A., II-15 Frieman, J. A., II-171, II-269, II-316 Fritschel, P., I-575 Frittelli, S., II-539 Fr¨ ob, M. B., II-409 Frodden, E., II-463 Frolov, A. V., I-179 Frolov, V. P., I-179, II-346, II-463, II-464 Frœschl´e, M., I-404 Fu, L., II-224

XXI

Fujimoto, M., II-40 Fujishima, Y., II-40 Fukugita, M., II-221 Fukushima, T., I-402 Fulling, S. A., II-406, II-460 Fumin, Y., I-366 Furlanetto, S., II-409 Fursaev, D. V., II-346, II-463 Fuskeland, U., II-139 Futamase, T., I-108, II-15, II-173, II-221–II-224 G Gabadadze, G., II-346 Gair, J. R., I-502, I-503, II-347 Galaverni, M., I-328 Galilei, G., I-80, I-104, I-311, I-397 Gallagher, J. S., II-171 Gambini, R., II-342, II-346, II-538, II-539, II-543, II-544 Gamov, G., II-137, II-268 Gao, F., I-626 Gao, L., II-223, II-224 Gao, X., II-270, II-407 Gao, Y., I-108, II-15, II-151, II-171 Garay, L. J., II-542 Garbrecht, B., II-410, II-412 Garc´ıa, A., I-179 Garcia-Bellido, J., I-504, II-406 Garcia-Compean, H., I-179, I-181 Garfinkle, D., II-344, II-461 Garnavich, P. M., II-169, II-171 Garriga, J., II-270, II-414 Gasperini, M., I-504 Gauntlett, J. P., II-344, II-461 Geier, S., II-170 Geiger, R., I-501, I-629 Geiller, M., II-463, II-542 Geng, C.-Q., I-314 Geng, J. J., I-504 G´enova-Santos, R., II-145 Georgi, H., II-268 Gerard, E., I-367 G´erard, J. M., I-312 Germani, C., II-270 Geroch, R. P., I-179 Gerstenlauer, M., II-408, II-409 Gertsenshtein, M. E., I-574, I-628 Geshnizjani, G., II-322, II-322, II-413, II-414

May 4, 2017 10:56

XXII

Ghosh, A., II-463 Giambiagi, J. J., II-340 Giard, M., II-140 Gibbons, G. W., II-344, II-460, II-461, II-462 Gibbs, J. W., I-105 Giblin, J. T., I-504 Giddings, S. B., II-316, II-344, II-408, II-409, II-461, II-464, II-465, II-544 Giesel, K., II-343, II-543 Giffard, R. P., I-574 Gil Pedro, F., II-318 Gill, P., I-367 Gillespie, A., I-576 Gilmore, R., I-80 Ginzburg, V. L., II-140 Giocoli, C., II-223 Giovannini, M., I-504 Gispert, R., II-137 Giveon, A., II-341 Gladders, M. D., II-224 Glashow, S. L., II-268 Glauber, R. J., I-576 Glavan, D., II-411 ´ B., II-269 Gliner, E. Glymour, C., I-106, I-399 Goda, K., I-576 Godier, S., I-403 Gold, B., II-148 Goldberg, D. M., II-223 Goldberg, J. N., I-256 Goldhaber, A. S., II-138, II-140 Goldhaber, M., I-328 Goldstone, J., II-464 Gong, X., I-501, I-627 Gonzalez, M. E., I-455, II-320 Goodkind, J. M., I-401 Gorenstein, M. V., II-41 Goroff, M. H., II-340, II-404 Gorski, K. M., I-328, II-318 Goss, W. M., I-313 Gossler, S., I-575 Goto, T., II-341 Gott III, J. R., II-140, II-221 Gottardi, L., I-500 G¨ otz, D. I-313, I-328 Goudsmit, S., I-312 Gour, G., II-464 Governato, F., II-41 Gra˜ na, M., II-342

Author Index

Grabmeier, J., I-179 Grain, J., II-139, II-347, II-544, II-545 Grainge, K., II-146 Granata, M., I-576 Grasso, D., II-412 Gray, J., I-83 Graziani, R., II-322 Green, M. B., II-341 Green, R. G., I-314 Greenstein, J., II-138 Gregory, R., II-318, II-322 Griffiths, J. B., I-179 Grimani, C., I-630 Grishchuk, L. P., I-500, I-504, II-405 Grogin, N. A., II-221 Gronwald, F., I-181, I-256 Gross, M., I-403 Grosseteste, R., I-80 Grossmann, M., I-106, I-256, I-311, I-399 Grot, N., II-539 Grote, H., I-576, I-577 Groten, E., I-402 Gu´ena, J., I-367 Gubitosi, G., I-328 Gubser, S. S., II-318, II-342, II-462 Guica, M., II-464 Guillemot, L., I-455 Guillemot, P., I-368, I-369 Guillen, L. C. T., I-255 Guillochon, J., II-170 Gukov, S., II-342 Gullstrand, A., I-179 Gundlach, J. H., I-104 Gunn, J. E., II-17, II-140 Gunzig, E., II-405 Guo, Z.-K., II-405 Gupta, R. R., II-171 Gurevich, L. E., II-269 G¨ urses, M., I-177 Guth, A. H., I-502, II-40, II-140, II-268, II-269, II-315, II-405–II-407 Guy, J., II-169 Guy, R., II-139 Gwinn, C. R., I-502 Gwyn, S. D. J., II-222 H H¨ oflich, P., Haack, M.,

II-171, II-172 II-317

May 4, 2017 10:56

Author Index

Haag, R., II-460 Haaga, J. L., II-223 Habib, S., II-412 Hachisu, I., II-170 Hackmann, E., I-179 Haggard, H. M., II-544 Hahn, O., II-224 Hailu, G., II-317 Haiman, Z., II-141 Hall, J. L., I-577, I-626 Hall, J. S., II-140 Halverson, N. W., I-503 Hamal, K., I-366 Hamana, T., II-221, II-222 Hamilton, P. A., I-313 Hamuy, M., II-171 Han, J. L., II-140 Han, M., II-543 Han, Z., I-108, II-15, II-151, II-170, II-171 Hanany, S., II-140 Hand, N., II-140 Hansen, R. O., I-179 Hanson, A., I-256 Hao, B.-L., II-341, II-407 Hara, O., II-341 Harari, D., II-319 Hargreaves, R., I-105 Harigaya, K., II-318 Harry, G. H., I-574–I-576 Hartle, J. B., II-460 Hartman, T., II-464 Harvey, A., I-179 Harvey, J. A., II-346 Haslam, C. G. T., II-140 Hattori, M., II-221 Hauchecorne, A., I-403 Haugan, M., I-312 Hauser, I., I-179, I-180 Hauser, M. G., II-140 Hawking, S. W., I-180, I-502, II-140, II-269, II-344, II-405, II-459–II-461, II-464, II-465, II-541 Hayashi, C., II-268 Hayashi, K., I-256, I-312 Hayden, B. T., II-171 Haynes, M. P., II-41 Hayward, S. A., II-464 Hazumi, M., II-140 Heaps, W., I-402

XXIII

Hearn, A. C., I-180 Heavens, A. F., II-223 Heaviside, O., I-82 Hebecker, A., II-318, II-408, II-409 Hechler, F., I-629 Hecht, R. D., I-256 Heckel, B. R., I-315 Heflin, E. G., I-575 Hehl, F. W., I-107, I-109, I-177, I-179–I-184, I-254, I-256, I-310, I-312, I-313, II-539 Heiles, C., I-313 Heimpel, K., II-318 Heinicke, C., I-107, I-109, I-179, I-180 Heinzel, G., I-576 Heisenberg, W., II-461 Heitmann, H., I-575 Helfer, A. D., II-460 Hellings, R., I-455, I-499, I-501, I-627, I-628 Hello, P., I-576 Helmholtz, H., I-81 Henderson, A., II-543, II-544 Hennawi, J. F., II-224 Henneaux, M., II-461, II-462 Henry, R. C., I-180 Hepp, K., II-409 Herdeiro, C., I-177, I-185, II-317 Hernquist, L., II-148 Hertz, H., I-81, I-82 Herzog, C. P., II-318 Hetterscheidt, M., II-222 Heusler, M., I-178, I-180 Hewitt, J. N., II-220 Heymans, C., II-223 Hezaveh, Y., II-42 Hibbs, A. R., II-460 Hicken, M., II-170 Higaki, T., II-318 Higuchi, A., II-409, II-410 Hilbert, D., I-107, I-180, I-256, I-398 Hild, S., I-577 Hildebrandt, S. R., II-140 Hill, C. T., II-461 Hill, G. W., I-630 Hill, H. A., I-402 Hill, J. C., II-139, II-316, II-406 Hils, D., I-498, I-577, I-630 Hiltner, W. A., II-141 Hindmarsh, M., II-320

May 4, 2017 10:56

XXIV

Hinoue, K., I-180 Hinshaw, G., I-328, I-502, II-147, II-148, II-271, II-316, II-405 Hirakawa, H., I-573, I-574 Hirano, S., II-317 Hirata, C. M., II-137, II-140, II-146, II-223 Hirata, K., II-269 Hirose, E., I-576 Hiscock, B., I-627 Hiscock, W. A., II-461 Ho, F.-H., I-256 Hoˇrava, P., II-342 Hobbs, G., I-455, I-457, I-502, II-320 Hod, S., II-462 Hodges, H. M., I-502 Hoedl, S. A., I-314 Hoekstra, H., II-222, II-223, II-224 Hoffman, L., I-454 Hofmann, F., I-404, I-455 Hofmann, S., II-543 Hogan, C. J., II-319–II-322 Hogan, J. M., I-501, I-629 Hogg, D. W., II-41 Hojman, R., II-539 Holder, G., II-42, II-138, II-141 Hollberg, L., I-367 Holley-Bockelmann, K., I-455 Holman, K. W., I-369 Holman, R., II-408, II-413 Holst, S., II-539 Hong, T., I-576 Horndeski, G. W., II-270 Horowitz, G. T., II-461 Hou, Z., II-405 Hough, J., I-575 Houri, T., I-180 Howell, D. A., II-170–II-172 Hoyle, F., II-170 Hsu, S. D. H., II-340, II-464 Hu, B. L., II-341, II-406, II-407 Hu, W., II-141, II-143, II-147 Huang, H.-W., I-313 Huang, Y. F., I-504 Huang, Y.-C., I-314 Hubble, E., II-40, II-171, II-268 Hubeny, V. E., II-346, II-462 Huchra, J., II-41 Hudson, D. D., I-369 Hudson, H. S., I-403

Author Index

Hudson, M. J., II-222 Hugenholtz, N. M., II-460 Hughes, S. P., I-629 Hull, C. M., II-341 Hulse, R. A., I-455, I-457 Humphreys, W. J., I-399 Hunter, C. J., II-461 Hunter, L., I-314 Hutchinson, J. R., I-576 Huterer, D., II-171, II-224 Hutsemekers, D., I-328, I-329 Huudson, M. J., II-222 Huygens, C., I-80 I Ibe, M., II-318 Iben, I., II-170 Iess, L., I-404, I-454, I-500, I-628 Iguchi, S., I-457 Ijjas, A., II-406 Ikushima, Y., I-577 Iliopoulos, J., II-412 Illarionov, A. F., II-141 Ingley, R. M. J., I-499 Inoue, S., II-271 Ionescu, A. D., I-180, I-404 Irbah, A., I-403 Irwin, A. W., I-402 Isaacson, R. A., I-499 Isenberg, J., I-257 Isham, C. J., II-539, II-540 Islam, J. N., I-180 Iso, S., II-344, II-460 Israel, W., I-180, I-183, II-459, II-461 Itin, Y., I-313 Ito, N., I-184 Iwakami, W., I-503 Iyer, B., I-626 J Jackiw, R., I-328, II-269 Jackson, J. D., I-180, I-310 Jackson, M. G., II-319 Jacobson, T., II-342, II-410, II-460–II-463, II-465, II-539, II-541 Jacoby, B. A., I-455 Jadczyk, A., I-313 Jaekel, M.-T., I-399 Jaffe, A. H., I-455

May 4, 2017 10:56

Author Index

Jaki, S. L., I-398 James, K. A., II-319 Janis, A. I., I-182 Janssen, M., I-257 Janssen, T. M., II-412 Jantzen, R. T., I-177 Jarnhus, P. R., II-407 Jarosik, N., II-141, II-148 Jebsen, J. T., I-180 Jee, M. J., II-224 Jenet, F. A., I-454, I-455, I-457, II-320 Jenkins, A., II-223 Jennrich, O., I-500, I-626 Jensen, S., I-501, I-628 Jewell, L., I-399 Jezierski, J., I-257 Jha, S., II-169 Jin, D., I-456 Jo, S., II-539 Johansen, N. V., I-180 Johansson, J., II-171 Johnson, B. R., I-328, II-140 Johnson, W. W., I-574 Jones, D. O., II-172 Jones, M., II-146 Jones, N., II-319 Jordan, R. D., II-341, II-407 Joseph, C., II-139 Joshi, S. A., I-328 Jouve, L., I-403 Ju, L., I-577 Julia, B., I-257 Jullo, E., II-221 Jungman, G., II-141 Justham, S., II-170 K Kachru, S., II-316, II-317, II-342 Kahya, E. O., II-405, II-406, II-410, II-411 Kaiser, D. I., II-406 Kaiser, N., II-221–II-223 Kajisawa, M., II-224 Kalemci, E., I-328 Kallosh, R., II-316–II-318, II-342 Kalogera, V., I-574 Kaltofen, E., I-179 Kamada, K., II-269, II-270 Kamali, V., II-318 Kamble, A., I-629

XXV

Kamenshchik, A. Yu., II-270 Kamionkowski, M., I-328, II-141, II-149 Kami´ nski, W., II-542, II-543 Kane, T. J., I-575 Kang, H.-S., II-147 Kaplinghat, M., II-41, II-141 Kappl, R., II-318 Kasai, M., II-221 Kasevich, M. A., I-501, I-629 Kashlinsky, A., II-269 Kasian, L. E., I-456 Kaspi, V. M., I-454, I-455, II-320 Kastor, D. A., II-344, II-461, II-462 Kato, M., II-170 Katz, J., I-257, I-259 Kauffman, L. H., II-539 Kauffmann, T., I-312 Kaufman, J. P., I-328, II-140 Kaufmann, W., I-82 Kawai, T., I-257 Kawamura, S., I-501, I-576, I-627, II-271 Kawasaki, M., II-270, II-271 Kawashima, N., I-575 Kawazoe, F., I-577 Kay, B. S., II-461 Kazanas, D., II-405 Keating, B. G., I-328, II-140 Keeler, C., I-180 Keeton, C. R., II-222 Kehagias, A., II-270 Keil, R., I-630 Keisler, R., II-141 Keith, M. J., I-455 Keldysh, L. V., II-341, II-407 Kennefick, D., I-107 Kepler, J., I-103, I-397 Kerlick, G. D., I-256, I-312 Kerner, R., II-460 Kerr, R. P., I-177, I-180 Kessler, R., II-170, II-172 Khalatnikov, I. M., II-142 Khalili, F. Y., I-576 Khan, F. M., I-455 Khanna, G., II-543 Khlopov, M. Yu., I-314 Kibble, T. W. B., I-257, I-312, II-268, II-321 Kiefer, C., II-340, II-459, II-538 Kijowski, J., I-257

May 4, 2017 10:56

XXVI

Kilbinger, M., II-224 Kim, A. G., II-169–II-171 Kim, C., I-503, I-574 Kim, H., II-317 Kim, J. E., I-313, II-317 Kimble, H. J., I-576 Kimura, S., I-574 King, L. J., II-223 King, M., I-183 Kinney, W. H., II-271, II-317 Kirchhoff, G., I-81 Kiriushcheva, N., I-257 Kirshner, R. P., II-169, II-172 Kirzhnits, D. A., II-269 Kisielowski, M., II-543 Kitamoto, H., II-411, II-412 Kitazawa, Y., II-411, II-412 Kitching, T. D., II-223 Kiziltan, B., I-456 Klainerman, S., I-180 Klebanov, I. R., II-318 Klein, F., I-105, I-257 Klein, O., II-339 Kleinw¨ achter, A., I-182 Klemm, D., II-462 Kleppe, G., II-410 Klypin, A. A., II-223 Kneib, J.-P., II-40, II-220–II-222, II-224 Knop, R. A., II-171 Ko, M., II-539 Kobayashi, S., I-257 Kobayashi, T., II-269–II-271 Kochanek, C. S., II-42, II-221, II-222 Kodama, H., II-141, II-269, II-269, II-270, II-271 Kodet, J., I-369 Koester, D., I-454 Kofman, L. A., II-269, II-270 Kogut, A., II-141, II-147, II-316 Kohlrausch, R., I-81 Kolb, E. W., II-271, II-405 Komatsu, E., I-502, II-141, II-148, II-271, II-316 Komossa, S., I-456 Konacki, M., I-458 Konopleva, N. P., I-257 Konopliv, A. S., I-456 Konstandin, T., I-504 Kopeikin, S. M., I-178, I-404, I-504

Author Index

Kopernik, M., I-80 Kornack, T. W., I-315 Kors, B., II-317 Koshti, S., I-629 Kosmann-Schwarzbach, Y., I-257 Kosowsky, A., II-141, II-149 Kosteleck´ y, V. A., I-313, I-328, II-346 Kotake, K., I-503, I-574 Kotera, K., II-346 Kottas, A., I-456 Kottler, F., I-105 Koul, R. K., II-345, II-462 Kovac, J. M., I-328, II-142 Kovalchuk, E. V., I-104 Krajewski, T., II-544 Kral, K., I-366 Kramer, D., I-181, I-184 Kramer, M., I-400, I-454–I-457, I-498, I-503 Krasi´ nski, A., I-178, I-181, I-183 Krasnov, K., II-345, II-463, II-539, II-541, II-543 Kraus, J., II-141 Krause, A., II-317 Krauss, L. M., I-502, II-404 Krippendorf, S., II-318 Krisher, T. P., I-313 Krishnan, B., II-345, II-459, II-465, II-541 Kronberg, P. P., I-328, I-329, II-412 Kroon, J. A. V., I-177 Krotkov, R., I-399 Krotov, D., II-412 Krueger, B. K., II-171 Kubiznak, D., II-462 Kubo, R., II-460 Kuchaˇr, K. V., I-257, II-540, II-543 Kuhn, J. R., I-402, I-403 Kuijken, K., II-222, II-224 Kulkarni, S. R., I-313 Kumar, J., II-408 Kundi´c, T., II-220 Kunimitsu, T., II-270 Kunstatter, G., II-462 Kunz, J., I-176 Kunz, M., II-320 Kuroda, K., I-105, I-314, I-400, I-461, I-500, I-505, I-573, I-576, I-577, I-626 Kuroyanagi, S., II-322 Kuzmin, L., II-142

May 4, 2017 10:56

Author Index

Kuzmin, S. V., I-257 Kwok, A., II-322 Kyutoku, K., I-574 L LaFave, N. J., II-343 Laas-Bourez, M., I-368 Laborde, A., I-83 Lacey, C. G., II-223 Laddha, A., II-543 Lamarre, J. M., II-142 L¨ ammerzahl, C., I-179, I-181, I-312, I-405, I-627, II-346, II-347 Lamy, H., I-328 Lanczos, C., I-257 Lanczos, K., I-181 Land, K., II-142 Landau, L. D., I-181, I-257, I-499, II-142 Langer, N., II-170 Langevin, P., I-83 Langlois, D., II-271 Lanyi, G., I-404 Larmor, J., I-82 Larsen, F., I-180 Larson, D., II-148, II-316 Larson, S. L., I-455, I-501, I-628, II-347 Lasenby, A., II-142 Laskar, J., I-401–I-403, I-630 Lass-Bourez, M., I-368 Latham, D. W., II-41 Lattimer, J. M., I-456 Lau, E. L., I-403 Lauber, P., I-367 Laue, M. V., I-310 Laurent, P., I-313 Lazarian, A., II-139, II-149 Le Verrier, U. J. J., I-103, I-397 Leach, S. M., II-142, II-271 Leahy, J. P., I-329 Lebach, D. E., I-404 Leblond, L., I-456, I-504, II-318, II-408 Lee, A. T., II-142 Lee, D. L., I-312, I-499 Lee, J., II-223, II-345, II-462, II-463 Lee, K. J., I-456 Lefor, A. T., II-222 Leger, A., II-142, II-145 Lehner, L., II-539 Lehto, H. J., I-457

XXVII

Leibundgut, B., II-169, II-170 Leitch, E. M., II-142 Lemaˆıtre, G., II-15, II-171, II-268, II-269 Lemonde, P., I-367 Lemson, G., II-41 Lense, J., I-108, I-181, I-314, I-405 Lentati, L., I-502, I-629 Leonard, A., II-223 Leonard, K. E., II-410, II-411 Lepora, N. F., I-329 Lesgourgues, J., II-142 Letellier, C., I-3, I-80, I-104 Levi-Civita, T., I-105, I-181 Levin, Y., I-576 Levinson, R. S., II-223 Lewandowski, J., II-343, II-345, II-538–II-541, II-543 Lewis, A., II-142 Lewis, G. F., II-221 Lewis, T., I-181 Lewkowycz, A., II-462 Li, D., I-456 Li, F., I-500 Li, G., I-630 Li, H., I-329 Li, L. F., II-542 Li, S., I-456 Li, W., II-171, II-172 Li, X., I-366 Li, X. D., II-170 Li, Z., II-171 Liang, Y.-R., I-626 Liao, A.-C., I-627 Libbrecht, K. G., I-402 Liberati, S., I-312, I-329, II-346, II-459, II-460, II-545 Lichtenegger, H. I. M., I-180, I-181 Liddle, A. R., II-140, II-269, II-271, II-317, II-405 Lidsey, J. E., II-270, II-405, II-407 Liebes, S., II-220, II-322 Liesenborgs, J., II-222 Lifshitz, E. M., I-181, I-257, II-142 Lightman, A. P., I-312, I-499 Lilje, P. B., II-222 Lim, E. A., I-504, II-408 Lin, R. P., I-403 Linde, A. D., I-502, II-41, II-268–II-270, II-315–II-318, II-342, II-405–II-407

May 4, 2017 10:56

XXVIII

Lindegren, L., I-404 Lindell, I. V., I-313 Linder, E. V., II-222 Lindquist, R. W., I-177 Link, F., II-220 Lins, S. L., II-539 Lintz, M., I-369 Liske, J., I-328 Liu, F. K., I-456 Liu, J.-L., I-255, I-257, I-258, I-260, I-261 Liu, K., I-456 Liu, X., I-457 Livine, E. R., II-463, II-540, II-543 Livio, M., II-170 Livne, E., II-170 Lloyd, G. E. R., I-80 Loeb, A., I-457, I-458, II-137, II-406, II-41 Logan, J. E., I-576 Loll, R., II-540 Lommen, A. N., I-455, I-501 Long, C., II-318 Longo, G., II-319 Lonsdale, C. J., II-409 Lopes Costa, J. L., I-178 Lopez, O., I-369 L´ opez-Caraballo, C. H., II-145 Loredo, T. J., I-329 Lorentz, H. A., I-82, I-104, I-397 Lorimer, D. R., I-455 Lozano, Y., II-341 Lu, T., I-504 L¨ uck, H., I-573, I-574 Ludlow, A. D., I-314, I-630 Ludvigsen, M., II-539 Lukierski, J., II-346 Lundmark, K., II-169 Luo, J., I-501, I-626 Luppino, G. A., II-221, II-222 Lupton, R., II-142 Luty, M. A., II-270, II-462 Luzum, B., I-402 Lydon, T. J., I-402 Lynch, R. S., I-456 Lynden-Bell, D., I-257, II-461 Lynds, R., II-220 Lyne, A. G., I-455, I-456

Author Index

Lyth, D. H., II-140, II-142, II-269, II-271, II-316, II-405, II-408 Lyutikov, M., I-456 M Ma, C.-P., II-142 Maartens, R., I-504 MacCallum, M. A. H., I-181, I-184 Maccione, L., I-329, II-346, II-545 Macdonald, D. A., II-464 Mac´ıas, A., I-180, I-184 Maddox, S. J., II-41, II-222 Maeda, K., II-170 Maeda, K.-I., II-408 Maggiore, M., I-499, II-462 Magliaro, E., II-343, II-540, II-541 Magueijo, J., II-138, II-142 Mahajan, N., II-407 Mahanthappa, K. T., II-341, II-407 Maischberger, K., I-575 Majid, S., II-346 Major, S. A., II-544 Majorana, E., I-576 Majumdar, M., II-316 Makino, N., II-221 Maldacena, J., II-142, II-342, II-405, II-462 Malebranche, N., I-80 Man, C. N., I-575 Manche, H., I-401–I-403, I-630 Manchester, R. N., I-399, I-407, I-456, I-457, I-501, I-503, I-626 Mandel, K. S., II-172 Mandel, L., II-142 Mandelstam, L., I-310 Mandic, V., I-457, II-320, II-321 Manko, V. S., I-179, I-181 Mann, R. B., II-344, II-460–II-462 Mannucci, F., II-170, II-171 Mao, D., I-368 Mao, S., II-221, II-222, II-322 Maoli, R., II-221 Maor, I., II-320 Maoz, D., II-170, II-171, II-222 Marchal, C., I-104 Marchesano, F., II-318 Marciano, A., II-346 Markevitch, M., II-16, II-42

May 4, 2017 10:56

Author Index

Marolf, D., II-343, II-407, II-410, II-413, II-459, II-464, II-539, II-540, II-543 Marozzi, G., II-407, II-411, II-414 Marriage, T. A., II-142 Mars, M., I-181 Marsh, G. E., I-181 Marshall, P. J., II-221 Mart´ınez-Gonz´ alez, E., II-146 Martin, I., I-576 Martin, J., II-406, II-413 Martin, N., I-368 Martin, P. C., II-460 Martin-Benito, M., II-542 Martin-de Blas, D., II-542 Martineau, P., II-413 Martinec, E. J., II-341 Martins, C. J. A. P., II-321, II-322 Marugan, G. A. M., II-542, II-543 Mashhoon, B., I-178, I-181, I-183 Masi, S., I-329 Massardi, M., I-329 Massey, R., II-223, II-224 Masters, K. L., II-41 Masui, K. W., II-409 Matarrese, S., II-137, II-271, II-408 Matassa, M., II-346 Mather, J. C., II-142, II-143 Mathews, J., I-456, I-499 Mathieu, P., II-460 Mathur, S. D., II-344, II-346, II-462 Matschull, H. J., II-539 Matsumoto, A., II-222 Mauceli, E., I-575 Maurice, N., I-369 Mauskopf, P., II-139 Maxwell, J. C., I-81, I-83 Mayer, L., I-454 Mazon, D., II-346 Mazumdar, A., II-317, II-404, II-409 Mazur, P. O., I-181, II-412, II-413 McAllister, L., II-316–II-318, II-342 McClintock, J. E., I-182 McClure, R. D., II-16 McCrea, J. D., I-256 McCulloch, P. M., I-313, I-457 McDonald, P., II-320 McGarry, J., I-368 McGaugh, S. S., II-16 McGlynn, T. A., II-149

XXIX

McGreevy, J., II-317 McGuirk, P., II-318 McHugh, M. P., I-501 McKenzie, K., I-576, I-630 McLaughlin, M. A., I-456, I-504 McNamara, P. W., I-626 McWilliams, S. T., I-456, I-501, I-627 Medezinski, E., II-222 Medved, A. J. M., II-462, II-464 Meers, B. J., I-575, I-576 Meftah, M., I-403 Mei, H.-H., I-313, I-328, I-329, I-503, II-346 Meinel, R., I-181, I-182, I-183 Meisel, L. V., I-576 Meissner, K. A., II-345, II-463, II-541 Melchior, P., II-223 Melia, F., I-182 Mellier, Y., II-220, II-223 Melnikov, K., II-461 Men, J. R., I-630 Mendes, L. E., II-269 Meneghetti, M., II-224 Meng, W., I-366 Meng, X., I-108, II-15, II-151, II-170, II-171 Meny, C., II-143 Mercati, F., II-346 Merkowitz, S. M., I-574 Merritt, D., I-456 Merten, J., II-224 Messager, V., I-3, I-80, I-104 Mester, Z., I-314 M´etivier, L., I-366 M´etris, G., I-104, I-406 Mewes, M., I-328, I-313, I-313, I-328 Meyer, D., II-463 Miao, S.-P., II-404, II-407, II-409–II-412 Michell, J., I-81, I-182 Michelson, A. A., I-82, I-83, I-104, I-397 Mie, G., I-182 Mielczarek, J., II-347, II-545 Mielke, E. W., I-180, I-256, I-258 Mignard, F., I-404 Mijic, M. B., II-271 Miknaitis, G., II-171 Milani, A., I-405, I-628 Milgrom, M., I-456, II-16, II-40 Miller, A. D., II-143 Miller, L., II-223

May 4, 2017 10:56

XXX

Author Index

Mills, R., I-261 Milosavljevi´c, M., I-456 Minkowski, H., I-104, I-105, I-310 Minkowski, R., II-169 Mio, N., I-626 Misner, C. W., I-105, I-182, I-254, I-258, I-311, I-400, I-499, I-574, II-15, II-340, II-406 Mitra, P., II-463 Miyamoto, K., II-321, II-322 Miyamoto, Y., II-270 Miyoki, S., I-575, I-577 Mizuno, J., I-576 Mo, H. J., II-223 Modesto, L., II-542 Moffat, J. W., I-313 Mohler, J., I-399 Mohr, P. J., I-314 Molina-Paris, C., II-412, II-413 Møller, C., I-258 Montesinos, M., II-540 Moodley, K., II-138 Moore, B., II-223 Moore, C. J., I-456, I-499, I-502 Moore, P., I-103, II-16 Mora, P. J., II-409, II-410, II-412 Morduch, G. E., I-106 Morita, T., II-344, II-460 Morley, E. W., I-82, I-104, I-397 Morris, M. S., II-271 Morrison, I. A., II-407, II-410, II-413 Mortonson, M. J., II-316 Moss, A., II-321 Mottola, E., II-412, II-413 Mouchet, A., II-463 Mourao, J. M., II-539 Mueller, G., I-576 Mukhanov, V. F., I-502, II-143, II-269, II-270, II-345, II-405, II-406, II-414, II-541 Mukherjee, P., II-406 Mukku, C., II-539 Mukohyama, S., II-271 M¨ uller, E., I-503 M¨ uller, H., I-312 M¨ uller, J., I-404, I-455, I-456 Muller, R. A., II-41 Mulryne, D. J., I-504 Murata, Y., I-457 Murphy, T. W., I-628

Myers, R. C., I-182, I-329, II-318 Myhrvold, N. P., II-412 N Naess, S., I-329, I-502 Nagel, M., I-104 Nakagawa, N., I-576 Nakamura, T., I-577, II-269, II-271 Nakayama, K., II-269, II-270, II-321 Nam, S. W., II-142 Nambu, Y., II-341 Nan, R., I-456, I-502 Narayan, G., II-172 Narayan, R., I-182, II-41, II-221, II-319 Narayanan, V. K., II-41 Narihara, K., I-573, I-574 Natarajan, P., I-454, II-222, II-224 Navarro, J. F., II-224 Navarro-Lerida, F., I-176 Navarro-Salas, J., II-406, II-407 Naylon, J., II-139 Neill, J. D., II-171 Neiman, Y., II-461 Nelemans, G., II-170 Nelson, W., II-347, II-544 Nester, J. M., I-107, I-187, I-255–I-261 Netterfield, C. B., II-143 Neugebauer, G., I-181, I-182 Neumann, F. E., I-81 Neveu, A., II-341 Newcomb, S., I-103, I-398 Newell, D. B., I-314 Newhall, X. X., I-401, I-404 Newman, A. B., II-223 Newman, E. T., I-182, II-539 Newton, I., I-80, I-103, I-182, I-311, I-397, I-398 Ni, W.-T., I-85, I-104–I-108, I-182, I-258, I-265, I-310–I-314, I-328, I-329, I-368, I-369, I-371, I-398–I-401, I-403, I-405, I-456, I-461, I-499–I-501, I-503, I-573, I-579, I-626–I-630, II-3, II-17, II-143, II-325, II-346, II-347, II-544 Nice, D. J., I-456, I-458, I-498 Nicholson, T. L., I-367 Nicolai, H., II-539 Nicolas, J., I-369 Nicolis, A., II-270 Nieto, M., II-140 Nightingale, M. P., II-464

May 4, 2017 10:56

Author Index

Nikoli´c, I. A., I-254 Nilles, H. P., II-317, II-318 Nishizawa, A., I-500 Nitsch, J., I-312 Niven, C., I-105 Nobili, A. M., I-312 Nodland, B., I-329 Noether, E., I-258 Nojiri, S., II-405 Nolta, M., II-148 Nolte, D., II-316 Nomizu, K., I-257, I-258 Nomoto, K., II-170 Nomura, Y., II-406 Nordsieck, A., II-407 Nordstr¨ om, G., I-106, I-182 Nordtvedt, K., I-312, I-400, I-401, I-403, I-405, I-456, I-458, II-346 Norton, J. D., I-257, I-258 Noui, K., II-345, II-463, II-541–543 Novikov, I. D., II-40 Nugent, P. E., II-170 Nunes, N. J., I-504 Nurmi, S., II-408 Nusser, A., II-40, II-41 Nussinov, S., II-464 O Oates, C. W., I-367 Obukhov, Yu. N., I-179, I-180, I-182, I-184, I-256, I-310, I-313, I-314 O’Callaghan, E., II-322 O’Connell, R. F., I-454 Odintsov, S. D., II-405 Ofek, E. O., II-222 Ogawa, Y., I-576 Ogle, P. M., I-329 Oguri, M., II-221–II-223 Oh, S. P., II-409 Ohanian, H. C., I-183 Ohashi, M., I-575 Ohnishi, N., I-503 Ohta, N., II-405 Okabe, N., II-221, II-223, II-224 Okamoto, T., II-141, II-143 Okolow, A., II-539 Okura, Y., II-222, II-223 Olinto, A. V., II-316, II-346 Olive, K. A., II-16 Olmedo, J., II-544

XXXI

¨ Olmez, S., II-321 Olmo, G. J., II-406, II-407 Olum, K. D., II-321 ´ Murchadha, N., I-254 O O’Neill, B., I-183 Onemli, V. K., II-405, II-406, II-410 Oppenheimer, J. R., I-183 Ord, K., II-146 O’Raifeartaigh, L., I-258 Ord´ on ˜ez, C. R., II-464 Oriti, D., II-343, II-538 Ort´ın, T., I-183 Ostriker, J. P., I-456, II-16, II-41, II-221, II-223 P Paci, F., I-328 Paczynski, B., II-221, II-322 Padmanabhan, T., II-460 Pagano, L., I-329 Page, D. N., II-459, II-461 Page, L., II-143, II-147, II-148 Pai, A., I-629 Paik, H. J., I-574 Painlev´e, P., I-183 Pais, A., I-258 Pajer, E., II-317, II-318 Pakmor, R., II-170 Pallua, S., II-344, II-460 Pan, H.-W., I-576 Pan, S.-s., I-627 Pan, W.-P., I-105, I-313, I-314, I-328, I-329, I-400, I-461, I-503, I-573, I-626, II-346 Panda, S., II-317 Panek, P., I-369 Papadodimas, K., II-465 Papapetrou, A., I-183, I-258 Paradis, D., II-143 Parasiuk, O. S., II-409 Pardo, J. R., II-143 Parikh, M. K., II-344, II-459, II-464 Park, S., II-404, II-410, II-411 Parker, L., II-270, II-404, II-406, II-407, II-459, II-460 Parker, S. R., I-104 Parsons, A. R., II-409 Pastor, S., II-142 Patanchon, G., II-138 Patil, S. P., II-316

May 4, 2017 10:56

XXXII

Pauli, W., I-258, II-340 Pavlis, E. C., I-178, I-314, I-405 Pawlowski, T., II-542, II-543 Payez, A., I-329 Peacock, J. A., II-143, II-224 Pearlman, M. R., I-368 Pearson, T. J., II-142 Peccei, R. D., I-313 Peebles, P. J. E., II-15, II-41, II-139, II-143, II-149, II-15, II-268 Peet, A. W., II-344, II-462 Peiris, H. V., II-147, II-269 Pelgrims, L., I-328 Peloso, M., II-317 Pen, U.-L., I-454, II-143, II-143, II-409 Penarrubia, J., II-41 Peng, G.-S., I-627 Penn, S. D., I-576 Penrose, R., I-183, I-258, II-342, II-539 Penzias, A. A., II-143, II-268 Pereira, J. G., I-255 Pereira, R., II-543 Perera, B. B. P., I-456 Perez, A., II-343, II-345, II-463, II-464, II-538, II-539, II-541, II-542, II-543 P´erez-Lorenzana, A., II-317 Perez-Nadal, G., II-411 Perini, C., II-343, II-540, II-541 Perley, R. A., I-328, I-329 Perlmutter, S. II-143, II-169, II-224, II-406 Perry, M. J., I-182, II-341, II-344, II-344, II-460 Peskin, M. E., II-464 Peter, A. H. G., II-41 Peters, A., I-312 Peters, P. C., I-456, I-499 Peterson, B. A., II-140 Petit, G., I-367 Petley, B. W., I-314 Petroff, D., I-183 Petrosian, V., II-16, II-220 Petrov, A. N., I-259 Petrov, A. Z., I-183 Pfister, H., I-183 Phillips, M. M., II-169 Phillips, P. R., I-315 Phinney, E. S., I-457, I-503, I-629, II-322 Phung, D. H., I-369

Author Index

Pi, S.-Y., I-502, II-140, II-269, II-405 Piazza, F., I-405 Pietroni, M., II-408 Pijpers, F. P., I-402 Pimentel, G. L., II-408 Pinto, R. F., I-403 Pirandola, S., II-465 Pirani, F., I-259 Pirani, F. A. E., I-183 Pirtskhalava, D., II-270 Pitjev, N. P., I-402, I-403 Pitjeva, E. V., I-401–I-403, I-630 Pitois, S., I-369 Pitts, J. B., I-259 Planck, M., I-104, I-105, I-311, I-398 Pleba´ nski, J., I-183 Plissi, M. V., I-577 Pober, J. C., II-409 Podolsk´ y, J., I-179 Podolsky, D., II-408 Podsiadlowski, Ph., II-170, II-171 Pogosian, L., II-320, II-321 Poincar´e, H., I-81–I-83, I-104, I-105, I-397 Poisson, E., II-465 Polarski, D., II-270, II-406 Polchinski, J., II-316, II-318, II-319, II-321, II-341, II-459 Poli, N., I-367 Pollock, M., II-149 Polyakov, A. M., II-268, II-341, II-412 Pontzen, A., II-41 Popov, F. K., II-413 Popov, V. N., I-257, II-340 Popper, K., I-83 Porrati, M., II-341, II-346 Pospelov, M., I-329 Pospieszalski, M., II-145 Post, E. J., I-311 Postman, M., II-224 Pound, R. V., I-399 Preskill, J., II-145, II-268 Press, W. H., II-171 Prestage, J. D., I-500, I-628 Pretorius, F., I-183, I-456, II-459 Price, R. H., II-464 Primack, J. R., I-503, II-41, II-223 Prince, T. A., II-322 Pritchet, C., II-171 Prochazka, I., I-366, I-367, I-369

May 4, 2017 10:56

Author Index

Prodi, G. A., I-575 Prokopec, T., II-407, II-408, II-410, II-411, II-412 Pshirkov, M. S., I-504 Pshirkov, M. S., II-322 Pskovskii, Y. P., II-169 Puchwein, E., II-411, II-412 Puget, J.-L., II-142, II-145 Pujolas, O., II-270 Pulido Pat´ on, A., I-629 Pullin, J., II-342, II-346, II-538, II-539, II-544 Puntigam, R. A., I-183 Punturo, M., I-574 Purcell, E. M., II-145 Purdue, P., I-259 Pustovoit, V. I., I-574 Puzio, R., II-539 Pyne, T., I-502 Q Quevedo, F., II-317, II-342 Quevedo, H., I-183 Quinn, H. R., I-313 Quinn, T. J., I-314 R Raab, F., I-576 Raab, S. A., I-314 Rabinovici, E., II-341 Radu, E., I-177, I-185 Rahman, S., I-183 Raine, D. J., II-410 Rajaraman, A., II-408 Rajesh Nayak, K., I-629 Raju, S., II-465 Ralston, J. P., I-329 Ramond, P., II-341 Randall, L., II-318, II-342 Rando, N., I-626 Randono, A., I-314 Rangamani, M., II-346, II-462 Rankine, W. J. M., I-105 Ransom, S. M., I-456, I-457 Rattazzi, R., II-270 Ravi, V., I-457 Ravndal, F., I-180 Rawlings, S., I-454, I-502 Ray, S., II-462

XXXIII

Raychaudhury, S., II-405 Readhead, A. C. S., II-142 Rebka, G. A., I-399 Rebolo, R., II-140 Reeb, D., II-464 Rees, M. J., I-328, II-321 Reese, E. D., II-138 Refregier, A. R., II-221, II-224 Refsdal, S., II-220, II-221 Regehr, M. W., I-575 Regge, C., II-461 Regge, T., I-256, I-259 Reichardt, C. L., II-145 Reissner, H., I-183 Renaux-Petel, S., II-271 Renn, J., I-257, I-259 Rephaeli, Y., II-145 Reynaud, S., I-399 Reynolds, M. T., I-457 Rhee, G., II-222 Ricci, G., I-105 Richard, J., II-221 Richards, P. L., II-142 Rickles, D., II-341 Rideout, D., II-463 Riemann, B., I-105 Riess, A. G., II-145, II-169, II-171, II-405 Rigault, M., II-171 Rinaldi, M., II-407 Rindler, W., I-183, II-15 Ringeval, C., II-321 Riotto, A., II-142, II-271, II-408 Risquez, D., I-630 Rix, H.-W., I-457, II-222 Robertson, D. I., I-575 Robertson, D. S., I-404 Robertson, H. P., II-15 Robinson, D. C., I-183 Robinson, S. P., II-344, II-460 Rocha, J. V., II-321 Rocha, M., II-41 Rodney, S. A., II-172 Rodrigues, M., I-104, I-406 Rodriguez, C., I-457 Rodriguez, Y., II-271 Roedig, C., I-457 Roemer, O., I-80 Roll, P. G., I-399, II-139 Romani, R. W., I-458

May 4, 2017 10:56

XXXIV

Romania, M. G., II-406, II-407 Romano, J. D., I-499, I-501, I-629, II-539, II-543 Roselot, J.-P., I-403 Rosenfeld, L., I-259 Roseveare, N. T., I-103, II-16 Ross, S. F., II-344, II-461 R¨ oser, H.-J., I-329 Roth, N., II-406 Roura, A., II-409, II-411, II-413 Rovelli, C., II-340, II-342–II-344, II-346, II-463, II-538–II-541, II-543–II-545 Rovera, D., I-367, I-368 Rowan, S., I-575 Rowe, B., II-223 Rowe, D. E., I-259 Rowland, H., I-399 Rubakov, V. A., II-149, II-270 Rubi˜ no-Mart´ın, J. A., II-140, II-145 Rubin, V., II-16, II-40 Rubinstein, H. R., II-412 Rudenko, V. N., I-504 Ruegg, H., II-346 Ruffini, R., I-183, II-460 Ruggiero, M. L., I-180 Rugina, C., I-180 Ruhl, J., II-145 Ruiter, A. J., II-170 Ruiz-Lapuente, P., II-170 Rust, B. W., II-169 Ryba, M. F., I-455, II-320 Ryckman, T. A., I-254 Ryu, S., II-345, II-462 S Sachs, I., II-345, II-461 Sachs, R. K., I-502, II-145 Sadofyev, A. V., II-413 Saffin, P. M., II-321 Sagnotti, A., II-340, II-404 Saha, P., I-177, II-222 Sahlmann, H., II-539, II-540 Sahni, V., II-17, II-405, II-406 Saikia, D. J., I-329 Saini, T. D., II-17, II-405 Saio, H., II-16, II-170 Sakellariadou, M., II-319, II-321 Sakharov, A. D., II-346, II-463 Sako, M., II-172

Author Index

Salam, A., II-269, II-464 Salomon, C., I-367 Salopek, D. S., II-269 Salter, C. J., I-329 ´ Samain, E., I-331, I-367–I-369, I-628, I-630 Sami, M., I-629 Samtleben, D., II-149 Samuel, S., II-346 Sanchez, C. A., I-314 Sanchez, N. G., II-413 Sanders, G. H., I-329 Sanders, R. H., I-184 Sanidas, S. A., I-457, II-321 Santoni, L., II-270 Santos, M. R., II-221 Sarangi, S., II-319 Sasaki, M., II-141, II-148, II-269, II-270, II-406 Sasselov, D., II-145, II-146 Sathyaprakash, B. S., I-629 Sato, H., II-269 Sato, K., I-108, I-502, I-503, II-15, II-225, II-268, II-269, II-405 Sato, S., I-575, I-577 Satz, A., II-541 Sauer, T., I-259 Saulson, P. R., I-574, I-576, I-577 Sawicki, I., II-270 Sayed, W. A., II-539 Sazhin, M. V., I-457, II-149, II-270, II-319 Sch¨ afer, G., I-455 Schafer, W., I-366 Schechter, P. L., II-223 Scheel, M. A., I-184 Schemmel, M., I-259 Schiff, L. I., I-312, I-314, I-329, I-405 Schiffer, M., I-176 Schild, A., I-259 Schive, H.-Y., II-318 Schlamminger, S., I-104 Schmidt, M., II-224 Schneider, D. P., II-220 Schneider, J., II-149 Schneider, P., II-221–II-224 Schnier, D., I-575 Schoen, R., II-340 Sch¨ onberg, M., I-313 Schouten, J. A., I-184

May 4, 2017 10:56

Author Index

Schrabback, T., II-224 Schramm, D. N., I-502 Schreiber, U., I-366, I-367 Schr¨ odinger, E., I-259, I-313 Schr¨ ufer, E., I-183 Schuldt, T., I-314 Schunck, F. E., II-317 Schutz, B. F., I-629, II-459 Schutzhold, R., II-460 Schwarz, D. J., II-406 Schwarz, J. H., II-341 Schwarzschild, K., I-107, I-184, I-259 Schwinger, J. S., II-341, II-407, II-460, II-461 Sciama, D. W., I-259, I-312 Scott, D., II-141, II-145, II-146, II-413 Scott, S. M., I-185 Scully, S. T., II-346 Seager, S., II-145, II-146 Seahra, S. S., I-504 Seery, D., II-270, II-407, II-408 Seitz, C., II-222, II-224 Seitz, S., II-223 Sekiguchi, T., II-322 Sekiguchi, Y., I-574 Sekiwa, Y., II-462 Selig, H., I-405, I-627 Seljak, U., II-143, II-146, II-149, II-223, II-316, II-320 Sellgren, K., II-146 Semboloni, E., II-224 Sen, A., II-464 Sen, S., II-345, II-461 Senatore, L., II-408 S´en´echal, D., II-460 Senovilla, J. M. M., I-181 Sepehri, A., II-318 Serabyn, E., II-143 Serra, G., II-140 Servin, M., I-504 Sesana, A., I-454, I-457, I-503, I-629 Setare, M. R., II-318 Sethi, S. K., II-17 Seto, N., I-501, I-627, II-271 Seto, O., II-271 Shafer, R. A., II-143 Shafi, Q., II-317 Shandera, S. E., II-319 Shannon, R. M., I-457, I-502, I-629 Shao, L., I-400, I-401, I-457

XXXV

Shapiro, I. I., I-103, I-399, I-402, I-457, II-16 Shapiro, S. S., I-404 Shapley, H., II-15, II-40 Shaposhnikov, M., II-270 Shaul, D., I-630 Shaw, L. D., II-223 Shellard, E. P. S., I-504, II-146, II-316, II-319, II-322 Shellard, E. P., II-321, II-321 Shen, K. J., II-170 Shen, Y., I-457 Sherwood, R., I-368 Sheth, R. K., II-223 Shibata, M., I-574 Shifman, M. A., I-313 Shimon, M., I-328 Shiomi, S., I-401 Shirafuji, T., I-256, I-312 Shiu, G., II-318, II-321 Shklovskii, I. S., I-457 Shlaer, B., I-456, I-504, II-317, II-321 Shoemaker, D., I-575 Shy, J.-T., I-368, I-627 Sibiryakov, S. M., II-462 Siday, R. E., II-414 Siemens, X., I-454, I-456, I-457, I-504, II-320, II-321 Sievers, J. L., II-405 Sigg, D., I-576 Sikivie, P., II-340 Silenko, A. J., I-182, I-314 Silk, J., II-141, II-146 Silva, S., I-257 Silverman, J. M., II-170 Silverstein, E., II-271, II-317, II-318 Sim, S. A., II-170 Simard-Normandin, M., I-328 Simon, J.-L., I-402 Simon, W., I-177, I-184 Singh, P., II-343, II-538, II-542 Singleton, D. A., II-413 Siopsis, G., II-462 Siuniaev, R. A., II-141 Sj¨ ors, S., II-318 Skenderis, K., II-345, II-463 Skillman, D. R., I-368 Skinner, R., I-259 Slepukhin, V. M., II-413 Slosar, A., II-320

May 4, 2017 10:56

XXXVI

Author Index

Sloth, M. S., II-407–II-409 Smail, I., II-40, II-220, II-222 Smit, D. J., II-342 Smit, J., II-408 Smith, D. E., I-367 Smith, F. G., I-313 Smith, G. P., II-221 Smith, J. R., I-574 Smith, K. M., II-146, II-316 Smolin, L., II-138, II-340, II-342, II-343, II-347, II-539, II-540, II-545 Smoot, G. F., I-502, II-41, II-146, II-269, II-316 Smullin, S. J., I-315 Snyder, H., I-183 So, L. L., I-259, I-260 Socorro, J., I-184 Soffel, M., I-401 Sofia, S., I-402 Soldner, J., II-220 Solganik, S., II-317 Sonego, S., I-312, II-459 Song, W., II-464 Soo, C., II-17 Sorbo, L., II-146 Sordre, L., II-222 Sorkin, R. D., II-345, II-462, II-463 Soucail, G., II-220 Speake, C. C., I-629 Spergel, D. N., II-41, II-139, II-141, II-143, II-147, II-148, II-271, II-316, II-318, II-406 Speziale, S., II-346, II-540, II-545 Spitzer, Jr., L., II-146, II-149 Spivak, M., I-260 Splaver, E. M., I-456 Springel, V., I-454, I-457 Springob, C. M., II-41 Squires, G., II-221–II-223 Srednicki, M., I-313, II-345, II-407, II-462 Srinivasan, K., II-460 Stachel, J., I-259, I-260, II-340 Stadel, J., II-223 Stadnik, Y. V., I-315 Staggs, S., II-149 Stairs, I. H., I-400, I-455–I-574 Standish, E. M., I-401, I-402 Stappers, B. W., I-457, II-321

Stark, C., II-41 Stark, J., I-184 Starobinsky, A. A., I-502, II-146, II-149, II-268–II-270, II-405, II-406, II-414, II-459 Statler, T. S., II-140 Stauffer, D., I-184 Steadman, B. R., I-177 Stebbins, A., II-141 Stebbins, R. T., I-402 Stecker, F. W., II-346 Steer, D. A., II-270, II-271 Stefani, F., I-369 Steigman, G., II-147 Steinhardt, P. J., I-502, II-16, II-41, II-137, II-138, II-268, II-270, II-316, II-405, II-406 Stelle, K. S., II-404 Stephani, H., I-184 Stern, O., I-312 Sternberg, S., I-184 Stewart, E. D., II-270, II-406 Stewart, K. R., I-457 Stoica, H., II-317, II-319 Stokes, G. G., I-83 Stoney, G. G., I-81 Strain, K. A., I-576 Strassler, M. J., II-318 Straumann, N., I-184, I-260 Straus, E. G., I-313 Strominger, A., II-342, II-344, II-345, II-461, II-462 Stuart, A., II-146 Sturgeon, R. E., I-314 Su, Y., I-404 Su, Z.-B., II-341, II-407 Sucher, J., II-340 Sudarsky, D., II-345, II-541 Sudou, H., I-457 Suen, W.-M., II-271 Sullivan, M., II-170, II-171 Sully, J., II-465 Sumitomo, Y., II-316 Sun, G., I-260 Sun, K.-X., I-629 Sun, X., I-368 Sundermeyer, K., I-260 Sundrum, R., II-318, II-342 Sunyaev, R. A., II-138, II-146

May 4, 2017 10:56

Author Index

Surdej, J., II-221 Susskind, L., II-342, II-462, II-464, II-465 Sutherland, W. J., I-458, II-41 Suyama, T., II-271 Suyu, S. H., II-221 Suzuki, N., II-17 Suzuki, T., I-574, I-575 Sweetser, T. H., I-629 Swiderski, R., II-544 Switzer, E. R., II-140, II-146 Syrovatsk, S. I., II-140 Szabados, L. B., I-260 Szil´ ard, L., II-139 T ’t Hooft, G., II-268, II-340, II-342, II-404, II-462, II-464 Tada, M., II-222 Takada, M., II-223, II-224 Takahashi, F., II-270, II-318 Takahashi, R., I-575, I-577 Takahashi, T., II-270 Takalo, L. O., I-457 Takamizu, Y., II-271 Takayanagi, T., II-345, II-346, II-462 Takeno, K., I-575 Tamai, R., I-328 Tamm, I. E., I-310 Tamm, J., I-310 Tammann, G. A., II-169 Tanaka, K. I., II-40 Tanaka, T., II-148, II-408, II-409, II-414 Tang, A., II-544 Tang, C.-J., I-401, I-630 Tangherlini, F. R., I-184 Tasinato, G., II-409 Tatarshi, D. C., I-313 Tate, R. S., II-539 Tatsumi, D., I-575 Tavel, M. A., I-260 Taveras, V., II-544 Taylor, A., II-146 Taylor, B. N., I-314 Taylor, G. B., I-457 Taylor, J. E., II-223 Taylor, J. H., I-455, I-457, I-458, I-498, II-320 Taylor, S. R., I-502

XXXVII

Tegmark, M., II-41 Teitelboim, C., I-256, I-259, II-344, II-345, II-461, II-462 Terno, D. R., II-463 Teryaev, O. V., I-182, I-314 Testi, L., I-329 Tetradis, N., II-269 Teyssier, R., I-454 Theiss, D. S., I-181 Thielemann, F.-K., II-170 Thiemann, T., II-342, II-343, II-538, II-540, II-543, II-545 Thirring, H., I-108, I-181, I-314, I-405 Thomas, R. M., II-138 Thompson, C., I-456 Thomson, R., I-254 Thomson, W., I-105 Thonnard, N., II-16, II-40 Thorlacius, L., II-465 Thorne, K. S., I-105, I-182, I-184, I-258, I-311, I-312, I-400, I-499, I-503, I-574, II-463, II-464 Thorsett, S. E., I-455, I-457 Thrane, E., I-499, I-501, I-629 Thurn, A., II-545 Timmes, F. X., II-171 Tingay, S. J., I-455 Tinsley, B. M., II-17 Tinto, M., I-500, I-501, I-627–I-629 Tkachev, I. I., II-269 Todorov, I. T., I-260 Tolman, R. C., I-260 Tomaras, T. N., II-412 Tomaru, T., I-577 Tomlin, C., II-543 Tong, D., II-271, II-317 Tonry, J. L., II-169 Tormen, G., II-223 Tornkvist, O., II-407, II-411, II-412 Torre, C. G., II-464 Torre, J.-M., I-368, I-369 Tortora, P., I-404, I-454, I-500, I-628 Touboul, P., I-104, I-406 Toupin, R., I-313 Tourain, C., I-367 Tournier, M., I-369 Townsend, P. K., II-341, II-342 Tran, H. D., I-329 Traschen, J. H., II-270, II-459, II-462 Trautman, A., I-183, I-260

May 4, 2017 10:56

XXXVIII

Treister, E., I-503 Tremaine, S., II-322 Tresguerres, R., I-184 Trimble, V., I-328 Trincherini, E., II-270 Trodden, M., II-405 Trujillo-Gomez, S., II-223 Truran, J. W., II-171 Tsamis, N. C., II-404–412, II-414 Tsao, H.-S., II-340, II-404 Tseng, H.-H., I-314 Tseng, S.-M., I-627 Tsubono, K., I-576 Tsujikawa, S., I-629, II-271 Tukey, J. W., II-146 Tulczyjew, W. M., I-257 Tullney, K., I-314 Tung, R.-S., I-187, I-255 Tuntsov, A. V., II-322 Turk, G. C., I-314 Turner, E. L., II-171, II-220–II-222 Turner, M. S., I-502, I-503, II-16, II-137, II-171, II-270, II-405 Turner, W., I-80 Turok, N., II-138, II-143, II-146, II-41 Turyshev, S. G., I-368, I-403, I-404, I-456, I-458 Tutukov, A. V., II-170 Twigg, L. W., I-402 Tye, S.-H. H., I-108, I-504, II-16, II-273, II-316–II-321 Tyson, J. A., II-221, II-222, II-224 U Uchiyama, T., I-577 Uehara, N., I-575 Uglum, J., II-462, II-465 Uhlenbeck, G., I-312 Uhrich, P., I-368 Umeda, H., II-170, II-171 Umetsu, H., II-344, II-460 Umetsu, K., II-221–II-224 Unruh, W. G., II-344, II-414, II-460, II-464 Unwin, S. C., I-502 Urakawa, Y., II-408, II-409 Uranga, A. M., II-318 Utiyama, R., I-260, I-312 Uzan, J., II-319

Author Index

V Vacca, G. P., II-411, II-414 Vachaspati, T., II-319, II-321, II-405 Vafa, C., II-342, II-344, II-462 Vagenas, E. C., II-464 Vaidya, S., II-462 Vainshtein, A. I., I-313 Valat, D., I-368 Valdes, F., II-221, II-222 Vallisneri, M., II-347 Valls-Gabaud, D., II-220 Valtonen, M. J., I-457 van Buren, D., I-400 van de Ven, A. E. M., II-340, II-404 van den Bosch, F. C., II-223 Van Den Broeck, C., II-345, II-463, II-541 van den Heuvel, E. P. J., I-454, II-170 van der Meulen, M., II-408 van Engelen, A., II-146 van Haarlem, M. P., II-409 van Haasteren, R., I-457 van Kerkwijk, M. H., I-454, II-170 van Leeuwen, J., I-457, I-503 van Nieuwenhuizen, P., II-340, II-404 van Stockum, W. J., I-184 van Straten, W., II-320 Van Waerbeke, L., II-221–II-224 Vanchurin, V., II-321 Vandersloot, K., II-542 Vanzo, L., II-344, II-460 Varadarajan, M., II-540, II-543, II-544 Vasili´c, M., I-254 Vaulin, R., II-412 Vecchiato A., I-404 Vecchio, A., I-457, I-503, I-629 Veillet, C., I-366–I-368 Velhinho, J. M., II-540 Veltman, M. J. G., II-340, II-404 Venemans, B. P., I-458 Veneziano, G., I-405, I-504, II-407 Verbiest, J. P. W., I-454, I-455, I-458, I-504 Vercnocke, B., II-318 Verdaguer, E., II-409, II-411 Verdoes Kleijn, G., II-224 Verma, A. K., I-403 Vernet, J., I-329 Vernotte, F., I-501

May 4, 2017 10:56

Author Index

Veryaskin, A. V., II-149, II-270 Viaggiu, S., I-177 Vidotto, F., II-342, II-538, II-544 Viel, M., II-41 Vielva, P., II-146 Vilenkin, A., I-455, I-458, II-146, II-269, II-270, II-271, II-316, II-319–II-321, II-407, II-408 Vilkovisky, G. A., II-464 Villani, D., I-405, I-628 Villar-Martin, M., I-329 Villasenor, E. J. S., II-345, II-541 Vincent, M. A., I-629 Vinet, J.-Y., I-576Viola, M., II-223 Visser, M., I-183,–I-185, II-346, II-460 Vizgin, V. P., I-260 Vlachynsky, E. J., I-184 Vocke, R. D., I-314 Vogelsberger, M., II-41 Voigt, W., I-82 Vokrouhlick´ y, D., I-405, I-628 Vollick, D., II-459 Volonteri, M., I-457, I-503, I-629 Vorontsov, Y. I., I-576 Vrancken, P., I-367, I-368 von Helmholtz, H., I-83 von Laue, M., I-106 von Soldner, J. G., I-105, I-398 von Westenholz, C., I-261 von der Heyde, P., I-256, I-312 W Wagner, T. A., I-104 Wagoner, R. V., I-574 Wahlquist, H. D., I-500, I-628 Wald, R. M., II-459–II-461, II-464, II-539 Walker, A. G., II-15 Walker, M. G., II-41 Walker, T. P., II-147 Wall, A. C., II-459, II-541 Wall´en, K. H., I-313 Wallner, R. P., I-258 Walsh, D., II-42, II-220 Walsh, J., I-329 Wands, D., II-406 Wang, B., II-170 Wang, G., I-401, I-627, I-628, I-630 Wang, L., II-171

XXXIX

Wang, L.-M., II-406 Wang, M.-T., I-255, I-261 Wang, T.-G., I-458 Wang, X., I-329 Wang, X. F., II-170 Wang, Y., II-406 Warburton, R. J., I-401 Ward, R. L., I-576 Wardle, J. F. C., I-329 Ware, B., I-630 Warrington, B., I-367 Wasserman, I., I-329, II-320, II-321 Watson, R. A., II-147 Wayth, R. B., II-222 Weaver, T. A., II-170 Webbink, R. F., II-170 Weber, J., I-573, I-574 Weber, W., I-81 Webster, R. L., II-222 Wechsler, R. H., II-224 Weems, L. D., II-461 Wehus, I. K., II-139 Weick, J., I-367 Weinberg, D. H., II-41 Weinberg, S., I-261, I-313, II-147, II-269, II-341, II-404, II-406, II-464 Weiner, N., II-41 Weingartner, J. C., II-139, II-147 Weinstein, M., II-461 Weisberg, J. M., I-454, I-458, I-498 Weispfenning, V., I-179 Weiss, M., I-367 Weller, J., II-223 Welton, T. A., I-576 Wen, L., I-502 Wenk, R. A., II-221, II-222 Westphal, A., II-317, II-318 Wex, N., I-400, I-401, I-455–I-458, I-498 Weyl, H., I-184, I-261 Weymann, R. J., II-42, II-220 Wheeler, J. A., I-105, I-178, I-182, I-258, I-311, I-400, I-499, I-574, II-344, II-459 Wheeler, J. C., II-171 Whelan, J., II-170 White, M., I-502, II-141, II-141, II-147, II-223 White, S. D. M., I-454, II-223, II-224 Whitrow, G. J., I-106 Whittaker, E. T., I-103, II-147

May 4, 2017 10:56

XL

Author Index

Wichmann, E. H., II-344, II-460 Wiersema, K., I-329 Wilczek, F., I-313, II-344, II-404, II-459, II-460, II-464 Wilkinson, D. T., II-41 Will, C. M., I-184, I-312, I-367, I-398–I-401, I-458, I-628 Willey, R. S., II-465 Williams, D. A., II-139 Williams, J. G., I-401–I-404, I-455, I-456, I-458, I-630 Williams, J. R., I-367 Williams, L. L. R., II-221, II-222 Williamson, R., II-147 Willke, B., I-575 Wilson, C., I-103 Wilson, G., II-221 Wilson, R. W., II-143, II-268 Wilson-Ewing, E., II-542 Wiltshire, D. L., I-185 Wiltshire, R. S., I-630 Winkler, W., I-577 Winnink, M., II-460 Winstein, B., II-149 Wise, M. B., II-137, II-270 Witten, E., II-318, II-321, II-341, II-342, II-461, II-463 Wittman, D. M., II-221, II-224 Wolf, E., II-137, II-142 Wolf, P., I-312, I-367, I-369, I-405 Wolfe, A. M., I-502, II-145 Wolfram, S., I-185 Wollack, E. J., II-145 Wolszczan, A., I-458 Wong, S. S. C., II-317 Wong, W. W., I-185 Woo, R., I-628 Wood, B. M., I-314 Wood, R., II-461 Wood-Vasey, W. M., II-171 Woodard, R. P., II-325, II-349, II-340, II-341, II-404–II-408, II-410–II-412, II-414, II-538 Woosley, S. E., II-170 Wootters, W. K., II-465 Wright, C. O., II-221 Wu, A.-M., I-368, I-500, I-627, I-630 Wu, H.-Y., II-224 Wu, M.-F., I-261

Wu, X.-N., I-261 Wyithe, J. S. B., I-457, I-458 Wyman, M., II-318, II-320 X Xia, J.-Q., I-329 Xie, N., I-256 Xu, X., I-627 Xue, W., II-407 Y Yadav, A. P. S., I-328 Yagi, K., I-458, I-629 Yamaguchi, M., II-269–II-271 Yamamoto, K., I-576, II-148 Yamamoto, T., II-346 Yan, Q.-Z., I-626 Yan, W. M., I-458 Yanagida, T., II-270 Yang, C. N., I-261 Yang, D., II-465 Yang, F. M., I-366 Yang, L., I-314 Yang, N., I-500 Yang, S. T., I-575 Yang, W., II-171 Yang, X., II-223 Yariv, A., I-576 Yau, S.-T., I-255, I-261, II-340 Ye, J., I-314, I-367, I-630 Yee, H. K. C., II-224 Yeh, H.-C., I-626 Yeomans, D. K., I-628 Yi, Z., I-630 Yokoi, K., II-170 Yokoyama, J., I-108, I-503, II-15, II-225, II-269–II-271, II-414 York, J. W., II-461 York, Jr., J. W., I-255 Yoshimura, M., II-269 Young, T., I-80 Yu, H., II-460 Yu, H.-L., II-17 Yu, J. T., II-143 Yu, Y.-W., II-321 Y¨ uksel, H., I-503 Yun, S., II-317 Yurtsever, U., II-464

May 4, 2017 10:56

Author Index

Z Zahn, O., II-146, II-148 Zakharov, V. I., I-313 Zakrzewski, W. J., II-346 Zalamansky, G., I-501 Zaldarriaga, M., II-41, II-137, II-146, II-148, II-149, II-271, II-408 Zanelli, J., II-345, II-461 Zavala, J., II-41 Zavala, R. T., I-457 Zavattini, E., I-329 Zehavi, I., II-41 Zel’dovich, Ya. B., II-40, II-146, II-270, II-459 Zelnikov, A. I., I-179, II-346, II-463 Zhang, X., I-329 Zhang, X.-G., I-458

Zhang, Y.-Z., II-405 Zheng, H., II-409 Zheng, Z., II-41 Zhou, L., I-314 Zhou, W., II-460 Zhou, Z. B., I-626 Zhu, X.-J., I-458, I-502 Zhu, Z.-H., II-222 Zibin, J. P., II-413 Zimmermann, M., I-182, II-409 Zitrin, A., II-221, II-224 Zohren, S., II-463 Zuber, M. T., I-367 Zucker, M. E., I-576 Zurek, W. H., II-463, II-465 Zwicky, F., II-16, II-40, II-220 Zyczkowski, K., II-465

XLI