The Encyclopedia Of Cosmology (In 4 Volumes) 9814656194, 9789814656191

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Table of contents :
Volume 1: Galaxy Formation and Evolution
Preface
Contents
Part I: Basic Theory of Galaxy Formation
1. Introduction
2. Review of Cosmology
3. Statistics of Galaxy Formation
4. Linear Perturbation Theory and the Power Spectrum
5. Non-linear Processes and Dark Matter Halos
6. Stellar Dynamics and the Virial Theorem
7. Galactic Disks and Spiral Structure
8. Gravitational Lensing
9. Summary and Conclusions
Part II: Early Galaxies and 21-cm Cosmology
10. Overview
11. Galaxy Formation: High-redshift Highlights
12. 21-cm Cosmology
13. The Supersonic Streaming Velocity
14. Cosmic Milestones of Early Radiative Feedback
15. 21-cm Signatures of the First Stars
16. Summary and Conclusions
Volume 2: Numerical Simulations in Cosmology
Preface
Contents
List of Videos
1. Overview: Cosmological Framework and the History of Computational Cosmology • Kentaro Nagamine
2. Cosmological N-Body Simulations • A. Klypin
3. Hydrodynamic Methods for Cosmological Simulations • Klaus Dolag
4. First Stars in Cosmos • Hajime Susa
5. First Galaxies and Massive Black Hole Seeds • Volker Bromm
6. Galaxy Formation and Evolution • Kentaro Nagamine
7. Secular Evolution of Disk Galaxies • Isaac Shlosman
8. Cosmic Gas and the Intergalactic Medium • Greg L. Bryan
9. Computational Modeling of Galaxy Clusters • Daisuke Nagai and Klaus Dolag
Index
Volume 3: Dark Energy
Preface
Contents
1. Introduction
2. Expanding Universe
3. General Relativity
4. Cosmic Expansion History
5. Observational Evidence of Dark Energy at the Background Level
6. Cosmological Perturbation Theory
7. Physics of CMB Temperature Anisotropies
8. Observational Probes for Dark Energy from CMB, Galaxy Clusterings, BAO, Weak Lensing
9. Cosmological Constant
10. Modified Matter Models of Dark Energy
11. Modified Gravity Models of Dark Energy
12. Horndeski Theories and Cosmological Perturbations
13. Second-order Massive Vector Theories
14. Screening Mechanisms of Fifth Forces
15. Effective Field Theory of Dark Energy
16. Conclusions
Appendix A: Equations of Motion in Horndeski Theories
Appendix B: Effective Mass Term in Horndeski Theories
Index
Volume 4: Dark Matter
Dedication
Preface
Acknowledgments
Contents
1. Introduction
2. Dark Matter Production in the Universe
3. Dark Matter and Large Scale Structures in the Universe
4. Symmetry Principles
5. Extended Objects
6. Bosonic Collective Motion
7. WIMPs and E-WIMPs
8. Baryogenesis and ADM
9. Detection
Index
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The

Encyclopedia of

Cosmology Volume 1 Galaxy Formation and Evolution

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World Scientific Series in Astrophysics ISSN: 2529-7511 Series Editor: Amiel Sternberg (Tel Aviv University, Israel) The field of observational and theoretical Astrophysics today spans an astonishing array of subjects, including extrasolar planets, star formation, interstellar and intergalactic medium, galaxy evolution, neutron stars and black holes, high energy phenomena, cosmology, and early Universe studies. Astrophysics is intrinsically interdisciplinary, bringing together knowledge in physics, chemistry, biology, computer science, mathematics, engineering and instrumentation technology, all for the goal of exploring and understanding the Universe at large. With the high-resolution and sensitive observations now possible with advanced telescopes on the ground and in space operating across the entire electromagnetic spectrum we are now in a golden era of discovery. There is tremendous interest in the results of world-wide research in Astrophysics across many domains, among scientists, engineers, and of course the general public. Published The Encyclopedia of Cosmology (In 4 Volumes) edited by Giovanni G Fazio (Harvard Smithsonian Center for Astrophysics, USA) Star Formation by Mark R Krumholz (Australian National University, Australia)

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World Scientific Series in A s tr

p h y s ic s

The

Encyclopedia of

Cosmology Volume 1 Galaxy Formation and Evolution

Rennan Barkana Tel Aviv University

Editor

Giovanni G Fazio

Harvard Smithsonian Center for Astrophysics, USA

World Scientific NEW JERSEY



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LONDON



SINGAPORE



BEIJING



SHANGHAI



HONG KONG



TA I P E I



CHENNAI

1/2/18 9:11 AM

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Names: Fazio, Giovanni G., 1933– editor. Title: The encyclopedia of cosmology / editor, Giovanni G. Fazio (Harvard Smithsonian Center for Astrophysics, USA). Description: Singapore ; Hackensack, NJ : World Scientific, [2018] | Series: World Scientific series in astrophysics | Includes bibliographical references and index. Contents: volume 1: Galaxy formation and evolution / by Rennan Barkana (Tel Aviv University) - volume 2: Numerical simulations in cosmology / edited by Kentaro Nagamine (Osaka University / University of Nevada) -- volume 3: Dark energy / by Shinji Tsujikawa (Tokyo University of Science) -- volume 4: Dark matter / by Jihn Kim (Seoul National University). Identifiers: LCCN 2017033919| ISBN 9789814656191 (set ; alk. paper) | ISBN 9814656194 (set ; alk. paper) | ISBN 9789814656221 (v.1 ; alk. paper) | ISBN 9814656224 (v.1 ; alk. paper) | ISBN 9789814656238 (v.2 ; alk. paper) | ISBN 9814656232 (v.2 ; alk. paper) | ISBN 9789814656245 (v.3 ; alk. paper) | ISBN 9814656240 (v.3 ; alk. paper) | ISBN 9789814656252 (v.4 ; alk. paper) | ISBN 9814656259 (v.4 ; alk. paper) Subjects: LCSH: Cosmology--Encyclopedias. Classification: LCC QB980.5 .E43 2018 | DDC 523.103--dc23 LC record available at https://lccn.loc.gov/2017033919 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Cover image credit: Vol. 1 - (front) NASA/JPL-Caltech, (back) NASA, ESA, S. Beckwith (STScI) and the HUDF Team; Vol. 2 - Illustris Collaboration; Vol. 3 - NASA, ESA, R. O'Connell (University of Virginia), F. Paresce (National Institute for Astrophysics, Bologna, Italy), E. Young (Universities Space Research Association/Ames Research Center), the WFC3 Science Oversight Committee, and the Hubble Heritage Team (STScI/AURA); Vol. 4 - NASA, ESA, E. Jullo (JPL/LAM), P. Natarajan (Yale) and J-P. Kneib (LAM). Copyright © 2018 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Desk Editor: Ng Kah Fee Typeset by Stallion Press Email: [email protected] Printed in Singapore

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Winston Churchill said: “My most brilliant achievement was to persuade my wife to marry me”. This volume is dedicated to my family — Riki, Or, Ariel, and my parents, who have been my companions on my career journey. It is also dedicated to Richard Feynman, whose writings first ignited my interest in physics. Feynman knew how to eloquently express the beauty of science1 : “Poets say science takes away from the beauty of the stars — mere globs of gas atoms. I too can see the stars on a desert night, and feel them. But do I see less or more? The vastness of the heavens stretches my imagination — stuck on this carousel my little eye can catch onemillion-year-old light. A vast pattern — of which I am a part. . . What is the pattern, or the meaning, or the why? It does not do harm to the mystery to know a little about it. For far more marvelous is the truth than any artists of the past imagined it. Why do the poets of the present not speak of it? What men are poets who can speak of Jupiter if he were a man, but if he is an immense spinning sphere of methane and ammonia must be silent? ” Feynman also wrote about the excitement of science2 : “We are very lucky to live in an age in which we are still making discoveries. It is like the discovery of America — you only discover it once”. When he wrote this in 1965, Feynman was referring to the then golden age of particle physics. I believe that today we are living in a golden age of cosmology, particularly on the topics in Part II of this volume. May the reader experience the joy of discovery! 1 R.

2 M.

Feynman, The Character of Physical Law (1965). Modern Library. ISBN 0-679-60127-9. Sands, R. Feynman, R. B. Leighton, The Feynman Lectures on Physics (1964). AddisonWesley.

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Preface

This volume lays out the subjects of galaxy formation and evolution based on the current paradigm in cosmology. Part I presents the theoretical understanding and modeling of galaxy formation, including a brief treatment of galactic structure. While not intended to be completely comprehensive, it is meant to cover background knowledge that is important for graduate students and researchers working in cosmology or galaxy formation, and is written mostly in textbook style. It assumes pre-knowledge of cosmology (which is only briefly reviewed), and thus can be used as a source of advanced topics in a cosmology course, or as the basis for a follow-up course in advanced cosmology. The approach is astrophysical, focusing on galaxy formation and making only limited use of general relativity where necessary. When working through some of the more complicated sections, the reader may find encouragement in two famous quotes by Einstein: “Things should be made as simple as possible, but not any simpler”; and, “In the middle of every difficulty lies opportunity.” Part II builds on Part I by presenting the exciting subject of highredshift galaxy formation, including topics such as cosmic dawn, the first stars, cosmic reionization, and 21-cm cosmology. Combining a review of progress on these topics with some detailed physics, it is meant to bring active researchers up to speed on recent work on galaxy formation at early times. R. Barkana

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Contents

Preface

vii

Part I: Basic Theory of Galaxy Formation

1

1.

Introduction

3

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

Review of Cosmology

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2.

2.1

The Friedmann–Robertson–Walker (FRW) metric 2.1.1 The metric . . . . . . . . . . . . . . . . . . 2.1.2 Using the FRW metric . . . . . . . . . . . 2.2 Cosmic expansion: dynamics . . . . . . . . . . . . 2.2.1 Hubble’s law . . . . . . . . . . . . . . . . . 2.2.2 Redshift of light . . . . . . . . . . . . . . . 2.2.3 Luminosity distance . . . . . . . . . . . . . 2.3 Cosmic expansion: kinematics . . . . . . . . . . . 2.3.1 Friedmann equation . . . . . . . . . . . . . 2.3.2 Distribution functions and pressure . . . . 2.3.3 Equation of state . . . . . . . . . . . . . . 2.3.4 Einstein–de Sitter (EdS) limit . . . . . . . 2.4 Redshifting of peculiar velocity . . . . . . . . . . . 2.5 Temperature evolution of gas and radiation . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . 3.

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Statistics of Galaxy Formation

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3.1

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Random fields and correlation functions . . . . . . . . . . . . . . . 3.1.1 Continuous and discrete fields . . . . . . . . . . . . . . . . ix

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3.1.2 The correlation function . . . . . . . . . . . . . . 3.1.3 Shot noise and the discrete correlation function . 3.1.4 Higher-order correlation functions . . . . . . . . 3.1.5 Random walks and mean free paths . . . . . . . 3.2 The power spectrum . . . . . . . . . . . . . . . . . . . 3.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Relation to the correlation function . . . . . . . 3.2.3 The discrete power spectrum . . . . . . . . . . . 3.3 Detailed correlation function example . . . . . . . . . . 3.3.1 Cox process . . . . . . . . . . . . . . . . . . . . . 3.3.2 Analytical calculation of ξ(r) . . . . . . . . . . . 3.3.3 Numerical Cox process with extensions . . . . . 3.4 Statistical topics in galaxy formation . . . . . . . . . . 3.4.1 Gaussian random fields . . . . . . . . . . . . . . 3.4.2 Window functions . . . . . . . . . . . . . . . . . 3.4.3 Model for biased galaxy formation . . . . . . . . 3.4.4 Limber’s equation . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.

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Linear Perturbation Theory and the Power Spectrum 4.1

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Preview of perturbation theory . . . . . . . . . . . . . . 4.1.1 Mass-radius relations for initial halo mass . . . . 4.1.2 Linear spherical collapse . . . . . . . . . . . . . 4.1.3 Examples of linear perturbation growth . . . . . Distribution functions . . . . . . . . . . . . . . . . . . . 4.2.1 Quantum statistical mechanics . . . . . . . . . . 4.2.2 Fluid frame and stress tensor . . . . . . . . . . . The collisionless Boltzmann equation and its moments 4.3.1 Liouville equation and collisionless Boltzmann equation (CBE) . . . . . . . . . . . . . . . . . . 4.3.2 Moments of the CBE . . . . . . . . . . . . . . . Eulerian fluid equations in comoving coordinates . . . . 4.4.1 Coordinate transformation . . . . . . . . . . . . 4.4.2 Poisson equation . . . . . . . . . . . . . . . . . . 4.4.3 Continuity equation . . . . . . . . . . . . . . . . 4.4.4 Euler equation . . . . . . . . . . . . . . . . . . . Linear perturbation theory . . . . . . . . . . . . . . . . 4.5.1 Longitudinal isentropic (“adiabatic”) modes . . 4.5.2 Transverse modes . . . . . . . . . . . . . . . . . 4.5.3 GR modes . . . . . . . . . . . . . . . . . . . . . 4.5.4 No growth . . . . . . . . . . . . . . . . . . . . . 4.5.5 The Jeans length . . . . . . . . . . . . . . . . . .

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4.5.6 Super-horizon perturbations . . . Cosmic history of perturbations . . . . . The power spectrum . . . . . . . . . . . Baryons . . . . . . . . . . . . . . . . . . . 4.8.1 The baryon-photon fluid . . . . . 4.8.2 Silk damping . . . . . . . . . . . . 4.8.3 Early history of the Jeans mass . 4.8.4 Mean molecular weight . . . . . . 4.8.5 Optical depth due to reionization References . . . . . . . . . . . . . . . . . . . . 4.6 4.7 4.8

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Stellar Dynamics and the Virial Theorem Collisionless stellar systems . . . . . . . . . . . . . . . . . . . . The collisionless Boltzmann equation in spherical coordinates The spherical Jeans equation . . . . . . . . . . . . . . . . . . . The virial theorem . . . . . . . . . . . . . . . . . . . . . . . . . Constants and integrals of motion . . . . . . . . . . . . . . . . The isothermal sphere . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Isothermal gas analogy . . . . . . . . . . . . . . . . . . 6.6.2 The singular isothermal sphere (SIS) . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Galactic Disks and Spiral Structure

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Cylindrical coordinates and disks Theory of epicycles . . . . . . . . 7.2.1 Oort constants . . . . . . . Spiral structure: derivation . . . . 7.3.1 The winding problem . . . 7.3.2 Spiral density waves . . . . 7.3.3 Equation of a spiral arm .

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6.1 6.2 6.3 6.4 6.5 6.6

7.1 7.2

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Spherical non-linear collapse . . . . . . . . . . . . . . . . . . 5.1.1 Setup and solution . . . . . . . . . . . . . . . . . . . . 5.1.2 Linear limit . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Non-linear stages and the critical density for collapse 5.2 Scaling relations for halos . . . . . . . . . . . . . . . . . . . . 5.3 Virialization . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 The Press–Schechter model . . . . . . . . . . . . . . . . . . . 5.5 Mass-radius relations for virialized halos . . . . . . . . . . . 5.6 The galaxy luminosity function . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Non-linear Processes and Dark Matter Halos 5.1

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Tight-winding approximation . . . . . Surface density . . . . . . . . . . . . . Plane-wave potential . . . . . . . . . Response of the disk to the potential structure: result . . . . . . . . . . . . Basic solution . . . . . . . . . . . . . Dispersion relation . . . . . . . . . . . Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Gravitational Lensing 8.1 The lens equation . . . . . . . . . . . . 8.2 Point-mass lens . . . . . . . . . . . . . 8.3 General lens . . . . . . . . . . . . . . . 8.4 Magnification and shear . . . . . . . . . 8.5 Axisymmetric lens . . . . . . . . . . . . 8.6 The singular isothermal sphere lens . . 8.7 The time delay and Fermat’s principle References . . . . . . . . . . . . . . . . . . .

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Summary and Conclusions

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Part II: Early Galaxies and 21-cm Cosmology 10.

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10.1 Brief outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 10.2 Detailed introduction . . . . . . . . . . . . . . . . . . . . . . . . . 144 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 11.

Galaxy Formation: High-redshift Highlights 11.1 Halos and their gas content . . . . . . . . . . . . . . . . 11.1.1 Halos: profiles and biased clustering . . . . . . . 11.1.2 Baryons: linear evolution, pressure, and cooling . 11.2 Large fluctuations in the galaxy number density . . . . 11.3 Simulations at high redshift: challenges and approaches 11.4 The very first stars . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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21-cm Cosmology

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12.4 Observational aspects . . . . . . . . . . . . . . . . . . . . . . . . . 184 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 13.

The Supersonic Streaming Velocity 13.1 Cosmological origins . . . . . . . 13.2 Effect on star formation in early 13.3 Consequences . . . . . . . . . . References . . . . . . . . . . . . . . .

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21-cm Signatures of the First Stars

Summary and Conclusions

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15.1 21-cm signatures of reionization . . . . . . . . . . . 15.2 21-cm signatures of Lyα coupling and LW feedback 15.3 Large 21-cm fluctuations from early cosmic heating 15.4 Late heating and reionization . . . . . . . . . . . . 15.5 The global 21-cm spectrum . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . 16.

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Cosmic Milestones of Early Radiative Feedback 14.1 Reionization . . . . . . . . . . . . . . . . . 14.2 Lyα coupling and Lyman–Werner feedback 14.3 Cosmic heating . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

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Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

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Part I Basic Theory of Galaxy Formation

This part begins with a brief review of basic cosmology and of relevant topics in statistics, and then develops in detail the theory of galaxy formation as understood in modern cosmology; this includes linear perturbation theory, spherical collapse, and baryonic effects. After deriving some essential elements of stellar dynamics and of galactic structure, this part ends with a brief presentation of gravitational lensing.

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Chapter 1

Introduction

The theoretical basis of modern cosmology was laid out soon after Einstein’s 1915 discovery of general relativity. Friedmann first worked out models of the expanding Universe, with Lemaitre adding the concepts of redshift and the initial Big Bang. Hubble’s 1929 discovery of cosmic expansion was the first observational milestone. The hot Big Bang theory accumulated more substance as the first theoretical predictions of the cosmic microwave background (CMB) and of Big-Bang nucleosynthesis were made by Alpher, Gamow, and Herman in the 1940’s. A watershed moment in observational cosmology came with the 1965 discovery of the CMB [1], which was the first direct evidence for the hot and dense initial state of the Universe. Meanwhile, astronomical evidence accumulated for the evolution of the Universe over time, as increasingly distant galaxies were discovered. Soon after quasars were identified as high-redshift objects, Gunn & Peterson [2] used their spectra (in 1965) to show that the inter-galactic gas around them was highly ionized; this was the first sign that the gas had undergone cosmic reionization, likely by the stars in early galaxies. For a long time, the most fundamental questions about our Universe remained unanswered, including the energy contents of the Universe and the fate of its expansion. Observational evidence for dark matter emerged as early as 1933, from Zwicky’s analysis of the Coma cluster of galaxies. This was not, however, taken seriously until the 1970’s, when galactic rotation curves were shown by Vera Rubin to indicate the presence of massive amounts of unseen matter. Meanwhile, comparison of the observed abundances of the lightest elements with the predictions of Big Bang nucleosynthesis increasingly indicated a low cosmic baryon density, thus requiring the dark matter to be mostly non-baryonic. An early period of cosmic inflation was proposed by Guth in 1981 [3], helping to explain large-scale features of the Universe and the initial conditions needed for galaxy formation. Those initial conditions were finally discovered observationally in 1992, in the form of temperature anisotropy in the cosmic microwave background [4]; NASA’s COBE satellite also confirmed the near-perfect black-body spectrum of the CMB [5], as expected in the hot Big Bang 3

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model. A major surprise1 was the 1998 discovery, using Type Ia supernovae as distance indicators, of accelerated expansion consistent with a large cosmological constant dominating the present energy density [6, 7]. Meanwhile, increasingly precise measurements were made of the CMB over a wide range of angular scales. The peak of the spectrum of temperature fluctuations at an angular scale of one degree was detected in 2000 by experiments including BOOMERanG [8] and MAXIMA [9]; these experiments also showed hints of higher-order acoustic peaks, which were unambiguously confirmed in 2003 by NASA’s WMAP satellite [10]. The fact that hundreds of individual points of the CMB angular fluctuation spectrum match the precise shape predicted by a model with 6 parameters (in the simplest ΛCDM model, i.e. cosmological constant Λ plus cold dark matter) is a scientific triumph that solidifies the entire theoretical framework of modern cosmology. Meanwhile, large galaxy redshift surveys detected the corresponding features in the power spectrum of the galaxy distribution [11, 12], including the fluctuation peak as well as baryon acoustic oscillations (the after-effects of baryons being carried along with the photons in sound waves in the early Universe). CMB polarization measurements have further confirmed the standard cosmological model, and have also detected the signature of cosmic reionization, although the originally high value of the optical depth [10] has been revised substantially, and the recent value from ESA’s Planck satellite is much lower [13]. Thus, the scientific study of the history of the Universe has undergone a tremendous acceleration in recent decades. Today it is expanding in many new directions. A major effort is directed towards increasingly large surveys of galaxies, one goal of which is to determine whether the cosmic equation of state is consistent with a cosmological constant. Other work focuses on further measurements of the CMB, including its polarization and spectral distortions. In between these two regimes, of the early Universe on the one hand and the recent one on the other, is the story of the formation of the first stars and quasars and their cosmic effects. This largely unmapped chapter in cosmic history is the subject of Part II of this volume. Part I is meant to be a largely self-contained introduction to galaxy formation. To make it self-contained, it includes reviews of basic cosmology and of relevant statistical methods, and a brief derivation of the collisionless Boltzmann equation and its moments. The emphasis is on a physical understanding of galaxy formation, so linear perturbation theory, spherical collapse, and related topics are covered in detail. Extensive presentations of theoretical cosmology and galaxy formation can be found in many books, e.g. [14–19]. These books are good sources for subjects that are only briefly touched on here, such as particle cosmology (including cosmology of the very early Universe), perturbation theory in general relativity (including tensor 1 The author recalls attending a major cosmology conference as a student in the early 1990’s, where a vote was taken on the most likely cosmological model. Theorists voted for a flat matter-only Universe, observers for an open, low-density, matter-only Universe; almost no-one was in favor of a Universe dominated by a cosmological constant.

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perturbations and the issue of gauge), and CMB physics (including the calculation and analysis of the anisotropies). Some of the material here, in particular in Chap. 4, was influenced by (my Ph.D. adviser) Ed Bertschinger’s Les Houches lecture notes [20]. Chapters 6–8 present a selection of other topics that, I submit, any cosmologist should know. Anyone who studies galaxy formation should know something about the internal properties of galaxies, including our basic understanding of spiral structure. More generally, stellar dynamics is an important basis for understanding galaxies, galaxy clusters, and dark matter halos. The material in Chaps. 5 and 6 was mostly influenced by the standard advanced textbook of Binney & Tremaine [21]. Finally, gravitational lensing is both a physically and mathematically beautiful subject, and an important method with many astrophysical and cosmological applications. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

[19] [20] [21]

A. A. Penzias, R. W. Wilson, ApJ 142 (1965) 419. J. E. Gunn, B. A. Peterson, Astroph. J. 142 (1965) 1633. A. H. Guth, Phys. Rev. D 23 (1981) 347. G. F. Smoot, C. L. Bennett, A. Kogut, et al., Astroph. J. Lett. 396 (1992) L1. J. C. Mather, E. S. Cheng, D. A. Cottingham, et al., Astroph. J. 420 (1994) 439. S. Perlmutter, G. Aldering, G. Goldhaber, et al., Astroph. J. 517 (1999) 565. A. G. Riess, A. V. Filippenko, P. Challis, et al., Astron. J. 116 (1998) 1009. P. de Bernardis, P. A. R. Ade, J. J. Bock, et al., Nature 404 (2000) 955. S. Hanany, P. Ade, A. Balbi, et al., Astroph. J. Lett. 545 (2000) L5. C. L. Bennett, M. Halpern, G. Hinshaw, et al., 2003, Astroph. J. Supp. 148 (2003) 1. W. J. Percival, C. M. Baugh, J. Bland-Hawthorn, et al., Mon. Not. R. Astron. Soc. 327 (2001) 1297. D. J. Eisenstein, I. Zehavi, D. W. Hogg, et al., Astroph. J. 633 (2005) 560. (Planck Collaboration:) Aghanim, N., Ashdown, M., et al. 2016, arXiv:1605.02985 S. Weinberg, Gravitation and Cosmology, Wiley, New York, 1972. P. J. E. Peebles, The Large-Scale Structure of the Universe, Princeton University Press, Princeton, 1980. P. J. E. Peebles, Principles of Physical Cosmology, Princeton University Press, Princeton, 1993. E. W. Kolb, M. S. Turner, The Early Universe, Addison-Wesley, Redwood City, CA, 1990. Padmanabhan, T. 2002, Theoretical Astrophysics — Volume 3, Galaxies and Cosmology, by T. Padmanabhan, pp. 638. Cambridge University Press, December 2002. ISBN-10: 0521562422. ISBN-13: 9780521562423, 638 Dodelson, S. 2003, Modern cosmology / Scott Dodelson. Amsterdam (Netherlands): Academic Press. ISBN 0-12-219141-2, 2003, XIII + 440 p., E. Bertschinger, 1994, arXiv:astro-ph/9503125. J. Binney, S. Tremaine, Galactic Dynamics: Second Edition, by James Binney and Scott Tremaine. ISBN 978-0-691-13026-2 (HB). Published by Princeton University Press, Princeton, NJ USA, 2008.

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Chapter 2

Review of Cosmology

This volume assumes pre-requisite knowledge of basic cosmology and of standard methods of mathematical physics. This chapter presents just a brief review of essential elements of the expansion history of the Universe (more exhaustive expositions are available in standard textbooks such as those cited in the previous chapter). 2.1. 2.1.1.

The Friedmann–Robertson–Walker (FRW) metric The metric

In general relativity, the metric for a space which is spatially homogeneous and isotropic is the Friedmann–Robertson–Walker (hereafter FRW) metric, which can be written in the form   dr2 2 2 2 2 2 (2.1) + r dΩ , ds = dt − a (t) 1 − k r2 where (r, θ, φ) are spherical comoving coordinates, t is the proper/physical time, a(t) is the cosmic scale factor that describes expansion in time, and the angular area element is dΩ2 = dθ2 + sin2 θ dφ2 .

(2.2)

Here a(t) is dimensionless, r is called a “comoving” position coordinate since the scale factor has been factored out, and k has units of an inverse squared length. Unless otherwise indicated, we use units in which the speed of light c ≡ 1 (sometimes we still include factors of c in equations when this adds clarity). The constant k determines the geometry of the metric; it is positive in a closed Universe, zero in a flat Universe, and negative in an open Universe. While the closed Universe has a finite volume, the flat or open ones are infinite.1 1 This is a good opportunity to mention Einstein’s clever quote: “Only two things are infinite, the Universe and human stupidity, and I’m not sure about the former.”

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We can obtain an alternative form of the metric by factoring a(t) out of the time variable as well, yielding the comoving or “conformal” time variable τ . If we also transform to a new radial variable χ in order to simplify the radial element, the metric becomes    (2.3) ds2 = a2 (τ ) dτ 2 − dχ2 + sin2k χ dΩ2 , where

⎧   −1/2 ⎪ sin k 1/2 χ ⎪ ⎨k sink χ = χ ⎪ ⎪ ⎩(−k)−1/2 sinh (−k)1/2 χ

if k > 0 (closed) if k = 0 (flat)

(2.4)

if k < 0 (open).

These cases correspond to a closed, flat, or open spacetime geometry, as indicated. Here (χ, θ, φ) are another set of spherical comoving coordinates. In the important case of a spatially flat (k = 0) Universe, r = χ and the spatial parts of the two just-presented forms of the metric become identical. We follow the convention of setting the scale factor to unity today, i.e. a(t0 ) = a(τ0 ) = 1. 2.1.2.

Using the FRW metric

The FRW metric can be used to understand various properties of the space-time and the dynamics of its residents. In particular, stationary (“comoving”) observers at rest at a fixed (r, θ, φ) remain at rest,2 with their physical separation increasing with time in proportion to a(t). This case yields the relation between t and τ : dr = dχ = dΩ = 0 =⇒ ds = dt = a dτ.

(2.5)

If we consider a radial displacement at a fixed time, we obtain the physical (or “proper”) distance in the radial direction: dr = a(τ ) Δχ. (2.6) dt = dΩ = dτ = 0 =⇒ Δs = a(t) √ 1 − kr2 The tangential (azimuthal) direction describe the physical area in the angular direction (e.g. on a spherical shell): dt = dτ = dr = dχ = 0 =⇒ Area = 4πa2 r2 = 4πa2 sin2k χ.

(2.7)

An equivalent statement is that the angular diameter distance is DA = ar = a sink χ, where a small angle θ corresponds to a transverse physical distance DA θ. 2 This

must formally be demonstrated by solving the geodesic equation, e.g. [1].

(2.8)

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Another important special case is that of light, described by ds = 0. If, for simplicity, we consider a radial light ray, then dr ds = dΩ = 0 =⇒ Δt = a √ = physical distance, (2.9) 1 − kr2 and also Δτ = Δχ = comoving distance.

(2.10)

The comoving horizon (also the “particle horizon”) is the largest comoving distance from which light (or other causal influences) could have reached an observer at time t since the Big Bang (at which t and τ were zero): t dt = Δχ = τ. (2.11) Comoving horizon =  t =0 a(t ) 2.2. 2.2.1.

Cosmic expansion: dynamics Hubble’s law

The “Hubble constant” (constant in space but varying in time) is defined as H(t) ≡

d ln a(t) 1 da a˙ = = , dt a dt a

(2.12)

where in general p˙ denotes the time derivative of a variable p. In terms of the scale factor and the Hubble constant, the time variables are then da da ; τ= . (2.13) t = dt = 2 a H(a) a H(a) We will often use x for vector comoving positions (corresponding to the comoving spherical coordinates (χ, θ, φ) above), and r for the corresponding physical positions (unrelated to the r coordinate in Equation 2.1, which we will avoid using). Consider now two comoving observers, one at the origin and another at a fixed comoving position x. The comoving displacement between them is x, while the proper/physical displacement is r = a(t)x. The physical velocity is thus v =

da dr = x = Hax = Hr. dt dt

(2.14)

This is Hubble’s law. 2.2.2.

Redshift of light

The cosmological redshift of light looks locally just like the classical Doppler shift corresponding to the recession velocity according to Hubble’s law. Imagine a photon emitted at the origin of the coordinates, and observed by a comoving observer

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at x. Then the velocity of the observer relative to the emitter is v = x(da/dt) (where x = |x|) as in Eq. (2.14), and the relation between the emitted and observed wavelength of the photon is λobs = λemit (1 + v). Here we have used the Newtonian Doppler shift, assuming a small distance x (small compared to the horizon cH −1 ) and thus v  c. The redshift z is defined through 1+z ≡

λobs =⇒ z = v. λemit

(2.15)

We can relate this to the cosmic expansion. The photon travels for a time Δt = aΔτ = ax, so by the time it reaches the observer, the scale factor has increased to

 da da da Δt = a + ax = a 1 + x = a(1 + v) = a(1 + z). a ˜=a+ dt dt dt Finally, for an observer at a ˜ = 1, we obtain 1 λ(˜ a = 1) =1+z = . λ(a) a

(2.16)

This result (λ ∝ a) is sometimes expressed intuitively as imagining that the wavelength of each photon simply expands along with the Universe. Note that we can subdivide a cosmological distance into many small segments, each of which can be analyzed as we have done here, together yielding λ ∝ a also over cosmologically large distances.3 2.2.3.

Luminosity distance

Imagine a light emitter of luminosity L at a seen by a present observer at a0 = 1. The luminosity distance DL is defined so that the observed flux is F ≡

L . 4πDL2

(2.17)

Using the metric of Eq. (2.3), we center the coordinates on the emitter and place the observer at a comoving radial distance χ. Then the emitted light spreads out over a sphere of physical area 4πa20 sin2k χ. With a0 = 1, we obtain F =

L a2 , 4π sin2k χ

(2.18)

where we have added two factors of a due to the redshifting of energy and of the photon rate (which are both lower due to the cosmological redshift). The energy of 3 Strictly speaking, in order to complete this argument, the statement must first be shown to be true locally including corrections from special relativity.

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each photon is hν ∝

1 , λ

(2.19)

where ν is the physical frequency, so that the photon energy changes by a factor of a from emission to observation (see the previous subsection). For the interval between two photons, note that the interval dτO at the observer equals the interval dτE at the emitter, since each photon travels the same Δτ = χ [Eq. (2.10)]. Thus, the relation between the physical time intervals is dtO = dτO = dτE =

dtE , a

i.e. the photon rate at the observer is multiplied by a factor of a. If we denote the comoving distance by DC , we can summarize the various distances as: DA = a sink χ ; 2.3. 2.3.1.

DC = χ ;

DL =

1 sink χ. a

(2.20)

Cosmic expansion: kinematics Friedmann equation

The Einstein field equations of general relativity, Gμν = 8πGTμν , yield the Friedmann equation H 2 (t) =

k 8πG ρ− 2, 3 a

(2.21)

which relates the expansion of the Universe (through H) to its matter-energy content (through the energy density ρ) and curvature (through k). For an intuitive interpretation of this equation, consider an expanding Newtonian shell enclosing a fixed mass M with a radius increasing ∝ a. Then Newtonian conservation of energy for the shell yields precisely the Friedmann equation [see Sec. 5.1], where H 2 comes from the kinetic energy term, ρ from the gravitational potential energy (which is negative so becomes positive on the right-hand side), and −k is proportional to the conserved total energy of the shell (i.e. a negative total energy corresponds to a positive k, which is the case of an expansion that only reaches a finite radius and then re-collapses). The field equations also yield a second independent equation: d(ρa3 ) = −pd(a3 ),

(2.22)

which is analogous to dE = −pdV in classical thermodynamics (where E is the energy of a gas of pressure p in a volume V ). This equation can be combined with the Friedmann equation to yield the acceleration of the expansion: 4πG a ¨ =− (ρ + 3p). a 3

(2.23)

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Cosmic densities are often expressed relative to the critical density ρc , which is the density corresponding to a flat Universe. From Eq. (2.21), ρc (t) ≡

3H 2 (t) , 8πG

Ω(t) ≡

ρ . ρc

(2.24)

This definition yields an alternate form of the Friedmann equation: k = H 2 (t) [Ω(t) − 1] , a2

k = H02 [Ω0 − 1] ,

(2.25)

where the expression on the right is the same equation at the present. This equation displays the relation between k (which sets the geometry of the Universe in the metric of Eq. (2.1) or Eq. (2.3)) and the contents of the Universe. It is an example of the deep geometry–density connection in general relativity. 2.3.2.

Distribution functions and pressure

The subject of the distribution functions of particles will be dealt with extensively below. Here we only present a few basic results that are needed for describing the evolution of various components of the energy density of the Universe. Define the distribution function f (x, q) for a collection of particles (such as a gas, a fluid, or particles in an N-body simulation) so that the number of particles in a phase-space volume d3 x d3 q at position x and momentum q is dN = f (x, q) d3 x d3 q. Then the number density is

n(x) =

f (x, q) d3 q

(2.26)

(2.27)

 and the total number of particles is N = n d3 x. The energy density (where as before c = 1) is (2.28) ρ = E(q) f (x, q) d3 q,  where E(q) = (qc)2 + (mc2 )2 is the energy of a particle of momentum q. Now consider the case of isotropic pressure. To calculate the pressure, consider the force dF in the z direction exerted by the particles on a piston (or the side of a box containing the particles) of small area dA that is perpendicular to the z direction (note: in this subsection z denotes a Cartesian coordinate, not redshift). The pressure is defined as p=

dqz /dt dF = , dA dA

where dqz is the z component of momentum imparted to the piston, and we used the z component of Newton’s law in the form that is also valid in special relativity.

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Particles with z momentum qz contribute 2qz per particle to dqz (with the factor of 2 due to elastic recoil). Particles with z velocity vz can reach the piston during the time dt from as far away as dz = vz dt. Thus, dqz = 2qz × f d3 q × dA vz dt. Now, in√ general (in special relativity) q = γmv, where m is the particle mass and γ = 1/ 1 − v 2 . Also, isotropy implies: vx2  = vy2  = vz2  =

1 2 v , 3

where the averaging here is over all the particles at a given position. Thus, qz vz  = 1 3 q v, but we must then add a factor of 1/2 since in this averaging we must count only the half of particles headed towards the piston (and not in the opposite z direction). Thus, 1 dqz = q v f d3 q dA dt, 3 so that p=

1 3



q v f d3 q.

(2.29)

Since E = γmc2 , another way to write this result is q 2 c2 f d3 q. p= 3E(q)

(2.30)

In the case of non-relativistic particles of mass m: Non-relativistic: E = mc2 ,

q = mv,

p∼ρ

v 2  , c2

(2.31)

so that p is usually negligible when it competes with ρ (as in the cosmic expansion). For radiation: Photons: v = c, 2.3.3.

E = qc,

p=

1 ρ. 3

(2.32)

Equation of state

The equation of state is the relation between the pressure and density for each component of the cosmic energy density. Many components are described by the simple equation of state p = wρ,

(2.33)

where w is constant. Eq. (2.22) applies separately to each such component (when different components do not interact and exchange energy), which yields a simple

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first-order differential equation whose solution is a power law in a: ρ �