New Phenomena and New States of Matter in the Universe: From Quarks to Cosmos 9811220905, 9789811220906

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NEW PHENOMENA AND NEW STATES OF MATTER IN THE UNIVERSE F rom Q ua rks to C osmos

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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 Control Number: 2022945021 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

NEW PHENOMENA AND NEW STATES OF MATTER IN THE UNIVERSE From Quarks to Cosmos Copyright © 2023 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.

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ISBN 978-981-122-090-6 (hardcover) ISBN 978-981-122-091-3 (ebook for institutions) ISBN 978-981-122-092-0 (ebook for individuals) For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/11847#t=suppl Desk Editor: Nur Syarfeena Binte Mohd Fauzi Typeset by Stallion Press Email: [email protected] Printed in Singapore

We dedicate this book to our families and to our wifes Andrea Hess (m. to Peter Hess), Birgit Boller (m. to Thomas Boller), and Monica Estr´ azulas (m. to Cesar Zen) Peter Otto Hess Bechstedt Institute of Nuclear Science, Universidad Nacional Aut´ onoma de M´exico (UNAM), M´exico City, M´exico

Thomas Boller Max Planck Institute for Extraterrestrial Physics (MPE), Garching, Germany

C´ esar Augusto Zen Vasconcellos Instituto de F´ısica, Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, Brazil and International Center for Relativistic Astrophysics Network (ICRANet), Rome, Pescara, Italy

The Editors

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c 2023 World Scientific Publishing Company  https://doi.org/10.1142/9789811220913 fmatter

Foreword

Recent discoveries in astronomy and relativistic astrophysics as well as experiments on particle and nuclear physics have blurred the traditional boundaries of physics. It is believed that at the birth of the universe, a whirlwind of matter and antimatter, of quarks, leptons and exotic particles, briefly appeared and merged into a sea of energy. In more recent years, the discovery of new phenomena and new states of matter in the universe, only achievable under extreme laboratory conditions, reveal the deep connection between quarks and the Cosmos. This book addresses related topics that are at the root of these phenomena and the new states of matter: gravitational waves, dark matter, dark energy, exotic contents of compact stars, high-energy and gamma-ray astrophysics, heavy ion collisions and the formation of the quark-gluon plasma in the early universe. The book presents some of the latest research on these fascinating themes and is useful for experts and students in the field.

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Preface

A better understanding of the mysteries of the physical universe in its two extreme domains, — the very small, realm of particle physics, — and the very large, domain of cosmology — , has become one of the most challenging problems of our time. As for the first domain, as a result of decades of theoretical and experimental studies, our understanding of the fundamental structure of matter, made up of nature’s building blocks — quarks, leptons and hadrons (baryons and mesons) —, and the four forces of nature, is synthesized in the standard model of particle physics. Concerning the second domain, great achievements of modern cosmology have increased our understanding of the genesis and evolution of the universe and of the intimate and profound connections between elementary particles and the cosmos. In this process, modern cosmology brought into focus a new set of fundamental questions and topics involving gravitational waves, gamma-ray bursts, exotic content of compact stars, dark matter, dark energy, gamma ray astrophysics, and perspectives in the nascent multi-messenger astronomy, among others, which are discussed in this book. In the following, we briefly present the summary of the chapters of the book. About the content of the book Chapter 1. Author: Jos´e Antonio de Freitas Pacheco. Abstract: Gravitational waves are the only messenger able to probe physical processes that have occurred in the early universe, since the cosmic plasma is opaque to electromagnetic radiation at high redshift (z > 1100). Quantum fluctuations of a putative scalar field that drives inflation are able to produce a stochastic gravitational wave background. Additional

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mechanisms contributing to such a background are bubble collisions, sound waves and turbulence generated during the electroweak or the QCD phase transitions. Gravitational waves produced by these mechanisms could be detected by the planned Big Bang Observer space-based interferometer (phase 2) but unfortunately their signals are immersed in the astrophysical background generated by binary systems constituted by compact objects, despite different techniques proposed for separation of the signals. If the QCD transition is crossover and if the turbulence level is high enough, the expected signal is above the astrophysical background and detection would be possible. Modifications of the standard model leading to a strong electroweak transitions would also render possible the detection of the signal. A cosmic string network is able to generate a detectable gravitational wave background if the string tension is larger than Gμ/c2  1.4 × 10−17 . Chapter 2. Authors: Jorge A. Rueda and Juan D. Uribe. Abstract: In the binary-driven hypernova model of long gamma-ray bursts, a carbon-oxygen star explodes as a supernova in presence of a neutron star binary companion in close orbit. Hypercritical (i.e. highly superEddington) accretion of the ejecta matter onto the neutron star sets in, making it reach the critical mass with consequent formation of a Kerr black hole. We have recently shown that, during the accretion process onto the neutron star, fast neutrino flavour oscillations occur. Numerical simulations of the above system show that a part of the ejecta keeps bound to the newborn Kerr black hole, leading to a new process of hypercritical accretion. We address here the occurrence of neutrino flavour oscillations given the extreme conditions of high density (up to 1012 g cm−3 ) and temperatures (up to tens of MeV) inside this disk. We estimate the evolution of the electronic and non-electronic neutrino content within the two-flavour formalism (νe νx ) under the action of neutrino collective effects by neutrino self-interactions. We find that neutrino oscillations inside the disk have frequencies between ∼ (105 –109 ) s−1 , leading the disk to achieve flavour equipartition. This implies that the energy deposition rate by neutrino annihilation (ν + ν¯ → e− + e+ ) in the vicinity of the Kerr black hole, is smaller than previous estimates in the literature not accounting by flavour oscillations inside the disk. The exact value of the reduction factor depends on the νe and νx optical depths but it can be as high as ∼ 5.

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Chapter 3. Author: Ulisses Barres de Almeida. Abstract: Thirty years since the detection of the first TeV source, thanks to the successful observations by the Whipple Observatory in 1989, very-high-energy (VHE) gamma-ray astronomy has finally reached full maturity. The recent observational breakthroughs in the detection of subTeraelectronvolt emission from Gamma-ray Bursts, the searches for counterparts of the VHE neutrinos from IceCube, as well as the fast expansion of the sensitivity of observations at the highest energies, in its pursuit of Galactic PeVatrons and the origin of high-energy cosmic rays – all of these accomplishments reveal a picture of the high-energy cosmos that goes much beyond previous expectations, and demonstrate how far the field has evolved in terms of its technological capabilities. Groundbased gamma-ray astronomy is in fact a pivotal player in the nascent Multi-messenger Astronomy, bridging the non-electromagnetic events with the electromagnetic signals and consequently their associated astrophysical sources. In this contribution I will review the main recent observational results of the field, which are closely connected to the search for the origin of cosmic-rays, concentrating on detailing the efforts towards establishing the counterparts to multi-messenger events. I will also sketch the landscape of future experiments and observatories that will enter operation in the next decade, and the perspective they lay out for the field of Astro-particle Physics. Chapter 4. Author: Daniela P´erez. Abstract: The current view of the universe is the result of two major theories elaborated in the 20th Century: General Relativity and Quantum Field theory. According to such theories the basic stuff of the universe is formed by space-time and matter fields existing on space-time. Such a view, articulated in the standard cosmological model of the Hot Big Bang, is under question because of the tension created by the discrepancies between the observed dynamics of visible matter and the cinematic expectations from General Relativity. Chapter 5. Authors: Delaney Farrell, Aksel Alp, William Spinella, Fridolin Weber, Germ´an Malfatti, Milva G. Orsaria, and Ignacio F. Ranea-Sandoval. Abstract: In this chapter, we investigate the structure and composition of hot neutron star matter and proto-neutron stars. Such objects are made of

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baryonic matter that is several times denser than atomic nuclei and tens of thousands times hotter than the matter in the core of our Sun. The relativistic finite-temperature Green function formalism is used to formulate the expressions that determine the properties of such matter in the framework of the density-dependent mean field approach. Three different sets of nuclear parametrizations are used to solve the many-body equations and to determine the models for the equation of state of ultra-hot and dense stellar matter. The meson-baryon coupling scheme and the role of the Δ(1232) baryon in proto-neutron star matter are discussed in great detail. General relativistic models of non-rotating as well as rotating proto-neutron stars are presented in part two of our study. Chapter 6. Author: Peter Otto Hess. Abstract: A review will be presented on the algebraic extension of the standard Theory of Relativity (GR) to the pseudo-complex formulation (pc-GR). The pc-GR predicts the existence of a dark energy outside and inside the mass distribution, corresponding to a modification of the GRmetric. The structure of the emission profile of an accretion disc changes also inside a star. Discussed are the consequences of the dark energy for cosmological models, permitting different outcomes on the evolution of the universe. Chapter 7. Author: Pedro H. R. S. Moraes. Abstract: According to the Standard Model of Cosmology, ∼70% of the universe is composed by dark energy, ∼25% of dark matter and only the remaining ∼5% of known baryonic matter. Although we have some clues about the main properties of dark energy and dark matter, we still do not know exactly what they are, neither have we detected them in laboratory. The effects of dark energy and dark matter can be explained in an alternative form, by modifying the Einstein–Hilbert gravitational action, such that the resulting field equations contain new terms which, in principle, are capable of describing such dark sector effects. A well-behaved alternative gravity theory must also work in the stellar scales. As long as we still do not know the equation of state of super dense matter within neutron stars, a possibility to explain some recently detected massive pulsars comes exactly from those new terms in the field equations of Alternative Gravity Theories. Moreover, the recently emerged gravitational wave astrophysics

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may also be a field of applications to Alternative Gravity. In this chapter we review the importance of Alternative Gravity and its applications in the equilibrium configurations of neutron stars. We show how the recent gravitational wave astrophysics can be used to constraint among different Alternative Gravity Theories. Chapter 8. Author: David Blaschke, Oleksii Ivanytskyi and Mahboubeh Shahrbaf Abstract: In this contribution, we present for the first time a scenario according to which early quark deconfinement in compact stars is triggered by the Bose condensation of a light sexaquark (S) (mS < 2054 MeV) that has been suggested as a candidate particle to explain the baryonic dark matter in the Universe. The onset of S Bose condensation marks the maximum mass of hadronic neutron stars and it occurs when the condition for the baryon chemical potential μ = mS /2 is fulfilled in the center of the star, corresponding to Monset < 0.7M . In the gravitational field of the star the density of the Bose condensate of S increases until a new state of the matter is attained, where each of the S-states got dissociated into a triplet of color-flavor-locked (CFL) diquark states. These diquarks are the Cooper pairs in the color superconducting CFL phase of quark matter, so that the developed scenario corresponds to a BEC-BCS transition in strongly interacting matter. For the description of the CFL phase, we develop here for the first time the three-flavor extension of the densityfunctional formulation of a chirally symmetric Lagrangian model of quark matter where confining properties are encoded in a divergence of the scalar self-energy at low densities and temperatures.

Thomas Boller, Peter Otto Hess, and C´esar Zen Vasconcellos Garching (Germany), Mexico City (Mexico), and Porto Alegre (Brazil) May 2021 The Editors

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Contents

Foreword

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Preface

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List of Figures

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List of Tables

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1. Cosmological Stochastic Gravitational Wave Background

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Jos´e Antonio de Freitas Pacheco 2. Neutrino Flavor Oscillations in Gamma-Ray Bursts

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Jorge A. Rueda and Juan D. Uribe 3. Gamma-Rays and the New Multi-messenger Astrophysics

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Ulisses Barres de Almeida 4. Dark Matter and Dark Energy vs. Modified Gravity: An Appraisal

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Daniela P´erez 5. Hot Neutron Star Matter and Proto-neutron Stars Delaney Farrell, Aksel Alp, Fridolin Weber, William Spinella, Germ´ an Malfatti, Milva G. Orsaria and Ignacio F. Ranea-Sandoval xv

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6. Review on the Pseudo-complex General Relativity and Dark Energy

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Peter O. Hess 7. Alternative Gravity Neutron Stars in the Gravitational Wave Era

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Pedro H. R. S. Moraes 8. Quark Deconfinement in Compact Stars Through Sexaquark Condensation David Blaschke, Oleksii Ivanytskyi and Mahboubeh Shahrbaf

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List of Figures

Chapter 1 Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Expected dimensionless energy density spectrum of primordial gravitational waves produced during inflation with some parameters constrained by CMB data. . . . . . . . . Expected primordial gravitational wave spectrum generated post-bounce and computed for the following parameters: τ0 = 0.1 and j = 100. The black curve corresponds to  = 0.1 and the magenta curve to  = 0.75. . . . . . . . Expected primordial gravitational wave spectrum generated in a cyclic universe — the black curve corresponds to case Tr = 1013 GeV while the magenta curve corresponds to Tr = 1011 GeV. . . . . . . . . . . . . . . . . . . . . . . . Cartoon illustrating the formation of low-temperature bubbles inside the high-temperature phase (adapted from Allen (1997)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expected gravitational wave spectrum generated during a strong EW phase transition, including modifications of the Standard Model. The cyan curve corresponds to bubble collisions, the magenta curve shows the contribution of damped sound waves, the yellow curve shows the contribution of the MHD turbulence and the black curve is the sum of all these processes. . . . . . . . . . . . . . . . . . . . . .

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Fig. 6

Fig. 7

Fig. 8 Fig. 9

Fig. 10

Fig. 11

Fig. 12

List of Figures

Pressure of the confined (black curve) and deconfined (cyan curve) phases plotted as a function of the temperature. The deconfined phase includes the flavors u, d, s while the confined phase consists of mesons π and K. The crossing point occurs at Td = 174 MeV. . . . . . . . . . . . . . . . . . . . Expected spectra of gravitational waves generated during the QCD phase transition by different mechanisms: bubble collisions (cyan curve), sound waves (magenta curve), turbulence (yellow curve) and sum of the different processes (black curve). . . . . . . . . . . . . . . . . . . . . . . . . . Velocity spectrum at 1 μs after the beginning of the simulation (adapted from Mour˜ ao-Roque and Lugones (2013)). Gravitational wave spectrum expected for a crossover QCD transition, parameterized by the initial mean root square fluid velocity according to Mour˜ ao-Roque and Lugones (2013). The red curve corresponds to σt = 10−3 c and the black curve to σt = 10−5 c. . . . . . . . . . . . . . . . . . . Sketch of the orbital configuration proposed for the Big Bang Observer (phase 2), adapted from Harry et al. (2006) — satellites in each cluster are distant from each other by arms of 5 × 104 km. . . . . . . . . . . . . . . . . . Black curves show the sensitivity curves for different planned space-based laser interferometers. Magenta curves show the predicted spectrum for different mechanisms able to produce a cosmological background labeled as: 1 — QCD crossover transition, 2 — GWs from inflation and 3 — expected background spectrum from the EW phase transition. The cyan curve indicates the astrophysical background expected from inspiraling black hole binaries. . . . Predicted gravitational wave spectrum from a QCD firstorder phase transition (magenta curve) compared with the planned sensitivity curve (black) of the Big Bang Observer (phase 2). The astrophysical background due to black hole binaries is also shown (cyan curve). The upper limit for a GW background signal derived from pulsar timing array NANOGrav is also indicated. . . . . . . . . . . . . . . . .

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List of Figures

Fig. 13

Black curves indicate the sensitivity of different space-based interferometers as labeled on the panel. The magenta curve shows the expected gravitational wave signal for a string tension of Gμ/c2 = 5 × 10−15 and α = 0.1 (see Cui et al., 2017). The cyan curve, as before, shows the predicted astrophysical background due to black hole binaries. . . . . . .

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Chapter 2 Fig. 1

Fig. 2

Fig. 3 Fig. 4 Fig. 5

Fig. 6

Fig. 7

Total number emissivity for electron and positron capture (p + e− → n + νe , n + e+ → p + ν¯e ) and electron–positron annihilation (e− + e+ → ν + ν¯) for accretion disks with M˙ = 0.1M s−1 between the inner radius and the ignition radius. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distribution of mass fractions in accretion disks in the absence of oscillations with M = 3M , α = 0.01, a = 0.95. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermodynamic properties of accretion disks in the absence of oscillations with M = 3M , α = 0.01, a = 0.95. . . . . . Properties of neutrinos in accretion disks in the absence of oscillations with M = 3M , α = 0.01, a = 0.95. . . . . . . Total optical depth (left scale) and mean free path (right scale) for neutrinos and anti-neutrinos of both flavors for accretion disks with M˙ = 1M s−1 , 0.1M s−1 , 0.01M s−1 between the inner radius and the ignition radius. . . . Oscillation potentials as functions of r with M = 3M , α = 0.01, a = 0.95 for accretion rates M˙ = 1M s−1 , M˙ = 0.1M s−1 and M˙ = 0.01M s−1 , respectively. The vertical line represents the position of the ignition radius. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Survival provability for electron neutrinos and antineutrinos for the accretion disk with M˙ = 0.1M s−1 at r = 10rs . The survival probabilities for neutrinos and anti-neutrinos coincide. The black plot corresponds to inverted hierarchy and the red plot corresponds to normal hierarchy. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Fig. 8

Fig. 9

Fig. 10

List of Figures

Survival provability for electron neutrinos and antineutrinos for the accretion disk with M˙ = 0.1M s−1 at r = 9rs , 10rs , 11rs , 12rs . The survival probabilities for neutrinos and anti-neutrinos in both plots coincide. . . . . . . Comparison between the main variables describing thin disks with and without neutrino flavor equipartition for osc and T osc each accretion rate considered. ρosc , ηeosc − , Ye are the properties of the disk under flavor equipartition. Together with Fig. 3, these plots completely describe the profile of a disk under flavor equipartition. . . . . . . . . . Comparison of the neutrino annihilation luminosity per  unit volume ΔQνi ν¯i = ¯i kk between disk k,k ΔQνi ν without (left column) and with (right column) flavor equipartition. . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 3 Fig. 1

Fig. 2

Multi-wavelength view of the Crab Nebula. Radio: M. Bietenholz, J.M. Uson and T.J. Cornwell (Very Large Array — NRAO/AUI); Infrared: R. Gehrz (Spitzer — NASA/JPL-Caltech); Visible: J. Hester and A. Loll (Hubble — NASA, ESA); Ultraviolet: E. Hoversten (UVOT — NASA/Swift); X-ray: F. Seward et al. (Chandra — NASA/CXC/SAO); Gamma: R. Buehler (LAT — NASA/DOE/Fermi). Image composition from Wikipedia: Hubble Space Telescope (HST, 2021). . . . . . . . . . . . . 119 Multi-wavelength light curve of GRB 180720B. At the top (a) is shown the energy-flux light curve detected by Fermi-GBM (band fit; green), Fermi-LAT (power law; blue), H.E.S.S. (power-law intrinsic; red) and the optical r -band (purple). The Swift-BAT spectra (15 keV 150 keV) are extrapolated to the XRT band (0.310 keV) to produce a combined light curve (gray) and an upper limit (95% confidence level) for the second H.E.S.S. observation window (power-law intrinsic, red arrow). The black dashed line indicates a temporal decay with = 1.2. In panel (b), the photon index of the Fermi-LAT, Swift and H.E.S.S. spectra is shown — error bars correspond to 1. Image from Abdalla et al. (2019). . . . . . . . . . . . . . . . . . . . . . . . . . . 125

List of Figures

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Gamma-ray and X-ray light curves for GRB 190114C. In red, the photon flux light curve above 0.3 TeV measured by MAGIC (from T0 + 62 s to T0 + 210 s), compared with the emission between 15 and 50 keV measured by Swift-BAT in gray (from T0 to T0 + 210 s) is shown. The blue dashed line shows the photon flux above 0.3 TeV of the Crab Nebula for reference. Image from MAGIC Collaboration (2019). . . . Multi-wavelength energy flux vs. time for GRB 190114C on 14 January 2019. The MAGIC light curve for the energy range 0.3 TeV (green circles) is compared with light curves at lower frequencies: in radio, the measurements by VLA, ATCA, ALMA, GMRT and MeerKAT have been multiplied by 109 for clarity. The vertical dashed line marks the end of the prompt-emission phase, identified as the end of the last flaring episode. Image reproduced from MAGIC Collaboration et al. (2019). . . . . . . . . . . . . . . . . . . . . The first-ever multi-messenger SED in astrophysics! Broadband spectral energy distribution (SED) for the blazar TXS 0506+056. The SED is based on observations obtained within 14 days of the detection of the IceCube-170922A event. Image reproduced from IceCube Collaboration et al. (2018). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Artist’s rendering of the southern CTA site, showing the three types of telescopes which will form the array. Image credits: The CTA Consortium. . . . . . . . . . . . . . . . . Differential flux sensitivity of CTA at selected energies as function of observing time, in comparison with the Fermi LAT instrument (Pass 8 analysis, extragalactic background, standard survey observing mode). Image credit: CTA Consortium; available from: (HST, 2021). . . . . . . . . . . . . (Top) HAWC significance map of the Galactic plane above 56 TeV. Black triangles denote the high-energy sources, which are labeled. (Bottom) The same as above but for emission energy above 100 TeV. Figure adapted from Abeysekara et al. (2020). . . . . . . . . . . . . . . . . . . . . . .

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Fig. 9

List of Figures

Point-source differential sensitivity of the SWGO baseline design (yellow), as compared with Fermi-LAT Pass 8 (gray), HAWC (cyan) and LHAASO (black). The sensitivity of current IACT facilities (MAGIC, blue) and the expected performance of CTA South (red) are also shown as reference. Image from La Mura et al. (2019). . . . . . . . . . . 141

Chapter 4 Fig. 1

Constraints on the fraction f (M ) ≡ ΩPBH /ΩCDM of the halo in primordial black holes. The constraints come from evaporation (red), lensing (magenta), dynamical effects (green), accretion (light blue), CMB distortions (orange), large-scale structure (dark blue) and background effects (gray). For details on this plot, see Carr et al. (2020). . . . 158

Fig. 2

Experimental constraints on spin–nucleon cross-sections as a function of WIMP mass from Schumann (2019) (Direct Detection of WIMP Dark Matter: concepts and status, J. Phys. G Nucl. Phys. 46(10), 103003). The dashed line denotes the “neutrino floor” produced by the neutrinonucleus coherent scattering of solar and atmospheric neutrinos (Billard, Figueroa-Feliciano and Strigari, 2014). . . . 160

Chapter 5 Fig. 1

Schematic illustration of different temporal stages in the evolution of proto-neutron stars to neutron stars (Prakash et al., 1997). They are characterized by different values of entropy (s) and lepton number (YL ). The formation of black holes (solid black spheres) is possible during different evolutionary stages, depending on the interplay between gravity and pressure. The transition of a hot PNS to a cold NS takes less than a minute. During the first few hundred years NSs cool quickly via neutrino emission from the core. Photon emission becomes the dominant cooling mechanism thereafter (Page et al., 2006). . . . . . . . . . . . . . . . . 202

Fig. 2

Pressure as a function of energy density for the DD2 parameter set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

List of Figures

Fig. 3 Fig. 4 Fig. 5 Fig. 6 Fig. 7 Fig. 8 Fig. 9 Fig. 10

Fig. 11

Fig. 12

Fig. 13

Fig. 14

Fig. 15

Fig. 16 Fig. 17

Pressure as a function of energy density for the GM1L parameter set. . . . . . . . . . . . . . . . . . . . . . . . . . Pressure as a function of baryon number density for the DD2 parameter set. . . . . . . . . . . . . . . . . . . . . . . Same as Fig. 4, but for the GM1L parameter set. . . . . . Composition of dense stellar matter computed for DD2 parameterization and a temperature of T = 1 MeV. . . . . Same as Fig. 6, but for a stellar temperature of T = 10 MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . Same as Fig. 6, but for a stellar temperature of T = 25 MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . Same as Fig. 6, but for a stellar temperature of T = 50 MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . Baryon–lepton composition of PNS matter obtained for DD2 model with s = 1 and YL = 0.4 (Malfatti et al., 2019). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Baryon–lepton composition of PNS matter obtained for GM1L model with s = 1 and YL = 0.4 (Malfatti et al., 2019). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Baryon–lepton composition of PNS matter obtained for DD2 model with s = 2 and YL = 0.2 (Malfatti et al., 2019). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Baryon–lepton composition of PNS matter obtained for GM1L model with s = 2 and YL = 0.2 (Malfatti et al., 2019). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Baryon–lepton composition of PNS matter obtained for DD2 model with s = 2 and Yνe = 0 (Malfatti et al., 2019). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Baryon–lepton composition of PNS matter obtained for GM1L model with s = 2 and Yνe = 0 (Malfatti et al., 2019). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature as a function of energy density of PNS matter obtained for the DD2 model. . . . . . . . . . . . . . . . . . Same as Fig. 16, but for the GM1L model. . . . . . . . . .

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Fig. 23

List of Figures

Pressure as a function of the energy density for different values of the vector coupling constant ζv (see text) (Malfatti et al., 2019). The black line represents the hadronic DD2 EOS and the dash-dotted and dashed lines are the EOSs of the quark (3nPNJL) phase for different ζv values. The horizontal lines mark the hadron–quark phase transitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . Same as Fig. 18, but for the hadronic GM1L EOS (Malfatti et al., 2019). . . . . . . . . . . . . . . . . . . . . . . . . . . Particle population of stellar quark matter at zero temperature as a function of baryon number density (Malfatti et al., 2019). The gray area indicates the density regime where matter described by the hadronic DD2 model exists. The hadron phase ends abruptly at the vertical line slightly above 0.6 fm−3 . The population of muons is increased by a factor of 100 to make it visible. The strength of the vector repulsion among quarks is ζv = 0.328. . . . . . . . . . . . . Same as Fig. 20, but for a vector repulsion among quarks of ζv = 0.331 (Malfatti et al., 2019). . . . . . . . . . . . . . The relative number density of particles in cold NS matter as a function of baryon number density (in units of the saturation density) (Spinella, 2017). The meson–Δ coupling constants are xσΔ = xωΔ = 1.1, and xρΔ = 1.0, and the vector meson–hyperon coupling constants are given by the SU(3) ESC08 model. The gray shading indicates baryon number densities beyond the maximum for the given parameterization. . . . . . . . . . . . . . . . . . . . . . . . Nuclear saturation potential (in MeV) of Δs in symmetric nuclear matter in the σωΔ coupling space (Spinella, 2017). Hyperons were included with the vector meson–hyperon given by the SU(3) ESC08 model. The star marker indicates the location of xσΔ = xωΔ = 1.1 and xρΔ = 1.0. Dashed contours are lines of constant potential as labeled and represent possible constraints. Gray pixels indicate that no Δs were populated for the given set of couplings. White pixels indicate couplings for which the effective mass of at least one baryon became negative before the maximum baryon number density of the NS was reached. . . . . . . .

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Fig. 24

Fig. 25

Fig. 26

Fig. 27

Fig. 28 Fig. 29 Fig. 30

Maximum mass (in solar mass units M ) in the σωΔ coupling space (Spinella, 2017). Hyperons were included with the vector meson–hyperon coupling constants given by the SU(3) ESC08 model. Solid lines are maximum mass contours for the associated hyperonic EOS (no Δs) in the ESC08 model. Colorbar tick marks represent the maximum mass constraints set by PSR J0348+0432 (1.97 − 2.05 M at 1σ, and 1.90 − 2.18 M at 3σ). Markers, contours, and pixels are as described for Fig. 23. . . . . . . . . . . . . . . Delta isobar fraction (percentage) of the maximum mass NS in the σωΔ coupling space (Spinella, 2017). Hyperons were included with the vector meson–hyperon coupling constants given by the SU(3) ESC08 model. Markers, contours, and pixels are as described for Fig. 23. The Δ fracSWL = 8.41%, tions for xσΔ = xωΔ = 1.1 are as follows: fΔ GM1L DD2 = 6.31%, and fΔ = 10.2%. . . . . . . . . . . . . . fΔ Critical baryon number density (in units of n0 ) for the appearance of Δs in the σωΔ coupling space (Spinella, 2017). Hyperons were included with the vector meson– hyperon coupling constants given by the SU(3) ESC08 model. Markers, contours, and pixels are as described for Fig. 23. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Radius (in km) of the canonical 1.4 M NS in the σωΔ coupling space (Spinella, 2017). Hyperons were included with the vector meson–hyperon coupling constants given by the SU(3) ESC08 model. The solid contour in the bottom (DD2) panel represents the 13.2 km upper limit of the radial constraint from Lattimer and Steiner (2014) represented as L90% on the colorbar. The 1σ and 2σ upper limits from Steiner et al. (2010) are represented on the colorbar as S1σ and S2σ, respectively. Markers, dashed contours, and pixels are as described for Fig. 23. . . . . . . . . . . . . . . Mass–radius relationships of PNS computed for the DD2 parameterization. . . . . . . . . . . . . . . . . . . . . . . . Same as Fig. 28, but computed for the GM1L nuclear parameterization. . . . . . . . . . . . . . . . . . . . . . . . Gravitational mass of PNSs as a function of central energy density in units of the energy density of ordinary nuclear matter ( 0 = 140 MeV/fm3 ) computed for GM1L. . . . . .

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239 241 242

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Fig. 31 Fig. 32 Fig. 33 Fig. 34

List of Figures

Gravitational mass of rotating PNSs vs. equatorial speed for GM1L EOS. . . . . . . . . . . . . . . . . . . . . . . . . Kepler periods of rotating PNs vs. gravitational mass for the GM1L EOS. . . . . . . . . . . . . . . . . . . . . . . . . Moment of inertia vs. gravitational mass, for the DD2 parameter set. . . . . . . . . . . . . . . . . . . . . . . . . . Same as Fig. 33, but for the GM1L parameter set. . . . . .

246 246 248 249

Chapter 6 Fig. 1

Fig. 2

Fig. 3

Fig. 4

The orbital frequency of a particle in a circular orbit for the case GR (upper curve) and for n = 3 (long dashed curve) and n = 4 (short dashed curve) [Boller, Hess, M¨ uller and St¨ocker (2019); Hess, Boller, M¨ uller and St¨ocker (2019)]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The position of the Innermost Stable Circular Orbit (ISCO) is plotted versus the rotational parameter a. The upper curve corresponds to GR and the lower curves to pc-GR. The light gray shaded region corresponds to a forbidden area for circular orbits within pc-GR. For small values of a the ISCO in pc-GR follows more or less the one of GR, but at smaller values of r. From a certain a on stable orbits are allowed until to the surface of the star (for n = 3 this limit is approximately 0.4 m and for n = 4 it is at 0.5 m). . . . Infinite, counter clockwise rotating geometrically thin accretion disk around static and rotating compact objects viewed from an inclination of 80◦ . The left panel shows the disk model by [Page and Thorne (1974)] in pc-Gr, with a = 0. The right panel shows the modified model, including pc-GR correction terms as described in the text. . . . . . . Infinite, counter clockwise rotating geometrically thin accretion disk around static and rotating compact objects viewed from an inclination of 60◦ and a = 0.6. The left panel is GR and the right one pc-GR. A resolution of 20 μas was assumed. A resolution of 20 μas is assumed. . . . . .

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Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Fig. 10

Axial gravitational modes in pc-GR. The vertical axis gives the real part of ω ˜ = mω while the horizontal axis depicts the negative of its imaginary part. . . . . . . . . . . . . . . The panel shows a case where the acceleration of the universe approaches a constant value for t → ∞. The figures are taken from Hess, Maghlaoui and Greiner (2010); Hess, Sch¨ afer and Greiner (2015). . . . . . . . . . . . . . . . . . The panel shows a case where the acceleration slowly approaches zero for t → ∞. The figures are taken from Hess, Maghlaoui and Greiner (2010); Hess, Sch¨ afer and Greiner (2015). . . . . . . . . . . . . . . . . . . . . . . The figure shows the dependence of the mass of the star as a function of its radius (figure taken from Rodr´ıguez, Hess, Schramm and Greiner (2014)). . . . . . . . . . . . . . . . . Dark energy density as a function of Rr , where R is the radius of the star. The upper curve depicts the result of the monopole approximation and the lower curve is an approximation for the upper one. The figure is taken from Caspar, Rodr´ıguez, Hess and Greiner (2016). . . . . . . . . . . . . The mass of a star as a function on its radius R. With the modified coupling of the dark-energy to the mass density the maximal mass possible is now about 200. The figure is taken from Caspar, Rodr´ıguez, Hess and Greiner (2016). . . . . .

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Chapter 7 Fig. 1

Fig. 2

Mass vs. radius relation for neutron stars in the f (R, T ) = R+ 2λT gravity for different values of λ and p = ωρ5/3 with ω = 1.475 × 10−3 [fm3 /MeV]2/3 . The full circles indicate the maximum masses for each λ value. . . . . . . . . . . . 300 Mass vs. radius relation for neutron stars in the energy– momentum tensor conserved f (R, T ) gravity for different values of α. The full circles indicate the maximum masses for each α value. . . . . . . . . . . . . . . . . . . . . . . . . 302

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Fig. 3

List of Figures

Mass vs. radius relation in General Relativity for several equations of state. The blue and orange regions are the constraints for the mass–radius obtained from GW170817 (LIGO Scientific Collaboration and VIRGO Collaboration, 2018). The continuous red line refers to the minimum mass value of the compact object in GW190814 (Abbott et al., 2020) (presumably a neutron star). The other horizontal lines, namely dotted yellow, dotted-dashed purple and continuous blue lines, refer to other massive neutron stars reported in the literature. . . . . . . . . . . . . . . . . . . 305

Chapter 8 Fig. 1

Fig. 2

Fig. 3

Energy per particle ε/nb − m as a function of baryon density nb for DD2, DD2Y-T and DD2Y-T + S2054 . The red solid line shows the hybrid EoS resulting from a Maxwell construction between DD2 and the used quark matter EoS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 Pressure P for DD2, DD2Y-T and DD2Y-T + S2054 as a function of the baryochemical potential potential μb (upper panel) and as a function of energy density ε = μb nb − P (lower panel) ε, where we emphasize that the region between highest energy densities in neutron stars and applicability of perturbative QCD cannot be probed with neutron stars. Line styles are as in Fig. 1. The red solid line corresponds to a hybrid EoS for which the sexaquark onset triggers a first-order phase transition to a color superconducting quark matter phase that is fitted by a CSS form of EoS (Shahrbaf et al., 2022). We show as grey hatched region the EoS constraint from (Hebeler et al., 2013) and by black dashed lines the one from (Miller et al., 2020). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 Mass vs radius (upper panel) and mass vs. baryon density (below panel) for compact stars in which the sexaquark S particle is assumed with a constant mass mS = 2054 MeV (dotted green line). The blue solid line corresponds to the hadronic EoS model without sexaquark while the

List of Figures

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

blue dashed line stands for the hadronic model without hyperons. The red solid line corresponds to a hybrid EoS for which the sexaquark onset triggers a first-order phase transition to a color superconducting quark matter phase that is fitted to CFL phase EoS (Shahrbaf et al., 2022). For a comparison the new 1.0 − σ mass-radius constraints from the NICER analysis of observations of the massive pulsar PSR J0740 + 6620 (Fonseca et al., 2021) are indicated in red (Riley et al., 2021) and blue (Miller et al., 2021) regions. Additionally, the magenta bars mark the excluded regions for a lower limit (Bauswein et al., 2017) and an upper limit (Annala et al., 2018) on the radius deduced from the gravitational wave observation GW170817. The green region is from the NICER mass-radius measurement on PSR J0030 + 0451 (Miller et al., 2019). . . . . . . . . . . . . . . . . . Diquark pairing gap Δ (left panel) and pressure p (right panel) of color superconducting quark matter as functions of baryonic chemical potential μb . . . . . . . . . . . . . . . Illustration of the transition from a BEC of sexaquarks as 3-diquark bound state (left panel) to a BCS condensate of diquarks in the CFL phase (right panel). . . . . . . . . . . Pressure P of hybrid quark–hadron matter as a function of the baryonic chemical potential μb (left panel) and as function of the baryon density nb (right panel). . . . . . . Mass-radius relation of hybrid star with the quark-hadron EoS presented on Fig. 6. The empty circle on the hadronic curve indicates the hyperon onset. Quark and hadron branches split at the sexaquark onset mass of about 0.7 M . The astrophysical constraints depicted by the colored bends and shaded areas are discussed in the text. . . . . . . . . . Tidal deformability Λ as a function of stellar mass M (left panel) and tidal deformability of the low mass component of the compact star merger Λ2 as a function of the corresponding parameter Λ1 of the high mass one found for the chirp mass M = 1.188 M (right panel). The calculations are performed for hybrid stars with the quark-hadron EoS presented on Fig. 6 (color curves) and compared to the results obtained for purely hadronic star with the DD2npY - T EoS

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(black curves). Dark and light green shaded areas on the right panel demonstrate the regions falling into the 50 % and 90 % confidence levels, while filled circles represent the configurations with equal masses of two components M1 = M2 = 1.3646 M . . . . . . . . . . . . . . . . . . . . . 337

c 2023 World Scientific Publishing Company  https://doi.org/10.1142/9789811220913 fmatter

List of Tables

Chapter 1 Table 1 Parameters of the primordial gravitational waves generated before the big “crunch” for two cases defined by the reheating temperature Tr . . . . . . . . . . . . . . . . . . . . . . .

18

Chapter 2 Table 1 Comparison of total neutrino luminosities Lν and annihilation luminosities Lν ν¯ in MeV s−1 between disks with and without flavor oscillations for selected accretion rates. . . .

92

Chapter 5 Table 1 Parameters of the SWL and GM1L (Spinella, 2017; Spinella et al., 2018) and DD2 (Typel et al., 2010) parameterizations used in this work. . . . . . . . . . . . . . . . . . . . . . . . Table 2 Properties of symmetric nuclear matter at saturation density for the SWL and GM1L (Spinella, 2017; Spinella et al., 2018) and DD2 (Typel et al., 2010) parameterizations. . . . Table 3 Saturation potentials of nucleons and Δs in symmetric nuclear matter with xσΔ = xωΔ = 1.1 and xρΔ = 1.0 (Spinella, 2017). . . . . . . . . . . . . . . . . . . . . . . . . Table 4 Properties of maximum mass NSs with Δs and hyperons with vector meson–hyperon coupling constants given in SU(3) symmetry with the ESC08 model (Spinella, 2017). .

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© 2023 World Scientific Publishing Company https://doi.org/10.1142/9789811220913 0001

Chapter 1

Cosmological Stochastic Gravitational Wave Background Jos´e Antonio de Freitas Pacheco Universit´e de la Cˆ ote d’Azur, Observatoire de la Cˆ ote d’Azur Bd de l’Observatoire, 06304 Nice Cedex, France [email protected] Gravitational waves are the only messenger able to probe physical processes that have occurred in the early universe, since the cosmic plasma is opaque to electromagnetic radiation at high redshift (z > 1100). Quantum fluctuations of a putative scalar field that drives inflation are able to produce a stochastic gravitational wave background. Additional mechanisms contributing to such a background are bubble collisions, sound waves and turbulence generated during the electroweak or the QCD phase transitions. Gravitational waves produced by these mechanisms could be detected by the planned Big Bang Observer space-based interferometer (phase 2) but unfortunately their signals are immersed in the astrophysical background generated by binary systems constituted by compact objects, despite different techniques proposed for separation of the signals. If the QCD transition is crossover and if the turbulence level is high enough, the expected signal is above the astrophysical background and detection would be possible. Modifications of the standard model leading to a strong electroweak transitions would also render possible the detection of the signal. A cosmic string network is able to generate a detectable gravitational wave background if the string tension is larger than Gµ/c2  1.4 × 10−17 . Keywords: Primordial gravitational waves; Cosmology; Primordial universe.

1. Introduction The detection of the first gravitational wave signal by the laser interferometers LIGO on September 14, 2015 (Abbott et al., 2016a) represents 1

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J. A. de Freitas Pacheco

a breakthrough on the experimental basis of General Relativity (GR), as well as on studies related to the physics of very compact objects like neutron stars and black holes. The first gravitational wave source dubbed GW 150914 was the consequence of a merger of two massive stellar black holes that constituted originally a binary system. Such a detection is a further experimental support to the General Relativity theory since it confirms the early predictions by Albert Einstein (Einstein, 1916). The initial masses of the black holes forming the pair as well as those of subsequent detections are substantially higher than masses estimated for galactic black hole candidates, most of them members of binary X-ray sources. These galactic counterparts have masses on the average around 8–9 M while values up to 50 M were found among the recent detected gravitational wave sources (Abbott et al., 2019). These massive stellar black holes raise a series of questions concerning the nature of the progenitors and their evolution, since very massive stars lose mass via strong stellar winds and mass-exchange processes are expected to occur in close binaries. These mechanisms modify the evolutionary timescale of the components of the pair as well as their initial masses. The problems concerning the evolution of the progenitors leading to these massive black holes were aggravated by the detection of the event GW 190521, which was interpreted as being produced by the merger of two black holes having masses respectively of 66 M and 85 M (Abbott et al., 2020). Core-collapse supernovas (Kuroda et al., 2016; Murphy et al., 2009; Sotani and Takiwaki, 2016) and magnetars (Regimbau and de Freitas Pacheco, 2006) have also been proposed as potential gravitational wave sources. However, the power released during the inspiral phase and final merger of two compact objects is by far the most important source of GWs. The gravitational wave emission of all these sources along the history of the universe produces a stochastic background (see, for instance, de Freitas Pacheco, 2020). The detection of such a background would be an important source of information about the origin and evolution of the sources and of the cosmic star formation history. Although this astrophysical gravitational background is of great interest, here we will be focused essentially on another possible background component whose origin dates from the early instants of the universe. Such a relic cosmological stochastic background is expected to be steady, isotropic, unpolarized and it arises from fundamental processes that are supposed to have occurred in the early universe. Among these it should be mentioned quantum vacuum fluctuations, cosmic

Cosmological Stochastic Gravitational Wave Background

3

phase transitions and cosmic strings (Caprini and Figueiroa, 2018). These relic gravitational waves are the only messenger able to carry information on the physical conditions prevailing at these early times, since beyond redshift z ∼ 1100 the universe is opaque to the electromagnetic radiation. Hence, the detection of such a relic background would have a considerable impact on the study of the physical processes that took place in those primordial epochs. In this chapter, different processes able to generate a gravitational wave and cosmological background will be analyzed. In Sec. 2, the generation of GWs during inflation will be discussed as well as during a possible preinflationary era; in Sec. 3, the production of GWs in a cyclic universe is examined; in Secs. 4 and 5, the production of GWs respectively during the electroweak and the quark–hadron phase transitions is discussed; in Sec. 6 detection techniques of such a background are presented and in Sec. 7 the final considerations are given, including suggestions for modifications of the standard model that increase considerably the amplitude of the signal generated during the electroweak phase transition. It will be also shown that a cosmic string network will be able to generate a detectable stochastic background if the string tension is larger than Gμ/c2  1.4 × 10−17 . 2. Inflation and Gravitational Waves Despite the successes of the standard big bang cosmology, some longstanding shortcomings remained, in particular the so-called horizon problem, the flatness problem and the abundance of unwanted relics like magnetic monopoles. In the early eighties inflation was proposed as a possible solution for all these problems (Guth, 1981; Linde, 1982). Since the inflation theory is developed in many textbooks (see, for instance, Kolb and Turner, 1990a) here only the basic principles will be recalled. In general, a single scalar field φ (christened inflation) is supposed to drive the inflationary process and the action Sφ characterizing such a field is given by  √ Sφ = [∂μ φ∂ μ φ + V (φ)] −g d4 x. (1) In the equation above V (φ) is the interaction potential and g is the determinant of the space–time metric. Adopting a Friedmann–Robertson–Walker (FRW) metric and performing a variation with respect to the field, the

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J. A. de Freitas Pacheco

resulting equation of motion is ∂V (φ) = 0, φ¨ + 3H φ˙ + ∂φ

(2)

where H = (a/a) ˙ is the Hubble–Lemaˆıtre parameter, a is the scale factor and upper dots indicate derivatives with respect to the proper time. The scalar field was assumed to be homogeneous. The energy–momentum tensor of the field is Tμν = ∂μ φ∂ν φ − gμν Lφ ,

(3)

where Lφ is the field Lagrangian associated to the action defined by Eq. (1). Assuming also that the homogeneous scalar field behaves like a perfect fluid, the associated pressure and energy density are respectively given by Pφ =

1 ˙2 φ − V (φ), 2

(4)

ρφ =

1 ˙2 φ + V (φ). 2

(5)

and

If the scalar field gives the principal contribution to the energy budget of the universe at that epoch, the Hubble and the Friedmann equations are respectively (in natural units) H2 =

8π ρφ , 3Mp2

(6)

and  2 a ¨ 4π =− (ρφ + 3 Pφ ). a 3Mp2

(7)

In the above equations Mp is the Planck mass (in energy units). During the inflation period, the kinetic term is expected to be negligible face to the potential, which should be sufficiently “flat” in order that an exponential growth of the scale factor could occur (this is known as the “slow-roll” approximation). Under these conditions the scale factor varies as a(t) ∝ eλt ,

(8)

where λ2 = 8πVI /3Mp2 and VI represents the effective value of the potential during the inflationary period.

Cosmological Stochastic Gravitational Wave Background

5

It is always useful to introduce the following dimensionless “slow-roll” parameters ε=

  Mp2 V  2 H˙ = , H2 16π V

(9)

Mp2 V  , 8π V

(10)

and η=

where V  and V  are respectively the first and the second derivatives of the potential with respect to the field φ. With these definitions, Eq. (7) can be recast as a ¨ = (1 − ε)H 2 . a

(11)

From this last equation, it can be easily verified that inflation requires ε  1 and that the “slow-roll” approximation ends when ε ∼ 1. Primordial gravitational waves can be described by the evolution of metric perturbations. The perturbed metric can be written as (0) gμν = gμν + hμν , (0)

(12)

where gμν is the unperturbed metric tensor and hμν is a small perturbation. Introducing Eq. (12) into Einstein equations, developing and keeping only first-order terms permit to obtain the equations describing the evolution of different perturbation modes (for details see, for instance, Turner et al., 1993). Here we are interested only in tensor modes that describe gravitational waves, which are characterized by the horizon crossing scale during radiation- and matter-dominated eras. Gravitational waves satisfy the Klein–Gordon equation   d ln a ∂hk,λ ∂ 2 hk,λ + k 2 hk,λ = 0. +2 (13) ∂τ 2 dτ ∂τ  In the equation above τ is the conformal time (τ = dt/a), k is the wave number characterizing the Fourier component of the perturbation amplitude and λ defines the polarization mode (“+” or “×”). For a given Fourier mode, before horizon crossing (kτ  1), amplitudes remain practically constant, i.e., hk,λ ∝ constant. After horizon crossing (kτ  1) the solution oscillates as hk,λ ∝ cos (kτ /a).

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J. A. de Freitas Pacheco

When τ  τeq , where τeq is the equipartition time (the instant corresponding to the equality between the energy densities of radiation and matter), the solution of Eq. (13) can be written as hk,λ (τ ) = hk,λ (0)T (k, τ ),

(14)

where T (k, τ ) is the transfer function derived, in general, by a numerical solution of the Klein–Gordon equation that can be suitably fitted by the equation     2 k k 3 j1 (kτ0 ) 1 + 1.34 . (15) + 2.5 T (kτ0 ) = kτ0 keq keq In the above equation j1 (x) is the first-order spherical Bessel function. In order to perform a numerical fit of the solution of the transfer function, the following scales were introduced: τ0 = 1.41 × 104 Mpc and the wave number corresponding to the horizon scale at equipartition, that is, keq = 0.073h20Ωm = 0.01033 Mpc−1 . Notice that in the precedent relation h0 = H/(100 km · s−1 · Mpc−1 ) and Ωm is the matter density parameter. Numerical values were computed by using the cosmological parameters given by the Planck collaboration (Aghamin et al., 2018). Once the solution of Eq. (13) is obtained, the gravitational wave energy density can be estimated from the equation   Mp2 h2k,λ , (16) d(ln k)k 2 ρgw = 16π λ

where the sum is over the polarization states. The amplitude of the fluctuations is essentially fixed by the Hawking–Gibbons temperature, that is

hk,λ

√   16π H  , Mp 2π

(17)

Using Eqs. (17) and (16), the energy density per logarithm interval of modes can be written as k 2 Mp2 k2 dρgw = 2 H2 = PT (k0 ), d ln k 2π 32π

(18)

where the amplitude of the tensor power spectrum at the pivot scale k0 was introduced. In general, the evolution of the power spectrum is given by PT (k, τ ) = PT (k0 , 0)T 2 (k, τ ),

(19)

Cosmological Stochastic Gravitational Wave Background

7

where the primordial power spectrum of gravitational waves is defined by 2  nT  k 16 HI PT (k, 0) = , (20) π Mp k0 where HI is the effective value of the Hubble–Lemaˆıtre parameter during the exponential expansion phase, nT is the power index exponent (not necessarily equal to that of the scalar power spectrum) and the adopted pivot scale is k0 = 0.002 Mpc−1 . Generally, in cosmology, energy densities are expressed in terms of the critical energy density ρcr required to close the universe. Hence, the density parameter of gravitational waves per logarithm interval of modes is defined by   dρgw 1 . (21) Ωgw = ρcr c2 d ln k Taking into account the definitions above, Eq. (21) can be rewritten as Ωgw =

k2 PT (k, 0)T 2 (k, τ0 ). 12H02

(22)

A further step can be done by expressing the primordial scalar and tensor power spectra in terms of the slow-roll parameters, that is  ns −1 k HI2 , (23) PS (k) = 2 πεMp k0 and PT (k) =

16HI2 πMp2



k k0

nT .

(24)

Hence, the ratio between both spectra at the pivot scale defines the parameter r, which can be constrained by observations as we shall see later. r=

PT (k0 ) = 16ε. PS (k0 )

(25)

Using Eqs. (9), (10) and (23), the running index of the scalar power spectrum can be expressed in terms of the scalar field potential, that is  3   Mp2 V  d V d ln Ps (k) =1− ln ns − 1 = . (26) d ln k 8π V dφ V 2

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J. A. de Freitas Pacheco

From these equations, one obtains easily relations between the scalar and tensor running indexes with the slow-roll parameters, i.e., ns − 1 = 2η − 6ε,

(27)

and r nT = −2ε = − . (28) 8 Using the precedent equations, the spectrum of the primordial gravitational waves expected to be produced during the inflationary process can be now expressed in terms of the slow-roll parameters as  2 c F (ν), (29) Ωgw (ν) = 12εAs τ0 H0 where As is the amplitude of the scalar power spectrum and the function F (ν) is given by  nT     2 ν ν ν 2 1 + 1.32 , (30) F (ν) = j1 (ν/ν1 ) + 2.5 ν0 ν2 ν2 where ν0 = 3.09 × 10−18 Hz, ν1 = 1.09 × 10−19 Hz and ν2 = 1.59 × 10−17 Hz. Planck-2018 data release (Aghamin et al., 2018) gives respectively for the amplitude of the scalar power spectrum at the assumed pivot scale As = 2.10 × 10−9 and for the scalar running index ns = 0.9649 ± 0.0042. For the ratio r between amplitudes of the tensor and scalar power spectra, Planck-2018 gives only a weak upper limit r < 0.1. The GW spectrum given by Eq. (29) can be now evaluated if the slowroll parameter ε is known and, in this case, a specific model for the scalar field is required. A plethora of potential models have been proposed to explain the exponential expansion phase and a summary of different categories of models can be found in Martin et al. (2013). Here a different class of model will be considered, i.e., the “inflationary-α attractor” based on the Poincar´e disk hyperbolic geometry. In Carrasco et al. (2015) this category of model was described by a Kahler potential with a built-inflaton shift symmetry. The potential adopted in the present calculation is a simplified version of the “T-model” defined by Carrasco et al. (2015) and is given by   φ 2 . (31) V (φ) = V0 tanh φ0 In general, the slow-roll parameters do not remain constant during the inflationary process and, in this case, the number of e-folds between the

Cosmological Stochastic Gravitational Wave Background

beginning and the end of the exponential phase is estimated from  φF 8π V (φ) N= 2 dφ, Mp φI V  (φ)

9

(32)

where φI and φF are respectively the initial and the final values of the field during the process. Notice that the adopted potential implies necessarily that φI > φF . The potential defined by Eq. (31) is now inserted into Eqs. (9), (26) and (32), which define respectively the first slow-roll parameter, the scalar running index and the number of e-folds. Using the Planck results mentioned above and taking N = 55 for a successful inflation, one obtains for the initial and final values of the field and for the scale parameter: φI /φ0 = 0.9098, φF /φ0 = 0.1917 and φ0 /Mp = 2.8926. These values permit to estimate the initial value of first slow-roll parameter, ε = 0.004216 and the running index of the tensor power spectrum nT = −0.008432. The ratio between the amplitudes of the tensor and scalar spectra is r = 0.067, which does not violate the Planck-2018 upper limit. For a question of completeness, recall that the Hubble parameter during inflation HI can be expressed in term of the first slow-roll parameter ε and of the amplitude of scalar power spectrum as HI2 = πεMp2 As . Inserting the derived and measured values, one obtains HI = 6.3 × 1013 GeV. Similarly, after estimating the mean value of the potential during inflation (VI = 0.26V0 ) and inserting this result into 1/4 the Hubble equation, one obtains V0 = 6.3 × 1016 GeV1/4 . The maximum observable (redshifted) frequency in the spectrum or the high-energy cutoff corresponds to the scale that has crossed outside the horizon just at the end of the inflation. Such a cutoff frequency is given by (Bin´etruy et al., 2012a). 2/3 

1/3 1/4 Trh VI 6 Hz, (33) νmax = 1.54 × 10 1014 GeV 1014 GeV where Trh is the reheating temperature that can be estimated by the following simple argument: at the end of the inflationary process the scalar field oscillates and its kinetic energy density ρK begins to decrease due to the expansion of the universe and due to the field decay into particles and/or radiation. The later process becomes dominant when the field decay rate is comparable to the expansion rate, that is  8πρK . (34) Γφ = H = 3Mp2

10

J. A. de Freitas Pacheco

When the condition above is satisfied, one assumes that all kinetic energy of the field is converted into radiation or, in other words ρK = ρr =

π2 4 g∗ Trh , 30

(35)

where g∗ = 106.75 is the effective number of degrees of freedom for the Standard Model. Combining Eqs. (34) and (35), one obtains  Trh =

90 8π 3 g∗

1/4

M p Γφ .

(36)

Under these approximations, Eq. (36) tell us that the reheating temperature is fixed essentially by the decay rate of the field, which is a quite uncertain quantity. Possible decay models and bounds on Γφ are given in Ellis et al. (2015). According to those authors, the field decay rate must be such that the resulting reheating temperatures should be in the range 10 MeV ≤ Trh ≤ 1014 GeV. The lower limit is required in order to have a successful nucleosynthesis whereas the upper limit is a necessary condition to avoid overproduction of gravitinos. In general, the decay rate can be written as Γφ = (g 2 /4π)mφ , where g is the coupling constant and mφ is the mass of the scalar particle. For the adopted potential (see Eq. (31)) and the values of the parameters estimated previously, the mass of the associated scalar particle is mφ = 2V0 /φ20  1.6 × 1016 GeV. If the scalar field decays into a lepton–anti-lepton pair, the coupling constant is of the order of mφ /Mp  1.3×10−5, which corresponds to Γφ  2.0×109 GeV and a reheating temperature of Trh  3.7×1013 GeV. This decay model leads to a high reheating temperature but does not violate the upper bound mentioned above. The reheating temperature can be considerably reduced if gravitational effects are included in the decay process (Bastero-Gil et al., 2016; Watanabe, 2011). In this case, the decay rate is given approximately by Γφ  m3φ /Mp2  2.7 × 104 GeV, leading to a reheating temperature of about 1011 GeV. Inserting these two reheating temperature estimates into Eq. (33), one obtains respectively for the high-frequency cutoff values of 81 MHz and 11 MHz. The resulting energy density spectrum of gravitational waves generated during inflation is shown in Fig. 1. Notice the existence of a maximum at the frequency of about 2 × 10−19 Hz with an amplitude of Ωgw h20  10−12 . For frequencies higher than 2×10−16 Hz the energy density decreases slowly and approximately as ν −0.0084 . Notice also that the spectrum is not

Cosmological Stochastic Gravitational Wave Background

11

Fig. 1. Expected dimensionless energy density spectrum of primordial gravitational waves produced during inflation with some parameters constrained by CMB data.

completely flat since oscillations are present in all frequency range. The amplitude at millimeter frequencies is quite small (Ωgw h20 ∼ 5 × 10−16) but, as we shall see in Sec. 4, such a gravitational signal could be detected by some proposed space-based interferometers. 2.1. Pre-inflationary gravitational waves General Relativity is expected to breakdown at Planck scales and, in the absence of a complete quantum gravity approach, effective field theories are adopted to provide a description of the universe evolution during the Planck era. Loop Quantum Gravity (LQG) is an example of such a class of theories. LQG introduces important modifications to the standard evolution of the early universe, in particular the avoidance of the initial singularity (Ashtekar et al., 2006; Bojowald, 2008, 2015). As a consequence, the dynamic equations describing the expansion lead to a bouncing solution on the semi-classical level where the singularity is replaced by an state of maximum energy density ρ∗ given by the equation √ 3 c ρ∗ = , (37) 16π 2 γI 4p

12

J. A. de Freitas Pacheco

where γ√I is the Immirzi–Barbero parameter defined by the relation γI = ln 2/(π 3). However, this phase is not long enough to explain the observed flatness of the universe but, in general, after the bounce the universe can attain the adequate initial conditions to start the standard slow-roll inflation as described in the previous section. The main features of the evolution near the bounce can be summarized as follows: near the energy density maximum, the kinetic energy Kφ of the inflaton field dominates the dynamics. Then, the field decays very rapidly and the total energy density is now essentially potential (but the field is not in a “true” vacuum state) and the slow-roll phase begins (see, for details, Artymowski et al., 2008). It should be mentioned that at the end of the bounce the slow-roll parameters are slightly modified as (Artymowski et al., 2008) ε=

Mp2 V 2 , 2 16π V (1 − V /ρ∗ )

(38)

η=

Mp2 V  . 8π V (1 − V /ρ∗ )

(39)

and

Notice that the pre-inflationary phase in LQG is a generic property that does not depend on the particular field (or fields) driving the expansion. Primordial gravitational waves generated during the bounce itself were investigated by Mielczarek (2008) and will not be considered here since the expected signal is very weak and well below the sensitivity of the planned space-based interferometers. The predicted spectrum has a maximum around the frequency of 7 × 10−14 Hz with an amplitude of about Ωgw h20  10−23 . In order to compute the power spectrum of the tensor modes generated after the bounce but prior to the “classical” inflation regime, the approach by Mielczarek and Szydlowski (2007) will be here adopted. In that work, the transition from the quantum to the classical description occurs when the scale factor parameter is equal to a critical prior specified value acrit . Super-horizon modes are those that first cross out the horizon (k ∼ aH) and then remain “frozen” during the expansion of the universe until reentering at late epochs. These modes keep information on the physical conditions prevailing in the very first evolutionary stages of the universe. Since the metric fluctuations are expected to have a quantum origin, the “classical” variables must be changed into their corresponding operators. Then,

Cosmological Stochastic Gravitational Wave Background

13

the field and the conjugate momentum can be decomposed into Fourier modes from which the power spectrum at horizon crossing can be computed, that is PT (k) = AT k nT , where the amplitude is defined by the equation  2β 3Γ2 (β + 1/2)24+2β β AT = , π 2 γI jMp2 2p τ0

(40)

(41)

and the running index nT =

6 . (2 + )

(42)

In the equations above, the following parameters were introduced: j is a semi-integer quantization parameter; is a “ill-defined” quantization parameter that satisfies the constraint 0 < < 1 (see, for instance, Bojowald, 2002) and β = 2(1 − )/(2 + ). The remaining parameter τ0 is the (dimensionless) conformal time corresponding to the scale parameter acrit . In the classical approximation describing the evolution of the tensor modes, the physical picture of the graviton creation process is preserved and, in this case, the maximum frequency corresponds to the scale νmax ∝ 1/τ0 or, in other words kmax =

1 β(β + 1 . τ0

(43)

The maximum frequency observed today, i.e., corrected for the redshift is   acrit kmax νmax = . (44) 2πacrit a0 Using the approximation a0 /acrit  Tp /T0 , the maximum frequency can be recast as     3β(1 + β) T0 β(1 + β) 11 νmax = Hz. (45) = 2.81 × 10 4π 2 γjτ02 t2p Tp jτ02 Then, in a second step, the Fourier modes of the field operator μ ˆk and its conjugate momentum π ˆk should be expressed in terms of annihilation and creation operators as it follows ∗ ˆ1,2 (k)f1,2 (k, τ ) + a ˆ†1,2 (−k)f1,2 (k, τ ), μ ˆk (τ ) = a

(46)

J. A. de Freitas Pacheco

14

where the indices 1, 2 indicate solutions respectively for τ < −τ0 and τ > −τ0 . The distinct functions f and g will be explicated below. Creation of gravitons can be now estimated from the Bogoliubov transformation and B(k) is the resulting non-zero coefficient that gives the production of gravitons in the final state, that is B(k) =

f1 (−τ0 )g2 (−τ0 ) − f2 (−τ0 )g1 (−τ0 ) . f2∗ (−τ0 )g2 (−τ0 ) − g2∗ (τ0 )f2 (−τ0 )

(47)

Hence, the energy density of gravitons is 8πhν 3 | B(k) |2 dν, (48) c3 where the two-polarization states were taken into account. Therefore using the definition of the density parameter (see Eq. (21)), one obtains dρgw =

128π 3 t2p 4 ν | B(z) |2 , (49) 3H02 where tp is the Planck time and z = −kτ0 = β(1 + β)(ν/νmax ). The constant factor in the equation above is given numerically by Ωgw =

128π 3 t2p = 3.7 × 10−49 Hz−4 . 3H02

(50)

Notice that the Hubble parameter was normalized as usually to 100 km · s−1 ·Mpc−1 . In order to compute the spectrum from Eq. (49), the functions appearing in Eqs. (46) or (47) must be defined, i.e.,

τ0 iHs(1) (z), (51) f1 (z) = A1 2   A1 (1 + 2s) (1) iHs(1) (z) , i zHs+1 (z) + (52) g1 (z) = √ 2 2τ0 (2) (53) f2 (z) = A2 1 + 4β H0 (qz)ei(q−1)z , and

  A2 β (2) (2) √ g2 (z) = H (qz) − z 1 + 4β H1 (qz) ei(q−1)z , τ0 1 + 4β 0

(54)

where H (1,2) are the Hankel functions of first and second kind and the following parameters were introduced

π i(s+1/2) π 2 , e (55) A1 = 2

Cosmological Stochastic Gravitational Wave Background

15

Fig. 2. Expected primordial gravitational wave spectrum generated post-bounce and computed for the following parameters: τ0 = 0.1 and j = 100. The black curve corresponds to  = 0.1 and the magenta curve to  = 0.75.

and

A2 =

π 8

τ0 −i π e 4, β

(56)

where s = β + 1/2 and q = 1 + 3(2 − )/4(1 − ). Figure 2 shows the post-bounce spectrum computed from Eq. (49) with the same parameters adopted in Mielczarek and Szydlowski (2007), that is τ0 = 0.1 and j = 100. Two values of the parameter were considered:

= 0.1 (black curve) and = 0.75 (blue curve). For the first case the maximum frequency is about 350 GHz and the energy density attains a value of Ωgw h20 ≈ 0.01. The spectrum in the frequency interval 100 Hz up to 10 GHz can be represented by a power law, that is Ωgw ∝ ν 1.19 . Adopting a higher value, i.e., = 0.75, the maximum frequency decreases to about 130 GHz and the amplitude to Ωgw h20 ≈ 4.8 × 10−4. The resulting spectrum varies also as a power law but with a more important slope, i.e., Ωgw h20 ∝ ν 2.39 . However, it should be emphasized that these spectra depend critically on very uncertain quantum parameters. It should be mentioned that string cosmology scenarios (to be discussed later) generate a quasi-thermal spectrum of gravitons during the so-called

16

J. A. de Freitas Pacheco

dilaton-driven phase, which is tilted towards high frequencies (Brustein, 1995) and similar to the expected post-bounce spectrum discussed here. The slope of the spectrum in the former scenario changes for the different modes crossing the horizon, but remains as a consequence of an enhanced production of high-frequency gravitons.

3. Gravitational Waves from a Cyclic Universe The so-called standard cosmological model combines the big-bang scenario first proposed by Georges Lemaˆıtre around 1930 and the inflationary model described in Sec. 2. This scenario explains quite well the observed large scale homogeneity and isotropy as well as the spatial flatness of the universe. Moreover, quantum fluctuations of the inflaton field, expected to occur during the very short inflationary period, explain the small temperature fluctuations present in the cosmic microwave background. As we have seen in the precedent section, Loop Quantum Cosmology avoids the big-bang singularity and predicts the existence of a contracting phase prior to the maximum density state and to the subsequent expansion. As a consequence of such a picture a natural question arises — could the contracting and expanding phases of the universe be repetitive? In fact, a cosmological model in which endless sequences of “bangs” and “crunches” occur was proposed by Steinhardt and Turok (2002a,b). The energy density, as in LQC, is finite on “bangs” and “crunches” and this cyclic model is able to explain homogeneity, flatness and density fluctuations that are responsible for the structures observed today. Between a “bang” and a “crunch”, the scalar field φ rolls back and forth in the effective potential and it is possible to identify different evolutionary stages: (1) the present phase, that is an expanding phase in which the cosmic acceleration prevails and the potential acts as the “dark energy” component; (2) the field kinetic energy becomes significant and a deceleration phase begins until the Hubble parameter passes by zero, initiating the contracting stage; (3) density fluctuations appear just before the “big crunch”; (4) the bounce occurs reverting the contraction into expansion; (5) end of the field kinetic energy domination and beginning of the radiation followed by the matter era; (6) expansion continues until the potential energy begins to dominate again and stage (1) is recovered. Tensor fluctuations obey the same Klein–Gordon equation already defined by Eq. (13). As we have seen, inflation predicts a primordial gravitational wave spectrum nearly “scale-invariant” but with an increasing

Cosmological Stochastic Gravitational Wave Background

17

power at very low frequencies (see Fig. 2). According to Boyle, Steinhardt and Turok (2003), in a cyclic universe the spectrum of primordial gravitational waves is exponentially suppressed at long wavelengths and is tilted towards high frequencies. Solutions of the Klein–Gordon equation give the following regimes for the primordial gravitational wave power spectrum: for low frequencies, which correspond to modes that re-enter the horizon after matter-radiation equality, that is k < keq , one has PT (k) ∝ k (α−1) ,

(57)

where the exponent α is related to the running index of the scalar power spectrum by α = (1−ns )/(1+ns ) and keq = aeq Heq . Here, aeq and Heq are respectively the scale factor and the Hubble parameter at matter-radiation equality. The intermediate frequency regime corresponds to modes that reenter the horizon between equality and the beginning of the radiation era, i.e., keq < k < kr , where kr = ar Hr (note that Boyle, Steinhardt and Turok (2003) assume that ar = 1 and not a0 = 1 as usually). In the intermediate frequency regime PT (k) ∝ k α .

(58)

Here we will focus on the high-frequency regime, corresponding to modes re-entering the horizon during the expanding kinetic phase (kr < k < kend ) since most of the gravitational wave energy is concentrated in these modes. The present power spectrum is given by Boyle, Steinhardt and Turok (2003)

√ 3  2 k04 (Γs Hr )1−2α  π 2 1+2α , PT (k) = cos kτr −  k 2 2 2 π Mp keq kr 4

(59)

where kend = Γs kr , k0 = a0 H0 and we have introduced  Γs =

1 1+χ



2α 1 − 2α



Vend Mp2 Hr2

1/3 .

(60)

In the equation above Vend = −V (φend ) is the potential depth at the bounce and χ  1 is a small positive parameter that measures the amount of radiation created at the density maximum. Using Eq. (16), the energy density under the form of gravitational waves can be written as k2 dρgw = M 2 PT (k), d ln k 16π p

(61)

J. A. de Freitas Pacheco

18

and the corresponding density parameter 2  k 1 Ωgw = PT (k). 6 H0

(62)

Using the above equations, the primordial gravitational waves generated before the big “crunch” were calculated for two cases defined by the reheating temperature Tr . The resulting parameters are summarized in Table 1. The second column gives the adopted reheating temperature, the third and fourth columns give respectively the resulting Hubble parameter when T = Tr and the coefficient Γs whereas the last column gives the maximum expected frequency. Figure 3 shows the resulting spectra. Table 1. Parameters of the primordial gravitational waves generated before the big “crunch” for two cases defined by the reheating temperature Tr . Model 1 2

Tr (GeV)

Hr (GeV)

Γs

νmax (GHz)

1013 1011

1.4 × 108 1.4 × 104

5.01 × 102 2.32 × 105

2.5 11.5

Fig. 3. Expected primordial gravitational wave spectrum generated in a cyclic universe — the black curve corresponds to case Tr = 1013 GeV while the magenta curve corresponds to Tr = 1011 GeV.

Cosmological Stochastic Gravitational Wave Background

19

4. Gravitational Waves from the Electroweak Phase Transition One important aspect related to the physics of the early universe relies on the assumption that the baryonic charge of the primordial plasma is zero. However, the present day universe is largely dominated by matter. In order to explain the observed baryon asymmetry several possibilities were envisaged (see, for instance, Riotto, 1998): (i) violation of the baryon number conservation; (ii) C and CP violation and (iii) processes occurring far from equilibrium conditions. The observed asymmetry could be also the consequence of a strong first-order electroweak (EW) phase transition that allows a CP violation process (Sakharov, 1991), a mechanism dubbed EW baryogenesis. The symmetry between weak and electromagnetic interactions is expected to be restored at the very first evolutionary stages of the universe for temperatures higher than a critical value around 100–200 GeV. The EW phase transition can be described according to the following scenario: initially the universe is in a metastable high-temperature phase, the “symmetric” state. As the universe expands and cools, there is a probability of quantum tunnelling to the “true” vacuum state and the symmetry is broken. Such a transition occurs via nucleation of bubbles that develop in the background constituted by the metastable state. The bubbles appear, expand and coalesce until the universe be completely in the broken symmetric state. The vacuum energy drives the expansion of the bubbles, which combined with their bulk motions generate gravitational waves. This process is illustrated in Fig. 4. Past investigations assumed that the EW phase transition is of second order (Kirzhnits, 1972). However, later studies suggested that gauge theories including particles heavier than the Higgs boson and including high-temperature corrections may lead to a first-order phase transition (Kirzhnits and Linde, 1975). Further investigations have shown that phase transitions in grand unified theories are always of first order (Linde, 1981). In a first-order phase transition, the bubble nucleation process is fixed by the tunnelling probability between the two vacua states of the effective potential that, in general, can be modelled by the expression V (T, φ) =

σ λ γ 2 (T − T02 )φ2 − T φ3 + φ4 . 2 3 4

(63)

J. A. de Freitas Pacheco

20

Fig. 4. Cartoon illustrating the formation of low-temperature bubbles inside the hightemperature phase (adapted from Allen (1997)).

In the equation above, the coupling parameters can be expressed in terms of masses of the gauge bosons W , Z, the top quark and the Higgs boson as γ=

2 (2MW + MZ2 + 2Mt2 ) , 4ν02

(64)

σ=

3 (2MW + MZ3 ) , 2πν03

(65)

λ=

   2 2 MW MH 3 4 − ln 2M W 2ν02 16π 2 ν04 αB T 2     MZ2 Mt2 4 + MZ4 ln ln − 4M . t αB T 2 αF T 2

(66)

The temperature scale T0 can be also expressed in terms of masses of gauge bosons and that of the top quark as    1 3  2 2 4 4 4 (67) T0 = MH − 2 2 2MW + MZ − 4Mt . 2γ 8π ν0 In these equations the following parameters appear — the EW scale given −1/2 = 246 GeV, in terms of the Fermi coupling constant, i.e., ν0 = 2−1/4 GF

Cosmological Stochastic Gravitational Wave Background

21

αB = 33.448 and αF = 3.127. Particle masses were taken from the CERN particle database, namely: MH = 125.18 ± 0.16 GeV, MW = 80.379 ± 0.012 GeV, MZ = 91.1876 ± 0.0021 GeV and Mt = 173.0 ± 0.4 GeV. Using these values, one obtains for the coupling constants: σ = 0.01921; γ = 0.3350; λ = 0.11428 and for the temperature scale, T0 = 163.1 GeV. Notice that the coupling parameter λ of the quartic term of the effective potential depends weakly on the temperature. Here this parameter was evaluated at the temperature T0 and small variations around such a value were ignored. As it will be discussed later in some more detail, a significant generation of gravitational waves during the EW transition occurs only in the case of a strong transition. A model-independent condition necessary to have a strong phase transition in the case of the standard model is φc /Tc ≥ 1 (Shaposhnikov, 1987), where φc is the value of the field at the critical temperature Tc (see below). In general this condition is not fulfilled for the standard model because the thermal induced cubic term appearing in Eq. (63) is not large enough. Modifications of the standard model have been suggested in different investigations (see, for instance, Ahriche, 2007; Huang et al., 2016; Leitao and M´egevand, 2016, among others). In these approaches, an additional term proportional to T φ3 is included in the effective potential, whose coupling constant depends on the adopted interaction model with the Higgs field. In Leitao and M´egevand (2016), for instance, two extra bosons were included. In our approach, we will assume that these modifications are equivalent to take an effective coupling constant σef for the cubic term in Eq. (63). Therefore, in all calculations, instead of the value estimated previously, i.e., σ = 0.01921, we will adopt σef = 0.30, a value compatible with those adopted in the aforementioned investigations. 4.1. Thermodynamics of the EW transition The two vacua states are derived from the condition ∂V (T, φ) = 0. ∂φ

(68)

Inserting Eq. (63) into Eq. (68), one obtains that the symmetric state corresponds to φmin,a = 0 with V (T, φmin,a ) = 0, while the broken symmetric phase corresponds to the field value φmin,b

σef T = 2λ



 1+

4λγ(T 2 − T02 ) 1− . 2 T2 σef

(69)

J. A. de Freitas Pacheco

22

The stability of the symmetric state is given by the condition ∂ 2 V (T, φ)/∂φ2 > 0, where the partial derivative should be evaluated at φmin,a = 0 (the symmetric state). Notice that this stability condition is equivalent to say that the effective mass of the field at φ = 0 must be positive. It is trivial to show that the symmetric vacuum is stable as long as the temperature satisfies the condition T > T0 and when the inequality goes in the opposite way the vacuum becomes unstable. The symmetric and the broken symmetry states are characterized by their free-energy density Fφ (T ) that is essentially the finite temperature effective potential or, in other words ∂V (φ, T ) , (70) ∂T where ρφ is the energy density and the second term on the right side corresponds to the latent heat. The pressure of each phase is given by Pφ = −Fφ (T ) and at the transition temperature Tc the pressure of both phases are equal, which is equivalent to say that at Tc the free energy of the broken and of the symmetric phases are equal. From this condition and using Eq. (63) one obtains after some algebra

 2 2σef 2 (71) Tc 1 − = T02 . 9γλ V (φ, T ) = Fφ (T ) = ρφ + T

Inserting the values of the coupling parameters mentioned above, one obtains Tc = 236 GeV. Notice that despite pressure equality at Tc , the energy density of both phases are different and such a difference corresponds to the latent heat of the transition. Once the temperature drops below Tc , the symmetric state becomes unstable and a first-order phase transition occurs at the temperature T∗ satisfying Tc > T∗ ≥ T0 . The vacuum energy density ρφ associated with the transition can be estimated from the energy difference between the two vacua states and the latent heat of the transition (Kolb and Turner, 1990a)    ∂V (T, φmin,b )   , (72) ρφ = −V (Tc , φmin,b ) + T∗   ∂T T∗ where the field value φmin,b corresponds to the broken symmetry state. Inserting Eq. (63) into the above equation, one obtains after some lengthy calculations ρφ =

4 2 T∗4 γ 2 2 σef 4σef f (x) + T0 T∗ , 1 3 2 24λ 9λ

(73)

Cosmological Stochastic Gravitational Wave Background

23

where the new variable x is given by x=

9λγ (T∗ − T0 ) , 2 σef T∗

and the function f1 (x) is defined by   3/2 4x 8x 8x2 − + 1− . f1 (x) = 1 + 27 3 9

(74)

(75)

The value of the variable x and, consequently, the value of the nucleation temperature T∗ can be estimated according to the following procedure. As mentioned previously, the transition from the metastable to the true vacuum state via tunnelling occurs through nucleation of bubbles that expand nearly the velocity of light. In this case, the probability of bubble nucleation per unit of volume and per unit of time is Γn = Ae−S(t) , where S(t) is the Euclidean action associated to the equation of motion of the φ-field. In the high-temperature limit, the action is approximately given by (Dine et al., 1992) S(x) = where f2 (x) = 1 +

41 σef λ−3/2 x3/2 f2 (x), 9   0.26 x 24 + . 1+ 4 (1 − x) (1 − x)2

(76)

(77)

Previous estimates (Huang et al., 2016) suggest that the EW phase transition occurs for S(x)  142, implying that x∗ = 0.582944 from the numerical solution of Eq. (76). This corresponds to a nucleation temperature T∗  165.8 GeV, quite close to the temperature scale T0 . Replacing this value of x∗ into Eq. (73) one obtains ρφ  1.21T∗4 and for the ratio between the vacuum to the thermal energy αT = ρφ /ρT  3.68 × 10−2 (100/g∗), where g∗ is the number of degrees of freedom of particles present in the cosmic plasma at the EW transition. Notice that from Eq. (69) one obtains trivially that φ∗ /T∗  2.35, confirming that the condition for a strong EW phase transition is satisfied. 4.2. Generation of gravitational waves In a strong first-order EW phase transition, gravitational waves are generated by three different mechanisms: from colliding bubbles during their

24

J. A. de Freitas Pacheco

expanding phase, since they generate a quadrupole contribution to the stress–energy tensor, which sources GWs; from the decay of magnetohydrodynamic turbulence produced by bulk motions of the bubbles that convert their kinetic energy into turbulence and from the propagation of damped sound waves. These two last mechanisms can continue to work for several multiple Hubble times, even after the phase transition is completed, providing additional sources of gravitational waves. The intensity of gravitational waves depends essentially on the following parameters: the duration scale of the transition β −1 , the expansion velocity uw of the bubble wall and the ratio αT between the vacuum to the thermal energy densities (Kamionkowski et al., 1994). The timescale of the transition in terms of the bubble nucleation rate is defined by     1  dΓn   dS  = . (78) β= Γn  dt   dt  This expression is, in general, very difficult to compute and here the approximate result derived by Turner et al. (1992) will be adopted, that is   β Mp  4 ln , (79) H∗ T∗ where H∗ is the Hubble parameter at the EW phase transition. As we shall see, the smaller the ratio β/H∗ is, the stronger the phase transition as well as the GW signal. In the strong regime (uw ≥ 0.8), the φ-field can couple to the thermal plasma and, consequently, friction effects cannot be neglected. In this case the wall velocity tends to a constant value. Nevertheless, in certain conditions such a velocity limit can be surpassed and a runaway process occurs depending if the parameter αT is larger or not than a critical value α∞ defined by Caprini et al. (2016) α∞  4.9 × 10−3



φ∗ T∗

2 .

(80)

Using the previous results, α∞  2.71 × 10−2 < αT and, consequently, we expect to be in the runaway regime, i.e., uw → 1. An important parameter defining the amplitude of the signal is the efficiency factor κ∞ of conversion of the latent heat into bulk motions.

Cosmological Stochastic Gravitational Wave Background

25

In the runaway regime, this is given approximately by Espinosa et al. (2010) κ∞ = 

α∞ . √ 0.73 + α∞ + 0.083 α∞

(81)

Gravitational waves generated by bubble collisions have been investigated by numerical simulations and a convenient representation of the resulting spectrum is given by (Huber and Konstandin, 2008) h20 Ωc (ν) = 1.67 × 10−5



H∗ β

2 

κ∞ αT 1 + αT

2 

100 g∗

1/3

0.11u3w 0.42 + u2w

 Sc (ν/νc ), (82)

where the function Sc (ν) corresponds to the spectral shape of the signal. An adequate fit of simulated data gives (Huber and Konstandin, 2008) Sc (ν/νc ) =

3.8(ν/νc )2.8 , 1 + 2.8(ν/νc )3.8

(83)

where νc is a characteristic frequency depending on the duration β −1 of the transition that corrected conveniently for the redshift is given by      β T∗ 0.62 g∗ 1/6 Hz. νc = 1.65 × 10−5 H∗ 100 GeV 100 1.8 + u2w − 0.1uw (84) For the generation of GW from the decay of MHD turbulence, the conversion efficiency is given by κt = εκ∞ where ε is the fraction of the bulk motion energy converted into turbulent motions. This is a quite uncertain quantity that is expected to be in the range 0.01 < ε < 0.10. In the present calculations, it was assumed ε = 0.05. Using the results by Kosowsky et al. (2002), the expected GW spectrum due to MHD turbulence is given by h20 Ωt (ν)

= 3.35 × 10

−4



H∗ β



κt αT 1 + αT

3/2 

100 g∗

1/3 uw St (ν/νt ), (85)

where again the function St (ν) defines the spectral shape. From an analytical study by Bin´etruy et al. (2012b) one obtains St (ν) = 

(ν/νt )3 , 1 + (ν/νt )11/3 (1 + ν/ν∗ )

(86)

26

J. A. de Freitas Pacheco

where the characteristic frequencies are respectively    β T∗ g∗ 1/6 −5 1 Hz, νt = 2.7 × 10 u w H∗ 100 GeV 100 and ν∗ = 6.56 × 10−7



T∗ 100 GeV



g∗ 1/6 Hz. 100

(87)

(88)

Finally, the third mechanism able to generate GWs during the EW phase transition concerns the production of sound waves by the expansion and motion of the bubbles. According to Hindmarsh et al. (2014), the compression waves in the plasma continue to produce GWs even after the bubbles have merged. The power spectrum of GWs generated by this process was derived from 3D simulations of bubble nucleation. In the runaway regime, the conversion efficiency κs of the kinetic energy of the bubbles into sound waves can be approximately given by   α∞ (89) κ∞ . κs = αT In this case, the GW spectrum can be expressed by Caprini et al. (2016)  2  1/3  H∗ 100 κs αT 2 −6 h0 Ωs (ν) = 2.65 × 10 uw Ss (ν/νs ). (90) β 1 + αT g∗ The function defining the spectral shape was taken from Kamionkowski et al. (1994) and it is given by  3  7/2 ν 7 Ss (ν) = , (91) νs 4 + (ν/νs )2 where the characteristic frequency is     1 β T∗ g∗ 1/6 Hz. νs = 1.9 × 10−5 uw H∗ 100 GeV 100

(92)

The resulting GW spectrum generated during the EW phase transition was estimated from the numerical solution of the equations above and it is shown in Fig. 5. The maximum of the emission occurs at a frequency of about 8.4 mHz while the amplitude attains a value of approximately h20 Ωgw  8 × 10−14, a value higher than that expected at the same frequencies for a GW signal originated during inflation. Notice that the largest contribution comes from the sound waves mechanism and that bubble collisions do not contribute significantly in the runaway regime.

Cosmological Stochastic Gravitational Wave Background

27

Fig. 5. Expected gravitational wave spectrum generated during a strong EW phase transition, including modifications of the Standard Model. The cyan curve corresponds to bubble collisions, the magenta curve shows the contribution of damped sound waves, the yellow curve shows the contribution of the MHD turbulence and the black curve is the sum of all these processes.

5. Gravitational Waves from the QCD Phase Transition After the EW phase transition episode, the universe continues to expand and to cool. The cosmic plasma is now constituted essentially by leptons, the associated neutrinos, quarks, gluons and photons. Quarks and gluons remain in a state of “asymptotic freedom” until the conditions for the existence of such a state are no more satisfied, leading to the confinement and, consequently, to the appearance of hadrons. The confining property of quarks and gluons manifests itself in the long-range behavior of the quark potential. At zero temperature the potential rises approximately linearly with the particle separation. The resulting force obliges quarks and gluons to be confined into a hadronic bag. On the other side, chiral symmetry breaking leads to a non-vanishing quark anti-quark condensate in the vacuum but inside the hadronic bag this condensate disappears. At high temperatures the individual bags are expected to merge and quarks and gluons can move freely. This bag picture is closely related to percolation models for the QCD phase transition. It provides an

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J. A. de Freitas Pacheco

intuitive argument for the occurrence of the deconfinement and chiral symmetry restoration. Two distinct aspects of the phase transition must be considered — one associated to chiral symmetry restoration, occurring at the temperature Tχ and related to models like the SU(3) gauge theory; the other related to the deconfinement at the temperature Td > Tχ . However, in “pure” QCD there seems to be only one transition from the hightemperature quark–gluon plasma to the low-temperature hadronic regime. The thermodynamics of the process is discussed in the next section. 5.1. Thermodynamics of the QCD phase transition From a thermodynamical point of view, the deconfinement can be treated as a first-order phase transition. Under this assumption, the transition occurs when the free energy of both phases are equal. However, the equation of state (EoS) for the quark–gluon plasma is still quite uncertain. One of the possible methods to handle this problem is lattice gauge theory that has emerged as a successful non-perturbative tool to investigate QCD. Such an approach permits an estimate of the EoS of the quark–gluon plasma and a rich literature on this subject is available (see, for instance, Brown et al., 1988; Fodor and Katz, 2004; Karsch, 1995; Khan et al., 2001). An important result derived from lattice calculations is that the phase transition from a phase of confined color degrees of freedom to a deconfined regime of free gluons is in fact of first order. Lattice calculations are performed in a compact Euclidean space–time of temperature and three-volume, which is discretized on a hypercubic lattice with Nt and Ns points respectively in the temporal and spatial directions. Therefore the lattice size is Nt × Ns3 . In order to estimate the EoS of the quark–gluon plasma using the lattice method, one computes the so-called trace anomaly I(T ), which is essentially the trace of the energy–momentum tensor that corresponds to the quantity (ρ − 3P ), where ρ is the energy density and P is the pressure. The trace anomaly satisfies also the following equation:   P d . (93) I(T ) = T 5 dT T 4 Hence, if the trace anomaly as a function of the temperature is known, the pressure can be derived from integration of the equation above. Besides the aforementioned references, calculations of the trace anomaly and other thermodynamic quantities are reported in Cheng et al. (2010), whose results

Cosmological Stochastic Gravitational Wave Background

29

will be used to estimate the transition temperature Td . Lattice calculations takes generally into account two or three flavors. The computations by Cheng et al. (2010) adopted here include u, d, s quarks. Thus, in order to be consistent, the confined phase represented usually by π-mesons must include also strange mesons like the Kaon. In this case, their total pressure Pm is given by ( = c = 1) Pm =

T4  gi Fi (T ), 2π 2

(94)

i

where the index i stands either for pions or kaons, gi is the number of degrees of freedom (taken equal to one for each meson, since there are one neutral and two charged pions and kaons) and the function Fi (T ) is given by    ∞ m2 Fi (T ) = − x (95) x2 − 2i lg(1 − e−x )dx, T mi /T where the meson masses were taken as: mπ = 140 MeV and mK = 494 MeV (small mass differences between charged and neutral species were neglected). Figure 6 shows the quantity P/T 4 as a function of the temperature for both phases. The values of the quantity P/T 4 for the deconfined phase were

Fig. 6. Pressure of the confined (black curve) and deconfined (cyan curve) phases plotted as a function of the temperature. The deconfined phase includes the flavors u, d, s while the confined phase consists of mesons π and K. The crossing point occurs at Td = 174 MeV.

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taken from Table III by Cheng et al. (2010) while for the confined phase they were obtained from numerical solution of Eq. (94). The pressure of both phases are equal at the temperature Td = 174 MeV. This value is slightly higher than the critical temperature at μ = 0 derived from lattice computations by Fodor and Katz (2004) (Td = 162 ± 2 MeV), but is within the range estimated from heavy-ion collisions, i.e., 150 < T < 180 MeV. The deconfinement pressure indicated by the crossing in Fig. 6 is Pd = 44.6 MeV · fm−3 . Notice that this is the common pressure of the quark– gluon and pion + kaon gases but not that of the cosmic plasma at this moment, since the contribution of the other particles present in the plasma must be taken into account. At this temperature, interpolation of lattice data by Cheng et al. (2010) gives an energy density for the quark–gluon plasma of ρqg = 463.4 MeV · fm−3 while that of the pion + kaon gas is ρm  163.4 MeV · fm−3 . The energy difference is the latent heat of the transition, i.e., L = ρqg − ρm = 300 MeV · fm−3 . 5.2. Generation of gravitational waves The QCD phase transition begins with the appearance of small bubbles of hadrons inside the cosmic plasma constituted by leptons, neutrinos, quarks, photons and gluons. The bubbles grow and coalesce until all quarks and gluons be confined. Such a picture is similar to that described previously for the EW phase transition. Therefore, one should expect that GWs will be generated by bubble collisions, sound waves and turbulence as before. However, the parameters involved in all these processes are now characterized by the physical conditions of QCD phase transition itself. Here the parameter αT is given by the ratio between the latent heat of the transition and the total thermal energy, that is αT = L/ρT = 0.292. Moreover, the efficiency factor of energy transfer from the latent heat to kinetic motions, based on numerical simulations, was taken from Kamionkowski et al. (1994) 

1 4 3αT 0.715αT + = 0.254, (96) κT = (1 + 0.715αT ) 27 2 where the numerical factor was obtained by inserting the value of αT estimated previously. The bubble wall velocity, in the detonation approximation, was also taken from Kamionkowski et al. (1994) 1/3 + α2T + 0.666αT  0.856. (97) uw = (1 + αT ) √ Notice that the wall velocity of bubbles are supersonic, i.e., uw > 1/ 3.

Cosmological Stochastic Gravitational Wave Background

31

In the present case, the parameter β/H is more difficult to evaluate. A rough estimate can be obtained by the following argument: considering adiabatic conditions, the ratio between the scale factor after and before the transition is given by a2 = a1



s1 s2

1/3 ,

(98)

where s1 and s2 are respectively the total entropy density before and after the transition. Inserting the values of the corresponding energy and pressure of the cosmic plasma, one obtains a2 /a1  1.0947. This is comparable to the result by Schmid et al. (1998), who estimated a2 /a1  1.1 from the lattice model adopted in that work. The duration of the transition Δt can be now estimated from the relation d lg a/dt = H that gives HΔt  0.0865. This value is adopted in the present computations. Another important aspect concerns GWs sourced by sound waves and the efficiency of the process. When the bubbles begin to merger together,

Fig. 7. Expected spectra of gravitational waves generated during the QCD phase transition by different mechanisms: bubble collisions (cyan curve), sound waves (magenta curve), turbulence (yellow curve) and sum of the different processes (black curve).

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J. A. de Freitas Pacheco

a stage of linear fluid evolution in which sound waves appear and produce shocks. Following Ellis et al. (2020), the efficiency parameter characterizing the energy transfer from sound (shock) waves to GW is given by   αeff αeff , (99) κs = √ αT (0.73 + αeff + 0.083 αeff ) where αeff = (1 − κT )αT . Notice the similitude between the equation above with Eq. (81), which defines the parameter κ∞ . The GW spectra due to these three mechanisms were taken from the approach by Ellis et al. (2020) and adapted conveniently to the case of the QCD phase transition. Hence, the contribution due to bubble collisions is given by  2  1/3 κT αT 100 2 −6 2 Sc (ν), (100) h0 Ωc (ν) = 5.01 × 10 (HΔt) 1 + αT g where the average number of degrees of freedom during the QCD phase transition was taken as g = 41 and the spectral shape is  3   2.07 −2.18 ν ν Sc (ν) = . (101) 1+2 νc νc The spectrum generated by sound waves after the coalescence of the bubbles is given by  2  1/3 κs αT 100 uw Ss (ν), (102) h20 Ωs (ν) = 2.65 × 10−6 (HΔt) 1 + αT g where the spectral shape is defined by  3   2 −7/2 ν 3 ν . 1+ Ss (ν) = νs 4 νs Similarly, the GW spectrum generated by turbulence is  3/2  1/3 εκT αT 100 2 −4 h0 Ωt (ν) = 3.35 × 10 (HΔt)uw St (ν), 1 + αT g

(103)

(104)

where ε = 0.05 is, as before, the fraction of the bubble kinetic energy converted into turbulent motions while the spectral shape is given by  3 

 11/3    −1 ν ν ν . (105) 1+ 1+ St (ν) = νt νt ν∗

Cosmological Stochastic Gravitational Wave Background

The peak frequency appearing in each spectral shape is given by   Td g 1/6 −1 −5 νi = ki × 10 [uw (HΔt)] Hz, 100 GeV 100

33

(106)

where i = c, s, t and the coefficient ki is equal to 1.82, 1.91 and 2.88 respectively for collision, sound and turbulent mechanisms. The characteristic frequency ν∗ in Eq. (105) is to be evaluated from Eq. (88) given in Sec. 4.2. As expected, since the QCD phase transition occurs later than the EW transition, the peak frequencies are considerably lower. Figure 7 shows the resulting spectra of these different mechanisms as well as the estimated total GW spectrum generated during the QCD phase transition. The spectrum peaks at the frequency of 0.33 μHz and has an amplitude of h20 Ωgw  8.9 × 10−11 . Most of the signal, as in the case of the EW transition, comes from GWs generated by sound waves. The expected total amplitude is almost three orders of magnitude larger than that predicted for the EW signal but occurs at lower frequencies, having consequences for its detection as we shall see later. 5.3. A “crossover” QCD transition? Previously, it was mentioned that the QCD phase transition is of first order. However, some lattice calculations suggest that when two light quarks (u, d) and one heavy quark (s) are considered, it is possible to find a situation in which the transition is “crossover”, i.e., quarks are confined smoothly. This would occur if the mass of the s-quark is above a certain critical value (see, for instance, Fodor, 2003). In this scenario, temperature and/or velocity fluctuations in the fluid are able to produce GWs with frequencies ranging from μHz up to mHz. These fluctuations are expected to be generated in the very early universe (Dolgov, Grasso and Nicolis, 2002). They survive because the plasma viscosity is extremely low as heavy-ions collisions seem to indicate (Song et al., 2011). In a first-order phase transition, the energy feeding turbulence is continuously injected into the fluid at the scale of bubbles and the cascade process leads to a Kolmogorov like spectrum. However, if the transition is crossover there is no continuous injection of energy because the viscosity is very tiny and cascading accumulates energy at small scales. Consequently turbulence develops a spectrum with a positive slope that is, the power is more concentrated in small eddies. Simulations aiming the evaluation of the GW power generated by turbulence in this case were performed by Mour˜ ao-Roque and Lugones (2013).

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The authors adopted an EoS based on lattice computations including two light quarks and one heavy as well as a background formed by leptons, neutrinos, their antiparticles and photons. They found a confining temperature of 170 MeV, in agreement with the results of Sec. 5.1. The GWs satisfy the non-homogeneous wave equation given by ¨ ij (k, τ ) + 2 a˙ h˙ ij (k, τ ) + k 2 hij (k, τ ) = 8πGa2 T T T (k, τ ), h ij a

(107)

where over dots indicate derivatives with respect to the conformal time τ , a is the metric scale factor and the term on the right side of the equation is the transverse traceless part of the stress–energy tensor sourcing the GWs, which is defined as TijT T (k, τ ) = (ρ + P )Πij (k, τ ),

(108)

where Πij (k, τ ) depends on the power spectrum of the velocity fluctuations. The spectral GW emission is defined as before, i.e., dρgw k 3 | h˙ |2 = , d ln k 16π 3 Ga2 where the GW power spectrum is   ˙ h˙ ij (k, τ )h˙ ∗ij (k  , τ ) = 2π 3 δ(k − k  ) | h(k, τ ) |2 ,

(109)

(110)

and the ensemble average . . . is over the stochastic process generating GWs. In Mour˜ ao-Roque and Lugones (2013) velocity fluctuations present at the QDC epoch were generated in the early universe and are supposed to survive until the transition due to the extremely low viscosity of the cosmic plasma. In the simulations the fluid is assumed to be at rest and the initial fluctuations are assumed to have a random distribution of amplitudes that are parameterized by a mean square root value. Figure 8 shows the velocity spectrum of turbulence after 1 μs of the beginning of the transition. Since the viscosity is very low, the energy of small eddies does not dissipate and accumulates in these low scales, producing the observed growth of power towards small scales. Figure 9 shows the computed GW spectrum by Mour˜ ao-Roque and Lugones (2013) for two values of the initial mean root square fluid velocity. Both spectra display a broad maximum around 10−4 Hz and have

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J. A. de Freitas Pacheco

6. Detection Techniques The analysis of the cosmological background is based on the assumption that the signal is isotropic, stationary and Gaussian. In this case, the signal is completely specified by its spectrum. Are all these assumptions justified? In the case of the cosmological microwave background (CMB), observations indicate that this component is highly isotropic. Deviations from isotropy arise from the proper motion of our local group of galaxies, which induces a dipolar distribution of the temperature with an amplitude of about 3.36 mK (Adam et al., 2016). Consequently, there is no major reason to expect important deviations from isotropy for the cosmological gravitational wave background. The stationary approximation is probably acceptable since the age of the universe is many orders of magnitude larger than the period of GWs relevant to the background mechanisms here investigated. Finally, the assumption of a Gaussian field seems to be justified considering the processes in the early universe responsible for such a signal (Allen, 1997). In general, the signal levels produced by different mechanisms that generate the GW background are quite small and, consequently, the output of detectors is dominated by their intrinsic noise. Even so, it is possible to detect the background signal by correlating the output of two or more detectors, under the assumption that they have independent noise (Michelson, 1987). The noise is fully characterized by the one-sided power spectrum density Sn (ν) defined by ˜ n(ν)˜ n∗ (ν  ) =

1 δ(ν − ν  )Sn (ν). 2

(111)

The factor 1/2 is introduced by convenience to indicate that the power spectrum density (PSD) is defined only for positive frequencies. As usually, n ˜ (ν) denotes the Fourier transform of n(t) (or F [n(t)] = n ˜ (ν) and its n(ν)] = n(t)), the upper “ ∗ ” indicates the complex conjugate inverse, F −1 [˜ and the angle brackets means an ensemble average over many noise realizations. In fact, we have only a single realization to work with, but the ensemble average can be replaced by a time average for stationary stochastic noise (Moore et al., 2015). Notice that Sn (ν) has units of inverse frequency. In the case of a single detector, the output can be expressed as the sum of the signal and the noise, i.e., s(t) = h(t) + n(t). The best strategy to extract the signal consists in the introduction of a Wiener filter W (ν) as   ˜ +n s˜(ν)W (ν) = h(ν) ˜ (ν) W (ν). (112)

Cosmological Stochastic Gravitational Wave Background

37

The signal can be recovered via the inverse Fourier transform or ˜ +n R(t) = F −1 [˜ s(ν)W (ν)] = F −1 [(h(ν) ˜ (ν))W (ν)].

(113)

The optimum Wiener filter should give the best representation of the signal and therefore we require that the quantity  (114) A = [R(t) − h(t)]2 dt, be a minimum. Using the Parseval’s theorem, we have  ˜ ˜ A= | R(ν) − h(ν) |2 dν.

(115)

˜ Replacing the definition of R(ν) given in Eq. (113) into Eq. (115), expanding and rearranging the terms  ˜ A= | (W (ν) − 1)h(ν) +n ˜ (ν)W (ν) |2 dν. (116) Under the assumption that the signal and the noise are not correlated, one obtains finally    h(ν) |2 + | n ˜ (ν) |2 W 2 (ν) dν. (117) A= (1 − W (ν))2 | ˜ Minimizing the equation above with respect to the filter, i.e., ∂A/∂W (ν) = 0; gives immediately the condition W (ν) =

˜ ˜ | h(ν) |2 | h(ν) |2  . ˜ |n ˜ (ν) |2 | h(ν) |2 + | n ˜ (ν) |2

(118)

The last term on the right-hand side of Eq. (118) is a consequence of the fact that the noise dominates the detector output. Note that the filter is essentially given by the ratio between the PSDs of the signal and noise. Moreover, since the spectral shape of the signal appears in the expression for the filter, good templates are necessary for an optimized extraction of the signal. In the case of two detectors, the situation is more complex since a possible misalignment between the detectors and a phase difference due to their separation are effects that must be taken into account. In this sense, the overlap reduction function (ORF) was introduced in Michelson (1987) aiming to take into account these effects affecting the resulting signal. The ORF depends on the GW frequency, the detectors configuration,

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J. A. de Freitas Pacheco

their relative positions and orientations. A closed form for the ORF was derived by Flanagan (1993), that is    5 x/c  F + F + + F × F × e2πiν Ω. , (119) γ(ν) = dΩ 1 2 2 2 8π  is a vector on the 2-sphere, x is the distance between the detectors where Ω and the F ’s are the form factors for the two polarization states “+” and “×”. The ORF can be expressed as a linear combination of Bessel functions (Flanagan, 1993) γ(ν) = Aj0 (α) + B

j2 (α) j1 (α) +C 2 , α α

(120)

where A, B, C are constants depending only on the location and the arm orientation of the detectors. The other parameter is given by the relation α = 2πν | x | /c. Performing the correlation between the output of the two detectors, one obtains for the PSD of the signal (see Allen, 1997; Flanagan, 1993, for details) Sh (ν) =

3H02 Ωgw γ(ν), 10π 2 ν 3

(121)

and for the signal-to-noise ratio (correlation between two interferometers)  2  ∞ Ω2gw (ν) S 9H04 2 dν, (122) = T γ (ν) i N 50π 4 ν 6 Sn1 (ν)Sn2 (ν) 0 where Ti is the integration time and Sn1 (ν), Sn2 (ν) are respectively the PSD of the first and second detector. 6.1. Detectors After several years of operation, the first generation of ground-based laser interferometers like GEO (0.6 km), TAMA (0.3 km), LIGO (4.0 km) and VIRGO (3.0 km) is being replaced by a second generation of instruments like Ad-LIGO or Ad-VIRGO, which were responsible for the detection of the first GW signals originated from merger events of two compact objects. However these antennas do not have an adequate sensitivity to detect a cosmological GW background with amplitudes estimated previously. A 2.5-generation of interferometers like KAGRA (3.0 km) (Kagra collaboration, 2019) or a still more advanced third generation like the Einstein Telescope (ET) with an arm length of 10 km (Ponturo et al., 2010) are been

Cosmological Stochastic Gravitational Wave Background

39

planned and, at least for the later, a detection of the astrophysical background produced by inspiraling black hole binaries is expected (de Freitas Pacheco, 2020). However ground-based interferometers are limited by the seismic noise and hence they are unable to detect signals with frequencies below few Hz. Therefore, space instruments are necessary to search for a cosmological signal which are important at sub-millimetric frequencies. Space-based interferometers have a structure similar to ground-based instruments. In the case of space-based antennas, the test masses are located inside of independent and widely separated satellites. Different projects have already been proposed like the “Laser Interferometer Space Antenna (LISA)” and its modified version “eLISA” or “New Gravitational Observatory (NGO)” (Danzmann et al., 2011), the “Advanced Laser Interferometer Antenna (ALIA)” (Bender et al., 2005), the “Deci-Hertz Interferometer Gravitational Wave Observatory (DECIGO)” (Seto et al., 2001) and the ambitious project “Big Bang Observer (BBO)” (Harry et al., 2006). The basic configuration of space-based detectors consists of clusters of satellites in a 1.0 AU orbit around the Sun. Figure 10 shows a sketch of the orbital configuration for the BBO2 (stage 2) project including three clusters of satellites. The twelve satellites form four interferometers consisting of three satellites in an equilateral triangle configuration with sides of 5 × 104 km and whose plane is tilted of 60◦ with respect to the ecliptic. LISA original configuration consisted in a triangular cluster with sides of 5 × 106 km, having the same tilt angle and trailing the Earth’s orbit by

Fig. 10. Sketch of the orbital configuration proposed for the Big Bang Observer (phase 2), adapted from Harry et al. (2006) — satellites in each cluster are distant from each other by arms of 5 × 104 km.

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an angle of 20◦ . The revised version (Evolved Laser Interferometer Space Antenna — eLISA) proposed in 2013 and approved in 2017 has the same configuration but the arms are now long of 2.5×106 km and the detector has a maximum sensitivity around 0.1–1.0 Hz. The Japanese project DECIGO consists of four clusters of satellites with a triangular configuration but with shorter arms (1000 km). DECIGO intends to cover the frequency gap between LISA-eLISA and ground-based detectors, i.e., 0.1–10 Hz. The mission ALIA is planned to cover lower frequencies since the arms of the triangle constituted by the spacecrafts have a length of 5 × 105 km. Finally, as mentioned above, the BBO project that was designed primarily to detect GWs generated during inflation. Two phases are planned: in the first (BBO1), the space antenna will be constituted by a cluster of three spacecrafts in a triangular configuration with arm lengths of 5 × 104 km while in the second (BBO2), three clusters are planned — two consisting of three satellites while the third will be composed by six satellites. In phase 1 the configuration is expected to optimize the signal around ∼1 Hz. In the second phase, two of the clusters will be centered on a 20◦ Earth-trailing orbit, rotated of 60◦ with respect to each other in the plane of the constellations. The remaining constellation including six satellites is to be placed in an Earth-like orbit 120◦ ahead and behind the two other clusters (see Fig. 10). This configuration is expected to provide greater angular resolution for foreground sources and a maximum cross-correlation between constellations, with a minimal correlated noise. 6.2. Comparison with theoretical predictions Theoretical amplitudes for the different sources of the cosmological background of GWs are given in terms of the relative density parameter Ωgw , which expresses the ratio between the gravitational wave energy density and the critical energy density required to have a closed space–time universe. Thus, it is interesting to express the equivalent minimal value of the density parameter that can be detected by a given space-based instrument in terms of its noise PSD (Sn (ν)), i.e., Ωgw (ν) =

4π 2 3 ν Sn (ν). 3H02

(123)

Figure 11 shows the expected sensitivity curves for different planned space-based antennas expressed in terms of the critical energy density parameter. It is clear from this figure that the best project is the BBO2,

Cosmological Stochastic Gravitational Wave Background

43

7. Final Considerations Gravitational waves are the unique probe of physical processes that have occurred in the early universe. The GW background produced during the inflation era has different parameters constrained by observations of the cosmic microwave background, which fix the amplitude and the value of the running index of the scalar power spectrum of the primordial quantum fluctuations and impose upper limits to the ratio between the amplitudes of the tensor and the scalar power spectrum. The expected spectrum is almost flat in a wide range of frequencies with an amplitude given in terms of a density parameter of about h20 Ωgw ∼ 5 × 10−16 . Despite such a small amplitude, this signal would be accessible to the planned spacebased Big Bang Observer laser interferometer (phase 2) in the frequency range −2.3 < log ν < 0.0 but unfortunately these GWs are immersed in the astrophysical background produced by black hole binaries. Bounce cosmologies lead to GWs generated before inflation having a positive power law spectrum that becomes important only at high frequencies (ν > 1 kHz). However, the spectrum estimates are very uncertain since the results depend on several ill-defined parameters. A similar analysis for cyclic universe scenarios lead to the same conclusions but with spectra with significant amplitudes only at still higher frequencies. A cosmological gravitational wave background can be also generated during the electroweak and the QCD phase transitions. Both depend on parameters still uncertain like the energy injected into gravitational waves produced via bubble collisions, sound waves and turbulence. Despite these uncertainties, theoretical estimates suggest that the electroweak phase transition produces a spectrum that peaks around 8 mHz with an amplitude of about h20 Ωgw ∼ 8 × 10−14 . It should be emphasized that such an amplitude is obtained only if modifications of the standard model are introduced. Here an additional term proportional to T φ3 was included in the effective potential that permits to satisfy the condition required to have a strong transition. Even so the resulting signal, despite of being detectable by the planned BBO2, remains immersed in the background generated by black hole binaries. If the QCD phase transition is crossover, the resulting spectrum has a broad maximum around 100 μHz, amplitudes above the astrophysical background of black hole binaries and above the sensitivity of the planned BBO2 space interferometer. The estimated spectrum amplitude depends critically on the adopted initial mean square velocity for turbulence. However, these values are relatively high and such turbulence levels

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probably have not been able to develop at that epoch. On the other side, if the QCD phase transition is of first order, the spectrum peaks at lower frequencies, i.e., around 0.068 μHz and with an amplitude above the aforementioned astrophysical background. However, the maximum occurs now at lower frequencies where the planned sensitivity of the BBO2 interferometer is not enough to detect the signal. Taking into account the points discussed above, it is clear that the predicted gravitational wave background signal from inflation and that from the electroweak transition, even taking into account particular modifications of the standard model, cannot be detected since they are immersed in the astrophysical background of black hole binaries. This is not the case of a first-order QCD phase transition whose signal is above the BHB background but requires an improvement of the BBO2 sensitivity at sub-μHz frequencies to be detected. A possible issue to such an impasse is to consider additional modifications of the standard model that affect the electroweak transition. As an example, in Kobakhidze et al. (2017) the authors included an anomalous Higgs-self coupling that affects the cubic term of the potential, leading to a strong first-order electroweak transition. In the most favorable case, the estimated GW background has a broad maximum around 0.6 mHz, which is displaced to frequencies lower than that estimated in this work (see Fig. 5). This is a consequence of the resulting lower temperature that defines the onset of the appearance of bubbles (T∗ = 86.8 GeV). In this case, the estimated amplitude of the GW signal surpasses the astrophysical background level and would be detected by the planned BBO2 experiment. A network of cosmic strings can also produce a cosmological GW background, a proposition first formulated by Vachaspati and Vilenkin (1985). Cosmic strings are one-dimensional spatial objects associated to topological defects that decay mainly by the emission of gravitational waves. The simplest string structures are characterized essentially by their linear mass density μ (given in mass per unit of length). In Grand Unified Theories (GUT) strings can be formed during a phase transition at the energy scale of about 1016 GeV and they can also emerge as vortex-like solutions of field theory similar to vortex lines in super-fluid helium. Cosmic strings have the property of keeping a fixed fraction of the total energy density of the universe since the early times until today. A string network produces GWs from oscillating closed string loops, whose normal modes have frequencies inversely proportional to their

Cosmological Stochastic Gravitational Wave Background

45

length . The size of a given loop created at instant ti evolves as

= αcti − γ

Gμ (t − ti ), c

(124)

where α and γ are dimensionless constants determined from numerical simulations. Typical values are α  0.1 and γ  50. The equation above says that a given loop disappears at instant td given by   αc2 (125) td = 1 + ti = βti . γGμ The cosmological GW background depends on the rate of loop production and on their decay fixed by the string tension Gμ/c2 that determines also the energy rate P under the form of GWs released by a typical loop, i.e. (see, for instance, Vachaspati and Vilenkin, 1985), P = γGμ2 c.

(126)

The spectrum of GWs produced by a cosmic string network was computed among others by Blanco-Pillado et al. (2013); Caldwell and Allen (1992); Vachaspati and Vilenkin (1985); Vilenkin and Shellard (1994) and more recently by Cui et al. (2017). In this last reference, the string network was described in the context of the standard cosmological model and based on the Nambu–Goto action, which describes the dynamics of a (classical) relativistic string, combined with results of recent numerical simulations of string networks. At high frequencies satisfying the condition ν  2.5 × 10−19 (γGμ/c2 )−1 Hz, the spectrum is flat and is given approximately by  1/2 Gμ (h20 Ωm ) h20 Ωgw  2.38 , (127) 2 γc (1 + zeq ) where Ωm is the present matter energy density parameter and zeq is redshift at which occurs the equality between the energy densities of matter and radiation. For lower frequencies the spectrum varies as h20 Ωgw ∝ ν 3/2 and can be approximated by the expression  2 Gμ (h20 Ωm ) 3/2 2 28 ν . (128) h0 Ωgw  1.9 × 10 γ c2 (1 + zeq ) It is worth mentioning that at high frequencies gravitational waves come mainly from loops that have been formed during the radiation era while loops that have appeared in the matter era contribute mainly to the lowfrequency regime.

Cosmological Stochastic Gravitational Wave Background

47

the signal is well above the background and could be detected either by BBO2 or LISA-DECIGO. Inspection of Fig. 13 suggests also that a GW wave background produced by a string network requires that the string tension must satisfy the condition Gμ/c2 > 1.4 × 10−17 in order to be above the astrophysical background produced by black hole binaries and be detected by the planned space-based interferometers. In conclusion, the expected background produced during the inflationary era could be detected only at frequencies below 0.1 μHz since in this spectral domain the signal would have an amplitude above that of the astrophysical background. Future investigations will tell us if the orbital period distribution of black hole binaries can be extended up to few months or a cutoff exists at shorter orbital periods, implying that the astrophysical background cannot be extrapolated down to μHz frequencies. This would change dramatically the spectral region in which the detection of the signal would be possible. Concerning the background generated during the electroweak phase transition, it seems that a detection would be possible only if modifications of the Standard Model are introduced. In this case, future positive or negative detection results will be testing such modifications. The background produced during the QCD phase transition is in a similar situation. If the transition is crossover and a high level of turbulence is present at that moment, then the signal is expected to be strong enough to surpass the astrophysical background and to be detected by BBO2. Finally, in a cosmological scenario including the presence of a string network, a cosmological background produced by string loops would be detected if the string tension is the range 1.4 × 10−17 < Gμ/c2 < 3.0 × 10−12 .

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© 2023 World Scientific Publishing Company https://doi.org/10.1142/9789811220913 0002

Chapter 2

Neutrino Flavor Oscillations in Gamma-Ray Bursts Jorge A. Rueda∗,‡ and Juan D. Uribe†,§ ∗ ICRANet, P.zza della Repubblica 10, I–65122 Pescara, Italy ICRANet-Ferrara, Dipartimento di Fisica e Scienze della Terra Universit` a degli Studi di Ferrara, Via Saragat 1, I–44122 Ferrara, Italy INAF, Istituto di Astrofisica e Planetologia Spaziali, Via Fosso del Cavaliere 100, 00133 Rome, Italy † ICRANet, P.zza della Repubblica 10, I–65122 Pescara, Italy Facultad de Ciencias B´ asicas, Universidad Santiago de Cali, Campus Pampalinda, Calle 5 No. 6200, 760035 Santiago de Cali, Colombia ‡ [email protected] § [email protected]

In the binary-driven hypernova model of long gamma-ray bursts, a carbon–oxygen star explodes as a supernova in presence of a neutron star binary companion in close orbit. Hypercritical (i.e. highly superEddington) accretion of the ejecta matter onto the neutron star sets in, making it reach the critical mass with consequent formation of a Kerr black hole. We have recently shown that, during the accretion process onto the neutron star, fast neutrino flavor oscillations occur. Numerical simulations of the above system show that a part of the ejecta keeps bound to the newborn Kerr black hole, leading to a new process of hypercritical accretion. We address here the occurrence of neutrino flavor oscillations given the extreme conditions of high density (up to 1012 g cm−3 ) and temperatures (up to tens of MeV) inside this disk. We estimate the evolution of the electronic and non-electronic neutrino content within the two-flavor formalism (νe νx ) under the action of neutrino collective effects by neutrino self-interactions. We find that neutrino oscillations inside the disk have frequencies between ∼ (105 − 109 ) s−1 , leading the disk to achieve flavor equipartition. This implies that the energy deposition rate by neutrino annihilation (ν + ν¯ → e− + e+ ) in the vicinity of the Kerr black hole, is smaller than previous estimates in

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the literature not accounting for flavor oscillations inside the disk. The exact value of the reduction factor depends on the νe and νx optical depths but it can be as high as ∼5. Keywords: Gamma-ray bursts; Neutrino oscillations; Black hole.

Hypernova;

1. Introduction Neutrino flavor oscillations are now an experimental fact, and in recent years, its study based only on Mikheyev–Smirnov–Wolfenstein (MSW) effects (Mikheyev and Smirnov, 1986; Wolfenstein, 1978) has been transformed by the insight that refractive effects of neutrinos on themselves due to the neutrino self-interaction potential are essential. Their behavior in vacuum, matter or by neutrino self-interactions has been studied in the context of early universe evolution (see, e.g., Abazajian et al., 2002; Dolgov et al., 2002; Kirilova, 2004; Wong, 2002), solar and atmospheric neutrino anomalies (see, e.g., Dighe, 2010; Giunti, 2004; Haxton et al., 2013; Vissani, 2017), and core-collapse supernovae (SN) (see, e.g., Dasgupta and Dighe, 2008; Duan et al., 2008, 2010; Horiuchi and Kneller, 2018; Kneller, 2015; Mirizzi et al., 2016; Sawyer, 2009; Volpe, 2016; Wu and Qian, 2011; Zaizen et al., 2018). We are interested in astrophysical situations when neutrino self-interactions become more relevant than the matter potential. This implies systems in which a high density of neutrinos is present and in fact, most of the literature on neutrino self-interaction dominance is concentrated on supernova neutrinos. It has been there shown how collective effects, such as synchronized and bipolar oscillations, change the flavor content of the emitted neutrinos when compared with the original content deep inside the exploding star. This chapter aims to explore the problem of neutrino flavor oscillations in the case of long gamma-ray bursts (GRBs) within the binary-driven hypernova (BdHN) scenario. The GRB progenitor is a binary system composed of a carbon–oxygen star (COcore ) and a companion neutron star (NS) (Fryer et al., 2014; Ruffini et al., 2015). The COcore explodes as SN, ejecting matter that produces a hypercritical accretion process onto the NS companion. The NS reaches the critical mass for gravitational collapse, hence forming a rotating black hole (BH). The emission of neutrinos is a crucial ingredient since they act as the main cooling process that allows the accretion onto the NS to proceed at very high rates of up to 1 M s−1 (Becerra et al., 2015, 2016, 2019).

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In Becerra et al. (2018), we studied the neutrino flavor oscillations in the aforementioned hypercritical accretion process onto the NS, all the way to BH formation. We showed that the density of neutrinos on top of the NS, in the accreting “atmosphere”, is such that neutrino self-interactions dominate the flavor evolution leading to collective effects. The latter induces in this system quick flavor conversions with a short oscillation length as small as (0.05 − 1) km. Far from the NS surface, the neutrino density decreases and so the matter potential and MSW resonances dominate the flavor oscillations. The main result has been that the neutrino flavor content emerging on top of the accretion zone was completely different compared to the one created at the bottom of it. However, in the BdHN scenario, part of the SN ejecta keeps bound to the newborn Kerr BH, forming an accretion disk onto it. In Becerra et al. (2018) we discussed this possibility and mentioned that the oscillation behavior can be markedly different. In this context, the study of accretion disks and their nuances related to neutrinos is of paramount importance to shed light on this aspect of the GRB central engine. In most cases, the exchanged mass in close binaries has enough angular momentum so that it cannot fall radially. As a consequence, the gas will start rotating around the star or BH forming a disk. However, the magneto-hydrodynamics that describe the behavior of accretion disks are too complex to be solved analytically and a full numerical analysis is timeconsuming and costly. To bypass this difficulty, different models make approximations that allow casting the physics of an accretion disk as a two- or even one-dimensional problem. The specific tuning of these approximations breeds one of the known disk models (see, e.g., Abramowicz et al., 1999; Abramowicz and Fragile, 2013; Blaes, 2004, 2014; Frank et al., 2002; Kato et al., 2008; Lasota, 2016; Narayan and McClintock, 2008; Pringle, 1981 and references therein). The options are numerous, and each model is full of subtleties making accretion flows around a given object a rich field of research. Of particular interest for GRB physics are Neutrino Cooled Accretion Disks (see Liu et al., 2017 and references therein) (NCADs). NCADs are hyperaccreting slim disks, optically thick to radiation that can reach high densities ρ ≈ 1010 –1013 g cm−3 and high temperatures T ≈ 1010 –1011 K around the inner edge. Under these conditions, the main cooling mechanism is neutrino emission since copious amounts of (mainly electron) neutrinos and anti-neutrinos are created by electron–positron pair annihilation, URCA and nucleon–nucleon bremsstrahlung processes, and

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later emitted from the disk surface. These ν ν¯ pairs might then annihilate above the disk producing an e− e+ dominated outflow. NCADs were proposed as a feasible central engine for GRBs in Popham et al. (1999) and have been studied extensively since (Chen and Beloborodov, 2007; Di Matteo et al., 2002; Gu et al., 2006; Kawanaka and Mineshige, 2007; Kawanaka et al., 2013; Kohri and Mineshige, 2002; Narayan et al., 2001; Xue et al., 2013). In Di Matteo et al. (2002) and later in Chen and Beloborodov (2007), it was found that the inner regions of the disk can be optically thick to νe ν¯e , trapping them inside the disk, hinting that NCADs may be unable to power GRBs. Yet, the escape of neutrinos from the disk involves propagation through dense media and, consequently, an analysis of neutrino oscillations, missing in the above literature, must be performed. The dominance of the self-interaction potential induces collective effects or decoherence. In either case, the neutrino flavor content of the disk changes. Some recent articles are starting to recognize their role in accretion disks and spherical accretion (Becerra et al., 2018; Frensel et al., 2017; Malkus et al., 2012; Tian et al., 2017; Wu and Tamborra, 2017). The energy deposition rate above and accretion disk by neutrino-pair annihilation as a powering mechanism of GRBs in NCADs can be affected by neutrino oscillation in two ways. The neutrino spectrum emitted at the disk surface depends not only on the disk temperature and density but also on the neutrino flavor transformations inside the disk. Also, once the neutrinos are emitted they undergo flavor transformations before being annihilated. The chapter is organized as follows. In Sec. 2, we outline the features of NCADs making emphasis on the assumptions needed to derive the equations. In Sec. 3, we discuss the general details of the equations that drive the evolution of neutrino oscillations and use the information from the previous section to build a simple model that adds this dynamic to NCADs. In Sec. 4, we give some details on the initial conditions needed to solve the equations of accretion disks and neutrino oscillations. In Sec. 5, we discuss the main results of our calculations and analyze the neutrino oscillation phenomenology in accretion disks. Finally, we present in Sec. 6 the conclusions of this work. Additional technical details are presented in a series of appendices at the end. 2. Hydrodynamics 2.1. Units, velocities and averaging Throughout this article we will use Planck units c = G =  = kB = ke = 1. To describe the space–time around a Kerr BH of mass M we use the metric

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gμν in Boyer–Lindquist coordinates, with spacelike signature, and with a dimensionless spin parameter a = J/M 2 so that the line element is   ds2 = gtt − ω 2 gφφ dt2 + gφφ (dφ − ω dt)2 + grr dr2 + gθθ dθ2 ,

(1)

in coordinates (t, r, θ, φ). The covariant components (g)μν of the metric are   2M r Σ gtt = − 1 − , grr = , gθθ = Σ, Σ Δ   3 2 a r 2 M sin2 θ sin2 θ, gφφ = r2 + M 2 a2 + (2) Σ 2 M2 a r sin2 θ, gtφ = − Σ and its determinant is g = −Σ2 sin2 θ,

(3)

with the well-known functions Σ = r2 + M 2 a2 cos2 θ,

Δ = r2 − 2M r + M 2 a2 .

(4)

We denote the coordinate frame by CF. Note that these coordinates can be used by an observer on an asymptotic rest frame. The angular velocity of the locally non-rotating frame (LNRF) is ω=−

2 a M2 gtφ , = 3 gφφ (r + M 2 a2 r + 2M 3 a2 )

(5)

and in Eq. (2) it can be seen explicitly that if an observer has an angular velocity dφ/dt = ω,

(6)

it would not measure any differences between the ±φ directions. The LNRF is defined by orthonormality and the coordinate change (Bardeen, 1970a; Bardeen et al., 1972) φLNRF = φ˜ = φ − ω t.

(7)

We assume that the disk lies on the equatorial plane of the BH, i.e., θ = π/2. This way we represent the average movement of the fluid by

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geodesic circular orbits with angular velocity Ω = dφ/dt = uφ /ut ,

(8)

plus a radial velocity so that the local rest frame (LRF) of the fluid is ˆ obtained by performing, first, an azimuthal Lorentz boost with velocity β φ to a corotating frame (CRF) (Gammie and Popham, 1998), and then a radial Lorentz boost with velocity β r˜. Clearly, the metric on the LNRF, CRF and LRF is diag(−1, 1, 1, 1). The expression for the angular velocity of circular orbits is obtained by setting r˙ = r¨ = 0 in the r-component of the geodesic equation √ M ± , (9) Ω = ±  3/2 r ± M 3/2 a where (+) is for prograde orbits and (−) is for retrograde orbits. We will limit our calculations to prograde movement with 0 ≤ a ≤ 1 but extension to retrograde orbits is straightforward. Finally, we can get the components of the 4-velocity of the fluid by transforming uLRF = (1, 0, 0, 0) back to the CF   γr˜γφˆ γr˜γφˆΩ γr˜β r˜ μ , √ , 0,  , (10) u =  grr ω 2 gφφ − gtt ω 2 gφφ − gtt leaving β r˜ to be determined by the conservation laws. In Eq. (10) we have ˆ replaced β φ with Eq. (A.2). A discussion on the explicit form of the transformations and some miscellaneous results are given in Appendix A.1. We will also assume that the disk is in a steady-state. This statement requires some analysis. There are two main ways in which it can be false: (i) As matter falls into the BH, its values M and a change (Bardeen, 1970b; Thorne, 1974), effectively changing the space–time around it. For the space–time to remain the same, we require Ω−1  tacc = ΔM0 /M˙ acc , where ΔM0 is the total mass of the disk and M˙ acc is the accretion rate. The characteristic accretion time must be larger than the dynamical time of the disk so that flow changes due to flow dynamics are more important than flow changes due to space–time changes. Equivalent versions of this condition that appear throughout disk accretion articles are tdym  tvisc and β r  β φ < 1,

(11)

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where it is understood that the accretion rate obeys M˙ acc ≈ ΔM0 /tacc .

(12)

To put this numbers into perspective, consider a solar mass BH and a disk with mass between one and ten solar masses. For accretion rates up to one solar mass per second, we obtain the values tacc  (1 − 10) s,

Ω−1 ∼ (10−5 − 10−1 ) s,

(13)

between r = rISCO and r = 2000M. Consequently, a wide range of astrophysical systems satisfy this condition and it is equivalent to claiming that both ∂ t and ∂ φ are Killing fields. (ii) At any point inside the disk, any field Ψ(t, r, θ, φ) that reports a property of the gas may variate in time due to the turbulent behavior of the flow. So, to assume that any field is time-independent and smooth enough in r for its flow to be described by Eq. (10) means replacing such field by its average over an appropriate space– time volume. The same process allows us to choose a natural set of variables that split the hydrodynamics into r-component equations and θ-component equations. The averaging process appears in Gammie and Popham (1998); Novikov and Thorne (1973) and Page and Thorne (1974). We include the analysis here and try to explain it in a self-consistent manner. The turbulent motion is characterized by the eddies. The azimuthal extension of the largest eddies can be 2π, like waves crashing around an island, but their linear measure cannot be larger than the thickness of the disk, and, as measured by an observer on the CRF, their velocity is of the order of β r˜ so that their period along the r component is Δt˜ ≈ (Thickness)/β r˜ (e.g. Sec. 33 of Landau and Lifshitz (1959)). If we denote by H the average half-thickness of the disk as measured by this observer at r over the time Δt˜, then the appropriate volume V contains the points (t, r, θ, φ) such that t ∈ [t∗ − Δt/2, t∗ + Δt/2] , θ ∈ [θmin , θmax ] , and φ ∈ [0, 2π),

(14)

where we have transformed Δt˜ and Δ˜ r back to the CF using Eq. (A.3) as approximations. The values θmin and θmax correspond to the upper and lower faces of the disk, respectively. Then, the average takes

J. A. Rueda and J. D. Uribe

60

the form t∗ +Δt/2 2π ψ (t, r, θ, φ) → ψ (r, θ) =

t∗ −Δt/2

0

ψ (r, t, θ, φ)

t∗ +Δt/2 t∗ −Δt/2

2π 0

−g grr gθθ dtdφ

−g grr gθθ dtdφ

. (15)

The steady-state condition is achieved by requiring that the Lie derivative of the averaged quantity along the Killing field ∂ t vanishes: L∂ t ψ = 0. Note that the thickness measurement performed by the observer already has an error ∼ M 2 a2 H 3 /6r4 since it extends the Lorentz frame beyond the local neighborhood but, if we assume that the disk is thin (H/r  1), and we do, this error remains small. At the same time, we can take all metric components evaluated at the equator and use Eq. (10) as the representative average velocity.  Under these conditions, we have θmax − θmin ≈ 2H/r and the term −g/grr in Eq. (15) cancels out. It becomes clear that an extra θ integral is what separates the radial and polar variables. In other words, the r-component variables are the vertically integrated fields  θmax √ ψ (r, θ) gθθ dθ. (16) ψ (r, θ) → ψ (r) = θmin

The vertical equations of motion can be obtained by setting up Newtonian (with relativistic corrections) equations for the field ψ (r, θ) at each value of r (see, e.g., Abramowicz et al., 1996, 1997; Liu et al., 2017; Novikov and Thorne, 1973). 2.2. Conservation laws The equations of evolution of the fluid are contained in the conservation laws ∇μ T μν = 0,

∇μ (ρuμ ) = 0.

(17)

The most general stress–energy tensor for a Navier–Stokes viscous fluid with heat transfer is (Mihalas and Mihalas, 1984; Misner et al., 1973) Ideal Fluid

  T = (ρ + U + P ) u ⊗ u + P g Viscous Stress

  Heat flux  + (−2ησ − ζ (∇ · u) P ) + q ⊗ u + u ⊗ q,

(18)

Neutrino Flavor Oscillations in Gamma-Ray Bursts

61

where ρ, P , U , ζ, η, q, P and σ are the rest-mass energy density, pressure, internal energy density, dynamic viscosity, bulk viscosity, heat-flux 4-vector, projection tensor and shear tensor, respectively, and thermodynamic quantities are measured on the LRF. We do not consider electromagnetic contributions and ignore the causality problems associated with the equations derived from this stress–energy tensor since we are not interested in phenomena close to the horizon (Gammie and Popham, 1998). Before deriving the equations of motion and to add a simple model of neutrino oscillations to the dynamics of disk accretion we must make some extra assumptions. We will assume that the θ integral in Eq. (15) can be approximated by 

√ ψ gθθ dθ ≈ ψr (θmax − θmin ) ≈ 2Hψ,

θmax

(19)

θmin

for any field ψ. Also, we use Stokes’ hypothesis (ζ = 0). Since we are treating the disk as a thin, differentially rotating fluid, we will assume that, on average, the only non-zero component of the shearing stress on the CRF is σr˜φ˜ (there are torques only on the φ direction), and qθ˜ is the only non-zero component of the energy flux (on average the flux is vertical). By uμ σμν = 0 and Eq. (A.5), we have σrφ =

γφ3ˆ

gφφ  ∂r Ω, σrt = −Ωσrφ . 2 2 ω gφφ − gtt

(20)

Finally, the turbulent viscosity is estimated to be ∼ lΔu where l is the size of the turbulent eddies and Δu is the average velocity difference between points in the disk separated by a distance l. By the same arguments in Sec. 33 of Landau and Lifshitz (1959) and in Sec. 2.2, l can be, at most, equal  to 2H and Δu can be at most equal to the isothermal sound speed cs = ∂P/∂ρ or else the flow would develop shocks (Frank et al., 2002). The particular form of cs can be derived from Eq. (24). This way we get η = Πνturb = 2αΠHcs ,

(21)

with α ≤ 1 and Π = ρ + U + P . In a nutshell, this is the popular α-prescription put forward in Shakura and Sunyaev (1973). As we mentioned at the end of Sec. 2.1, on the CRF for a fixed value of r, the polar equation takes the form of Euler’s equation for a fluid at rest where the acceleration is given by the tidal gravitational acceleration,

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J. A. Rueda and J. D. Uribe

that is, the θ component of the fluid’s path-lines relative acceleration in the θ direction.     1 ∂θ P ≈ ρr cos θ R u, ∂ θ˜, u · ∂ θ˜ , θ=π/2 r

(22)

with R the Riemann curvature tensor. With uμ˜ ≈ (1, 0, 0, 0), Eq. (19), Eq. (A.6) and assuming that there is no significant compression of the fluid under the action of the tidal force, integration of this equation yields the relation up to second order in π/2 − θ    2  1  θ˜ 2 2 π −θ H −r , (23) P = ρ R t˜θ˜t˜  2 2 θ=π/2 where we used the condition P = 0 at the disk’s surface. Hence, the average pressure inside the disk is (cf., Abramowicz et al., 1997; Chen and Beloborodov, 2007; Liu et al., 2017) P =

 1 ˜  ρH 2 R θ t˜θ˜t˜  . 3 θ=π/2

(24)

The equation of mass conservation is obtained by directly inserting into Eq. (A.11) the averaged density and integrating vertically 0 = ∂r (2rHρur ) ⇒ 2Hrρur = constant ˙ M ⇒ 2Hrρur = − , 2π

(25)

where the term 2Hrρur is identified as the average inward mass flux through a cylindrical surface of radius r per unit azimuthal angle and thus must be equal to the accretion rate divided by 2π. The same process applied to Eq. (A.10) yields the energy conservation equation   U +P r ∂r (Hρ) = 2ηHσ rφ σrφ − H, (26) u ∂r (HU ) − ρ where factors proportional to H/r were ignored and we assume Π ≈ ρ to integrate the second term on the left-hand side.  is the average energy density measured on the LRF (see the discussion around Eq. (A.14)). The first term on the right-hand side is the viscous heating rate Fheat and the second term is the cooling rate Fcool . The last constitutive equation is

Neutrino Flavor Oscillations in Gamma-Ray Bursts

63

obtained by replacing the density in Eq. (25) using Eq. (A.19) ur = −

4αHcs σφr . M f (x, x∗ )

(27)

2.3. Equations of state We consider that the main contribution to the rest-mass energy density of the disk is made up of neutrons, protons and ions. This way, ρ = ρB = nB mB with baryon number density nB and baryon mass mB equal to the atomic unit mass. NCADs reach densities above ∼ 107 g cm−3 and temperatures above ∼ 5 × 109 K. For these temperatures, forward and reverse nuclear reactions are balanced and the abundances in the plasma are determined by the condition μi = Zi μp + Ni μn , that is, the Nuclear Statistical Equilibrium (NSE). We denote the mass fraction of an ion i by Xi = ρi /ρB (if i = p or n we are referring to proton or neutrons), and it can be calculated by the Saha equation (Calder et al., 2007; Clifford and Tayler, 1965)

Ai mB Gi Xi = ρ



T Ai mB 2π



3/2 exp

   C Z i μp + μC p + N i μn − μi + B i . T (28a)

With constraints  i

Xi = 1,



Zi Yi = Ye .

(28b)

i

In these equations, T , Ai , Ni , Zi , Ye , Yi , Gi , μi and Bi are the temperature, atomic number, neutron number, proton number, electron fraction (electron abundance per baryon), ion abundance per baryon, nuclear partition function, chemical potential (including the nuclear rest-mass energy) and ion binding energy, respectively. For densities above 106 g cm−3 , the electron screening of charged particle reactions can affect the nuclear reaction rates. For this reason, to obtain an accurate NSE state, it is necessary to include Coulomb corrections to the ion chemical potential. The Coulomb corrections for the NSE state in a dense plasma are represented by the terms μC i (see Potekhin and Chabrier, 2000, for further details). The binding energy data for a large collection of nuclei can be found in Mavrodiev and Deliyergiyev (2018), and the temperature-dependent partition functions are found in Rauscher (2003) and Rauscher and Thielemann (2000).

64

J. A. Rueda and J. D. Uribe

Even though we take into account Coulomb corrections in NSE we assume that the baryonic mass can be described by an ideal gas1 and   Xi 3 Pi = nB T , UB = PB . (29) PB = Ai 2 i i The disk also contains photons, electrons, positrons, neutrinos and antineutrinos. As is usual in neutrino oscillations analysis, we distinguish only νe ) and x (anti-)neutrinos νx (¯ νx ), between electron (anti-)neutrinos νe (¯ where x = μ + τ is the superposition of muon neutrinos and tau neutrinos. Photons obey the usual relations Pγ =

π2 T 4 , Uγ = 3Pγ , 45

while, for electrons and positrons we have √   2 ne± = 2 ξ 3/2 F1/2,0 (ξ, ηe± ) + ξF3/2,0 (ξ, ηe± ) , π √   2 Ue± = 2 ξ 5/2 F3/2,0 (ξ, ηe± ) + ξF5/2,0 (ξ, ηe± ) , π√   2 2 5/2 ξ Pe± = ξ F3/2,0 (ξ, ηe± ) + F5/2,0 (ξ, ηe± ) , 3π 2 2

(30)

(31a) (31b) (31c)

with ξ = T /me and written in terms of the generalized Fermi functions  ∞ k x 1 + xy/2 dx. (32) Fk, (y, η) = exp (x − η) + 1  In these equations, ηe± = (μe± − me )/T is the electron (positron) degeneracy parameter without rest-mass contributions (not to be confused with η in Sec. 2.2). Since electrons and positrons are in equilibrium with photons due to the pair creation and annihilation processes (e− + e+ → 2γ) we know that their chemical potentials are related by μe+ = −μe− , which implies ηe+ = −ηe− − 2/ξ. From the charge neutrality condition, we obtain nB Ye = ne− − ne+ .

(33)

For neutrinos, the story is more complicated. In the absence of oscillations and if the disk is hot and dense enough for neutrinos to be trapped 1 Since

bulk viscosity effects appear as a consequence of correlations between ion velocities due to Coulomb interactions and of large relaxation times to reach local equilibrium, the NSE and ideal gas assumptions imply that imposing Stokes’ hypothesis becomes de rigueur (Buresti, 2015; Mihalas and Mihalas, 1984; Vincenti and Kruger, 1965)

Neutrino Flavor Oscillations in Gamma-Ray Bursts

65

within it and in thermal equilibrium, nν , Uν , Pν can be calculated with Fermi–Dirac statistics using the same temperature T   T3 F2,0 ην(¯ν ) !, 2 π   T4 = 2 F3,0 ην(¯ν ) , π trapped Uν(¯ ν) , = 3

= ntrapped ν(¯ ν)

(34a)

trapped Uν(¯ ν)

(34b)

trapped Pν(¯ ν)

(34c)

where it is understood that F (η) = F (y = 0, η) with ην(¯ν ) = μν(¯ν ) /T and the ultra-relativistic approximation mν  1 for any neutrino flavor is used. If thermal equilibrium has not been achieved, Eq. (34) cannot be used. Nevertheless, at any point in the disk and for a given value of T and ρ, (anti-)neutrinos are being created through several processes. The processes we take into account are pair annihilation e− + e+ → ν + ν¯, electron or positron capture by nucleons p + e− → n + νe or n + e+ → p + ν¯e , electron capture by ions A + e− → A + νe , plasmon decay γ˜ → ν + ν¯ and nucleon–nucleon bremsstrahlung n1 + n2 → n3 + n4 + ν + ν¯. The emission rates can be found in Appendix A.2. The chemical equilibrium for these processes determines the values of ην(¯ν ) . In particular,  ηνe = ηe− + ln

Xp Xn

 +

1−Q , ξ

(35a)

ην¯e = −ηνe ,

(35b)

ηνx = ην¯x = 0,

(35c)

satisfy all equations. Here, Q = (mn − mp )/me ≈ 2.531. And once the (anti-)neutrino number and energy emission rates (Ri , Qi ) are calculated for each process i, the (anti-)neutrino thermodynamic quantities are given by nfree ν(¯ ν) = H



Ri,ν(¯ν ) ,

(36a)

Qi,ν(¯ν ) ,

(36b)

.

(36c)

i

free Uν(¯ ν) = H

free Pν(¯ ν) =



i free Uν(¯ ν)

3

Remember we are using Planck units so in these expressions there should be an H/c instead of just an H. The transition for each (anti-)neutrino

J. A. Rueda and J. D. Uribe

66

flavor between both regimes occurs when Eqs. (34b) and (36b) are equal and it can be simulated by defining the parameter wν(¯ν ) =

free Uν(¯ ν) free + U trapped Uν(¯ ν) ν(¯ ν)

.

(37)

With this equation, the (anti-)neutrino average energy can be defined as trapped free Uν(¯  Uν(¯  ν) ν)

Eν(¯ν )  = 1 − wν(¯ν ) free + wν(¯ν ) trapped . nν(¯ν ) nν(¯ν )

and the approximated number and energy density are  if wν(¯ν ) < 1/2, nfree ν(¯ ν) nν(¯ν ) = trapped nν(¯ν ) if wν(¯ν ) ≥ 1/2,  free if wν(¯ν ) < 1/2, Uν(¯ν ) Uν(¯ν ) = trapped if wν(¯ν ) ≥ 1/2, Uν(¯ν ) Pν(¯ν ) =

Uν(¯ν ) . 3

(38)

(39a) (39b) (39c)

Note that both Eqs. (36c) and (39c) are approximations since they are derived from equilibrium distributions, but they help make the transition smooth. Besides, the neutrino pressure before thermal equilibrium is negligible. This method was presented in Chen and Beloborodov (2007) where it was used only for electron (anti-)neutrinos. The total (anti-)neutrino number and energy flux through one of the disk’s faces can be approximated with  nνj (¯νj ) , (40a) n˙ νj (¯νj ) = 1 + τνj (¯νj ) j∈{e,x}

Fνj (¯νj ) =



j∈{e,x}

Uνj (¯νj ) . 1 + τνj (¯νj )

(40b)

Here, τνi is the total optical depth for the (anti-)neutrino νi (¯ νi ). Collecting all the expressions we write the total internal energy and total pressure    (41a) Uνj + Uν¯j + UB + Ue− + Ue+ + Uγ , U= j∈{e,x}

P =

   Pνj + Pν¯j + PB + Pe− + Pe+ + Pγ .

j∈{e,x}

(41b)

Neutrino Flavor Oscillations in Gamma-Ray Bursts

67

The (anti-)neutrino energy flux through the disk faces contributes to the cooling term in the energy conservation equation but it is not the only one. Another important energy sink is photodisintegration of ions. To calculate it we proceed as follows. The energy spent knocking off a nucleon of an ion i is equal to the binding energy per nucleon Bi /Ai . Now, consider a fluid element of volume V whose moving walls are attached to the fluid so that no baryons flow in or out. The total energy of photodisintegration contained within this volume is the sum over i of (energy per nucleon of ion i)×(the number of freed nucleons of ion i inside V ). This can be written as 

(Bi /Ai )nf,i V,

(42)

i

or, alternatively, nB V



(Bi /Ai )Xf,i .

(43)

i

¯ If we approximate Bi /Ai by the average binding energy per nucleon B (which is a good approximation save for a couple of light ions) the expression becomes ¯ nB V B



¯ f = nB V B(X ¯ p + Xn ). Xf,i = nB V BX

(44)

i

The rate of change of this energy as measured by an observer on the LRF with proper time λ is  d  ¯ (Xp + Xn ) = nB V B ¯ d (Xp + Xn ). nB V B dλ dλ

(45)

The derivative of nB V vanishes by baryon conservation. Transforming back to CF and taking the average we find the energy density per unit time used in disintegration of ions ¯ r H∂r (Xp + Xn ). ions = nB Bu

(46)

The average energy density measured on the LRF  appearing in Eq. (26) is  = ions +

1  (Fνi + Fν¯i ). H i∈{e,x}

(47)

68

J. A. Rueda and J. D. Uribe

Finally, a similar argument allows us to obtain the equation of lepton number conservation. For any lepton , the total lepton number density is  (n − n¯ + nν − nν¯ ). (48) ∈{e,μ,τ }

So, with Eq. (33), calculating the rate of change as before, using Gauss’ theorem and taking the average we get ⎡ ⎤   (nν − nν¯ )⎦ = (n˙ ν¯ − n˙ ν ), (49) ur H ⎣nB ∂r Ye + ∂r ∈{e,x}

∈{e,x}

where the right-hand side represents the flux of lepton number through the disk’s surface. 3. Equations of Oscillation The equations that govern the evolution of an ensemble of mixed neutrinos are the Boltzmann collision equations iρ˙ p,t = C (ρp,t ), iρ¯˙ p,t = C (¯ ρp,t ).

(50)

The collision terms should include the vacuum oscillation plus all possible scattering interactions that neutrinos undergo through their propagation. For free streaming neutrinos, only the vacuum term and the forward-scattering interactions are taken into account so that the equations become   ¯ p,t , ρ¯p,t . iρ˙ p,t = [Hp,t , ρp,t ], iρ¯˙ p,t = H

(51)

¯ p,t ) is the oscillation Hamiltonian for (anti-)neutrinos and Here, Hp,t (H ρp,t ) is the matrix of occupation numbers: (ρp,t )ij = a†j ai p,t for neuρp,t (¯ trinos ((¯ ρp,t )ij = ¯ a†i a ¯j p,t for anti-neutrinos), for each momentum p and flavors i, j. The diagonal elements are the distribution functions fνi (¯νi ) (p) such that their integration over the momentum space gives the neutrino number density nνi of a determined flavor i at time t. The off-diagonal elements provide information about the overlapping between the two neutrino flavors. Taking into account the current–current nature of the weak interaction in the standard model, the Hamiltonian for each equation is

Neutrino Flavor Oscillations in Gamma-Ray Bursts

69

(Dolgov, 1981; Hannestad et al., 2006; Sigl and Raffelt, 1993)  √   d3 q 2GF lq,t − ¯lq,t (1 − vq,t · vp,t ) (2π)3  √ d3 q + 2GF (ρq,t − ρ¯q,t ) (1 − vq,t · vp,t ) 3, (2π)  √   d3 q = −Ωp,t + 2GF lq,t − ¯ lq,t (1 − vq,t · vp,t ) 3 (2π)  √ d3 q + 2GF (ρq,t − ρ¯q,t ) (1 − vq,t · vp,t ) , (2π)3

Hp,t = Ωp,t +

¯ p,t H

(52a)

(52b)

where GF is the Fermi coupling constant, Ωp,t is the matrix of vacuum oscillation frequencies, lp,t and ¯lp,t are matrices of occupation numbers for charged leptons built in a similar way to the neutrino matrices, and vp,t = p/p is the velocity of a particle with momentum p (either neutrino or charged lepton). As stated before, we will only consider two neutrino flavors: e and x = μ + τ . Three-flavor oscillations can be approximated by two-flavor oscillations as a result of the strong hierarchy of the squared mass differences |Δm213 | ≈ |Δm223 |  |Δm212 | (see Particle Data Group, 2018, for their numerical values). In this case, only the smallest mixing angle θ13 is considered. We will drop the suffix for the rest of the discussion. Consequently, the relevant oscillations are νe  νx and ν¯e  ν¯x , and each term in the Hamiltonian governing oscillations becomes a 2 × 2 Hermitian matrix. Now, consider an observer on the LRF (which is almost identical to the CRF due to Eq. (11) at a point r. In its spatial local frame, the unit ˆ φˆ of the CF, respectively. vectors x ˆ, yˆ, zˆ are parallel to the unit vectors rˆ, θ, Solving Eq. (51) in this coordinate system would yield matrices ρ, ρ¯ as functions of time t. However, in our specific physical system, the matter density and the neutrino density vary with the radial distance from the BH. This means that the equations of oscillations must be written in a way that makes explicit the spatial dependence, i.e., in terms of the coordinates x, y, z. For a collimated ray of neutrinos, the expression dt = dr would be good enough, but for radiating extended sources or neutrino gases, the situation is more complicated. In Eq. (51), we must replace the matrices of occupation numbers by the space-dependent Wigner functions ρp,x,t (and ρ¯p,x,t ) and the total time

J. A. Rueda and J. D. Uribe

70

derivative by the Liouville operator (Cardall, 2008; Strack and Burrows, 2005)

ρ˙ p,x,t

Explicit Time   Drift External Forces

 

  ∂ρp,x,t + vp · ∇x ρp,x,t + p˙ · ∇p ρp,x,t . = ∂t

(53)

In this context, x represents a vector in the LRF. In the most general case, finding ρp,x,t and ρ¯p,x,t means solving a 7D neutrino transport problem in the variables x, y, z, px , py , pz , t. Since our objective is to construct a simple model of neutrino oscillations inside the disk, to obtain the specific form of Eq. (51) we must simplify the equations by imposing on it conditions that are consistent with the assumptions made in Sec. 2. (i) Due to axial symmetry, the neutrino density is constant along the z direction. Moreover, since neutrinos follow null geodesics, we can set p˙ z ≈ p˙φ = 0. (ii) Within the thin disk approximation (as represented by Eq. (19)), the neutrino and matter densities are constant along the y direction, and the momentum change due to curvature along this direction can be neglected, that is, p˙y ≈ 0. (iii) In the LRF, the normalized radial momentum of a neutrino can be written as r . (54) px = ± √ 2 r − 2M r + M 2 a2 Hence, the typical scale of the change of momentum with radius is      d ln px −1 r r2 − 2M r + M 2 a2   Δrpx ,eff =  , (55) = dr  M (M a2 − r) which obeys Δrpx ,eff > rs for r > 2rin . This means that we can assume p˙ x ≈ 0 up to regions very close to the inner edge of the disk. (iv) We define an effective distance    d ln (Ye nB ) −1  .  (56) Δrρ,eff =   dr For all the systems we evaluated we found that is comparable to the height of the disk (Δrρ,eff ∼ 2 − 5 rs ). This means that at any point of the disk we can calculate neutrino oscillations in small regions assuming that both the electron density and neutrino densities are constant.

Neutrino Flavor Oscillations in Gamma-Ray Bursts

71

(v) We neglect energy and momentum transport between different regions of the disk by neutrinos that are recaptured by the disk due to curvature. This assumption is reasonable except for regions very close to the BH but is consistent with the thin disk model (see, e.g., Page and Thorne, 1974). We also assume initially that the neutrino content of neighboring regions of the disk (different values of r) do not affect each other. As a consequence of the results discussed above, we assume that at any point inside the disk and at any time, an observer can describe both the charged leptons and neutrinos as isotropic gases around small enough regions of the disk. This assumption is considerably restrictive but we will generalize it in Sec. 5. The purpose of these approximations is twofold: (1) We can reduce the problem considerably since they allow us to add the neutrino oscillations to a steady-state disk model by simply studying the behavior of neutrinos at each point of the disk using the constant values of density and temperature at that point. We will see in Sec. 5 that this assumption would correspond to a transient state of an accretion disk since, very fast, neighboring regions of the disk start interacting. (2) The approximations allow us to simplify the equations of oscillation considering that all but the first term in Eq. (53) vanish, leaving only a time derivative. In addition, both terms of the form vq,t · vp,t in Eq. (52) average to zero so that ρp,x,t = ρp,t and ρ¯p,x,t = ρ¯p,t . We are now in a position to derive the simplified equations of oscillation for this particular model. Let us first present the relevant equations for ¯ p,t , the corresponding neutrinos. Due to the similarity between Hp,t and H equations for anti-neutrinos can be obtained analogously. For simplicity, we will drop the suffix t since the time dependence is now obvious. In the twoflavor approximation, ρp is a 2 × 2 Hermitian matrix and can be expanded in terms of the Pauli matrices σi and a polarization vector Pp = (Px , Py , Pz ) in the neutrino flavor space, such that  ρp =

ρee ρex ρxe ρxx

 =

1 (fp I + Pp · σ ), 2

(57)

where fp = Tr[ρp ] = fνe (p) + fνx (p) is the sum of the distribution functions for νe and νx . Note that the z component of the polarization vector obeys Pzp = fνe (p) − fνx (p).

(58)

J. A. Rueda and J. D. Uribe

72

Hence, this component tracks the fractional flavor composition of the system. Appropriately normalizing ρp allows to define survival and mixing probabilities  1 1 + Pzp , 2  1 1 − Pzp . = 2

Pp,νe →νe =

(59a)

Pp,νe →νx

(59b)

The Hamiltonian can be written as a sum of three interaction terms: H = Hvacuum + Hmatter + Hνν .

(60)

The first term is the Hamiltonian in vacuum (Qian and Fuller, 1995): Hvacuum =

ωp 2

  ωp − cos 2θ sin 2θ B · σ , = sin 2θ cos 2θ 2

(61)

where ωp = Δm2 /2p, B = (sin 2θ, 0, − cos 2θ) and θ is the smallest neutrino mixing angle in vacuum. The other two terms in Eq. (52) are special since they make the evolution equations nonlinear. Since we are considering that the electrons inside the disk form an isotropic gas, the vector vq in the first integral is distributed uniformly on the unit sphere and the factor vq · vp averages to zero. After integrating the matter Hamiltonian is given by Hmatter =

λ 2



1 0 0 −1

 =

λ L · σ , 2

(62)

√ where λ = 2GF (ne− − ne+ ) is the charged current matter potential and L = (0, 0, 1). Similarly, the same product disappears in the last term and after integrating we get Hνν =

√   ¯ · σ , 2GF P − P

(63)

where  P=

Pp dp/(2π)3 .

(64)

Introducing every Hamiltonian term in Eq. (51), and using the commutation relations of the Pauli matrices, we find the equations of oscillation

Neutrino Flavor Oscillations in Gamma-Ray Bursts

for neutrinos and anti-neutrinos for each momentum mode p √   ¯ ] × Pp , P˙ p = [ωp B + λL + 2GF P − P √   ¯˙ = [−ω B + λL + 2G P − P ¯ ]×P ¯ , P p

p

F

p

73

(65a) (65b)

where we have assumed that the total neutrino distribution remains constant f˙p = 0. In this form, it is clear how the polarization vectors can be ¯ p /f¯p → P ¯p normalized. Performing the transformation Pp /fp → Pp and P and, multiplying and dividing the last term by the total neutrino density Eq. (65) can be written as P˙ p = [ωp B + λL + μD] × Pp , ¯˙ p = [−ωp B + λL + μD] × P ¯p P    1 ¯ q dq . fq Pq − f¯q P D= nνe + nνx (2π)3

(66a) (66b) (66c)

This is the traditional form of the equations in terms of the vacuum, matter and self-interaction potentials ωp , λ and μ with μ=

 √ 2GF nνi .

(67)

i∈{e,x}

Different normalization schemes are possible (see, e.g., Dasgupta et al., 2008; Esteban-Pretel et al., 2007; Hannestad et al., 2006; Mirizzi et al., 2016). Assuming that we can solve the equation of oscillations with constant potentials λ and μ simplifies the problem even further. Following Duan et al. (2006), with the vector transformation (a rotation around the z-axis of flavor space) ⎛

⎞ cos (λt) sin (λt) 0 Rz = ⎝− sin (λt) cos (λt) 0⎠ . 0 0 1

(68)

Equation (66) become P˙ p = [ωp B + μD] × Pp , ¯˙ = [−ω B + μD] × P ¯ , P p

p

p

(69a) (69b)

eliminating the λ potential but making B time dependent. Defining the ¯ p and, adding and subtracting Eqs. (69a) and (69b) vector Sp = Pp + P

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we get S˙ p = ωp B × Dp + μD × Sp ≈ μD × Sp , ˙ p = ωp B × Sp + μD × Dp ≈ μD × Dp . D

(70a) (70b)

The last approximation is true if we assume that the self-interaction potential is larger than the vacuum potential ωp /μ  1. We will show later that this is the case for thin disks (see Fig. 6). The first equation implies that all the vectors Sp and their  integral S evolve in the same way, suggesting the relation Sp = fp + f¯p S. By replacing in Eq. (70b) and integrating S˙ = μD × S, ˙ = ωB × S, D where



ω =

  ωp fp + f¯p dp/(2π)3

(71a) (71b)

(72)

is the average vacuum oscillation potential. The fact that in our model, the equation of oscillations can be written in this way has an important consequence. Usually, as it is done in supernovae neutrino oscillations, to solve Eq. (66), we would need the neutrino distributions throughout the disk. If neutrinos are trapped, their distribution is given by Eq. (34). If neutrinos are free, their temperature is not the same as the disk’s temperature. Nonetheless, we can approximate the neutrino distribution in this regime by a Fermi–Dirac distribution with the same chemical potential as defined by Eq. (35) but with an effective temperature Tνeff . This temperature can be obtained by solving the equation     (73)

Eν  = U Tνeff , ην /n Tνeff , ην , which gives Tνeff νx  νx = Eνx ,¯ x ,¯

Tνeff νe = e ,¯

180 ζ(3) , 7π 4

Eνe ,¯νe  Li3 (− exp (ηνe ,¯νe )) , 3 Li4 (− exp (ηνe ,¯νe ))

(74a)

(74b)

where ζ(3) is Ap´ery’s constant (ζ is the Riemann zeta function) and Lis (z) is Jonqui`ere’s function. For convenience, and considering the range

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of values that the degeneracy parameter reaches (see Sec. 6), we approximate the effective temperature of electron neutrinos and anti-neutrinos with the expressions = Tνeff e



Eνe   2 aηνe + bηνe + c , 3

(75a)

Eν¯e  . 3

(75b)

Tν¯eff = e

with constants a = 0.0024, b = −0.085, c = 0.97. However, Eq. (71) allows us to consider just one momentum mode, and the rest of the spectrum behaves in the same way. 4. Initial Conditions and Integration In the absence of oscillations, we can use Eqs. (26), (24) and (49) to solve for the set of functions ηe− (r), ξ (r), Ye (r) using as input parameters the accretion rate M˙ , the dimensionless spin parameter a, the viscosity parameter α and the BH mass M . From Chen and Beloborodov (2007) and Liu et al. (2017) we learn that neutrino cooled disks require accretion between 0.01 M s−1 and 1 M s−1 (this accretion rate range vary depending on the value of α). For accretion rates smaller than the lower value, the neutrino cooling is not efficient and, for rates larger than the upper value, the neutrinos are trapped within the flow. We also limit ourselves to the above accretion rate range since it is consistent with the one expected to occur in a BdHN (see, e.g., Becerra et al. (2019); Fryer et al. (2014)). We also know that high spin parameter, high accretion rate, and low viscosity parameter produce high density and high temperature disks. This can be explained using the fact that several variables of the disk, like pressure, density and height are proportional to a positive power of the quotient M˙ /α. To avoid this semi-degeneracy in the system, reduce the parameter space, and considering that we want to study the dynamics of neutrino oscillations inside the disk, we fix the BH mass at M = 3M , the viscosity parameter at α = 0.01 and the spin parameter at a = 0.95 while changing the accretion rate. Equations (26) and (49) are first-order ordinary differential equations, and since we perform the integration from an external (far away) radius rout up to the innermost stable circular orbit rin , we must provide two boundary conditions at rout . Following the induced gravitational collapse (IGC) paradigm of gamma-ray bursts (GRBs) associated with type Ib/c supernovae we assume that at the external edge of the disk, the infalling matter

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is composed mainly by the ions present in the material ejected from an explosion of a carbon–oxygen core, that is, mainly oxygen and electrons. This fixes the electron fraction Ye (rout ) = 0.5. The second boundary con√ dition can be obtained by the relation (T η + mB ) gtt = constant (Klein, 1949a,b; Tolman, 1934), with η the degeneracy parameter of the fluid. If we require the potentials to vanish at infinity and using Euler’s theorem we arrive at the relation in the weak field limit  ρ + U + P − T S  M = . (76)  rout ρ r=rout For a classical gas composed of ions and electrons this relation becomes  M U   . (77) rout ρ r=rout Equation (77) can be used with Eqs. (24) and (41) to solve for ηe− (rout ), ξ (rout ). The value of rout is chosen to be at most the circularization radius of the accreting material as described in Becerra et al. (2015, 2016). We can estimate this radius by solving for r in the expression of the angular momentum per unit mass for a equatorial circular orbits. So using Eq. (10), we need to solve uφ = M

x2 − 2x + a2 √ ∼ 3 × 107 cm, x3/2 x3 − 3x + 2a

(78)

 where x = r/M , which yields rout ∼ 1800rs , and the expression is in geometric units. Finally, for the initial conditions to be accepted, they are evaluated by the gravitational instability condition (Paczynski, 1978)   √  R θ˜t˜θ˜t˜  Ω ≥ 2 3πρ. (79) θ=π/2

Integration of the equations proceeds as follows, with the initial conditions we solve Eq. (49) to obtain the electron fraction in the next integration point. With the new value of the electron fraction, we solve the differential-algebraic system of Eqs. (26) and (24) at this new point. This process continues until the innermost stable circular orbit rin is reached. To add the dynamics of neutrino oscillations we proceed the same as before but at each point of integration, once the values of Ye , η and ξ are found, we solve Eq. (66) for the average momentum mode to obtain the survival probabilities as a function of time. We then calculate the new neutrino

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within the inner regions of the disk. Given this fact, to solve the equations of oscillations, we can approximate the initial conditions of the polarization vectors with ¯ ≈ (0, 0, 1). P=P

(81)

5. Results and Analysis In Figs. 2–5 we present the main features of accretion disks for the parameters M = 3M , α = 0.01, a = 0.95, and three different accretion rates M˙ = 1M s−1 , M˙ = 0.1M s−1 and M˙ = 0.01M s−1 . It exhibits the usual characteristics of thin accretions disks. High accretion rate disks have higher density, temperature, and electron degeneracy. Also, for high accretion rates, the cooling due to photodisintegration and neutrino emission kicks in at larger radii. For all cases, as the disk heats up, the number of free nucleons starts to increase, enabling the photodisintegration cooling at r ∼ (100 − 300)rs . Only the disintegration of alpha particles is important, and the nucleon content of the infalling matter is of little consequence for the dynamics of the disk. When the disk reaches temperatures ∼1.3 MeV, the electron capture switches on, the neutrino emission becomes significant and the physics of the disk is dictated by the energy equilibrium between Fheat and Fν . The radius at which neutrino cooling becomes significant (called ignition radius rign ) is defined by the condition Fν ∼ Fheat /2. For the low accretion rate M˙ = 0.01M s−1 , the photodisintegration cooling finishes before the neutrino cooling becomes significant, leading to fast heating of the disk. Then the increase in temperature triggers a strong neutrino emission that carries away the excess heat generating a sharp spike in Fν , surpassing Fheat by a factor of ∼3.5. This behavior is also present in the systems studied in Chen and Beloborodov (2007). But there, it appears for fixed accretion rates and high viscosity (α = 0.1). This demonstrates the semi-degeneracy mentioned in Sec. 5. The evolution of the fluid can be tracked accurately through the degeneracy parameter. At the outer radius, ηe− starts to decrease as the temperature of the fluid rises. Once neutrino cooling becomes significant, it increases until the disk reaches the local balance between heating and cooling. At this point, ηe− stops rising and is maintained (approximately) at a constant value. Very close to rin , the zero torque condition of the disk becomes important, and the viscous heating is reduced drastically. This is reflected in a sharp decrease in the fluid’s temperature and an increase in the degeneracy parameter. For high accretion rates, additional effects have to be taken into account. Due to high

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Fig. 2. Distribution of mass fractions in accretion disks in the absence of oscillations with M = 3M , α = 0.01, a = 0.95.

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Fig. 3. Thermodynamic properties of accretion disks in the absence of oscillations with M = 3M , α = 0.01, a = 0.95.

νe optical depth, neutrino cooling is less efficient, leading to an increase in temperature and a second dip in the degeneracy parameter. This dip is not observed in low accretion rates because τνe does not reach high enough values. With the information in Figs. 3 and 4, we can obtain the oscillation potentials which we plot in Fig. 6. Since the physics of the disk for r < rign is independent of the initial conditions at the external radius and for r > rign the neutrino emission is negligible, the impact of neutrino oscillations is significant only inside rign . We can see that the discussion at the end

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Fig. 4. Properties of neutrinos in accretion disks in the absence of oscillations with M = 3M , α = 0.01, a = 0.95.

of Sec. 3 is justified since for rin < r < rign the potentials obey the relation

ω  μ  λ.

(82)

Generally, the full evolution of neutrino oscillations is a rather complex interplay between the three potentials, yet it is possible to understand the neutrino response in the disk using some numerical and algebraic results obtained in Esteban-Pretel et al. (2007); Fogli et al. (2007) and Hannestad et al. (2006) and references therein. Specifically, we know that if μ  ω, as long as the MSW condition λ  ω is not met (precisely our case), collective effects should dominate the neutrino evolution even if λ  μ. On the other hand, if μ  ω, the neutrino evolution is driven by the

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Fig. 5. Total optical depth (left scale) and mean free path (right scale) for neutrinos ˙ = 1M s−1 , 0.1M s−1 , and anti-neutrinos of both flavors for accretion disks with M 0.01M s−1 between the inner radius and the ignition radius.

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Fig. 6. Oscillation potentials as functions of r with M = 3M , α = 0.01, a = 0.95 for ˙ = 0.1M s−1 and M ˙ = 0.01M s−1 , respectively. ˙ = 1M s−1 , M accretion rates M The vertical line represents the position of the ignition radius.

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relative values between the matter and vacuum potentials (not our case). With Eq. (71), we can build a useful analogy. These equations are analogous to the equations of motion of a simple mechanical pendulum with a vector position given by S, precessing around with angular momentum D, subjected to a gravitational force ωμB with mass μ−1 . Using Eq. (81), we obtain the expression |S| = S ≈ 2 + O( ω/μ).

(83)

Calculating ∂t (S · S) it can be checked that this value is conserved up to fluctuations of order ω/μ. The analogous angular momentum is D = ¯ = 0. Thus, the pendulum moves initially in a plane defined by B P−P and the z-axis, i.e., the plane xz. Then, it is possible to define an angle ϕ between S and the z-axis such that S = S (sin ϕ, 0, cos ϕ).

(84)

Note that the only non-zero component of D is y-component. From Eq. (71) we find ϕ˙ = μD, ˙ D = − ωS cos(ϕ + 2θ).

(85a) (85b)

The above equations can be equivalently written as ϕ¨ = −k 2 sin(2θ + ϕ),

(86)

where we have introduced the inverse characteristic time k by k 2 = ωμS,

(87)

which is related to the anharmonic oscillations of the pendulum. The role of the matter potential λ is to logarithmically extend the oscillation length by the relation (Hannestad et al., 2006)    k

ω −1 τ = −k ln 1+ . (88) 1/2 Sμ θ (k 2 + λ2 ) The total oscillation time can then be approximated by the period of an harmonic pendulum plus the logarithmic extension tosc =

2π + τ. k

(89)

Neutrino Flavor Oscillations in Gamma-Ray Bursts

The initial conditions of Eq. (81) imply  

ω ϕ (t = 0) = arcsin sin 2θ . Sμ

85

(90)

so that ϕ is a small angle. The potential energy for a simple pendulum is 2

V (ϕ) = k 2 [1 − cos (ϕ + 2θ)] ≈ k 2 (ϕ + 2θ) .

(91)

If k 2 > 0, which is true for normal hierarchy Δm2 > 0, we expect small oscillations around the initial position since the system begins in a stable region of the oscillation potential. The magnitude of flavor conversions is of the order ∼ ω/Sμ  1. We stress that normal hierarchy does not mean an absence of oscillations but rather imperceptible oscillations in Pz . No strong flavor oscillations are expected. On the contrary, for the inverted hierarchy Δm2 < 0, k 2 < 0 and the initial ϕ indicates that the system begins in an unstable position and we expect very large anharmonic oscillations. Pz ¯ z ) oscillates between two different maxima passing through a (as well as P ¯ z ) several times. This implies total flavor conversion: minimum −Pz (−P all electronic neutrinos (anti-neutrinos) are converted into non-electronic neutrinos (anti-neutrinos) and vice versa. This phenomenon is called bipolar oscillations in the literature (Duan et al., 2010). If the initial condition are not symmetric as in Eq. (81), the asymmetry is measured by a constant ¯ z < Pz or ς = Pz /P ¯ z if P ¯ z > Pz so that 0 < ς < 1. Bipolar ¯ z /Pz if P ς=P oscillations are present in an asymmetric system as long as the relation 1+ς μ τνx , then Fνeq = Fν and the equipartition is unnoticeable. But if 1 < τνx < τνe then Fνeq /Fν > 1. In our simulations, this fraction reaches values of 1.9 for M˙ = 1M s−1 to 2.5 for M˙ = 0.01M s−1 . The disk variables do not change at each point beyond a factor of order 5 in the most obvious case. However, these changes can be significant for cumulative quantities, e.g., the total neutrino luminosity and the total energy deposition rate into electron–positron pairs due to neutrino antineutrino annihilation. To see this, we perform a Newtonian calculation of these luminosities following (Janka, 1991), Kawanaka and Kohri (2012), Liu et al. (2017), Popham et al. (1999), Ruffert et al. (1997), Rosswog et al. (2003) and Xue et al. (2013), and references therein. The neutrino luminosity is calculated by integrating the neutrino cooling flux throughout both faces of the disk  rout Ccap Fνi rdr. (97) Lνi = 4π rin

The factor 0 < Ccap < 1 is a function of the radius (called capture function in Thorne (1974)) that accounts for the proportion of neutrinos that are re-captured by the BH and, thus, do not contribute to the total luminosity. For a BH with M = 3M and a = 0.95, the numerical value of the capture function as a function of the dimensionless distance x = r/rs is well fitted by  −1 0.3348 , Ccap (x) = 1 + 3/2 x

(98)

with a relative error smaller than 0.02%. To calculate the energy deposition rate, the disk is modeled as a grid of cells in the equatorial plane. Each cell k has a specific value of differential neutrino luminosity Δkνi = Fνki rk Δrk Δφk and average neutrino energy Eνi k . If a neutrino of flavor i is emitted from

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the cell k and an anti-neutrino is emitted from the cell k  , and, before interacting at a point r above the disk, each travels a distance rk and rk , then, their contribution to the energy deposition rate at r is (see Appendix A.3 for details) 2   Δkνi Δkν¯i  rk · rk k k 

Eνi  + Eν¯i  ΔQνi ν¯i kk = A1,i 2 1− rk rk2 rk rk       Δk Δk

Eνi k + Eν¯i k rk · rk 1 − +A2,i 2νi 2ν¯i . (99) rk rk

Eνi k Eν¯i k rk rk The total neutrino annihilation luminosity is simply the sum over all pairs of cells integrated in space   Lνi ν¯i = 4π ΔQνi ν¯i kk d3 r, (100) A k,k

where A is the entire space above (or below) the disk. In Table 1, we show the neutrino luminosities and the neutrino annihilation luminosities for disks with and without neutrino collective effects. In each case, flavor equipartition induces a loss in Lνe by a factor of ∼3, and a loss in Lν¯e luminosity by a factor of ∼2. At the same time, Lνx and Lν¯e are increased by a factor ∼10. This translates into a reduction of the energy deposition rate due to electron neutrino annihilation by a factor of ∼7, while the energy deposition rate due to non-electronic neutrinos goes Table 1. Comparison of total neutrino luminosities Lν and annihilation luminosities Lν ν¯ in MeV s−1 between disks with and without flavor oscillations for selected accretion rates. 1 M s−1

0.1 M s−1

0.01 M s−1

Without oscillations

Lν e Lν¯e Lν x Lν¯x Lνe ν¯e Lνx ν¯x

6.46 × 1058 7.33 × 1058 1.17 × 1058 1.17 × 1058 1.25 × 1057 1.05 × 1055

9.19 × 1057 1.08 × 1058 8.06 × 1055 8.06 × 1055 1.62 × 1055 1.27 × 1050

1.05 × 1057 1.12 × 1057 2.43 × 1055 2.43 × 1055 1.78 × 1053 8.68 × 1048

With oscillations

Lν e Lν¯e Lν x Lν¯x Lνe ν¯e Lνx ν¯x

1.87 × 1058 4.37 × 1058 7.55 × 1058 5.44 × 1058 1.85 × 1056 2.31 × 1056

2.47 × 1057 4.89 × 1057 7.75 × 1057 5.27 × 1057 1.78 × 1054 1.64 × 1054

4.29 × 1056 5.48 × 1056 6.71 × 1056 5.70 × 1056 3.53 × 1052 1.23 × 1052

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from being negligible to be of the same order of the electronic energy deposition rate. The net effect is to reduce the total energy deposition rate of neutrino annihilation by a factor of ∼ (3 − 5) for the accretion rates considered. In particular we obtain factors 3.03 and 3.66 for M˙ = 1 M s−1 and M˙ = 0.01 M s−1 , respectively and a factor 4.73 for M˙ = 0.1 M s−1 . The highest value corresponds to the intermediate value of the accretion rate because, in this case, there is a νe cooling suppression (τνe > 1), and the quotient τνe /τνx is maximal. By Eq. (96), the difference between the respective cooling terms is also maximal. In Fig. 10 we show the energy deposition rate per unit volume around the BH for each flavor with accretion rates M˙ = 1 M s−1 and M˙ = 0.1 M s−1 . There we can see the drastic enhancement of the non-electronic neutrino energy deposition rate and the reduction of the electronic deposition rate. Due to the double peak in the neutrino density for M˙ = 0.01 M s−1 case (see Fig. 4), the deposition rate per unit volume also shows two peaks. One at rs < r < 2rs and the other at 10 rs < r < 11 rs . Even so, the behavior is similar to the other cases.

6. Concluding Remarks The generation of an energetic e− e+ plasma seems to be a general prerequisite of GRB theoretical models for the explanation of the prompt (MeV) gamma-ray emission. The e− e+ pair annihilation produces photons leading to an opaque pair-photon plasma that self-accelerates, expanding to ultrarelativistic Lorentz factors of the order of 102 –103 (see, e.g., Preparata et al., 1998; Ruffini et al., 1999, 2000). The transparency of MeV-photons at large Lorentz factor and corresponding large radii is requested to solve the so-called compactness problem posed by the observed non-thermal spectrum in the prompt emission (Meszaros et al., 1993; Piran et al., 1993; Shemi and Piran, 1990). There is a vast literature on this subject, and we refer the reader to Berger (2014), Kumar and Zhang (2015), M´esz´aros (2002, 2006), and Piran (1999, 2004) and references therein for further details. Neutrino-cooled accretion disks onto rotating BHs have been proposed as a possible way of producing the above-mentioned e− e+ plasma. The reason is that such disks emit a large amount of neutrino and antineutrinos that can undergo pair annihilation near the BH (Chen and Beloborodov, 2007; Di Matteo et al., 2002; Gu et al., 2006; Janiuk and Yuan, 2010; Kawanaka and Mineshige, 2007; Kawanaka et al., 2013; Kohri

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and Mineshige, 2002; Kohri et al., 2005; Lee et al., 2005; Luo and Yuan, 2013; Narayan et al., 2001; Popham et al., 1999; Xue et al., 2013). The viability of this scenario clearly depends on the energy deposition rate of neutrino–anti-neutrinos into e− e+ and on the local (anti-)neutrino density and energy. We have here shown that, inside these hyperaccreting disks, a rich neutrino oscillations phenomenology is present due to the high neutrino density. Consequently, the neutrino/antineutrino emission and the corresponding pair annihilation process around the BH leading to electron– positron pairs are affected by neutrino flavor conversion. Using the thin disk and α-viscosity approximations, we have built a simple stationary model of general relativistic neutrino-cooled accretion disks around a Kerr BH that factors in a wide range of neutrino emission processes, nucleosynthesis and the dynamics of flavor oscillations. The main assumption relies on considering the neutrino oscillation behavior within small neighboring regions of the disk as independent from each other. Although a first approximation, this has allowed us to set the main framework to analyze the neutrino oscillations phenomenology inside neutrino-cooled disks. In the absence of oscillations, a variety of neutrino-cooled accretion disks onto Kerr BHs, without neutrino flavor oscillations, have been modeled in the literature (see, e.g., Chen and Beloborodov, 2007; Gammie and Popham, 1998; Liu et al., 2017; Popham et al., 1999; Xue et al., 2013; for a recent review). The physical setting of our disk model follows closely the ones considered in Chen and Beloborodov (2007), but with some extensions and differences in some aspects: (i) The equation of vertical hydrostatic equilibrium, Eq. (24), can be derived in several ways (Abramowicz et al., 1997; Gammie and Popham, 1998; Novikov and Thorne, 1973). We followed a particular approach consistent with the assumptions in Novikov and Thorne (1973), in which we took the vertical average of a hydrostatic Euler equation in polar coordinates. The result is an equation that leads to smaller values of the disk pressure when compared with other models. It is expected that the pressure at the center of the disk is smaller than the average density multiplied by the local tidal acceleration at the equatorial plane. Still, the choice between the assortment of pressure relations is tantamount to a fine-tuning of the model. All these approaches are equivalent within the thin disk approximation since they all assume vertical equilibrium and neglect self-gravity.

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(ii) Following the BdHN scenario for the explanation of GRBs associated with Type Ic SNe (see Sec. 2), we considered a gas composed of 16 O at the outermost radius of the disk and followed the evolution of the ion content using the Saha equation to fix the local NSE. In Chen and Beloborodov (2007), only 4 He is present, and, in Xue et al. (2013), ions up to 56 Fe are introduced. The affinity between these cases implies that this particular model of disk accretion is insensible to the initial mass fraction distribution. This is explained by the fact that the average binding energy for most ions is very similar. Hence, any cooling or heating due to a redistribution of nucleons, given by the NSE, is negligible when compared to the energy consumed by direct photodisintegration of alpha particles. Additionally, once most ions are dissociated, the main cooling mechanism is neutrino emission which is similar for all models, modulo the supplementary neutrino emission processes included in addition to electron and positron capture. However, during our numerical calculations, we noticed that the inclusion of non-electron neutrino emission processes reduces the electron fraction by up to ∼ 8%. This effect is observed again during the simulation of flavor equipartition, alluding to the need for detailed calculations of neutrino emissivities when establishing NSE state. We obtain similar results to Chen and Beloborodov (2007) (see Figs. 2–4), but by varying the accretion rate and fixing the viscosity parameter. This suggests that a more natural differentiating set of variables in the hydrodynamic equations of an α-viscosity disk is the combination of the quotient M˙ /α and either M˙ or α. This result is already evident in, for example, Figs. 11 and 12 of Chen and Beloborodov (2007) but was not mentioned there. Concerning neutrino oscillations, we showed that the conditions inside the ignition radius, the oscillation potentials follow the relation ω  μ  λ, as illustrated in Fig. 6. We also showed that within this region, the number densities of electron neutrinos and anti-neutrinos are very similar. As a consequence of this particular environment, very fast pair conversions νe ν¯e  νx ν¯x , induced by bipolar oscillations, are obtained for the inverted mass hierarchy case with oscillation frequencies between 109 s−1 and 105 s−1 . For the normal hierarchy case, no flavor changes are observed (see Figs. 7 and 8). Bearing in mind the magnitude of these frequencies and the low neutrino travel times through the disk, we conclude that an accretion disk under our main assumption cannot represent a steadystate. However, using numerical and algebraic results obtained in EstebanPretel et al. (2007), Fogli et al. (2007), and Raffelt and Sigl (2007), and references therein, we were able to generalize our model to a more realistic

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picture of neutrino oscillations. The main consequence of the interaction between neighboring regions of the disk is the onset of kinematic decoherence in a timescale of the order of the oscillation times. Kinematic decoherence induces fast flavor equipartition among electronic and non-electronic neutrinos throughout the disk. Therefore, the neutrino content emerging from the disk is very different from the usually assumed (see, e.g., Liu et al., 2016; Malkus et al., 2012). The comparison between disks with and without flavor equipartition is summarized in Fig. 9 and Table 1. We found that flavor equipartition, while leaving anti-neutrino cooling practically unchanged, it enhances neutrino cooling by allowing the energy contained (and partially trapped inside the disk due to high opacity) within the νe gas to escape in the form of νx , rendering the disk insensible to the electron neutrino opacity. We give in Eq. (96) a relation estimating the change in Fν as a function of τνe τνx that describes the behavior of the disk under flavor equipartition. The variation of the flavor content in the emission flux implies a loss in Lνe and an increase in Lνx and Lν¯e . As a consequence, the total energy deposition rate of the process ν + ν¯ → e− + e+ is reduced. We showed that this reduction can be as high as 80% and is maximal whenever the quotient τνe /τνx is also maximal and the condition τνe > 1 is obtained. At this point, we can identify several issues which need to be investigated: Throughout the accretion disk literature, several fits to calculate the neutrino and neutrino annihilation luminosity can be found (see, e.g., Liu et al., 2017, and references therein). However, all these fits were calculated without taking into account neutrino oscillations. Since we have shown that oscillations directly impact luminosity, these results need to be extended. Additionally, the calculations of the neutrino and neutrino annihilation luminosities we performed ignore general relativistic effects and the possible neutrino oscillations from the disk surface to the annihilation point. In Salmonson and Wilson (1999), it has been shown that general relativistic effects can enhance the neutrino annihilation luminosity in a neutron star binary merger by a factor of 10. In Popham et al. (1999), however, it is argued that in BHs, this effect has to be mild since the energy gained by falling into the gravitational potential is lost by the electron–positron pairs when they climb back up. Nonetheless, this argument ignores the bending of neutrino trajectories and neutrino capture by the BH, which can be significant for r  10rs . In Birkl et al. (2007), the increment is calculated to be no more than a factor of 2 and can be less depending on the geometry of the emitting surface. But, as before, they assume a purely νe ν¯e emission and ignore oscillations after the emission. Simultaneously, the literature

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on neutrino oscillation above accretion disks (see, e.g., Malkus et al., 2012) does not take into account oscillations inside the disk and assumes only νe ν¯e emission. A similar situation occurs in works studying the effect of neutrino emission on r-process nucleosynthesis in hot outflows (wind) ejected from the disk (see, e.g., Caballero et al., 2012). It is still unclear how the complete picture (oscillations inside the disk → oscillations above the disk + relativistic effects) affect the final energy deposition. We are currently working on the numerical calculation of the annihilation energy deposition rate using a ray tracing code and including neutrino oscillations inside the disk, and after their emission from the disk surface. These results will be the subject of a future publication. Once the neutrino fluxes and luminosities are calculated, the question of detectability can be raised. Following Liu et al. (2016), for an event that emits 1059 MeV of total neutrino energy with an average neutrino energy of 30 MeV, we can estimate the radius of detection of Hyper-Kamiokande, LENA and JUNO to be less than 4.5 Mpc. Using the local BdHNe rate ∼ 1 Gpc−3 yr−1 (see, e.g., Ruffini et al., 2016) and the neutrino luminosities in Table 1, it is clear that the direct neutrino detection in these systems is quite improbable. However, it would still possible to indirectly identify the presence of oscillations. As we have seen, neutrino oscillations have an effect on the energy deposition rate of ν ν¯ annihilation. Additionally, as seen in Fig. 9, neutrino oscillations modify the electron fraction of the disk, which, in turn, leads to changes in the nucleosynthesis of disk winds (see, e.g., Caballero et al., 2012, 2014; Fujimoto et al., 2004; Janiuk, 2019). Both phenomena can be used as indirect probes of oscillation within the powering mechanisms of GRBs. An analysis of these effects will be presented elsewhere. Although the final behavior of a neutrino-dominated accretion disk with neutrino oscillations would be obtained by 8D neutrino transport simulations, since these simulations are already costly for systems of a high degree of symmetry, a first approximation is needed to identify key theoretical and numerical features involved in the study of oscillations in neutrino-cooled accretion disks. This chapter serves as a platform for such a first approximation. Considering that kinematic decoherence is a general feature of anisotropic neutrino gases, with the simplified model presented here, we were able to obtain an analytical result that agrees with the physics understanding of accretion disks. The unique conditions inside the disk and its geometry lend themselves to varied neutrino oscillation phenomena that

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can have an impact on a wide range of astrophysical systems: from e− e+ plasma production above BHs in gamma-ray bursts models to r-process nucleosynthesis in disk winds and possible MeV neutrino detectability. As such, this topic deserves appropriate attention since it paves the way for new astrophysical scenarios for testing neutrino physics. A. Appendix A.1. Transformations and stress–energy tensor For the sake of completeness, here we give explicitly the transformation used in Eq. (10) and the Christoffel symbols used during calculations. The coordinate transformation matrices between the CF and the LNRF on the tangent vector space are (Bardeen et al., 1972) ⎛ ⎞ √ 2 1 0 0 0 ω gφφ −gtt ⎜ ⎟ √1 ⎜ 0 0 ⎟ 0 grr ⎟, (A.1a) eνˆμ = ⎜ ⎜ 0 0 √g1θθ 0 ⎟ ⎝ ⎠ √ 2ω 0 0 √g1φφ ω gφφ −gtt

eνˆμ

⎛ 2 ⎞ ω gφφ − gtt 0 0 0 √ ⎜ grr 0 0 ⎟ 0 ⎟, =⎜ √ ⎝ gθθ 0 ⎠ 0 0 √ √ 0 0 gφφ −ω gφφ

(A.1b)

so that the basis vectors transform as ∂ νˆ = eμν˜ ∂ μ , that is, with eT . For clarity, coordinates on the LNRF have a caret (xμˆ ), coordinates on the ˜ CRF have a tilde (xμ˜ ) and coordinates on the LRF have two (xμ˜ ). An observer on the LNRF sees the fluid elements move with an azimuthal ˆ velocity β φ . This observer then can perform a Lorentz boost Lβ φˆ to a new frame. On this new frame, an observer sees the fluid elements falling radially with velocity β r˜, so it can perform another Lorentz boost Lβ r˜ to the LRF. Finally, the transformation between the LRF and the CF ˜ ˜ coordinates xμ = eρˆμ (Lβ φˆ )α˜ρˆ(Lβ r˜ )ν˜˜α˜ xν˜ = Aν˜˜μ xν˜ . To obtain the coordinate transformation between the CF and the CRF Aν˜μ and Aν˜μ we can simply ˜ set β r˜ = 0 in the expressions for Aν˜˜μ and Aν˜μ . With this, we can calculate ˆ

uμ eφμ dφˆ ˆ = βφ = = dtˆ uν etˆ ν



gφφ (Ω − ω), ω 2 gφφ − gtt

(A.2)

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d˜ r=

√ 1 √ grr dr, dt˜ =  dt, dθ˜ = gθθ dθ. 2 −gtt − 2Ωgtφ − Ω gφφ

(A.3)

Using the connection coefficients and the Kerr metric, both evaluated at the equatorial plane, we can collect several equations for averaged quantities. The expansion of the fluid world lines is θ = ∇μ uμ =

2 r u + ∂r u r . r

(A.4)

There are several ways to obtain an approximate version of the shear tensor (see, e.g., Gammie and Popham, 1998; Moeen, 2017; Moghaddas et al., 2012), but by far the simplest one is proposed by Novikov and Thorne (1973). On the CRF, the fluid 4-velocity can be approximated by uμ˜ = (1, 0, 0, 0) by Eq. (11). Both the fluid 4-acceleration aν = uμ ∇μ uν and expansion parameter, Eq. (A.4), vanish so that the shear tensor reduces to 2σμ˜ ν˜ = ∇μ˜ uν˜ + ∇ν˜ uμ˜ . In particular, the r–φ-component is   1 1  t˜ ˜ φ r˜ t˜ = − Γ φ˜ 2c + Γ + 2c ˜ ˜r ˜ t˜φ r˜φ t˜r˜ 2 4 2 √ gφφ 1 ˜ γφˆ  = cr˜t˜φ = √ ∂r Ω, 2 2 ω 2 gφφ − gtt grr

σr˜φ˜ = −

(A.5)

where cμ˜ν˜α˜ are the commutation coefficients for the CRF. Finally, of par˜ ticular interest is the θ-component of the Riemann curvature tensor  ˜  R θ t˜θ˜t˜ 

θ=π/2

=

M r2 − 4aM 3/2 r1/2 + 3M 2 a2 , r3 r2 − 3M r + 2aM 3/2 r1/2

(A.6)

which gives a measurement of the relative acceleration in the θ˜ direction of nearly equatorial geodesics. Here we present some equations related to the stress–energy tensor. Equation (18) for a zero bulk viscosity fluid in components is Tνμ = Πuμ uν + P δνμ − 2ησνμ + q μ uν + qν uμ .

(A.7)

Its covariant derivative vanishes and it is equal to     Π qν μ μ ∇μ Tν = u uν ∂μ Π − ∂μ ρ − ∂μ ρ + Πaν ρ ρ +∂ν P − 2η∇μ σνμ + q μ ∇μ uν + uν ∇μ q μ + uμ ∇μ qν ,

(A.8)

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where baryon conservation was used ρθ = −uμ ∂μ ρ. To get an equation of motion for the fluid, we project along the direction perpendicular to uν qβ Pβν ∇μ Tνμ = − uμ ∂μ ρ + Πaβ + ∂β P − 2η∇μ σβμ + q μ ∇μ uβ + uμ ∇μ qβ ρ +uβ (uν ∂ν P + 2ησ μν σμν − qν aν ),

(A.9)

where the identities qμ uμ = uμ aμ = σ μν uν = 0, uμ uν = −1, σ μν σμν = σ μν ∇μ uν are used. Combining the Eqs. (A.8) and (A.9) we get   U +P μ ∂μ ρ = 2ησ μν σμν − qμ aμ − ∇μ q μ . (A.10) u ∂μ U − ρ With Eq. (A.4), we can obtain an equation for mass conservation   2 r 0 =∇μ (ρuμ ) = uμ ∂μ ρ + ρθ = uμ ∂μ ρ + ρ u + ∂r u r r  2 r 2 j ⇒ ∂r r ρu + r u ∂j ρ = 0, for j ∈ {t, θ, φ}. (A.11) Finally, we reproduce the zero torque at the innermost stable circular orbit condition that appears in Page and Thorne (1974). Using the killing vector fields ∂ φ , ∂ t , and the approximation Π ≈ ρ, we can calculate √  1 0 = ∇ · (T · ∂ φ ) = ∇μ Tφμ = √ ∂μ −gTφμ , −g  1  r ≈ 2 ∂r ρu uφ r2 − 2ησφr r2 + uφ ∂θ q θ , r   ⇒ ∂r ρur uφ r2 − 2ησφr r2 = −r2 uφ ∂θ q θ ,   M˙ r uφ + 4rHησφ = 2Huφ , ⇒ ∂r (A.12) 2π where the last equality is obtained after integrating vertically and using Eq. (25). Analogously, using Eq. (20), for ∂ t we obtain the relation   M˙ ut − 4rHΩησφr = 2Hut . (A.13) ∂r 2π The vertical integration of the divergence of the heat flux is as follows: Since, on average, q = q θ ∂ θ , we have ∇μ q μ = ∂θ q θ and transforming, ˆ ˆ

q θ = rq θ . Vertically integrating yields  θmax θmax ˜ ˜ ∂θ q θ rdθ = r q θ  = 2q θ = 2H, θmin

θmin

(A.14)

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where q θ is the averaged energy flux radiating out of a face of the disk, as measured by an observer on the LRF, which we approximate as the halfthickness of the disk H times the average energy density per unit proper time  lost by the disk. With the variable change z = 8πrHησφr /M˙ and ˙ the equations reduce to y = 4πH/M, ∂r (uφ + z) = yuφ , ∂r (ut − Ωz) = yut .

(A.15)

Using the relation ∂r ut = −Ω∂r uφ (see Eq. (10.7.29) in Zeldovich and Novikov, 1971) and ∂r (ut + Ωuφ ) = uφ ∂r Ω, we can combine the previous equations to obtain z=−

  y (ut + Ωuφ ) , ∂r AB 2 = B∂r uφ , ∂r Ω

(A.16)

with A = y/∂r Ω and B = ut + Ωuφ . To integrate these equations, we use the zero torque condition z(r = r∗ ) = 0, where r∗ is the radius of the innermost stable circular orbit, which gives the relation  r ∂r Ω (ut + Ωuφ ) ∂r uφ dr y= 2 (ut + Ωuφ ) r∗    r ∂r Ω r = (A.17) ut uφ |r∗ − 2 uφ ∂r ut dr , 2 (ut + Ωuφ ) r∗ or, equivalently, with ρ ≈ Π 8πHrΠνturb σφr

   r M˙ r =− ut uφ |r∗ − 2 uφ ∂r ut dr . (ut + Ωuφ ) r∗

(A.18)

Using Eq. (10), the approximation γr˜ ≈ 1 and the variable change r = M x2 , the integral can be easily evaluated by partial fractions 8πHrρνturb σφr = M˙ M f (x, x∗ ), f (x, x∗ ) =

x3/2 

√

x3 + a x3 − 3x + 2a

 x  1  ax2 − 2x + a 3 i i ln × x − x − a ln ∗ + 2 x 2 x2i − 1 3

∗

(A.19a)



i=1

x − xi x∗ − xi

 ,

(A.19b) where x1 , x2 , x3 are the roots of the polynomial x3 − 3x + 2a.

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A.2. Neutrino interactions and cross-sections In this appendix, we include the neutrino emission rates and neutrino crosssections used in the accretion disk model. These expressions have been covered in Bruenn (1985), Burrows et al. (2006), Burrows and Thompson (2004), Dicus (1972), Ruffert et al. (1996), Tubbs and Schramm (1975), and Yakovlev et al. (2001). We also include the expression energy emission rate for ν ν¯ annihilation into electron–positron pairs. Whenever possible, we write the rates in terms of generalized Fermi functions since some numerical calculations were done following Aparicio (1998). We remind the reader that all expressions are written in Planck units. The numerical values of the constants can be found in Particle Data Group (2018). A.2.1. Neutrino emissivities • Pair annihilation e− + e+ → ν + ν¯ This process generates neutrinos of all flavors but around 70% are electron neutrinos (Becerra et al., 2018). This is because the only charged leptons in the accretion systems we study are electrons and positrons, so the creation of electron neutrinos occurs through charged or neutral electroweak currents, while the creation of non-electronic neutrinos can only occur through neutral currents. Using the electron or positron 4-momentum p = (E, p), the Dicus’ cross-section for a particular flavor i is (Dicus, 1972) σD,i =

G2F [C+,i (m4e + 3m2e pe− ·pe+ + 2 (pe− ·pe+ )2 ) 12πEe− Ee+   +3C−,i m4e + m2e pe− ·pe+ ]. (A.20)

The factors C±,i are written in terms of the weak interaction vector and axial-vector constants (Particle Data Group, 2018) 2 2 ± Ca,i , Cv,e = 2 sin2 θW + 1/2, C±,i = Cv,i

(A.21)

Ca,e = 1/2, Cv,x = CV,e − 1, Ca,x = CA,e − 1,

(A.22)

where the factor sin2 θW is the Weinberg angle. In the following, fe− (fe+ ) represent the Fermi–Dirac distribution for electrons (positrons) with ηe∓ denoting the electron (positron) degeneracy parameter including its rest mass. The number and energy emission rates can be calculated by replacing

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Λ = 2 and Λ = Ee− + Ee+ in the integral (Yakovlev et al., 2001):  4 ΛσD fe− fe+ d3pe− d3pe+ , 6 (2π)

(A.23)

giving the expressions Rνi +¯νi =

Qνi +¯νi =

G2F m8e [C+,i (8U1 V1 + 5U−1 V−1 + 9U0 V0 − 2U−1 V1 − 2U1 V−1 ) 18π (A.24a) +9C−,i (U−1 V−1 + U0 V0 )], G2F m9e [C+,i (8 (U2 V1 + U1 V2 ) + 7 (U1 V0 + U0 V1 ) 36π +5 (U−1 V0 + U0 V−1 ) − 2 (U2 V−1 + U−1 V2 )) +9C−,i (U0 (V1 + V−1 ) + V0 (U1 + U−1 ))].

(A.24b)

The functions U, V can be written in terms of generalized Fermi functions  j+1  √ 3/2  j+1 k Uj = 2ξ ξ Fk+1/2,0 (ξ, ηe− ), k k=0  j+1  √ 3/2  j+1 k Vj = 2ξ ξ Fk+1/2,0 (ξ, ηe+ ). k

(A.25a)

(A.25b)

k=0

It is often useful to define the function 4  2G2F (me ) = fe− fe+ (Eem− + Eem+ ) σD,i d3 pe− d3 pe+ . εm i 7 3 (2π)

(A.26)

For m = 0 and m = 1, Eq. (A.26) gives the neutrino and anti-neutrino number emissivity (neutrino production rate), and the neutrino and antineutrino energy emissivity (energy per unit volume per unit time) for a certain flavor i, respectively (that is, Eq. (A.24)). Hence, we can calculate the total number and energy emissivity and the neutrino or anti-neutrino εm energy moments with Eνmi (¯νi )  = εi0 , for m ≥ 1. i

• Electron capture and positron capture p+e− → n+νe and n+e+ → p+ ν¯e Due to lepton number conservation this process generated only electron (anti-)neutrinos. The number and energy emission rates for electron and

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positron capture by nucleons are as follows: Rνe =

 m5e G2F cos2 θc  2 √ 1 + 3gA Δnp ξ 3/2 [ξ 3 F7/2,χ (ξ, ηe− ) 2π 3 + (3 − 2Q) ξ 2 F5/2,χ (ξ, ηe− ) + (1 − Q) (3 − Q) ξF3/2,χ (ξ, ηe− ) 2

+ (1 − Q) F1/2,χ (ξ, ηe− )], Qνe =

(B.6a)

 m6e G2F cos2 θc  2 √ 1 + 3gA Δnp ξ 3/2 [ξ 4 F9/2,χ (ξ, ηe− ) 3 2π + ξ 3 (4 − 3Q) F7/2,χ (ξ, ηe− ) + 3 (Q − 1) (Q − 2) ξ 2 F5/2,χ (ξ, ηe− ) 2

3

+ (1 − Q) (4 − Q) ξF3/2,χ (ξ, ηe− ) + (1 − Q) F1/2,χ (ξ, ηe− )], (B.6b) Rν¯e =

 m5e G2F cos2 θc  2 √ 1 + 3gA Δpn ξ 3/2 [ξ 3 F7/2,0 (ξ, ηe+ ) 2π 3 + (3 + 2Q) ξ 2 F5/2,0 (ξ, ηe+ ) + (1 + Q) (3 + Q) ξF3/2,0 (ξ, ηe+ ) 2

+ (1 + Q) F1/2,0 (ξ, ηe+ )], Qν¯e =

(B.6c)

 m6e G2F cos2 θc  2 √ 1 + 3gA Δnp ξ 3/2 [ξ 4 F9/2,0 (ξ, ηe+ ) 3 2π + ξ 3 (4 + 3Q) F7/2,0 (ξ, ηe+ ) + 3 (Q + 1) (Q + 2) ξ 2 F5/2,0 (ξ, ηe+ ) + (1 + Q)2 (4 + Q) ξF3/2,0 (ξ, ηe+ ) + (1 + Q)3 F1/2,0 (ξ, ηe+ )]. (B.6d)

where Δij = (ni − nj ) / (exp (ηi − ηj ) − 1),

i, j ∈ {p, n},

(A.27)

are the Fermion blocking factors in the nucleon phase spaces, Q = (mn − mp )/me ≈ 2.531 is the nucleon mass difference, and cos2 θc is the Cabibbo angle. Since we do not consider ions above oxygen, the (anti-)neutrino production for electron and positron capture by an ion i is zero (Burrows et al., 2006; Burrows and Thompson, 2004). • Plasmon decay γ˜ → ν + ν¯

Rνe +¯νe =

Cv,e σ0 T 8 6 γ˜ (˜ γ + 1) exp (−˜ γ ), 96π 3 m2e α∗

(A.28a)

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Qνe +¯νe =

 Cv,e σ0 T 9 6  2 γ˜ γ˜ + 2˜ γ + 2 exp (−˜ γ ), 192π 3 m2e α∗

(A.28b)

Rνx +¯νx =

Cv,x σ0 T 8 6 γ˜ (˜ γ + 1) exp (−˜ γ ), 48π 3 m2e α∗

(A.28c)

Qνx +¯νx =

 Cv,x σ0 T 9 6  2 γ˜ γ˜ + 2˜ γ + 2 exp (−˜ γ ), 3 2 ∗ 96π me α

(A.28d)

where α∗ = 1/137 is the fine structure constant, σ0 = 4G2F m2e /π, with γ˜ = γ˜0

and γ˜0 = 2

α∗ 3π

2 (π 2 + 3 (ηe− + 1/ξ) )/3,

(A.29)

≈ 5.565 × 10−2 .

• Nucleon–nucleon bremsstrahlung n1 + n2 → n3 + n4 + ν + ν¯ The nucleon–nucleon bremsstrahlung produces the same amount of neutrinos of all three flavors. The number and energy emission rates can be approximated by (see, e.g., Burrows et al., 2006) Rνi +¯νi Qνi +¯νi

    28 13 2 2 = 2.59 × 10 Xp + Xn + Xp Xn n2B ξ 9/2 , 3     28 −9 2 2 = 4.71 × 10 Xp + Xn + Xp Xn n2B ξ 10/2 . 3

(A.30a) (A.30b)

A.2.2. Cross-sections We consider four interactions to describe the (anti-)neutrino total crosssection. • Neutrino annihilation (ν + ν¯ → e− + e+ ).

σνe ν¯e =

Eνe  Eν¯e  4 1 + 4 sin2 θW + 8 sin4 θW Kνe ν¯e σ0 , with Kνe ν¯e = 2 3 me 12 (A.31a)

σνx ν¯x =

4 1 − 4 sin2 θW + 8 sin4 θW

Eνx  Eν¯x  Kνx ν¯x σ0 , with K = ν ν ¯ x x 3 m2e 12 (A.31b)

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• Electron (anti-)neutrino absorption by nucleons (νe + n → e− + p and ν¯e + p → e+ + n).  σνe n = σ0

1 + 3ga2 4



 σν¯e p = 3.83 × 10

22

Eνe  +Q me

℘ Eν¯e  −Q me

2 & ' '1 −  (

 g(Eν¯e ) = −0.07056 + 0.02018 ln

2 & ' '1 −  (

℘ Eν¯e  me

1 Eνe

me

+Q



1 ℘ Eν¯e

me

2 ,

−Q

2



(A.32a) ℘ Eν¯e  me

 − 0.001953 ln3

g(Eν¯e ) ,

(A.32b)  ℘ Eν¯e  , me (A.32c)

where ℘ = 0.511. • (anti-)neutrino scattering by baryons (ν + Ai → ν + Ai and ν¯ + Ai → ν¯ + Ai ).   σ0 E2 1 + 3ga2 4 2 σp = 4 sin θW − 2 sin θW + , (A.33a) 4m2e 4 σ0 E2 1 + 3ga2 , (A.33b) σn = 4m2e 4    Zi σ0 A2i E2  Zi σAi = +1− 4 sin2 θW − 1 . (A.33c) 2 16me Ai Ai • (anti-)neutrino scattering by electrons or positrons (ν + e± → ν + e± and ν¯ + e± → ν¯ + e± ).    3

E 1 ηe + 1/ξ 2 2 σe = σ0 ξ 1+ (Cv,i + n Ca,i ) + (Cv,i − n Ca,i ) . 8 me 4 3 (A.34) Here, n is the (anti-)neutrino lepton number (that is, 1 for neutrinos and −1 for anti-neutrinos, depending on the cross-section to be calculated), and, in the last four expressions, E is replaced by the average (anti-)neutrino energy of the corresponding flavor. With these expressions, the total opacity for neutrinos or anti-neutrinos is ) σi ni , (A.35) κνi (¯νi ) = i ρ

J. A. Rueda and J. D. Uribe

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where ni is the number density of the target particle associated with the process corresponding to the cross-section σi . The (anti-)neutrino optical depth appearing in Eq. (40) can then be approximated as  τνi (¯νi ) = κνi (¯νi ) ρdθ ≈ κνi (¯νi ) ρH. (A.36) A.3. Neutrino–anti-neutrino pair annihilation Since the main interaction between ν ν¯ is the annihilation into e− e+ , this process above neutrino-cooled disks has been proposed as the origin of the energetic plasma involved in the production of GRBs. Once the (anti-)neutrino energy emissivity and average energies are calculated it is possible to calculate the energy deposition rate of the process νi + ν¯i → e− + e+ for each flavor i. Ignoring Pauli blocking effects in the phase spaces of electrons and positrons, the local energy deposition rate Qνi ν¯i at a position r by ν ν¯ annihilation can be written in terms of the neutrino and anti-neutrino distributions fνi = fνi (r, Eν ), fν¯ i = fν¯i (r, Eν¯ ) and the total intensity (energy integrated intensity) Iν = Eν3 fν dEν as (Janka, 1991; Ruffert et al., 1997)   2 Qνi ν¯i = A1,i dΩνi Iνi dΩν¯i Iν¯i ( Eνi  + Eν¯i ) (1 − cos θ) S2

S2



+A2,i



dΩνi Iνi S2

dΩν¯i Iν¯i S2

Eνi  + Eν¯i  (1 − cos θ), (A.37)

Eνi  Eν¯i 

where we have introduced the constants appearing in Eq. (99) * + 2 2 σ0 (Cv,i − Ca,i ) + (Cv,i + Ca,i ) A1,i = 12π 2 m2e  2  2 σ0 2Cv,i − Ca,i A2,i = . 6π 2 m2e

(A.38)

In Eq. (A.37), θ is the angle between the neutrino and anti-neutrino momentum, and dΩ is the differential solid angle of the incident (anti-)neutrino at r. The incident radiation intensity passing through the solid differential angle dΩ at r is the intensity Ird ,ν emitted from the point on the disk rd diluted by the inverse square distance rk = |r − rd | between both points. Finally, assuming that each point rd on the disk’s surface acts as a half-isotropic radiator of (anti-)neutrinos, the total flux emitted at rd is

Neutrino Flavor Oscillations in Gamma-Ray Bursts

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π/2 2π       Frd ,ν = 0 0 Ird ,ν cos θ sin θ dθ dφ = πIrd ,ν , with θ , φ the direction angles at rd . Collecting all yields  Qνi ν¯i = A1,i

 drd,νi

rd,νi ∈disk



+ A2,i

drd,¯νi

rd,¯ νi ∈disk

Frd ,νi Frd ,¯νi ( Eνi  + Eν¯i ) (1 − cos θ)2 2 2 rk,ν r k,¯ ν i i



drd,νi

rd,νi ∈disk

drd,¯νi

rd,¯ νi ∈disk

Frd ,νi Frd ,¯νi Eνi  + Eν¯i  (1 − cos θ). 2 2 rk,ν rk,¯

Eνi  Eν¯i  νi i (A.39)

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Chapter 3

Gamma-Rays and the New Multi-messenger Astrophysics Ulisses Barres de Almeida Brazilian Center for Physics Research (CBPF) Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, Brazil [email protected] Thirty years since the detection of the first TeV source, thanks to the successful observations by the Whipple Observatory in 1989, very-high-energy (VHE) gamma-ray astronomy has finally reached full maturity. The recent observational breakthroughs in the detection of sub-Teraelectronvolt emission from gamma-ray bursts, the searches for counterparts of the VHE neutrinos from IceCube, as well as the fast expansion of the sensitivity of observations at the highest energies, in its pursuit of Galactic PeVatrons and the origin of high-energy cosmic rays — all of these accomplishments reveal a picture of the high-energy cosmos that goes much beyond previous expectations, and demonstrate how far the field has evolved in terms of its technological capabilities. Ground-based gamma-ray astronomy is in fact a pivotal player in the nascent multi-messenger astronomy, bridging the non-electromagnetic events with the electromagnetic signals and consequently their associated astrophysical sources. In this contribution I will review the main recent observational results of the field, which are closely connected to the search for the origin of cosmic-rays, concentrating on detailing the efforts towards establishing the counterparts to multi-messenger events. I will also sketch the landscape of future experiments and observatories that will enter operation in the next decade, and the perspective they lay out for the field of Astroparticle Physics. Keywords: Ground-based gamma-ray astronomy; multi-messenger astronomy; astroparticle physics; high-energy astrophysics; gamma-ray bursts; astrophysical neutrinos.

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1. Introduction Thirty years since the detection of the first TeV source, very-high-energy (VHE) gamma-ray astronomy has finally reached its full maturity. The new and recent discoveries of pulsed TeV emission from the Crab pulsar, as well as the observation of photons of energies as high as 300 TeV from its associated nebula, offer solid testimony to how far the techniques for ground-based gamma-ray observations have evolved since the first successful observations by the Whipple Observatory in 1989. The recent breakthroughs in the detection of sub-teraelectronvolt emission from gamma-ray bursts (GRBs), crowning years of efforts in trying to observe these elusive sources with Cherenkov telescopes, are a major technological milestone, pushing the limits of the technique towards the lowest energies and the shortest timescales. Likewise, at the other extreme of the spectrum, the field is advancing fast in its pursuit of definite answers to the central question of the origin of high-energy cosmic rays. Here, the highest energy photons detected (Amenomori et al., 2021; Cao et al., 2021) start to hint at parent particle populations with energies of PeVs, pointing to sources (the so-called PeVatrons) which are capable of accelerating particles up to the knee of the cosmic-ray spectrum, providing a potentially complete explanation to their Galactic origins. Such accomplishments reveal a picture of the high-energy cosmos that goes much beyond previous expectations, with a large catalog of sources covering almost all types of astrophysical objects.1 The whole situation is even more exciting as this all happens amidst the dawn of the multi-messenger era in astronomy, and the identification of the first electromagnetic (astrophysical) counterparts to neutrinos and gravitational wave events. In this contribution I will first review the main recent observational results of the field, such as mentioned above, which are closely connected to the search for the origin of cosmic-rays, concentrating on detailing the efforts towards establishing the counterparts to multi-messenger signals. Then, I will sketch the landscape of future experiments and observatories that will enter operation in the next decade, and the perspective they lay out for a Roaring Twenties 2 in the field of Astroparticle Physics.

1 See

the complete, up-to-date TeV catalog at Wakel and Horan (2021). term refers to the post-War/post-Pandemic decade of the 1920s in Western Society which was an unparalleled period of economic prosperity with implications to a particularly distinctive cultural dynamism in large cities in Europe and the United States. Beyond gamma-ray astrophysics, are we set to see something similar in society on this decade, once we are free from the COVID-19 Pandemic? 2 The

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2. The Crab Anniversary The pulsar-powered wind nebula (PWNe) are bright sources from the radio to the gamma-rays, which take their luminosity from the (very high) spindown power of pulsars: Lspin = IΩΩ˙ ∼ 105 L , for I ≈ 1045 g.cm2 . They constitute the closest relativistic objects to Earth, presenting strong relativistic jets, and are the most numerous and brightest class of galactic TeV sources, being important e± cosmic-ray accelerators. Generally speaking (e.g. see SNR G21.5-0.9), PWNe ate initially confined structures within their parent supernova remnants (SNRs), whose winds expand nonrelativistically into the medium (Reynolds et al., 2017). The first astrophysical source detected at TeV energies, little over 30 years ago, was the Crab Nebula, the brightest, and one of the youngest (a few thousand years, with a luminosity Lspin > 1036 erg.s−1 ) among the known PWNe (Weekes et al., 1989). Its observational history in very-high energies is a testimony to the evolution of ground-based gamma-ray astronomy in the past decades. The Crab, whose nebular emission is used as a standard candle in TeV gamma-rays, has also had its pulsed emission directly detected in sub-TeV gamma-rays by the MAGIC Collaboration (Ansoldi et al., 2016), thanks to new trigger techniques that have pushed the energy threshold of the observations well below 100 GeV. Another recent result from the Crab, which exemplifies the major technological achievements of the field, and in particular of the imaging atmospheric Cherenkov technique (IACT), which has achieved impressive resolving power, was the measure, by the H.E.S.S. Collaboration, of its extension at TeVs, at the sub-arcminute level: 52.2 ± 2.9 (stat) ± 6.6 (sys) (H.E.S.S. Collaboration, 2020).

Fig. 1. Multi-wavelength view of the Crab Nebula. Radio: M. Bietenholz, J.M. Uson and T.J. Cornwell (Very Large Array — NRAO/AUI); Infrared: R. Gehrz (Spitzer — NASA/JPL-Caltech); Visible: J. Hester and A. Loll (Hubble — NASA, ESA); Ultraviolet: E. Hoversten (UVOT — NASA/Swift); X-ray: F. Seward et al. (Chandra — NASA/CXC/SAO); Gamma: R. Buehler (LAT — NASA/DOE/Fermi). Image composition from Wikipedia: Hubble Space Telescope (HST, 2021).

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The Crab nebula dossier (for which an excellent dedicated review was produced by Buehler and Blandford (2014)), as briefly presented below, serves indeed as a proxy for the evolution of the field itself: • 1989: Whipple discovery of TeV gamma-ray emission from the Crab Nebula (Weekes et al., 1989); • 2008: MAGIC discovery of pulsed gamma-rays above 25 GeV from the Crab Pulsar (Aliu et al., 2008); • 2010: AGILE and Fermi-LAT detection of enhanced gamma-ray emission from the Crab Nebula (Tavani et al., 2011); • 2011: VERITAS detection of pulsed gamma-rays from the Crab Pulsar above 100 GeV (VERITAS Collaboration, 2011); • 2016: MAGIC detection of Teraelectronvolt pulsed emission from the Crab Pulsar (Ansoldi et al., 2016); • 2019: HAWC and Tibet measurement of the Crab Nebula Spectrum past 100 TeV (Abeysekara et al., 2019; Amenomori et al., 2019), which is well fit by a log parabola shape with emission up to at least 100 TeV, without the presence of any cutoff; • 2020: H.E.S.S. resolves the Crab PWNe at TeV energies (H.E.S.S. Collaboration, 2020), reporting an angular extension at gamma-ray energies of 52 arcseconds, significantly larger than at X-rays; • 2021: LHAASO observations of PeV photons from the Crab Nebula (Cao et al., 2021); 2.1. The success of the imaging atmospheric technique Very-high-energy (VHE) gamma-ray astronomy is the most recent field of observational Astrophysics, being firmly established in the past couple of decades. Over the last 15 years, the so-called third generation of ground-based gamma-ray observatories (MAGIC (Rico et al., 2016), H.E.S.S. (H.E.S.S. Collaboration, 2018a) and VERITAS (Park et al., 2015)) have made a number of exciting new discoveries, which have truly consolidated this new Teraelectronvolt window into the universe. Today, the field counts more than 200 detected sources, and a number of different classes of gamma-ray emitters have been identified. Ground-based gamma-ray astronomy is based mainly on two technologies, the most developed of which is the Imaging Atmospheric Cherenkov Technique (IACT), a very matured method responsible for the almost totality of the detections in the TeV energy range as of today, and which has reached the level of precision measurements. The basic concept of the

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technique is based on the fact that, when a gamma-ray of very-high energy impinges the Earth’s atmosphere, it creates a cascade of highly energetic charged particles, which move super-luminally through the atmosphere, producing Cherenkov radiation as a result — i.e. light in the visible range of the spectrum that can be collected by the large reflective surfaces of the telescopes onto large cameras based on photo-multiplier detectors. These telescopes are characterized by having a very fast electronics readout system, which allows them to be able to detect the very fast Cherenkov flashes from the showers, which last for only a few nanoseconds. Current instrumentation is based on arrays of such imaging Cherenkov telescopes working together for stereoscopic observations of the air-showers, which was proved to improve significantly the angular resolution, thanks to the “multi-perspective” view it provides in reconstructing the shower’s arrival direction. On the other hand, the presence of multiple telescopes observing a same event allows to reach √ a deeper sensitivity, which improves, at first approximation, with the N , N being the number of telescopes triggered. Some of these most recent breakthrough results, such as the detection of TeV emission from GRBs or the detection of a potential counterpart to very-high energy neutrinos, will be detailed in this contribution, as we look forward to the potential contributions of gamma-ray observations to the new multi-messenger astrophysics. But before we proceed to that, let us briefly outline a few general results that are central to the field, and testimony to the fact that it has finally reached the level of “real astronomy”, meaning that precision measurements at the TeV energy scale are now possible, even if only for the brightest sources. The first of these recent breakthroughs we would like to highlight here is associated to the search for PeVatrons. The cosmic-ray spectrum as measured from Earth shows two distinct features, the so-called knee, located at energies of a few-PeV, and the ankle, at still higher energies. Cosmic-ray particles below the knee are thought to originate from within our Galaxy, and PeVatrons, the conjectured cosmic-ray factories that would be able to accelerate particles up to these energies, are therefore crucial to understand the origin of Galactic cosmic-rays. Generally speaking, cosmic-ray sources can be detected by observing gamma-ray emission coming from their direction. The accelerated particles, upon leaving the sources, interact with target material (usually in Molecular Clouds), and gamma-rays are then produced as a result of proton–proton interactions. Grosso modo, 1 PeV protons would leave a

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gamma-ray signature at around 100 TeV, so that, phenomenologically, PeVatron sources are expected to produce hard gamma-ray power-law spectra extending up to 50 TeV and beyond. The first such PeVatron candidate firmly detected is at the Galactic Center region, where diffuse gamma-rays were measured with energies above 50 TeV coming from the dense molecular Cloud region around Sgr A* (H.E.S.S. Collaboration, 2016). The Galactic Center region in fact harbors a number of potential PeVatron candidates beyond the central supermassive black hole, that could explain this diffuse emission, such as supernova remnants (SNRs), stellar clusters and starforming regions, amongst others. In the standard scenario, Supernova Remnants were assumed to be the sources of cosmic-rays, up to PeV energies, but the currently-running facilities have shown that at least the bright known supernovae show a cutoff around 10–20 TeV (Cristofari et al., 2018), posing a stringent question as to what are the nature of PeVatrons. Possibilities still include SNR, but perhaps only during a limited period, at the earliest (100 yr) stages of their lives (SN1987A would be an optimal test case in this sense); or maybe the signature for PeV particles should be looked for within molecular clouds illuminated by the escaping CRs (Aharonian, 2001; Casanova et al., 2010). Other alternative sources could involve star-forming regions (Aharonian et al., 2019), as already mentioned, or even, and more excitingly, unknown sources that could be detected by means of unbiased surveys, such as the one CTA plans to carry out for the Galactic Plane (Anguner et al., 2019), or by wide-field instruments, such as LHAASO (Cao et al., 2021). In addition to the PeVatrons and the extreme energies, the opening up of the angular resolution, and the detection of extended/diffuse emission is another important developmental milestone of the field. Old pulsar wind nebulae, such as Vela X (Aharonian et al., 2006), and Monogem and Geminga (Abeysekara et al., 2017), hosting middle-aged (>10 kyr) energetic rotation-powered pulsars (with Lspin > 1036 erg.s−1 ) — some of which are primarily observed in TeV gamma-rays — are shown to be much larger (extended) in these VHE than in X-rays, for example, pointing to the existence of long-hypothesized so-called TeV halos, that only now tart being uncovered by observations (Sudoh et al., 2019) thanks to instruments such as HAWC. Such morphological characteristics make old PWNe very interesting as the most promising candidates for being the counterparts to the TeV unidentified sources, thanks also to their very high TeV to X-ray flux. In fact, a large number (47 out of 78 objects reported by H.E.S.S. in its galactic

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plane survey (H.E.S.S. Collaboration, 2018a)) of TeV sources are unidentified, in the sense that they do not have established resolved counterparts in lower energies, showing a lot of potential remains in store for future studies by instruments like CTA. But before we discuss future prospects, we would like to spend some time describing two sets of observational results from the side of transient astrophysics which have put VHE gamma-ray astronomy right at the center of the multi-messenger landscape. The first of these concerns the TeV detection of GRBs, which are interesting not only for representing the opening up the field to extreme and fast transients, but also because short-GRBs are counterparts to gravitational wave events; the second set of results is related to the search for astrophysical counterparts to VHE neutrinos, such as detected by IceCube.

3. The First Gamma-ray Bursts at sub-TeV Energies A very recent, and long-awaited revolution in VHE Gamma-ray Astronomy came from the recent observation of GRBs by ground-based observatories. Tremendous efforts in the building of fast-slewing telescopes, and aggressive operation strategies for following-up triggers, as well as remote operation techniques, have paid off, with the detection of four GRBs by the H.E.S.S. and MAGIC observatories in little more than two years. Two of them, GRB 180720B (Abdalla et al., 2019), and GRB 190829A (Chand et al., 2020; de Naurois, 2019) were detected deep in the afterglow phase, several hours after the initial burst — t0 + 10h for the case of GRB 180720B and t0 + 4h20 for GRB 190829A — which was unexpected by most models. In contrast, another source, GRB 190114C (MAGIC Collaboration, 2019), was detected only 50 s after the burst and showed a very intense signal during the first 300 s of observations. In fact, this GRB was so strong that could be detected even with a relatively high observation energy threshold of 300 GeV (due to low elevation conditions during moonlight). The very large collection area that ground-based observatories offer meant that many thousands of photons have been detected from this source, resulting in sub-minute timescale light-curve and unprecedented gamma-ray spectral resolution. Finally, the fourth and last among the long-GRB detected at TeVs (which are most likely sources associated with the collapse of a massive star), GRB 201216C, has recently been announced by the MAGIC Collaboration, at similar observational conditions, around 50 s after the burst (Blanch et al., 2020).

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It is relevant to mention as well that MAGIC has obtained a 3σ hint from the short GRB 160821B (MAGIC Collaboration, 2021). Despite the low statistical significance, the result is of great observational significance, as short GRBs are associated to the mergers of binary neutron star systems, and therefore connected to gravitational wave emission events. In the following, we will briefly discuss the late-afterglow GRB observations made by H.E.S.S., before presenting the near-prompt results obtained by the MAGIC collaboration. 3.1. Late-afterglow GRB detections by H.E.S.S. GRB 180720B was an extremely bright3 long burst, at a redshift z = 0.65. Due to observational constraints, data taking by H.E.S.S. started only 10 hours after the burst, but even so yielded a surprising ∼5.3σ post-trials detection (Abdalla et al., 2019). After taking into account absorption by pair-creation on the extragalactic radiation field, the high-energy spectrum of the burst appeared to be very hard, with a derived photon index of Γ = 1.6±0.2. The VHE emission is currently interpreted as an inverse-Compton component. As can be seen from Fig. 2, the energy flux of the VHE emission appears similar to the Fermi-LAT 100 MeV–10 GeV flux, and the FermiGBM 8 keV–10 MeV emission. Likewise, the afterglow falling rate is similar in all measured bands, both in gamma-rays as well as in X-rays and optical, implying that all emission components are directly connected. The low-luminosity GRB 190829A was another long gamma-ray burst, but with completely different properties to the previous source. Despite being rather modest in terms of its released energy, with very low-luminosity in prompt gamma-rays, it was a very nearby event, with a redshift z = 0.078, which makes it one of the closest GRBs ever observed. Here, again, H.E.S.S. observations started deep in the afterglow phase, 4 hours after the burst, and yielded a very intense signal extending to above 1 TeV, which lasted for three nights, until it faded below 3σ afterwards. Differently therefore from the prompt emission, the afterglow was very energetic, and it was proposed by some authors that the peculiar event can be explained by the off-axis jet scenario (Sato et al., 2021). Such a model can explain why the prompt gamma-ray emission was weak, and still reconcile that with the afterglow light-curves and an SSC flux of ∼10−11 erg/cm2 /s, dominating for the first few 10 ks of the event. Once again, the afterglow falling rate was 3 In fact, the second brightest afterglow ever measured by Swift-XRT, and the seventh brightest prompt emission detected by Fermi-GBM.

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Fig. 3. Gamma-ray and X-ray light curves for GRB 190114C. In red, the photon flux light curve above 0.3 TeV measured by MAGIC (from T0 + 62 s to T0 + 210 s), compared with the emission between 15 and 50 keV measured by Swift-BAT in gray (from T0 to T0 + 210 s) is shown. The blue dashed line shows the photon flux above 0.3 TeV of the Crab Nebula for reference. Image from MAGIC Collaboration (2019).

only detectable for little over half an hour, after which it faded beyond the instrumental sensitivity. In addition to the detection by MAGIC, the source was observed simultaneously at many wavelengths (MAGIC Collaboration et al., 2019). Figure 3 shows the multi-wavelength light-curve for the source covering 20 orders of magnitude in energy and showing, in addition to MAGIC data, observations by various instruments in radio, optical and X-rays. The multi-band data was used to confirm that the gamma-ray flux seen by MAGIC was coming from inverse-Compton emission in the afterglow component, an emission feature that had never before been firmly identified from a GRB, and which was later confirmed by the H.E.S.S. detections as mentioned earlier. According to the detailed modeling presented by the MAGIC Collaboration, the VHE gamma-rays were the result of synchrotron self-Compton emission (SSC), and this recent discovery has opened a new window to probe the particle acceleration and radiation mechanisms of GRBs. The data allowed to constrain the acceleration efficiency, characterized by the Bohm factor4 — which measures the deviation from the acceleration in the Bohm limit η = 1, that is, for which particle scattering occurs elastically 4 The

Bohm factor η = λmfp /rg is the ratio of the particle mean free path and its giroradius.

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off turbulence in the local fluid frame — that was shown to be required to be η < 100. Numerical simulations by another group of authors (Asano et al., 2020) also showed that the energy fraction carried by thermal electrons was >10%, demonstrating that the VHE observations are very effective in allowing to probe the particle populations and acceleration mechanisms in GRBs. 3.3. The short Gamma-ray burst GRB 160821B Finally, a last important example concerns the short gamma-ray burst GRB 160821B which was tentatively detected by MAGIC with a significance of about 3σ. Despite the low-significance result, a closer look on this source is of interest, as short GRBs are the potential electromagnetic counterparts of gravitational wave events.

Fig. 4. Multi-wavelength energy flux vs. time for GRB 190114C on 14 January 2019. The MAGIC light curve for the energy range 0.3 TeV (green circles) is compared with light curves at lower frequencies: in radio, the measurements by VLA, ATCA, ALMA, GMRT and MeerKAT have been multiplied by 109 for clarity. The vertical dashed line marks the end of the prompt-emission phase, identified as the end of the last flaring episode. Image reproduced from MAGIC Collaboration et al. (2019).

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For this particular case, despite the synchrotron self-Compton flux had to be low to explain the gamma-ray data, the source showed extended emission, which again suggested a long-lasting activity of the central engine. In this case, it is reasonable to expect significant external inverse-Compton high-energy emission, which was found to be able to explain the MAGIC data (Murase et al., 2018). Such result is very encouraging to the search for the electromagnetic counterpart of gravitational wave events in the future, especially with CTA, as demonstrated by Zhang et al. (2021). 4. The Era of Multi-messenger Astrophysics Multi-messenger studies have been greatly increasing in activity within the Gamma-ray Astronomy community over the past few years, as the observatories embark in numerous follow-up programs of both high-energy neutrino triggers from IceCube, and of gravitational wave events from LIGO–VIRGO, and below we will detail some of the discoveries and perspectives in this new and rich field of astrophysics. 4.1. Blazars as potential counterparts to VHE neutrinos Active galactic nuclei (AGN), especially blazars, are the most numerous extragalactic gamma-ray sources. Thanks to their brightness and cosmological distances, they are important for a number of studies such as the measurement of the extragalactic background light (EBL) and intergalactic magnetic fields, as well as for probing fundamental physics. Despite being well studied, there remains various fundamental open questions about AGNs, such as what is the composition of the jet, which are the mechanisms for flares, and what are the precise locations and physical conditions at the gamma-ray emitting regions. In particular, among AGNs, recently detected extreme blazars are of special interest. These are an extreme flavor of high-synchrotron peaked blazars (HBLs), generally very faint, and where the peak of the synchrotron emission component is beyond soft X-rays, implying an IC peak at 100s GeV, and in some cases reaching TeV energies. EHBLs are particularly interesting for the study of multi-messenger physics, thanks to their hard spectra at the TeV range (Biteau et al., 2020). The most relevant observational aspect, for us here, is the recent association of AGN, and particularly blazars, as astrophysical sources of ultrahigh-energy neutrinos (Smith, 2018). In this regard, the main event, which resulted in a large multi-messenger campaign by a number of collaborations and observatories, was the electromagnetic follow-up of the IceCube

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neutrino IC 170922A (IceCube Collaboration, 2018), detected in 2017. At the time, the source TXS 0506+056, which was spatially coincident with the direction of the neutrino event, was flaring in GeV gamma-rays, as seen by Fermi-LAT, and the spatial-temporal correlation between the two events was established at a level of 3σ, hinting at a possible multimessenger association — the first ever for a VHE astrophysical neutrino — and which prompted a very large follow-up multi-wavelength observational campaign (IceCube Collaboration et al., 2018). A few days after the measurement of the neutrino, MAGIC announced the detection of VHE gamma-rays from the direction of TXS 0506+056, which was varying at the time. In fact, the first observation by MAGIC of the source was 32 hours after the neutrino alert, but detection was only successful 4 days later, above 90 GeV, which suggests that, if indeed correlated, the neutrino emission preceded the VHE flare. For a few months after that first detection, until February 2018, both MAGIC and VERITAS continued to observe the source, but were only able to redetect it at the end of this period, when it was measured to present a sharp spectral cutoff (soft spectrum ΓVHE = 4.8 ± 1.3) in the VHE gamma-rays. In comparison, Fermi observations at GeV presented a very hard spectrum. Since, then, and up until 2020, there has been no further evidence of variability from this source at VHEs, the same being true for the GeVs as observed by Fermi-LAT observations. The lack of variability outside the period of the detection of the VHE neutrino may be interpreted as reinforcing the multi-messenger connection between both, but clearly more observational evidence is needed to firmly point blazars as the putative sources of extragalactic energetic neutrinos, and consequently acceleration sites to ultrahigh-energy cosmic rays (UHECR). In any case, the study of extreme blazars by the future generation of ground-based instruments will be of great importance to this question. In fact, the first EHBL catalog by the MAGIC Collaboration (Acciari et al., 2020) shows that the VHE emission from these sources can, among other possibilities, be modeled by proton-synchrotron radiation, pointing to a potential direct link to neutrino astronomy and of AGNs as UHECR accelerators (Cerruti et al., 2015).

4.2. Gravitational wave follow-ups Ground-based gamma-ray observatories have also been performing very ambitious follow-up programs of gravitational wave events detected by

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Telescope Array. But beyond CTA, additional potential exists for the new frontier of wide-field monitoring ground-based gamma-ray facilities. The direct detection of astrophysical gamma-rays is only possible with satellite-based detectors, such as the Fermi Telescope, but the restrictions of space instrumentation limit the collection areas and sensitivity. As fluxes become too small around 100 GeV, satellite observations are no longer an option, and one needs to use the showers of particles created as a result of the gamma-ray interaction with the atmosphere for an indirect detection. The most advanced technique to measure the atmospheric showers is the imaging atmospheric Cherenkov technique (IACT), which directly observes the Cherenkov light from the showers, and of which the next major facility will be the CTA, described in more detail below.

5.1. The Cherenkov Telescope Array Despite the enormous evolution seen in recent years in the field of groundbased gamma-ray astronomy, a great deal of exploring power is still available for the field, both for deep morphological and spectral studies, as well as in expanding the overall observational reach (both of the Galactic and extragalactic skies) beyond what can be done with current facilities. Leading the way for the group of new gamma-ray facilities set to revolutionize the field of Astroparticle Physics on this decade, is the Cherenkov Telescope Array (CTA), which is the upcoming atmospheric Cherenkov observatory, working from several 10s of GeV up to 100s of TeV. CTA is expected to unleash the full exploratory power of ground-based gammaray astrophysics. In fact, CTA’s evolution in the TeV range will be similar to that experienced by the high-energy gamma-ray astronomy (MeV–GeV range) at the time of transition from EGRET5 in the late 1990s, to the Fermi Large Area Telescope6 in the late 2010s, two decades later. The CTA will be an array of many tens of imaging atmospheric Cherenkov telescopes, designed to have a large energy coverage, from 20 GeV to 300 GeV, with a slightly larger field of view if compared to current facilities, of ∼8◦ , and an angular resolution at the level of ∼ arcmin. In addition, CTA has a requirement of achieving 10% angular resolution,

5 See the EGRET catalog, with many hundreds of sources, available online at (EGRET, 2021). 6 See, in comparison, the LAT catalog available online at (LAT, 2021), showing more than four thousand sources.

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and to improve sensitivity, throughout the energy range, by a significant amount of 5–10× what is available today. CTA will be constituted by three different telescope types, necessary to cover its wide energy range and will constitute de facto sub-arrays covering three major spectral regions: sub-TeV, TeV, and multi-TeV energies. The large sized telescopes (LSTs), with about 23 m in diameter, will cover the lowest energy range, driven by the study of GRBs and cosmological objects, such as distant AGN, whereas the medium sized telescope (MSTs), with 12 m, will cover the central energies around 1 TeV, for a deep and very extended survey of the Galaxy and the search for dark matter. Finally the 4 m mirror diameter small sized telescopes (SSTs), centered around the study of very energetic sources, such as PeVatrons, will cover from 10 TeV up to 100s of TeV, and will be the first-ever dual-mirror Cherenkov telescope, with a large field of view of ∼9◦ , ideal for the study of extended objects. The Cherenkov Telescope Array will also have two different sites, one in the Southern Hemisphere, at Cerro Paranal, in Chile, and another one in the North, at the Canary Island of La Palma, and it will operate as an open observatory, providing data access to all scientists from participating countries, as well as high-level data products and analysis tools for the community. The Northern Hemisphere site will initially contain only large and medium sized telescopes, whereas the Southern Site is expected to contain all three types of instruments. CTA will be therefore an integral part of the multi-wavelength and multi-messenger field, responding to external alerts and generating alerts to the global community in the timescale of minutes. CTA will have an expected operation lifetime of at least 30 years. An artist’s rendering of the CTA southern site is presented in Fig. 6. Concerning the project status, CTA is about to enter its construction phase, scheduled for start in late 2021, and lasting for a period of 5 years, when we will see the bulk of the telescopes being constructed and deployed on the sites. Afterwards, the operations phase is set to commence, with the conduction of the so-called Key Science Projects, to be managed by the CTA Consortium, as well as Announcement of Opportunities for the community. In terms of performance, CTA is expected to have a flux sensitivity one order of magnitude better than the current generation of instruments above 1 TeV, for both observatory sites, with the difference that CTA-South will have a generally better high-energy sensitivity (above 10 TeV) because of the Small-Sized telescopes (Knoedlseder, 2020). The current status of the

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Fig. 6. Artist’s rendering of the southern CTA site, showing the three types of telescopes which will form the array. Image credits: The CTA Consortium.

CTA instrumentation is that there are working prototypes for each different type of telescope, all of which have already had their first light. In fact, the first LST, given its dimensions and cost, is not strictly a prototype, but actually the first telescope of the Northern array, which is already taking commissioning data from the Crab Nebula and has also successfully detected the Crab pulsar (Cortina, 2019). The CTA Key Science Projects are discussed in great detail in a dedicated recent publication (CTA Consortium, 2019). The KSPs are crucial for CTA science because they will utilize about 40% of the observatory’s time in the first decade to the conduction of major legacy projects, such as complete surveys of the Galactic Plane (with greater depth at the Galactic Center) and the Large Magellanic Cloud, the first survey of the Extragalactic sky at VHE energies (covering circa 1/4 of the full sky), as well as two dedicated observational programs, for transient sources, and for dark matter searches. In terms of survey performance, CTA will be able to reach all the way across the Galaxy, compared with current facilities that were limited to the detection of VHE sources within about 4–5 kpc radius around the Sun’s position (equivalent to about 1/5 of the Galaxy). Likewise, the survey speed of CTA will be around 300 times that of H.E.S.S., thanks to both its wider field of view, greater sensitivity, and much larger number of telescopes. CTA’s angular resolution will reach ∼ arcmin, which is a factor of three better than the sharpest gamma-ray image so far, produced by H.E.S.S. from the SNR RXJ1713.7-3946 (H.E.S.S. Collaboration, 2018b). Both an extended survey capability, which allows to scan large regions

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of the sky quickly, as well as a good angular resolution and source localization/counterpart identification are very important for multi-messenger studies. Although CTA will conduct four large surveys — of the Galactic plane, the extragalactic sky, a deep survey of the Galactic Center, and a survey of the Large Magellanic Cloud — the Galactic plane one will be the largest and most important in scientific terms. It will provide a new census of the VHE sky (expecting a factor of 5× more sources than currently known), unveiling the diffuse high-energy cosmic-ray sea of the Galaxy at TeV, and disclosing an unprecedented high-resolution view of a number of extended objects (e.g., CTA Consortium, 2017). Another important feature of CTA, which is of great relevance to the study of variable astrophysical phenomena associated to multi-messenger events, will be the observatory’s excellent transient sensitivity. Figure 7 illustrates CTA’s flux sensitivity for variable sources as a function of event time or duration. It is very clear that, for low-energies, below 100 GeV, which overlap the coverage of Fermi-LAT, CTA will achieve a sensitivity at least 10,000 times better. A clear caveat, nevertheless, is that, while

Fig. 7. Differential flux sensitivity of CTA at selected energies as function of observing time, in comparison with the Fermi LAT instrument (Pass 8 analysis, extragalactic background, standard survey observing mode). Image credit: CTA Consortium; available from: (HST, 2021).

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Fermi-LAT offers a very large field of view (of about 2π sr), CTA will have a much smaller field of view, of up to 70 deg2 . This limitation can be at least partially compensated by CTA’s fast pointing capabilities, of about 30 sec to any point in the sky, to catch flares and follow-up alerts quickly. These timing capabilities, which are at the center of multi-messenger astrophysics, open up the way for unique studies of GRBs and AGN, as well as to the follow-up of GW and neutrino events.

5.2. Coming of age of wide-field facilities There are two distinct observational approaches to mapping the universe at the highest energies. In addition to the pointed observations from IACTs, as explained previously, there are the wide field-of-view surface arrays, of which the most relevant current examples are HAWC (Springer et al., 2016), LHAASO (Cao et al., 2019), Tibet-ASγ (Zhang et al., 2017) and ARGO-YBJ (Di Girolamo et al., 2016). Both approaches have very unique advantages, with Imaging Air-Cherenkov Telescopes being able to achieve excellent angular resolution and measuring spectra down to 100 GeV, the downside being their operational uptime which is limited to nighttime. On the other hand, with the wide-field arrays, it is possible to achieve a very high duty-cycle, at the expense of the angular resolution, which is considerably inferior to that of pointing instruments. Additionally, there exists another trade-off between sensitivity and sky coverage of the two techniques, which in turn impacts in the kind of science chiefly pursued by each one, and makes them essentially complementary in observational terms. The general detection principle of these wide-field arrays is based on water-filled detector units which produce Cherenkov radiation when a charged particle passes through the water, and is then detected by photosensors. When an air-shower sweeps through the array, a number of tanks are triggered in sequence, and the timing information of the triggers is used to reconstruct the arrival direction of the air-shower. The technique also allows for an event-by-event energy reconstruction, as the water tanks function as calorimeters sampling the air-shower front energy deposition at the ground. As with the imaging atmospheric technique, the crucial step of the analysis is in differentiating between gamma-ray and cosmic-ray showers (present in the proportion of 1:10,000 at these energies), which is done based on the different footprints that both types of particle-induced airshowers leave on the array, with the gamma-ray ones being considerably more compact.

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5.3. Mapping the sky with HAWC The High-Altitude Water Cherenkov (HAWC) Observatory is housed in Mexico, near the city of Puebla, at an altitude of 4100 m above sea level. It is composed of an array of 300 steel water tanks, each tank containing 200 thousand liters of pure water, and instrumented with 4 PMTs at the bottom. The tanks jointly cover an area of 22,000 m2 , which has recently been increased to 100,000 m2 with the addition of 350 smaller outrigger tanks. Thanks to its high-altitude, HAWC is sensitive to gamma-rays between 300 GeV up to beyond 100 TeV — and for cosmic rays this range extends to PeV energies. Since HAWC is not a pointed telescope, its field of view is basically the overhead sky, and every 24 hours the observatory scans 2/3 of the sky. In five years of data-taking, HAWC has achieved a sensitivity down to a few-percent of the Crab Nebula flux above 10 TeV, and accumulated over 100 billion cosmic-ray showers, which has allowed it to perform a number of cosmic-ray studies, such as measuring the all-particle cosmic-ray spectrum up to 500 TeV (Alfaro et al., 2017), as well as a full-sky measurement of the cosmic-ray anisotropy, in a joint analysis with IceCube, which was used to map the local interstellar magnetic field (Abeysekara et al., 2019). HAWC has recently published its third source catalog (3HWC Survey), using 1523 days of data taking (Albert et al., 2020a). The catalog, which is the deepest and most complete above 10 TeV, contains 65 sources, 20 of which had not been seen at TeV energies before, thus showing the complementarity between HAWC observations of the inner Galaxy and the H.E.S.S. Galactic Plane Survey, for example. Within the richness of the science contained in the catalog, a few results deserve special highlight. The first of these is that HAWC has substantially added to its previous detection of extended emission from PWNe. As previously explained, it has been predicted that leptonically powered TeV halos should be common around old pulsars (Sudoh et al., 2019), and indeed in the 3HWC eight new candidate pulsars were detected which are predicted to belong to this class. HAWC has also performed detailed spectral and morphological studies of several regions in the Galaxy, exploring whether most of emissions detected are leptonic or hadronic in origin. For some of the sources, such as HAWC J1825-134 (Albert et al., 2020c) and HAWC J2227+610 (Boomerang SNR) (Albert et al., 2020b), the spectra has been measured to extend past 100 TeV. Protons being accelerated to PeV energies are highly

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likely to be the parent particle population of the observed radiation, making these objects excellent candidates for IceCube to search for neutrino emission in the future, and try to provide unambiguous signs of hadronic acceleration within the Galaxy. Figure 8 shows a map of the inner Galactic plane as observed by HAWC, where we see eight sources detected with emission above 56 TeV, three of which extend past 100 TeV without a cutoff in their spectra (Abeysekara et al., 2020). Besides the question of cosmic-ray acceleration, for which these sources are excellent PeVatron candidates, the direct observation of photons of such high energies can also be used to test for new theories of Physics beyond the Standard Model. One example is constraining Lorentz Invariance Violation (LIV), under which the standard dispersion , allowing photons to relation is modified to a form Eγ2 − p2γ = |αn |pn+2 γ decay above a certain energy scale, producing a cutoff in the highest energy spectra as a result. The fact that HAWC finds evidence of 100 TeV photons emitted from multiple astrophysical sources can therefore be used to exclude LIV above energies 1800 times above the Planck Energy scale, an improvement of an order of magnitude over previous limits (Albert et al., 2020d).

5.4. The upcoming LHAASO observatory LHAASO is a high-altitude (4,410 m a.s.l.) Northern Hemisphere observatory located at the edge of the Himalayan plateau, close to the city of Chengdu. It is the world’s largest ground-based gamma-ray observatory, built for the observation of ultra-high energy gamma and cosmic-rays up to PeV energies. It consists of a hybrid detector system, formed by a large, km2 array of scintillators to measure the shower front (5,195), and water Cherenkov detector units (1,171) for muon detection (the so-called KM2A detector); at the center of the array, a large, 80,000 m2 surface water Cherenkov detector pond is installed (the WCD detector). The observatory also hosts 18 wide-field (16◦ ×16◦ ) air-Cherenkov telescopes for complementary observations of ultra-high-energy cosmic rays, and cross-calibration of the ground particle array (Cao et al., 2019). Although only partially operational at the moment of writing, LHAASO has already been able to achieve very good measurements, thanks to its excellent sensitivity and gamma-hadron separation capability above 1 TeV, both at the WCD and the KM2A detectors. In fact, the WCD detector, with peak performance below 10 TeV, has reported a sensitivity of

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60 mC.U. (milli-Crab Units), and an angular resolution of 0.26◦, and in its first 300 days of partial operation survey data has discovered six sources with a significance above 10σ, including the extragalactic source Mkn 421. The KM2A detector array has a sensitivity of 0.1 C.U. above the threshold energy of 100 TeV, in which range it has detected three sources above 10σ in its first 173 days of partial survey operation, with an angular resolution of 0.26◦ as well. In particular, the detector array has detected emission well-above 100 TeV (without any apparent cutoff) from all of the detected objects, posing challenges to acceleration models that limit the power of Galactic sources (Cao et al., 2021). In addition to these early results, which testimony to the great science expectations of the observatory, LHAASO has also been successful in registering the most energetic astrophysical EM signal ever recorded, a 0.9 ± 0.2 PeV gamma-ray photon signal (with chance probability −1), and varies with time. Many quintessence potentials have been proposed. They can be classified into two broad classes: (a) freezing models and (b) thawing models (Caldwell and Linder, 2005). In the former, the quintessence field has a density which closely tracks (but is less than) the radiation density until matter-radiation equality, when the field density eventually catches up the background fluid (Ratra and Peebles, 1988; Zlatev, Wang and Steinhardt, 1999). A representative potential that belongs to this class is the inverse power-law potential V (φ) = M 4+n φ−n (n > 0) which appears into the fermion condensate model as a dynamical supersymmetry breaking (Bin´etruy, 1999). Another example of freezing model is   V (φ) = M 4+n φ−n exp αφ2 /m2pl . This potential can be constructed in the framework of supergravity (Brax and Martin, 1999). In thawing models, from early times until recently, the field with mass ˙ mφ is frozen due to the Hubble friction, characterized by the term H φ. After, the field begins to evolve when H drops below mφ . The equation of state is wφ −1 at early times and then starts to grow for H < mφ . The representative potentials that belongs to this class are: (i) V (φ) = V0 + M 4−n φn , (n > 0), (ii) V (φ) = M 4 cos4 (φ/f ). The potential (i) with n = 1 was originally proposed by Linde (1987) to replace the cosmological constant by a slowly evolving scalar field. The potential (ii) was introduced by Frieman et al. (1995). Class (ii) appears as a potential for the Pseudo-Nambu–Goldstone Boson (PNGB). The field acts as an effective cosmological constant before relaxing into a condensate of non-relativistic bosons.

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The allowed parameter space for freezing and thawing models are 3wφ (1 + wφ )  wφ  0.2wφ (1 + wφ ) and 1 + wφ  wφ  3 (1 + wφ ), respectively (see Caldwell and Linder, 2005; Linder, 2006, for details). At the moment the observational data is not precise enough to distinguish between these two models, though it is expected that future high precision observations will be able to detect variations of wφ . 4.3.2. k-essence Quintessence is based on a canonical scalar field with a slowly varying potential. Another possibility is to consider scalar fields with non-canonical terms. The action for such models is in general given by   1 4 √ R + P (φ, X) + Sm , (31) S = d x −g 2 κ2 where P (φ, X) is a function in terms of the scalar field φ and its kinetic energy 1 X = − g μν ∂μ φ∂ν φ. 2

(32)

As usual, Sm is a matter action. These type of scalar fields often appear in the context of particle physics. Armend´ariz-Pic´ on, Damour and Mukhanov (1999) first suggested that scalar fields with non-canonical terms could drive an inflationary evolution starting from rather generic initial conditions in the early universe. Chiba, Okabe and Yamaguchi (2000) applied this scenario to the problem of the current accelerated expansion of the universe. They showed that a scalar field with non-canonical terms enables to model a component of the cosmological fluid that violates the weak energy condition. More general models based on the action (31) were later developed and were called “k-essence” (Armendariz-Picon, Mukhanov and Steinhardt, 2000, 2001). The most relevant are as follows: (i) Low-energy string theory (Gasperini and Veneziano, 1993, 2003), (ii) Ghost condensate (ArkaniHamed et al., 2004), (iii) Tachyon field (Abramo and Finelli, 2003; Aguirregabiria and Lazkoz, 2004; Copeland et al., 2005; Padmanabhan, 2002), (iv) Dirac-Born-Infeld (DBI) theory (Guo and Ohta, 2008; Martin Yamaguchi, 2008). The dark energy field can take a variety of forms. However, there is an essential condition it must satisfied: its energy density should remain well below the energy density of radiation and matter in the past, and should become dominant only recently. k-essence models fulfill this requirement by

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letting the energy density of the dark energy field decline (track) at the same rate as the dominant energy component of the universe.8 During cosmic time, k-essence undergoes two transitions in its behavior, one beginning at the onset of matter domination and a second when k-essence overtakes the matter density. At the radiation-dominated era, the k-essence energy tracks the radiation, falling as 1/a(t)4 , where a(t) is the scale factor. When the density of matter and radiation become equal, k-essence begins to act as an energy component with negative pressure. This effect is achieved by introducing a kinetic energy density which is nonlinear in X (see Eq. (32)). Because the energy density of k-essence declines much more slowly than that of matter, there is a moment, roughly at the present epoch, when k-essence begins to dominate well after the matter epoch, and the universe begins to accelerate after structure has formed. Despite the different properties of dark matter and dark energy, there are some models that intend to unify these dark components in one entity using a single fluid or a single scalar field (Bento, Bertolami and Sen, 2002; Kamenshchik, Moschella and Pasquier, 2001; Scherrer, 2004). We present next an example of a single fluid model: the Generalized Chaplygin gas.9 4.3.3. Unified models of dark energy and dark matter: The Generalized Chaplygin gas model The Generalized Chaplygin gas is modeled as a perfect fluid where the pressure P and the energy density ρ are related as P = −Aρ−α ,

(33)

where A is a positive constant. The original Chaplygin gas model corresponds to α = 1 (Kamenshchik, Moschella and Pasquier, 2001). If we substitute this equation into the continuity equation ρ˙ + 3H (ρ + p) = 0,

(34)

the integrated solution yields  ρ(t) = A +

B a3(1+α)

1 1+α

,

(35)

8 This feature is not restricted to k-essence; some quintessence models also display a tracking behavior. 9 Another possibility to construct unified models of dark matter and dark energy is by using a single scalar field in the context of k-essence (Scherrer, 2004).

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where B is an integration constant. The density ρ evolves as ρ ∝ a−3 in 1 the early epoch (a 1) and ρ ∝ A 1+α in the late epoch (a  1), respectively. The corresponding equation of state takes the form (Amendola and Tsujikawa, 2010) −1  Ω∗m 3(1+α) (1 + z) , (36) w(z) = − 1 + 1 − Ω∗m Ω∗m ≡

B , A+B

(37)

where Ω∗m is interpreted as an effective matter density, different from Ω0m . At high redshift (z  1), w ≈ 0. In the present epoch, w(0) = − (1 − Ω∗m ), while w → −1 in the future. In this way, the Generalized Chaplygin gas can model both dark matter and dark energy at the background level. The proposals we have just described are the most relevant. The reader is referred, for instance, to the book by Amendola and Tsujikawa (2010) where an exhaustive list of candidates for the dark energy field is presented. 4.4. Current limits Dark energy field models are essentially characterized by a dynamical equation of state w(z). The reconstruction of the equation of state through cosmological observations is, in principle, possible. Using SN Ia observations, the Hubble parameter H(z) is estimated by measuring the luminosity distance dL (z) (Amendola and Tsujikawa, 2010)  (0) 2 2 (1 + z) c2 (1 + z) + ΩK H02 d2L (z) 2 E (z) = , (38) [(1 + z) H0 dL (z) − H0 dL (z)]2 where E(z) ≡

H(z) . H0

(39)

The prime denotes the derivative with respect to z. This allows to reconstruct the equation of state of dark energy   (0) (0) 4 2 (1 + z) E 2 (z) − 3E 2 (z) − Ωr (1 + z) + ΩK (1 + z) . wDE (z) =   (0) (0) (0) 4 3 2 3 E 2 (z) − Ωr (1 + z) − Ωm (1 + z) − ΩK (1 + z) (40) In practice, this task is quite complex. For instance, the luminosity distance dL (z) is known at discrete values of the redshift. So, it is not

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possible to directly differentiate dL (z) with respect to z in Eq. (38) to obtain H(z). One way to solve this problem is to assume parametric forms of dL (z), or H(z), or w(z). For example, if we postulate a parameterization of w(z), the Hubble parameter can be directly computed as  4 3 2 H 2 (z) = H02 Ω0r (1 + z) + Ω0m (1 + z) + Ω0DE γ(z) + Ω0K (1 + z) . (41)   z 3 (1 + wDE ) d˜ z . (42) γ(z) = exp 1 + z˜ 0 Thus, integration of Eq. (38) with respect to z yields the luminosity distance that can be directly compared with observations. The parametric reconstruction of the equation of state is based on the hypothesis that dL (z), H(z) and wDE (z) vary sufficiently slowly with redshift and therefore can be approximated by a fitting formula that depends on a few number of parameters. There are several parameterizations for wDE (z) (Amendola and Tsujikawa, 2010):  wn xn (z), (43) wDE (z) = n=0

where the expansions can take one of these forms (1) xn (z) = z n ,

(2) xn (z) = 1 −

a a0

n

= n

z 1+z

n

,

(3) xn (z) = [ln (1 + z)] . The scale factor parameterization (2) with n ≤ 1 was proposed by Chevallier and Polarski (2001), and Linder (2003) wDE (z) = w0 + w1 (1 − a) = w0 + w1

z . 1+z

(44)

The value of w at z = 0 is denoted w0 , and a ≡ (1 + z)−1 . In the limit z → ∞, wDE (z) → w0 + w1 , and when z → 0, wDE (z) → 0. The fit given by (44) can be written in a more general form (Sahni and Starobinsky, 2006): w(a) = wp + (ap − a) wa .

(45)

Here, ap is the value of the scale factor where the equation of state w(a) is most tightly constrained.

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Thought parametric equations of state of the dark energy field are purely phenomenological, they should be able to mimic a wide class of dark energy models. On 17 July 2018, ESA and the Planck Collaboration released to the public a new and improved version of the data acquired by the Planck satellite, which constitutes the final official release from Planck. The latest 12 collaboration papers include the cosmological parameter results from the final full-mission Planck measurements of the CMB anisotropies, combining information from the temperature and polarization maps and the lensing reconstruction (Planck Collaboration et al., 2020). The collaboration also explored extensions to the ΛCDM model. In particular, they obtained constraints on the parameters w0 and w1 of the equation of state given by (44). If only Planck data is used, the allowed values for w0 and w1 are −3 < w0 < 1 and −5 < w1 < 5. However, most of these values correspond to phantom models for which H0 is very high; these models are inconsistent with the late-time evolution constrained by Type Ia supernovae (SNe) and Baryon Acoustic Oscillations (BAO). When external data coming from BAO, SNe, Redshift-space distortions (RSDs) and Weak gravitational Lensing (WL) are taken into account, the constraints on w0 and w1 narrow towards the ΛCDM values of w0 = −1 and w1 = 0. This is consistent with a cosmological constant. Another collaboration that aims to set constraints on the equation of state of the dark energy field is the Dark Energy Survey (DES). The project is dedicated to the measurements of Type Ia supernova light curves, gravitational lensing, BAO and galaxy clustering via a photometric survey. Constraints on the dark energy field equation of state (Abbott et al., 2019) were obtained by assuming a cosmological model, named wCDM model, with fixed curvature (Ωk = 0). The value obtained was w = −0.80+0.09 −0.11 . In 2020, the Sloan Digital Sky Survey (SDSS) provided the final extended Baryon Oscillation Spectroscopic Survey (eBOSS) cosmology analysis (eBOSS Collaboration et al., 2020). They used the BAO and RSD data from SDSS, the Planck CMB data, SNe Ia from the Pantheon sample, and weak lensing and clustering data from DES to provide the tightest available constraints on the parameters within the standard ΛCDM model and its extensions. They specifically explored several cosmological models where the equation of state for the dark energy component could have the following three forms: (1) w(a) = −1, (2) w(a) = w, (3) w(a) = w0 + wa (1 − a).

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Case (1) corresponds to a cosmological constant; case (2) represents a constant equation of state while case (3) refers to the Chevallier–Polarski– Linder (CPL) parameterization (see Eq. (44)). In all these models, Ωk = 0 and were denoted as ΛCDM, wCDM and w0 wa CDM. Models where Ωk was taken as a free parameter were also considered and were referred as oΛCDM, owCDM and ow0 wa CDM. The eBOSS Collaboration showed that the best fitting parameters in extended models remain in accordance with a ΛCDM cosmology; the most flexible ow0 wa CDM model is constrained by Ωk = −0.0023 ± 0.0022, w0 = −0.912 ± 0.081 and wa = −0.48+0.36 −0.30 ; these values correspond to an equation of state wp = −1.020 ± 0.032 at a pivot redshift zp = 0.29 and a Dark Energy Figure of Merit of 92.10 The results of all these observations point out that models that consider the validity of General Relativity and a non-vanishing cosmological constant best describe the cosmic acceleration. 5. Alternative Theories of Gravitation An alternative approach to explain the current dynamical state of the universe consist of modifying the left-hand side of the field equations of General Relativity. Einstein himself employed this approach when he attempted to obtain static cosmological models. He was the first that modified the gravitational law by introducing an additional term on the left-hand side of his equations 1 8πG Rμν − gμν R + Λgμν = 4 Tμν . 2 c

(46)

The cosmological constant Λ sets the scale where gravity becomes repulsive. We stress that even though from a mathematical point of view “writing” the cosmological constant either on the left or on the right-hand side is equivalent, the physical meaning is completely different. In this interpretation the cosmological constant Λ is associated with a modified gravitational law and not with vacuum energy or a dark energy field. The standard cosmological model, also known as ΛCold Dark Matter (ΛCDM) is constructed assuming the validity of Einstein field equations in the form given by (46). The model also postulates the existence of cold dark matter. Furthermore, the ΛCDM model resorts to another model, 10 The Figure of Merit (FoM) for dark energy experiments is the combination of error [σ(ωp )σ(ωa )]−1 (Albrecht et al., 2006; Mortonson, Weinberg and White, 2013).

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inflation, to account for the seeds that gave rise to the large-scale structure of the cosmos. As mentioned in the last section, the ΛCDM model provides the best fits to the cosmological observations. Despite its achievements, it presents major problems derived from the theories, assumptions and models that laid on its foundations. Maybe, the observed dynamics of visible matter are governed by some gravitational interaction that General Relativity fails to describe. Maybe, a different approach should be followed, a method that might be fairly called “Einstein’s approach”; instead of introducing unknown entities into the world (additional components to the energy–momentum tensor), we could change the gravitational law. This strategy involves the modification of the left-hand side of Einstein field equations, that is, it requires a new relativistic theory of gravitation. This is the road of alternative theories of gravity. 5.1. The landscape of modified gravity theories Modifications to General Relativity can be broadly implemented in the following ways (Clifton et al., 2012): (1) The gravitational interaction could be mediated by extra scalar, vector and tensor fields. (2) The field equations of the theory could have derivatives of the metric higher than second order. (3) Dimensions higher than four could be considered. The first approach encompasses scalar–tensor theories, vector–tensor theories in the form of Einstein-æther theories, bimetric theories of gravity, Tensor–Vector–Scalar theory (TeVeS), the Einstein–Cartan–Sciama–Kibble theory (ECSK), and Scalar–Tensor–Vector Gravity theory (STVG). Among the many families of alternatives theories of gravity, scalar– tensor theories have been the most extensively studied. In these theories, the gravitational interaction is mediated by both the metric tensor gμν and a dynamical scalar field denoted φ. Brans–Dicke theory (Brans and Dicke, 1961)(BD) is perhaps the most important of scalar–tensor theories. The original motivation for BD was to implement the idea of Mach that the phenomenon of inertia was due to the acceleration of a given system with respect to the general mass distribution of the universe. An essential feature of BD theory is that the gravitational constant G varies with time because it is determined by the scalar field φ.

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The variation of G would affect the orbits of planets, the stellar evolution, and many other astrophysical phenomena. Vector–tensor theories introduce, instead, a space–time 4-vector field Aμ ; in particular, Einstein-æther theories consider Aμ to have a time-like direction. Some of these theories were constructed to provide modifications on the Newtonian law of gravitation at galactic scales that could explain the dynamics of some astrophysical system without dark matter (Clifton et al., 2012). A severe drawback of generalized Einstein-æther theory cosmological models is that they provide poor fits of CMB data at large scales. As the name suggests, Tensor–Vector–Scalar theory (TeVeS) has two additional field, besides the metric tensor gμν , to account for the gravitational interaction: a scalar field φ, and a (dual) vector field Aμ . TeVeS was originally proposed by Bekenstein (2004) and was developed as a relativistic generalization of Milgrom’s Modified Newtonian Dynamics (Milgrom, 1983), also known as MOND. In 1983, Milgrom proposed a phenomenological modification of Newtonian gravity on galactic scales that could correctly predict the rotational velocity curves of galaxies without dark matter. In MOND, the spherically symmetric gravitational potential has two regimes corresponding to high and low accelerations. In regions of high accelerations, the dynamics is described by Newtonian gravity while in regions of low acceleration, Newton’s second law is modified. Milgrom’s prescription could successfully fit a large range of spiral galaxies observations, and could also account for Tully–Fisher relation.11 Notice that MOND is restricted to non-relativistic regimes, and cannot be considered a complete theory of gravity. Instead, TeVeS incorporates MOND in its non-relativistic weak acceleration limit and is Newtonian in the non-relativistic strong acceleration regime. TeVeS passes the usual solar system tests of GR, predicts gravitational lensing in agreement with the observations (without requiring dark matter), does not exhibit superluminal propagation, and provides a specific formalism for constructing cosmological models (Bekenstein, 2004). A year later after Bekenstein’s paper, Moffat introduced Scalar–Tensor– Vector Gravity (STVG), where not one but three dynamical scalar fields G, w and μ ˜ are postulated; we will discuss STVG in some details in the next section. 11 The Tully–Fisher relation is an empirical relationship between the mass or intrinsic luminosity of a spiral galaxy and its asymptotic rotation velocity or emission line width.

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Another approach to modify GR is to allow for field equations higher than second order. This is achieved by supplying additional scalar curvature invariants to the Einstein–Hilbert (EH) action, or by making the EH action a more general function of the Ricci scalar. The latter choice leads to fourth-order field equations. These theories are all under the name of f (R)-gravity. In Sec. 5.1.2 we offer a general picture of these theories. Other generalizations of the EH action correspond to the L = f (T ) theories, where T is a contraction of the torsion tensor T =

1 1 μνρ T Tμνρ + T μνρ Tρνμ − Tμν μ T νρ ρ , 4 2

(47)

where T μ νρ is the torsion tensor defined in terms of the vierbien from β gμν = ηαβ hα μ hν . If f (T ) = T , Einstein field equations are recovered in the teleparallel approach to GR (Combi and Romero, 2018; Unzicker and Case, 2005). However, if f = T the field equations of the theory are of four order. We dedicate a few words to the third strategy. The idea of extra dimensions was originally introduced by Kaluza (1921) with the aim of unifying gravitation and electromagnetism. Kaluza proposed the action (Romero and Vila, 2014)   1 ˆ −ˆ R g d4 x dy. (48) S= ˆ 16π G R The coordinate y corresponds to the extra-dimension and the hats denote five-dimensional (5D) quantities. The extra dimension should have no effect over gravitation. This is realized by imposing the condition ∂ˆ gμν = 0. ∂y

(49)

Since in GR the effects of gravity manifest through derivatives of the metric, condition (49) ensures that the extra dimension does not affect the predictions of GR. The price paid for the unification of gravity and electromagnetism is the introduction of a scalar field φ called the dilaton (Kaluza fixed φ = 1) and an extra dimension which is not observed. Klein (1926) argued that the fifth dimension was not observable because it is compactified on a circle. The size of the extra dimension is so extremely small that cannot be detected in experiments. The Kaluza–Klein theory is not consistent with some observed features of particle physics as described by the standard model. This problem

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is addressed in a radical alternative way in the braneworld scenario. In Kaluza–Klein theory the extra dimension must be small and compact. On the contrary, in the braneworld scenario the extra dimensions can be much larger, even infinite. Among the braneworld models, the Arkani-Hamed, Dimopulos, Dvali model (ADD) is one of the most well known (Arkani-Hamed, Dimopoulos and Dvali, 1998). It was originally proposed as a solution to the socalled hierarchy problem: it is very difficult to explain why the characteristic energy scale of gravity, the Planck energy MP c2 ≈ 1019 GeV, is 16 orders of magnitude larger than the electro-weak scale, Mex c2 ≈ 1 TeV. The ADD model postulates an n flat compact extra dimensions of size R, and the standard model fields are confined to a 4D brane embedded in some higher dimensional space–time known as the bulk ; gravity only propagates in the bulk. Randall and Sundrum (1999) suggested that the bulk geometry might be curved and the brane could have a tension. In this way, the brane becomes a gravitating object, interacting dynamically with the bulk. A Randall– Sundrum (RS) universe consists of two branes of torsion σ1 and σ2 bounding a slice of anti-de Sitter space. The two branes are separated by a distance L and the fifth dimension y is periodic with period 2L. The three approaches here presented to construct alternative theories of gravity do not exhaust all the possibilities. Additionally, each of these approaches give rise to many family of theories. Even a brief presentation of all of them largely exceeds what can be contained in a single section of a book chapter. Our intention, in this section, is to provide the reader a very general picture on how General Relativity can be modified and the complexity of the task. For more references and other approaches, the reader is referred to the excellent review by Clifton et al. (2012). In what follows, we choose two modified theories of gravitation, that are not only good representatives of the possible generalization to GR, but have also been extensively explored with some success as alternatives to dark matter and dark energy. These are Scalar–Tensor–Vector Gravity and f (R)-gravity. 5.1.1. Scalar–Tensor–Vector Gravity In 2006, Moffat (2006) postulated the Scalar–Tensor–Vector Gravity theory (STVG), also called MOdified Gravity theory (MOG). In STVG theory, gravity is not only an interaction mediated by a tensor field, but by scalar and vector fields.

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The full action12 in STVG theory is (Moffat, 2006): S = SGR + Sφ + SS + SM , where

 √ 1 1 SGR = d4 x −g R, 16π G    √ 1 μν 1 2 μ Sφ = − d4 x −g B Bμν − μ ˜ φ φμ , 4 2    1 μν 1 4 √ g ∇μ G∇ν G − V (G) SS = d x −g 3 G 2    1 μν 1 g ∇μ μ + d4 x 2 ˜∇ν μ ˜ − V (˜ μ) . μ ˜ G 2

(50)

(51) (52) (53) (54)

Here, gμν is the space–time metric, R denotes the Ricci scalar, and ∇μ is the ˜ is covariant derivative; φμ stands for a Proca-type massive vector field, μ ˜(x) are scalar fields that vary its mass, and Bμν = ∂μ φν − ∂ν φμ ; G(x), and μ in space and time, and V (G), and V (˜ μ) are the corresponding potentials. We adopt the metric signature ημν = diag(−1, +1, +1, +1). The term SM in the action refers to possible matter sources. The full energy–momentum tensor for the gravitational sources is M φ S Tμν = Tμν + Tμν + Tμν ,

(55)

2 δSM M Tμν = −√ , −g δg μν

(56)

2 δSφ φ Tμν = −√ , −g δg μν

(57)

2 δSS S Tμν = −√ . −g δg μν

(58)

where

M Following the notation introduced above, Tμν denotes the ordinary matter φ energy–momentum tensor and Tμν the energy–momentum tensor of the S considers the scalar contributions to the energy–momentum field φμ ; Tμν tensor. 12 As suggested in Moffat and Rahvar (2013) and Moffat and Toth (2009), we dismiss the scalar field ω, and we treat it as a constant, ω = 1.

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The core motivation for constructing the theory was to be able to correctly predict the observed dynamics of rotation curves of galaxies, galaxies clusters and cosmology without dark matter. This is accomplished by a modified acceleration law for weak gravitational fields. Such a law is derived assuming that the fields G and μ take constant values, that is, these parameters do not depend on the temporal or spatial coordinates: μ ˜ ≈ μ˜0 ,

(59)

G = GN (1 + α) .

(60)

Here, GN denotes Newton’s gravitational constant and α is a free dimensionless parameter. Given these two hypotheses, the action (50) takes the form,    √ R 1 − B μν Bμν . (61) S = d4 x −g 16πG 4 Variation of the latter expression with respect to gμν yields the STVG field equations: φ , Gμν = 8πGTμν

(62)

where Gμν is the Einstein tensor, and the energy–momentum tensor for the vector field φμ is given by13   1 1 φ Tμν =− (63) Bμ α Bνα − gμν B αβ Bαβ . 4 4 If we vary the action (61) with respect to the vector field φμ , we obtain the dynamical equation for such field: ∇ν B μν = 0. The equation of motion for a test particle in coordinates xμ is  2 μ  α β d x q μ dxν μ dx dx B ν , + Γ = αβ dτ 2 dτ dτ m dτ

(64)

(65)

where τ denotes the particle proper time, and q is the coupling constant with the vector field. φ (2015) set the potential V (φ) equal to zero in the definition of Tμν given in Moffat (2006).

13 Moffat

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In STVG, for large values of the coordinate r, a static spherically symmetric metric field should approximate the Schwarzschild solution. Furthermore, under the assumptions 2GM/r 1 and the slow motion approximation dr/ds ≈ dr/dt 1, the equation of motion (65) reduces to (Moffat, 2006): GM q dφ0 d2 r JN2 , − 3 + 2 = dt2 r r m dr

(66)

where JN is the Newtonian orbital angular momentum, and φ0 is the t-component of the gravitational vector field φμ . In the presence of weak gravitational fields, to first order, the static equations for φ0 for the source-free case are (see Eq. (64)):  2 φ0 − μ ∇ ˜20 φ0 = 0,

(67)

 2 φ0 is the Laplacian operator, and the contribution from the selfwhere ∇ interaction potential W (φ) has been neglected. If the static field φ0 is spherically symmetric, Eq. (67) yields 2 φ0 + φ0 − μ ˜20 φ0 = 0, r

(68)

which has the Yukawa solution: φ0 (r) = −GN M α

exp(−˜ μ0 r) . μ ˜0 r

(69)

If we replace the latter equation in Eq. (66), we get d2 r JN2 GM exp(−˜ μ0 r) − 3 + 2 = GN M α (1 + μ ˜0 r) . dt2 r r μ ˜0 r2

(70)

We immediately obtain the radial acceleration a(r) = −

GN (1 + α) exp(−˜ μ0 r) + GN α (1 + μ ˜0 r) . r2 r2

(71)

The first term of Eq. (71) represents an enhanced gravitational attraction, and serves to explain galaxy rotation curves, light bending phenomena, and cosmological data, without dark matter. The second term describes gravitational repulsion and becomes relevant when μ ˜0 r 1. This Yukawatype force counteracts the enhanced attraction, and from the interplay, the Newtonian acceleration law is recovered at μ ˜ 0 r 1 scales. In this limit, STVG is consistent with observational data from the Solar System.

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Moffat (2015) also postulates that the gravitational source charge q of the vector field φμ is proportional to the mass of the source particle  (72) q = ± αGN m. The positive value for the root is chosen (q > 0) to maintain a repulsive, gravitational Yukawa-like force when the mass parameter μ ˜ is non-zero. Then, we see that in STVG theory the nature of the gravitational interaction has been modified with respect to GR in two ways: an enhanced gravitational constant G = GN (1 + α), and a vector field φμ that exerts a gravitational Lorentz-type force on any material object through Eq. (65). STVG has two free parameters α and μ that can be computed as (Moffat and Toth, 2009) M , α = α∞ √ ( M + E)2 α∞ =

(G∞ − GN ) , GN

D μ= √ . M

(73)

(74) (75)

The quantity G∞ is the asymptotic limit of G for very large mass concentrations. The expressions for α and μ were derived from a point source solutions of the STVG field equations and thus are only valid for particular situations. The value of the constants D and E are universal and independent of the mass of the central source M . These are given by (Moffat and Toth, 2009) 1/2

D = 6.25 × 103 M

1/2

E = 2.5 × 104 M .

kpc−1 ,

(76) (77)

Models constructed from STVG theory provide successful fits of rotation curves of a large sample of galaxies (Brownstein and Moffat, 2006b; Moffat and Rahvar, 2013), including the Milky Way (Davari and Rahvar, 2020; Moffat and Toth, 2015). Furthermore, the dynamical masses of galaxy clusters derived using the modified acceleration law match the cluster gas masses without the need of introducing a non-baryonic dark matter component (Brownstein and Moffat, 2006a, 2007; Moffat and Rahvar, 2014).

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Recently, observations of groups of clusters-like objects in the diffuse galaxy NGC 1052-DF2 showed that the velocity dispersion of this object is consistent with its measured stellar mass without dark matter (van Dokkum et al., 2018). Moffat and Toth (2019) demonstrated that models constructed using STVG are able to reproduce the observed velocity dispersion of NGC 1052-DF2. The theory is also able to correctly predict the lensing and Einstein ring observed of the Bullet Cluster (1E0657-558), Abel 520 (Moffat, Rahvar and Toth, 2018) and Abell 3827 galaxy clusters (Moffat and Toth, 2021). The parameters α and μ ˜ were assumed to be universal when the modified gravitational potential was used to fit the galaxy rotation curves and the mass profiles of galaxy clusters, both without dark matter. This strong hypothesis was put into test by a series of authors (De Martino and De Laurentis, 2017; Haghi and Amiri, 2016). Using STVG, De Martino and De Laurentis (2017) predicted the thermal Sunyaev–Zeldovich temperature anisotropies for the Coma cluster, and compared their results with those obtained using the Planck 2013 Nominal maps. They found α to be consistent at the 68 percent CL with its universal value (Moffat and Rahvar, 2013), while the scale length, μ ˜, was not compatible with such assumption at more than 3.5 σ. There is further evidence, as we present in Sec. 5.2, that the value of the parameter α is not universal from studies of the strong field regime of the theory. 5.1.2. f (R)-gravity A general way of introducing changes in the geometric sector of Einstein field equations is to modify the relativistic action. GR is obtained from the action:  1 √ R −g d4 x, (78) S[g] = 2κ which can be generalized to S[g] =



√ 1 f (R) −g d4 x, 2κ

(79)

where g is the determinant of the metric tensor and f (R) is some function of the curvature (Ricci) scalar. The generalized field equations are derived by varying (79) with respect to the metric   √  √ 1 (80) δS[g] = δf (R) −g + f (R) δ −g d4 x. 2κ

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The variation of the determinant is δ

√

 1√ −g = − −g gμν δg μν . 2

(81)

The Ricci scalar is defined as R = g μν Rμν .

(82)

Then, its variation with respect to the inverse metric g μν is given by δR = Rμν δg μν + g μν δRμν = Rμν δg μν + g μν (∇ρ δΓρνμ − ∇ν δΓρρμ ).

(83)

Since δΓλμν is the difference of two connections, it should transform as a tensor. Therefore, it can be written as δΓλμν =

1 λa g (∇μ δgaν + ∇ν δgaμ − ∇a δgμν ) , 2

(84)

and substituting in the equation above we get δR = Rμν δg μν + gμν δg μν − ∇μ ∇ν δg μν .

(85)

The variation in the action results in:    √ 1√ 1 −g gμν δg μν f (R) d4 x δS[g] = F (R) δR −g − 2κ 2   √ 1 −g F (R)(Rμν δg μν + gμν δg μν − ∇μ ∇ν δg μν ) = 2κ 1 μν − gμν δg f (R) d4 x, 2 (R) where F (R) = ∂f∂R . Integrating by parts on the second and third terms we get   √ 1 1 δS[g] = −gδg μν F (R)Rμν − gμν f (R) + (gμν  − ∇μ ∇ν ) F (R) d4 x. 2κ 2 (86)

By demanding that the action remains invariant under variations of the metric, i.e. δS[g] = 0, we find the field equations in generic f (R)-gravity: 1 F (R)Rμν − f (R)gμν + [gμν  − ∇μ ∇ν ] F (R) = κTμν , 2

(87)

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where Tμν is the energy–momentum tensor defined as √ 2 δ( −g Lm ) Tμν = − √ , −g δg μν

(88)

and Lm is the matter Lagrangian. If F (R) = 1, i.e. f (R) = R, we recover Einstein’s theory. Equation (87) is a system of nonlinear partial differential equations of four order in the coefficients of the metric tensor field gμν . A full description of f (R)-gravity can be found in the book by Capozziello and Faraoni (2011). An interesting feature is that the Ricci scalar R and the trace of the energy–momentum tensor T = g μν Tμν are related in a differential way14 : F (R)R − 2f (R) + 3F (R) = κT,

(89)

This implies that for some prescriptions of f (R), the Ricci scalar can be different from zero even if T = 0. One of the primary motivations for developing f (R)-gravity was that it could account for the effects of the accelerated expansion of the universe without invoking a dark energy field. In what follows, we describe how this is achieved. Consider the FLRW line element in comoving coordinates (t, r, θ, φ)   2  dr2 2 2 2 dθ + r + sin θdφ , (90) ds2 = −dt2 + a2 (t) 1 − kr2 where a(t) is the scale factor and k = −1, 0, 1 is the spatial curvature. As usual, we model the cosmological fluid as a perfect fluid with energy– momentum tensor given by T μν = (ρ + p) uμ uν + pg μν .

(91)

We denote uμ the 4-velocity of an observer comoving with the fluid; ρ and p are the energy density and pressure, respectively. Assuming a spatially flat universe, k = 0 and the energy–momentum tensor (91), the cosmological field equations in f (R) are obtained by inserting the line element (90) into Eq. (87). After some algebraic manipulations, 14 Equation

(89) is derived by taking the trace of Eq. (87).

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we get  κ R F (R) − f (R) − 3 H R˙ F  (R) , ρ+ 3F (R) 2  −κ 2H˙ + 3H 2 = p + R˙ 2 F  (R) + 2 H R˙ F  (R) F (R) 1  ¨ + R F (R) + (f (R) − R F (R)) . 2 H2 =

(92)

(93)

It is possible to define an effective energy density and pressure of the geometry as ρeff =

R F (R) − f (R) 3 H R˙ F  (R) − , 2 F (R) F (R)

peff =

¨ F  (R) + R˙ 2 F  (R) + 2 H R˙ F  (R) + R F (R)

(94) 1 2

(f (R) − R F (R))

. (95)

Inspection of Eq. (92) reveals that in the limit ρ → 0, ρeff is necessarily positive. Given these definitions, Eqs. (92) and (93) in vacuum are κ ρeff , 3 κ a ¨ = − (ρeff + 3 peff ). a 6

H2 =

(96) (97)

The effective energy density and pressure are related by an effective equation of state of the form: weff ≡

¨ F  (R) + 1 (f (R) − R F (R)) R˙ 2 F  (R) + 2 H R˙ F  (R) + R peff 2 . = ρeff R F (R) − f (R) − 3 H R˙ F  (R) (98)

Since ρeff > 0, the sign of weff is determined by the numerator of the right-hand side of the equation above. In order to reproduce a de Sitter cosmological background, i.e. weff = −1, the function f (R) should satisfy ¨ H R˙ − R F  (R) = . F  (R) R˙ 2

(99)

For instance, the function f of the form f (R) ∝ Rn with n = 2 yields weff = −1. The reader is referred to Sotiriou and Faraoni (2010) and Capozziello, Carloni and Troisi (2003) for details on this calculation.

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Another way to visualize how cosmological solutions with accelerating expansion can be obtained in f (R)-gravity is considering the case in which the Ricci scalar is constant R = R0 . In the absence of matter fields, Eqs. (87) and (89) become Rμν −

1 R0 1 R0 gμν = 0 ⇒ Rμν = gμν , 2 F (R0 ) 2 F (R0 )

F (R0 )R0 − 2f (R0 ) = 0 ⇒ R0 =

2f (R0 ) . F (R0 )

(100) (101)

If we replace the expression for R0 in Eq. (101) into Eq. (100), it yields Rμν =

f (R0 ) gμν . F 2 (R0 )

(102)

Recalling that Einstein field equations with a cosmological constant in vacuum are Rμν = Λgμν ,

(103)

f (R0 ) . F 2 (R0 )

(104)

we can make the identification Λ=

Thus, the gravitational law in f (R)-gravity can produce cosmic acceleration without introducing a cosmological constant or invoking dark energy fields. A viable f (R) cosmological model should not only predict the accelerated expansion on the universe. In addition, it has to properly describe the sequence of eras of the standard cosmological model. In the framework of f (R)-gravity, it is always possible to find a class of f (R) models that correspond to the observed evolution of a(t). However, the information derived from observational data on the evolution of a(t) is not sufficient to completely specify the function f (R). A possible solution may be provided by studying the growth of cosmological perturbations. The evolution of the growth of cosmological perturbations changes depending on the underlying theory of gravitation. This, in turn, should be reflected in the CMB and in the large-scale structure of the universe. There is an extensive literature regarding the growth and evolution of perturbations in f (R) gravity. The reader is referred to the review by De Felice and Tsujikawa (2010), where perturbation theory in f (R)-gravity is developed, and the comparison of the predictions of some f (R) models with CMB and large-scale structure observations is offered. Additional

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references can be found in Capozziello and Faraoni (2011); Clifton et al. (2012); Sotiriou and Faraoni (2010). Many natural questions arise in regards to the viability of f (R) as a complete theory of gravity. There are specific requirements it should fulfill; just to mention some (Sotiriou and Faraoni, 2010): (a) it must have the correct weak-field limit at the Newtonian and post-Newtonian level; (b) it must be stable at the classical and semiclassical level; (c) it must not contain ghost fields; (d) it must admit a well-posed Cauchy problem. A detailed treatment of all these issues largely exceeds the possibilities of this chapter. Instead, we discuss in Sec. 5.2.2 if f (R)-gravity models can provide an alternative to the ΛCDM model, in particular if f (R)-gravity can substitute the effects of dark matter in the early universe, and predict the observed dynamics of galaxies and galaxies clusters. 5.2. Theoretical challenges 5.2.1. Scalar–Tensor–Vector Gravity Up to now, STVG has been able to correctly predict several astronomical phenomena without invoking the existence of dark matter. Most of the verifications mentioned previously were restricted to the weak gravitational field regime. However, the viability of the theory needs to be explore in many other contexts; for instance, cosmological predictions, the strong gravity field regime, gravitational waves, just to mention a few. Regarding the strong field regime, STVG admits black hole solutions (Moffat, 2015). The astrophysical implications of these solutions were explored in two ways. Thin accretion disks around STVG black holes were constructed and the luminosity of such disks were computed for different values of the α parameter (P´erez, Armengol and Romero, 2017). The disks are colder and under luminous in comparison with thin relativistic accretion disks in GR. The spectral energy distributions predicted by STVG theory are not in contradiction with current astronomical observations given our ignorance of the details of the accretion regime. STVG also offers an alternative mechanism for jet formation of purely gravitational origin that is compatible with observational data (Lopez Armengol and Romero, 2017a). Neutron stars models were also constructed in STVG theory. Lopez Armengol and Romero (2017b) found a new restriction for the α parameter when stellar mass sources are taken into account: α < 10−2

1.5 × 105 c2 −1 M cm. GN

(105)

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All these investigations in the strong gravity field regime showed that the value of the parameter α is not universal and certainly depends on the mass of the central source. The detection of gravitational waves by the LIGO Scientific Collaboration and VIRGO Collaboration has been used as an additional test on alternative theories of gravity. Specifically, the observation in gravitational waves of the neutron star merger GW170817 (Abbott et al., 2017a,b) and the detection of the electromagnetic counterpart, GRB170817A (Abbott et al., 2017c) 1.7 s after GW170817 provided strong evidence that the speed of gravitational waves cGW and the speed of light c are identical. In a large class of alternative theories of gravitation, dubbed “Dark Matter Emulators” by Kahya and Woodard (2007) and Boran and coworkers (Boran et al., 2018), photons suffer an additional Shapiro time delay, and hence cGW = c. Consequently, all these theories were discarded. We emphasize that this was not the case for STVG, where gravitational waves move at the speed of light (Green, Moffat and Toth, 2018). In regards to cosmology, STVG possesses acceptable cosmological epochs (Jamali, Roshan and Amendola, 2018). However, the extra fields of the theory cannot mimic the dynamical effects of the cosmological constant when Λ = 0. Jamali, Roshan and Amendola (2018) also computed the angular size of the sound horizon θs and found that θs is 19 percent smaller than the observed value. In Jamali, Roshan and Amendola (2020), it was shown that the growth of matter perturbations in STVG is slower than in ΛCDM model, thus posing a further challenge to the theory. A way to avoid this problem is to consider that the density ρφ associated with the spin 1 Proca vector graviton (VG) field φμ dominates in the early universe over the baryon density ρb before the epoch of reionization. After this period, it is postulated that a transition to modified gravity occurs when ρb > ρφ , that could be caused by a matter–energy phase transition. Moffat (2020) showed that in this scenario there is an enhanced growth of matter perturbations. Additionally, the angular power spectrum calculations matches the ΛCDM model fit to the Planck Collaboration 2018 data. Besides the promising results, no explanation is offer that justifies the strong assumptions regarding the behavior of the vector field through cosmic time. We conclude that although STVG is not yet ruled out, a full analysis of CMB and lensing data will provide a strong challenge to the theory.

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5.2.2. f (R)-gravity Dark matter has been essentially postulated to account for: (a) the formation of structure in the early universe; (b) the dynamics of large astrophysical systems such as galaxies, clusters of galaxies, and gravitational lensing. Here, we briefly discuss if f (R) models are able to correctly predict such a variety of phenomena. It can be shown (Capozziello, Stabile and Troisi, 2009) that a general gravitational potential with a Yukawa correction can be derived in the Newtonian limit of an analytic f (R)-gravity model. This Yukawa-like correction to the Newtonian potential acts on certain scales allowing to reproduce the rotation curves of spiral and elliptical galaxies and galaxies clusters. The Yukawa correction considered in Capozziello and De Laurentis (2012) Φ(r) = −

r  GM  1 + δ exp− L , (1 + δ) r

(106)

depends on the parameters δ and L. The latter is related to the length scale of the effective scalar field introduced in the theory. Notice that the Yukawa correction should be negligible at small distances and in the so-called “chameleon mechanism” (Khoury and Weltman, 2004). There are values for δ and L such that rotation curves of galaxies and the mass profile of galaxy clusters are in accordance with observations (Capozziello and De Laurentis, 2012). However, these values are far from universal and depend on the properties of the astrophysical object under study. According to the ΛCDM model, dark matter is needed to provide the additional gravitational assistance for the growth of structure as we observe today in the universe. This process should have left an imprint in the CMB. In fact, the angular power spectrum of the CMB temperature anisotropies provides the strongest support for the existence of dark matter. Could f (R) models predict the observed properties of the CMB? Some early works on the subject (Bean et al., 2007; Song, Hu and Sawicki, 2007) show that the evolution of density fluctuations in f (R)-gravity leads to predictions that are inconsistent with CMB observations. To summarize, it is apparent that the modifications introduced by f (R)gravity on the gravitational law successfully account for the dynamics of the

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universe at large scale. These modifications, however, fail on smaller scales and regimes where dark matter models have a superior explanatory power.

6. Final Remarks In this review, we have examined two approaches proposed to explain the discrepancies between the observed dynamics of visible matter and the cinematic expectations from General Relativity. Dark matter belongs to the first approach. Though dark matter was originally postulated to account for the dynamics of galaxies and galaxy clusters, it was later found that it is an essential ingredient to explain the formation of structure in the universe. Unfortunately, numerical simulations using cold dark matter predict certain features at galactic and extragalactic scales, i.e. cuspy halo density profiles, an overabundance of satellite galaxies, among others, that are not observed. But the greatest problem the dark matter hypothesis faces today is that it has never been detected. Its nature remains elusive. Modifying the gravitational law is an alternative approach and certainly more economical from an ontological point of view. We have presented a general picture of the ways the gravitational interaction can be modified and focused on two alternative theories of gravitation. Scalar–Tensor–Vector Gravity theory correctly predicts the flat rotation curves of galaxies, and the observed dynamics of galaxy clusters, as well as gravitational lensing without dark matter. In the early universe, however, STVG cosmological models seem to fail: the growth of matter perturbations in STVG is lower than when cold dark matter is assumed; additionally, the theory does not reproduce some features of the CMB. The accelerated expansion of the universe can also be understood in the framework of alternative theories of gravitation. The simplest modification to Einstein field equations is adding a term on the left-hand side of the equations that is a constant multiple of the metric tensor gμν . The cosmological constant sets the scale in which gravity becomes repulsive. More general modifications of the Einstein–Hilbert action, for instance by introducing a function f (R) of the Ricci scalar, also lead to cosmological models with accelerated expansion. This is the case of f (R)-gravity theory, that we have described in some detail. Some of the problems of this theory, as STVG, are that its predictions are inconsistent with CMB observations. The accelerated expansion of the universe could instead be caused by a so-called “dark energy field”, an extraneous form of matter with a negative

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energy density. Differently from the cosmological constant, the equation of state of the dark energy field changes with time. Up to date, all cosmological observations point out that General Relativity and a non-vanishing cosmological constant best describe the cosmic acceleration. This extremely brief summary makes evident that none of the approaches, on its own, seems to provide a consistent solution valid at different scales and in different epochs of the universe. Maybe, we are just dealing with a false dichotomy. Maybe, a possible solution could emerge combing these two strategies. The question is how. The inextricable nature of cold dark matter is not the only challenge to the ΛCDM model. The initial singularity problem points out to an unsolvable deficiency of the model that has its roots in General Relativity. Cosmological models that display a bounce solve by construction this problem. In bouncing cosmological models, the universe starts from a very diluted phase and proceeds to contract. The contraction then smoothly evolves into a bounce that leads to the current phase of expansion as described by the ΛCDM model. As the cosmic fluid contracts most structure is erased and the universe becomes smooth (Rees, 1969). Black holes, however, might survive the bounce and play some role in the subsequent expanding universe (Carr and Coley, 2011; Clifton, Carr and Coley, 2017; Sikkema and Israel, 1991). It is plausible that black holes that survive the contracting phase might become the seeds that trigger the process of structure of formation in the early universe. The so-called “primordial black holes”, in this context, would come from the contracting epoch. As showed by Carr, K¨ uhnel and Sandstad (2016), primordial black holes in the mass range 10 M < M < 102 M could be relevant to provide a fraction of the dark matter in the universe as well to explain the observed LIGO/VIRGO coalescence events in the mass range O(10) M . Future experiments might detect dark matter, and conclusive evidence on the existence of some form of dark energy field might emerge. Meanwhile, some alternative explanations deserve to be explored; in the way, maybe, new physics might be unveiled.

Acknowledgments D. P´erez is most grateful to Gustavo E. Romero not only for the many discussions and insights on these topics, but also for his guidance, encouragement and support in all aspects of life along many years. This work was supported by CONICET (PIP2014-0338).

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Chapter 5

Hot Neutron Star Matter and Proto-neutron Stars Delaney Farrell∗, , Aksel Alp∗,∗∗ , Fridolin Weber∗,†,††,‡‡ , an Malfatti§,¶¶ , Milva G. Orsaria§,¶, William Spinella‡,§§ , Germ´ and Ignacio F. Ranea-Sandoval§,¶,∗∗∗ ∗

Department of Physics, San Diego State University 5500 Campanile Drive, San Diego, CA 92182, USA † Center for Astrophysics and Space Sciences University of California at San Diego La Jolla, CA 92093, USA ‡ Department of Physical Sciences, Irvine Valley College Irvine Center Drive, Irvine, CA 92618, USA § Grupo de Gravitaci´ on, Astrof´ısica y Cosmolog´ıa Facultad de Ciencias Astron´ omicas y Geof´ısicas Universidad Nacional de La Plata Paseo del Bosque S/N, La Plata (1900), Argentina ¶ Consejo Nacional de Investigaciones Cient´ıficas y T´ecnicas (CONICET) Godoy Cruz 2290, Buenos Aires (1425), Argentina  [email protected] ∗∗ [email protected] †† [email protected] ‡‡ [email protected] §§ [email protected] ¶¶ [email protected]  [email protected], ∗∗∗ [email protected] In this chapter, we investigate the structure and composition of hot neutron star matter and proto-neutron stars. Such objects are made of baryonic matter that is several times denser than atomic nuclei and tens of thousands of times hotter than the matter in the core of our Sun. The relativistic finite-temperature Green function formalism is used to formulate the expressions that determine the properties of such matter in the framework of the density-dependent mean-field approach. Three

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different sets of nuclear parameterizations are used to solve the manybody equations and to determine the models for the equation of state of ultra-hot and dense stellar matter. The meson–baryon coupling schemes and the role of the Δ(1232) baryon in proto-neutron star matter are discussed in great detail. The use of the non-local three-flavor Nambu– Jona-Lasinio model to describe quark matter, the hadron-quark composition of dense baryonic matter at zero temperature is discussed. General relativistic models of non-rotating as well as rotating proto-neutron stars are presented in part two of our study. Keywords: Proto-neutron stars; Neutron stars; Nuclear equation of state; Nuclear field theory; Quark Matter.

1. Introduction Within a few million years after a massive star ( 8 M ) is born, its core undergoes nuclear fusion reactions that will result in a dense, heavyion center. Up until the formation of an iron core, the massive star has been supported from collapsing by the energy released from fusing lighter elements into iron and electron degeneracy pressure (Burrows and Vartanyan, 2021; Foglizzo, 2016; Janka, 2012; Mezzacappa, 2005). When an iron core is formed, the fusion processes and subsequent energy cease; at this point, the star can no longer support its mass against the force of gravity and will begin to rapidly collapse in the span of just a few milliseconds. At this moment, the core’s temperature skyrockets and the density surpasses the point of electron degeneracy, sparking the formation of neutrons through electron capture, p + e − → n 0 + νe ,

(1)

where p, a proton, and e− , an electron, combine to form a neutron, n0 , and an electron neutrino, νe . These neutrinos are released carrying large quantities of energy, contracting the core further. The density of the core increases until it reaches nuclear density (baryon number density of around 3 0.16 fm−3 , mass density of or 2.65×1014 g/cm ), where nucleon degeneracy pressure halts the collapse. Parts of the core surpassing nuclear density, like the inner most part of the core, will rebound to create a shock wave as the exterior core layers are expelled. Over the next tens of seconds, the shock wave reverses the inward trajectory of the collapsing stellar material as it moves through the stellar envelope, partially cooling the extreme temperature and contracting the material it passes through. The shock wave alone does not possess enough energy to pass through the entire stellar envelope

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and complete the supernova explosion; the shock wave is revived by the massive quantities of neutrinos created alongside neutrons. While most neutrinos are expelled, some remain trapped behind the shock wave, increasing the pressure and pushing the wave outward (Camelio et al., 2017). This portion of the star’s collapse is referred to as the Kelvin–Helmholtz phase, and the contracting core during this phase is called a proto-neutron star (PNS) (Camelio et al., 2017; Pons et al., 1999; Prakash et al., 1997; Strobel et al., 1999). Depending on the final mass of the core after the short-lived life of a PNS, a black hole or neutron star (NS) is left behind. This chapter will focus on the structure and evolution of the compact stellar objects (proto-neutron stars) produced by the collapse of massive (8 M to around 20 M ) stars. The macroscopic evolution of the PNS during the Kelvin–Helmholtz phase, where a hot, lepton-rich PNS turns into a cold, deleptonized neutron star (Becker, 2009; Glendenning, 2012; Orsaria et al., 2019; Rezzolla et al., 2019; Sedrakian, 2007; Weber, 1999, 2005), is dependent on the microphysical ingredients of the star, the equation of state (EOS) of the dense matter comprising the core, and neutrino opacity (Pons et al., 1999). Immediately (in a matter of 0.1–0.5 s) following the core bounce during a massive star’s collapse and just prior to the Kelvin–Helmholtz phase, the PNS radius rapidly decreases from over 150 km to less than 20 km as pressure decreases as a result of neutrinos being released from the outer envelope of the star (Pons et al., 1999). While the star’s original matter rapidly compresses, the supernova’s shock causes accretion which results in a substantial increase in mass and total neutrino emission. These conditions make it so the copious amount of neutrinos cannot escape freely, and instead diffuse over the course of about a minute (the deleptonization stage) while a large fraction of the gravitational binding energy is released during the contraction of the stellar envelope (Foglizzo, 2016). After this minute-long period, neutrinos can escape freely, and the PNS enters a cooling stage where the entropy steadily decreases (Pons et al., 1999). The completion of the deleptonization and cooling stages signifying the end of the Kelvin–Helmholtz phase and the beginning of the life of a neutron star. The different stages in the evolution of hot proto-neutron stars to cold neutron stars, as described above, are schematically illustrated in Fig. 1. Proto-neutron stars are the compact remnants produced at the end of the evolution of intermediate-mass stars with masses of M  10M (see Mariani, 2020, and references therein). Their structure and composition passes through different physical stages within just a few

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Fig. 1. Schematic illustration of different temporal stages in the evolution of protoneutron stars to neutron stars (Prakash et al., 1997). They are characterized by different values of entropy (s) and lepton number (YL ). The formation of black holes (solid black spheres) is possible during different evolutionary stages, depending on the interplay between gravity and pressure. The transition of a hot PNS to a cold NS takes less than a minute. During the first few hundred years NSs cool quickly via neutrino emission from the core. Photon emission becomes the dominant cooling mechanism thereafter (Page et al., 2006).

seconds (see, for example, Pons et al., 1999; Prakash et al., 1997). Stars with M  10M are known to evolve in a complex fashion via nuclear burning. At the end of their lives, when most of the nuclear fuel has been consumed and massive cores of Fe (or O-Ne/Mg) have been built up, gravitational collapse occurs. During this phase (stage “1” in Fig. 1) a rebound of the outer mantle of the star occurs. The core is surrounded by a mantle characterized by low density but high entropy of the matter. The mantle extends for around 200 km and is stable until it explodes due to the aforementioned rebound. At this point, two different evolutionary tracks of the star are possible, which essentially depend on how powerful the explosion was. If the explosion was not strong enough to deleptonize the outer mantle,

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continued accretion of matter onto the star would give way to the formation of a black hole. The other alternative is that the star explodes successfully as a supernova (i.e. the mantle collapses and accretion of matter becomes less important), giving birth to a hot PNS where neutrinos are trapped in the stellar core (stage “2” in Fig. 1). During the next stage of evolution, the star begins to rapidly lose neutrinos. This leads to a reduction of the pressure due to deleptonization, which would be followed by the formation of a black hole if the gravitational pull on the matter overcomes the pressure provided by the matter. If this does not happen, the star will continue to deleptonize itself as it is being heated-up by the Joule effect of the escaping neutrinos (stage “3” in Fig. 1). It is assumed that the maximum heating of the star occurs immediately after the neutrinos have left the star. Continued cooling via neutrino emission from the stellar core (stage “4” in Fig. 1) (Malfatti, 2020) quickly reduces the star’s temperature to just a few MeV or less (Page et al., 2006). At such temperatures the matter in the core can be described by a cold nuclear EOS and the corresponding star is referred to as an NS. Understanding the physics behind a core collapse supernova and the subsequent formation of a PNS has been of interest in the particle and astrophysics communities for decades (see Burrows and Vartanyan, 2021; Janka, 2012; Mezzacappa, 2005, and references therein). The well-documented explosion of a type II supernova in the Large Magellanic Cloud in 1987 (SN1987a) was the first supernova event of this kind that could be studied in detail. Nineteen neutrinos have been detected from this event, which may be too few to provide a significant constraint on our understanding of the particle composition and physics of the supernova, but do provide an important milestone for these types of events. Since then, physicists have made great strides using numerical models to simulate supernova explosions (Camelio et al., 2017). More difficult to describe through numerical codes is the lifespan of a PNS, but recent efforts as in Fischer et al. (2010), H¨ udepohl et al. (2010) and Camelio et al. (2017) have been able to more accurately describe the quasi-stationary evolution of a PNS. In this chapter we investigate the structure and composition of (hot) proto-neutron stars. In part one of the paper, we introduce the fieldtheoretic Lagrangian that is used to compute models for the EOS of the matter in the cores of such stars. The relativistic mean-field approach is used to describe the interactions among nucleons mediated by scalar, vector and iso-vector mesons. In our calculations, we focus on the densitydependent SWL, DD2, and GM1L nuclear models (Malfatti et al., 2019;

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Spinella, 2017; Spinella and Weber, 2020; Typel et al., 2010). All three models account for the presence of hyperons as well as of Δ baryons in hot and dense matter. The possible existence of deconfined quarks in such matter will be briefly discussed in this paper too. Investigations of this topic have also been carried out by Steiner et al. (2000), Shao (2011), Mariani et al. (2017) and Malfatti et al. (2019). The relativistic finite-temperature Green function formalism is used to derive the equations that characterize ultra-hot and dense stellar matter (Weber, 1999). In part two of our study, the properties of non-rotating as well as rotating proto-neutron stars are studied by solving Einstein’s field equation using the models for the EOS derived in part one of the paper. The rotating stellar models are computed fully self-consistently, as required by the general relativistic expression for the Kepler (mass shedding) frequency. 2. Modeling Hot and Dense Neutron Star Matter 2.1. The nonlinear nuclear Lagrangian While an NS does get its namesake from the large quantities of neutrons created in the core during its birth, a more accurate depiction of interior composition is a mixture of neutrons and protons whose electric charge is balanced by leptons (L = e− , μ− ). Other particles may also exist in the core like hyperons B = [n, p, Λ, Σ± , Σ0 , Ξ0 , Ξ− ] (Glendenning, 1985) and the electrically charged states of the Δ isobar (Boguta, 1982; Pandharipande, 1971; Sawyer, 1972). The existence of these particles is made possible only if their Fermi energies become large enough that existing baryon populations need to be rearranged so that a lower energy state can be reached (Glendenning, 1985). To understand how the baryons within the core interact, we shall make use of the nonlinear density-dependent relativistic mean-field (DDRMF) theory. This theory describes the interactions between baryons in terms of meson exchange. These mesons include a scalar meson (σ) which describes attraction between baryons, a vector meson (ω) which describes repulsion, and an isovector meson (ρ) which is important to describe the baryon–baryon interactions in isospin asymmetric matter such as NS matter (Glendenning, 1985; Spinella, 2017). Due to the pion’s odd parity, this particle does not contribute at the mean-field description of dense matter. The nuclear Lagrangian of the theory is therefore given

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by (see also Glendenning, 2012; Sedrakian et al., 2022; Spinella and Weber, 2020; Weber, 1999)    L= ψ¯B γμ (i∂ μ − gωB ω μ − gρB τ · ρμ ) − (mB − gσB σ) ψB B

1 1 1 + (∂μ σ∂ μ σ − m2σ σ 2 ) − ˜bσ mN (gσN σ)3 − c˜σ (gσN σ)4 2 3 4 1 1 1 1 − ωμν ω μν + m2ω ωμ ω μ + m2ρ ρ μ · ρ μ − ρ μν · ρ μν , 4 2 2 4

(2)

where ψB stands for the various baryon fields, gσB , gωB and gρB are (density-dependent) meson–baryon coupling constants, and ˜bσ and c˜σ denote two additional coupling parameters associated with nonlinear (cubic and quartic) self-interactions introduced by (Boguta and Bodmer, 1977). The density-dependent coupling constants are given by Typel (2018)   −1 , (3) giB (n) = giB (n0 )ai 1 + bi (n/n0 + di )2 1 + ci (n/n0 + di )2 for σ and ω mesons (i = σ, ω), and by gρB (n) = gρB (n0 ) exp [ −aρ (n/n0 − 1) ] ,

(4)

for ρ mesons. Here the choice of parameters ai , bi , ci , and di account for nuclear medium effects, and are fixed by the binding energies, charge, and diffraction radii, spin–orbit splittings, and the neutron skin thickness of finite nuclei. The quantities mB , mσ , mω , mρ in Eq. (2) denote the masses of baryons and mesons and mN is the nucleon mass. The quantity τ = (τ1 , τ2 , τ3 ) is the Pauli isospin matrices. The quantities ω μν and ρ μν denote meson field tensors, where ω μν = ∂ μ ω ν − ∂ ν ω μ and ρ μν = ∂ μ ρ ν − ∂ ν ρ μ . The field equations of the baryon and meson fields are obtained by evaluating the Euler–Lagrange equations for the fields in Eq. (2). This leads for the baryon fields to   (5) (iγ μ ∂μ − mB ) ψB = gωB γ μ ωμ + 12 gρB γ μ τ · ρµ − gσB σ ψB . The field equation of the scalar σ-meson is given by    μ 2 3 gσB ψ¯B ψB − bσ mn gσN (gσN σ) − cσ gσN (gσN σ) , ∂ ∂μ + m2σ σ = B

(6)

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and the field equations of the vector mesons have the form  gωB ψ¯B γν ψB , ∂ μ ωμν + m2ω ων =

(7)

B

∂ μ ρμν + m2ρ ρν =



gρB ψ¯B τ γν ψB .

(8)

B

In relativistic mean-field approximation, the field equations (5)–(8) become  m2σ σ ¯= gσB nsB − ˜bσ mN gσN (gσN σ ¯ )2 − c˜σ gσN (gσN σ ¯ )3 , (9) B

¯ m2ω ω

=



gωB nB ,

(10)

gρB I3B nB ,

(11)

B

m2ρ ρ¯ =

 B

where I3B is the 3-component of isospin and nsB and nB are the scalar and particle number densities for each baryon B. The latter terms are given by † (x) ψB (x) , nB = ψB s ¯ n = ψB (x) ψB (x) , B

(12) (13)

† † 0 denotes the conjugate Dirac spinor and ψ¯ ≡ ψB γ respectively, where ψB stands for the adjoint Dirac spinor. To keep the notation to a minimum, we use the definition x ≡ (x0 , x).

2.2. Baryonic field theory at finite density and temperature To calculate the densities (12) and (13) for NS matter at finite temperature, we use the finite-temperature Green function formalism. The starting point is the spectral function representation of the two-point Green function given by (Dolan and Jackiw, 1974; Weber, 1999) g B (p0 , p) =

 dω

aB (ω, p) ω − (p0 − μB )(1 + iη)

− 2iπ sign(p0 − μB )

1 aB (p0 − μB , p), exp(|p0 − μB |/T ) + 1 (14)

where μB denotes the chemical potential of a baryon of type B and aB stands for the spectral function of that baryon (η > 0 and infinitesimally

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small). The spin and isospin dependences of g B and aB are not shown explicitly. The spectral function is obtained by evaluating aB (ω, p) =

 1  B g˜ (ω + iη, p) − g˜B (ω − iη, p) . 2iπ

(15)

Here g˜B denotes the analytically continued two-point Green function, which obeys the analytically continued Dyson equation, ˜ B (z, p))˜ g B (z, p) = −1. (γ 0 (z + μB ) − γ · p − mB − Σ

(16)

The spectral function has a scalar, vector and a time-like contribution generally written as (Weber, 1999) 0 B ˆ aB aB (p) = aB S (p) + γ · p V (p) + γ a0 (p),

(17)

∗ ∗ B ∗ B where aB S (p) = mB /(2EB (p)), aV (p) = −|p|/(2EB (p)), and a0 (p) = 1/2. For thermally excited anti-baryon states one has 0 B ¯B ˆa ¯B ¯0 (p), a ¯B (p) = a S (p) + γ · p V (p) + γ a

(18)

∗ ∗ ∗ B where a ¯B ¯B S (p) = −mB /(2EB (p)), a V (p) = |p|/(2EB (p)), and a0 (p) = 1/2. ∗ and effective baryon mass, m∗B , are The effective single-baryon energy, EB given by  ∗ (p) = p2 + m∗B 2 (19) EB

and m∗B = mB − gσB σ ¯,

(20)

respectively. In terms of the two-point Green function, the expression for the baryon number density (12) becomes    0 (21) d3 x g B (x, x+ ) + g B (x, x− ) , nB = i Tr γ where the trace is to be taken over the spin and isospin matrix indices. Transformation of Eq. (21) to momentum space leads to  0 d4 p iηp0 0 nB = i Tr γ (e + e−iηp )g B (p). (22) (2π)4

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Next we note that   3 d4 p iηp0 0 B 0 d p B e p g (p , p) = −i a (p)ωB (p)fB − (p), 4 (2π) (2π)3

(23)

and 

d4 p −iηp0 0 B 0 e p g (p , p) = i (2π)4



d3 p B a ¯ (p)¯ ωB (p)fB + (p), (2π)3

(24)

which leads for Eq. (22) to  n B = γB

d3 p (fB − (p) − fB + (p)), (2π)3

(25)

where γB ≡ (2JB + 1) accounts for the spin-degeneracy. The quantities fB ± in Eq. (25) denote Fermi–Dirac distribution functions given by fB − (p) =

1 ∗ ∗ e(EB (p)−μB )/T

+1

,

(26)

and fB + (p) =

1 ∗ ¯∗ e(−EB (p)+μB )/T

+1

.

(27)

The quantity μ∗B in Eqs. (26) and (27), given by ˜, ¯ − gρB ρ¯I3B − R μ∗B = μB − gωB ω

(28)

defines the effective baryon chemical potential in terms of the standard ˜ chemical potential and the mean-fields of σ and ρ mesons. The quantity R is the rearrangement term given by (Fuchs et al., 1995; Spinella and Weber, 2020)

 ∂gωB (n) ∂gρB (n) ∂gσB (n) s ˜ nB ω I3B nB ρ¯ − nB σ R= ¯+ ¯ . (29) ∂n ∂n ∂n B

This term is mandatory for thermodynamic consistency, proven with the Hugenholtz–van Hove theorem that relates the total baryonic pressure (which contains the rearrangement term) of a particle to its chemical potential (Hofmann et al., 2001). The expression of the total baryonic pressure of the standard nonlinear relativistic mean-field theory therefore contains ˜ (see Eq. (54)). the additional term nR

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The single-baryon energies, ωB (p), are given in terms of these meson ∗ , according to the relation fields plus the effective single-baryon energies, EB ∗ ωB (p) = gωB ω ¯ + gρB I3B ρ¯ + EB (p).

(30)

Similarly, the single-particle energies of thermally excited anti-baryon states, ω ¯ B (p), are given by ∗ ω ¯ B (p) = gωB ω ¯ + gρB I3B ρ¯ − EB (p) .

(31)

From the above relations, one sees that for baryons ∗ ωB (p) − μB = EB (p) − μ∗B ,

(32)

and for states outside the Fermi sea of anti-particles ∗ −¯ ωB (p) + μB = EB (p) + μ∗B .

(33)

With these definitions, the traces in Eq. (22) and in the expressions for the energy density and pressure to be discussed below can be calculated. In particular, one obtains ∗ Tr aB = γB m∗B /EB ,

Tr γ 0 aB = γB ,

∗ Tr a ¯B = −γB m∗B /EB ,

(34)

Tr γ 0 a ¯B = −γB .

(35)

Next we turn to the scalar density, nsB , defined in Eq. (13). Expressed in terms of the two-point Green function, Eq. (13) reads    (36) nsB = i Tr d3 x g B (x, x+ ) + g B (x, x− ) . Transforming this expression to momentum space gives  0 d4 p iηp0 nsB = i Tr (e + e−iηp )g B (p). 4 (2π)

(37)

By making use of Eq. (24), the integration of p0 can be carried out analytically. The Green functions then get replaced by the baryon spectral functions and the Fermi–Dirac distribution functions, leading to the final result for the scalar density given by  3 d p m∗B (38) nsB = γB ∗ (p) (fB − (p) + fB + (p)). (2π)3 EB

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3. Composition and EOS of Hot and Dense (Proto-) Neutron Star Matter The total energy density and pressure of the stellar matter are calculated from the energy–momentum tensor Tμν (x) = gμν L(x) +

 B

∂L(x) ∂ν ψB (x), ∂ ∂ μ ψB (x)

(39)

with the Lagrangian L given by Eq. (2). The energy density and pres sure are given by = T 00  and P = 13 k T kk , respectively. Using the Green function formalism, the expression for the energy density is given by (Weber, 1999)   0 0 d4 p Tr (eiηp + e−iηp ) = i 4 (2π) B  1 × p0 γ 0 − gσB σ ¯ + γ 0 (gωB ω ¯ + gρB I3B ρ¯) g B (p) 2  3 1  4 1˜ − bσ mN gσN σ − c˜σ gσN σ . (40) 6 4 The integration over p0 in Eq. (40) can be carried out analytically via contour integration, which leads to  3  d p B d4 p iηp0 B 0 e g (p , p) = −i a (p)fB − (p) (41) (2π)4 (2π)3 and



d4 p −iηp0 B 0 e g (p , p) = i (2π)4



d3 p B a ¯ (p)fB + (p). (2π)3

(42)

The energy density is then given as a momentum integral over single-baryon energies, baryon spectral functions, and Fermi–Dirac distribution functions, as shown below:  3  d p Tr (ω B (p)γ 0 aB (p)fB − (p) − ω ¯ B (p)γ 0 a ¯B (p)fB + (p)) = (2π)3 B  3  1 d p  − Tr ( −gσB σ ¯ + γ 0 (gωB ω ¯ + gρB I3B ρ¯) aB (p)fB − (p) 2 (2π)3 B  B  ¯ + γ 0 (gωB ω ¯ + gρB I3B ρ¯) a ¯ (p)fB + (p)) − −gσB σ 3 1  4  1˜ − bσ mN gσN σ − c˜σ gσN σ . (43) 6 4

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By making use of Eqs. (31), (34) and (35), this expression can be written as  3  d p ∗ γB E (p) (fB − (p) + fB + (p)) = (2π)3 B B  3  d p + γB (gωB ω ¯ + gρB I3B ρ¯) (fB − (p) − fB + (p)) (2π)3 B

 3 1 d p m∗B 0 − gσB σ γB ¯ + γ (gωB ω ¯ + gρB I3B ρ¯) − ∗ 2 (2π)3 EB (p) B

× (fB − (p) − fB + (p))  3 1  4 1 − ˜bσ mN gσN σ − c˜σ gσN σ . 6 4

(44)

It is customary to express Eq. (44) in a more compact way. This is accomplished by noticing that, according to Eq. (38), the integral over the first term in the third line above can be written as  3   d p m∗B gσB σ γB ¯ (fB − (p) + fB + (p)) = gσB nsB σ ¯ . (45) ∗ 3 (2π) EB (p) B B  s Making use of the σ-meson-field equation (9) to replace B gσB nB in Eq. (45) leads after some algebra to the final result for the energy density given by (Weber, 1999)  3  1 1 d p ∗ γB EB (p) (fB − (p) + fB + (p)) + m2σ σ ¯ 2 + m2ω ω ¯2 = 3 (2π) 2 2 B

 3 1  4 1 1 + m2ρ ρ¯2 + ˜bσ mN gσN σ + c˜σ gσN σ . 2 3 4

(46)

The expression for the pressure of hot NS matter has the form (Weber, 1999)   0 0 d4 p P = i Tr (eiηp + e−iηp ) (2π)4 B

 1 1 × γ · pˆ + −gσB σ ¯ + γ 0 (gωB ω ¯ + gρB I3B ρ¯) g B (p) 3 2 3 1  4  1˜ + bσ mN gσN σ + c˜σ gσN σ . (47) 6 4 As for the energy density, the integration of p0 can be carried out analytically using the mathematical relations shown in Eqs. (41) and (42).

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This leads to 1 P = Tr 3



d3 p (γ · pˆ aB (p)fB − (p) − γ · pˆ a ¯B (p)fB + (p)) (2π)3 B  3  1 d p  + Tr ( −gσB σ ¯ + γ 0 (gωB ω ¯ + gρB I3B ρ¯) aB (p)fB − (p) 2 (2π)3 B   B − −gσB σ ¯ + γ 0 (gωB ω ¯ + gρB I3B ρ¯) a ¯ (p)fB + (p))     1 1 3 4 + ˜bσ mN gσN σ + c˜σ gσN σ . (48) 6 4

With the help of Eqs. (17) and (18) for the spectral functions and Eqs. (34) and (35) for the traces, Eq. (48) can be written as 1 γB P = 3



p2 d3 p ∗ (p) (fB − (p) + fB + (p)) (2π)3 EB B

 3 1 d p m∗B gσB σ γB ¯ + gωB ω ¯ + gρB I3B ρ¯ + − ∗ 2 (2π)3 EB (p) B

× (fB − (p) − fB + (p))  3 1  4 1 + ˜bσ mN gσN σ + c˜σ gσN σ . 6 4

(49)

The second line in this equation can be written in terms of the scalar and baryon number densities. To see this we begin with Eq. (38), from which it follows that 



 d3 p m∗B gσB σ ¯ (fB − (p) − fB + (p)) = gσB nsB σ ¯. ∗ 3 (2π) EB (p)

γB

B

(50)

B

On the other hand, it is known from the σ-meson-field equation (9) that 

3 4   gσB nsB σ ¯ = m2σ σ ¯ 2 + ˜bσ mN gσN σ + c˜σ gσN σ .

(51)

B

Similarly, for the ω-meson-dependent term in Eq. (49) we have  B

 γB

 d3 p − (p) − fB + (p)) = g ω ¯ (f gωB nB ω ¯ ωB B (2π)3 =

B m2ω ω ¯2

,

(52)

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and for the ρ-meson-dependent term    d3 p γB gρB I3B ρ¯ (fB − (p) − fB + (p)) = gρB I3B nB ρ¯ 3 (2π) B

=

B m2ρ ρ¯2 .

(53)

Substituting Eqs. (50)–(53) into Eq. (49) leads for the pressure of NS matter to (Weber, 1999)  3 1 p2 1 1 d p P = (f − (p) + fB + (p)) − m2σ σ γB ¯ 2 + m2ω ω ¯2 ∗ 3 (2π)3 EB (p) B 2 2 B

1 + m2ρ ρ¯2 − 2

3 1  4  1˜ ˜. bσ mN gσN σ − c˜σ gσN σ + nR 3 4

(54)

3.1. Leptons and neutrinos Leptons are treated as free Fermi gases with the grand canonical potential given by (Malfatti et al., 2019; Weber, 1999)  γL  d3 p p2 (f − (p) + fL+ (p)), ΩL = − (55) 3 (2π)3 EL (p) L L

where γL = (2JL + 1) is the lepton degeneracy factor. The sum over L in Eq. (55) runs over e− and μ− , with masses mL , and massless neutrinos, νe , in the case they are trapped in a PNS (see Secs. 3.2 and 3.3). The lepton distribution function is given by fL∓ (p) = where EL (p) = leptons.

1 , e(EL (p)∓μL )/T + 1

(56)

 p2 + m2L denotes the energy–momentum relation of free

3.2. Chemical equilibrium and electric charge neutrality Three important constraints must be taken into account when determining the EOS of PNS matter: electric charge neutrality, baryon number conservation, and chemical equilibrium. Neutron star matter must be charge neutral, satisfying (Glendenning, 1985; Malfatti et al., 2019; Weber, 1999)   qB nB + qL nL = 0, (57) B

L

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where qB and qL are baryon and lepton electric charge, respectively. Baryon number must also be conserved, which leads to  nB − n = 0. (58) B

Finally, the constraint of chemical equilibrium for hadronic matter can be defined as (Prakash et al., 1997) μB = μn + qB (μe − μνe ) ,

(59)

where μn , μe and μνe are the neutron, electron and neutrino chemical potentials, respectively. The chemical potential of the latter follows from the equilibrium reaction e− ↔ μ− + νe + ν¯μ ,

(60)

which leads for the corresponding chemical potentials to the condition μe = μμ + μνe + μν¯µ .

(61)

Neutrinos are trapped inside of a proto-neutron star immediately after its formation. Mathematically this is expressed as (Malfatti et al., 2019; Prakash et al., 1997) ne + nνe , n nμ + nνµ = 0, Yμ = n Ye =

(62)

where ne , nμ , nνe , and nνµ denote the number densities of electrons, muons, electron neutrinos, and muon neutrinos, respectively. During this very early stellar phase, the matter is opaque to neutrinos and the composition of the matter is characterized by three independent chemical potentials, namely μn , μe , and μνe . The condition YLµ = 0 expressed in Eq. (62) reflects the fact that only very few muons are present in PNS matter right after core bounce, when neutrinos are still trapped. The value of YLe ( 0.4) depends on the efficiency of electron capture reactions during the initial state of the formation of proto-neutron stars (Prakash et al., 1997). The quantity YL , defined as YL = Ye + Yνe ,

(63)

is used in the figures to show the relative fractions of electrons and neutrinos for which the respective curves have been computed.

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other dense-matter properties presented in this chapter) shown in Figs. 2 and 3 have been computed for B = n, p, Λ, Σ± , Σ0 , Ξ0 , Ξ− , all electrically charged states of the Δ(1232) baryon, and L = e, μ, νe . The values of the baryon–hyperon coupling constants will be discussed in detail in Sec. 5.1. The values chosen for the Δ–hyperon couplings are xσΔ = xωΔ = 1.1 and xρΔ = 1.0 as described in Sec. 5.2. A general investigation of the Δ(1232) coupling spaces is provided in Sec. 5.3.

3.3. Composition of hot and dense matter In this section we show the composition of hot and dense matter as it exists in the cores of proto-neutron stars. Following the core bounce post supernova explosion, PNSs experience a deleptonization stage where hot, lepton-rich matter becomes lepton-poor over the course of about a minute. During this time, the entropy per baryon and lepton fraction of the dense matter within the PNS core change quickly. These values start at around s = 1 and YL = 0.4, change to s = 2 and YL = 0.2 after around 0.5 to 1 seconds, and take values of s = 2 and Yνe = 0 about 15 to 30 seconds after the birth of a proto-neutron star (Malfatti et al., 2019; Prakash et al., 1997; Strobel et al., 1999). As neutrinos and photons diffuse from the object the stellar temperature drops to less than 1 MeV and a hot PNS becomes a cold NS. Figures 6–9 illustrate how drastically the particle composition in the core of a neutron star changes with temperature. In fact, as can be seen by comparing the compositions shown in Figs. 6 and 8 with each other, the particle composition at a temperature of 25 MeV no longer resembles the zero-temperature (i.e. 1 MeV) composition at all. Moreover, in matter at even higher temperatures the threshold densities of all the baryons have changed so much that all baryonic particle states taken into account in our calculations are present at all densities, as shown in Fig. 9. The next set of figures show the composition of proto-neutron star matter for different combinations of entropy and lepton number, which characterize several different stages in the evolution of a hot, newly formed PNS to a cold NS. Proto-neutron stars in their earliest phases of evolution have s = 1 and YL = 0.4 followed by s = 2 and YL = 0.2. The particle compositions of such matter are shown in Figs. 10–13 for the DD2 and GM1L parameterizations. The matter in proto-neutron stars with s = 2 and YL = 0.2 undergoes deleptonization and becomes lepton poor. Such matter is characterized by s = 2 and Yνe = 0 (neutrinos are no longer

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4. The Hadron–Quark Phase Transition In this chapter, we briefly turn to the study of quark matter in compact stars. The possible existence of such matter in compact stars was already discussed in the 1960s by Ivanenko and Kurdgelaidze (1965) and in the 1970s by Baym and Chin (1976); Chapline and Nauenberg (1977); Fechner and Joss (1978); Fritzsch et al. (1973); Itoh (1970); Keister and Kisslinger (1976). Since then, a large number of scientific papers have been published describing the possible existence of quark matter in neutron stars with increasingly improved theoretical models (see, for instance, Alford et al., 2008; Baym et al., 2018; Blaschke and Chamel, 2018; Bonanno and Sedrakian, 2012; Burgio and Plumari, 2008; Orsaria et al., 2013, 2014; Page and Reddy, 2006; Tolos and Fabbietti, 2020, and references therein). In the following, we concentrate on the hadron–quark phase transition as described by the Nambu–Jona-Lasinio (NJL) model (Buballa, 2005; Fukushima and Hatsuda, 2011; Fukushima and Sasaki, 2013; Hatsuda and Kunihiro, 1994; Klevansky, 1992). We shall use a non-local variant of the NJL model, denoted 3nPNJL, which includes vector interactions as well as the Polyakov loop. The Lagrangian of this model is given by [Malfatti et al. (2019)]  GV μ μ GS  s s ν ¯ ja ja − ja ja + jap jap Dν + m)ψ ˆ + L = ψ(−iγ 2 2  s s s  H p − Aabc ja jb jc − 3 jas jb jcp + U [ A ] , 4

(64)

where U[A] accounts for the Polyakov loop dynamics and the H-dependent term is the ’t Hooft term responsible for quark flavor mixing. The quark ˆ = diag(mu , md , ms ) is the fields are described by ψ ≡ (u, d, s)T and m current quark mass matrix. The quantities jaμ , jas , and jap denote scalar (s), pseudo-scalar (p), and vector (μ) interaction currents, respectively, and GS and GV are the scalar and vector coupling constants. It is customary to express GV in multiples of GS and to write their ratio as ζv ≡ GV /GS . The covariant derivative is given by Dν ≡ ∂ν − igAaν ta , where Aaν are the gluon fields and ta = λa /2 the generators of SU(3) (for more details, see Malfatti et al., 2019). To model the phase equilibrium between hadronic matter and quark matter in a neutron star, we assume here that this equilibrium is of first order and Maxwell-like, that is, the pressure in the mixed hadron–quark phase is constant. Theoretically the transitions could be Gibbs-like as well, depending on the surface tension at the hadron–quark interface. The value

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of the surface tension is however only poorly known. Lattice gauge calculations, for instance, predict surface tension values in the range of 0 − 100 MeV fm−2 (Kajantie et al., 1991). Using different theoretical models for quark matter, a range of values for the hadron–quark surface tension have been obtained in the literature (see, for example, Alford et al., 2001; Ke and Liu, 2014; Lugones et al., 2013, and references therein). According to theoretical studies, surface tensions above around 70 MeV fm−2 favor the occurrence of a sharp (Maxwell-like) hadron–quark phase transition rather than a softer Gibbs-like transition (Sotani et al., 2011; Yasutake et al., 2014). The EOS of both the hadronic phase and the quark phase is obtained from the Gibbs relation  μi n i , (65) = −P + T S + i

where pressure, entropy, and the particle number densities are given by P = ∂P −Ω, S = ∂P ∂T , and ni = ∂μi , respectively. To construct the hadron–quark phase transition we adopt the Gibbs condition for equilibrium between both phases, expressed as GH (P, T ) = GQ (P, T ),

(66)

where GH and GQ are the Gibbs free energies per baryon of the hadronic (H) and the quark (Q) phase, respectively, to be determined at a given pressure and transition temperature. The crossing of GH and GQ in the G − P plane then determines the pressure and density at which the phase transition occurs for a given transition temperature. The expressions of GH and GQ are given by  nj μj , (67) Gi (P, T ) = n j where i = H or Q and the sum over j is over all the particles present in each phase. For the hadron–quark phase transition, the particle chemical potentials in each phase are different, so that is becomes necessary to calculate the Gibbs free energy as a function of pressure to construct the phase transition. Results for the hadron–quark phase transitions are shown in Fig. 18 for the DD2 nuclear model and in Fig. 19 for the GM1L nuclear model. Two phase transitions are visible in each figure, depending on the value of the vector coupling constant, ζv (= GV /GS ). The solid black and gray lines in these figures represent the hadronic DD2 and GM1L EOSs, respectively,

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and the dash-dotted and dashed lines are the EOSs of the quark phase computed for the 3nPNJL model. The horizontal lines indicate the locations of the hadron–quark phase transitions where GH (P, T = 0) = GQ (P, T = 0) according to Eq. (66). The hadronic and the quark matter EOS are very similar at pressures where GH (P, T = 0) ≈ GQ (P, T = 0) (Malfatti et al., 2019). This makes it difficult to distinguish between the two phases in the relevant pressure regions, P ∼ 100−400 MeV/fm3 . This can be interpreted as a masquerading behavior of dense matter, different from pure deconfined quark matter (see Malfatti et al., 2019, and references therein for details). The 2M constraint of PSR J1614-2230 and PSR J0348+043 (Antoniadis et al., 2013; Arzoumanian et al., 2018; Demorest et al., 2010; Lynch et al., 2013) and the assumption that quark matter exists in the cores of neutron stars have been used to determine the range of the vector coupling constant ζv in the quark matter phase. This leads to 0.331 < ζv < 0.371 for GM1L, and 0.328 < ζv < 0.385 for DD2, where the lower bounds are determined by the 2 M mass constraint and the upper bounds by the requirement that quark matter exists in the cores of neutron stars. In Figs. 20 and 21 we show the quark compositions of cold neutron stars

Fig. 20. Particle population of stellar quark matter at zero temperature as a function of baryon number density (Malfatti et al., 2019). The gray area indicates the density regime where matter described by the hadronic DD2 model exists. The hadron phase ends abruptly at the vertical line slightly above 0.6 fm−3 . The population of muons is increased by a factor of 100 to make it visible. The strength of the vector repulsion among quarks is ζv = 0.328.

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Fig. 21. Same as Fig. 20, but for a vector repulsion among quarks of ζv = 0.331 (Malfatti et al., 2019).

computed for GM1L in combination with 3nPNJL and DD2 in combination with 3nPNJL, respectively.

5. The Parameters of the Hadronic Theory For this study, we will consider three popular nuclear parameterization sets which are denoted SWL, GM1L and DD2 (Spinella, 2017; Spinella et al., 2018; Typel et al., 2010). The parameter values of these sets are shown in Table 1 and the corresponding saturation properties of symmetric nuclear matter are shown in Table 2 (Malfatti et al., 2019). These are the nuclear saturation density n0 , energy per nucleon E0 , nuclear compressibility K0 , effective nucleon mass m∗N /mN , asymmetry energy J, asymmetry energy slope L0 , and the value of the nucleon potential UN . The values of L0 listed in Table 2 are in agreement with the value of the slope of the symmetry energy deduced from nuclear experiments and astrophysical observations (Oertel et al., 2017). The DD2 parameterization is designed such that it eliminates the need for the nonlinear self-interactions of the σ meson shown in Eqs. (2) and (6) (Malfatti et al., 2019). The nonlinear terms are therefore only considered for the GM1L model. As already mentioned in Sec. 2.1, the baryons considered in this study to populate NS matter include all states of the spin- 12 baryon octet comprised of the nucleons (n, p) and hyperons

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Table 1. Parameters of the SWL and GM1L (Spinella, 2017; Spinella et al., 2018) and DD2 (Typel et al., 2010) parameterizations used in this work. Parameters

Units

SWL

mσ mω mρ gσN gωN gρN ˜bσ c˜σ aσ bσ cσ dσ aω bω cω dω aρ

GeV GeV GeV — — — — — — — — — — — — — —

0.550 0.783 0.763 9.7744 10.746 7.8764 0.003798 −0.003197 0 0 0 0 0 0 0 0 0.3796

GM1L 0.550 0.783 0.7700 9.5722 10.6180 8.1983 0.0029 −0.001068 0 0 0 0 0 0 0 0 0.3898

DD2 0.5462 0.783 0.7630 10.6870 13.3420 3.6269 0 0 1.3576 0.6344 1.0054 0.5758 1.3697 0.4965 0.8177 0.6384 0.5189

Table 2. Properties of symmetric nuclear matter at saturation density for the SWL and GM1L (Spinella, 2017; Spinella et al., 2018) and DD2 (Typel et al., 2010) parameterizations. Saturation property

Units

SWL

GM1L

DD2

n0 E0 K0 m∗N /mN J L0 UN

fm−3 MeV MeV — MeV MeV MeV

0.150 −16.0 260.0 0.70 31.0 55.0 −64.6

0.153 −16.3 300.0 0.70 32.5 55.0 −65.5

0.149 −16.02 242.7 0.56 31.67 55.04 −75.2

(Λ, Σ+ , Σ0 , Σ− , Ξ0 , Ξ− ). In addition, all states of the spin- 32 delta isobar Δ(1232) (Δ++ , Δ+ , Δ0 , Δ− ) are taken into account as well. 5.1. The meson–hyperon coupling space A detailed discussion of the meson–hyperon coupling constants giH (where i = σ, ω, ρ) can be found in Malfatti et al. (2019); Spinella (2017),

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Malfatti et al. (2020) and Spinella and Weber (2020). As usual we express the values of the meson–hyperon coupling constant, giH , in terms of the meson–nucleon coupling strength, giN , that is, xiH = giH /giN . The meson–hyperon couplings are not well constrained experimentally compared to those of the nucleons. However, the scalar meson–hyperon couplings (xσH ) can be constrained by the available experimental data on hypernuclei, but their calculation first requires the determination of the vector meson-hyperon couplings (xωH ). The coupling scheme used in our study is based on the Nijmegen extended-soft-core (ESC08) model (Rijken et al., 2010). The scalar meson–hyperon coupling constants (xσH ) can be fit to the hyperon potential depths, UH at nuclear saturation density, n0 . Our parameters sets are fitted to potential depths of UΛ = −28 MeV, UΞ = −18 MeV, and UΣ = +30 MeV (see Friedman and Gal, 2021; Schaffner-Bielich and Gal, 2000; Spinella and Weber, 2020; Tolos and Fabbietti, 2020, and references cited therein). The values of the isovector meson– hyperon coupling constants are chosen as xρH = 2|I3H | (Maslov et al., 2016; Miyatsu et al., 2013; Weissenborn et al., 2012). 5.2. Δ(1232) isobars The potential presence of the delta isobar Δ(1232) in neutron star matter (Boguta, 1982; Huber et al., 1998; Pandharipande, 1971; Sawyer, 1972) has been relatively ignored, especially when compared to the attention that hyperons have received in the literature. It is reasonable to assume Δs would not be favored in NS matter for a number of reasons. First, their rest mass is greater than both the Λ and Σ hyperons. Second, negatively charged baryons are generally favored as their presence reduces the high Fermi momenta of the leptons, but the Δ− has triple the negative isospin of the neutron (I3Δ− = −3/2), and thus its presence should be accompanied by a substantial increase in the isospin asymmetry of the system. However, these arguments now appear to be largely invalid since recent many-body calculations paint a different picture (Cai et al., 2015; Chen et al., 2009; Dexheimer and Schramm, 2008; Drago et al., 2014; Lavagno, 2010; Li et al., 2018; Malfatti et al., 2020; Sch¨ urhoff et al., 2010; Spinella, 2017; Zhu et al., 2016). Recent theoretical works have suggested conflicting constraints on the saturation potential of the Δs in symmetric nuclear matter given by ˜, ¯ − xσΔ gσN σ ¯+R UΔ (n0 ) = xωΔ gωN ω

(68)

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˜ denotes the rearrangement term of Eq. (29). Drago et al. (2014) where R incorporated a number of experimental and theoretical results to deduce the following range for UΔ at n0 , −30 MeV + UN (n0 )  UΔ (n0 )  UN (n0 ) ,

(69)

indicating a slightly more attractive potential than that of the nucleons. Included in these analyses was an analysis of the photo-excitation of nucleons to Δs that suggested the following relation between the scalar and vector couplings, 0 < xσΔ − xωΔ < 0.2 .

(70)

Kolomeitsev et al. (2017) cited numerous studies of Δ production in heavy-ion collisions to suggest a less attractive potential in the range UN (n0 )  UΔ (n0 )  23 UN (n0 ), finally settling on UΔ (n0 ) ≈ −50 MeV as a best estimate (Riek et al., 2009). However, it is worth noting that constraining the potential does not directly constrain xσΔ or xωΔ , rather the relationship between the two. The meson–Δ coupling space will be systematically investigated in Sec. 5.3, but first the particle number densities in the presence of both hyperons and Δs will be examined with the following set of couplings, xσΔ = xωΔ = 1.1, xρΔ = 1.0 .

(71)

These lead to saturation potentials more attractive than that of the nucleons as shown in Table 3. The scalar and isovector meson–hyperon coupling constants will continue to be determined as described in Sec. 5.1 and the vector meson–hyperon couplings will be given by the SU(3) ESC08 model. The properties of maximum mass NSs made of hyperonic matter with and without the Δ states are shown in Table 4. The properties include the stellar mass M , the radius R, and the baryon number density nc at the center of the stars. Also shown are the radii R1.4 of neutron stars with a Table 3. Saturation potentials of nucleons and Δs in symmetric nuclear matter with xσΔ = xωΔ = 1.1 and xρΔ = 1.0 (Spinella, 2017). Potential

SWL

UN (n0 ) (MeV) −64.6 UΔ (n0 ) (MeV) −71.1

GM1L −65.5 −72.1

DD2 −75.2 −86.0

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Table 4. Properties of maximum mass NSs with Δs and hyperons with vector meson–hyperon coupling constants given in SU(3) symmetry with the ESC08 model (Spinella, 2017). Hyperons EOS SWL GM1L DD2

Hyperons plus Δs

M (M )

R (km)

nc (1/fm3 )

R1.4 (km)

M (M )

R (km)

nc (1/fm3 )

R1.4 (km)

2.01 2.04 2.09

11.5 11.6 12.1

0.98 0.95 0.89

12.80 12.82 13.45

2.02 2.04 2.11

11.4 11.5 11.9

1.00 0.97 0.92

12.85 12.90 13.28

canonical mass of 1.4 M. As can be seen, including the Δ baryon actually leads to equal or marginally greater maximum masses in both cases. While the maximum masses are very similar, the mass–radius curves differ slightly due to the low density appearance of the Δ− that causes a bend toward lower radii reducing the radius of the canonical 1.4 M NS. We also note that specifying the vector meson–hyperon couplings with the SU(3) ESC08 model rather than SU(6) is necessary in order to satisfy the ∼ 2 M mass constraint with the GM1L and DD2 parameterizations (Spinella, 2017). The relative particle number densities for the SU(3) coupling scheme are presented in Fig. 22 for the GM1L and DD2 parameterizations. For GM1L the Δ− is the first additional baryon to be populated at ∼ 2.3 n0 and reaches nearly the same number density as the proton before it starts being replaced by the Ξ− at around 4 n0 . In DD2 the Δ− again precedes the onset of hyperonization but appears at an extremely low baryon number density of around 1.8 n0 and again reaches densities comparable to that of the proton before beginning to decline due to the population of the Ξ− at around 4 n0 . At low densities the Σ− and Ξ− may be disfavored in comparison to the Δ− due to a repulsive potential and significantly higher rest mass respectively. However, the low Δ− critical density in DD2 is primarily due to the density dependence of the isovector meson–baryon coupling that greatly reduces the isovector contribution to the Δ chemical potential compared to standard relativistic mean-field calculations. The early appearance of the Δ− is directly related to the slope of the asymmetry energy, L0 , as discussed by Drago et al. (2014). The extreme low density appearance of the Δ− has important consequences for the mass-radius curve of an NS, since it bends toward smaller radii much more substantially compared to EOSs where Δs are absent. The effect is the most drastic for the DD2 parameterization, where the presence of Δs reduces the

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5.3.1. The σωΔ coupling space The exploration of the σωΔ coupling space begins with a heatmap for the Δ saturation potential UΔ (n0 ) in symmetric nuclear matter given in Fig. 23. First, it is important to note that including the Δ baryon can result in a rapid increase of the scalar field (¯ σ ) causing a correspondingly rapid decrease in the effective baryon masses (see Eq. (20)), some becoming negative before the maximum central density is reached. This invalidates the EOS and the associated couplings; as a result, much of the σωΔ coupling space is not accessible with a given EOS model and parameterization. These areas are identified in Fig. 23 and the heatmaps to follow as empty (white) pixels. Further, Δs do not populate for a significant region of the coupling space due to the presence of hyperons, and these couplings are identified by the gray pixels. In particular, we find that for the chosen parameterizations Δs are largely absent when xσΔ − xωΔ  −0.1, and do not populate at all when xσΔ − xωΔ < −0.2. A study by Zhu et al. (2016) investigated Δs in the density-dependent relativistic Hartree–Fock (DDRHF) approach, and much of the analysis therein was conducted with xσΔ = 0.8 and xωΔ = 1.0. However, they did not account for hyperonization and our results suggest that Δs may not even appear with the given choice of couplings, illustrating the importance of simultaneously considering hyperons. Finally, our investigation of the coupling range spanning −0.25  xσΔ − xωΔ  0.25 appears sufficient, as outside this range Δs either do not populate or their presence results in an EOS that is erroneous due to either a negative effective baryon mass or a pressure that is not monotonically increasing. Thus, if Δs are to appear in NS matter, xσΔ and xωΔ are likely relatively close in value. Figure 23 indicates that an increase in either xσΔ or xσΔ − xωΔ results in a decrease in UΔ , the potential becoming more attractive. The region between the top two contours is consistent with the potential constraint suggested by Drago et al. (2014) given in Eq. (69). Satisfaction of this constraint requires that 1.0  xσΔ  1.7 in SWL and DD2, and 0.9  xσΔ  1.6 in GM1L. Requiring that Eq. (70) be simultaneously satisfied completely excludes the bottom half of the coupling space and leaves only a limited region in the top-middle that is consistent with the constraints, this region including the previously employed couplings indicated by the star marker and given in Eq. (71). The region between the bottom two contours is consistent with the potential constraint suggested in Kolomeitsev et al. (2017). However, if we simultaneously require the

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Fig. 23. Nuclear saturation potential (in MeV) of Δs in symmetric nuclear matter in the σωΔ coupling space (Spinella, 2017). Hyperons were included with the vector meson– hyperon given by the SU(3) ESC08 model. The star marker indicates the location of xσΔ = xωΔ = 1.1 and xρΔ = 1.0. Dashed contours are lines of constant potential as labeled and represent possible constraints. Gray pixels indicate that no Δs were populated for the given set of couplings. White pixels indicate couplings for which the effective mass of at least one baryon became negative before the maximum baryon number density of the NS was reached.

satisfaction of Eq. (70) here the SWL and DD2 parameterizations are completely excluded, and the Δ couplings are limited to a very small range in the GM1L parameterization. The maximum mass of NSs in the σωΔ coupling space is shown in Fig. 24. The maximum mass constraint is satisfied by the majority of the meson–Δ coupling space in all parameterizations, with large regions producing a maximum mass greater than that of the purely hyperonic EOS with ESC08 vector couplings indicated by the solid contours. Consequently, the maximum mass constraint alone does not serve to constrain xσΔ and xωΔ significantly. The highest maximum masses appear where both the Δ saturation potential is the most attractive and the difference between the scalar and vector meson–Δ couplings is the greatest. Satisfaction of both Eq. (70) and the mass constraint requires xσΔ > 1.0 for SWL, xσΔ > 0.9 for GM1L, and xσΔ > 0.975 for DD2. Kolomeitsev et al. (2017) concluded that the most likely value for UΔ (n0 ) ≈ −50 MeV, and we find that the

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Fig. 24. Maximum mass (in solar mass units M ) in the σωΔ coupling space (Spinella, 2017). Hyperons were included with the vector meson–hyperon coupling constants given by the SU(3) ESC08 model. Solid lines are maximum mass contours for the associated hyperonic EOS (no Δs) in the ESC08 model. Colorbar tick marks represent the maximum mass constraints set by PSR J0348+0432 (1.97−2.05 M at 1σ, and 1.90−2.18 M at 3σ). Markers, contours, and pixels are as described for Fig. 23.

maximum mass constraint can only be satisfied with this potential provided xωΔ > xσΔ , violating Eq. (70) (Kolomeitsev et al., 2017; Riek et al., 2009). The total number NΔ of delta isobars present in a given NS model can be calculated from  4πr2 dNΔ = nΔ (r), dr 1 − 2m(r)/r Δ

(72)

which is to be solved in combination with the TOV equation that will be introduced in Sec. 6. NΔ is given in Fig. 25 as a fraction of the total baryon number, fΔ = NΔ /NB . The Δ fraction varies considerably in the range 2%  fΔ  18% when the σωΔ couplings are consistent with the constraints given in Eqs. (69) and (70). However, a quick examination of the same region in Fig. 24 reveals that this variance in fΔ has little to no effect on the maximum stellar mass. It appears that there is a fΔ hot-spot that is centered in a region of the σωΔ coupling space inaccessible to the GM1L and DD2 EOS models, but the predictable result is that fΔ increases

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Fig. 25. Delta isobar fraction (percentage) of the maximum mass NS in the σωΔ coupling space (Spinella, 2017). Hyperons were included with the vector meson–hyperon coupling constants given by the SU(3) ESC08 model. Markers, contours, and pixels are as described for Fig. 23. The Δ fractions for xσΔ = xωΔ = 1.1 are as follows: SWL = 8.41%, f GM1L = 6.31%, and f DD2 = 10.2%. fΔ Δ Δ

with an increase in xσΔ −xωΔ when xσΔ  0.8. The DD2 parameterization presents with the highest fΔ for the smallest difference xσΔ − xωΔ , followed by SWL and then GM1L. The critical density ncr for the appearance of Δs is shown in Fig. 26 for the σωΔ coupling space. As long as Eq. (70) is satisfied, Δs appear  2.3 n0 , nGM1L  2.3 n0 , prior to the onset of hyperonization, and nSWL cr cr DD2 and ncr  1.9 n0 . If we also enforce simultaneous satisfaction of Eq. (69) ≈ 2 n0 , nGM1L ≈ 1.9 n0 , and the critical densities could be as low as nSWL cr cr DD2 ncr ≈ 1.6 n0 . Increasing xσΔ leads to a gradual decrease in ncr when xσΔ − xωΔ  −0.1, and a gradual increase in ncr when xσΔ − xωΔ  −0.1, the increase in the repulsive vector coupling overcoming the increasingly attractive potential in the latter case. However, increasing xσΔ − xωΔ leads to an obvious and rapid decrease in nc for the entire σωΔ coupling space, and if Eq. (70) is simultaneously satisfied an increase in xσΔ − xωΔ also leads to a significant reduction in the radius of the canonical 1.4 M NS as shown in Fig. 27. For example, the DD2 parameterization with nucleons (and hyperons) produces a 13.5 km radius for the canonical 1.4 M NS and

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Fig. 26. Critical baryon number density (in units of n0 ) for the appearance of Δs in the σωΔ coupling space (Spinella, 2017). Hyperons were included with the vector meson– hyperon coupling constants given by the SU(3) ESC08 model. Markers, contours, and pixels are as described for Fig. 23.

thus fails to satisfy the 13.2 km upper-radial constraint from Lattimer and Steiner (2014), but if Δs are included and xσΔ > xωΔ the radius is reduced sufficiently to satisfy the constraint. If we allow Eq. (69) to be violated resulting in a very attractive UΔ , the inclusion of Δs makes it possible for all three parameterizations to satisfy at least part of the 2σ upper limit on the radial constraints from Steiner et al. (2010). 5.3.2. The xρΔ coupling To examine the dependence of the NS mass, the Δ critical density ncr , and the Δ fraction fΔ on the isovector meson–Δ coupling xρΔ we set xσΔ = xωΔ = 1.1 and varied xρΔ in the range 0.5 < xρΔ < 2.5 (Spinella, 2017). (Note that the saturation potential of the Δ is determined in symmetric nuclear matter and is therefore independent of xρΔ .) The NS maximum mass was found to not be terribly sensitive to xρΔ , decreasing over the entire range by less than 1% for the GM1L and DD2 parameterizations. However, ncr and fΔ turned out to be much more sensitive to changes in xρΔ for the GM1L parameterization due to the fact that the isovector contribution to

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Fig. 27. Radius (in km) of the canonical 1.4 M NS in the σωΔ coupling space (Spinella, 2017). Hyperons were included with the vector meson–hyperon coupling constants given by the SU(3) ESC08 model. The solid contour in the bottom (DD2) panel represents the 13.2 km upper limit of the radial constraint from Lattimer and Steiner (2014) represented as L90% on the colorbar. The 1σ and 2σ upper limits from Steiner et al. (2010) are represented on the colorbar as S1σ and S2σ, respectively. Markers, dashed contours, and pixels are as described for Fig. 23.

the chemical potential is much higher than for DD2. The critical density for GM1L increases from ∼ 2 n0 to ∼ 3 n0 across the entire xρΔ range, with a corresponding drop in fΔ of ∼ 8−9% as this EOS reverts back to a nearly purely hyperonic EOS. The ncr of the DD2 parameterization increases very little from around 1.7 n0 to 1.9 n0 , but with an accompanying drop in fΔ of almost 3% down to about 8%. Overall, lower values of xρΔ lead to a lower critical density, resulting in higher fractions of Δs to replace hyperons and lower the strangeness fraction, increasing the NS maximum mass. 6. General Relativistic Stellar Structure Equations Neutron stars are objects of highly compressed matter so that the geometry of surrounding space–time is changed considerably from flat space. Einstein’s theory of general relativity is therefore to be used when modeling the properties of NSs rather than Newtonian mechanics. Einstein’s field equation is given by (we use units where the gravitational constant and the

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speed of light are G = c = 1) 1 Rμν − g μν R = 8πT μν , 2

(73)

where Rμν is the Ricci tensor, g μν the metric tensor, R the scalar curvature, and T μν the energy–momentum tensor of matter. The latter is given by T μν = ( + P ( )) uμ uν + g μν P ( ).

(74)

Models for the EOS, P ( ), which are input quantities in the energymomentum tensor equation, have been derived in Sec. 2. These models will be used in this section to study the properties of NSs. 6.1. Non-rotating proto-neutron stars We begin with non-rotating, spherically symmetric NSs. They are relatively easy to study since the metric of such objects depends only on the radial coordinate. The line element ds2 in this case is given by the Schwarzschild metric (Misner et al., 1973; Schwarzschild, 1916; Shapiro and Teukolsky, 2008) ds2 = −e2 Φ(r) dt2 + e2 Λ(r) dr2 + r2 (dθ2 + sin2 θ dφ2 ) ,

(75)

where Φ(r) and Λ(r) denote unknown metric functions whose mathematical form is determined by Einstein’s field equation (73) and the conservation of energy–momentum, ∇μ T μν = 0, and have the form

−1 2m(r) 1− r

2m(r) = 1− r

e2 Λ(r) =

(inside and outside of star),

(76)

e2 Φ(r)

(only outside of star).

(77)

The solution of Φ(r) for the stellar interior is given by 1 dP (r) dΦ(r) =− , dr + P ( ) dr

(78)

where the pressure gradient is given by the Tolman–Oppenheimer–Volkoff (TOV) equation (Misner et al., 1973; Oppenheimer and Volkoff, 1939; Tolman, 1939),   ( (r) + P (r)) m(r) + 4πr3 P (r) dP =− . (79) dr r2 (1 − 2m(r)/r)

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set by the Kepler frequency ΩK , at which mass shedding from a star’s equator sets in. The expression of the general relativistic Kepler frequency is derived from the line element shown in Eq. (81), evaluated at the equator of a compact stellar object. Since dr = 0 and dθ = 0 for a mass element rotating at the equator, one obtains from Eq. (81) for the proper time dτ 2 (= −ds2 ) the relation 1/2  dt. (82) dτ = e2 ν(r,θ,Ω) − e2 ψ(r,θ,Ω) (Ω − ω(r, θ, Ω))2 The equatorial orbit, which is the circular path with the maximum possible distance from the center of a gravitating body, is obtained from Eq. (82) by determining the extremum of the functional J(r) associated with Eq. (82), that is,  1/2  . (83) J(r) ≡ dt e2 ν(r,θ,Ω) − e2 ψ(r,θ,Ω) (Ω − ω(r, θ, Ω))2 Applying the extremal condition δJ(r) = 0 to this functional leads to   1/2 δ dt e2 ν(r,θ,Ω) − e2 ψ(r,θ,Ω) (Ω − ω(r, θ, Ω))2 = 0, (84) from which it follows that (Weber, 1999)  ν,r e2 ν − (ψ,r (Ω − ω) − ω,r ) (Ω − ω) e2 ψ = 0. dt δr  1/2 e2 ν − e2 ψ (Ω − ω)2

(85)

For the sake of brevity, we suppress all arguments here and in the following. The subscripts ,r on the metric functions and the frame dragging frequency in Eq. (85) denote partial derivatives with respect to the radial coordinate, r. Next, we introduce the orbital velocity V of a comoving observer at the star’s equator relative to a locally non-rotating observer with zero angular momentum in the φ-direction. This velocity is given by V = eψ−ν (Ω − ω).

(86)

This relation is suggested by the expression of the time-like component ut of the four-velocity of a mass element rotating in the equatorial plane, ut =

 −1/2 dt = e−ν 1 − V 2 , dτ

(87)

where V is given by Eq. (89). Substituting Eq. (86) into Eq. (85) then leads to ψ,r e2 ν V 2 − ω,r eν+ψ V − ν,r e2 ν = 0 ,

(88)

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which guarantees that the integrand in Eq. (85) vanishes identically for arbitrary variations δr. Equation (88) represents a quadratic equation for the velocity V . The solutions are given by (Friedman et al., 1986)

2 1/2 ν,r ω,r 2(ψ−ν) ω,r ψ−ν e ± + e . (89) V = 2 ψ,r ψ,r 2 ψ,r The general relativistic Kepler frequency, ΩK , is then obtained (cf. Eq. (86)) as ΩK = eν−ψ V + ω.

(90)

We note that Eqs. (89) and (90) need to be computed self-consistently together with Einstein’s field equations, which determined the metric functions ν and ψ and the frame dragging frequency ω at an (initially unknown) equatorial distance. The result of classical Newtonian mechanics for  the Kepler frequency and the velocity of a particle in a circular orbit, ΩK = M/R3 and V = RΩ respectively, are recovered from Eqs. (89) and (90) by neglecting the curvature of space–time geometry, the rotational deformation of a rotating star, and the dragging effect of the local inertial frames. Figure 30 shows the impact of rapid rotation on the gravitational masses of proto-neutron stars. The solid lines in this figure show the masses of non-rotating (i.e. TOV) stars. The dashed lines reveal by how much these masses increase if the stars are rotating at the highest possible spin rate, which is the Kepler frequency given by (90). The increase in mass of cold NSs is typically at the 20% level, depending on the EOS (Friedman et al., 1986). The same is the case for the gravitational masses of proto-neutron stars, as can be inferred from Fig. 30. We also note that the stars’ central energy density, c , decreases with rotation speed, because of the additional rotational pressure in the radial outward direction created by rotation. As mentioned just above, to find the Kepler frequency ΩK (Kepler period, P = 2π/ΩK ) of a compact star, Eqs. (89) and (90) are to be computed self-consistently in combination with the differential equations for the metric and frame-dragging functions in Eq. (81), which follow from Einstein’s field equation (73). The entire set of coupled equations is to be evaluated at the equator of the rotating star (Friedman et al., 1986; Weber and Glendenning, 1992), which is not known at the beginning of the computational procedure. Here, the results of stars rotating at the Kepler period are computed in the framework of Hartle’s perturbative rotation formalism (Weber and Glendenning, 1992). The latter constitute a perturbative

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radiation which inevitably accompanies the aspherical transport of matter does not damp the modes, but rather drives them (Chandrasekhar, 1970; Friedman, 1983a,b). Bulk and shear viscosity play the important role of damping such gravitational-wave radiation–reaction instabilities at a sufficiently reduced rotational frequency such that the viscous damping rates and power in gravity waves are comparable (Andersson and Kokkotas, 2001; Lindblom and Detweiler, 1977). Theoretical studies suggest that either the f -modes or the r-modes determine the maximum rotation frequency of neutron stars. 6.3. The moment of inertia Another very important stellar quantity, which will be discussed in this section, is the moment of inertia, I. This quantity is given by (Hartle, 1973)  1 dr dθ dφ T φ t (r, θ, φ; Ω) (−g(r, θ, φ; Ω))1/2 , (91) I(Ω) = Ω A where A denotes the region inside of a compact stellar object rotating at a uniform angular velocity, Ω. The quantity g denotes the determinant of the metric tensor gμν , whose components can be read off from Eq. (81). One obtains (Weber, 1999) √ −g = eλ+μ+ν+ψ . (92) The energy–momentum tensor component Tφ t is given by (see Eq. (74)) Tφ t = ( + P ) uφ ut ,

(93)

with the 4-velocities uφ and ut given by ut =

e−ν (1 − (ω − Ω)2 e2ψ−2ν )

1/2

,

uφ = (Ω − ω) e2 ψ ut .

(94) (95)

Substituting Eqs. (94) and (95) into Eq. (93) leads for Tφ t to Tφ t =

( + P ) (Ω − ω) e2 ψ . e2 ν − (ω − Ω)2 e2 ψ

(96)

Substituting the expression given in Eqs. 92 and 96 into Eq. (91) leads for the moment of inertia of a rotationally deformed compact stellar object to

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investments made in international nuclear physics facilities such as FAIR, FRIB, NICA, CERN, BNL, J-Park and multiple new instruments for sky surveys that have become operational in recent years, such as FAST, eROSITA, NICER and the gravitational-wave detectors LIGO, VIRGO, KAGRA. In particular, the observation of the first binary neutron star merger, GW170817, using LIGO and VIRGO (Abbott, 2017) have led the scientific community into the new era named multi-messenger astronomy with gravitational waves. Depending on the combined masses of two merging NSs, there are in principle four possible outcomes to a merger (Chirenti et al., 2019): (1) prompt formation of a black hole, (2) formation of a hypermassive NS (HMNS) (Baumgarte et al., 2000; Shapiro, 2000), (3) formation of a supramassive rotating NS (Falcke and Rezzolla , 2014), or (4) the formation of a stable NS. Numerical relativity simulations have shown that the threshold masses related to these scenarios depend strongly on the properties and the EOS of hot and dense NS matter. (For a comprehensive review of the physics of NS mergers, see Baiotti and Rezzolla (2017), and references therein.) The same is true for the lifetime of HMNSs, which depends strongly on the total mass of the binary system and, thus, on the nuclear EOS. The post-merger emissions are typically characterized by two distinct frequency peaks, one at high and the other at lower frequencies. The EOS-dependent high-frequency peak is believed to be associated with the oscillations of the HMNS produced in a merger, while the low-frequency peak is understood to be related to the merger process and to the total compactness (i.e. mass–radius ratio) of the merging objects (Takami et al., 2015), which is inexorably linked to the EOS of dense nuclear matter. The EOS of cold and dense nuclear matter is sufficient to describe NS matter prior to a NS merger. After contact, however, large shocks develop which considerably increase the internal energy of the colliding NSs. Numerical simulations have shown that overall matter in NS collisions reaches densities that are several times higher than the nuclear saturation density and temperatures that are roughly as high as 50 MeV (Baiotti and Rezzolla, 2017; Hanauske et al., 2019a; Perego, Albino et al., 2019). As shown in this chapter, such extreme conditions of density and pressure modify the EOS and in particular the baryon–lepton composition of the matter tremendously. Very recently, it has been shown that a strong first-order phase transition in NS mergers may register itself in the gravitational-wave frequency, fpeak , and the stellar tidal deformability, Λ [Bauswein et al. (2019)]. Since

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both the tidal deformability during inspiral and the oscillation frequencies of the post-merger remnant can be determined very reliably (Baiotti and Rezzolla, 2017; Duez and Zlochower, 2018; Faber and Rasio, 2012; Friedman, 2018; Paschalidis and Stergioulas, 2017), this finding relates NS merger simulations to the general science question whether or not phase transitions occur in dense nuclear matter. Signatures of possible hadron–quark phase transitions in NS mergers have also been studied by Most et al. (2019). This study shows that changes in the pressure of the quark phase can produce a decisive signature in the post-merger gravitational-wave signal and spectrum. It was also shown that a hadron–quark phase transition may lead to a hot and dense quark core which could produce a ring-down signal different from what is expected for a pure hadronic core. The possibility of detecting the hadron–quark phase transition with gravitational waves has been discussed recently by Hanauske et al. (2019b). A great deal of experimental, theoretical as well as computational work will need to be carried out over the coming years to determine a comprehensive class of state-of-the-art models for the EOS of ultra-hot and dense nuclear matter for use in binary NS merger simulations and PNS simulations (Banik et al., 2014; Hanauske et al., 2019b; Rezzolla and Olindo, 2013; Shen et al., 2011).

Acknowledgments The results of this paper contribute to the research projects of the NP3M collaboration on the Nuclear Physics of Multi-Messenger Mergers. This research was supported by the National Science Foundation (USA) under Grants No. PHY-1714068 and PHY-2012152. MO and IFR-S thank CONICET, UNLP, and MinCyT (Argentina) for financial support under grants PIP-0714, G157, G007 and PICT 2019-3662.

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© 2023 World Scientific Publishing Company https://doi.org/10.1142/9789811220913 0006

Chapter 6

Review on the Pseudo-complex General Relativity and Dark Energy Peter O. Hess Instituto de Ciencias Nucleares Universidad Nacional Aut´ onoma de M´exico Circuito Exterior, C.U., A.P. 70-543, 04510 M´ exico D.F., Mexico Frankfurt Institute for Advanced Studies Johann Wolfgang Goethe Universit¨ at Ruth-Moufang-Str. 1, 60438 Frankfurt am Main, Germany [email protected] A review will be presented on the algebraic extension of the standard Theory of General Relativity (GR) to the pseudo-complex formulation (pc-GR). The pc-GR predicts the existence of a dark energy outside and inside the mass distribution, corresponding to a modification of the GRmetric. The structure of the emission profile of an accretion disk changes also inside a star. Discussed are the consequences of the dark energy for cosmological models, permitting different outcomes on the evolution of the universe. Keywords: General Relativity; Pseudo-complex General Relativity; Dark Energy.

1. Introduction The Theory of General Relativity (GR) (Misner, Thorne and Wheeler, 1973) is one of the best tested known theories (Will, 2006), mostly in solar system experiments. Also the loss of orbital energy in a binary system (Weisberg, Taylor and Fowler, 1981) was the first indirect proof for gravitational waves, which were finally detected in Abbott et al. (2016). On April 10, The Event Horizon Telescope collaboration announced the

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first picture taken from the black hole in M87. This gives us an opportunity to compare results from the pc-GR to GR. Nevertheless, the limits of GR may be reached when strong gravitational fields are present, which can lead to different interpretations of the sources of gravitational waves (Hess, 2016, 2019). A first proposal to extend GR was attempted by A. Einstein (Einstein, 1945, 1948) who introduced a complex valued metric Gμν = gμν +iFμν , with G∗μν = Gνμ . The real part corresponds to the standard metric, while the imaginary part defines the electromagnetic tensor. With this, A. Einstein intended to unify GR with Electrodynamics. Another motivation to extend GR is published in Born (1938, 1949), where M. Born investigated on how to recover the symmetry between coordinates and momenta, which are symmetric in Quantum Mechanics but not in GR. To achieve his goal, he introduced also a complex metric, where the imaginary part is momentum dependent. In Caianiello (1981) this was more elaborated, leading to the square of the length element (c = 1)   (1) dω 2 = gμν dxμ dxν + l2 duμ duν , which implies maximal acceleration (see also Hess, Sch¨ afer and Greiner, 2015). The interesting feature is that a minimal length “l” is introduced as a parameter and Lorentz symmetry is, thus, automatically maintained, no deformation to small lengths is necessary! In Kelly and Mann (1986) the GR was algebraically extended to a series of variables and the solutions for the limit of weak gravitational fields were investigated. As a conclusion, only real and pseudo-complex coordinates (called in Kelly and Mann (1986) hyper-complex) make sense, because all others show either tachyon or ghost solutions, or both. Thus, even the complex solutions don’t make sense. This was the reason to concentrate on the pseudo-complex extension. In pc-GR, all the extended theories, mentioned in the last paragraph, are contained and the Einstein equations require an energy-momentum tensor, related to vacuum fluctuations (dark energy), described by an asymmetric ideal fluid (Hess, Sch¨ afer and Greiner, 2015). Due to the lack of a microscopic theory, this dark energy is treated phenomenological. One possibility is to choose it such that no event horizon appears, or barely still exists. The reason to do so is, that in our philosophical understanding, no theory should have a singularity, even a coordinate singularity of the type of an event horizon encountered in a black hole. Though it is only a coordinate singularity, the existence of an event horizon implies that even a

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black hole in a nearby corner cannot be accessed by an outside observer. Its event horizon is a consequence of a strong gravitational field. Because no quantized theory of gravitation exists yet, we are lead to the construction of models for the distribution of the dark energy. In Nielsen and Birnholz (2018) the pc-GR was compared to the observation of the amplitude for the inspiral process. As found, the fall-off in r of the dark energy has to be stronger than suggested in earlier publications. We will discuss this and what will change, in the main body of the text. We also will compare EHT observations with pc-GR, taking into account the low resolution of 20μas, as obtained by the EHT. The main question is if one can discriminate between GR and pc-GR. A general principle emerges, namely that mass not only curves the space (which leads to the standard GR) but also changes the space-(vacuum-) structure in its vicinity, which in turn leads to an important deviation from the classical solution. The consequences will be discussed in section 3. There, also the cosmological effects are discussed, with different outcomes for the evolution of the dark energy as function of time/radius of the universe. Another application treats the interior of a stars, where first attempts will be reported on how to stabilize a large mass. In section 4 Conclusions will be drawn. 2. Pseudo-complex General Relativity (pc-GR) An algebraic extension of GR consists in a mapping of the real coordinates to a different type, as for example complex or pseudo-complex (pc) variables X μ = xμ + Iy μ ,

(2)

with I 2 = ±1 and where xμ is the standard coordinate in space-time and y μ the complex component. When I 2 = −1 it denotes complex variables, while when I 2 = +1 it denotes pseudo-complex (pc) variables. This algebraic mapping is just one possibility to explore extensions of GR. In Kelly and Mann (1986) all possible extensions of real coordinates in GR where considered. It was found that only the extension to pseudocomplex coordinated (called in Kelly and Mann (1986) hyper-complex) makes sense, because all others lead to tachyon and/or ghost solutions, in the limit of weak gravitational fields. In what follows, some properties of pseudo-complex variables are resumed, which is important to understand some of the consequences.

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• The variables can be expressed alternatively as μ μ σ+ + X− σ− , X μ = X+

1 (1 ± I). 2

(3)

2 = σ± , σ+ σ− = 0. σ±

(4)

σ± = • The σ± satisfy the relations

• Due to the last property in (4), when multiplied one variable proportional to σ+ by another one proportional to σ− , the result is zero, i.e. there is a zero-divisor. The variables, therefore, do not form a field but a ring. • In both zero-divisor component (σ± ) the analysis is very similar to the standard complex analysis. In pc-GR the metric is also pseudo-complex + − σ+ + gμν σ− . gμν = gμν

(5)

Because σ+ σ− = 0 in each zero-divisor component one can construct independently a GR theory. For a consistent theory, both zero-divisor components have to be connected! One possibility is to define a modified variational principle, as done in Hess and Greiner (2009). Alternatively, one can implement a constraint, namely that a particle should always move along a real path, i.e. that the pseudo-complex length element should be real. The infinitesimal pc length element squared is given by (see also Hess and Greiner, 2017) dω 2 = gμν dX μ dX ν μ μ + ν − ν = gμν dX+ dX+ σ+ + gμν dX− dX− σ− ,

(6)

as written in the zero-divisor components. In terms of the pseudo-real and pseudo-imaginary components, we have s a (dxμ dxν + dy μ dy ν ) + gμν (dxμ dy ν + dy μ dxν ) dω 2 = gμν  a  s + I gμν (dxμ dxν + dy μ dy ν ) + gμν (dxμ dy ν + dy μ dxν ) , (7)  +   +  s − a − and gμν . The upper indices s with gμν = 12 gμμ + gμν = 12 gμμ − gμν and a refer to a symmetric and anti-symmetric combination of the metrics.

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+ − a For the case when gμν = gμν = gμν , i.e. gμν = 0, leads to

gμν (dxμ dxν + dy μ dy ν ) + Igμν (dxμ dy ν + dy μ dxν ).

(8)

Identifying y μ = luμ , where l is an infinitesimal length and uμ the 4-velocity, one obtains the length element defined in Caianiello (1981). It also contains the line element as proposed in Born (1938, 1949), where the y μ is proportional to the momentum component pμ of a particle. However, this identification of y μ is only valid in a flat space, where the second term in (8) µ is just the scalar product of the 4-velocity (uμ = dx dτ to the 4-acceleration 2 µ y μ = ddτx2 ). The connection between the two zero-divisor components is achieved, requiring that the infinitesimal length element squared in (7) is real, i.e., in terms of the σ± components it is   + μ μ ν − ν = 0. (9) dX+ dX+ − gμν dX− dX− (σ+ − σ− ) gμν Using the standard variational principle with a Lagrange multiplier, to account for the constraint, leads to an additional contribution in the Einstein equations, interpreted as an energy-momentum tensor. The action of the pc-GR is given by Hess and Greiner (2017)  √ (10) S = dx4 −g (R + 2α), where R is the Riemann scalar. The last term in the action integral allows to introduce the cosmological constant in cosmological models, where α has to be constant in order not to violate the Lorentz symmetry. This changes when a system with a uniquely defined center is considered, which has spherical (Schwarzschild) or axial (Kerr) symmetry. In these cases, the α is allowed to be a function in r, for the Schwarzschild solution, and a function in r and ϑ, for the Kerr solution. ± leads to the The variation of the action with respect to the metric gμν equations of motion 1 ± Λ R± μν − gμν R± = 8πT± μν 2 with ± T±Λ μν = λuμ uν + λ (y˙ μ y˙ ν ± uμ y˙ ν ± uν y˙ μ ) + αgμν ,

(11)

in the zero-divisor component, denoted by the independent unit-elements σ± . These equations still contain the effects of a minimal length parameter

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l, as shown in Caianiello (1981). because the effects of a minimal length scale is difficult to measure, maybe not possible at all, we neglect it, which corresponds to map the above equations to their real part, giving 1 ± Λ R = 8πTμν . Rμν − gμν 2

(12)

Λ , R± The Tμν μν is real and is now given by Hess and Greiner (2017)   Λ Λ Λ Λ = (Λ + pΛ (13) Tμν,R ϑ )uμ uν + pϑ gμν + pr − pϑ kμ kν , Λ where pΛ ϑ and radial pr are the tangential and pressure respectively. For Λ Λ μ an isotropic fluid we have pΛ ϑ = pr =p . The u are the components of the 4-velocity of the elements of the fluid and k μ is a space-like vector (kμ k μ = 1) in the radial direction. It satisfies the relation uμ k μ = 0. The fluid is anisotropic due to the presence of yμ . The λ and α are related to the pressures as (Hess and Greiner, 2017)

˜ α = 8π α λ = 8π λ, ˜,  Λ  ˜ = p + Λ , α λ ˜ = pΛ ϑ ϑ,

  ˜ μ yν = pΛ − pΛ kμ kν . λy r ϑ

(14)

The reason, why the dark energy outside a mass distribution has to be an anisotropic fluid, is understood contemplating the Tolman-OppenheimerVolkov (TOV) equations (Adler, Bazin and Schiffer, 1975) for an isotropic fluid: The TOV equations relates the derivative of the dark-energy pressure with respect to r (for an isotropic fluid, the tangential pressure has to be Λ Λ the same as the radial pressure, i.e., pΛ ϑ = pr = p ) to the dark energy density Λ . Assuming the isotropic fluid and  equation of state for  that the the dark energy is pΛ = −Λ , the factor pΛ + Λ in the TOV equation Λ

for dp dr is zero, i.e., the pressure derivative is zero. As a result the pressure is constant and with the equation of state also the density is constant, which leads to a contradiction. Thus, the fluid has to be anisotropic, due to an additional term, allowing the pressure to to fall off as a function on dpΛ r increasing distance. The additional term in the radial pressure dr , added   2 2 Λ to the TOV equation, is given by r ΔpΛ = r pΛ (Rodr´ıguez, Hess, ϑ − pr Schramm and Greiner, 2014). For the density one has to apply a phenomenological model, due to the lack of a quantized theory of gravity. What helps is to recall one-loop calculations in gravity (Birrell and Davies, 1986), where vacuum fluctuations result due to the non-zero back ground curvature (Casimir effect). Results are presented in Visser (1996), where at large distances the density falls

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off approximately as 1/r6 . The semi-classical Quantum Mechanics (Birrell and Davies, 1986) was applied, which assumes a fixed back-ground metric and is thus only valid for weak gravitational fields (weak compared to the solar system). Near the Schwarzschild radius the field is very strong which is exhibited by a singularity in the energy density, which is proportional to 1 2 , with m a constant mass parameter (Visser, 1996). (1− 2m r ) Because we treat the vacuum fluctuations as a classical ideal anisotropic fluid, we are free to propose a different fall-off of the negative energy density, which is finite at the Schwarzschild radius. In earlier publications the density did fall-off proportional to 1/r5 . However, in Nielsen and Birnholz (2018) it is shown that this fall-off has to be stronger. Thus, in this contribution we will also discuss a variety of fall-offs as a function of a parameter n n, i.e., proportional to rB n+2 , where Bn describes the coupling of the dark energy to the mass. With the assumed density, the metric for the Kerr solution changes to Caspar, Sch¨onenbach, Hess, Sch¨afer and Greiner (2012); Schoenenbach (2014) g00 = − g11 =

r2 − 2mr + a2 cos2 ϑ + r2 +

,

r2 + a2 cos2 ϑ , Bn r2 − 2mr + a2 + (n−1)(n−2)r n−2

g22 = r2 + a2 cos2 ϑ, 2

2

2

g33 = (r + a ) sin ϑ + g03 =

Bn (n−1)(n−2)r n−2 a2 cos2 ϑ

 a2 sin4 ϑ 2mr −

Bn (n−1)(n−2)r n−2

r2 + a2 cos2 ϑ

2 Bn −a sin2 ϑ 2mr + a (n−1)(n−2)r n−2 sin ϑ

r2 + a2 cos2 ϑ

(15)

 ,

,

where 0 ≤ a ≤ m is the spin parameter of the Kerr solution and n = 3, 4, .... For n = 2 the old Ansatz is achieved. The Schwarzschild solution is obtained, setting a = 0. The parameter Bn = bn mn measures the coupling of the dark energy to the central mass. The definition of n here is related to the nN in Nielsen and Birnholz (2018) by nN = n − 1. When no event horizon is demanded, the parameter Bn has a lower limit given by

n−1 2(n − 1)(n − 2) 2(n − 1) mn = bmax mn . (16) Bn > n n

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268

For the equal sign, an event horizon is located at rh = e.g..,

4 3

for n = 3 and

3 2

2(n − 1) m, n

(17)

for n = 4.

3. Applications 3.1. Motion of a particle in a circular orbit In Sch¨onenbach, Caspar, Hess, Boller, M¨ uller, Sch¨afer and Greiner (2013) the motion of a particle in a circular orbit was investigated. This section were first discussed in Boller, Hess, M¨ uller and St¨ocker (2019); Hess, Boller, M¨ uller and St¨ocker (2019). The main results is resumed in the Figs. 1 and 2. In Fig. 1 the orbital frec , is depicted versus the radial distance r, in units of m, quency, in units of m for a rotational parameter of 0.9m. The function for the orbital frequency,

0.4 0.35 0.3 ω [m/c]

0.25 0.2

0.15 0.1 0.05 0

1

2

3

r [m]

4

5

6

Fig. 1. The orbital frequency of a particle in a circular orbit for the case GR (upper curve) and for n = 3 (long dashed curve) and n = 4 (short dashed curve) [Boller, Hess, M¨ uller and St¨ ocker (2019); Hess, Boller, M¨ uller and St¨ ocker (2019)].

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6 last stable orbit in GR "last" stable orbit in pc−GR constraint for general orbits "first" stable orbit in pc−GR 5 stable orbits in GR/pc−GR

r[m]

4

3

no stable orbits in GR/pc−GR

no stable orbits in GR stable orbits in pc−GR

2

1 0

0.2

0.4

0.8

0.6

1

a[m]

Fig. 2. The position of the Innermost Stable Circular Orbit (ISCO) is plotted versus the rotational parameter a. The upper curve corresponds to GR and the lower curves to pc-GR. The light gray shaded region corresponds to a forbidden area for circular orbits within pc-GR. For small values of a the ISCO in pc-GR follows more or less the one of GR, but at smaller values of r. From a certain a on stable orbits are allowed until to the surface of the star (for n = 3 this limit is approximately 0.4 m and for n = 4 it is at 0.5 m).

in pro-grade orbits, is given by ωn =

hn (r) =

1  a+

2r hn (r)

,

2 nBn − . r2 (n − 1)(n − 2)rn+1

(18)

The upper curve in Fig. 1 corresponds to GR while the two lower ones to pc-GR with n = 3 (dashed curve) and n = 4 (dotted curve). The curve shows a maximum at

rωmax

n(n + 2)bmax = 6(n − 1)(n − 2)

1

n−1

m,

(19)

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which, for bn = bmax as given in (16), is independent of the value of a, after which it falls off toward the center and reaches zero at rh (Eq. (17)), which is independent on the rotational parameter a. After the maximum the curve falls off toward smaller r. These features will be important for the understanding of the emission structure of an accretion disk (see next subsection). As one can see, the difference between n = 3 and n = 4 is minimal and, thus, will not change the qualitative results as obtained for n = 3 in former publications. The position of the maximum, which gives the position of the dark ring discussed below, is approximately the same in both cases. For bn → 0 the curve approaches the one for GR. In Fig. 2 the last stable orbit, for n = 3, is plotted versus the rotational parameter a. The solid enveloping curve is the result for GR. For a = 0 the last stable orbit in GR is at 6m, while in pc-GR it is further in. The dark gray shaded area describes stable orbits in pc-GR and the light gray area unstable orbits. The pc-GR follows closely the GR with a greater deviation for larger a. At about a = 0.45m (for n = 3, for n = 4 its value is a little bit larger) all orbits in pc-GR are stable up to the surface of the star, which is estimated to ly at approximately 43 m. For a = m, in GR the last stable orbit is at r = m. 3.2. Accretion disks In order to connect to actual observations (EHT, 2019a,b,c,d,e,f), one possibility is to simulate accretion disks around massive objects as the one in the center of the elliptical galaxy M87. The underlying theory was published by D. N. Page and K. S. Thorne (Page and Thorne, 1974) in 1974. The basic assumptions are (see also Sch¨onenbach, Caspar, Hess, Boller, M¨ uller and Greiner, 2014) • A thin, infinitely extended accretion disk. This is a simplifying assumption. A real accretion disk can be a torus. Nevertheless, the structure in the emission profile will be similar, as discussed here. These disks are easier to calculate • An energy-momentum tensor is proposed which includes all main ingredients, as mass and electromagnetic contributions. • Conservation laws (energy, angular momentum and mass) are imposed in order to obtain the flux function, the main result of Page and Thorne (1974).

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• The internal energy of the disk is liberated via shears of neighboring orbitals and distributed from orbitals of higher frequency to those of lower frequency.

How to deduce finally the flux is described in detail in Hess and Greiner (2017). In order to understand within pc-GR the structure of the emission profile in the accretion disk, we have to get back to the discussion in the last subsection. The local heating of the accretion disk is determined by the gradient of orbital frequency, when going further inward (or outward). At the maximum, neighboring orbitals have nearly the same orbital frequency, thus, friction is low. On the other hand, above and especially below the position of the maximum the change in orbital frequency is large and the disk gets heated. At the maximum the heating is minimal which will be noticeable by a dark ring. Further inside, the heating increases again and a bright ring is produced. The above consideration is relevant for a larger than approximately 0.4, as can be seen from Fig. 2 (for explanations, see the figure caption) and (Sch¨ onenbach, Caspar, Hess, Boller, M¨ uller, Sch¨afer and Greiner, 2013). For lower values of a, in pc-GR the last stable orbit follows the one of GR, but with lower values for the position of the ISCO. As a consequence, the particles reach further inside and due to the decrease of the potential, more energy is released, producing a brighter disk. However, the last stable orbit in pc-GR does not reach rωmax . This changes when a is a bit larger than 0.4. Now, rωmax is crossed and the existence of the maximum of ω has to be taken into account as explained above. Some simulations are presented in Fig. 3. The line of sight of the observer to the accretion disk is 80◦ (near to the edge of the accretion disk), where the angle refers to the one between the axis of rotation and the line of sight. Two rotation parameters of the Kerr solution are plotted, namely a = 0 (no rotation of the star, corresponding to the Schwarzschild solution) and nearly the maximal rotation a = 0.9 m. As a global feature, the accretion disk in pc-GR appears brighter, which is due to the fact that the disk reaches further inside where the potential is deeper, thus releasing more gravitational energy, which is then distributed within the disk. The reason for the dark fringe and bright ring was explained above due to the variability of the friction. The dark ring is the position of the

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Fig. 3. Infinite, counter clockwise rotating geometrically thin accretion disk around static and rotating compact objects viewed from an inclination of 80◦ . The left panel shows the disk model by [Page and Thorne (1974)] in pc-Gr, with a = 0. The right panel shows the modified model, including pc-GR correction terms as described in the text.

maximum of the orbital frequency. An observed position of a dark ring can, thus, be used to determine n. The differences in the structure of an accretion disk give us a clear observational criteria to distinguish between GR and pc-GR. There are still others, maybe more realistic disk models, e.g., a thick disk as described in Kluzniak and Rappaport (2007). In case there is no disk present, as is probably the case in SgrA∗ , then the synchrotron model of infalling and emitted gas (Dodds-Eden, Porquet, Trap, et al., 2009) maybe more realistic. However, in all of those models the above discussed ring structure of the disk will not change. Unfortunately, this is for the moment the only clear prediction to differentiate pc-GR from GR. In the next subsection we will discuss gravitational waves and we will see pc-GR and GR give different interpretations of the source, though, the final outcome is the same. Finally, in Fig. 4 we compare a disk simulation for GR (left panel) with pc-GR (right panel), for a = 0.6 and a 60o inclination angle. The intensity in GR is smaller while in pc-GR it is much stronger. Also, the maximum of the intensity is more in line with the EHT data, which reports the maximum at approximately 3-4m. Otherwise, the ring structure in pc-GR is lost due to the low resolution of 20 μas and the cross-structure of pc-Gr is the same as in GR.

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Fig. 4. Infinite, counter clockwise rotating geometrically thin accretion disk around static and rotating compact objects viewed from an inclination of 60◦ and a = 0.6. The left panel is GR and the right one pc-GR. A resolution of 20 μas was assumed. A resolution of 20 μas is assumed.

3.3. Gravitational waves in pc-GR In Abbott et al. (2016) the first observed gravitational wave event was reported. In Hess (2016) this gravitational event was investigated within the pc-GR, for n = 3. Using GR and the mass-point approximation for the two black holes, before the merging, a relation is obtained between the observed frequency and its temporal change to the chirping mass Mc , namely (Maggiore, 2008)

35 5 dfgw − 11 c3 3 fgw . (20) Mc = Mc Fω (r) = G 96π 83 dt substituting on the right hand side the observed frequency and its change and using Fω (r) = 1 for GR, the interpretation of the source of the gravitational waves is of two black holes of about 30 solar masses each which fuse to a larger one of less than 60 solar masses. The difference in energy is radiated away as gravitational waves. However, these changes in pc-GR, where the two black holes can come very near to each other. Unfortunately, the point mass approximation is not applicable, though in Hess (2016) this approximation was still used in order to show in which direction the interpretation

 2  , which of the source changes. In pc-GR (n = 3) Fω (r) = 1 − 3b43 m r for bn given by the right hand side of Eq. (16) is exactly zero. Therefore, a range of the last possible distances of the two black holes before merging

274

P. O. Hess

was assumed. On the left hand side of Eq. (20), the function Fω (r) becomes very small near where the two in-spiraling black holes merge. Thus, the c must be much larger than the chirping mass M deduced chirping mass M

 3  , thus, the in GR. For n = 4, the function Fω (r) changes to 1 − b34 m r main conclusions are the same, though the r-dependence has changed. We have not yet made explicit calculations, for one reason: The model applied in Hess (2016) has to be modified, because the point approximation is not very good. The main result is that the source in pc-GR corresponds to two black holes with several thousand solar masses. This may be related to the merger of two primordial galaxies whose central black hole subsequently merges. One way to distinguish the two predictions is to look for light events very far way. If for observed gravitational wave events in future, there is a consistent appearance of light events much father away as the distance deduced from GR, then this might be in favor for pc-GR. However, all the prediction depends on the assumption that the point mass approximation is still more or less valid when the two black holes are near together, which is not very good! In Nielsen and Birnholz (2018) the inspiral frequency was determined within pc-GR, for various values of n, which is related to the one used in Nielsen and Birnholz (2018) by nN = n − 1. As demonstrated, the wave form cannot be reproduced satisfactorily for n = 3, thus, it has to be increased and let us to investigate the dependence of the results as a function in n. In Hess (2019) the Schwarzschild case was considered and the ReggeWheeler, for negative parity solutions, (Regge and Wheeler, 1957) and Zerrilli equations, for positive parity solutions, (Chandrasekhar, 1983) were solved, using an iteration method (Cho, Cornell, Doukas, Huang and Naylor, 2012). Due to a symmetry, in GR the two type of solutions have the same frequency spectrum (Chandrasekhar, 1983), which unfortunately is lost in pc-GR. For pc-GR, the spectrum of frequencies for axial modes show a convergent behavior for the frequencies, which is shown in Fig. 5. A negative imaginary part indicates a stable mode, which turns out to be the case. For an increasing imaginary part the convergence is less sure. Unfortunately, for the polar modes no convergence for the polar modes were obtained up to now. Another problem is to distinguish between GR and pc-GR. It depends very much on the observation of the ring-down frequency of the merger (Hess, 2019), which is not very well measured yet. Without it,

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Fig. 5. Axial gravitational modes in pc-GR. The vertical axis gives the real part of ω  = mω while the horizontal axis depicts the negative of its imaginary part.

we are not able to distinguish between both theories and various possible scenarios can be obtained in pc-GR (Hess, Sch¨afer and Greiner, 2015; Hess, 2019). 3.4. Dark energy in the universe The pc-Robertson-Walker model is presented in detail in Hess, Maghlaoui and Greiner (2010); Hess, Sch¨afer and Greiner (2015). The main results will be resumed in this subsection. The line element in Gaussian coordinates has the form 1

dω 2 = (dt)2 − a(t)2  1+

ka(t)2

 2  2 2 2 2 2 2 dR + R dϑ + a(t) sin ϑdϕ , (21)

4a20

where R is the radius of the universe and k is a parameter and the energy density of matter was assumed to be homogeneous. The value k = 0 corresponds to a flat universe, which will be taken here.

P. O. Hess

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The corresponding Einstein equations were solved and an equation for the radius a(t) if the universe was deduced (Hess, Sch¨ afer and Greiner, 2015): 

a(t) =

4πG 4πG −3(1+α)+1 (3β − 1)Λa(t)3(β−1)+1 − (1 + 3α)ε0 a(t) , 3 3 (22)

where G is the gravitational constant and β, Λ are parameters of the theory. The equation of state is set as p = αε, where ε is the matter density and α is set to zero for dust. Two particular solutions are shown in Figs. 6 and 7. Shown is the acceleration of the universe as a function of the radius a(t). The left panel shows the result for β = 12 and Λ = 3 and on the right hand side the parameters β and Λ are set to 23 and 4, respectively. The left figure correspond to a solution where the acceleration tends to a constant, i.e., the universe will expand for ever with an increasing acceleration, while in the right figure the acceleration tends slowly to zero for very large a(t). In both examples the universe expands for ever. These are not the only solutions, also one where the universe collapses again is possible. These results are not very predictive, because one can obtain several possible outcomes, depending on the values of β and Λ. Nevertheless, they show that possible scenarios for the future of our universe are still possible.

Fig. 6. The panel shows a case where the acceleration of the universe approaches a constant value for t → ∞. The figures are taken from Hess, Maghlaoui and Greiner (2010); Hess, Sch¨ afer and Greiner (2015).

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Fig. 7. The panel shows a case where the acceleration slowly approaches zero for t → ∞. The figures are taken from Hess, Maghlaoui and Greiner (2010); Hess, Sch¨ afer and Greiner (2015).

3.5. Interior of stars For the description of the interior of a star one needs the equation of state of matter and the coupling of the dark-energy with the matter. For the equation of state one can use the model presented in Dexheimer and Schramm (2008), which also takes into account nuclear and meson resonances. However, these approximations will loose their validity when the matter density is too large. The situation is worse for the dark-energy contribution and it is twofold: i) one has to know how the dark-energy evolves within the star (presence of matter) and ii) how it is coupled to the matter itself. Both are not known and we have to rest on incomplete models. Alternatively, one can approach the problem with a very interesting and distinct model to simulate the dark energy, as done in Hadjimichef, Volkmer, Gomes and Vasconcellos (2018); Razeira, Hadjimichef, Machado, K¨ opp, Volkmer and Vasconcellos (2017); Volkmer and Hadjimichef (2017); Volkmer (2018), where compact and dense objects were investigated within the pc-GR and maximal masses were also deduced. In Rodr´ıguez, Hess, Schramm and Greiner (2014) a simple coupling model of dark-energy to the mass density was proposed Λ = αρm ,

(23)

where the index Λ refers to the dark energy and ρm to the mass density. In this proposal the dark energy follows neatly the mass distribution. The Tolman-Oppenheimer-Volkoff (TOV) equations have to be solved, which is doubled in number, one treating the mass part and the other

280

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masses cannot be obtained due to the limits the model of Dexheimer and Schramm (2008) reaches. This model also suffers from the approximations made and a complete description cannot be given. Nevertheless, that now stars with up to 200 solar masses can be stabilized shows that the inclusion of dark-energy in massive stars may lead to stable stars of any mass! (Though, only within a phenomenological model.) 4. Conclusions A report on the recent advances of the pseudo-complex General Relativity (pc-GR) was presented. The theory predicts a non-zero energy-momentum tensor on the right hand side of the Einstein equation. The new contribution is related to vacuum fluctuations, but due to a missing quantized theory of gravitation one recurs to a phenomenological Ansatz. Calculations in one-loop order, with a constant back-ground metric, shows that the dark energy density has to increase toward smaller r. Consequences of the theory were presented; i) The appearance of a dark ring followed by a bright one in accretion disks around black holes, ii) a new interpretation of the source of the first gravitational event observed, iii) possible outcomes of the future evolving universe and iv) attempts to stabilize stars with large masses. The only robust prediction is the structure in the emission profile of an accretion disk. Acknowledgment Peter O. Hess acknowledges the financial support from DGAPA-PAPIIT (IN100418). Very helpful discussions with T. Boller (Max Planck Institute for Extraterrestrial Physics, Garching, Germany) and T. Sch¨onenbach are also acknowledged. Bibliography Abbott, B.P. et al. (2016). Observation of gravitational waves from a binary black hole merger, LIGO scientific collaboration and VIRGO collaboration, Phys. Rev. Lett. 116, 061102. Adler, R., Bazin, M. and Schiffer, M. (1975). Introduction to General Relativity (McGraw-Hill, N.Y., USA). Birrell, N.D. and Davies, P.C.W. (1986). Quantum Field in Curved Space (Cambridge University Press, Cambridge, UK).

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Boller, T., Hess, P.O., M¨ uller, A., and St¨ ocker, H. (2019). Predictions of the pseudo-complex theory of gravity for EHT observations - I. Observational tests, Mon. Not. R. Astron. Soc. Lett. 485, L34. Born, M. (1938). A suggestion for unifying quantum theory and relativity, Proc. Roy. Soc. A 16, 291. Born, M. (1949). Reciprocity theory of elementary particles, Rev. Mod. Phys. 21, 463. Caianiello, E.R. (1981). Nuovo Cim. Lett. 32, 65. Caspar, G., Sch¨ onenbach, T., Hess, P.O., Sch¨ afer, M. and Greiner, W. (2012). Pseudo-complex general relativity: Schwarzschild, REISSNER¨ and Kerr solutions, Int. J. Mod. Phys. E 21, 1250015. NORDSTROM Caspar, G., Rodr´ıguez, I., Hess, P.O. and Greiner, W. (2016). Vacuum fluctuation inside a star and their consequences for neutron stars, a simple model, Int. J. Mod. Phys. E 25, 1650027. Chandrasekhar, S. (1983). The Mathematical Theory of Black Holes, 1st edn. (Claredon Pres, Oxford, UK). Cho, H., Cornell, A.S., Doukas, J., Huang, T.-R. and Naylor, W. (2012). A new approach to black hole quasinormal modes: A review of the asymptotic iteration method, Adv. Math. Phys. 2012, 281705. Dexheimer, V. and Schramm, S. (2008). Proto-neutron and neutron stars in a chiral SU(3) model, Astrophys. J. 683, 943. Dodds-Eden, K., Porquet, D., Trap, G. et al. (2009). Evidence for X-ray synchrotron emission from simultaneous mid-infrared to X-ray observations of a strong sgr A* Flare, Astrophys. J. 698, 676. Einstein, A. (1945). A generalization of the relativistic theory of gravitation, Ann. of Math. 46, 578. Einstein, A. (1948). A generalized theory of gravitation, Rev. Mod. Phys. 20, 35. EHT — The Event Horizon Telescope collaboration (2019). First M87 event horizon telescope results. I. The shadow of the supermassive black hole, Astrophys. J. 875, L1. EHT — The Event Horizon Telescope collaboration (2019). First M87 event horizon telescope results. II. Array and instrumentation, Astrophys. J. 875, L2. EHT — The Event Horizon Telescope collaboration (2019). First M87 event horizon telescope results. III. Data processing and calibration, Astrophys. J. 875, L3. EHT — The Event Horizon Telescope collaboration (2019). First M87 event horizon telescope results. IV. Imaging the central supermassive black hole, Astrophys. J. 875, L4. EHT — The Event Horizon Telescope collaboration (2019). First M87 event horizon telescope results. V. Physical origin of the asymmetric ring, Astrophys. J. 875, L5. EHT — The Event Horizon Telescope collaboration (2019). First M87 event horizon telescope results. VI. The shadow and mass of the central black hole, Astrophys. J. 875, L6. Hadjimichef, D., Volkmer, G.L., Gomes, R.O. and Zen Vasconcellos, C.A. (2018). Dark Matter Compact Stars in Pseudo-Complex General Relativity, in

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P.O. Hess and H. St¨ ocker (eds.), Memorial Volume: Walter Greiner, (World Scientific, Singapore). Hess, P.O. and Greiner, W. (2009). Pseudo-complex general relativity, Int. J. Mod. Phys. E 18, 51. Hess, P.O., Maghlaoui, L. and Greiner, W. (2010). The Robertson-Walker metric in a pseudo-complex general relativity, Int. J. Mod. Phys. E 19, 1315. Hess, P.O., Sch¨ afer, M. and Greiner, W. (2015). Pseudo-complex General Relativity (Springer, Heidelberg, Germany). Hess, P.O. (2016). The black hole merger event GW150914 within a modified theory of general relativity, Mon. Not. R. Astron. Soc. 462, 3026. Hess, P.O. and Greiner, W. (2017). Pseudo-Complex General Relativity: Theory and Observational Consequences in C.A. Zen Vasconcellos (ed.), Centennial of General Relativity: A Celebration (World Scientific, Singapore), p. 97. Hess, P.O. (2019). Regge-Wheeler and Zerilli equations within a modified theory of general relativity, Astr. Nachr. 340, 89. Hess, P.O., Boller, T., M¨ uller, A. and St¨ ocker, H. (2019). Mon. Not. R. Astron. Soc. Lett. 482, L121. Kelly, P.F. and Mann, R.B. (1986). Classical and quantum gravity ghost properties of algebraically extended theories of gravitation, Class. Quant. Grav. 3, 705. Kluzniak, W. and Rappaport, S. (2007). Magnetically torqued thin accretion disks, Astrophys. J. 671, 1990. Maggiore, M. (2008). Gravitational Waves 1 (Oxford Univ. Press, Oxford, UK). Misner, C.W., Thorne, K.S. and Wheeler, J.A. (1973). Gravitation (W. H. Freeman and Company, San Francisco, USA). Nielsen, A. and Birnholz, O. (2018). Testing pseudo-complex general relativity with gravitational waves, Astr. Nachr. 339, 298. Page, D.N. and Thorne, K.S. (1974). Disk-accretion onto a black hole. Timeaveraged structure of accretion disk, Astrophys. J. 191, 499. Razeira, M., Hadjimichef, D., Machado, M.V.T., K¨ opp, F., Volkmer, G.L. and Vasconcellos, C.A.Z. (2017). Effective field theory for neutron stars with WIMPS in the pc-GR formalism, Astr. Nachr. 338, 1073. Regge, T. and Wheeler, J.A. (1957). Stability of a Schwarzschild singularity, Phys. Rev. 108, 1063. Rodr´ıguez, I., Hess, P.O., Schramm, S. and Greiner, W. (2014). Neutron stars within pseudo-complex general relativity, J. Phys. G 41, 105201. Sch¨ onenbach, T., Caspar, G., Hess, P.O., Boller, T., M¨ uller, A., Sch¨ afer, M. and Greiner, W. (2013). Experimental tests of pseudo-complex general relativity, Mon. Not. R. Astron. Soc. 430, 2999. Sch¨ onenbach, T., Caspar, G., Hess, P.O., Boller, T., M¨ uller, A. and W. Greiner, W. (2014). Ray-tracing in pseudo-complex general relativity, Mon. Not. R. Astron. Soc. 442, 121. Sch¨ onenbach, T. (2014). Title of the Thesis, PhD thesis (Universit¨ at Frankfurt am Main, Germany). Visser, M. (1996). Gravitational vacuum polarization. II. Energy conditions in the Boulware vacuum, Phys. Rev. D 54, 5116.

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Volkmer, G.L. and Hadjimichef, D. (2017). Mimetic dark matter in pseudocomplex general relativity, Int. J. Mod. Phys. Conf. Series 45, 1760012. Volkmer, G.L. (2018). Um objeto compacto ex´ otico na relatividade geral pseudocomplexa, Ph.D. thesis (UFRGS, Porto Alegre, Brasil). Weisberg, J.M., Taylor, J.H. and Fowler, L.A. (1981). Gravitational waves from an orbiting pulsar, Scientific American, 245(4), 74. Will, C.M. (2006). The confrontation between general relativity and experiment, Living Rev. Relativ. 9, 3.

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Chapter 7

Alternative Gravity Neutron Stars in the Gravitational Wave Era Pedro H. R. S. Moraes Universidade de S˜ ao Paulo (USP) Instituto de Astronomia, Geof´ısica e Ciˆencias Atmosf´ericas (IAG) 05508-090 S˜ ao Paulo, SP, Brazil [email protected] According to the Standard Model of Cosmology, ∼70% of the universe is composed by dark energy, ∼25% of dark matter and only the remaining ∼5% of known baryonic matter. Although we have some clues about the main properties of dark energy and dark matter, we still do not know exactly what they are, neither have we detected them in laboratory. The effects of dark energy and dark matter can be explained in an alternative form, by modifying the Einstein–Hilbert gravitational action, such that the resulting field equations contain new terms which, in principle, are capable of describing such dark sector effects. A well-behaved alternative gravity theory must also work in the stellar scales. As long as we still do not know the equation of state of super dense matter within neutron stars, a possibility to explain some recently detected massive pulsars comes exactly from those new terms in the field equations of Alternative Gravity Theories. Moreover, the recently emerged gravitational wave astrophysics may also be a field of applications to Alternative Gravity. In this chapter we review the importance of Alternative Gravity and its applications in the equilibrium configurations of neutron stars. We show how the recent gravitational wave astrophysics can be used to constraint among different Alternative Gravity Theories. Keywords: Alternative gravity; Gravitational waves; Neutron stars.

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1. Introduction The end of the last century was marked by an outstanding discovery that forever changed Cosmology: the expansion of the universe is accelerating. Such a milestone in history of Physics came from the observation of Type Ia Supernovae whose brightness was weaker than expected, indicating that those phenomena were taking place further than expected (Perlmutter et al., 1999; Riess et al., 1998). Over time, other observations came to corroborate the cosmic acceleration, as one can see, for instance, Weinberg et al. (2013), for a broad review. The cause behind the cosmic acceleration is still a mystery. In fact, it is certainly among the biggest mysteries in Physics nowadays. We normally refer to it as dark energy. In the Standard Model of Cosmology or ΛCDM Model (see Ryden, 2003, for instance), for which Λ is the cosmological constant and CDM stands for cold dark matter, Λ, as a matter of fact, can predict the cosmic acceleration in accordance with observations (Perlmutter et al., 1999; Riess et al., 1998). That is to say that for some Λ value, there is agreement between ΛCDM Cosmological Model theoretical predictions and observational data. The ΛCDM Cosmological Model is obtained from the consideration of the Cosmological Principle, which states that the universe is homogeneous and isotropic for scales >100 Mpc (see Ryden, 2003), within the Einstein’s field equations of General Relativity (GR) in the presence of the cosmological constant (Einstein, 1917): 1 Rμν − Rgμν = 8πTμν − Λgμν , 2

(1)

with Rμν being the Ricci tensor, R the Ricci scalar, gμν the metric tensor, Tμν the energy–momentum tensor and natural units are assumed. Then, for some cosmological constant value, it is possible to match the theoretical predictions arising from (1) with cosmological observational data (Conley et al., 2006; Perlmutter et al., 1999; Planck, 2014; Riess et al., 1998). However, the cost of this match is very high. Λ is physically interpreted as the quantum energy associated to vacuum and when one calculates this energy via Particle Physics, the result (Weinberg, 1989) strongly diverges from the observational value (Conley et al., 2006; Perlmutter et al., 1999; Planck, 2014; Riess et al., 1998). This is the well-known cosmological constant problem (Weinberg, 1989) (see also Ng, 1982, among many others), sometimes referred to as “the worst theoretical prediction in history of physics” (Hobson et al., 2006).

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Remarkably, the cosmological constant problem is not the only ΛCDM model shortcoming nowadays. Recent Planck observations of fluctuations of temperature on the cosmic microwave background radiation (Planck, 2014) point to Ωm = 0.3089 ± 0.0062, with Ωm representing the matter density parameter. A small fraction of this value is related to baryonic or ordinary matter, while dark matter is by far the dominant component in Ωm . There are other indications of the existence of a huge portion of dark matter in the universe such as galaxy clusters weak lensing (Clowe et al., 2004, 2006, 2007; Hoekstra et al., 2004) and numerical simulations (Moore et al., 1999). Finally, another important and reputed dark matter existence indication comes from rotation curves of galaxies (Blitz, 1979; de Blok et al., 2008; Honma and Sofue, 1997; Sofue, 1996). It is of vital importance to remark that the aforementioned dark matter existence indications are based only on the gravitational effects observed. No dark matter particle was detected so far (Baudis et al., 1998; Bernabei et al., 2003; Monroe et al., 2012). This fact is sometimes referred to as dark matter problem. A third important problem surrounding ΛCDM model is the recent Hubble tension, which is the 4–6σ disagreement between predictions of the Hubble constant H0 observational value made by early-time probes and a number of late-time local measurements of the same. There are a plethora of references quite recently treating the Hubble tension, among which I quote (Berghaus and Karwal, 2020; Di Valentino et al., 2021; Lin et al., 2019; Poulin et al., 2019; Smith et al., 2020). There are still other problems surrounding ΛCDM cosmology, such as the missing satellites problem (Brooks et al., 2013; Kravtsov et al., 2004; Pe˜ narrubia et al., 2012; Simon and Geha, 2007; Strigari et al., 2007), the coincidence problem (Arkani-Hamed et al., 2000; Del Campo et al., 2008) and the Big-Bang singularity (Alexander and Biswas, 2009; Battisti and Montani, 2007; Elgenhardt et al., 2015; Klinkhamer, 2019), and important reviews on this subject can be found in Bull et al. (2016) and LopezCorredoira (2017). Finally, it is worth mentioning that GR Theory, which as it was aforementioned is the underlying theory of gravity in ΛCDM cosmological model, apparently cannot be quantized (Ashtekar, 2000; Mac´ıas, 2008; Padmanabhan, 2002; Rovelli, 2000). The problems surrounding ΛCDM model can be treated and sometimes evaded in a different perspective, from Alternative Gravity Theories (AGTs). The Einstein’s field equations of GR (1) are obtained from the

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variational principle applied to the Einstein–Hilbert action in the presence of a cosmological constant, namely,    √ R − 2Λ + Lm , SGRΛ = d4 x −g (2) 16π with g = detgμν and Lm the matter Lagrangian density. On the other hand, AGTs field equations can be obtained from the variational principle applied to a generalized version of (2), usually in the absence of Λ, as it is going to be explained below. The simpler and most popular AGT is the f (R) theory (De Felice and Tsujikawa, 2010; Hu and Sawicki, 2007; Nojiri and Odintsov, 2011), in which f (R) is a generic function of R to substitute R − 2Λ in (2). The variational principle applied to the resulting action (see Sec. 2) yields field equations with new terms “replacing” Λ and sometimes able to describe the cosmological constant effect of cosmic acceleration (Amendola et al., 2007; Hu and Sawicki, 2007; Navarro and Van Acoleyen, 2007; Nojiri and Odintsov, 2011; Song et al., 2007). Remarkably, the dark matter issues are also investigated in the f (R) theory of gravity frame, as one can see, for instance, Arnalte-Mur et al. (2017); Shi et al. (2015) and Tsujikawa et al. (2009). Note that besides nulling Λ, the f (R) theory effectively changes the lhs of Einstein’s field equations (1), keeping the energy–momentum tensor intact. Anyhow it is possible to generalize both sides of Einstein’s field equations through the f (R, T ) theory (Harko et al., 2011). Keeping the previous line of thought, the function f (R, T ) substitutes R − 2Λ in (1) and the variational principle yields new field equations whose new terms can describe dark energy effects (Baffou et al., 2015; Chakraborty, 2013; Houndjo, 2012; Reddy et al., 2013). While the extra terms proportional to R are motivated by the fact that Einstein took the simplest R-dependence in its gravitational action, the extra terms proportional to T can be interpreted as due to the existence of imperfect fluids (see Sec. 3). There are a lot of other AGTs, such as teleparallel gravity (TG) (Arcos and Pereira, 2004; Maluf, 2013) and symmetric teleparallel gravity (STG) (Adak et al., 2006; Adak, 2006). Even braneworld models (Maartens, 2004; Maartens and Koyama, 2010) can be seen as AG. Each of these gravity models has been used to construct cosmological models that evade the shortcomings of ΛCDM cosmology (Bamba et al., 2013; Barros et al., 2020; Campos et al., 2003; Golovnev and Koivisto, 2018; Sahni and Shtanov, 2003; Xu et al., 2012) and their formalism will be visited in Sec. 2.

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It is clear that there is a plethora of gravity formalisms in the present literature. In fact, besides the aforementioned models, there is a lot more, as one can see, for instance, the reviews (Langlois, 2019; Nojiri et al., 2017). An important way to constrain these models nowadays is through gravitational waves (GWs) astrophysics. In 2015, the first sign of GW, namely GW150914, was ever detected (LIGO Scientific Collaboration and VIRGO Collaboration, 2016). The GW sign detected by LIGO (Laser Interferometer Gravitational-wave Observatory) was emitted by a coalescing black hole binary with 36+5 −4 M +0.03 and 29 ± 4 M at redshift z = 0.09−0.04 . GW150914 has been used to constrain AGTs, as one can see, for instance, Hess (2016) and Yarman et al. (2019). In 2017, the first observation of GWs from a binary neutron star (NS) inspiral happened (LIGO Scientific Collaboration and VIRGO Collaboration, 2017). The event, called GW170817, was observed by LIGO and VIRGO detectors. The mass of the components is in the range 1.17– 1.60 M . GW170817 was the first GW event to have its electromagnetic counterpart detected (Abbott et al., 2017), therefore giving birth to the illustrious multi-messenger era of Astronomy. GW170817 is probably the most studied event in Astronomy history and it has also been effectively used to constrain AGTs (Baker et al., 2017; Ezquiaga and Zumalac´ arregui, 2017; Green et al., 2018; Nojiri and Odintsov, 2018; Sakstein and Jain, 2017). It is quite interesting to constrain AGTs with GW events since these events are generated by compact astrophysical objects that are also studied or analyzed within AGTs. In fact, AGTs must also predict healthy solutions in stellar astrophysics and not only in the cosmological regimes. For instance, black holes have been exhaustively studied in braneworld models of gravity, as one can see, for instance, Casadio et al. (2002); Chamblin et al. (2000); Frolov and Stojkovi´c (2002) and Kofinas et al. (2002). The same happens for NSs, not only in braneworld gravity but in each AGT, as one can see, for instance, Doneva et al. (2015); Germani and Maartens (2001); Momeni et al. (2015); Orellana et al. (2013); Staykov et al. (2016) and Yazadjiev et al. (2015). Particularly, the AGTs application to neutrons is well motivated by the ¨ fact that we still do not know the NS equation of state (EoS) (Ozel and Psaltis, 2009; Radice et al., 2018; Read et al., 2009,b; Watts et al., 2016) so that AGTs become a valuable tool to increase the maximum mass of such objects (Rahaman et al., 2020; Wiseman, 2002; Yazadjiev et al., 2015)

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(further references on this subject will be given below) to get in touch with recent observations (Abbott et al., 2020; Antoniadis et al., 2013; Cromartie et al., 2020; Demorest et al., 2010; Linares et al., 2018). Let me remark some important points regarding two of the references just mentioned. In Radice et al. (2018), the GW170817 event together with its electromagnetic counterpart was used to constraint the NS EoS. In Abbott et al. (2020), the GW190814 event was reported, as consisting of a binary system with a 22.2–24.3 M black hole and a 2.5–2.67 M NS at a distance of 241+41 −45 Mpc. This is the most massive NS known to date and to describe it theoretically can be a challenge. It is valuable to mention that AGTs have been applied even to white dwarfs (Banerjee et al., 2017; Carvalho et al., 2017; Das and Mukhopadhyay, 2015a,b; Kalita and Mukhopadhyay, 2018; Wojnar, 2021), mainly with the same purpose of increasing their maximum masses (massive white dwarfs have also been observed, as one can see, for instance, Howell et al. (2006); Scalzo et al. (2010) and Silverman (2011)). It was recently shown that the GR effects are indeed important to describe white dwarfs structure (Carvalho et al., 2018). From this starting point, Banerjee et al. (2017); Carvalho et al. (2017); Das and Mukhopadhyay (2015a,b); Kalita and Mukhopadhyay (2018) and Wojnar (2021) show that effects beyond GR could also be relevant, therefore considerable and possibly necessary. Finally, other less popular astrophysical objects such as gravastars and wormholes have also been analyzed within AGTs context (Arba˜ nil et al., 2019; Bronnikov and Kim, 2003; Das et al., 2020; Moraes and Sahoo, 2019; Moraes et al., 2017; Sahoo et al., 2018). In this chapter I will review some recent advances on AGTs applied to stellar astrophysics. After presenting the main properties of the AGTs formalism and some particular gravitational models, I will show some results and, when it is possible, confront such results with GW observational data, which has shown to be an important tool for constraining AGTs, as mentioned before. Finally, I will discuss the prospects for AGTs for the next decades. Before properly starting, let me quote that GR is in accordance with a great number of observations as one can see, for instance, Abbott et al. (2020); Johannsen (2016); LIGO Scientific Collaboration and VIRGO Collaboration (2016, 2017); Nobili and Will (1986); Shapiro et al. (1971) and Turyshev et al. (2013, 2016), among others. It is not my purpose here to try to convince the reader that GR is not right (this would not be the purpose of any reasonable author). In some paragraphs above I have shown that at

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least in some regimes of gravity fields and length scales, GR may not be the final answer. In view of that, in such regimes, some corrections may be necessary, but always in direct contact with GR such that Einstein’s theory is always recoverable. 2. Alternative Gravity Theories Einstein’s theory of GR (D’Inverno, 1992; Einstein, 1915) is certainly one of the biggest humanity achievements of the last century. However, as it was shown in the last section, some apparent deficiencies on it have appeared and accumulated, which led theoretical physicists to question if GR was the unique and basic theory to describe gravitational interactions. In this section I am going to review some important features of AGTs and how one can distinguish among them. 2.1. Geometric trinity of gravity GR has introduced geometry into physics. The equivalence principle envisioned by Albert Einstein dictates the universality of gravity, which yields its geometrical description. GR geometrization of gravity is made in terms of metric and curvature. In fact, gravitational phenomena have been recognized as a manifestation of curved space–time (these important fundaments of GR can be revised, for instance, in Hobson et al. (2006) and also in D’Inverno (1992), among many others). The metric tensor, however, cannot define curvature by itself so that a connection is required (Aldrovandi and Pereira, 1995). The connection can have either vanishing or non-vanishing curvature and torsion. There is also the possibility of describing gravity in a flat and torsion-free space–time. The space–time manifold can be described from three different geometrical objects: curvature, torsion and non-metricity. We can refer to it as geometric trinity of gravity. In the case of non-null curvature and null torsion and non-metricity we have GR (Einstein, 1915) and its extensions (f (R) (Amendola et al., 2007; Arnalte-Mur et al., 2017; De Felice and Tsujikawa, 2010; Hu and Sawicki, 2007; Navarro and Van Acoleyen, 2007; Nojiri and Odintsov, 2011; Shi et al., 2015; Song et al., 2007; Tsujikawa et al., 2009), f (R, T ) theories (Baffou et al., 2015; Chakraborty, 2013; Harko et al., 2011; Houndjo, 2012; Reddy et al., 2013), . . .). In the case with non-null torsion and null curvature and non-metricity we have the TG (Arcos and Pereira, 2004; Maluf, 2013) and its extensions (f (T ) theories (Daouda et al., 2012; Wei et al., 2012; Wu and Yu, 2010,b; Yang, 2011; Zhang et al., 2011), with

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T being the torsion scalar). Lastly, in the case in which the non-metricity is non-null but both curvature and torsion are null, we have the STG (Adak et al., 2006; Adak, 2006) and its extensions (f (Q) theories (Jim´enez et al., 2020; Mandal et al., 2020a,b), with Q being the non-metricity scalar). 2.1.1. Curvature theories GR in the presence of the cosmological constant is described by action (2), which yields the field equations (1). For the aforementioned reasons, mainly related to the dark sector of the universe, namely dark energy and dark matter, it is common to see extensions of GR, that consider higherorder dependency on R instead of Λ. The action of such theories, called f (R) theories, reads (De Felice and Tsujikawa, 2010; Hu and Sawicki, 2007; Nojiri and Odintsov, 2011)    √ f (R) + Lm . (3) Sf (R) = d4 x −g 16π By varying Eq. (3) with respect to the metric gμν yields the f (R) gravity field equations, which read as follows: 1 fR Rμν − f gμν + (gμν  − ∇μ ∇ν )fR = 8πTμν . 2

(4)

In Eq. (4), fR ≡ df /dR and throughout this book chapter, the matter energy–momentum tensor reads as follows: 2 δLm . Tμν = − √ −g δgμν

(5)

The Bianchi identities (see, for instance, D’Inverno (1992)) applied to Eq. (4) yields the conservation equation ∇μ Tμν = 0.

(6)

Let me remark some important features of Eq. (4). In the case f (R) = R one naturally retrieves Einstein’s GR field equations with no cosmological constant, namely 1 Rμν − Rgμν = 8πTμν , 2

(7)

as fR = 1. The extra terms appearing in Eq. (4) when compared to Eq. (7) can predict the cosmic acceleration (or dark energy effects), according to Amendola et al. (2007); Hu and Sawicki (2007); Navarro and Van Acoleyen (2007); Nojiri and Odintsov (2011) and Song et al. (2007).

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It is also possible to extend both sides of Einstein’s field equations of GR, namely the geometrical and material sectors, through the f (R, T ) theories. To be more precise, the f (R, T ) theories would fit in the so-called curvaturematter coupling theories (see, for instance, Bertolami and Sequeira (2009); Bertolami et al. (2013) and Gomes et al. (2017)). However, by taking the “geometric trinity” into consideration, it is perfectly plausible to consider the f (R, T ) theories as belonging to the “curvature theories sector”. From the f (R, T ) gravity action (Harko et al., 2011)    √ f (R, T ) + Lm , (8) Sf (R,T ) = d4 x −g 16π one obtains, by applying the variational principle application, the f (R, T ) gravity field equations, which read as 1 fR Rμν − f gμν + (gμν  − ∇μ ∇ν )fR = 8πTμν + fT (Tμν − Lm gμν ). (9) 2 Now, in Eq. (9), fR ≡ ∂f /∂R and fT ≡ ∂f /∂T . Moreover, one should be aware that while in Eq. (4), f is a function of R only, in Eq. (9), it is a function of both R and T , which is also the reason for which fR carries ordinary derivatives in the f (R) gravity and fR and fT carry partial derivatives in the f (R, T ) theory. The extra terms in (9) can also describe the dark energy effects, as one can see, for instance, Baffou et al. (2015); Chakraborty (2013); Houndjo (2012) and Reddy et al. (2013). The Bianchi identities applied to (9) yields the non-conservation equation     fT T (Lm gμν − Tμν )∇μ ln fT + ∇μ Lm − gμν . (10) ∇μ Tμν = 8π + fT 2 Equation (10) above shows the non-conservation of the energy–momentum tensor. In a cosmological perspective, this is related to creation of matter through the universe evolution (Kumar and Singh, 2015; Singh and Singh, 2016). In the formalism of the curvature theories, there are also the braneworld models (Maartens, 2004; Maartens and Koyama, 2010). Braneworld models were not necessarily motivated by cosmological issues but rather by the hierarchy problem (Arkani-Hamed et al., 1998, 2001; Cohen and Kaplan, 1999; Vissani, 1998), which is the weakness of the gravity force when compared to the other fundamental forces. In the braneworld models, departing from the other fundamental forces, gravity is capable of interacting with an extradimensional bulk. In other words, gravity “leaks” through the extra

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dimension, what explains the weakness of gravity in our 3 + 1 observational world. On the other hand, the other fundamental forces of nature remain confined to the brane. 2.1.2. Torsion theories For a long time, gravity was purely identified with curvature, that is, gravitational phenomena were seen as curvature in space–time. In torsion theories, gravity is described by torsion rather than curvature, with no non-metricity. Although different as it is (see below), TG is equivalent to the description of GR. Most importantly, it seems to be a much more appropriate theory to deal with the quantization of the gravitational field (Marsh, 2018), an important problem of Physics mentioned in Introduction. In fact, TG was considered by Einstein in 1928 as a possible geometrical set up for the unification of gravitational and electromagnetic fields (Einstein, 1928). We can say that TG is a gravitational theory that uses the curvaturefree Weitzenbock connection (Weitzenbock, 1923) (see below) to define the covariant derivative, instead of the conventional torsionless Levi-Civita connection of GR theory. In its simplest form, it is equivalent to GR, however, motivated by attempts to explain the observed acceleration of the universe in a natural way, that is, free of a cosmological constant or dark energy, there has been a great deal of recent interest in a generalization of this theory, in which the gravitational Lagrangian is an arbitrary function of T . Those are the f (T ) theories (Daouda et al., 2012; Wei et al., 2012; Wu and Yu, 2010,b; Yang, 2011; Zhang et al., 2011). In fact, the cosmic acceleration is predicted by f (T ) theories, for instance, in Cardone (2012); Farajollahi (2012); Qi et al. (2016); Ren et al. (2021) and Shaikh et al. (2021). Even dark matter effects are predicted in f (T ) theories (Jamil et al., 2012; Rahaman et al., 2014). Let me finally introduce TG and its extensions. The dynamical variables are the tetrad fields ha (xμ ), that form an orthonormal basis for the tangent space at each point of the manifold with space–time coordinates xμ . The space–time metric reads as (Li et al., 2011) gμν = ηab haμ hbν ,

(11)

ηab = diag(+1, −1, −1, −1).

(12)

in which

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While GR uses the Levi-Civita connection (D’Inverno, 1992) Γλμν ≡

g λσ (∂ν gσμ + ∂μ gσν − ∂σ gμν ), 2

(13)

which leads to non-null space–time curvature and zero torsion, TG uses the Weitzenbock connection (Weitzenbock, 1923) λ γμν ≡ hλb ∂ν hbμ = −hbμ ∂ν hλb ,

(14)

which yields to null curvature and non-null torsion. Now, the torsion tensor reads λ λ λ ≡ γνμ − γμν = hλb (∂μ hbν − ∂ν hbμ ). Tμν

(15)

The difference between the Levi-Civita and the Weintzenbock connections is the contorsion tensor ρ ρ Kμν ≡ γμν − Γρμν = hρa ∇ν haμ .

(16)

U ρμν ≡ K μνρ − g ρν T σμσ + g ρμ T σνσ ,

(17)

Finally, by defining

the TG lagrangian density reads LT G ≡

h ρμν h T = U Tρμν , 16π 32π

(18)

√ in which h = dethλa = −g and from which it is clear that T ≡ U ρμν Tρμν /2. Now one can write the TG total action as (Li et al., 2011)    T 4 ST G = d xh + Lm , (19) 16π which when varied with respect to the tetrad haλ yields the TG field equations 1 1 1 δLM ∂σ (hhρa Uρλσ ) − hσa U μνλ Tμνσ + hλa T = 8π . h 2 h δhaλ

(20)

The TG generalization or extension is naturally obtained by making T → f (T ) in (18), so that (Li et al., 2011) Lf (T ) =

h f (T ). 16π

(21)

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Similarly, the f (T ) gravity field equations read   1 1 1 δLM ρ λσ σ μνλ ∂σ (hha Uρ ) − ha U Tμνσ + fT T hρa Uρλσ ∂ξ T + hλa f = 8π , fT h 2 h δhaλ (22) in which fT ≡ df /dT , fT T ≡ d2 f /dT 2 . Clearly, when f = T one recovers (20). Let me remark that it is also possible to consider extra material terms in the f (T ) gravity, giving rise to the f (T , T ) gravity theories (Junior et al., 2016). 2.1.3. Non-metricity theories In non-metricity theories, the connection has neither curvature nor torsion. This approach was firstly considered in Nester and Yo (1999) and will be presented in the following. In STG, gravity is ascribed to the non-metricity Qαμν ≡ ∇α gμν ,

(23)

and is materialized in a flat and torsion free geometry. In order to present the STG, it is important to invoke (23) together with the redefinition of the torsion scalar simply as α α Tμν = 2(Γα μν − Γνμ ).

(24)

α , It is important to note that the most general affine connection Cμν comprehending all possible contributions (curvature, torsion and nonmetricity), allows the decomposition (Ortin, 2004) α α α = Γα Cμν μν + Kμν + Lμν ,

(25)

with the first term on the rhs of the above equation being the Levi-Civita connection, while the contorsion tensor is rewritten as 1 αλ g (Tμλν + Tνλμ + Tλμν ), (26) 2 and the disformation tensor is defined in terms of the non-metricity tensor as 1 αλ Lα (Qλμν − Qμνλ − Qνμλ ). (27) μν ≡ g 2 Now, it has to be clear that GR is constructed from the consideration α α α = Lα that Kμν μν = 0. Meanwhile, TG is obtained when Γμν = Lμν = 0 α ≡ Kμν

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α and STG when Γα μν = Kμν = 0. The three formalisms are equivalent at the field equations level. By paralleling transport a vector around in a loop, if the vector twists around by comparing its initial and final orientations (at the same position), we can say the space in which the vector was transported is curved. This is also referred to as holonomy. The torsion tensor acts on two vector fields and gives us the separation vector that results when the two vector fields are parallel transported along each other. If the region is torsionless, then the separation between the parallel transported vectors is zero. In such a case, we have a completed closed four-sided shape from the parallel transports. Therefore, for a connection to be torsion-free, the parallel transported vectors close properly. Finally, in a space–time with no curvature nor torsion, a vector transported along a curve will change its side and that represents the nonmetricity. From Hohmann et al. (2019), for instance, the STG total action reads    √ Q + Lm . (28) SST G = d4 x −g 16π

By varying (28) with respect to the metric yields the STG field equations: √ 1 2 γ γι √ ∇γ ( −gPμν ) + gμν Q + Pμγι Qγι ν − 2Qγιμ Pν = −8πTμν , −g 2

(29)

in which the non-metricity conjugate is defined as follows: λ ˜ λ )gμν − 1 (δ λ Qν + δ λ Qμ ), (30) 4Pμν = −Qλμν + Qμλν + Qμλν + (Qλ − Q ν 2 μ with

Qα = Qααμ , ˜ α = Qμ , Q αμ

(31) (32)

and such that the trace of the non-metricity tensor reads as Q = −Qλμν P λμν .

(33)

Action (28) can be generalized to an arbitrary function of Q, giving rise to the f (Q) gravity, whose action reads (Jim´enez et al., 2020; Mustafa et al., 2021)    f (Q) 4 √ + Lm . Sf (Q) = d x −g (34) 16π

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By varying (34) with respect to the metric yields the f (Q) gravity field equations √ 2 1 γ γι √ ∇γ ( −gfQ Pμν ) + gμν f + fQ (Pμγι Qγι ν − 2Qγιμ Pν ) = −8πTμν , −g 2 (35) with fQ ≡ df /dQ. Similarly as before, it is possible to also insert terms dependent on T in Eq. (34), yielding the f (Q, T ) gravity (Xu et al., 2019). 3. Stellar Equilibrium Configurations We have now finally arrived to the stellar astrophysics subject. In this section I am going to show the hydrostatic equilibrium equation or Tolman– Oppenheimer–Volkoff (TOV) equation (Oppenheimer and Volkoff, 1939; Tolman, 1939) in GR and then in AGTs. From the hydrostatic equilibrium configurations one obtains several important information about stellar features such as mass × radius relation, maximum mass, central density, etc. In order to derive the TOV equation one starts from a spherically symmetric metric such as ds2 = ea dt2 − eb dr2 − r2 (dθ2 + sin2 dφ2 ),

(36)

in which a and b are functions of r only. The Einstein tensor non-null components for such a metric are G00 =

e−b  (b r + eb − 1), r2

e−b (−a r + eb − 1), r2   e−b a (a − b ) (b − a ) + G22 = −a − , 2 2 r

G11 =

G33 = G22 ,

(37) (38) (39) (40)

with primes indicating radial derivatives. The energy–momentum tensor of a perfect fluid is Tμν = diag(ρ, −p, −p, −p).

(41)

In the equation above, ρ is the matter-energy density and p is the pressure inside the star.

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The quantity m is introduced representing the gravitational mass within the sphere of radius r and such that e−b = 1 −

2m . r

(42)

The 00 and 11 components of (7) are e−b  (b r + eb − 1) = 8πρ, r2

(43)

e−b (−a r + eb − 1) = −8πp. r2

(44)

By substituting Eq. (42) in Eq. (43) yields the following: m = 4πr2 ρ.

(45)

Moreover, from Eq. (6), which is the conservation of the energy– momentum tensor equation, we now have 2p + (ρ + p)a = 0. Finally, by replacing Eq. (44) into Eq. (46) yields   (ρ + p) 4πpr + rm2  , p =− 1 − 2m r

(46)

(47)

which is the renowned TOV equation. It is clear from the above calculations that the TOV equation is modeldependent, that is, it depends on the field equations one is working with. Since different AGTs present different field equations, consequently a different TOV equation (or, more precisely, TOV-like equation) shall be obtained for each gravitational formalism. Let us take, for instance, the f (R, T ) theory of gravity. The TOV equation in f (R, T ) gravity was first derived in Moraes et al. (2016). In Moraes et al. (2016), and also in Carvalho et al. (2017) and Moraes et al. (2018), the TOV equation was derived for f = R + 2λT,

(48)

with λ a constant parameter. This functional form has been used in the literature with different purposes for applications, such as cosmology (Kumar and Singh, 2015; Moraes, 2016; Moraes et al., 2018b) and wormholes (Moraes and Sahoo, 2017; Moraes et al., 2017). The f (R, T ) gravity TOV

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equation for this functional form reads (Carvalho et al., 2017; Moraes et al., 2018)

(ρ + p) 4πpr + rm2 + λ(ρ−3p)r 2

. (49) p = −   λ  dρ 2m 1 − + 1 1− r 8π+2λ dp From (49) it is clear that the usual TOV equation (47) is recovered simply by taking λ = 0. Recall that we still do not know the NS EoS. In Moraes et al. (2016, 2018), the authors have used the polytropic EoS p = ωρ5/3 ,

(50)

following the work by Tooper (1964), with ω = 1.475 × 10−3 [fm3 /MeV]2/3 (Ray et al., 2003). The mass vs. radius relation obtained for the above configuration can be seen below in Fig. 1 (Moraes et al., 2018). From Fig. 1 we can see that when the λ value is decreased, the NS becomes larger and more massive. However, while the radius of the NS can increase 43.41% when the λ = 0 (GR) and λ = −0.2 cases are compared, the mass increases only 5.83%. This clearly shows us that the referred scenario is not a good alternative to explain the massive NSs recently reported.

Fig. 1. Mass vs. radius relation for neutron stars in the f (R, T ) = R + 2λT gravity for different values of λ and p = ωρ5/3 with ω = 1.475 × 10−3 [fm3 /MeV]2/3 . The full circles indicate the maximum masses for each λ value.

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301

Note that the functional form f (R, T ) = R + 2λT was used not only in Carvalho et al. (2017) and Moraes et al. (2016, 2018) in the context of stellar equilibrium configurations but also e.g. in Rocha et al. (2020) and Sharif and Siddiqa (2017), for charged polytropic stars and charged white dwarfs, respectively. The steps to obtain Eq. (49) are carefully explained in Carvalho et al. (2017) and Moraes et al. (2018). Generally speaking, it is obtained from the substitution of Eqs. (36) and (41) into Eqs. (9) and (10). It has been previously pointed that Eq. (10) is a non-conservative equation for the energy–momentum tensor and that in a cosmological perspective it would imply in creation of matter through the universe evolution. How can one interpret (10) in a static case such as the hydrostatic equilibrium configurations? The absence of an answer to the above question has led to the development of dos Santos et al. (2019). The development of dos Santos et al. (2019) was based on the following line of thought: one can force Eq. (10) to be a conservation equation by making its lhs to vanish and solving the resulting equation for fT . The solution, when integrated in T , reveals the T dependence of the f (R, T ) function that conserves the energy–momentum tensor. By constructing the field equations for such an f (R, T ) functional form and developing its TOV equation, one finds a conservative approach. Such a development not only evades the issue of physically interpreting a non-conservative hydrostatic equation. Remarkably it also reveals the possibility of getting in touch with the observations of massive pulsars, as it is going to be shown below. Assuming the polytropic EoS (50), the development of the steps explained above yields (dos Santos et al., 2019) α , fT = √ 2/3 ρ(ωρ + 1)3

(51)

with constant α. Equation (51) together with its integration in ρ yield to the f (R, T ) gravity TOV equation with conserved energy–momentum tensor. In Fig. 2 the NS mass vs. radius relation for such a formalism is presented. Note that for α = −0.15 it is possible to explain the massive pulsars of Antoniadis et al. (2013); Cromartie et al. (2020); Demorest et al. (2010) and Linares et al. (2018), whose masses are ∼2 M or even greater. Now let me introduce a quite recent contribution to the literature in what concerns the f (T ) gravity TOV equation. In Lin et al. (2021) we

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Fig. 2. Mass vs. radius relation for neutron stars in the energy–momentum tensor conserved f (R, T ) gravity for different values of α. The full circles indicate the maximum masses for each α value.

have an example of how to construct and solve the f (T ) gravity TOV equation. In Lin et al. (2021), motivated by the Starobinsky model (Starobinsky, 1987) within the f (R) gravity, the functional form f (T ) = T + αT 2 ,

(52)

was taken. Naturally, in the case of Eq. (52), α is a different constant with different units when compared to the cases of the f (R, T ) gravity (51) and the Starobinsky model itself, besides the case of Eq. (59) below. Still in Lin et al. (2021), two kinds of EoS were used, namely: SLy (Douchin and Haensel, 2001) and BSk (Potekhin et al., 2013). Those read respectively as

(a3 ξ + a2 )ξ + a1 a5 (ξ − a6 ) + a9 (a8 ξ + a7 )(a10 − ξ) ζ=k a4 ξ + 1  + a13 (a12 ξ + a11 )(a14 − ξ) + a17 (a16 ξ + a15 )(a18 − ξ) ,

(53)

Alternative Gravity Neutron Stars in the Gravitational Wave Era

ζ=k

(a3 ξ + a2 )ξ + a1 a5 (ξ − a6 ) + a9 (a8 ξ + a7 )(a6 − ξ) a4 ξ + 1

303



+ a12 (a11 ξ + a10 )(a13 − ξ) + a16 (a15 ξ + a14 )(a17 − ξ) + X ,

(54)

with ζ = log[p/(dyn/cm2 )], k = k(x) = (ex + 1)−1 , ξ = log[ρ/(g/cm3 )], X =

a18 a21 + , 2 [a19 (ξ − a20 )] + 1 [a22 (ξ − a23 )]2 + 1

(55)

and the parameters a1 , a2 , . . . , a23 are listed in the Appendix A of Lin et al. (2021). The stable curves in the mass vs. radius diagram obtained in Lin et al. (2021) point to the existence of NSs a little more massive than 2 M . Finally, to the knowledge of the present author, spherically symmetric configurations have been developed within the f (Q) gravity context only very recently in Lin et al. (2021b). For metric (36), the non-metricity scalar reads Q=

(e−b − 1)(a + b ) . r

(56)

The 00 and 11 components of the f (Q) gravity field equations (35) for Eqs. (36) and (41) are given by the following       ea−b 2 2fQQ Q (eb − 1) + fQ (eb − 1) a + + (eb + 1)b + f eb r = 8πρ, 2r r (57)       1 2 2fQQ Q (eb − 1) + fQ (eb − 1) a + b + − 2a + f eb r = 8πp, 2r r (58) with fQQ ≡ d2 f /dQ2 . In Lin et al. (2021b), for the Q-dependency of the formalism, the functional form f = Q + αQ2 ,

(59)

was used, also motivated by the Starobinsky model (Starobinsky, 1987). The EoS assumed in Lin et al. (2021b) was similar to (50), but with the polytropic index being 2 rather than 5/3. It was shown to be possible to achieve, depending on the value of α, stars as massive as ∼4 M , with radius of ∼12.5 km. As pointed out in Lin

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et al. (2021b), this model rises as a good alternative to explain massive NSs (even the 2.5–2.67 M NS of Abbott et al. (2020)) with no need of posing challenges for nuclear physics.

4. Gravitational Wave Constraints It is clear that there is a plethora of AGTs in the present literature. It should be remarked that in the aforementioned AGTs, the f function is a general function of the argument, with each functional form giving rise to a new gravity formalism. Moreover, the argument can be a function of one, two or even more scalars (see, for instance, Sharif and Zubair, 2013; Yousaf et al., 2020). Finally, there are further AGTs that have not been mentioned above and will not be contemplated in the present chapter, such as f (G) gravity (Nojiri and Odintsov, 2005) with G being the Gauss–Bonnet scalar, Brans–Dicke theory (Agnese and La Camera, 1995), Chern–Simons model (Jackiw and Pi, 2003) and Born–Infeld theory (Gibbons and Herdeiro, 2001). To discriminate among these theories is a hard quest. Cosmological tests (Jain and Khoury, 2010; Koyama, 2016) are one of the main alternatives, though the deep investigation of stellar configurations is, by itself, a possibility (see also Hakimov, 2013; Novak, 1998). With the advent of the GWs detection (LIGO Scientific Collaboration and VIRGO Collaboration, 2016), a new tool for constraining AGTs has arisen (recall the references given in the Introduction section). Particularly, in LIGO Scientific Collaboration and VIRGO Collaboration (2018), measurements of NS radii and EsoS were obtained in view of GW170817 (LIGO Scientific Collaboration and VIRGO Collaboration, 2017). Figure 3 (Lobato et al., 2020) shows the constraints for NS mass–radius obtained in LIGO Scientific Collaboration and VIRGO Collaboration (2018) from GW170817. We can also see the mass vs. radius relation predicted curve for several EsoS using GR. In Lobato et al. (2020), NSs were investigated in the f (R, T ) = R + 2λT gravity in light of GW170817. Firstly, it was shown that the main contribution from such a theory is an increase in the radius of the star. In f (R, T ) gravity, larger NSs can be obtained with smaller central energy densities compared to GR outcomes. Remarkably, it was shown in Lobato et al. (2020) that, in view of GW170817, only the following EsoS are suitable for f (R, T ) = R + 2λT

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Fig. 3. Mass vs. radius relation in General Relativity for several equations of state. The blue and orange regions are the constraints for the mass–radius obtained from GW170817 (LIGO Scientific Collaboration and VIRGO Collaboration, 2018). The continuous red line refers to the minimum mass value of the compact object in GW190814 (Abbott et al., 2020) (presumably a neutron star). The other horizontal lines, namely dotted yellow, dotted-dashed purple and continuous blue lines, refer to other massive neutron stars reported in the literature.

gravity: APR3-4 (Akmal et al., 1998), WFF1-2 (Wiringa et al., 1988), ENG (Engvik et al., 1994) and MPA1 (M¨ uther et al., 1987). Further constraints on AGTs from GW detection can be seen in Karimi and Karami (2020); Kase and Tsujikawa (2019) and Odintsov and Oikonomou (2020). In Lin et al. (2021), GW constraints were put to the f (T ) gravity NSs. It was shown that in order to be able to predict the existence of GW190814 using the SLy EoS (53), the α parameter in (52) needs to be negative, while GR simply cannot reach GW190814 mass. Finally, in Langlois et al. (2018), it was shown how to constrain AGT parameters based on the fact that the speed of GWs has shown to be the same as the speed of light within deviations of 10−15 order (Abbott et al., 2017).

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5. Discussion and Perspectives We live in a time of unprecedented interest in Astronomy. For the nonscientific community this is reflected in the increasing number of movies, television shows and novels approaching the subject. The scientific community witnesses the birth of multi-messenger astronomy era (recall Abbott et al., 2017), which is a milestone in the history of science. The GW astronomy will soon be implemented with novel detectors, such as the Einstein Telescope (Punturo et al., 2010), which having an improved sensitivity, will permit more detailed measurements of the physical parameters of the GW source. Also, as the Einstein Telescope will have a greater detectable horizon, a higher event rate is expected to be detected. The third generation of GW detectors naturally allied to the electromagnetic counterpart of such events will certainly be a fundamental tool in the field discussed in the present chapter, i.e. AGTs. We expect that GW astronomy will lead us to a better understanding of NS microphysics (Forbes et al., 2019; Orsaria et al., 2019; Zhu et al., 2018). Optimistically speaking, one could expect it to be able to solve even the high degeneracy involving NS macroscopical features with different AGTs/EsoS. The AGT realm will certainly be affected by a higher GW event statistics. For now, besides the constraining methods presented above, the GW polarization state is also an important tool (see, for instance, Hou et al., 2018). On this regard, one should keep in mind that different AGTs may predict the existence of a combination of different polarization states for GWs. In parallel to the further NS physics predictions, cosmology is also affected by GW astronomy. Recalling the main current cosmology problems, namely dark energy and dark matter problems and Hubble tension, they have been visited in light of GW detection respectively in Ezquiaga and Zumalac´ arregui (2017, 2018), Bird et al. (2016) and Boran et al. (2018). Finally, the GW astronomy is and will be fundamental to constraint among the high number of AGTs, and possibly provide the ultimate theory of gravity. The importance of achieving an attempt like this is unmeasurable, as its consequences appear not only in the astrophysical and cosmological realms, but also in the search for a quantum theory of gravity (Agullo et al., 2021; Calcagni et al., 2019). Acknowledgment PHRSM thanks CAPES for financial support.

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© 2023 World Scientific Publishing Company https://doi.org/10.1142/9789811220913 0008

Chapter 8

Quark Deconfinement in Compact Stars Through Sexaquark Condensation David Blaschke∗ , Oleksii Ivanytskyi† and Mahboubeh Shahrbaf‡ Institute of Theoretical Physics, University of Wroclaw pl. M. Borna 9, 50-204 Wroclaw, Poland ∗ [email protected] † [email protected] ‡ [email protected] In this contribution, we present for the first time a scenario according to which early quark deconfinement in compact stars is triggered by the Bose-Einstein condensation (BEC) of a light sexaquark (S) with a mass mS < 2054 MeV, that has been suggested as a candidate particle to explain the baryonic dark matter in the universe. The onset of S BEC marks the maximum mass of hadronic neutron stars and it occurs when the condition for the baryon chemical potential µb = mS /2 is fulfilled in the center of the star, corresponding to Monset < 0.7M . In the gravitational field of the star the density of the BEC of S increases until a new state of the matter is attained, where each of the S-states got dissociated into a triplet of color-flavor-locked (CFL) diquark state. These diquarks are the Cooper pairs in the color superconducting CFL phase of quark matter, so that the developed scenario corresponds to a BEC–BCS transition in strongly interacting matter. For the description of the CFL phase, we develop here for the first time the three-flavor extension of the density-functional formulation of a chirally symmetric Lagrangian model of quark matter where confining properties are encoded in a divergence of the scalar self-energy at low densities and temperatures. Keywords: Sexaquark; Compact stars; Diquarks.

1. Introduction The discussion of quark matter in compact stars has a long history. It started with the early works by Ivanenko and Kurdgelaidze (1965) and Itoh 317

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(1970), but a proper foundation for the necessity of quark deconfinement in QCD existed only after asymptotic freedom was proven. Based on this argument Baym and Chin (1976) developed the concept of a thermodynamic bag model in order to describe macroscopic volumes of deconfined quark matter, for instance inside neutron stars. At that time also the idea of a third family of compact stars was already formulated by Gerlach (1968) but found practically no resonance. Another idea was that of “collapsed nuclei” (Bodmer, 1971), namely that much smaller and deeper bound nuclei, possibly electrically neutral due to their hyperchange, could exist in nature, separated from ordinary nuclei by a sufficiently high barrier to guarantee stability of the latter on cosmological time scales. After the MIT bag model was conceived (Chodos et al., 1974b) and used to explain the structure of baryons (Chodos et al., 1974a), a hypothetical hypernucleus with baryon number 6 and strangeness −6 (hexalambda) was proposed by Terazawa (1979) within the MIT bag model. The idea of an absolutely stable strange quark matter state with astrophysical applications was formulated by Witten as “Cosmic Separation of Phases” (Witten, 1984) and immediately developed further by Farhi and Jaffe (1984) as “Strange Matter”. Despite intense searches for remnants of strange matter at different length scales, in the Cosmos and in Laboratory experiments, no conclusive evidence for the existence of strange quark matter nuggets has been found yet. About a decade ago it was even argued that strangeness is likely absent from neutron star interiors (Blaschke et al., 2010). The reasoning for this was based on the fact that a standard solution for the hyperon puzzle was early quark deconfinement (Baldo, Burgio and Schulze, 2003; Burgio et al., 2002) (for a recent update fulfilling the 2 M constraint, see (Shahrbaf et al., 2020a; Shahrbaf, Blaschke and Khanmohamadi, 2020b)) which eliminated the appearance of hyperons as a scenario for compact star interiors. When for the quark matter phase a Nambu-Jona-Lasinio (NJL)-type model with its sequential occurrence of quark flavors (Blaschke et al., 2009) was adopted, then after the possible one- and two-flavor color superconducting phases, the occurence of the strange quark flavor in the CFL color superconducting phase was accompanied with the gravitational instability of the corresponding hybrid star configurations, see also (Kl¨ ahn et al., 2007; Kl¨ahn, L  astowiecki and Blaschke, 2013). With this line of reasoning and the demonstration of numerous examples, there was no place for strangeness in compact stars, neither in the hadronic nor in the quark matter phases.

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Concerning the idea of absolutely stable strange quark matter (Bodmer, 1971; Farhi and Jaffe, 1984; Witten, 1984) and its realization in the form of strange stars (Alcock, Farhi and Olinto, 1986; Haensel, Zdunik and Schaeffer, 1986), one could even formulate a “No-Go” conjecture for the absolute stability of strange quark matter, at least for a description within NJL-type models (Kl¨ahn and Blaschke, 2018). In our contribution to this book we want to reopen the chapter of strange dark matter (DM) in compact stars, based on the new, multi-messenger phenomenology of compact stars and the possibility of a light sexaquark as a dark matter particle that evaded detection in laboratory experiments due to its stability against decays on cosmological timescales. This new perspective is made possible by a recent development of a density functional approach to quark matter in compact stars which allows to address confining effects (Kaltenborn, Bastian and Blaschke, 2017) that were absent in the formulation using NJL models of quark matter. In (Ivanytskyi, Blaschke and Maslov, 2022; Ivanytskyi and Blaschke, 2022), this approach was refined so that its Lagrangian was manifestly chirally symmetric and the important diquark interactions resulting in color superconducting phases were added. In the present contribution, for the first time, this approach will be generalized to three quark flavors and we will show that the light sexaquark acts as a trigger for entering the CFL quark matter phase at an onset mass below 0.7 M , while fulfilling the modern mass-radius constraints on pulsars. 2. Bose-Einstein Condensation of Sexaquarks as a Trigger for Strange Quark Matter in Compact Stars The discussion of a light and compact sexaquark (S) state with the quark content (uuddss) and the consequences for neutron star phenomenology that follow from its possible existence is of much interest in nuclear physics, particle physics as well as astrophysics. The S is an electrically neutral spinless boson with baryon number BS = 2 and strangeness SS = −2 in a flavorsinglet state. For mS ≤ 2(mp +me ) = 1877.966 MeV, due to baryon number conservation the S is stable, while for mS ≤ mp +me +mΛ = 2054.466 MeV, it should decay with a lifetime exceeding the age of the universe (Farrar and Zaharijas, 2004). In a recent work by Shahrbaf et al. (2022) it was shown that the existence of a light S in the mass range 1885 MeV < mS < 2054 MeV leads to S BEC in neutron stars. For the upper bound of this mass

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range, i.e. mS = 2054 MeV, the condensation happens at masses as low as M ≈ 0.7 M limiting thus the maximum mass of neutron stars. It was also shown that a plausible positive mass shift with increasing density allows for a stable sequence of neutron stars with sexaquark enriched matter in their cores that eventually undergoes a deconfinement transition to color superconducting (two-flavor) quark matter. Such hybrid stars with cores of deconfined, color superconducting quark matter would then populate the mass range from the typical neutron star masses ∼1.4 M up to the maximum mass Mmax > 2 M . The precise value of the maximum mass and the radius where it is reached depend on the details of the choice of parameters for the quark matter model. Here we will examine a new scenario where the S does not undergo a mass shift and therefore its BEC at a low star mass is inevitable and leads to a collapse of the hadronic neutron star, which gets stopped by quark deconfinement and the formation of a hybrid star. This new, stable hybrid star sequence will fulfill the recent constraints on neutron star radii at 1.4 M from the gravitational wave signal of the binary neutron star merger GW170817 by the LIGO-Virgo Collaboration (Abbott et al., 2018) and at 2.0 M by the recent NICER measurement on PSR J0740 + 6620 (Miller et al., 2021; Riley et al., 2021). This new scenario is built on the existence of the S as a deeply bound state with low enough mass to be stable on cosmological time scales (age of the universe) and is therefore a DM candidate. The observed DM to baryon ratio is ΩDM /ΩB = 5.3 ± 0.1 (Ade et al., 2016; Tanabashi et al., 2018) and a successful model for DM has to reproduce this value. An abundance of S DM (SDM) in agreement with this observation has been obtained within a statistical model on the basis of assumptions for the quark masses and an effective temperature Teff = 156 MeV1 of the transition from the quark–qluon plasma to the hadronic phase when mS = 1860 MeV (Farrar, Wang and Xu, 2020). When mS = 2mp = 1876.54 MeV, the observed ratio ΩSDM /ΩB = 5.3 is obtained for Teff = 150 MeV. It was first proposed in (Farrar, 2003) that S is a candidate of DM and this idea was followed during the next years as well (Farrar, 2017, 2018a, 2018b; Farrar, Wang and Xu, 2020). The fact that the light S cannot decay and that it is electrically neutral explains why it has so far evaded detection in laboratory experiments (Farrar, Wang and Xu, 2020). 1 This value is motivated by the recent result of T = 156.5 ± 1.5 MeV for the pseudoc critical temperature obtained in lattice QCD simulations (Bazavov et al., 2019).

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In the present contribution, the mass of the S is considered to be constant and set equal to the upper bound of its stability range, mS = 2054 MeV. At this mass, the S is light enough to be metastable on cosmological scales so as to serve as a DM candidate, and sufficiently heavy on the other hand to not disturb the stability of nuclei (Farrar and Zaharijas, 2003, 2004; Gross et al., 2018). We want to consider the possible relevance of S for the properties of neutron stars based on the recent multi-messenger observations that constrain their regions of accessibility in the mass-radius diagram. When the S is found in cold dense baryonic matter in neutron stars, it may form a BEC as soon as the baryochemical potential in the center of the star fulfills μB = mS /2. For mS = 2054 MeV, this occurs for a star with M = 0.7 M and, due to a saturation of the pressure with the BEC, this value marks the maximum mass that can be reached. This would be in clear contradiction with the observation of pulsars as massive as 2 M like PSR J0740 + 6620 (Fonseca et al., 2021) or PSR J0348 + 0432 (Antoniadis et al., 2013). Therefore, we introduce the sexaquark dilemma in this section and its solution by quark deconfinement. The EoS of hadronic matter is obtained from a generalized relativistic density functional (GRDF) with baryon–meson couplings that depend on the total baryon density of the system. The original density functional for nucleonic matter considers the isoscalar σ and ω mesons and the isovector ρ meson as exchange particles that describe the effective in-medium interaction. The density dependence of the couplings is adjusted to describe properties of atomic nuclei (Typel, 2005; Typel and Wolter, 1999). It has been confirmed that such GRDFs are successful in reproducing the properties of nuclear matter around nuclear saturation (Kl¨ ahn et al., 2006). GRDFs with different parameterization of the density-dependent couplings have been studied in (Typel, 2018). In the present work, the parameterization DD2 (Typel et al., 2010) is used for the σ, ω, and ρ couplings. It predicts characteristic nuclear matter parameters that are consistent with recent constraints (Oertel et al., 2017) and leads to a rather stiff EoS at high baryon densities with a maximum neutron star mass of 2.4 M in a pure nucleonic scenario of the strongly interacting system without hyperons. In the description of neutron-star matter, one has to consider that new baryonic degrees of freedom can become active with increasing density as the chemical potentials rise. Neglecting any interaction, a new species appears when the corresponding chemical potential crosses the particle mass.

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With the GDRF for hadronic matter at hand, it is possible to explore the EoS and corresponding properties of neutron stars. There are different scenarios to be distinguished in the following. The most simple case with only nucleons and leptons corresponds to the original DD2 model as presented in (Typel et al., 2010). Adding hyperons the model is called DD2YT to distinguish it from the similar model DD2Y introduced in (Marques et al., 2017). The DD2Y-T predictions were compared already to other EoS models with hyperons in (Stone et al., 2021). Finally, after including also the sexaquark, there is the full model which will be denoted DD2Y-TS in the following. When the mass of the S is = 0.25 fm−3 (corremS = 2054 MeV, the onset of the S occurs at nonset,S b sponding to more than 1.5 times the saturation density) with an immediate appearance of the BEC and a collapse of the neutron star because with increasing central density the pressure remains constant. To show that the sexaquark onset triggers a first-order phase transition to a color superconducting quark matter, we need the EoS for quark matter. There is a phenomenological formulation of the EoS of quark matter in use which has been introduced and motivated in Alford et al. (2005). In that work, the quark matter EoS consists of the first three terms of a series in even powers of the quark chemical potential ΩQM = −

3 3 a4 μ4 + 2 a2 μ2 + Beff , 4π 2 4π

(1)

where a4 , a2 , and Beff are coefficients independent of μ. The quartic coefficient a4 = 1−c is well defined for an ideal massless gas for which c = 0. Perturbative QCD corrections in lowest order, i.e. O(αs ) for massless quark matter lead to a reduction of a4 , e.g., accounted for by c = 0.3 so that a4 = 0.7. The quadratic μ2 term arises from an expansion in the finite strange quark mass ms and the diquark pairing gap Δ, so that a2 = m2s − 4Δ2 . In CFL quark matter, they are almost in the same order of about 100 MeV so that the coefficient a2 is almost zero and this corresponds to a constant speed of sound c2s = 1/3, the conformal limit [Alford et al. (2005)]. For simplicity we set ms = 0, providing electric neutrality of the considered quark matter EoS without leptons. The values of coefficients in (1), which are used in this work, are taken from Anti´c et al. (2021) and 1/4 amount to a4 = 0.22, a2 = −(299.6 MeV)2 , Beff = 174.2 MeV. In the present work, devoted to the application to neutron stars, we can restrict ourselves to the case of matter at zero temperature. At densities below saturation, there are no hyperons or sexaquarks and the unified crust EoS of the original GRDF-DD2 model with clusters is used. It contains the

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Fig. 1. Energy per particle ε/nb − m as a function of baryon density nb for DD2, DD2Y-T and DD2Y-T + S2054 . The red solid line shows the hybrid EoS resulting from a Maxwell construction between DD2 and the used quark matter EoS.

well-known sequence of nuclei in a body-centered cubic lattice with a uniform background of electrons and a neutron gas above the neutron drip line. The transition to homogeneous matter just below the nuclear saturation density is described consistently within the same approach. The main modification is to include the new degrees of freedom at supersaturation densities in the GRDF model. We show the energy per particle ε/nb − m, where m is the nucleon mass, as a function of baryon density nb for different scenarios in hadronic matter as well as a used quark matter EoS in CFL phase in Fig. 1. The pressure P = −ΩQM as a function of baryonic chemical potential μb = 3μ and energy density for DD2, DD2Y-T, DD2Y-T + S as well as the deconfined quark matter are shown in Fig. 2. The corresponding massradius and mass-density curves are shown in Fig. 3. We recognize the instability of the hadronic model that includes the S by the fact that the square of the eigenmode of radial oscillations of the . Here spherical star, given by ω 2 = dM/ncb , is negative for ncb > nonset,S b M denotes the mass of the star, ncb is the baryon density at its center and is the baryon density at the sexaquark onset. The red solid line in nonset,S b Figs. 2 and 3 corresponds to a hybrid EoS for which the sexaquark onset

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Fig. 2. Pressure P for DD2, DD2Y-T and DD2Y-T + S2054 as a function of the baryochemical potential potential μb (upper panel) and as a function of energy density ε = μb nb −P (lower panel) ε, where we emphasize that the region between highest energy densities in neutron stars and applicability of perturbative QCD cannot be probed with neutron stars. Line styles are as in Fig. 1. The red solid line corresponds to a hybrid EoS for which the sexaquark onset triggers a first-order phase transition to a color superconducting quark matter phase that is fitted by a CSS form of EoS (Shahrbaf et al., 2022). We show as grey hatched region the EoS constraint from (Hebeler et al., 2013) and by black dashed lines the one from (Miller et al., 2020).

triggers a first-order phase transition to a color superconducting quark matter phase that is fitted by a CSS form of EoS (Shahrbaf et al., 2022). In the next section, we will consider a new density functional approach to color superconducting strange quark matter which is capable of addressing quark confinement by a divergent scalar self-energy (quark mass) following from the suggested chirally symmetric energy density functional model.

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Fig. 3. Mass vs radius (upper panel) and mass vs. baryon density (lower panel) for compact stars in which the sexaquark S particle is assumed with a constant mass mS = 2054 MeV (dotted green line). The blue solid line corresponds to the hadronic EoS model without sexaquark while the blue dashed line stands for the hadronic model without hyperons. The red solid line corresponds to a hybrid EoS for which the sexaquark onset triggers a first-order phase transition to a color superconducting quark matter phase that is fitted to CFL phase EoS (Shahrbaf et al., 2022). For a comparison the new 1.0 − σ mass-radius constraints from the NICER analysis of observations of the massive pulsar PSR J0740 + 6620 (Fonseca et al., 2021) are indicated in red (Riley et al., 2021) and blue (Miller et al., 2021) regions. Additionally, the magenta bars mark the excluded regions for a lower limit (Bauswein et al., 2017) and an upper limit (Annala et al., 2018) on the radius deduced from the gravitational wave observation GW170817. The green region is from the NICER mass-radius measurement on PSR J0030 + 0451 (Miller et al., 2019).

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3. Density Functional Approach to Strange Quark Matter Deconfinement Here we present the 3-flavor extension of the density functional approach to confining quark matter that was initiated by (Kaltenborn, Bastian and Blaschke, 2017) and then generalized to a chirally-symmetric Lagrangian formulation in (Ivanytskyi, Blaschke and Maslov, 2022; Ivanytskyi and Blaschke, 2022), where also the effect of color superconductivity was included. The confining aspect of quark matter is introduced to the consideration via a fast growth of the quark self-energy already at the mean-field level. This simple mechanism provides an efficient suppression of quark degrees of freedom in the confined region of phase diagram. In principle, the approach allows even divergence of the quark self-energy, which corresponds to an absolute “confinement”. However, the analysis performed in Ivanytskyi, Blaschke and Maslov (2022) as well as in Ivanytskyi and Blaschke (2022) suggests quite high but finite values of the mean-field selfenergy of confined quarks. The mathematical formulation of the density functional approach to three-flavor, three-color quark matter is based on the Lagrangian L = q(i∂/ − m)q + LV + LD − U,

(1)

written for three-component flavor spinor of quarks q T = (u, d, s). The diagonal matrix m = diag(mu , md , ms ) acts in the flavor space and represents current quark masses. The second term in Eq. (1) generates the vector repulsion LV = −GV (qγμ q)2 .

(2)

Its strength is controlled by the coupling constant GV . This repulsive interaction can be motivated by the non-perturbative gluon exchange (Song et al., 2019) and is phenomenologically important in order to provide enough stiffness of dense quark matter needed to support existence of hybrid quark–hadron stars of two solar masses. The third term in Eq. (1) stands for scalar color-flavor antitriplet attractive interaction among quarks  (qiγ5 Ta λb q c )(q c iγ5 Ta λb q), (3) LD = GD a,b=2,5,7

where GD is coupling constant of the color antitriplet scalar diquark channel, where the charge conjugate quark field is defined as q c = iγ2 γ0 q T ,

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while eight flavor Ta and eight color λb Gell-Mann matrices are connected to the generators of the corresponding SU(3) groups. For further con

2 venience, we also introduce T0 = 3 . We note that the summation in Eq. (3) is performed only over the antisymmetric generators with indices a, b = 2, 5, 7. The quark interaction in this channel is responsible for the diquark pairing and formation of the color superconducting phases of quark matter. The energy density functional U models the effects related to dynamical restoration or breaking of chiral symmetry of quark matter. Following the symmetries of the QCD Lagrangian, we require U to be chirally symmetric. The most obvious way to fulfill this requirement corresponds to choosing the argument of U to be chirally symmetric itself. The first part of such an argument is

 1  (qTa q)2 + (qiγ5 Ta q)2 . 2 a=0 8

O1 =

We also introduce the instanton induced term of the ’t Hooft form      O2 = ζ det q(1 + γ5 )q + det q(1 − γ5 )q ,

(4)

(5)

with ζ being a constant discussed below and determinant carried in the flavor space. Thus, similar to the two-flavor case (Ivanytskyi, Blaschke and Maslov, 2022; Ivanytskyi and Blaschke, 2022) we define the chirally symmetric density functional as 1

U = D0 [X − O1 − O2 ] 3 ,

(6)

where D0 and X are constants. The form of this potential is motivated by the string flip model (Horowitz et al., 1985; R¨opke et al., 1986). The present consideration is limited to the mean-field case, when expectation values of the operators are f f   = f f δf f  , f iγ5 Ta f   = 0.

(7) (8)

Within this approximation, flavor matrices q(1 ± γ5 )q become diagonal, i.e. q(1 ± γ5 )q = diag(uu, dd, ss).

(9)

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Using Eqs. (7)–(9) we find the mean-field interaction potential (6) ⎡ U (0) = D0 ⎣X −

 f

⎤13

f f 2 − 2ζ f f ⎦ .

(10)

f

Let us analyze this expression in the two-flavor case. For this the strange quark mass should be approached to infinity ms → ∞ leading to ss = ss0 and 

U (0) |ms →∞ = D0 X − ss20 −

13 uu2 + dd2 + 2uudd · ζss0

. (11)

Hereafter, the subscript index “0” denotes the quantities defined in the vacuum. Obviously, the potential given by Eq. (11) should coincide with the two-flavor one from Ivanytskyi, Blaschke and Maslov (2022); Ivanytskyi and Blaschke (2022)

(0) UNf =2

= D0

2  2  (1 + α) uu0 + dd0 − uu + dd

13 (12)

with α = const. From this we immediately express the parameters of the three-flavor potential through the ones of the two-flavor potential 2  X = (1 + α) uu0 + dd0 +ss20 , ζ=

1 . ss0

(13) (14)

Having the chirally symmetric interaction potential defined, we can analyze the modification of the single quark properties caused by the interaction described by U. For this we follow the strategy of Kaltenborn, Bastian and Blaschke, 2017 and perform the first-order expansion around the mean-field expectation values of scalar operators f f . We point out that the expansion around the expectation values of the pseudoscalar operators qiγ5 Ta q produces non-vanishing terms starting from the second order only (Ivanytskyi, Blaschke and Maslov, 2022; Ivanytskyi and Blaschke, 2022). The first-order terms include derivatives of the mean-field potential with respect to chiral condensates of different flavors, which are nothing but the

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flavor matrix of the mean-field self-energy of quarks   (0) (0) Σ(0) = diag Σ(0) , u , Σd , Σs

(0)

Σf ≡

∂U (0) . ∂f f 

(15)

With this notation the first-order expanded interaction potential and the corresponding Lagrangian become U (1) = U (0) + qΣ(0) q − qΣ(0) q,

(16)

L(1) = q(i∂/ − m∗ )q + LV + LD − U (0) + qΣ(0) q.

(17)

The latter includes the matrix of medium-dependent quark masses m∗ = m + Σ(0) .

(18)

In the region of small temperatures and densities, the mean-field value of the argument of the interaction potential can be approximated as X−O1 −O2 ∼ q + q. It is interesting to consider the effective quark mass in the vacuum ⎡ ⎤ − 23

2D0 α 1 ⎣ (19) m∗f = mf −  f  f  0 ⎦ .  43 f f 0 + ss 0  3 uu + dd 0

0

f =f

From this expression it becomes clear that a vanishing α leads to a divergent effective mass of quarks at f f  = f f 0 , consequently, to an absolute suppression of quarks in the confining region. A small but finite value of α leads to a large m∗f , which is sufficient for an efficient suppression of quarks at low temperatures and densities. In what follows, however, we consider the limiting case of α = 0, which allows us to demonstrate the confining mechanism of the present approach in the most radical way. Mean-field treatment of the vector and diquark pairing channels corresponds to the linearization of the Lagrangian around the expectation values of operators qγμ q and q c iγ5 Ta λb q. Only the μ = 0 component of the first of them yields a non-vanishing expectation value q + q being the quark number density. The diquark pairing operators generate three non-vanishing condensates q c iγ5 T2 iλ2 q,

q c 5γ5 T5 iλ5 q,

q c iγ5 T7 λ7 q.

(20)

They correspond to the formation of the following diquark pairs: (ur , dg ) with (ug , dr ), (dg , sb ) with (db , sg ), and (sb , ur ) with (sr , ub ), respectively.

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Here the subscript index running over (r, g, b) represents the color state of quarks. The resulting mean-field Lagrangian is LMF +q + μq = QS −1 Q+GV q + q2 −



|q c iγ5 Ta λa q|2 −U (0) +qΣ(0) q.

a=2,5,7

It is written through the Nambu–Gorkov bispinor QT = propagator ⎛ ⎜ S −1 = ⎝  a=2,5,7

i∂/ − m∗ + μ∗ γ0 iq c iγ5 Ta λa q∗ γ5 Ta λa

 a=2,5,7

√1 (q 2

(21) q c ) and the

iq c iγ5 Ta λa qγ5 Ta λa

i∂/ − m∗ − μ∗ γ0

⎞ ⎟ ⎠. (22)

We note that the vector repulsion renormalizes the quark chemical potentials as μ∗ = diag(μu , μd , μc ) − 2GV q + q.

(23)

The Lagrangian (21) is quadratic in the quark fields allowing their functional integration and leading to the thermodynamic potential Ω=−

 T |q c iγ5 Ta λa q|2 Tr ln(βS −1 ) − GV q + q2 + GD 2V a=2,5,7

+ U (0) − qΣ(0) q,

(24)

where β = T1 is the inverse temperature, V stands for the system volume, and the trace is performed over the Nambu–Gorkov, Dirac, color, flavor, momentum, and Matsubara indices. The latter appear since quark propagator in Eq. (24) is given in the momentum representation. Solving this trace requires eigenvalues of S −1 , which in the Nambu–Gorkov–Dirac colorflavor space is a 72 × 72 matrix. In the case of equal current quark masses and chemical potentials, these eigenvalues are highly degenerate provided the fact that the effective quark masses, the chiral condensates of three quark flavors, and the three non-vanishing diquark condensates coincide. For simplicity, we consider in what follows such a regime of the present model.

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In order to make the consideration as introductory as possible, we also neglect current quark masses. This leads to the effective quark mass

m∗ = σ ·



D0 σ02

23

·

 2 1+  3 5−

3σ2 σ0 2

σ σ0

−

 2σ3 σ03

23 ,

(25)

expressed through the reduced mass parameter σ≡−

uu dd ss =− =− . D0 D0 D0

(26)

In the color-flavor space, the eigenvalues of the inverse Nambu–Gorkov propagator split into an octet and a singlet. They are ± ωoct ± ωsing

 = (ω ± )2 + Δ2 ,  = (ω ± )2 + (2Δ)2 .

(27) (28)

√ where ω ± = ω ± μ∗, ω = k 2 + m∗2 , k is the quark momentum, μ∗ = μ∗u = μ∗d = μ∗s , and the diquark pairing gap is defined as Δ ≡ 2GD |q c iγ5 T2 iλ2 q| = 2GD |q c 5γ5 T5 iλ5 q| = 2GD |q c iγ5 T7 λ7 q|. (29) Hereafter, the index f is suppressed in order to stress the degeneracy of flavors. In the considered case, −2    dk   ω 2 + ω +2 ωn2 + ωoct T oct n −1 Tr ln(βS ) = T + ln 8 ln 2V (2π)3 T2 T2 n  +2 −2 ωn2 + ωsing ωn2 + ωsing + ln + ln . (30) T2 T2

Here ωn = πT (2n + 1) is a fermionic Matsubara frequency. Performing the summation over the corresponding index and taking the limit T → 0, which is of interest for the astrophysical applications, we arrive at the zero temperature thermodynamic potential of the CFL phase of massless

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interacting quark matter Ω = Ωq − GV q + q2 + with

 Ωq = −

3Δ2 + U (0) − qΣ(0) q, 4GD

  dk   + − + − + ω + ω 8 ω , + ω oct oct sing sing (2π)3

(31)

(32)

being a quark part. The momentum integrals in Eq. (32) are regularized by a sharp cutoff Λ. The definitions of the quark self-energy Σ and the diquark gap Δ provide stationarity of the thermodynamic potential, which is equivalent to the gap equations

    ω + μ∗ ω − μ∗ ω − μ∗ m ∗ dk ω + μ∗ 1 , 8 + + − + + 3σ = + − D0 (2π)3 ω ωoct ωoct ωsing ωsing

     1 1 dk 1 1 3Δ = 4GD Δ + − 8 . +4 + + − + (2π)3 ωoct ωoct ωsing ωsing

(33) (34)

Let us first analyze the equation for the reduced mass parameter. Since m∗ ∝ σ, then it obviously has a trivial solution representing the chirally restored CFL phase. For another one, the reduced mass parameter has its vacuum value. In this case, effective quark mass diverges and the integrand in Eq. (33) simplifies to 18. This solution corresponds to the chirally broken normal phase. Thus, σχSB = σ0 , σCFL = 0.

(35)

Note that σ0 = −f f 0 /D0 is connected to the vacuum value of a single flavor chiral condensate f f 0 which defines the momentum cutoff by the relation  6 Λ2 dk = . (36) σ0 = D0 (2π)3 π 2 D0 We used f f 0 = −(251 MeV)3 , which yields Λ = 538 MeV. This value of the momentum cutoff is used in order to parametrize vector and diquark couplings by ηV = GV Λ2 and ηD = GD Λ2 . The models considered in what follows are labelled by pairs of numbers (ηV , ηD ). For example, (1.0, 2.0) corresponds to GV = Λ−2 and GD = 2Λ−2 . ± ± and ωsing diverge in the The effective quark mass and, consequently, ωoct phase with broken chiral symmetry, leading to a vanishing diquark pairing

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gap. In the CFL phase quarks are massless and this pairing gap becomes finite. Thus, ΔχSB = 0,  3=

dk (2π)3

16GD 8GD 8GD 16GD + − + + + − + ωoct ωoct ωsing ωsing

(37)

 .

m∗ =0, Δ=ΔCFL

(38) The divergence of m∗ in the chirally broken phase leads to Ωq = −qΣ(0) q = 3σm∗ , while U (0) = 0 due to σ = σ0 . The quark number density and the diquark gap also vanish in this phase. Therefore, its pressure p being the negative of the thermodynamic potential vanishes. In the CFL phase quarks are massless, which is equivalent to Σ(0) = 0. Therefore, the pressure of two phases PχSB = 0,

(39)

PCFL = −Ωq |m∗ =0,

Δ=ΔCFL

+ GV q + q2 −

3Δ2CFL (0) + Uσ=0 . 4GD

(40)

It is worth mentioning that the last term in the previous expression is constant and negative. Therefore, it can be related to the effective bag pressure (0)

1

B ≡ −Uσ=0 = D0 (5σ02 ) 3 .

(41)

The quark number density of the CFL phase should be self-consistently found according to q + qCFL = −

∂Ωq ∂μ

m∗ =0, Δ=ΔCFL

.

(42)

This number density is related to the baryonic charge density as q + q = 3nb , while the chemical potential of the baryonic charge is μb = 3μ. It is also important to note that equality of masses and chemical potentials, for the three quark flavors leads to the equality of their partial number densities. As a result, the quark matter is electrically and color neutral. Two solutions of the present model should be merged at μB providing PχSB = PCFL . At low chemical potentials, the solution corresponding to the CFL phase has negative pressure PCFL < 0 signalling that superconducting chirally symmetric quark matter is disfavored in that region. At a certain μB the pressure PCFL attains a zero value and gets positive at

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related through the chirp mass [Peters and Mathews (1963)] 3

M=

(M1 M2 ) 5 1

(M1 + M2 ) 5

.

(44)

The best measured combination of the masses from GW 170817 yields M = 1.188+0.004 −0.002 M [Abbott et al. (2017)], which is used below. For each M1 and M2 providing this value of the chirp mass we calculated tidal deformabilities Λ1 and Λ2 . Relation between them is shown on the right panel of Fig. 8, which also depicts the areas corresponding to 50 % and 90 % confidence intervals [Abbott et al. (2018)]. At ηD = 2.0 agreement with the observational data at the 90 % confidence level is provided by ηV ≤ 2.0. Purely hadronic DD2npY-T EoS lies well beyond the corresponding interval. Agreement at the 50 % confidence level requires ηV ≤ 1.0, which, however, contradicts to the constraints on the mass-radius relation. 4. Conclusions In this contribution, we present for the first time a scenario according to which early quark deconfinement in compact stars is triggered by the BEC of a light sexaquark (mS < 2054 MeV) that has been suggested as a candidate particle to explain the baryonic DM in the Universe. The BEC onset of S marks the maximum mass of hadronic neutron stars and it occurs when the condition for the baryon chemical potential μb = MS /2 is fulfilled in the center of the star, corresponding to Monset < 0.7M . In the gravitational field of the star, the density of the BEC of S increases until a new state of the matter is attained, which consists of dissociated S states, the CFL phase with a diquark condensate, thus presenting a form of BEC–BCS transition in strongly interacting matter. For the description of the CFL phase, we have developed here for the first time the three-flavor extension of the density-functional formulation of a chirally symmetric Lagrangian model of quark matter with confining properties encoded in a divergence of the scalar self-energy at low densities and temperatures. As this density functional model does not show sequential but rather simultaneous deconfinement of quark flavors, the “No-Go” theorem of the NJL model against the possibility of absolutely stable strange quark matter does not apply here. However, since we had to make several approximations in this first evaluation of the three-flavor version of the density-functional model, we leave a discussion of the possibility of absolutely stable strange quark matter to a subsequent work.

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