,I» World Scientific INTRODUCTION TO THEORY THE OF THE EARLY UNIVERSE Cosmological Perturbations and Inflationary Theory t DM1TRY S GORBUNOV Institute for Nuclear Research of the Russian Academy of Sciences VALERY A RUBAKOV Institute for Nuclear Research of the Russian Academy of Sciences Et: Moscow State University INTRODUCTION TO THEORY THE OF THE EARLY UNIVERSE Cosmological Perturbations and Inflationary Theory ,~World Scientific NEW JERSEY. LONDON · SINGAPORE · BEIJING · SHANGHAI · HONG KONG · TAIPEI · CHENNAI 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 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. INTRODUCTION TO THE THEORY OF THE EARLY UNIVERSE Cosmological Perturbations and Inflationary Theory Copyright © 2011 by World Scientific Publishing Co. Pte. Ltd. All rigllls 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 informarion sTOroge and reTrieval system now known or to be invel1led, without wrillen permissionjrom The Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-981-4322-22-5 ISBN-I0 981-4322-22-9 ISBN-13 978-981-4343-78-7 (pbk) ISBN-IO 981-4343-78-1 (pbk) Typeset by Stall ion Press Email: [email protected] Printed in Singapore by Mainland Press Pte Ltd. To Olesya and Elvira Preface i\lodern cosmology is deeply connected to microphysics which studies elementary particles and their interactions at the most fundamental level. This book is written precisely from this perspective. It accompanies the book "Introduction to the Theory of the Early Universe: Hot Big Bang Theory" containing the material on the homogeneous and isotropic Universe at the hot stage of its evolution and at sub sequent stages. This book presents the theory of cosmological perturbations (inho mogeneities in the Universe), inflationary theory and theory of post-inflationary reheating. This book grew from the lecture course which is being taught for a number of years at the Department of Quantum Statistics and Field Theory, and now at the Department of Particle Physics and Cosmology of the Physics Faculty of the Lomonosov Moscow State University. This course is aimed at undergraduate stu dents specializing in theoretical physics. vVe decided, however, to add a number of more advanced Chapters and Sections which we mark by asterisks. For reading the main Chapters of this book, knowledge of material taught in general physics courses is in principle sufficient. So, the main Chapters must be understandable by undergraduate students. The necessary material on General Relativity and theory of random fields is collected in Appendices which, of course, do not pretend to give comprehensive presentation of these areas of physics. However, some parts labeled by asterisks make use of nonequilibrium statistical mechanics and specific methods of quantum field theory, so basic knowledge of these areas it> required for reading these parts. Some useful concepts and results are given in Chapter 8 and Appendices D and E. We tried to make Chapters 11, 12, 13, 14 and 15, where we discuss inflation and post-inflationary reheating, as self-contained as possible. We hope that the reader interested in the inflationary theory will be able to read these Chapters without constantly consulting the rest of the book. In appropriate places of this book we use the notations and results presented in the accompanying book "Introduction to the Theory of the Early Universe: Hot Big Bang Theory". References to that book begin with 1. As an example, 1.3.1 means Section 3.1 of that book, and (1.3.13) means equation (3.13) there. Literature on cosmology is huge, and presenting systematic and comprehensive bibliography would be way out of the scope of this book. To orient the reader, in the end of this book we give a list of monographs and reviews where the issues we vii Vlll Preface touch upon are considered in detail. Certainly, this list is by no means complete. '0/e refer to original literature in those places where we present concrete results without detailed derivation. Both observational cosmology and experimental particle physics develop very fast. Observational and experimental data we quote, the results of their compilations and fits (values of the cosmological parameters, limits on masses and couplings of hypothetical particles, etc.) will most probably get more precise even before this book is published. This drawback can be corrected, e.g., by using the regularly updated material of Particle Data Group at http://pdg.lbl.gov / We would like to thank our colleagues from the Institute for uclear Research of the Russian Academy of Sciences F. L. Bezrukov, S. V. Demidov, S. L. Dubovsky, A. A. Khmelnitsky, V. A. Kuzmin, D. G. Levkov, M. V. Libanov, E. Y. Nugaev, G. 1. Rubtsov, D. V. Semikoz, 1. 1. Tkachev and S. V. Troitsky for participation in the preparation of the lecture course and numerous helpful discussions and comments. vVe are deeply indebted to G. S. Bisnovaty-Kogan, V. N. Lukash, V. F. Mukhanov, 1. D. Novikov, K. A. Postnov, M. V. Sazhin, 1. L. Shapiro, M. E. Shaposhnikov, Yu. V. Shtanov, R. A. Sunyaev, A. V. Tikhonov and O. V. Verkhodanov for useful comments and criticism on the preliminary version of this book. Contents Preface vii 1. Jeans Instability in Newtonian Gravity 1 1.1 Jeans Instability in Static Background ....... . 2 1.2 Development of Instability in Expanding Universe . . 6 1.3 Linear Sizes of Perturbations and Masses of Objects. 11 2. Cosmological Perturbations in General Relativity. Equations of Linearized Theory 13 2.1 Background Metric . . . . . . . . . . . . . . . . . . . . 14 2.1.1 Metric in conformal time . .......... . 14 2.1.2 Cosmological parameters and conformal times of various epochs 15 2.2 Generalities . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.1 Gauge hOi = 0 ........ ... . . . ... . 20 2.2.2 Linearized energy-momentum tensor of ideal fluid 22 2.2.3 Helicity decomposition. 24 2.3 Equations in Helicity Sectors. 27 2.3.1 Tensor perturbations .. 27 2.3.2 Vector perturbations .. 29 2.3.3 Scalar perturbations: conformal Newtonian gauge. 30 2.4 Regimes of Evolution . . . . . . . . . . . . 33 2.5 *Scalar Field Condensate as Dark Matter. 37 3. Evolution of Vector and Tensor Perturbations 47 3.1 Vector Modes ..... ..... ... . 47 3.2 Tensor Modes: Relic Gravity Waves . . 48 3.2.1 Super horizon perturbations: constant and decaying modes . . . . . . . . . 48 3.2.2 Sub horizon perturbations. Matching to constant mode . . . . . . . . . . . 50 ix x Contents -1. Scalar Perturbations: Single-Component Fluids 53 4.1 Master Equation 53 4.2 Relativistic Matter .. 55 4.3 Non-relativistic Matter 57 4.4 Matter Perturbations at A Domination 58 5. Primordial Perturbations in Real Universe 61 5.1 Adiabatic and Isocurvature Modes 62 5.2 Adiabatic Mode in Superhorizon Regime 65 5.3 Initial Data for Isocurvature Modes . . . 71 5.4 Primordial Spectra: Results from Observations . 72 5.5 Evolution of Adiabatic Perturbations: A Preview 77 6. Scalar Perturbations before Recombination 81 6.1 Adiabatic Modes of Large Wavelengths 83 6.2 Adiabatic Modes Entering the Sound Horizon at Radiation Domination . . . ....... . 83 6.2.1 Dark matter perturbations ..... . 85 6.2.2 Perturbations in baryon-photon component 91 6.3 Adiabatic Perturbations of Intermediate Momenta . 95 6.4 CDM Isocurvature Perturbations . 97 6.5 Baryon Isocurvature Perturbations 101 7. Structure Formation 103 7.1 Matter Perturbations after Recombination: Linear Regime 104 7.1.1 Power spectrum P(k, z) . . 104 7.1.2 Baryon acoustic oscillations 114 7.2 Beginning of Non-linear Regime . 121 7.2.1 Preliminaries . . ..... . 121 7.2.2 Mass distribution of structures 122 8. *Beyond Ideal Fluid Approximation 133 8.1 Distribution Functions and Boltzmann Equation in Curved Space-Time 133 8.2 General Equations for Scalar Perturbations . 145 8.3 Warm Dark Matter . . . . . . . . . . . . . . 149 8.3.1 Suppressed growth of perturbations 149 8.3.2 Bound on WDM particle mass from phase space density . . . . . . . . . . . . . . 162 8.4 Neutrino Free Streaming .... . ... . . . . 165 8.4.1 Early evolution: relativistic neutrinos 167 8.4.2 Non-relativistic neutrinos .. . ... . 174 Contents xi 8.4.3 Neutrino effect on matter perturbations. Cosmological bound on neutrino masses . . . . . . . . 177 8.5 Photons and Baryons at Recombination Epoch 185 8.5.1 Thickness of last scattering sphere 186 8.5.2 Silk damping .......... . 189 9. Temperature of Cosmic Microwave Background 197 9.1 CMB Temperature Anisotropy . . . . . . 197 9.2 Temperature Anisotropy in Instant Photon Decoupling Approximation. 201 9.2.1 General formalism . .. . . 201 9.2.2 Large angular scales ... . 208 9.2.3 Intermediate angular scales 218 9.3 Small Angular Scales . . . . . . . . 228 9.4 Anisotropy Spectrum and Cosmological Parameters 231 9.5 Temperature Anisotropy Generated by Isocurvature Modes . 235 10. *CMB Polarization 239 10.1 Sources of CMB Polarization. 239 10.2 Polarization Tensor. E- and B-modes 242 10.3 Generation of CMB Polarization. 250 10.3.1 Generalities .. . . .. 251 10.3.2 Scalar perturbations . 256 10.3.3 Tensor perturbations. 266 10.4 Discussion .......... . 276 11. Drawbacks of the Hot Big Bang Theory. Inflation as Possible Way Out 279 11.1 Drawbacks of the Hot Big Bang Theory 279 11.1.1 Horizon problem 281 11.1.2 Flatness problem .. . . .... . 283 11.1.3 Entropy problem ....... . . 284 11.1.4 Primordial perturbation problem 285 11.2 Inflation: The Basic Idea .. ... .. . . 285 12. Inflation in Slow Roll Regime 291 12.1 Slow Roll Conditions . 291 12.2 Inflationary Models . . . . . . . . . . . 296 12.2.1 Large field inflation ("chaotic") 297 12.2.2 Small field inflation ("new") . 302 12.2.3 Hybrid inflation . . 305 12.2.4 Concluding remarks ... .. 309