Introduction to the Theory of the Early Universe: Cosmological Perturbations and Inflationary Theory

504 Pages·2011·6.614 MB·489\504

by  Gorbunov, D.S. and Rubakov, V.A.

World Scientific
INTRODUCTION TO
THEORY
THE OF THE
EARlY UNIVERSE
Cosmological Perturbations
and Inflationary Theory
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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
11»
World Scientific
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INTRODUCTION TO THE THEORY OF THE EARLY UNIVERSE
Cosmological Perturbations and Inflationary Theory
Copyright © 2011 by World Scientific Publishing Co. Pte. Ltd.
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To Olesya and Elvira
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Preface
Modern cosmology is deeply connected to microphysics which studies elementary
particles andtheir interactionsat the most fundamentallevel. 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
thehomogeneousandisotropicUniverseatthehotstageofitsevolutionandatsub-
sequent stages. This book presents the theory of cosmologicalperturbations (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. We 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
RelativityandtheoryofrandomfieldsiscollectedinAppendiceswhich,ofcourse,do
not pretend to give comprehensive presentationof 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
is required for reading these parts. Some useful concepts and results are given in
Chapter 8 and Appendices D and E.
WetriedtomakeChapters11,12,13,14and15,wherewediscussinflationand
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 appropriateplaces of this book we use the notations andresults presentedin
theaccompanyingbook“IntroductiontotheTheoryoftheEarlyUniverse:HotBig
Bang Theory”. References to that book begin with I. As an example, I.3.1 means
Section 3.1 of that book, and (I.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
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viii Preface
touchuponareconsideredindetail.Certainly,thislistisbynomeanscomplete.We
refertooriginalliteratureinthoseplaceswherewepresentconcreteresultswithout
detailed derivation.
Both observational cosmology and experimental particle physics develop very
fast.Observationalandexperimentaldatawequote,theresultsoftheircompilations
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 Nuclear 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. I. Rubtsov, D. V. Semikoz, I. I. Tkachev and S. V. Troitsky
for participation in the preparation of the lecture course and numerous helpful
discussions and comments. We are deeply indebted to G. S. Bisnovaty-Kogan,
V. N. Lukash, V. F. Mukhanov, I. D. Novikov, K. A. Postnov, M. V. Sazhin,
I. L. Shapiro,M. E. Shaposhnikov,Yu.V.Shtanov,R.A.Sunyaev,A.V.Tikhonov
andO.V.Verkhodanovforusefulcommentsandcriticismonthepreliminaryversion
of this book.
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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. CosmologicalPerturbations in General Relativity. Equations
of Linearized Theory 13
2.1 Background Metric . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.1 Metric in conformal time . . . . . . . . . . . . . . . . . . 14
2.1.2 Cosmologicalparameters and conformal times
of various epochs . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 Gauge h0i =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 Superhorizon perturbations: constant
and decaying modes . . . . . . . . . . . . . . . . . . . . . 48
3.2.2 Subhorizon perturbations. Matching
to constant mode . . . . . . . . . . . . . . . . . . . . . . . 50
ix