Cosmology in gauge field theory and string theory

317 Pages·2004·1.563 MB·English

by  D. Bailin, Alexander Love

GRADUATE STUDENT SERIES IN PHYSICS
Series Editor:
ProfessorDouglasF Brewer, MA,DPhil
EmeritusProfessorofExperimentalPhysics,UniversityofSussex
COSMOLOGY IN GAUGE
FIELD THEORY
AND STRING THEORY
DAVID BAILIN
DepartmentofPhysicsandAstronomy
UniversityofSussex
ALEXANDER LOVE
DepartmentofPhysics
RoyalHollowayandBedfordNewCollege
UniversityofLondon
INSTITUTE OF PHYSICS PUBLISHING
Bristol and Philadelphia
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ToEvaBailinand thememoryofWilliamBailin (1911–1994)
and
ToChristine
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Contents
Preface xi
1 Thestandardmodelofcosmology 1
1.1 Introduction 1
1.2 TheRobertson–Walkermetric 2
1.3 EinsteinequationsforaFriedmann–Robertson–Walkeruniverse 5
1.4 Scalefactordependenceoftheenergydensity 7
1.5 Timedependenceofthescalefactor 8
1.6 Ageoftheuniverse 8
1.7 Thecosmologicalconstant 10
1.8 Equilibriumthermodynamicsintheexpandinguniverse 17
1.9 Transitionfromradiationtomatterdomination 19
1.10 Cosmicmicrowavebackgroundradiation(CMBR) 21
1.11 Big-bangnucleosynthesis 21
1.12 Exercises 27
1.13 Generalreferences 27
Bibliography 28
2 Phasetransitionsintheearlyuniverse 29
2.1 Introduction 29
2.2 Partitionfunctions 30
2.3 Theeffectivepotentialatfinitetemperature 33
2.4 PhasetransitionsintheHiggsmodel 36
2.4.1 e4 (cid:1)λ 37
2.4.2 e4 (cid:2)λ 40
2.5 Phasetransitionsinelectroweaktheory 45
2.6 Phasetransitionsingrandunifiedtheories 48
2.7 PhasetransitionsinsupersymmetricGUTs 51
2.8 Phasetransitionsinsupergravitytheories 55
2.9 Nucleationoftruevacuum 59
2.10 Exercises 63
2.11 Generalreferences 63
Bibliography 63
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viii Contents
3 Topologicaldefects 65
3.1 Introduction 65
3.2 Domainwalls 66
3.3 Globalcosmicstrings 69
3.4 Localcosmicstrings 71
3.5 Gravitationalfieldsoflocalcosmicstrings 74
3.5.1 Doubleimages 75
3.5.2 Temperaturediscontinuities 76
3.5.3 Cosmicstringwakes 76
3.6 Dynamicsoflocalcosmicstrings 76
3.7 Magneticmonopoles 80
3.8 Monopoletopologicalquantumnumber 83
3.9 Magneticmonopolesingrandunifiedtheories 85
3.10 Abundanceofmagneticmonopoles 86
3.11 Exercises 89
3.12 Generalreferences 89
Bibliography 89
4 Baryogenesis 91
4.1 Introduction 91
4.2 Conditionsforbaryogenesis 94
4.3 Out-of-equilibriumdecayofheavyparticles 96
4.4 BaryogenesisinGUTs 99
4.5 BaryogenesisinSO(10)GUTs 110
4.6 StatusofGUTbaryogenesis 113
4.7 Baryon-numbernon-conservationintheStandardModel 114
4.8 Sphaleron-inducedbaryogenesis 120
4.9 CP-violationinelectroweaktheory 127
4.10 Phasetransitionsandelectroweakbaryogenesis 129
4.11 Supersymmetricelectroweakbaryogenesis 132
4.12 Affleck–Dinebaryogenesis 137
4.13 Exercises 142
4.14 Generalreferences 143
Bibliography 143
5 Relicneutrinosandaxions 147
5.1 Introduction 147
5.2 Relicneutrinos 150
5.3 Axions 151
5.3.1 Introduction:thestrongCPproblemandtheaxionsolution151
5.3.2 Visibleandinvisibleaxionmodels 156
5.3.3 Astrophysicalconstraintsonaxions 159
5.3.4 Axionsandcosmology 161
5.4 Exercises 169
5.5 Generalreferences 169
Copyright © 2004 IOP Publishing Ltd
Contents ix
Bibliography 170
6 Supersymmetricdarkmatter 172
6.1 Introduction 172
6.2 WeaklyinteractingmassiveparticlesorWIMPs 175
6.3 Thegravitinoproblem 177
6.4 Minimalsupersymmetricstandardmodel(MSSM) 179
6.5 Neutralinodarkmatter 181
6.6 Detectionofdarkmatter 187
6.6.1 Neutralino–nucleonelasticscattering 188
6.6.2 WIMPannihilationinthesunorearth 189
6.6.3 WIMPannihilationinthehalo 192
6.7 Exercises 192
6.8 Generalreferences 193
Bibliography 193
7 Inflationarycosmology 195
7.1 Introduction 195
7.2 Horizon,flatnessandunwantedrelicsproblems 195
7.2.1 Thehorizonproblem 195
7.2.2 Theflatnessproblem 197
7.2.3 Theunwantedrelicsproblem 198
7.3 Oldinflation 199
7.4 Newinflation 201
7.5 Reheatingafterinflation 206
7.6 Inflatonfieldequations 208
7.7 Densityperturbations 210
7.8 Aworkedexample 214
7.9 Complexinflatonfield 216
7.10 Chaoticinflation 217
7.11 Hybridinflation 220
7.12 Thespectralindex 221
7.13 Exercises 224
7.14 Generalreferences 224
Bibliography 224
8 Inflationinsupergravity 226
8.1 Introduction 226
8.2 Modelsofsupergravityinflation 227
8.3 D-termsupergravityinflation 232
8.4 Hybridinflationinsupergravity 234
8.5 Thermalproductionofgravitinosbyreheating 237
8.6 ThePolonyiproblem 238
8.6.1 InflatondecaysbeforePolonyifieldoscillation 240
8.6.2 InflatondecaysafterPolonyifieldoscillation 244
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x Contents
8.7 Exercises 248
8.8 Generalreferences 248
Bibliography 248
9 Superstringcosmology 249
9.1 Introduction 249
9.2 Dilatonandmodulicosmology 250
9.3 Stabilizationofthedilaton 255
9.4 Dilatonormoduliaspossibleinflatons 259
9.5 Ten-dimensionalstringcosmology 260
9.6 D-braneinflation 265
9.7 Pre-big-bangcosmology 269
9.8 M-theorycosmology—theekpyroticuniverse 272
9.9 Exercises 273
9.10 Generalreferences 273
10 Blackholesinstringtheory 275
10.1 Introduction 275
10.2 Black-holeeventhorizons 276
10.3 Entropyofblackholes 281
10.4 Perturbativemicrostatesinstringtheory 289
10.5 Extremeblackholes 291
10.6 TypeIIsupergravity 293
10.7 FormfieldsandD-branes 296
10.8 Blackholesinstringtheory 298
10.9 Countingthemicrostates 303
10.10Problems 305
10.11Generalreferences 307
Bibliography 307
Copyright © 2004 IOP Publishing Ltd
Preface
The new particle physics of the past 30 years, including electroweak theory,
quantum chromodynamics, grand unified theory, supersymmetry, supergravity
andsuperstringtheory,hasgreatlychangedourviewofwhatmayhavehappened
in the universe at temperatures greater than about 1015 K (100 GeV). Various
phasetransitionsmaybe expectedto haveoccurredas gaugesymmetrieswhich
were present at higher temperatureswere spontaneously broken as the universe
cooled. At these phase transitions topological defects, such as domain walls,
cosmic strings and magnetic monopoles, may have been produced. Various
types of relic particles are also expected. These may include neutrinos with
small mass and axions associated with the solution of the strong CP problem
in quantum chromodynamics. If supersymmetry exists, there should also be
relic supersymmetric partners of particles, some of which could be dark matter
candidates. If the supersymmetry is local (supergravity) these will include the
gravitino, the spin-3 partner of the graviton. Insight may also be gained into
2
the observedbaryonnumberof the universefrommechanismsfor baryogenesis
which arise in the context of grand unified theory and electroweak theory.
Supersymmetryandsupergravitytheoriesmayhavescopetoprovidetheparticle
physics underlyingthe inflationary universe scenario that resolves such puzzles
as the extreme homogeneity and flatness of the observed universe. Superstring
theory also gives insight into the statistical thermodynamicsof black holes. In
thecontextofsuperstringtheory,boldspeculationshavebeenmadeastoaperiod
ofevolutionoftheuniversepriortothebigbang(‘pre-big-bang’and‘ekpyrotic
universe’cosmology).
Thesematters,amongstothers,arethesubjectofthisbook. Thebookgives
a flavour of the new cosmology that has developed from these recent advances
inparticlephysics. Theaimhasbeentodiscussthoseaspectsofcosmologythat
are mostrelevantto particle physics. From some of these it may be possible to
uncovernew particle physicsthatisnotreadilydiscernibleelsewhere. Thisis a
particularlytimelyenterprise,since,ashasbeennotedbymanyauthors,therecent
data from WMAP and future data expected from Planck mean that cosmology
mayatlastberegardedasprecisionsciencejustasparticlephysicshasbeenfor
manyyears.
Copyright © 2004 IOP Publishing Ltd
We are grateful to our colleagues Nuno Antunes, Mar Bastero-Gil, Ed
Copeland, Beatriz de Carlos, Mark Hindmarsh, George Kraniotis, Andrew Liddle,
Andr´e Lukas and Paul Saffin for the particle and cosmological physics that we
have learned from them. Special thanks also to Malcolm Fairbairn for helping
us with the diagrams. Finally, we wish to thank our wives for their invaluable
encouragement throughout the writing of this book.
We intend to maintain an updated erratum page for the book at
http://www.pact.cpes.sussex.ac.uk/∼mpfg9/cosmobook.htm.
DavidBailinandAlexanderLove
June,2004
Copyright © 2004 IOP Publishing Ltd
Chapter 1
The standard model of cosmology
1.1 Introduction
The principal concern of this book is the way in which recent particle physics,
including electroweak theory, quantum chromodynamics, grand unified theory,
supersymmetry, supergravity and superstring theory, has changed our standpoint
on the history of the universe when its temperature was greater than 1015 K. This
will be studied in the context of the Friedman–Robertson–Walker solution of the
Einstein equations of general relativity. In this chapter, therefore, our first task
is the derivation of the field equations relating the scale factor R(t) that appears
in the metric to the energy density ρ and the pressure p that characterize the
(assumed homogeneous and isotropic) energy–momentum tensor. This is done
in the following two sections. In section 1.4 we show how, for a given equation
of state, energy–momentum conservation determines the scale dependence of the
energy density and pressure. The standard solutions for the time dependence of
the scale factor in a radiation-dominateduniverse, in a matter-dominated universe,
and in a cosmological constant-dominated universe are presented in section 1.5;
we give an estimate of the age of the universe in the matter-dominated case in
section 1.6. In section 1.7, we present the evidence that there is, in fact, a non-
zero cosmological constant and discuss why its size is so difficult to explain. The
discussion of phase transitions and of relics that is given in later chapters also
requires a description of the thermodynamics of the universe. So in the following
two sections we describe the equilibrium thermodynamics of the expanding
universe and derive the time dependence of the temperature in the various epochs.
In section 1.10, we discuss briefly the ‘recombination’ of protons and electrons
that left the presently observed cosmic microwave background radiation. Finally,
the synthesis of the light elements that commenced towards the end of the first
three minutes is discussed in section 1.11. The consistency of the predicted
abundances with those inferred from the measured abundances determines the
so-calledbaryonasymmetryoftheuniverse,whoseoriginisdiscussedatlength
in chapter 4.
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