GRADUATE STUDENT SERIES IN PHYSICS Series Editor: Professor Douglas F Brewer, MA, DPhil Emeritus Professor of Experimental Physics, University of Sussex COSMOLOGY IN GAUGE FIELD THEORY AND STRING THEORY DAVID BAILIN Department of Physics and Astronomy University of Sussex ALEXANDER LOVE Department of Physics Royal Holloway and Bedford New College University of London INSTITUTE OF PHYSICS PUBLISHING Bristol and Philadelphia Copyright © 2004 IOP Publishing Ltd c IOP Publishing Ltd 2004 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any mean s, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the pub lisher. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency under the terms of its agreement with Universities UK (UUK). British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN 0 7503 0492 8 Library of Congress Cataloging-in-Publication Data are available Commissioning Editor: John Navas Production Editor: Simon Laurenson Production Control: Leah Fielding Cover Design: Victoria Le Billon Marketing: Nicola Newey Published by Institute of Physics Publishing, wholly owned by The Institute of Physics, London Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK US Office: Institute of Physics Publishing, The Public Ledger Building, Suite 929, 150 South Independence Mall West, Philadelphia, PA 19106, USA Typeset in L A T E X2 by Text 2 Text Limited, Torquay, Devon Printed in the UK by MPG Books Ltd, Bodmin, Cornwall Copyright © 2004 IOP Publishing Ltd To Eva Bailin and the memory of William Bailin (1911–1994) and To Christine Copyright © 2004 IOP Publishing Ltd Contents Preface xi 1 The standard model of cosmology 1 1.1 Introduction 1 1.2 The Robertson–Walker metric 2 1.3 Einstein equations for a Friedmann–Robertson–Walker universe 5 1.4 Scale factor dependence of the energy density 7 1.5 Time dependence of the scale factor 8 1.6 Age of the universe 8 1.7 The cosmological constant 10 1.8 Equilibrium thermodynamics in the expanding universe 17 1.9 Transition from radiation to matter domination 19 1.10 Cosmic microwave background radiation (CMBR) 21 1.11 Big-bang nucleosynthesis 21 1.12 Exercises 27 1.13 General references 27 Bibliography 28 2 Phase transitions in the early universe 29 2.1 Introduction 29 2.2 Partition functions 30 2.3 The effective potential at finite temperature 33 2.4 Phase transitions in the Higgs model 36 2.4.1 e 4 λ 37 2.4.2 e 4 λ 40 2.5 Phase transitions in electroweak theory 45 2.6 Phase transitions in grand unified theories 48 2.7 Phase transitions in supersymmetric GUTs 51 2.8 Phase transitions in supergravity theories 55 2.9 Nucleation of true vacuum 59 2.10 Exercises 63 2.11 General references 63 Bibliography 63 Copyright © 2004 IOP Publishing Ltd viii Contents 3 Topological defects 65 3.1 Introduction 65 3.2 Domain walls 66 3.3 Global cosmic strings 69 3.4 Local cosmic strings 71 3.5 Gravitational fields of local cosmic strings 74 3.5.1 Double images 75 3.5.2 Temperature discontinuities 76 3.5.3 Cosmic string wakes 76 3.6 Dynamics of local cosmic strings 76 3.7 Magnetic monopoles 80 3.8 Monopole topological quantum number 83 3.9 Magnetic monopoles in grand unified theories 85 3.10 Abundance of magnetic monopoles 86 3.11 Exercises 89 3.12 General references 89 Bibliography 89 4 Baryogenesis 91 4.1 Introduction 91 4.2 Conditions for baryogenesis 94 4.3 Out-of-equilibrium decay of heavy particles 96 4.4 Baryogenesis in GUTs 99 4.5 Baryogenesis in SO(10) GUTs 110 4.6 Status of GUT baryogenesis 113 4.7 Baryon-number non-conservation in the Standard Model 114 4.8 Sphaleron-induced baryogenesis 120 4.9 CP-violation in electroweak theory 127 4.10 Phase transitions and electroweak baryogenesis 129 4.11 Supersymmetric electroweak baryogenesis 132 4.12 Affleck–Dine baryogenesis 137 4.13 Exercises 142 4.14 General references 143 Bibliography 143 5 Relic neutrinos and axions 147 5.1 Introduction 147 5.2 Relic neutrinos 150 5.3 Axions 151 5.3.1 Introduction: the strong CP problem and the axion solution 151 5.3.2 Visible and invisible axion models 156 5.3.3 Astrophysical constraints on axions 159 5.3.4 Axions and cosmology 161 5.4 Exercises 169 5.5 General references 169 Copyright © 2004 IOP Publishing Ltd Contents ix Bibliography 170 6 Supersymmetric dark matter 172 6.1 Introduction 172 6.2 Weakly interacting massive particles or WIMPs 175 6.3 The gravitino problem 177 6.4 Minimal supersymmetric standard model (MSSM) 179 6.5 Neutralino dark matter 181 6.6 Detection of dark matter 187 6.6.1 Neutralino–nucleon elastic scattering 188 6.6.2 WIMP annihilation in the sun or earth 189 6.6.3 WIMP annihilation in the halo 192 6.7 Exercises 192 6.8 General references 193 Bibliography 193 7 Inflationary cosmology 195 7.1 Introduction 195 7.2 Horizon, flatness and unwanted relics problems 195 7.2.1 The horizon problem 195 7.2.2 The flatness problem 197 7.2.3 The unwanted relics problem 198 7.3 Old inflation 199 7.4 New inflation 201 7.5 Reheating after inflation 206 7.6 Inflaton field equations 208 7.7 Density perturbations 210 7.8 A worked example 214 7.9 Complex inflaton field 216 7.10 Chaotic inflation 217 7.11 Hybrid inflation 220 7.12 The spectral index 221 7.13 Exercises 224 7.14 General references 224 Bibliography 224 8 Inflation in supergravity 226 8.1 Introduction 226 8.2 Models of supergravity inflation 227 8.3 D-term supergravity inflation 232 8.4 Hybrid inflation in supergravity 234 8.5 Thermal production of gravitinos by reheating 237 8.6 The Polonyi problem 238 8.6.1 Inflaton decays before Polonyi field oscillation 240 8.6.2 Inflaton decays after Polonyi field oscillation 244 Copyright © 2004 IOP Publishing Ltd x Contents 8.7 Exercises 248 8.8 General references 248 Bibliography 248 9 Superstring cosmology 249 9.1 Introduction 249 9.2 Dilaton and moduli cosmology 250 9.3 Stabilization of the dilaton 255 9.4 Dilaton or moduli as possible inflatons 259 9.5 Ten-dimensional string cosmology 260 9.6 D-brane inflation 265 9.7 Pre-big-bang cosmology 269 9.8 M-theory cosmology—the ekpyrotic universe 272 9.9 Exercises 273 9.10 General references 273 10 Black holes in string theory 275 10.1 Introduction 275 10.2 Black-hole event horizons 276 10.3 Entropy of black holes 281 10.4 Perturbative microstates in string theory 289 10.5 Extreme black holes 291 10.6 Type II supergravity 293 10.7 Form fields and D-branes 296 10.8 Black holes in string theory 298 10.9 Counting the microstates 303 10.10 Problems 305 10.11 General references 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 and superstring theory, has greatly changed our view of what may have happened in the universe at temperatures greater than about 10 15 K (100 GeV). Various phase transitions may be expected to have occurred as gauge symmetries which were present at higher temperatures were 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 2 partner of the graviton. Insight may also be gained into the observed baryon number of the universe from mechanisms for baryogenesis which arise in the context of grand unified theory and electroweak theory. Supersymmetry and supergravity theories may have scope to provide the particle physics underlying the 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 thermodynamics of black holes. In the context of superstring theory, bold speculations have been made as to a period of evolution of the universe prior to the big bang (‘pre-big-bang’ and ‘ekpyrotic universe’ cosmology). These matters, amongst others, are the subject of this book. The book gives a flavour of the new cosmology that has developed from these recent advances in particle physics. The aim has been to discuss those aspects of cosmology that are most relevant to particle physics. From some of these it may be possible to uncover new particle physics that is not readily discernible elsewhere. This is a particularly timely enterprise, since, as has been noted by many authors, the recent data from WMAP and future data expected from Planck mean that cosmology may at last be regarded as precision science just as particle physics has been for many years. 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. David Bailin and Alexander Love 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 10 15 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-called baryon asymmetry of the universe, whose origin is discussed at length in chapter 4. Copyright © 2004 IOP Publishing Ltd [...]... the Chandrasekhar limit and the supernova is born after the explosion The intrinsic luminosity of such supernovae is considered to be a constant That is, they are taken as standard candles and any variation in their apparent luminosity as measured on earth must be explicable in terms of their differing distances from the earth In a Euclidean space, the apparent luminosity l of a source with intrinsic... confronted with an unmitigated disaster However, including gravity in any supersymmetric theory inevitably leads to a supergravity theory, in which supersymmetry is a local, rather than a global, symmetry This is because in GR the momentum generator Pµ becomes a local field generating diffeomorphisms of spacetime Then, in a supersymmetric theory incorporating GR, the supersymmetry generators too become... The sparticles associated with the quarks and leptons, called respectively ‘squarks’ Copyright © 2004 IOP Publishing Ltd 16 The standard model of cosmology and ‘sleptons’, are (spin-0) scalar particles and, in a supersymmetric theory, they must have the same mass and quantum numbers as the original particles This has the important consequence that the vanishing cosmological constant result is unaffected... the low-energy limit of string theory The form of the potential in a supergravity theory is given in section 2.8 The main point to note is that, as in the case of global supersymmetry, supersymmetric vacua are generally stationary points of this potential but that at such points the vacuum energy density is now generally negative Non-supersymmetric (scalar) field configurations in which the energy density... have found most useful in preparing this chapter are: • • • • Kolb E W and Turner M S 1990 The Early Universe (Reading, MA: Addison-Wesley) Weinberg S 1972 Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (New York: Wiley) Weinberg S 1989 Rev Mod Phys 61 1 Sarkar S 1996 Rep Prog Phys 59 1493, arXiv:hep-ph/9602260 Copyright © 2004 IOP Publishing Ltd ... the redshifting of the photon energy discussed in section 1.2 Another interesting case is w = −1, which gives ρ = constant p = −ρ (1.51) This may be interpreted as vacuum energy and allows us to incorporate the cosmological constant into the discussion without introducing it explicitly, if we wish 1.5 Time dependence of the scale factor It is easy to solve the Friedmann equation (1.34) in the case... this is indeed the case requires at least one further input The The determination of m and recent data on the temperature anisotropies of the cosmic microwave background provide just such a constraint Photons originating at the ‘last scattering surface’, when matter and radiation decouple (see section 1.10), having a redshift z ∼ 1300, are seen now as the microwave background Quantum fluctuations in the... to matter domination As we have seen in (1.49) and (1.50), the energy density of radiation decreases as R −4 as the universe expands whereas the energy density of matter decreases as R −3 Thus, radiation domination gives way to matter domination at some point in the expansion of the universe For a matter-dominated universe, the energy density is given by (1.114) and for a radiation-dominated universe... a subtlety in the interpretation of N∗ which must be taken into account We shall assume that the transition temperature is sufficiently low that Copyright © 2004 IOP Publishing Ltd 20 The standard model of cosmology the only relativistic particles are the photon and three neutrinos Neutrinos drop out of thermal equilibrium below about 1 MeV when the (weak) interaction rate that keeps them in thermal... photon and three neutrinos 1.10 Cosmic microwave background radiation (CMBR) During the radiation-dominated era, the photons were in thermal equilibrium with matter (at the same temperature) because of interaction with the charge of the electrons and protons We are making the approximation here that all baryons in the universe at this time are in the form of protons However, eventually the electrons and . 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. 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. Publishing Ltd Preface The new particle physics of the past 30 years, including electroweak theory, quantum chromodynamics, grand unified theory, supersymmetry, supergravity and superstring theory,