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TLFeBOOK This page intentionally left blank TLFeBOOK AN INTRODUCTION TO MATHEMATICAL COSMOLOGY This book provides a concise introduction to the mathematical aspects of the origin, structure and evolution of the universe. The book begins with a brief overview of observational and theoretical cosmology, along with a short introduction to general relativity. It then goes on to discuss Friedmann models, the Hubble constant and deceleration parameter, singularities, the early universe, inflation, quantum cosmology and the distant future of the universe. This new edition contains a rigorous derivation of the Robertson–Walker metric. It also discusses the limits to the parameter space through various theoretical and observational constraints, and presents a new inflationary solution for a sixth degree potential. This book is suitable as a textbook for advanced undergradu- ates and beginning graduate students. It will also be of interest to cosmologists, astrophysicists, applied mathematicians and mathematical physicists. received his PhD and ScD from the University of Cambridge. In 1984 he became Professor of Mathematics at the University of Chittagong, Bangladesh, and is currently Director of the Research Centre for Mathematical and Physical Sciences, University of Chittagong. Professor Islam has held research positions in university departments and institutes throughout the world, and has published numerous papers on quantum field theory, general relativity and cosmology. He has also written and contributed to several books. TLFeBOOK TLFeBOOK AN INTRODUCTION TO MATHEMATICAL COSMOLOGY Second edition J. N. ISLAM Research Centre for Mathematical and Physical Sciences, University of Chittagong, Bangladesh TLFeBOOK The Pitt Building, Trumpington Street, Cambridge, United Kingdom The Edinburgh Building, Cambridge CB2 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarcón 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa http://www.cambridge.org First published in printed format ISBN 0-521-49650-0 hardback ISBN 0-521-49973-9 p a p erback ISBN 0-511-01849-5 eBook Cambrid g e University Press 1992, 2004 2001 (netLibrary) © TLFeBOOK Contents Prefacetothefirsteditionpageix Prefacetothesecondeditionxi 1Somebasicconceptsandanoverviewofcosmology1 2Introductiontogeneralrelativity12 2.1Summaryofgeneralrelativity12 2.2Somespecialtopicsingeneralrelativity18 2.2.1Killingvectors18 2.2.2Tensordensities21 2.2.3GaussandStokestheorems24 2.2.4Theactionprincipleforgravitation28 2.2.5Somefurthertopics32 3TheRobertson–Walkermetric37 3.1 A simple derivation of the Robertson–Walker metric37 3.2 Some geometric properties of the Robertson– Walkermetric42 3.3 Some kinematic properties of the Robertson– Walkermetric45 3.4 The Einstein equations for the Robertson–Walker metric51 3.5 Rigorous derivation of the Robertson–Walker metric53 4TheFriedmannmodels60 4.1Introduction60 4.2Exactsolutionforzeropressure64 4.3Solutionforpureradiation67 4.4Behaviourneartϭ068 4.5 Exact solution connecting radiation and matter eras68 v TLFeBOOK 4.6Thered-shiftversusdistancerelation71 4.7Particleandeventhorizons73 5TheHubbleconstantandthedecelerationparameter76 5.1Introduction76 5.2 Measurement of H 0 77 5.3 Measurement of q 0 80 5.4Furtherremarksaboutobservationalcosmology85 AppendixtoChapter590 6Modelswithacosmologicalconstant94 6.1Introduction94 6.2 Further remarks about the cosmological constant98 6.3Limitsonthecosmologicalconstant100 6.4 Some recent developments regarding the cosmologicalconstantandrelatedmatters102 6.4.1Introduction102 6.4.2 An exact solution with cosmological constant104 6.4.3Restrictionofparameterspace107 7Singularitiesincosmology112 7.1Introduction112 7.2Homogeneouscosmologies113 7.3 Some results of general relativistic hydrodynamics115 7.4Definitionofsingularities118 7.5Anexampleofasingularitytheorem120 7.6Ananisotropicmodel121 7.7Theoscillatoryapproachtosingularities122 7.8Asingularity-freeuniverse?126 8Theearlyuniverse128 8.1Introduction128 8.2Theveryearlyuniverse135 8.3Equationsintheearlyuniverse142 8.4 Black-body radiation and the temperature of the earlyuniverse143 8.5Evolutionofthemass-energydensity148 8.6Nucleosynthesisintheearlyuniverse153 8.7Furtherremarksaboutheliumanddeuterium159 8.8Neutrinotypesandmasses164 vi Contents TLFeBOOK 9Theveryearlyuniverseandinflation166 9.1Introduction166 9.2Inflationarymodels–qualitativediscussion167 9.3Inflationarymodels–quantitativedescription174 9.4Anexactinflationarysolution178 9.5Furtherremarksoninflation180 9.6Moreinflationarysolutions183 AppendixtoChapter9186 10Quantumcosmology189 10.1Introduction189 10.2Hamiltonianformalism191 10.3 The Schrödinger functional equation for a scalarfield195 10.4Afunctionaldifferentialequation197 10.5Solutionforascalarfield199 10.6Thefreeelectromagneticfield199 10.7TheWheeler–DeWittequation201 10.8Pathintegrals202 10.9Conformalfluctuations206 10.10Furtherremarksaboutquantumcosmology209 11Thedistantfutureoftheuniverse211 11.1Introduction211 11.2Threewaysforastartodie211 11.3Galacticandsupergalacticblackholes213 11.4Black-holeevaporation215 11.5Slowandsubtlechanges216 11.6Acollapsinguniverse218 Appendix220 Bibliography238 Index247 Contents vii TLFeBOOK TLFeBOOK [...]... additional topics and comments are considered in the Appendix at the end of the book Needless to say, in the limited size and scope of the book an exhaustive treatment of any topic is not possible, but we hope enough ground has been covered for the serious student of cosmology to benefit from it As this book was going to press, Fred Hoyle passed away Notwithstanding the controversies he was involved in,... greatest contributors to cosmology in the twentieth century The controversies, more often than not, led to important advances Hoyle’s prediction of a certain energy level of the carbon nucleus, revealed through his studies of nucleosynthesis, confirmed later in the laboratory, was an outstanding scientific achievement A significant part of my knowledge of cosmology, for what it is worth, was acquired through... have discussed plans for my books with Mrs Mary Wraith, who kindly typed the manuscript for my first book For more than three decades she has been friend, philosopher and mentor for me and my wife and in recent years a very affectionate godmother (‘Goddy’) to my daughters This book is fondly dedicated to this remarkable person Jamal Nazrul Islam Chittagong, 1991 TLFeBOOK Preface to the second edition The... observational cosmology has I believe improved considerably as a result of these comments I am grateful to ix TLFeBOOK x Preface to the first edition F J Dyson for his ideas included in the last chapter I thank Maureen Storey of Cambridge University Press for her efficient and constructive subediting I am grateful to my wife Suraiya and daughters Nargis and Sadaf and my son-in-law Kamel for support and encouragement... earlier edition, to which there appears to have been a favourable response, has been kept intact as far as possible in this new edition except for minor changes A number of new additions have been made Some standard topics have been added to the introduction to general relativity, such as Killing vectors Not all these topics are used later in the book, but some may be of use to the beginning student for mathematical. .. the galaxies A and B Similarly, the distance between galaxies C and D is fCDR, where the constant fCD depends on the galaxies C and D Thus if the distance between A and B changes by a certain factor in a definite period of time then the distance between C and D also changes by the same factor in that period of time The largescale structure and behaviour of the universe can be described by the single... distance between two points in the universe It is helpful to consider the analogy of a spherical balloon which is expanding and which is uniformly covered on its surface with dots The dots can be considered to correspond to ‘galaxies’ in a two-dimensional universe As the balloon expands, all dots move away from each other and from any given dot all dots appear to move away with speeds which at any... distance between A and B changes by a certain factor in any period of time, the distance between any pair of dots changes by the same factor in that period of time From the rate at which galaxies are receding from each other, it can be deduced that all galaxies must have been very close to each other at the same time in the past Considering again the analogy of the balloon, it is TLFeBOOK Scale factor... the term ‘open’ to mean a model which expands forever, and ‘closed’ for the opposite Sometimes the expression ‘closed’ is used to mean a universe with a finite volume, but, as mentioned earlier, it is only in the Friedmann models that a universe has infinite volume if it expands forever, etc The standard big-bang model of the universe has had three major successes Firstly, it predicts that something... relativity becomes untenable and one has to resort to the quantum theory of gravitation However, a satisfactory quantum theory for gravity does not yet exist Some progress has been made in what is called ‘quantum cosmology , in which quantum considerations throw some light on problems to do with initial conditions of the universe We shall attempt to provide an introduction to this subject in this book If the . contributed to several books. TLFeBOOK TLFeBOOK AN INTRODUCTION TO MATHEMATICAL COSMOLOGY Second edition J. N. ISLAM Research Centre for Mathematical and Physical. of distant galaxies by study- ing their red-shifts, which will be described quantitatively later. The red- shift can be caused by other processes than the