This page intentionally left blank AN INTRODUCTION TO MATHEMATICAL COSMOLOGY Thisbookprovidesaconciseintroductiontothemathematical aspectsof theorigin,structureandevolutionof theuniverse.The bookbeginswithabrief overviewof observationaland theoreticalcosmology,alongwithashortintroductiontogeneral relativity.ItthengoesontodiscussFriedmannmodels,the Hubbleconstantanddecelerationparameter,singularities,the earlyuniverse,inflation,quantumcosmologyandthedistant futureof theuniverse.Thisneweditioncontainsarigorous derivationof theRobertson–Walkermetric.Italsodiscussesthe limitstotheparameterspacethroughvarioustheoreticaland observationalconstraints,andpresentsanewinflationary solutionforasixthdegreepotential. Thisbookissuitableasatextbookforadvancedundergradu- atesandbeginninggraduatestudents.Itwillalsobeof interestto cosmologists,astrophysicists,appliedmathematiciansand mathematicalphysicists.    receivedhisPhDandScDfromthe Universityof Cambridge.In1984hebecameProfessorof MathematicsattheUniversityof Chittagong,Bangladesh,andis currentlyDirectorof theResearchCentreforMathematicaland PhysicalSciences,Universityof Chittagong.ProfessorIslamhas heldresearchpositionsinuniversitydepartmentsandinstitutes throughouttheworld,andhaspublishednumerouspaperson quantumfieldtheory,generalrelativityandcosmology.Hehas alsowrittenandcontributedtoseveralbooks. AN INTRODUCTION TO MATHEMATICAL COSMOLOGY Second edition J. N. ISLAM ResearchCentreforMathematicalandPhysicalSciences, UniversityofChittagong,Bangladesh           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 ©Cambridge University Press 1992, 2004 First published in printed format 2001 ISBN 0-511-01849-5 eBook (netLibrary) ISBN 0-521-49650-0 hardback ISBN 0-521-49973-9 paperback Contents Prefacetothefirstedition pageix Prefacetothesecondedition xi 1 Somebasicconceptsandanoverviewof cosmology 1 2 Introductiontogeneralrelativity 12 2.1 Summaryof generalrelativity 12 2.2 Somespecialtopicsingeneralrelativity 18 2.2.1 Killingvectors 18 2.2.2 Tensordensities 21 2.2.3 GaussandStokestheorems 24 2.2.4 Theactionprincipleforgravitation 28 2.2.5 Somefurthertopics 32 3 TheRobertson–Walkermetric 37 3.1 Asimplederivationof theRobertson–Walker metric 37 3.2 Somegeometricpropertiesof theRobertson– Walkermetric 42 3.3 Somekinematicpropertiesof theRobertson– Walkermetric 45 3.4 TheEinsteinequationsfortheRobertson–Walker metric 51 3.5 Rigorousderivationof theRobertson–Walker metric 53 4 TheFriedmannmodels 60 4.1 Introduction 60 4.2 Exactsolutionforzeropressure 64 4.3 Solutionforpureradiation 67 4.4 Behaviourneart(cid:2)0 68 4.5 Exactsolutionconnectingradiationandmatter eras 68 v vi Contents 4.6 Thered-shiftversusdistancerelation 71 4.7 Particleandeventhorizons 73 5 TheHubbleconstantandthedecelerationparameter 76 5.1 Introduction 76 5.2 Measurementof H 77 0 5.3 Measurementof q 80 0 5.4 Furtherremarksaboutobservationalcosmology 85 AppendixtoChapter5 90 6 Modelswithacosmologicalconstant 94 6.1 Introduction 94 6.2 Furtherremarksaboutthecosmological constant 98 6.3 Limitsonthecosmologicalconstant 100 6.4 Somerecentdevelopmentsregardingthe cosmologicalconstantandrelatedmatters 102 6.4.1 Introduction 102 6.4.2 Anexactsolutionwithcosmological constant 104 6.4.3 Restrictionof parameterspace 107 7 Singularitiesincosmology 112 7.1 Introduction 112 7.2 Homogeneouscosmologies 113 7.3 Someresultsof generalrelativistic hydrodynamics 115 7.4 Definitionof singularities 118 7.5 Anexampleof asingularitytheorem 120 7.6 Ananisotropicmodel 121 7.7 Theoscillatoryapproachtosingularities 122 7.8 Asingularity-freeuniverse? 126 8 Theearlyuniverse 128 8.1 Introduction 128 8.2 Theveryearlyuniverse 135 8.3 Equationsintheearlyuniverse 142 8.4 Black-bodyradiationandthetemperatureof the earlyuniverse 143 8.5 Evolutionof themass-energydensity 148 8.6 Nucleosynthesisintheearlyuniverse 153 8.7 Furtherremarksaboutheliumanddeuterium 159 8.8 Neutrinotypesandmasses 164 Contents vii 9 Theveryearlyuniverseandinflation 166 9.1 Introduction 166 9.2 Inflationarymodels–qualitativediscussion 167 9.3 Inflationarymodels–quantitativedescription 174 9.4 Anexactinflationarysolution 178 9.5 Furtherremarksoninflation 180 9.6 Moreinflationarysolutions 183 AppendixtoChapter9 186 10 Quantumcosmology 189 10.1 Introduction 189 10.2 Hamiltonianformalism 191 10.3 TheSchrödingerfunctionalequationfora scalarfield 195 10.4 Afunctionaldifferentialequation 197 10.5 Solutionforascalarfield 199 10.6 Thefreeelectromagneticfield 199 10.7 TheWheeler–DeWittequation 201 10.8 Pathintegrals 202 10.9 Conformalfluctuations 206 10.10 Furtherremarksaboutquantumcosmology 209 11 Thedistantfutureof theuniverse 211 11.1 Introduction 211 11.2 Threewaysforastartodie 211 11.3 Galacticandsupergalacticblackholes 213 11.4 Black-holeevaporation 215 11.5 Slowandsubtlechanges 216 11.6 Acollapsinguniverse 218 Appendix 220 Bibliography 238 Index 247