An introduction to mathematical cosmology

264 Pages·2004·1.214 MB·English

by  Islam J.N.

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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