RelativisticCosmology Cosmologyhasbeentransformedbydramaticprogressinhigh-precisionobservationsand theoreticalmodelling.Thisbooksurveyskeydevelopmentsandopenissuesforgraduate studentsandresearchers.Usingarelativisticgeometricapproach,itfocusesonthegeneral conceptsandrelationsthatunderpinthestandardmodeloftheUniverse. Part1coversfoundationsofrelativisticcosmology,whilstPart2developsthedynami- calandobservationalrelationsforallmodelsoftheUniversebasedongeneralrelativity. Part 3 focuses on the standard model of cosmology, including inflation, dark matter, dark energy, perturbation theory, the cosmic microwave background, structure formation and gravitational lensing. It also examines modified gravity and inhomogeneity as pos- sible alternatives to dark energy. Anisotropic and inhomogeneous models are described in Part 4, and Part 5 reviews deeper issues, such as quantum cosmology, the start of the universe and the multiverse proposal. Colour versions of some figures are available at www.cambridge.org/9780521381154. GeorgeF.R.Ellis FRS is Professor Emeritus at the University of CapeTown, SouthAfrica. Heisco-authorwithStephenHawkingofTheLargeScaleStructureofSpace-Time. RoyMaartens holds an SKAResearch Chair at the University of theWestern Cape, South Africa,andisProfessorofCosmologyattheUniversityofPortsmouth,UK. MalcolmA.H.MacCallumisDirectoroftheHeilbronnInstituteatBristol,andisPresidentof theInternationalSocietyonGeneralRelativityandGravitation. Relativistic Cosmology GEORGE F. R. ELLIS UniversityofCapeTown ROY MAARTENS UniversityofPortsmouthandUniversityoftheWesternCape MALCOLM A. H. MACCALLUM UniversityofBristol cambridge university press Cambridge,NewYork,Melbourne,Madrid,CapeTown, Singapore,SãoPaulo,Delhi,MexicoCity CambridgeUniversityPress TheEdinburghBuilding,CambridgeCB28RU,UK PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork www.cambridge.org Informationonthistitle:www.cambridge.org/9780521381154 ©CambridgeUniversityPress2012 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithout thewrittenpermissionofCambridgeUniversityPress. Firstpublished2012 PrintedintheUnitedKingdomattheUniversityPress,Cambridge AcataloguerecordforthispublicationisavailablefromtheBritishLibrary LibraryofCongressCataloguinginPublicationdata Ellis,GeorgeF.R.(GeorgeFrancisRayner) Relativisticcosmology/GeorgeEllis,RoyMaartens,MalcolmMacCallum. p. cm. Includesbibliographicalreferencesandindex. ISBN978-0-521-38115-4 1. Cosmology. 2. Relativisticastrophysics. 3. Relativisticquantumtheory. I. Maartens,R.(Roy) II. MacCallum,M.A.H. III. Title. QB981.E46542012 523.1–dc23 2011040518 ISBN978-0-521-38115-4Hardback Additionalresourcesforthispublicationatwww.cambridge.org/9780521381154. CambridgeUniversityPresshasnoresponsibilityforthepersistenceor accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto inthispublication,anddoesnotguaranteethatanycontentonsuch websitesis,orwillremain,accurateorappropriate. Contents Preface page xi Part1 Foundations 1 Thenatureofcosmology 3 1.1 Theaimsofcosmology 3 1.2 Observationalevidenceanditslimitations 5 1.3 Asummaryofcurrentobservations 9 1.4 Cosmologicalconcepts 17 1.5 Cosmologicalmodels 20 1.6 Overview 23 2 Geometry 25 2.1 Manifolds 26 2.2 Tangentvectorsand1-forms 28 2.3 Tensors 31 2.4 Liederivatives 34 2.5 Connectionsandcovariantderivatives 35 2.6 Thecurvaturetensor 37 2.7 Riemanniangeometry 39 2.8 Generalbasesandtetrads 51 2.9 Hypersurfaces 53 3 Classicalphysicsandgravity 56 3.1 Equivalenceprinciples,gravityandlocalphysics 56 3.2 Conservationequations 61 3.3 Thefieldequationsinrelativityandtheirstructure 64 3.4 RelationtoNewtoniantheory 69 Part2 Relativisticcosmologicalmodels 4 Kinematicsofcosmologicalmodels 73 4.1 Comovingcoordinates 73 4.2 Thefundamental4-velocity 74 4.3 Timederivativesandtheaccelerationvector 75 4.4 Projectiontogivethree-dimensionalrelations 76 v vi Contents 4.5 Relativepositionandvelocity 79 4.6 Thekinematicquantities 80 4.7 CurvatureandtheRicciidentitiesforthe4-velocity 86 4.8 Identitiesfortheprojectedcovariantderivatives 88 5 Matterintheuniverse 89 5.1 Conservationlaws 90 5.2 Fluids 95 5.3 Multiplefluids 101 5.4 Kinetictheory 104 5.5 Electromagneticfields 110 5.6 Scalarfields 115 5.7 Quantumfieldtheory 117 6 Dynamicsofcosmologicalmodels 119 6.1 TheRaychaudhuri–Ehlersequation 119 6.2 Vorticityconservation 124 6.3 TheotherEinsteinfieldequations 126 6.4 TheWeyltensorandtheBianchiidentities 132 6.5 Theorthonormal1+3tetradequations 134 6.6 Structureofthe1+3systemofequations 139 6.7 Globalstructureandsingularities 143 6.8 NewtonianmodelsandNewtonianlimits 147 7 Observationsincosmologicalmodels 153 7.1 Geometricalopticsandnullgeodesics 153 7.2 Redshifts 156 7.3 Geometryofnullgeodesicsandimages 159 7.4 Radiationenergyandflux 161 7.5 Specificintensityandapparentbrightness 167 7.6 Numbercounts 170 7.7 Selectionanddetectionissues 171 7.8 Backgroundradiation 172 7.9 Causalandvisualhorizons 173 8 Light-coneapproachtorelativisticcosmology 180 8.1 Model-basedapproach 180 8.2 Directobservationalcosmology 181 8.3 Idealcosmography 186 8.4 Fieldequations:determiningthegeometry 187 8.5 Isotropicandpartiallyisotropicobservations 190 8.6 Implicationsandopportunities 194 vii Contents Part3 Thestandardmodelandextensions 9 HomogeneousFLRWuniverses 201 9.1 FLRWgeometries 202 9.2 FLRWdynamics 210 9.3 FLRWdynamicswithbarotropicfluids 212 9.4 Phaseplanes 220 9.5 Kineticsolutions 225 9.6 Thermalhistoryandcontentsoftheuniverse 226 9.7 Inflation 238 9.8 OriginofFLRWgeometry 246 9.9 Newtoniancase 247 10 PerturbationsofFLRWuniverses 249 10.1 Thegaugeproblemincosmology 250 10.2 Metric-basedperturbationtheory 251 10.3 Covariantnonlinearperturbations 262 10.4 Covariantlinearperturbations 267 11 Thecosmicbackgroundradiation 282 11.1 TheCMBandspatialhomogeneity:nonlinearanalysis 282 11.2 Linearizedanalysisofdistributionmultipoles 287 11.3 TemperatureanisotropiesintheCMB 292 11.4 Thomsonscattering 294 11.5 Scalarperturbations 295 11.6 CMBpolarization 300 11.7 Vectorandtensorperturbations 303 11.8 Otherbackgroundradiation 303 12 Structureformationandgravitationallensing 307 12.1 Correlationfunctionsandpowerspectra 307 12.2 Primordialperturbationsfrominflation 309 12.3 Growthofdensityperturbations 317 12.4 Gravitationallensing 330 12.5 Cosmologicalapplicationsoflensing 339 13 ConfrontingtheStandardModelwithobservations 345 13.1 ObservationalbasisforFLRWmodels 346 13.2 FLRWobservations:probingthebackgroundevolution 351 13.3 AlmostFLRWobservations:probingstructureformation 355 13.4 Constraintsandconsistencychecks 363 13.5 Concordancemodelandfurtherissues 366 viii Contents 14 Accelerationfromdarkenergyormodifiedgravity 370 14.1 Overviewoftheproblem 370 14.2 DarkenergyinanFLRWbackground 373 14.3 ModifiedgravityinaRWbackground 376 14.4 Constrainingeffectivetheories 390 14.5 Conclusion 391 15 ‘Acceleration’fromlarge-scaleinhomogeneity? 395 15.1 Lemaître–Tolman–Bondiuniverses 395 15.2 Observablesandsourceevolution 399 15.3 Canwefitareadistanceandnumbercountobservations? 401 15.4 TestingbackgroundLTBwithSNIaandCMBdistances 403 15.5 PerturbationsofLTB 406 15.6 Observationaltestsofspatialhomogeneity 411 15.7 Conclusion:statusoftheCopernicanPrinciple 415 16 ‘Acceleration’fromsmall-scaleinhomogeneity? 416 16.1 Differentscaledescriptions 416 16.2 Cosmologicalbackreaction 421 16.3 Specificmodels:almostFLRW 423 16.4 Inhomogeneousmodels 426 16.5 Importanceofbackreactioneffects? 432 16.6 Effectsonobservations 435 16.7 Combinationofeffects:alteringcosmicconcordance? 440 16.8 Entropyandcoarse-graining 441 Part4 Anisotropicandinhomogeneousmodels 17 Thespaceofcosmologicalmodels 447 17.1 Cosmologicalmodelswithsymmetries 447 17.2 Theequivalenceproblemincosmology 452 17.3 Thespaceofmodelsandtheroleofsymmetricmodels 453 18 Spatiallyhomogeneousanisotropicmodels 456 18.1 Kantowski–Sachsuniverses:geometryanddynamics 457 18.2 BianchiIuniverses:geometryanddynamics 458 18.3 Bianchigeometriesandtheirfieldequations 462 18.4 Bianchiuniversedynamics 467 18.5 EvolutionofparticularBianchimodels 474 18.6 Cosmologicalconsequences 481 18.7 TheBianchidegreesoffreedom 486