Brane World Cosmology - Gauge-Invariant Formalism

arXiv:hep-th/0004160v37Aug2000BraneWorldCosmology—Gauge-InvariantFormalismforPerturbation—HideoKodama,AkihiroIshibashiandOsamuSetoYukawaInstituteforTheoreticalPhysics,KyotoUniversityKyoto606-8502,Japan(April24,2000)Inthepresentpaperthegauge-invariantformalismisdevelopedforperturbationsofthebrane-worldmodelinwhichouruniverseisrealizedasaboundaryofahigher-dimensionalspacetime.Forthebackgroundmodelinwhichthebulkspacetimeis(n+m)-dimensionalandhasthespatialsymmetrycorrespondingtotheisometrygroupofan-dimensionalmaximallysymmetricspace,gauge-invariantequationsarederivedforperturbationsofthebulkspacetime.Further,forthecasecorrespondingtothebrane-worldmodelinwhichm=2andthebraneisaboundaryinvariantunderthespatialsymmetryintheunperturbedbackground,relationsbetweenthegauge-invariantvariablesdescribingthebulkperturbationsandthoseforbraneperturbationsarederivedfromIsrael’sjunctionconditionundertheassumptionofZ2symmetry.Inparticular,forthecaseinwhichthebulkspacetimeisaconstant-curvaturespacetime,itisshownthatthebulkperturbationequationsreducetoasinglehyperbolicmasterequationforamastervariable,andthatthephysicalconditiononthegauge-invariantvariabledescribingtheintrinsicstressperturbationofthebraneyieldsaboundaryconditionforthemasterequationthroughthejunctioncondition.Onthebasisofthisformalism,itispointedoutthatitseemstobedifficulttosuppressbraneperturbationscorrespondingtomassiveexcitationsforabranemotiongivingarealisticexpandinguniversemodel.I.INTRODUCTIONMotivatedbytheM-theory[1,2],AdS-CFTcorrespondenceinstringtheories[3,4],andthehierarchyproblemintheparticletheory[5–8],brane-worldmodelsinwhichouruniverseisrealizedasaboundaryofahigher-dimensionalspacetimehavebeenactivelystudiedrecently[9–38].Inparticular,forthecaseinwhichthebulkspacetimeis5-dimensionalAnti-deSitterspacetimeandthebraneisrealizedasaflat4-dimensionalspacetime,thegravitationalinteractionbetweenmattersinthebraneiswell-describedbythestandardoneonscalesmuchlargerthanthescalecorrespondingtothebranetension[12–15].Further,asanextensionoftheanalysistoadynamicalsituation,theembeddingofRobertson-Walkeruniversemodelsinto5-dimensionalAnti-deSitterandAnti-deSitter-Schwarzschildspacetimeshasbeendiscussedbymanypeople[19–30].Insuchhigh-symmetrycases,althoughtheevolutionequationforthecosmicscalefactorismodifiedfromthestandardone,ouruniverseisstilladynamicallyclosedsystem,andthedifferenceintheevolutionequationcanbeneglectedwhentheenergydensityoftheuniversebecomesmuchsmallerthanthebranetension.Thusthebrane-worldmodelgivesanewworldmodelconsistentwiththepresentdayobservations.However,ifonegoesbeyondthislowest-levelapproximation,itisnotclearwhetherthebrane-worldmodelisconsistentwithallavailableobservationsbecauseouruniverseisnotdynamicallyclosedinthismodel[10].Oneofthesimplestwaystoanalyzethisproblemistoinvestigatethebehaviorofperturbationsofthebrane-worldmodel.Sinceperturbationsofthebraneareinevitablyassociatedwithperturbationsinthegeometryofthebulkspacetime,suchinvestigationwillmakeclearwhetherornottheopennatureoftheuniversedynamicsiscontrollable.Itwillalsomakepossibleanobservationaltestofthemodelintermsoftheanisotropyofthecosmicmicrowavebackground.Asthestartingpointofinvestigationsinthisline,inthepresentpaper,wedevelopagauge-invariantformalismforperturbationsofthebrane-worldmodel.Thebasicapproachisthesameasthatoriginallydevelopedfor4-dimensionalspacetimebyGerlachandSengupta[39–41]andutilizedbysomepeopleintheanalysisoftheinteractionbetweenadomainwallandgravitationalwavesin4-dimensionalspacetimes[42–44].Theformalismconsistsoftwoparts.Thefirstoneisagauge-invariantformalismforperturbationsinthegeometryofthebulkspacetime.Thisproblemhasalreadybeeninvestigatedbysomepeopleforthestandardcaseinwhichthebulkspacetimeisvacuumandmaximallysymmetric[45].Inthepresentpaper,takingaccountofthedevelopingnatureofthebrane-worldmodel,weextendtheformalismtothecaseinwhichthebulkspacetimeis(m+n)-dimensionalanditsunperturbedgeometryhasonlytheisometrycorrespondingtothemaximallysymmetricspaceofdimensionn(n≥1).Thissymmetryisutilizedtoexpandperturbationsintermsoftheharmonicfunctionsonn-dimensionalmaximallysymmetricspaceanddefinegauge-invariantvariables.1Thesecondpartestablishesrelationsbetweenthegauge-invariantvariablesdescribingperturbationsofthebraneandthoseforthebulkperturbations.Inthispartweassumethatm=2andthe(n+1)-dimensionalbraneisinvariantundertheisometrygroupofthebulkintheunperturbedmodel.ThusthebranerepresentsanexpandingRobertson-Walkeruniverseingeneral.Thepaperisorganizedasfollows.Inthenextsectionwefirstclassifyperturbationsintotensor,vector,andscalartypesintermsofthetensorialbehaviorwithrespecttothemaximallysymmetricn-dimensionalspacetime.Thenforeachtypewedefinethegauge-invariantvariablesdescribingperturbationsofthebulkgeometryandexpresstheEinsteinequationsintermsofthem.InSectionIII,afterintroducingangauge-invariantvariabledescribingthemotionofthebrane,weexpressIsrael’sjunctionconditioncorrespondingtotheZ2symmetryintermsofitandthebulkvariables.Wewillshowthatthisgivesexpressionsfortheintrinsicperturbationv