Generating Ekpyrotic Curvature Perturbations Befor

arXiv:hep-th/0702153v119Feb2007DAMTP-2007-14ITFA-2007-12GeneratingEkpyroticCurvaturePerturbationsBeforetheBigBangJean-LucLehners1,PaulMcFadden2,NeilTurok1andPaulJ.Steinhardt3,41DAMTP,CMS,WilberforceRoad,CB30WA,Cambridge,UK2ITFA,Valckenierstraat65,1018XEAmsterdam,theNetherlands3JosephHenryLaboratories,PrincetonUniversity,Princeton,NJ08544,USA4PrincetonCenterforTheoreticalPhysics,JadwinHall,PrincetonUniversity,Princeton,NJ08544,USAAbstractWeanalyzeageneralmechanismforproducinganearlyscale-invariantspectrumofcosmologicalcurvatureperturbationsduringacontractingphaseprecedingabigbang,thatcanbeentirelydescribedusing4deffectivefieldtheory.Themechanism,basedonfirstproducingentropicperturbationsandthenconvertingthemtocurva-tureperturbations,canbenaturallyincorporatedincyclicandekpyroticmodelsinwhichthebigbangismodelledasabranecollision,aswellasothertypesofcosmo-logicalmodelswithapre-bigbangphase.Weshowthatthecorrectperturbationamplitudecanbeobtainedandthatthespectraltiltnstendstorangefromslightlybluetored,with0.97ns1.02forthesimplestmodels,arangecompatiblewithcurrentobservationsbutshiftedbyafewpercenttowardsthebluecomparedtothepredictionofthesimplest,large-fieldinflationarymodels.E-mail:[e-mail],[e-mail],[e-mail],[e-mail]1IntroductionTheprimordialperturbationssodramaticallyimagedbyWMAPandothercosmologicalprobesoffersomeofourbestcluesastothefundamentalphysicsunderlyingthehotbigbang.Theobservedperturbationsseemextremelysimpleincharacter:small-amplitude,nearlyGaussian,growing-modeperturbationsthatareapproximatelyscale-invariantandadiabaticinform.Anyproposedcosmologicalmodelhastoexplainthesebasicfeatures,andanypredicteddeviationsfromGaussianity,scale-invarianceoradiabaticityarelikelytobecriticalinallowingustodiscriminatebetweenmodels.Forovertwodecades,ithasbeenwell-understoodthatsimplescalarfieldmodelsofinflationcan,withsomedegreeoffine-tuning,producedensityperturbationsoftherequiredform.Morerecently,analternativesetofcosmologicalmodelshasbeenproposed–theekpyroticandcyclicmodels[1,2]–which,ithasbeenargued,alsosolvethestandardhorizonandflatnesspuzzlesbeforethebigbangandgenerateanearlyscale-invariantspectrumofrealisticcosmologicalperturbations.Theekpyroticmechanismforgeneratingcosmologicalperturbationshasbeenthesub-jectofconsiderabledebateduetoakeyconceptualhurdle.Inmostcosmologicalmodelsofinterest,includingtheekpyroticandcyclicmodels,allormostofthecontractionphasepriortothebigbangcanbedescribedapproximatelyusing4deffectivefieldtheory.Inthisdescription,thegrowing-mode,adiabaticperturbationsinthecontracting,pre-bangphasehaveageometricalcharacterthatisverydifferentfromthegrowing-modeadiabaticperturbationsintheexpanding,post-bangphase.Inanexpandingphase,thegrowingadiabaticmodeisalocaldilatation(orcurvatureperturbation)oftheconstantdensityhypersurfaces.Butbytime-reversalinvariance,thismodeisgenericallythedecayingmodeinthecorrespondingcontractinguniverse.Instead,theadiabaticgrowingmodeinacontractinguniverseisatime-delayperturbationtothebigcrunchwhich,undertime-reversal,mapstoadecayingmodeperturbationinanexpandinguniverse.Consequently,anypre-bigbangscenarioforgeneratingtheprimordialperturbationsmustexplainhowgrowingmodetime-delayperturbationsbeforethebangconvertintogrowingmodecur-vatureperturbationsafterthebang.ThisissuehasbeenclearlyanalyzedbyCreminellietal.[3],forexample.Onepossibilityisthatitispreciselythebreakdownofthe4deffectivetheorynearthecrunch/bangtransitionthatallowsthegrowingtime-delaymodetotransformnaturallyintoagrowingmodecurvatureperturbation.Thispossibilityhasbeenstudiedextensively,2forexample,intheRandall-Sundrummodel[4,5],whereitwasshownthat,nearthebang(thecollisionbetweenbranes),thewarpingofthe5dbulkcausesthe4deffectivetheorytofailatorder(V/c)2,whereVisthebranecollisionspeed.Thisbreakdownallowsformixingbetweenthegrowinganddecayingmodes,whichisnormallyprohibitedwithinapurely4dcontext.Todate,theanalysishasreliedonmatchinga4deffectivedescriptionfarfromthebangwitha5ddescriptionnearthecollision,whichisnotcompletelysatisfying.Ultimately,thegoalistodescribethismechanismentirelyin5d,butthisrequiresincludingtheinterbranepotentialwithinafully5dcalculation,whichhasnotyetbeenachieved.Butevenifthe5dmechanismisprovenrigorouslytowork,onemayinquirewhethertheseinherently5deffectsaretheonlyapproachforobtainingnearlyscale-invariantfluctuationsduringacontractingphase.Inthispaper,weshowtheanswerisno:weanalyzehereanalternative“entropic”mechanism–describedentirelyintermsof4deffectivefieldtheory–thatconvertsgrowingmodescale-invariantperturbationsdevelopedinacontractingphaseintoscale-invariantcurvatureperturbationsjustbeforethebigcrunch/bigbangtransition.Thismechanism,whichhasbeensuggestedpreviouslybyNotariandRiotto[6](seealsoTsujikawaetal.[7]),useselementsthatarefamiliartomanycosmologistswhostudyentropicperturba-tionsininflationarycosmology.Hereweanalyzethepredictionsforthespectralamplitudeandtiltandshowthattheingredientsneededtogenerateentropicperturbationsandcon-vertthemintocurvatureperturb