Towards the Theory of Stationary Universe

TOWARDSTHETHEORYOFSTATIONARYUNIVERSE1

ArthurMezhlumian2

DepartmentofPhysics,StanfordUniversity,

Stanford,CA94305-4060

ABSTRACT

ThistalkpresentssomeprogressachievedincollaborationwithA.LindeandD.Linde1,2towardsunderstandingthetruenatureoftheglobalspatialstructureoftheUniverseaswellasthemostgeneralstationarycharacteristicsofitstime-dependentstatewitheternallygrowingtotalvolume.

Inouropinion,thesimplestand,simultaneously,themostgeneralversionofinflationarycosmologyisthechaoticinflationscenario3.Itcanberealizedinallmodelsweretheotherversions4,5ofinflationarytheorycanberealized.Severalyearsagoitwasrealizedthatinflationinthesetheorieshasaveryinterestingproperty6,7whichwillbediscussedinthistalk.IftheUniversecontainsatleastoneinflationarydomainofasizeofhorizon(h-region)withasufficientlylargeandhomogeneousscalarfieldφ,thenthisdomainwillpermanentlyproducenewh-regionsofasimilartype.DuringthisprocessthetotalphysicalvolumeoftheinflationaryUniverse(whichisproportionaltothetotalnumberofh-regions)willgrowindefinitely.

Fortunately,somekindofstationaritymayexistinmanymodelsofinflationaryUniverseduetotheprocessoftheUniverseself-reproduction8.Thepropertiesofinflationarydomains

formedduringtheprocessoftheself-reproductionoftheUniversedonotdependonthemomentoftimeatwhicheachsuchdomainisformed;theydependonlyonthevalueofthescalarfieldsinsideeachdomain,ontheaveragedensityofmatterinthisdomainandonthephysicallengthscale.Thiskindofstationarity,asopposedtothe“stationarydistribution”Pc(φ,t)(theprobabilitydensitytofindthefieldφinsideagivenHubbledomainattimet)cannotbedescribedintheminisuperspaceapproach.

InordertodescribethestructureoftheinflationaryUniversebeyondoneh-region(min-isuperspace)approachonehastoinvestigatetheprobabilitydistributionPp(φ,t),whichtakesintoaccounttheinhomogeneousexponentialgrowthofthevolumeofdomainsfilledbyfieldφ6.Solutionsforthisprobabilitydistributionwerefirstobtainedin6,7forthecaseofchaoticinflation.Itwasshownthatiftheinitialvalueofthescalarfieldφisgreaterthansomecriticalvalueφ∗,thentheprobabilitydistributionPp(φ,t)permanentlymovestolargerandlargerfieldsφ,untilitreachesthefieldφp,atwhichtheeffectivepotentialofthefield

becomesoftheorderofPlanckdensityMp4

(wewillassumeMp=1hereafter),wherethestandardmethodsofquantumfieldtheoryinacurvedclassicalspacearenolongervalid.Bythemethodsusedin6,7itwasimpossibletocheckwhetherPp(φ,t)asymptoticallyapproachesanystationaryregimeintheclassicaldomainφ<φp.

SeveralimportantstepstowardsthesolutionofthisproblemweremadebyAryalandVilenkin9,NambuandSasaki10andMiji´c11.Theirpaperscontainmanybeautifulresultsandinsights,andwewillusemanyresultsobtainedbytheseauthors.However,thegeneralizationoftheresultsof9tonon-trivialcaseofchaoticinflationappearedtobenotaneasyproblem1,2.Miji´c11didnothaveapurposetoobtainacompleteexpressionforthestationarydistributionPp(φ,t).ThecorrespondingexpressionswereobtainedforvarioustypesofpotentialsV(φ)in10.Unfortunately,accordingto10,thestationarydistributionPp(φ,t)isalmostentirelyconcentratedatφ≫φp,i.e.atV(φ)≫1,wherethemethodsusedin10areinapplicable.Inthistalkwewillarguethattheprocessoftheself-reproductionofinflationaryUniverseeffectivelykillsitselfatdensitiesapproachingthePlanckdensity.ThisleadstotheexistenceofastationaryprobabilitydistributionPp(φ,t)concentratedentirelyatsub-Planckianden-sitiesV(φ)<1inawideclassoftheoriesleadingtochaoticinflation.

Letusconsiderthesimplestmodelofchaoticinflationbasedonthetheoryofascalarfieldφminimallycoupledtogravity,withtheLagrangian

L=

1

2

∂µφ∂µφ−V(φ).

(1)

HereG=M−2

p=1isthegravitationalconstant,Risthecurvaturescalar,V(φ)istheeffectivepotentialofthescalarfield.IftheclassicalfieldφissufficientlyhomogeneousinsomedomainoftheUniverse(seebelow),thenitsbehaviorinsidethisdomainisgovernedbytheequations

φ¨+3Hφ˙=−dV/dφ,(2)

H2

+

k

3

󰀁

1

HereH=a/a,˙a(t)isthescalefactoroftheUniverse,k=+1,−1,or0foraclosed,openorflatUniverse,respectively.

ThemostimportantfactforinflationaryscenarioisthatformostpotentialsV(φ)(e.g.,inallpower-lawV(φ)=gnφn/nandexponentialV(φ)=geαφpotentials)thereisanin-termediateasymptoticregimeofslowrollingofthefieldφandquasi-exponentialexpansionoftheUniverse.Thisexpansion(inflation)endsatφ∼φewheretheslow-rollingregime¨≪3H(φ)φ˙breakesdown.φ

If,asitisusuallyassumed,theclassicaldescriptionoftheUniversebecomespossibleonlywhentheenergy-momentumtensorofmatterbecomessmallerthan1,thenatthismoment

<1andV(φ)∼<1.Therefore,theonlyconstraintontheinitialamplitudeofthe∂µφ∂µφ∼

<1.Thisgivesatypicalinitialvalueofthefieldφinthetheory(1):fieldφisgivenbyV(φ)∼

φ0∼φp,

whereφpcorrespondstothePlanckenergydensity,V(φp)=1.

Duringtheinflationalltheinhomogeneitiesarestretchedawayand,iftheevolutionof

theUniverseweregovernedsolelybyclassicalequationsofmotion(2),(3),wewouldendupwithextremelysmoothgeometryofthespatialsectionoftheUniversewithnoprimordialfluctuationstoinitiatethegrowthofgalaxiesandlarge-scalestructure.Fortunately,thesameJeansinstabilitywhichcausesthegrowthofgalaxiesduringtheHotBigBangeraleadstotheexistenceofthegrowingmodesofvacuumfluctuationsduringtheinflation.ThewavelengthsofallvacuumfluctuationsofthescalarfieldφgrowexponentiallyintheexpandingUniverse.WhenthewavelengthofanyparticularfluctuationbecomesgreaterthanH−1,thisfluctuationstopsoscillating,anditsamplitudefreezesatsomenonzerovalue

˙intheequationofmotionofthefieldφ.Theδφ(x)becauseofthelargefrictionterm3Hφ

amplitudeofthisfluctuationthenremainsalmostunchangedforaverylongtime,whereasitswavelengthgrowsexponentially.Therefore,theappearanceofsuchafrozenfluctuationisequivalenttotheappearanceofaclassicalfieldδφ(x)thatdoesnotvanishafteraveragingovermacroscopicintervalsofspaceandtime.

Becausethevacuumcontainsfluctuationsofallwavelengths,inflationleadstothecre-ationofmoreandmoreperturbationsoftheclassicalfieldwithwavelengthsgreaterthanH−1.TheaverageamplitudeofsuchperturbationsgeneratedduringatimeintervalH−1(inwhichtheUniverseexpandsbyafactorofe)isgivenby

|δφ(x)|≈

H

(4)

ofmotionofthelong-wavelengthfield12,13,14:

d

+H3/2(φ)

3H(φ)

∂

∂t

=

∂

8π2

3H(φ)

Pc

󰀆

,Theformalstationarysolution(∂Pc/∂t=0)ofequation(7)wouldbe12

Pc∼exp

󰀅

3

∂

∂

∂t

=

8π2

3H(φ)

Pp󰀆

+3H(φ)Pp

4

(7)

(9)

Themathematicalmodeldescribingsuchbehavioristherecentlydevelopedtheoryofanewtypeofbranchingdiffusionprocesses18.Eq.(9)is,fromthisperspective,thefor-wardKolmogorovequationforthefirstmomentofthegeneratingfunctionalofnumberofbranchingparticles.

Therearetwomainsetsofquestionswhichmaybeaskedconcerningsuchprocesses.Firstofall,onemaybeinterestedintheprobabilityPp(φ,t)tofindagivenfieldφatagiventimetundertheconditionthatinitialvalueofthefieldwasequaltosomeφ(t=0)=φ0.Inwhatfollowswewilldenoteφ0asχ.Ontheotherhand,onemaywishtoknow,whatistheprobabilityPp(χ,t)thatthegivenfinalvalueofthefieldφ(orthestatewithagivenfinaldensityρ)appearedasaprocessofdiffusionandbranchingofadomaincontainingsomefieldχ.Or,moregenerally,whatarethetypicalhistoryofbranchingBrowniantrajectorieswhichendupatahypersurfaceofagivenφ(oragivenρ)?

IthappensthatifwedonottakethePlanckboundaryseriously,thisequationalsodoesn’thaveatruestationarysolution(thesolutionsfoundbyNambuet.al.10areheavilyconcentratedatthesuper-Planckiandensitieswhichisjustanotherwaytostatethatthereisnorealstationarity).Thisconclusion,however,mayfailifwewilltreatthePlanckboundarymorecarefully.Therearedifferentreasonstodothis:

1.Eq.(9)mayhaveaslightlydifferentform,whichcorrespondstothedifferencebetweenstochasticapproachesofItoandStratonovich.Thisdifferenceisnotimportantatdensities

>1itmaybecomesignificant.(Note,however,thatthismuchsmallerthan1,butatV(φ)∼

isnotanunsolvableproblem.Onemayjustinvestigatemodifiedequationsaswell.Twootherproblemsaremorefundamental.)

2.Diffusionequationswerederivedinthesemiclassicalapproximationwhichbreaks

4

downnearthePlanckenergydensityρp∼Mp=1.3.Interpretationoftheprocessesdescribedbytheseequationsisbasedonthenotionofclassicalfieldsinaclassicalspace-time,whichisnotapplicableatdensitieslargerthan1becauseoflargefluctuationsofmetricatsuchdensities.Inparticular,ourinterpretationofPcandPpasofprobabilitiestofindclassicalfieldφinagivenpointatagiventimedoesnotmakemuchsenseatρ>1.

Thereisalsoanother,moregeneral,reasontoexpecttheexistenceofthestationarysolutions:aswewillarguenow,inflationkillsitselfasthedensityapproachesthePlanck

4

densityρp∼Mp.InourpreviousinvestigationweassumedthatthevacuumenergydensityisgivenbyV(φ)andtheenergy-momentumtensorisgivenbyV(φ)gµν.However,quantumfluctuationsofthescalarfieldgivethecontributiontotheaveragevalueoftheenergymomentumtensor,whichdoesnotdependonmass(form2≪H2)andisgivenby15

=

3H4

3V2gµν.

(10)

Theoriginofthiscontributionisobvious.Quantumfluctuationsofthescalarfieldφfreeze

5

outwiththeamplitude

H

(definedbytheequationsbelow):

Pp(φp,t|χ)=

∞󰀇

eλstψs(χ)πs(φ).

(12)

s=1

Indeed,thisgivesusasolutionofeq.(9)if

1

2π∂2

2π

∂

2π

∂

2π

πj(φ)

󰀆󰀆

∂

3H(χ)+∂

∂

󰀆

3H(φ)

πj(φ)+3H(φ)πj(φ)=λjπj(φ).(14)

Theorthonormalityconditionreads

󰀈

φpφe

ψs(χ)πj(χ)dχ=δsj(15)

Inourcase(withregularboundaryconditions)onecaneasilyshowthatthespectrumof

λjisdiscreteandboundedfromabove.ThereforetheasymptoticsolutionforPp(φ,t|χ)(inthelimitt→∞)isgivenby

Pp(φp,t|χ)=e

λ1t

ψ1(χ)π1(φ)·1+Oe

Hereψ1(χ)istheonlypositiveeigenfunctionofeq.(13),λ1isthecorresponding(real)eigenvalue,andπ1(φ)(invariantdensityofbranchingdiffusion)istheeigenfunctionoftheconjugateoperator(14)withthesameeigenvalueλ1.Note,thatλ1isthelargesteigenvalue,Re(λ1−λ2)>0.Wefound2thatinrealistictheoriesofinflationthetypicaltimeofrelax-ationtotheasymptoticregime,trel∼(λ1−λ2)−1,isextremelysmall.ItisonlyaboutafewthousandsPlancktimes,i.e.about10−40sec.Thismeans,thatthenormalizeddistribution

˜p(φ,t|χ)=e−λ1tPp(φp,t|χ)P

rapidlyconvergestothetime-independentnormalizeddistribution

˜p(φ|χ)≡P˜p(φ,t→∞|χ)=ψ1(χ)π1(φ).P

(18)(17)

󰀉󰀉

−(λ1−λ2)t

󰀊󰀊

.(16)

Itisthisstationarydistributionthatwewerelookingfor.Theremainingproblemisto

findthefunctionsψ1(χ)andπ1(φ),andtocheckthatallassumptionsabouttheboundaryconditionswhichwemadeonthewaytoeq.(16)areactuallysatisfied.

AftersomecalculationswecametothefollowingexpressionforPp(φ|χ)(notethatthefunctionsΦ(·)areessentiallythesame,theonlydifferenceistheargument):

Pp(φ|χ)=

C

16V(φ)

−

3

HerethenormalizationconstantCshouldbedeterminedfrom(15).UsingtheWKBap-proximation,wehavecalculated2Φ(z(φ))forawideclassofpotentialsusuallyconsideredinthecontextofchaoticinflation,includingpotentialsV∼φnandV∼eαφ.

Thesolutionwehavefoundfeaturesinterestingproperties.Firstofall,itisconcentratedheavilyatthehighestallowedvaluesoftheinflatonfield.Despitethepromisingexponentialprefactorin(19)whichlookslikeacombinationoftheHartle-Hawking16andtunneling19wavefunctions,thedependenceofΦ(z(φ))onφappearedtobeevenmoresteepthatthoseexponents.Thisfunctionfallsexponentially(withtherategovernedbythePlanckianener-gies)towardsthelowervaluesoftheinflatonfieldandstronglyoverwhelmsthedependenceoftheexponentinfrontofit.Onlynearthe“endofinflation”boundarythesolution(19)revealssomethingfamiliar—thedependenceontheinitialfieldχbecomessimilartothesquareoftunnelingwavefunction(simultaneously,thecorrespondingdependenceonthefinalfieldφcancelsout).And,aswehavealreadymentioned,therelaxationtimeisverysmall(whichisalsoaconsequenceofthefactthatthedynamicsoftheself-reproducingUniverseisgovernedbythemaximalpossibleenergies).Thestationarydistributionfoundin2isnotverysensitivetoourassumptionsconcerningtheconcretemechanismofsuppressionofpro->φp.Wehopethattheseresultsmayshowusawayductionofinflationarydomainswithφ∼

towardsthecompletequantummechanicaldescriptionofthestationarygroundstateoftheUniverse.

A.M.isgratefultoA.LindeandD.LindeforfruitfulandenjoyablecollaborationonthisprojectandtoA.Starobinsky,S.MolchanovandA.Vilenkinfordiscussions.

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