Quantum fluctuations of Cosmological Perturbations in Generalized Gravity

Jai-chanHwang

DepartmentofAstronomyandAtmosphericSciences,KyungpookNationalUniversity,Taegu,Korea

(February7,2008)Recently,wepresentedaunifiedwayofanalysingclassicalcosmologicalperturbationingeneralizedgravitytheories.Inthispaper,wederivetheperturbationspectrumsgeneratedfromquantumfluctuationsagaininunifiedforms.Weconsiderasituationwhereanacceleratedexpansionphaseoftheearlyuniverseisrealizedinaparticulargenericphaseofthegeneralizedgravity.Wetaketheperturbativesemiclassicalapproximationwhichtreatstheperturbedpartsofthemetricandmatterfieldsasquantummechanicaloperators.Ourgenericresultsincludetheconventionalpower-lawandexponentialinflationsinEinstein’sgravityasspecialcases.

arXiv:gr-qc/9607059v1 24 Jul 1996PACSnumbers:03.65.Sq,04.50.+h,04.62.+v,98.80.Cq

Thehighlyisotropiccosmicmicrowavebackgroundra-diationpermitsalinearparadigmoftreatingthestruc-tureformationprocessintheevolvinguniverse.Itiswidelyacceptedthatthecurrentlyobservablelargescalestructuresaredevelopedfromremnantsofthequantumfluctuationsimprintedduringearlyacceleratedexpan-sionstage.InthecontextofEinstein’sgravityboththequantumgenerationandclassicalevolutionprocessescanberigorouslyhandled;studiesin[1,2]canbecomparedwiththepreviousorderofmagnitudeanalyses[3],oranalysesindifferntgauges[4].Forascalarfield,selfcon-sistentandrigorousanalysesarepossiblemainlyduetoaspecialroleofparticulargaugecondition(orequivalently,gaugeinvariantcombination)whichsuitstheproblem:theuniform-curvaturegauge.

Varietyoftheoreticalreasonsalludepossiblegeneral-izationofthegravitysectorduetoquantumcorrectioninthehighenergylimit,andthusintheearlyuniverse[5,6].Thereweremanystudiesontheclassicalevolutionofstructuresinsomefavoredgeneralizedgravitytheories[7].However,again,intheclassesofgeneralizedgravityinvolvingthescalarfieldandscalarcurvature,wefoundthattheuniform-curvaturegaugesuitstheproblemal-lowingsimpleandunifiedtreatmentpossible[8].Forthegrowingmode,thelargescalesolutionknowninthemin-imallycoupledscalarfieldremainsvalideveninawideclassofgeneralizedgravity.Inthispaperweinvestigatethequantumgenerationprocessinthecontextofgeneral-izedgravity.Theproperlychosengaugeconditionagainallowsustopresentthegeneratedpowerspectrumingenericformswhichareapplicabletovariousgeneralizedgravitytheoriesandunderlyingbackgroundevolutions.Weconsiderthegravitytheorieswithanaction󰀆

√1

S=d4xf(φ,R)−

2

H

ϕ≡−

˙φ

2

󰀆

˙2−a3Zδφϕ

Ha3Z

󰀎

1

󰀁·󰀄·

󰀑

H

3

δφ2ϕdtdx.

(5)

Thenon-Einsteinnatureofthetheoryispresentina

parameterZwhichisdefinedas1

Z(t)≡ω+3F˙2󰀇

1+

F

˙δφ

˙ϕ−

󰀎a3Z

1

Ha3Z

H󰀁·󰀄·󰀑

δφϕ=0.

(7)

Inthelargescalelimit,ignoringtheLaplacianterm,we

haveageneralintegralformsolutionδφH2

ϕ(x,t)=−

φ

˙a3Z

φ

˙δφϕ,z(t)≡

aφ

˙Z,(9)

Eq.(7)canbewrittenasv′′−󰀊

∇2

z′′+

a

√√

H

ϕ.ˆ(12)

Anoverhatindicatesthequantumoperator.Theback-groundorderquantitiesareconsideredasclassicalvari-ables.Thisapproachhasadifferentspiritcomparedwith

thequantumfieldtheoryincurvedspacetime,whereinthelattercasethemetricsectorisregardedasclassicalandgiven(sometimesconsideringsomeprescribedback-reaction)[6,12].Ourapproachconsidersthefieldandthemetricinequalfooting[11].Sinceweareconsideringflatthree-spacebackgroundwemayexpandδφ

ˆa

(x,t)inthefollowingmodeexpansion

δφˆϕ(x,t)=

󰀆

d3ka3Z

δ3(x−x′).

(15)

InorderforEq.(14)tobeinaccordwithEq.(15),themodefunctionδφϕk(t)shouldfollowtheWronskiancondition

δφ˙∗−δφ∗δφi

ϕkδφϕkϕk

˙ϕk=π|η|

√

n+

1

adiabaticvacuum(indeSitterspaceitisoftencalledasBunch-Daviesvacuum,[13])choosesc2(k)≡1andc1(k)≡0whichcorrespondstothepositivefrequencysolutionintheMinkowskispacelimit.ThepowerspectrumbecomesPδφˆϕ(k,t)≡

k3

|δφϕk(t)|2

2π2

,

(20)

whereweusedaadiabaticvacuum,k|vacthe󰀐two-point≡0foreveryfunctionk.becomesAssumingtheG(x′

,x′′

)≡󰀏δφˆϕ(x′)δφˆϕ(x′′)󰀐(1vac=

×F󰀊16πa′a′′η′η′′

32

−ν;2;1+

∆η2−∆x2

√2;

x≡(x,t),∆η2∆x2≡(x′−x′′)2.

≡(η′−η′′)2and

Inthesmallscalelimit,thuskη≫1,(18)becomes

δφ1ϕk(η)=

2k

󰀌

c(k)eikη−i(ν+112+c2(k)e

−ikη+i(ν+12

Inthelarge-scalelimitwehave

󰀐1

Z

.(22)δφϕk(η)=i󰀅a2√2󰀏−ν󰀌c2(k)−c1(k)

󰀐1

Z

,(23)andthepowerspectrumbecomes

P1/2

Γ(ν)δφˆϕ

(k,η)=2󰀏3/2−ν󰀂󰀂󰀂c2(k)−c1(k)󰀂󰀂󰀂1Z

.(24)

InEqs.(18-24)noadditionaldependenceonkarises

fromthegeneralizednatureofthetheory.

LetusseetheimplicationoftheconditioninEq.(17).Introducethefollowingnotations

ǫH˙1≡Hφ˙,ǫ1

3≡

HF,ǫ14≡

HE

,

E≡F󰀓

ω+

3F

˙21aH

1

z

=

n

η2

3(1+w),

wehave

ǫ1=−1/qanda∝η2/(1+3w).InthelimitofEinstein’sgravity,thusZ=1,thesolutionsinEqs.(18-24)reducetotheonesderivedin§IIIof[1],[13,14].Forapower-lawexpansionstagesupportedbythescalarfieldgravity,wehaveǫ3=0=ǫ4andφ/H˙inEinstein’s

=constant.Thus,

ǫ1=ǫ2=−1/qandEqs.(18,27)leadto

ν=

3q−1

2(3w+1)

.(28)

w<−1

2;

inthis

casewehaveη=−1/(aH)whereHbecomesaconstant.Now,wederivesomeobservationallyrelevantclassicalpowerspectrumsgeneratedfromquantumfluctuationsastheinitialseeds.Ignoringthetransientmode,fromEq.(8)wehave

P1/2

C(k,t)=

H

2π2

Wehaveinmind󰀆

󰀏f(x+r,t)f(x,t)󰀐xe−ik·rd3r.

(30)

agenericscenariowheretheclassicalstructuresarisefromthequantumfluctuationspushedoutsidehorizonandclassicalizedduringacceleratedex-pansionstage.Asanansatzweidentify

Pδφϕ(k,t)=Q(k,t)×Pδφˆϕ(k,t),

(31)

wherePδφandPandthequantumδφˆarebasedontheclassicalvolumeaveragevacuumexpectationvalue,re-spectively,Eqs.(30,20).Q(k,t)isafactorwhichmaytakeintoaccountofthepossiblemodificationofthespec-trumduetotheclassicalizationprocessofthequantumfieldfluctuations.Wemaycallita“classicalizationfac-tor”.Ordinarilyitistakentobeunity,however,thedecoherence,noiseandnonlinearfieldeffectsmayaffectitsvalue,particularlyitsamplitude,[15].AssumingEq.(17)wehavederivedthequantumfluctuationsinthelargescalelimitinEq.(24).Thus,combiningEqs.(29,31,24)wehave

P1/2

=

H󰀊

k|η|C(k,η)π3/2a|η|

Q(k)

wherequantitiesintherighthandsideshouldbeevalu-atedwhenthescaleweareconsideringwasinthelargescalelimit(LS)duringanexpansionstagesupportedbyageneralizedgravitytheory(GGT).IntheEinsteingrav-itylimit(Z=1)andanexponentialexpansion(EXP)stage[ν=3

H2π

2

Wenotethatingeneralthepowerspectrumdependsonthechoiceofavacuumstateandpossiblyontheclassi-calizationfactorQ(k).

Now,inEq.(32)wehavethe“quantumfluctuationgeneratedpowerspectrum”imprintedinaconservedquantityC(x).FromthepowerspectrumofC(x)wecanderivethespectrumsofobservablequantitiesinthepresentuniverse,e.g.,densityfluctuations,velocityfluc-tuations,potentialfluctuations,andtemperaturefluctu-ationsinthecosmicmicrowavebackgroundradiationinthematterdominatederainEinstein’sgravity;theseare[8]

δ̺

5󰀊5k

5

δT

5C.

󰀊k

(34)

󰀂󰀂󰀂Q(k)󰀂

󰀂

theconditioninEq.(17).Constructingspecificmodelswithapplicationswillbeconsideredelsewhere.

WethankDr.H.Nohforusefuldiscussions.ThisworkwassupportedinpartbytheKoreaScienceandEn-gineeringFoundation,GrantsNo.95-0702-04-01-3andNo.961-0203-013-1,andthroughtheSRCprogramofSNU-CTP.

.(33)

LS,EXP

C,

Noticethatweareconsideringalineartheory.Inthelineartheory,allperturbedorderquantitiesarelinearlyrelatedwitheachotherwhichistrueevenbetweenvari-ablesindifferentgauges.C(x)isatemporallyconstant,butspatiallyvarying,coefficientofthegrowingmode.Thespatialcurvature(orpotential)fluctuationintheuniform-fieldgauge(whichcoincideswiththecomovinggauge[CG]inaminimallycoupledscalarfield,seeSec.IVCof[9])isconservedasC(x);i.e.,ϕδφ(x,t)=C(x)=ϕCG(x,t).C(x)encodesthespatialstructureofthefluc-tuationsandisconservedduringthelinearevolutioninthelargescalelimit.Indeed,thislinearityisonebasicunderlyingreasonwhyweweresuccessfulintracingthestructureevolutioninasimpleandunifiedway.How-ever,apparently,therearisesnostructureformationinthelineartheory[structuresarepreservedinC(x)],andoneshouldnotmissthatthegravitytheoriesarehighlynonlinear.

Inthispaperwehavenotusedtheconformaltransfor-mationwhichrelatesthegeneralizedgravityinEq.(1)toEinstein’sgravityinclassicallevel,[10].Quantumfluctu-ationsarederivedintheoriginalframeofthegeneralizedgravity.Still,theunderlyingconformalsymmetrywithEinstein’sgravitycanberegardedasanimportantfactorwhichallowstheunifiedandsimpleanalysespossibleintheclassicallevel,[10].

Wehaveimplictlyassumedtheexistenceofanacce-larationstagesupportedbythegeneralizedgravitywith

37,2099(1988);V.Sahni,Class.Quant.Grav.5,L113(1988).

[15]E.CalzettaandB.L.Hu,Phys.Rev.D52,6770(1995).

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