Inflation and nonequilibrium thermodynamics for the fluctuations in the infrared sector

sector

MauricioBellini∗

DepartamentodeF´ısica,FacultaddeCienciasExactasyNaturales

UniversidadNacionaldeMardelPlata,

Funes3350,(7600)MardelPlata,BuenosAires,Argentina.

IntheframeworkofinflationarycosmologyIstudysomeaspectsofnonequilibriumthermody-namicsforthematterfieldfluctuations.ThethermodynamicanalysisisdevelopedfordeSitterandpower-lawexpansionsoftheuniverse.Inbothcases,Ifindthattheheatcapacityisnegativeleadingrespectively,toexponentialandsuperexponentialgrowthforthenumberofstatesintheinfraredsectorfordeSitterandpower-lawexpansionsoftheuniverse.Thespectrumforthematterfieldfluctuationscanbeunderstoodfromthebackgroundeffectivetemperaturewhenthehorizonentry.

Pacsnumber(s):98.80.Hw,05.70.Ln

Sinceinflationstretchesmicroscopicscalesintoastro-nomicalones,itsuggeststhatthedensityperturbationswhichprovidetheseedsforgalaxyformationmighthaveoriginatedasmicroscopicquantumfluctuations[1,2].Apromisingapproachtowardsabetterunderstandingofthesephenomenaistheparadigmofstochasticinfla-tion.ThemostwidelyacceptedapproachassumethattheinflationaryphaseisdrivingbyaquantumscalarfieldϕwithapotentialV(ϕ).Withinthisperspective,thestochasticinflationproposestodescribethedynam-icsofthisquantumfieldonthebasisofasplittingofϕinahomogeneousandaninhomogeneouscomponents.Usuallythehomogeneousoneisinterpretedasaclassi-calfieldthatarisesfromacoarse-grainedaverageoveravolumelargerthanthehorizonvolume,andplaystheroleofaglobalorderparameter[3].Allinformationonscalessmallerthanthisvolume,suchasthedensityfluctuations,iscontainedintheinhomogeneouscompo-nent.DuringinflationvacuumfluctuationsonscaleslessthantheHubbleradiusaremagnifiedintoclassicalper-turbationsinthescalarfieldsonscaleslargerthantheHubbleradius.Theprimordialperturbationsarisesolelyfromthezero-pointfluctuationsofthequantizedfields.Althoughtheregionwhichultimatelyexpandedtobe-cometheobserveduniversemayhavecontainedexcita-tionsabovethevacuum,theseexcitationswouldnothaveanysignificanteffectonthepresentstateoftheuniversebecauseasufficientlylargeamountoftheinflationwouldhaveredshiftedtheseexcitationstoimmeasurablylongwavelengths.Thezero-pointfluctuations,ontheotherhand,havearbitrarilysmallwavelengths,indeedtheyaremostsignificantatverysmalllengthscales.Thezero-pointfluctuationsareimperceptiblebecauseoftheirshortwavelengths,buttheprocessofinflationcanstrechthesewavelengthstomacroscopicandeventuallytoas-tronomicaldimensions.Hence,thedensityperturbationsshouldberesponsibleforthelargescalestructureforma-tionintheuniverse.

InthispaperIaminterestedinthestudyofthetermo-dynamicalpropertiesofmatterfieldfluctuationsintheinfrared(IR)sector.Somethermodynamicalresearchsweredevelopedrecentlyintheframeworkoftheden-sitytopologyofthespacetimefoam[4]andthequantumstructureofSchwarschildblackholes[5].Duringinflationthissectorvariesthenumberofdegreesoffreedom.Thisisanunstablesectorwhichdescribestheuniverseonascalemuchlargerthantheobservableuniverse.Anaturalconsequenceofthisapproachistheself-reproductionofuniversesandthereturntoaglobalstationarypicture.Forinflationthesimplestassumptionisthattherearetwoscales:along-time,long-scaleassociatedwiththevacuumenergydynamics,andtheshort-time,short-distancescaleassociatedwitharandomforcecomponent.TheHubbletime1/H,separatesthetworegimes.

ThedynamicsofascalarfieldminimallycoupledtoaclassicalgravitationaloneisdescribedbytheLagrangian

L(ϕ,ϕ1

,µ)=−

√

16π

+

a2

∇2φ+3Hcφ

˙+V′′(φc)φ=0,(2)

whereV′′(φc)≡

∂2V(ϕ)

a

istheHub-bleparameterandaisthescalefactoroftheuniverse.

Duringinflation,theuniverseisaccelerateda¨>0andtheinflationendswhena¨∼0.Theequation(2)describesthematterfieldfluctuations(uptononlineartermsinφ),onagloballyhomogeneousandisotropicbackgroundspace-timedescribedbyaFriedmann-Robertson-Walkermet-ric

ds2=−dt2+a2(t)d󰀓x2.

(3)

Theeq.󰀆(2)canbesimplifiedbymeansofthemapφ=

e−3/2Hcdtχ

χ¨−a−2󰀁k2

0(t)−k2󰀃χ=0,(4)where

k2

0(t)=a2

󰀈

9

2

H

˙c−V′′[φc(t)]󰀉

(5)

isthetime-dependentwavenumberthatseparatesthe

infraredandtheultravioletsectors,whichdescribetworelevantsscalesduringinflation.Sincek

˙the

0(t)>0,duringinflationnewandnewmodesentersintheinfraredsectorsuchthatinthissectorthefrequencyωkholds

ω2

k(t)=−a−2󰀁k2

0

(t)−k2Asinpreviousworks[7,8]onecan󰀃<0.(6)writetheredefined

matterfieldfluctuationsintheinfraredsectorbymeansaFourierexpansionwhichselectsthelongwavelengths

modesχk=ei󰀐k.󰀐x

ξk(t)(fork≪k0)

χcg=

1

canbeconsideredasclassicalwhenthemodesofthissectorholdsthecondition󰀂󰀂

timedependent

󰀂Im[ξk(t)]

e−βωk(t)

=

󰀄

(2π)3

ωǫk0dωkρ(ωk)e−βωk,

(8)

ωk=0

whereβ−1playstheroleofthebackground“tempera-ture”andωkistherelevantfrequencygivenbyeq.(6).

Furthermore,Idenotethesquaredfrequencyoffwavenumberǫk0byω2ǫk=−a−2󰀁k2with2adimensionlessparametergivenby0(1−ǫ)󰀃thecut0.Here,ǫisk/k0theframeworkofstochasticinflationthebackground≪1.Inwellrepresentedbytheultravioletsector,whereω2

is

semiclassicallimitthefrequencyωthek>0.Inthekplaysroleoftheenergyforeachmodewithwavenumberk.Weareinterestedintheinfraredsector.Inthissectorthewavenumbersareverysmallwithrespecttok0(k≪k0),andthefrequencyωkisimaginarypure(ωk=±i|ωkAswewillseelater,theparameterβisalsoimaginary|).pureandthustheargumentoftheexponentialineq.(8)remainsreal.Thefunctionρ(ωk)givesthedensityofstateswithfrequencyωkontheinfraredsector

ρ(ω=1k)dωk󰀂󰀂󰀂2=k󰀂󰀂󰀂,(9)where󰀂󰀂󰀂d3k

󰀂dωk󰀂

dk

󰀂󰀂󰀂󰀁k2+a2ω2󰀂=0

k󰀃1/22π2

󰀁k2

0+a2(t)ω2󰀃1/2

k

|ωk|a2(t).(11)

Thethermodynamicsforsystemswithexponentially

growthofdensityofstateswasfirstconsideredbyHage-dornintheframeworkofthehadronmassspectruminbootstrapmodels[12,13].Energyaddedtoasystemcangoeitherintoincreasingtheenergyofexistingstatesorintocreatingnewstates.Inthecaseofthematterfieldfluctuationsintheinfraredsectornewandnewstatesarecreatedfromtheultravioletsector.

The“temperature”andtheheatcapacityaregivenby

β=󰀂

󰀂󰀂󰀂

∂ln[ρ]∂ω2k

󰀉−1󰀂󰀂

󰀂Theconditionthatthedensityof󰀂󰀂

.

(13)

k=ǫk0

statesrisessuperex-ponentiallyispreciselythatthesecondderivativeineq.

(13)bepositive,andCVthusbenegative.Systemswithnegativeheatcapacitiesarethermodynamicallyunsta-ble.Theyareplacedincontactwithaheatbathandwillexperiencerunawayheatingorcooling.Iftheden-sityofstatesgrowsexponentially,aninflowofenergyattheHagedorntemperaturegoesentirelyintoproduc-ingnewstates,leavingthetemperatureconstant.Ifthedensityofstatesgrowssuperexponentially,theprocessissimilar,buttheproductionofnewstatesissocopiousthataninflowofenergyactuallydrivesthetemperaturedown.Tosimplifythenotation,inthefollowingIwilldenoteωǫk0asω.Forinflationarymodelsoneobtainsintheinfraredsector

−µ4󰀋ω2µ2+µ4+2ω4

CV=

󰀇|ω|(µ2+ω2)

.(15)

Inthecasewearestudying,the“thermalbath”isde-scribedbytheultravioletsector,butitisnottrullyther-malized.Intheframeworkofsupercooledinflation,β−1itisnotatrullytemperature.Thus,itisimaginarypure.Furthermore,theparameterβdescribestheenvironmentoftheinfraredsector,herecharacterizedbytheultravi-oletsector.Thequantumnatureofthematterfieldfluc-tuationsintheultravioletsectorisanothermotivationfortheparameterβtobeimaginarypure.

Furthermore,theheatcapacitygivesinformationabouttheevolutionoftheinfraredsector,whichisanunstablesector.IfCV>0,thesystemdistributesitsenergyintheexistentstates.Theinversesituationde-scribesasystemwhichincrementsveryrapidlythenum-berofstates.

ThermodynamicsforadeSitterexpansion:AsafirsthexampleIwillconsiderascalefactora∼eH0t,whereH0istheHubbleparameter.ForadeSitterexpansionthisparameterisconstant.InadeSitterexpansionµ2=

ν2H02,whereν2=9H2.Thedensityofstatesisgiven

by

0

ρ(ωk)≃

1

β≃∓i

11−

ǫ2

≃∓

i

powerdensityperturbationsshouldbeafunctionofthebackgroundtemperature.Inotherwords,ifPφcg(t∗)=|δk|2isthepowerspectrumforthematterfieldfluc-󰀊󰀌󰀆ǫk0dk

tuations,suchthatφ2=cgIR0

2

(ǫ2νH(18)

0)4

(1−

ǫ2)

2

≃−2(ββ∗).

Notethattheheatcapacityisnegativebutconstant.Thismeansthat,asweputenergyintotheinfraredsec-tor,agreaterandgreaterproportionofitisemployedintheexponentialproductionofnewstatesratherthaninincreasingtheenergyofalreadyexistingstates.

Thermodynamicsforapower-lawexpansion:Nowweconsiderthecasewherethescalefactorevolvesasa∼tp.InthiscasetheHubbleparameterisHc=p/tandtheeffectivesquaredmassparameterbecomes[7]

µ2(t)=t−2

󰀈

9

2p+2󰀉.(19)Thedensityofstatesρ(ωk)is

ρ(ωk)≃

|ωk|

9

3

p+2.InflationholdswhenK>

0,i.e.,forp>3.04.Hence,theinverseoftheeffectivetemperatureoftheinfraredsectoris[seeeq.(12)]

β≃∓i

t

it1−ǫ2

≃∓∗(Kǫ2)

4

(1−

ǫ2)

2≃−2(ββ)2

.

(22)

Theexpression(22)forCVbecomesmoreandmoreneg-ativewithtime,duetotheunstabilityoftheinfraredsectorduringinflation.Aswasdemonstratedinapre-viouswork[7],inapower-lawexpansionfortheuni-versetheinflatonpotentialsuppressesthedispersionofthequantumfluctuationsinapower-lawexpansionfortheuniverse.ThisresultcoincideswiththequantumfieldpredictionandcouldberesponsiblefortheveryrapidlydecreasingoftheheatcapacityCV2.

Generalcomments:If(ββ∗)−1/isthezeromodetem-perature(orbackgroundtemperature),thesquaredin-fraredmatterfieldfluctuationswhenthehorizonentrywillbe

󰀊φ2󰀌

ǫ6

cgIR≃

ββ∗ξ0

=0,sothattheamplitudeforprimordial

4

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