tracelessmatterfield
arXiv:gr-qc/0204014v1 3 Apr 2002A.Bhadra∗
HighEnergyandCosmicRayResearchCentreUniversityofNorthBengal,Siliguri734430INDIA
Abstract
ω(φ)→∞limitofscalartensortheoriesarestudiedfortracelessmattersource.Itisshownthatthelimitω(φ)→∞doesnotreduceascalartensortheorytoGR.AnexactradiationsolutionofscalartensorcosmologyundermodifiedNordtvedtconditionsisobtainedforflatFriedmannuniverse.
PACSnumbers:04.50.+h
I.Introduction
RecentlyDamourandNordtvedt[1]hasbeendemonstratedquitegenerallythatscalarten-sor(ST)theoriesgenericallycontainanaturalattractormechanismtowardgeneralrelativity(GR).Suchapossibilitywasalsosuggestedpreviously[2,3]butonlyforsomeparticularclassesofmodels.ThismeansSTtheoriesarecosmologicallyevolvedtowardastatewithnoscalaradmixturetogravityduringmatterdominatedera[1]orevenduringradiationera[3,4,5].Thisattractorfeatureisimportantbecauseitdiminishesthedoubtontheexistenceofmass-lessscalarfieldwithgravitational-strengthcouplingbyprovidingreasonableexplanationforthesmalleffectofthescalarfields(incomparisonwithcurvatureeffect)inthepresentepochwithoutrequiringanyfine-tunedlargevalueofthecouplingparameterasneededinthecaseofcelebratedBrans-Dicke(BD)theory[6].Itisworthmentioningthatascalarpartnerto
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gravityisinevitableinmosteffortsatunifyinggravitywiththeotherinteractions,suchassu-perstringtheories[7]orKaluza-Klientheories[8].Incosmologyalso,thescalarfieldisfoundtoplayanappreciablerolebyprovidinganaturalwaytoovercomethefinetuningproblemof”new”inflationarymodelofuniversebyterminatingtheinflationaryerathroughnucleationofbubbles[9].Thepresentaccelerationoftheuniverse,asrevealedfromtheobservationaldataofthetype1asupernova[10],alsocanbeaccommodatewiththescalarfieldcoupledgravitytheorieswithouttheneedofanynewkindofmatterfield(socalled”Quintessencematter”)havingapositiveenergydensitybutanegativepressure[11,12].
TheattractormechanismisbasedontheideathataSTtheoryconvergestoGRforcertainvaluesofthecharacteristiccouplingparameterω(φ),whichrepresentsthestrengthofthecou-plingbetweenthescalarfieldandthecurvature.Duringcosmologicalevolution,theEinstein
˜)tendstozero[1]orequivalentlyinJordanframeω(φ)→∞framecouplingparameterα(φ
1˜)≡(α2(φ
Jordanframe,weshallworkinthisframethroughoutthepaper),thegeneralformoftheactiondescribingamasslessscalar-tensortheoryofgravityinnaturalunits(G=c=1)is[17,18]
A=
1
−gd4xφR−
ω(φ)
2
gµνR=
8π
φ28π
φ,µφ,ν−
1
φ
(φ,µ;ν−gµν2φ),
(2)
2φ=
2ω(φ)+3
gµνφ,µφ,ν;
(3)
withtheenergymomentumconservationequation
T;µνν=0,
µ
where2≡gµν∇µ∇ν,T=Tµisthetraceofthematterenergymomentumtensorandω′≡
(4)
dω
1−kr2
+r2dΩ2;
(5)
curvatureconstantk=−1,0,+1foranopen,flatorcloseduniverserespectively.Theconditionofhomogeneityimpliesthatthescalarfieldisonlyafunctionofthetimecoordinate,φ≡φ(t).Fortheabovelineelement,theEq.(4)reducesto
ρ˙+3
a˙
whereanoverdotdenotesd/dt.ForaradiationuniversewithequationofstateP=
1
8π
(2)and(3)thusyield
a˙2
φa4
+a˙2+k
φa4−k
Γa−4
(7)
whereΓisapositiveconstant.Γ=0correspondstovacuum(ρ=P=0).Thefieldequations
a˙φ−ω(φ)
φ2
6−2
˙φa−¨φ˙2φ
a
a
˙2ω′(φ)φ˙φ=−
φ
.
III.Non-convergenceofSTtheoriestoGR
Thefirstintegraloftheequation(10)immediatelygives,
˙=φ
A(3+2ω(φ))
(11)
whereAisaconstant.TheaboveequationistrueforanyradiationorvacuumcosmologicalFriedmannsolutionandisvalidforallSTtheories.ItistobenotedthatsinceAisindependentoftime,itcannotbeafunctionofω(φ)butforBrans-Dicketheory(ωisconstant)Amay˙fromEq.(11)intoEqs.(8)and(9)wegetunderthelimitω(φ)→∞dependonω.Insertingφ
a˙2+k
φa4
+
A2
a
+
a˙2+k
φa4
−
A2
ω2(φ)
).(13)
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andtheRicciscalarinthesamelimitisgivenby
R≃−
1
4ω2(φ)
−1
a6
,
(14)
Whenascalar-tensortheoryconvergestoGR,thescalarcurvaturemustapproachtozeroandatthesametime,thescalarfieldmustbecomeaconstantaswell.ItisevidentfromEq.(11)thatasω(φ)→∞,φistendingtoaconstantvalue(1).AshouldnotvanishinthislimitsinceAisindependentofω(φ).a(t)mustbeboundedforanyrealistictheoryofgravity.ThenitfollowsfromEq.(14)thatRisnotapproachingitsGRvalue(0)forlargevalueofω(φ)orevenfortheNordtvedtconditions[17]viz.ω(φ)→∞,
ω′(φ)
ω2(φ)
→0(15)
Butevenforsuchmodifiedconditions,Rremainsnon-zeroandthefieldequationsareclearlydifferentfromthecorrespondingGRequationswiththesameenergymomentumtensor.ThisshowsthatGRcannotberecoveredfromSTtheoriesbyimposingconstraintsonthecouplingparameterω(φ).
IV.Radiationsolution
Toobtainsolutionofthesimultaneousequations(12)and(13)undermodifiedNordtvedtconditions,wefirstwritetheseequationsinthesynchronousgauge
a+ka=Γ+
′2
2
A2
4a2
5
(17)
whereprimedenotesd/dηandtheconformaltimeηisdefinedthrough
adη=dt
Forflatuniverse(k=0),solutionof(16)and(17)is
a(η)=
(18)
A
η+Γη23
1/2
;(19)
andthecosmictime(t)isrelatedwiththeconformaltime(η)viat=
adη=
ηA+3Γ
√
12Γ3/2
Log2
3√
φ0
)−α;α>0,B1>0constants.ThetheoryreducestoBarker’s“constant”Gtheory
[20]forα=1andB1=−1/2andtoBrans-Dicketheory[6]forα=0.Thetheoryhasbeenstudiedextensivelyin[3,5,19].Theexactradiationsolutionsforα=1isgivenby[19]
φ(η)=
4φ0Kληλ(η+2η0)λ
φ(η)
(22)
whereλ=(3/2B1)1/2,andKandη0areconstants.Asω(φ)→∞,scalarfieldapproachesφ0andtheexpressionforscalefactor(22)indeedconvergesto(19).ItisclearfromEqs.(19)and(20)thatforlargeη,t→large,thescalefactordiffersfromthatofGR.
√2
η2anda∝t1/2i.e.,thesolution
approachestheusualradiationdominatedFriedmannmodelofGR.Butwhentisnotvery
V.Conclusion
Theevolutionofscalarfieldsuggests[3,4,5]thatthecouplingfunctionω(φ)mayevolvetoalargevalueduringradiationera.Someauthorshavediscussedtheeffectofaninflationary
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phaseinpushingω(φ)towardstheextremum[3].Consequently,thereisastandardbeliefthatSTtheoriesareindistinguishablefromGRatlateradiationepochandduringmatterdominatedera.Wehaveshownthatthelimitω(φ)→∞doesnotreduceaSTtheorytoGRfortrace-freestressenergy.ButthisisnotsurprisingasthePPNparametersdescribethepossibledeviationsfromGRinthelocalinteractionofmassivebodiesandtheNordtvedtconditionsreduceSTtheoriestoGRonlyintheweakfieldlimit.ItisalreadyknownthattheBrans-DicketheoryalsodoesnotalwaysreducetoGRwhenT=0.ButinthatcasethedependenceoftheBDscalarfieldφonthecouplingconstantessentiallyremainsarbitrary[16]andtheasymptoticbehaviorofφshouldbefixedrespectingMachiannatureoftheBDthe-ory(whichisthemotivationforthedevelopmentofthetheory)whichimmediatelysuggestsφ∼φ0+O
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