Bianchi Type I Cosmologies in Arbitrary Dimensional Dilaton

We study the low energy string effective action with an exponential type dilaton potential and vanishing torsion in a Bianchi type I space-time geometry. In the Einstein and string frames the general solution of the gravitational field equations can be exp

hep-th/0004096

BianchiTypeICosmologiesinArbitraryDimensionalDilatonGravities

Chiang-MeiChen

DepartmentofPhysics,NationalCentralUniversity,Chungli320,Taiwan

T.Harko andM.K.Mak

DepartmentofPhysics,UniversityofHongKong,Pokfulam,HongKong

(April10,2000)

arXiv:hep-th/0004096v1 13 Apr 2000

Westudythelowenergystringe ectiveactionwithanexponentialtypedilatonpotentialandvanishingtorsioninaBianchitypeIspace-timegeometry.IntheEinsteinandstringframesthegeneralsolutionofthegravitational eldequationscanbeexpressedinanexactparametricform.Dependingonthevaluesofsomeparameterstheobtainedcosmologicalmodelscanbegenericallypidedintothreeclasses,leadingtobothsingularandnonsingularbehaviors.Thee ectofthepotentialonthetimeevolutionofthemeananisotropyparameterisalsoconsideredindetail,anditisshownthataBianchitypeIUniverseisotropizesonlyinthepresenceofadilaton eldpotentialoracentralde citcharge.

PACSnumber(s):04.20.Jb,04.65.+e,98.80.-k

I.INTRODUCTION

Inanattempttoaddressthepotential,inheritedfromstringtheory,toeliminatetheinitialcosmologicalsingu-larity,fromwhichtimeandourUniversearesupposedtohavebegunabout15billionyearsago,GasperiniandVenezianoinitiatedaprogramknownasthepre-bigbangscenario[1].The eldequationsofthepre-bigbangcos-mologyarebasedonthelowenergye ectiveactionre-sultingfromstringtheory.InD-dimensions,themasslessbosonic eldsfromtheNS-NSsectorarethedilaton,φ,theantisymmetrictensor,Bµν,andthemetrictensor,g µν,whosedynamicsisdescribed,inthe“stringframe”,bythefollowingaction:

D2 =dx (φ),SH U

12[3]

(1)

whereH[3]=dB[2]andκ isageneralizeddilatoncouplingconstant( κ=4forsuperstringtheories).Moreover,we

(φ)ofthealsoallowfortheexistenceofapotentialU

dilaton eld.FromaphysicalpointofviewthemostimportantcandidateforthepotentialisacosmologicalconstantΛ,whichappearsinthemassiveextensionoftypeIIAsupergravityandisrestrictedtobepositive,Λ>0[4].

ForsimplicityinthefollowingweshallassumethatH[3]isvanishing.Inthiscircumstance,viaaconformalrescaling

gµν=e

4

.withU(φ)=eD 2 κ

Pre-bigbangin ationarycosmologicalmodels,basedontheactions(1)or(3)havebeenrecentlyintensivelyinvestigatedinthephysicalliterature[5–12].GasperiniandRicci[5]haveobtainedexactsolutionstothefour-dimensionallowenergystringe ectiveactionadoptingaspace-independentdilatonandvanishingKalb-Ramondanti-symmetrictensor eldsans¨atzfortheBianchitypeI,II,III,V,VI0andVIhgeometries.Theyhaveshownthatinsuchacontexttheinitialcurvaturesingularitiescannotbeavoided.Brandenberger,EastherandMaia[6]havefoundnon-singularspatiallyhomogeneousandisotropicsolutionsfordilatongravityinthepresenceofaspecialcombinationofhigherderivativetermsinthegravitationalaction.Someofthesesolutionscorrespond

4φ

gR κ( φ)2 U(φ),

(3)

We study the low energy string effective action with an exponential type dilaton potential and vanishing torsion in a Bianchi type I space-time geometry. In the Einstein and string frames the general solution of the gravitational field equations can be exp

toaspatially at,bouncingUniverseoriginatinginadilaton-dominatedcontractingphaseandemergingasanexpandingFRWUniverse.

Veryrecently,thestringcosmologyequationswithadilatonpotentialhavebeenexamined,inthestringframe,byEllisetal.[13],whoalsogiveagenerical-gorithmforobtainingsolutionswithdesiredevolution-aryproperties.Thepresenceofadilatonpotentialleadstotheviolationofthepre-bigbangsymmetrya(t)→1/a(t).Moreover,GarciadeAndrade[14]obtainedsev-eralclassesofsolutionsoftheEinstein-Cartandilatonicin ationarycosmology.Inthecaseswherethedilatonsareconstrainedbythepresenceofspin-torsione ectsarepulsivegravityisfound.Thetemperature uctuationhasalsobeencomputedfromthenearly atspectrumofthegravitationalwavesproducedduringin ation,withresultsagreeingwiththeCOBEdata.

Pre-bigbangcosmologicalmodels,inwhichthereisnoneedtointroducethein ationorto ne-tunepotentials,havemanyattractivefeatures[15].In ationisnatural,thankstothedualitysymmetriesofstringcosmology,andtheinitialconditionproblemisdecoupledfromthesingularityproblem.Finally,quantuminstability(paircreation)isabletoheatupaninitiallycoldUniverseandgenerateastandardhotbigbangwiththeadditionalfea-turesofhomogeneity, atnessandisotropy.

ItisthepurposeofthepresentpapertostudyBianchitypeIcosmologicalmodelsinthedilatongravity(1)and(3).Morespeci cally,weshallconsiderthee ectsofanexponentialtypepotential,U(φ)=U0eλφ,witharbitraryvaluesoftheconstantsU0,λ,onthedynamicsandevo-lutionofananisotropicspace-time,inboththeEinsteinandstringframes.Inthiscasethegeneralsolutionofthegravitational eldequationscanbeexpressedinanexactparametricform.Thephysicale ectsofthepo-tentialontheevolutionoftheanisotropicspace-timearealsoconsideredindetail.

Thepresentpaperisorganizedasfollows.ThebasicequationsdescribingthedilatonicBianchitypeIcosmo-logicalmodelareobtainedinSection2.Thegeneralsolu-tionofthe eldequationsforanexponentialtypedilatonpotentialisobtainedinSection3(Einsteinframe)andinSection4(stringframe).InSection5wediscussourresultsandconclusions.

II.EINSTEINFRAMEFIELDEQUATIONS,

GEOMETRYANDCONSEQUENCES

Inthispaper,weshallconsidertheD-dimensionalanisotropicgeneralizationofthe atFRWgeometry—theBianchitypeIspace-timedescribedbytheline-element

D 1ds2

= dt2

+

a2i(t)(dxi)2

.

(4)

i=1

Forthismetric,itisconvenienttointroducethefollow-ingvariables:volumescalefactorV,directionalHubblefactorsHiandmeanHubblefactorHas

DV:= 1ai,(5)

i=1

Ha˙i:=

i

D 1D 1

Hi,(7)i=1

Hi:=Hi H,

i=1,...,D 1.

(8)

Th …… 此处隐藏：19453字，全部文档内容请下载后查看。喜欢就下载吧 ……