Non-chaotic dynamics in general-relativistic and s

arXiv:gr-qc/0605050v19May2006Non-chaoticdynamicsingeneral-relativisticandscalar-tensorcosmologyV.Faraoni,M.N.JensenandS.A.TheuerkaufPhysicsDepartment,Bishop’sUniversityLennoxville,Qu`ebec,CanadaJ1M1Z7AbstractInthecontextofscalar-tensormodelsofdarkenergyandinflation,thedy-namicsofvacuumscalar-tensorcosmologyareanalysedwithoutspecifyingthecouplingfunctionorthescalarfieldpotential.AconformaltransformationtotheEinsteinframeisusedandthedynamicsofgeneralrelativitywithaminimallycou-pledscalarfieldarederivedforagenericpotential.Itisshownthatthedynamicsarenon-chaotic,thussettlinganexistingdebate.Keywords:scalar-tensorcosmology,darkenergy,inflation.PACS:98.80.-k,04.90.+e,04.50.+h1IntroductionRecentcosmologicalobservationshaveestablishedthattheuniverseisveryclosetobeingspatiallyflat,correspondingtovanishingcurvatureindexKintheFriedmann-Lemaitre-Robertson-Walker(“FLRW”)lineelementds2=−dt2+a2(t)dr21−Kr2+r2dθ2+sin2θdϕ2
(1.1)incomovingcoordinates(t,r,θ,ϕ).WeuseunitsinwhichthespeedoflightandNew-ton’sconstantassumethevalueunity,themetricsignatureis−,+,+,+,andtheothernotationsfollowthoseofRef.[1].IfKwereexactlyzero,itwouldnotbepossibletoestablishitobservationally,duetounavoidableexperimentalerrors.Notwithstandingthis,thecosmicmicrowavebackgroundexperiments[2]measuringatotalenergydensityρoftheuniverseclosetothecriticaldensityρc=3H28πG(whereH≡˙a/aistheHubbleparameterandanoverdotdenotesdifferentiationwithrespecttothecomovingtimet)canberegardedasareassuringvalidationofthetheoreticalpredictionthattheuniversewastakenextremelyclosetotheK=0statebyinflationearlyinitshistory[3].Anothersurprisingdiscovery,obtainedwiththestudyoftypeIasupernovaeathighredshifts[4],isthatthecosmicexpansionisaccelerated.InthecontextofEinstein’sgravitythisfactisexplainedbypostulatingthatthepressurePofthecosmicfluidsatisfiesP−ρ/3.Infact,thebestfittotheobservationaldatarequiresevenmoreexoticpropertiesforthedarkenergy,thefluidthataccountsfor70%oftheenergydensityoftheuniverseinEinstein’stheory;ifw≡P/ρdenotestheeffectiveequationofstateparameterofthedarkenergy,thereismarginalevidencethatw−1[5].ThisrangeofvaluesoftheparameterwcorrespondstoaHubbleparameterthatisincreasingwithtime(superacceleration):˙H0accordingtotheFriedmannequation˙H=−κ2(P+ρ),(1.2)whereκ≡8πGandGisNewton’sconstant.Mostdarkenergymodelsarebasedonscalarfieldsand,iftheuniversereallysuperaccelerates,modelsbasedongeneralrelativitywithacanonical,minimallycoupled,scalarfieldφareunviable.Infact,theenergydensityandpressureofsuchascalarfieldareρ=˙φ22+V(φ),P=˙φ22−V(φ),(1.3)whereV(φ)isthescalarfieldpotential,andeq.(1.2)reducesto˙H=−κ˙φ2/2≤0(thelimitingsituation˙H=0describesdeSittersolutions).Furthermore,thebestfittothe1observationaldatarequiresadynamicalformofdarkenergywithanequationofstateparameterchangingwithredshift,whichwoulddefinitelyexcludethecosmologicalcon-stantasanexplanationfordarkenergy(suchamodelisanywayunappealingbecauseofthecosmologicalconstantproblemandthecosmiccoincidenceproblem[6,7]).Forthisreason,alternativemodelsinwhichtheuniversecansuperacceleratehavebeencon-sidered,includingphantomfieldswithnegativekineticenergy[8],scalarfieldscouplednon-minimallytothecurvature[9,10],oralternativegravitytheories.Inthispaper,weconsiderscalar-tensorextensionsofgeneralrelativity.Thetheoriesinthisclass[11,12]generalizeBransandDicke’s[13]theory,exhibitmanyfeaturesincommonwithstring/Mtheory[14],arethearenaforextended[15]andhyperextended[16]inflationaryscenar-ios,andarewidelyusedincosmology[17,18].Scalar-tensorgravityisusedtomodelinflationintheearlyuniverseordarkenergyinthepresentlatetimeeraoftheuniverse.Bothphantomcosmologyandthetheoryofanon-minimallycoupledscalarfield,capableofproducingthesuperaccelerationphenomenon,canbeseenasscalar-tensortheories.Becauseofthewideuseofscalar-tensorgravity,motivatedbythebeliefthatthelattermaybemorefundamentalthanEinstein’sgravity,onewouldliketohaveaclearandgeneralpictureofscalar-tensorcosmology.Ingeneralrelativitythedynamicsofaparticular(darkenergyorinflationary)scenarioisusuallydeterminedoncethepoten-tialisfixed;inscalar-tensorgravityonealsohasfreecouplingfunctionsaddingextradegreesoffreedom.However,itwouldbedesirabletounderstandthedynamicswithasmuchgeneralityaspossiblewithoutfixingthesefunctions.Thisiswhatwesetouttodointhispaperand,onthebasisoftheobservationaldataandoftheoreticalprejudice,werestrictourselvestoconsideringaspatiallyflat(K=0)universe.Inparticular,wederiveconclusionsaboutthedynamicsbasedongeneralassumptionsaboutthepoten-tial(e.g.,monotonic,etc.).Tothebestofourknowledge,thisclassificationhasnotbeenexploredevenintherelativelysimplecaseofgeneralrelativitywithaminimallycoupledscalarfield,inwhichthepotentialV(φ)istheonlyunknownfunction(e.g.,ininflationaryscenarios).Moreover,thegeometryofthephasespaceisrarelydiscussed,eveninscenariosbasedongeneralrelativitywithspecificchoicesofthepotentialV(φ),forwhichphasespacestudiesexistintheliterature(seeRefs.[19]-[22]forBrans-Dicketheory)—usuallyonlyprojectionsofthephasespaceareconsidered.Theroleofchaosincosmologyhasreceivedmuchattentionsincetheearlyworkona