Cosmological models with dynamical lambda in scala

arXiv:gr-qc/9907016v211Nov1999CosmologicalmodelswithdynamicalΛinscalar-tensortheories.L.M.D´ıaz-RiveraandL.O.PimentelDepartmentofPhysics,UniversidadAut´onomaMetropolitana-IztapalapaApdo.Postal55-534,09340M´exico,D.F,M´exico(February7,2008)Inthecontextofafamilyofscalar-tensortheorieswithadynamicalΛ,thatisabinomialonthescalarfield,thecosmologicalequationsareconsidered.Ageneralbarotropicstateequationp=(γ−1)ρ,foraperfectfluidisusedforthemattercontentoftheUniverse.SomeFriedmann-Robertson-Walkerexactsolutionsarefound,theyhavescalefactorwhichshowsexponentialorpowerlawdependenceontime.Forsomemodelsthesingularitycanbeavoided.CosmologicalparametersasΩm,ΩΛ,q0andt0areobtainedandcomparedwithobservationaldata.I.INTRODUCTIONUnificationtheorieshaveanonzerocosmologicalconstantthatisabout120ordersofmagnitudelargerthantheobservedvalueforΛ,thisconstitutesthecosmologicalconstantproblem[1]-[2].Inordertoexplainandsolvesuchaproblem,andtomakecompatibletheactualobservationaldatawiththeinflationaryscenarioandparticlephysicsexpectations,atimedependentcosmologicalconstantwasproposed[3].Thisoldideahasreceivedalotofattention(seee.g.[4]-[43]).Whatpeoplehaveinmindistomakethevacuumenergydynamical.Insuchaway,duringtheevolutionoftheuniverse,theenergydensityofthevacuumdecaysintoparticles,thusleadingtothedecreaseofthecosmologicalconstant,obtainingasaresult,althoughsmall,acreationofparticles.Abroadsummaryofcosmologicalmodelswithtimedependentcosmological“constant”isgivenbyOverduinandCooperstock[44],re-examiningthere,theevolutionofthescalefactorwhenλisgivenasfunctionoft,a(t),H,orq.Afairlygeneralequationofstateisconsideredandnewnumericalsolutionsareobtained,butasinmostofthepreviousworks,thetimedependenceofthecosmologicaltermisintroducedadhoc.Analternativeisaneffectivetimedependentcosmological“constant”inthecontextofascalar-tensortheories,whichbecomesatrueconstantfort≫0[45].UsingJordan-Brans-Dicketheory(JBD)inparticular,the“gracefulexit”problemofoldinflationarycosmologymightbeimproved.ItremainstheproblemofdeterminingtheJBDparameterω,thataccordingtosolarsystemexperimentsitsvalueiskωk≈500,whichhasbeenderivedfromtimingexperimentsusingtheVikingspaceprobe[46].Abetterestimationofthisparametershouldbeobtainedfrommeasureofotherscosmologicalparametersinordertoconstrainωmorestronglythanbymeansofsolarsystemexperiments[47].However,theoriesofveryearlyuniverseasstringtheory,arebetterdescribedinthecontextofJBD,andshowsthatωcantakenegativevalues[48].Thus,scalar-tensortheories,andinparticularJBD,arebettertheories,inordertoget,inanaturalway,atimedependentcosmologicalconstant.Clearly,recentobservationalresultsrestrictthiskindoftheory,e.g.thetypeIasupernova(SNIa)results,whichin1998showthatΩΛ∼0.6[49]implyingthatouruniverseisspeedingup.Thus,amodelwhichattemptstodescribethecosmologicalconstantbehavior,shouldtakeintoaccounttheobservationalevidences.Inarecentwork[50]weinvestigatedtheeffectofatimedependentcosmologicalconstant,inafamilyofscalar-tensortheories.There,wegetcosmologicalmodelsinthecoastingperiod,wherethetimedependenceonthecosmologicalconstantoccursinanaturalway.Insuchamodelsweassumedasimplerelationλ(φ)=cφ(t)n,(withcandnconstants).Theexistenceofinflationaryphaseinscalar-tensortheories(STT)hasbeeninvestigatedbyPimentelandStein-Schabes[51],findinginflationaryphasesforapolynomialcosmologicalconstantinageneralSTT,whichincludesBrans-Dickemodelwithnon-zerocosmologicalconstant.Ontheotherhand,Guendelman[52]hasinvestigatedtherequirementsofthepotentialsinordertohavescaleinvariance.Therewasfoundtheformofthepotentialneededbytheglobalinvariance,whichinaddition,itsenergyintheconformalEinsteinframehasthecharacteristicsforasuitableinflationaryuniverseandΛdecayingscenarioforthelateuniverse.Motivatedbytheseideas,weshallconsiderageneralSTTasinourpreviouswork[50],butnowweshallconsiderabinomialλfunctiononφ(t),inordertoobtainexactsolutionsofthefieldequations,fromwhichweobtainsomekindofinflationarycosmologicalmodelsandrelatedcosmologicalparameters.Infact,weobtaininmostofoursolutions,a1powerlawgrowthforthecosmologicalscalefactora(t)∼tσ,whereσ∼1impliesinflationarymodels.Asitisknown,thisisagenericfeatureofaclassofmodelsthatattemptdynamicallytosolvethecosmologicalconstantproblem.Inourmodelsσisafreeparameter(atleastinmostofourmodels),inordertobeadjustedbyphysicalconditionsandtobeinagreementwithrecentdataforSNIa,thatimpliesσ≈1,andwhichisconsistentwiththenucleosynthesis[53].Mostofoursolutionspredictanacceleratedexpansion,suchsolutionsareinagreementwiththeSNIaresults,butΩmandΩΛ,dependonfreeparametersofourmodel.Insomespecificcaseswegetsolutionswithexponentialgrowthofthescalefactor.InsectionIIweobtainthefieldequationsandintroduceourmainansatz.SectionIIIisdevotedtoobtainanexpressionofthedensityparameterandthecorrespondingcontributionsofthematterandscalarfield.InsectionIVweconsiderthevacuumcaseandobtainasetofexactsolutionsaswellascosmologicalparameters.InsectionVweconsiderthegeneralcaseofabarotropicequationofstateandobtainexactsolutionsforthisgeneralcase.Asexample,wecalculatespecificallythecaseofadustfluidandastiffmat