Boson Stars in General Scalar-Tensor Gravitation E

BosonStarsinGeneralScalar-TensorGravitation:EquilibriumCongurationsDiegoF.TorresDepartamentodeFsica,UniversidadNacionaldeLaPlata,C.C.67,C.P.1900,LaPlata,BuenosAires,ArgentinaAbstractWestudyequilibriumcongurationsofbosonstarsintheframeworkofgeneralscalar-tensortheoriesofgravitation.Weanalyseseveralpossiblecou-plings,withacceptableweakeldlimitand,whenknown,nucleosynthesisbounds,inordertoworkinthecosmologicallymorerealisticcasesofthiskindoftheories.Wefoundthatforgeneralscalar-tensorgravitation,therangeofmassesbosonstarsmighthaveiscomparablewiththegeneralrel-ativisticcase.Wealsoanalysethepossibleformationofbosonstarsalongdierenterasofcosmicevolution,allowingfortheeectivegravitationalcon-stantfaroutformthestartodeviatefromitscurrentvalue.Inthesecases,wefoundthatthebosonstarsmassesaresensitivetothiskindofvariations,withinatypicalfewpercent.Wealsostudycasesinwhichthecouplingisimplicitlydened,throughthedependenceontheradialcoordinate,allowingittohavesignicantvariationsintheradiusofthestructure.PACSnumber(s):04.50.+h,04.40Dg,95.35.+dElectronicaddress:[e-mail],stellarstructuresrstproposedbyRuniandBonazzola[1],aregravita-tionallyboundmacroscopicquantumstatesmadeupofscalarbosons.Theydierfromneutronstars,theirfermioniccounterparts,inthattheirpressuresupportderivesfromtheuncertaintyrelationratherthanPauli’sexclusionprinciple.Althoughtheseminalworkofreference[1]waspublishedin1969,itwasfolloweduponlyinthepastdecade.Intheserecentsyears,cosmologyhasbeenrefurnishedwiththeintroductionofseveralideascon-cerningthecriticalrolescalareldmaywouldhaveintheevolutionoftheuniverse.Thisrevivedthepossibilityofconstructingstellarobjectsmadeupofthesescalarsinsteadofconventionalfermions.Consideringbosonsasdescribedbyanon-interacting,massive,complexscalareld,RuniandBonazzolasolvedtheequationsofmotiongivenbytheEinsteineldequationsandtheKlein-Gordonequation.Thegeneralsettingtheyusedisthesameastheoneexplainedbelow.TheyfoundthatthemassesofsuchbosonstarswereoforderM’M2pl=m,whereMplisthePlanckmassandmisthebosonmass.Thismodelservedtoopenthepossibilitythatbosonstarsmightindeedexistinnature,althoughtheirmassesweresmallenoughastodiscardthemasaviablesolutiontothedarkmatterproblem.Theorderofmagnitudeofthesebosonstarscoincideswithsimplecomputations.ForaquantumstateconnedintoaregionofradiusR,andwithunitsgivenbyh=c=1,thebosonmomemtumisp=1=R.Ifthestarismoderatelyrelativisticp’m,thenR’1=m.IfweequateRwiththeScharszchildradius2M2plM(recallthatG=M2pl)wegetM’M2pl=m.LaterworkmadebyColpi,ShapiroandWasserman[2]introducedaself-interactiontermforthescalareld.Withthisaddendum,stellarequilibriumcongurationshadmassesoforderM3pl=m2,whichareofthesameorderthantheChandrasekharmass(’1solarmasses).Thestabilityoftheseobjectshavealsobeenanalysedwithsimilarresultstothoseoftheneutroniccase[3].TakingRuniandBonazzolaandColpiet.al.worksasstartingpoints,severalextensionshavebeenproposed.JetzerandvanderBij[4]consideredtheinclusionofaU(1)gauge2chargeandJetzer[5]studiedthenitsstabilityproperties.Non-minimallycouplingsforthescalarshavealsobeenanalysedin[6].Theseandotherrelatedmodelsarereviewedin[7].Inaddition,providedmanyofthisobjectsmaywouldhaveaprimordialorigin,whilebeingformedinagasofbosonandfermions,itisexpectedthatboson-fermionsstarsmightexist.ThiswasstudiedbyHenriqueset.al.[8]withoutinteractionand,recently,bydeSousaet.al.withcurrent-currenttypeinteraction[9].Finally,thepossibleunderstandingofgalactichalopropertiesbymeansofbosonstarsmodelshavealsobeenproposed[10].Bosonstarssolutionshavebeen,however,scarcelyanalysedintheframeworkofalterna-tivetheoriesofgravitation.Weareparticularlyinterestedinscalar-tensorgravity,inwhichtheeectivegravitationalconstantisaeldvariable[11].Historically,mostinteresthasbeengiventoBrans-Dicke(BD)gravity,inwhichthecouplingfunction!(),freefunctionthesetheorieshave,isconstant.Toensurethattheweakeldlimitofthistheoryagreeswithcurrentobservations,!hastobebigenough[12].Butingeneral,when!varies,weneedthat!!1and!3d!=d!0whent!1,toallowtheweakeldlimitofscalar-tensorgravitationaccordwellwithgeneralrelativity(GR)testedpredictions.Afterwards,soonwasrealizedthatthesegeneralscalar-tensortheorieswouldadmitsignicantdeviationsfromGRinthepast[11]andthattheycouldbeausefultoolintheunderstandingofearlyuniversemodels.Theinterestonthemwasrecentlyrekindledbyinationaryscenarios[13]andfundamentaltheoriesthatseektoincorporategravitywithotherforcesofnature[14].Ingeneral,almostallstudiesmadeonscalar-tensorgravitationfocusinthecosmologicalmodelstheylead.Thisisinordertoputseveralconstraintsuponthecouplingfunction.Observationalbounds,mainlycomingfromweakeldtests[12]andnucleosynthesis[15{18]aremorerestrictiveifexactanalyticalsolutionsareknownforthecosmologicalequations.Afewyearsago,Barrow[19],BarrowandMimoso[20]andMimosoandWands[21]derivedalgebraicnumericalmethodsthatallowFriedmann-Robertson-Walker(FRW)solutionstobefoundinmodelswithmattercontentintheformofabarotropicuidforanykindofcoupling!().Someofthesemethod