Gravitational Theory, Galaxy Rotation Curves and C

arXiv:astro-ph/0412195v35May2005GravitationalTheory,GalaxyRotationCurvesandCosmologywithoutDarkMatterJ.W.MoffatThePerimeterInstituteforTheoreticalPhysics,Waterloo,Ontario,N2J2W9,CanadaandDepartmentofPhysics,UniversityofWaterloo,Waterloo,OntarioN2Y2L5,CanadaAbstractEinsteingravitycoupledtoamassiveskewsymmetricfieldFμνλleadstoanaccelerationlawthatmodifiestheNewtonianlawofattractionbetweenparticles.Weuseaframeworkofnon-perturbativerenormalizationgroupequationsaswellasobservationalinputtocharacterizespecialrenormaliza-tiongrouptrajectoriestoallowfortherunningoftheeffectivegravitationalcouplingGandthecouplingoftheskewfieldtomatter.Strongrenormal-izationeffectsoccuratlargeandsmallmomentumscales.ThelatterleadtoanincreaseofNewton’sconstantatlargegalacticandcosmologicaldistances.Forweakfieldsafittotheflatrotationcurvesofgalaxiesisobtainedintermsofthemass(mass-to-lightratioM/L)ofgalaxies.ThefitsassumethatthegalaxiesarenotdominatedbyexoticdarkmatterandthattheeffectivegravitationalconstantGrunswithdistancescale.TheequationsofmotionfortestparticlesyieldpredictionsforthesolarsystemandthebinarypulsarPSR1913+16thatagreewiththeobservations.Thegravitationallensingofclustersofgalaxiescanbeexplainedwithoutexoticdarkmatter.AnFLRWcosmologicalmodelwithaneffectiveG=G(t)runningwithtimecanleadtoconsistentfitstocosmologicaldatawithoutassumingtheexistenceofexoticcolddarkmatter.e-mail:jmoff[e-mail]1IntroductionAnonsymmetricgravitytheory(NGT)hasbeenappliedtoexplainrotationcurvesofgalaxiesandcosmology,withoutinvokingdominantdarkmatterandidentifyingdarkenergywiththecosmologicalconstant[1,2,3,4].Inthefollowing,wedevelopasimplergravitationaltheorybasedonEinstein’sgeneralrelativity(GR)anda1skewsymmetricrankthreetensorfieldFμνλ,formingametric-skew-tensor-gravity(MSTG)theory.Weshallapplythisgravitytheorytoexplaintheflatrotationcurvesofgalaxiesandclusterlensingwithoutpostulatingexoticdarkmatter.TheweakfieldapproximationoftheNGTfieldequationscorrespondstotheequivalentfieldequationsinMSTGand,therefore,tothisorderofapproximationwillyieldthesamepredictionsasMSTG.ThemorecomplicatedNGTwillhavedifferentpossibleconsequencesforstronggravitationalfields.However,inthefollowing,weshallbeconcernedwiththeobservationaldataforthesolarsystem,thebinarypulsarPSR1913+16,galaxies,clustersofgalaxiesandcosmology.TheMSTGtheoryprovidesamoredirectwayofderivingacomparisonwiththedata.Sincenodarkmatterhasbeendetectedsofar,itseemsimperativetoseekapossiblemodifiedgravitationaltheorythatcouldexplainthenowlargeamountofdataongalaxyrotationcurves.AcosmologicalmodelobtainedfromthefieldequationsandarunningoftheeffectivegravitationalcouplingconstantGinthematterepochcouldexplainthegrowthoflargescalestructureformationwithoutinvokingcolddarkmatter.Therunningofthecosmologicalconstantwouldpro-duceaquintessence-likedarkenergythatcouldaccountfortheaccelerationoftheexpansionoftheuniverse[5,6,7].Arenormalizationgroup(RG)framework[8]forMSTGisdevelopedtodescribetherunningoftheeffectivegravitationalcouplingconstantG,andtheeffectivecou-plingconstantγcthatmeasuresthestrengthofthecouplingoftheFμνλfieldtomatter.Amomentumcutoffidentificationk=k(x)associatestheRGscalestopointsinspacetime.Fortheweakfieldstatic,sphericallysymmetricsolutiontheRGflowequationsallowarunningwithmomentumkanddistanceℓ=1/kfortheeffectiveNewtoniancouplingconstantG=G(r),thecouplingconstantγc=γc(r),andtheeffectivemassoftheskewfieldμ=μ(r)whererdenotestheradialcoordi-nate.TheformofG(r)asafunctionofr,obtainedfromthemodifiedNewtonianaccelerationlaw,leadstoagreementwithsolarsystemobservations,terrestrialgrav-itationalexperimentsandthebinarypulsarPSR1913+16observations,whilestrongrenormalizationeffectsintheinfraredregimeatlargedistancesleadtofitstogalaxyrotationcurves.AfittobothlowsurfacebrightnessandhighsurfacebrightnessgalaxiesisachievedintermsofthetotalgalaxymassM(orM/L)withoutexoticdarkmatter.AsatisfactoryfitisachievedtotherotationalvelocitydatagenerictotheellipticalgalaxyNGC3379.FitstothedataofthetwospheroidaldwarfgalaxiesFornaxandDracoandtheglobularclusterωCentauriarealsoobtained.Thepredictedlightbendingandlensingcanleadtoagreementwithgalaxyclusterlensingobservations.Amodelofthemodifiedaccelerationlawthatincludesadescriptionofradialve-locitycurvesinthecoreofgalaxiesaswellasintheouterregionsofthegalaxyisshowntoyieldgoodfitstorotationalvelocitydata.Forahomogeneousandisotropicuniverse,basedontheFriedmann-Lemaitre-Robertson-Walker(FLRW)model,theaveragevalueoftheskewsymmetricfieldFμνλiszero.However,foraneffectivegravitationalconstantG=G(t)thatruns2withtime,wefindthattheconstraintG(t)∼G0(whereG0isNewton’sconstant,G0=6.673×10−8g−1cm3sec−2)mustbeimposedonG(t)atthetimeofbigbangnucleosynthesis(BBN)andthesurfaceoflastscattering(decoupling)whenthefractionofbaryonsisΩB∼0.04.ForarapidincreaseofG(t)afterthetimeofdecouplingwhenG(t)reachesavalueG(t)∼6G0,thebaryonicmatterfractionbecomesΩM∼6ΩB∼0.24andthereisnoneedforundetectednon-baryonicdarkmatter12TheFieldEquationsTheactionfortheMSTGtheoryisgivenbyS=SG+SF+SFM+SM,(1)whereSGistheEinstein-Hilbertaction(wechooseunitssuchthatc=