Cosmology with Modified Newtonian Dynamics (MOND)

arXiv:astro-ph/9710335v129Oct1997CosmologywithModifiedNewtonianDynamics(MOND)R.H.SandersKapteynAstronomicalInstitute,Groningen,TheNetherlandsABSTRACTItiswell-knownthattheapplicationofNewtoniandynamicstoanexpandingsphericalregionleadstothecorrectrelativisticexpression(theFriedmannequation)fortheevolutionofthecosmicscalefactor.Here,thecosmologicalimplicationsofMilgrom’smodifiedNewtoniandynamics(MOND)areconsideredbymeansofasimilarprocedure.EarlierworkbyFeltendemonstratedthatinaregiondominatedbymodifieddynamics,theexpansioncannotbeuniform(separationscannotbeexpressedintermsofascalefactor)andthatanysuchregionwilleventuallyre-collapseregardlessoftheinitialexpansionvelocityandmeandensity.HereIshowthat,becauseoftheaccelerationthresholdfortheMONDphenomenology,aregiondominatedbyMONDwillhaveafinitesizewhich,intheearlyUniverse(z3),issmallerthanthehorizonscale.Therefore,uniformexpansionandhomogeneityonhorizonscaleareconsistentwithMOND-dominatednon-uniformexpansionandthedevelopmentofinhomogeneitiesonsmallerscale.Intheradiation-dominatedera,theamplitudeofMOND-inducedinhomogeneitiesismuchsmallerthanthatimpliedbyobservationsofthecosmicbackgroundradiation,andthethermalanddynamicalhistoryoftheUniverseisidenticaltothatofthestandardBigBang.Inparticular,thestandardresultsforprimordialnucleosynthesisareretained.WhenmatterfirstdominatestheenergydensityoftheUniverse,thecosmologydivergesfromthatofthestandardmodel.Objectsofgalaxymassarethefirstvirializedobjectstoform(byz=10)andlargerstructuredevelopsrapidly.Atpresent,theUniversewouldbeinhomogeneousouttoasubstantialfractionoftheHubbleradius.1.IntroductionThemodifiedNewtoniandynamics(MOND)proposesthatthelawofinertiaorgravitytakesonaspecificnon-Newtonianformataccelerationswellbelowadefiniteuniversalvalue(Milgrom1983a,b,c).Asanalternativetodarkmatter,thesimpleMONDformulawithonenewfixedparameter(thecriticalaccelerationao)hasbeenquitesuccessfulinpredicting–2–theformofspiralgalaxyrotationcurvesfromtheobserveddistributionofdetectablematter(Begemanetal.1990,Sanders1996)aswellasthemagnitudeoftheconventionalmassdiscrepancyingalaxyclusters(Milgrom1983c)andinsuperclusters(Milgrom1997).MONDsubsumestheglobalscalingrelationshipsforgalaxies–theTully-FisherrelationforspiralsandtheFaber-Jacksonrelationforellipticals,aswellasanequivalentgastemperature-massrelationforclustersofgalaxies(Sanders1994).MONDstabilizesrotationallysupportedthindisks(Brada1996),explainsthepresenceofamaximumcriticalsurfacedensityinspiralgalaxiesandellipticals(Milgrom1983b),andpredictstheobservedlargeconventionalmassdiscrepancyinlow-surfacebrightnesssystems(Milgrom1983b,McGaugh&deBlok1997).However,anargumentoftendirectedagainstMONDisthat,asatheory,itisadhocandincomplete.Inparticular,MONDmakesnopredictionswithrespecttocosmologyandcosmogony.EventhoughthenearcoincidenceoftheempiricallydeterminedaowithcHoissuggestiveofacosmologicalbasisforMOND,thestructureofthatcosmologyisnotatallevident.ThenaiveexpectationisthatahypothesiswhichpositssucharadicaldeparturefromNewtoniandynamics(andpresumablyGeneralRelativity)onthescaleofgalaxiesandclustersofgalaxieswouldsurelyleadtoahighlyunconventionalcosmologywhichwouldbeinconsistentwiththephenomenologicalsuccessesofthestandardBigBang–mostnotablythenucleosynthesisofthelightelementsintheirobservedabundancesandthelargeandsmallscaleisotropyoftheUniverseattheepochofrecombination.Suchcriticismcannotbeaddressedbyanincompletetheory.ThereasonforthisincompletenessisthatMONDatthepresenttimelacksarelativisticextention;thereisnocrediblegeneraltheoryofgravitywhichpredictstheMONDphenomenologyintheappropriatelimit.Infact,non-standardscalar-tensortheorieshavebeenproposedasatheoreticalunderpinningofMOND(BekensteinandMilgrom1984,Bekenstein1988,Romatka1992,Sanders1997).Twoofthese,phase-couplinggravity(Bekenstein1988)andstratified(preferredframe)theorieswithaquadraticscalarfieldLagrangians(Bekenstein&Milgrom1984,Sanders1997)doyieldsensiblecosmologies–isotropicandhomogeneousandsimilartothelow-densityFriedmannmodels(Sanders1989,Sanders1997).Althoughsuchtheoriesdohavetheconsiderableadvantageofpredictivepoweronscalesotherthanextra-galactic,thefactisthattheyareunnatural;thesenon-standardscalar-tensortheoriesareascontrivedastheadhocmodificationofNewton’slawswhichtheypresumetoreplace.Thusitisperhapsprematuretoworkoutfullythecosmologicalimplicationsofsuchcomplicatedandtentativetheories.Must,then,considerationsofMONDcosmologythenbepostponeduntilthefinaltheoryisinplace?Evenbeforefurthertheoreticaldevelopments,itmaybepossibletodraw–3–somepreliminaryconclusionsaboutaMONDuniversebyconsideringafiniteexpandingsphericalregion.Itiswell-knownthattheapplicationofNewtoniandynamicstosucharegionleadstotheFriedmannequationfortheevolutionofthecosmicscalefactor.Eventhecosmologicalconstantcanbeincludedasanadditionalfluidwithnegativepressure.ItmightbeexpectedthatsomeinsightintoapureMONDcosmologymightbegainedbysuchaprocedureusingMilgrom’sformulainsteadofNewton’s.AninterestingstartinthisdirectionwasmadebyFelten(1984).HepointedoutthatwithMOND,thephysicalsizeoftheex