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arXiv:astro-ph/0601431v28Feb2006ModifiedgravitywithoutdarkmatterR.H.SandersKapteynAstronomicalInstitute,Groningen,TheNetherlandsSummary.Onanempiricallevel,themostsuccessfulalternativetodarkmatterinboundgravitationalsystemsisthemodifiedNewtoniandynamics,orMOND,proposedbyMilgrom.HereIdiscusstheattemptstoformulateMONDasamod-ificationofGeneralRelativity.IbeginwithasummaryofthephenomenologicalsuccessesofMONDandthendiscussthevariouscovarianttheoriesthathavebeenproposedasabasisfortheidea.Ishowwhytheseproposalshaveledinevitablytoamulti-fieldtheory.IdescribeinsomedetailTeVeS,thetensor-vector-scalartheoryproposedbyBekenstein,anddiscussitssuccessesandshortcomings.Thislectureisprimarilypedagogicalanddirectedtothosewithsome,butnotadeep,backgroundinGeneralRelativity1IntroductionThereisnowcompellingobservationalsupportforastandardcosmologicalmodel.Itismostimpressivethatthisevidenceisderivedfromverydifferentobservationaltechniquesappliedtoverydifferentphenomena:fromprecisemeasurementsofanisotropiesintheCosmicMicrowaveBackground(CMB)[1];fromsystematicphotometricobservationsofthelightcurvesofdistantsupernovae[2,3,4];fromredshiftsurveysmappingthedistributionofobserv-ablematteronlargescaleandinterpretingthatdistributioninthecontextofstructureformationbygravitationalcollapse[5,6].Usingthestandardpa-rameterisedFriedmann-Robertson-Walkermodels(FRW),alloftheseobser-vationsimplyaconvergencetoanarrowrangeofparametersthatcharacterisetheUniverse;thisconvergenceisrightlyheraldedasaremarkableachievementofthepastdecade.However,theUniversethatwearepresentedwithisstrangeinitscompo-sition:onlyfivepercentistheordinarybaryonicmatterthatwearefamiliarwith;twenty-fivepercentconsistsofpressurelessdarkmatterpresumedtobefundamentalparticlesthatareasyetundetectedbyothermeans;andaboutseventypercentistheevenstrangernegativepressuredarkenergy,possiblyidentifiedwithacosmologicalterminEinstein’sfieldequation,andemerging2R.H.SandersrelativelyrecentlyincosmichistoryasthedominatecontributertotheenergydensitybudgetoftheUniverse.Ageneralsenseofunease,primarilywiththisdarkenergy,hasledanum-berofpeopletoconsiderthepossibilitythatgravitymaynotbedescribedbystandardfour-dimensionalGeneralRelativity(GR)onlargescale(seeSami,thisvolume)–thatistosay,perhapstheleft-hand-sideratherthantheright-hand-sideoftheEinsteinequationshouldbereconsidered.Variouspossibil-itieshavebeenproposed–possibilitiesrangingfromtheadditionofascalarfieldwithanon-standardkineticterm,K-essence[7];togravitationalactionsconsistingofgeneralfunctionsoftheusualgravitationalinvariant,F(R)the-ories(Sotiriou,thisvolumeand[8,9]);tobraneworldscenarioswithleakageofgravitonsintoahigherdimensionalbulk([10]andMaartens,thisvolume).But,infact,thereisalongerhistoryofmodifyinggravityinconnectionwiththedarkmatterproblem–primarilythataspectoftheproblembroadlyde-scribedas“missingmass”inboundgravitationalsystemssuchasgalaxiesorclustersofgalaxies.Theobservationsofthisphenomenologyhaveanevenlongerhistory,goingbacktothediscoveryofasubstantialdiscrepancybe-tweenthedynamicalmassandtheluminousmassinclustersofgalaxies[11].Theprecisemeasurementofrotationcurvesofspiralgalaxiesinthe1970’sand1980’s,primarilyby21cmlineobservationswhichextendwellbeyondthevisiblediskofthegalaxy[12,13],demonstrateddramaticallythatthisdiscrepancyisalsopresentingalaxysystems.Afundamental,oftenimplicit,aspectofthecosmologicalparadigmisthatthisobserveddiscrepancyinboundsystemsisduetothecosmologicaldarkmatter–thatthecosmologicaldarkmatterclustersonsmallscaleandpro-motestheformationofvirializedsystemsviagravitationalcollapseintheexpandingUniverse.Thenecessityofclusteringonthescaleofsmallgalaxiesimpliesthattherearenophasespaceconstraintsonthedensityofthedarkmatterand,hence,thatitiscold,ornon-relativisticattheepochofmatter-radiationequality[14].Theexactnatureofthehypotheticalcolddarkmatter(CDM)isunknownbutparticlephysicstheorybeyondthestandardmodelprovidesanumberofcandidates.Thereareobservationalproblemsconnectedwiththeabsenceofphasespaceconstraintsinthisdarkmatterfluid,problemssuchastheformationofnumerousbutunseensatellitesoflargergalaxies[15]andthepredictionofcuspsinthecentraldensitydistributionsofgalaxies–cuspswhicharenotevidentintherotationcurves[16].Butitisusuallytakenasaarticleoffaiththat“complicatedastrophysicalprocesses”suchasstarformationandresultingfeed-backwillsolvetheseproblems.Themotivationbehindconsideringmodificationsofgravityasanalter-nativetoCDMisbasicallythesameasthatunderlyingmodifiedgravityasanalternativetodarkenergy:whenatheory,inthiscaseGR,requirestheexistenceofamediumwhichhasnotbeen,orcannotbe,detectedbymeansotherthanitsglobalgravitationalinfluence,i.e.,anether,thenitisnotun-reasonabletoquestionthattheory.Theprimarydriverforsuchproposalshasbeenthedirectobservationofdiscrepanciesinboundsystems–galaxiesModifiedgravitywithoutdarkmatter3andclustersofgalaxies–ratherthancosmologicalconsiderations,suchasthatofstructureformationinanexpandingUniverse.Themostsuccessfuloftheseveralsuggestions,modifiedNewtoniandynamicsorMOND,hasanentirelyphenomenologicalrathertha