Direct Dark Matter Detection with Velocity Distrib

arXiv:astro-ph/0603704v214Aug2006DIRECTDARKMATTEREVENTRATESWITHAVELOCITYDISTRIBUTIONINTHEEDDINGTONAPPROACHJ.D.Vergados1∗andD.Owen21UniversityofIoannina,Ioannina,GR45110,Greece†.2DepartmentofPhysics,BenGurionUniversity,Israel‡.Exoticdarkmattertogetherwiththevacuumenergy(associatedwiththecosmologicalconstant)seemtodominatetheUniverse.Thusitsdirectdetectioniscentraltoparticlephysicsandcosmology.Supersymmetryprovidesanaturaldarkmattercandidate,thelightestsupersymmetricparticle(LSP).OneessentialingredientinobtainingthedirectdetectionratesisthedensityandthevelocitydistributionoftheLSPinourvicinity.InthepresentpaperwestudysimultaneouslydensityprofilesandvelocitydistributionsinthecontextoftheEddingtonapproach.Insuchanapproach,unlikethecommonlyassumedMaxwell-Boltzmann(M-B)distribution,theupperboundofthevelocityarisesnaturallyfromthepotential.PACSnumbers:95.35.+d,12.60.JvI.INTRODUCTIONThecombinedMAXIMA-1[1],BOOMERANG[2],DASI[3]andCOBE/DMRCosmicMicrowaveBackground(CMB)observations[4]implythattheUniverseisflat[5],Ω=1.11±0.07andthatmostofthematterintheUniverseisDark[6].i.e.exotic.CombiningtheWMAPdatawithotherexperiments,crudelyspeakingonefinds:Ωb=0.05,ΩCDM=0.30,ΩΛ=0.65Sincethenonexoticcomponentcannotexceed40%oftheCDM[7],thereisroomfortheexoticWIMP’s(WeaklyInteractingMassiveParticles).Supersymmetrynaturallyprovidescandidatesforthedarkmatterconstituents[8]-[9].InthemostfavoredscenarioofsupersymmetrytheLSP(Light-estSupersymmetricParticle)canbesimplydescribedasaMajoranafermion,alinearcombinationoftheneutralcomponentsofthegauginosandhiggsinos[8]-[10].Inmostcalculationstheneutralinoisassumedtobeprimarilyagaugino,usuallyabino.Eventhoughthereexistsfirmindirectev-idenceforahaloofdarkmatteringalaxiesfromtheobservedrotationalcurves,itisessentialtodirectlydetect[8]-[11]suchmatter.Untildarkmatterisactuallydetected,weshallnotbeabletoexcludethepossibilitythattherotationcurvesresultfromamodificationofthelawsofnatureaswecurrentlyviewthem.Thismakesitimperativethatweinvestamaximumeffortinattemptingtodetectdarkmatterwheneveritispossible.Furthermoresuchadirectdetectionwillalsounravelthenatureoftheconstituentsofdarkmatter.Thepossibilityofsuchdetection,however,dependsonthenatureofthedarkmatterconstituents(WIMPs).SincetheWIMPisexpectedtobeverymassive,mχ≥30GeV,andextremelynonrel-ativisticwithaveragekineticenergyT≈50KeV(mχ/100GeV),itcanbedirectlydetected[8]-[11]mainlyviatherecoilingofanucleus(A,Z)inelasticscattering.Theeventrateforsuchaprocesscanbecomputedfromthefollowingingredients:1.AneffectiveLagrangianattheelementaryparticle(quark)levelobtainedintheframeworkofsupersymmetryasdescribed,e.g.,inRefs[10,12].2.AwelldefinedprocedurefortransformingtheamplitudeobtainedusingthepreviouseffectiveLagrangianfromthequarktothenucleonlevel,i.e.aquarkmodelforthenucleon.Thisstepisnottrivial,sincetheobtainedresultsdependcruciallyonthecontentofthenucleoninquarks∗Correspondingauthor†e-mail:[e-mail]‡e-mail:[e-mail]rthescalarcouplings,whichareproportionaltothequarkmasses[13]−[14],[15]aswellastheisoscalaraxialcoupling[15,16].3.Knowledgeoftherelevantnuclearmatrixelements[17]−[18],obtainedwithasreliableaspossiblemanybodynuclearwavefunctions.Fortunatelyinthecaseofthescalarcoupling,whichisviewedasthemostimportant,thesituationisabitsimpler,sincethenoneneedsonlythenuclearformfactor.4.KnowledgeoftheWIMPdensityinourvicinityanditsvelocitydistribution.Sincetheessentialinputherecomesfromtherotationalcurves,darkmattercandidatesotherthantheLSP(neutralino)arealsocharacterizedbysimilarparameters.Inthepastvariousvelocitydistributionshavebeenconsidered.Theonemostusedistheisother-malMaxwell-Boltzmannvelocitydistributionwithυ2=(3/2)υ20whereυ0isthevelocityofthesunaroundthegalaxy,i.e.220km/s.ExtensionsofthisM-Bdistributionwerealsoconsidered,inparticularthosethatwereaxiallysymmetricwithenhanceddispersioninthegalactocentricdirection[19,20].Insuchdistributionsanuppercutoffυesc=2.84υ0wasintroducedbyhand.Nonisothermalmodelshavealsobeenconsidered.Amongthoseoneshouldmentionthelateinfallofdarkmatterintothegalaxy,i.ecausticrings[21,22,23,24,25],darkmatterorbitingtheSun[26],Sagittariusdarkmatter[27].ThecorrectapproachinourviewistoconsidertheEddingtonproposal[28],i.e.toobtainboththedensityandthevelocitydistributionfromamassdistribution,whichdependsbothonthevelocityandthegravitationalpotential.OurmotivationinusingEddington[28]approachtodescribingthedensityofdarkmatterisfound,ofcourse,inhissuccessindescribingthedensityofstarsinglobularclusters.Sincethisapproachadequatelydescribesthedistributionofstarsinaglobularclusterinwhichthemaininteractionisgravitationalandbecauseofitsgenerality,weseenoreasonwhysuchanapproachshouldnotbeapplicabletodarkmatterthatalsointeractgravitationally.ItshouldbenotedthattheattempttouseMaxwellian(M-B)distributiontodescribethestardistributioninglobularclustersledtoresultsthatdidnotcorrespondtoobservations[28].SoitseemsthattheuseofM-Bdistributiontodescribedarkmatterisnotverywellmotivatedandadifferentapproachisrequired.Itseems,t