The phase-space structure of a dark-matter halo Implications for dark-matter direct detecti

Implicationsfordark-matterdirectdetectionexperiments

AminaHelmi,SimonD.M.WhiteandVolkerSpringel∗

Max-Planck-Institutf¨urAstrophysik,Karl-Schwarzschild-Str.1,85740GarchingbeiM¨unchen,Germany

(Dated:February5,2008)

Westudythephase-spacestructureofadark-matterhaloformedinahighresolutionsimulationofaΛCDMcosmology.Ourgoalistoquantifyhowmuchsubstructureisleftoverfromtheinho-mogeneousgrowthofthehalo,andhowitmayaffectthesignalinexperimentsaimedatdetectingthedarkmatterparticlesdirectly.Ifwefocusontheequivalentof“Solarvicinity”,wefindthatthedark-matterissmoothlydistributedinspace.Theprobabilityofdetectingparticlesboundwithindenselumpsofindividualmasslessthan107M⊙h−1issmall,lessthan10−2.ThevelocityellipsoidintheSolarneighbourhooddeviatesonlyslightlyfromamultivariateGaussian,andcanbethoughtofasasuperpositionofthousandsofkinematicallycoldstreams.Themotionsofthemostenergeticparticlesare,however,stronglyclumpedandhighlyanisotropic.WeconcludethatexperimentsmaysafelyassumeasmoothmultivariateGaussiandistributiontorepresentthekinematicsofdark-matterparticlesintheSolarneighbourhood.ExperimentssensitivetothedirectionofmotionoftheincidentparticlescouldexploittheexpectedanisotropytolearnabouttherecentmerginghistoryofourGalaxy.

PACSnumbers:95.35.+d,95.30.Cq,95.75.Pq,98.35.Gi,98.35.Df,98.35.Mp

arXiv:astro-ph/02012v2 23 Jul 2002I.INTRODUCTION

Oneofthemostfundamentalopenquestionsincos-mologyandparticlephysicstodayiswhatisthenatureofdark-matter.Thefirstindicationsofitsexistencecameinthe1930s,withthemeasurementsofthevelocitiesofgalaxiesinclusters.Theclustermassrequiredtograv-itationallybindthegalaxieswasfoundtoberoughlyanorderofmagnitudelargerthanthesumofthelu-minousmassesoftheindividualgalaxies[1,2].Inthe1970s,observationsoftherotationcurvesofspiralgalax-󰀂ies(Vc(r)=

galaxies,whichthenmergeandgiverisetothelargerscalestructuresweobservetoday.Thusstructurefor-mationoccursina“bottom-up”fashion[8,9].Thishierarchicalparadigmhasallowedastronomerstomakeverydefinitepredictionsforthepropertiesofgalaxiesto-dayandabouttheirevolutionfromhighredshift.DirectcomparisonstoobservationshaveshownthatthismodelisquitesuccessfulinreproducingboththelocalandthedistantUniverse.

Thecrucialtestofthisparadigmundoubtedlycon-sistsinthedeterminationofthenatureofdark-matterthroughdirectdetectionexperiments.Amongthemostpromisingcandidatesfromtheparticlephysicsperspec-tiveareaxionsandneutralinos.Axionshavebeenintro-ducedtosolvethestrong-CP(ChargeconjugationandParity)violations[10].Theycanbedetectedthroughtheirconversiontophotonsinthepresenceofastrongmagneticfield(e.g.[11,12]).Neutralinosarethelight-estsupersymmetricparticles,andmaybeconsideredasaparticularformofweaklyinteractingmassiveparticles(WIMPs).Themostimportantdirectdetectionpro-cessofneutralinosisthroughelasticscatteringonnu-clei.Theideaistodeterminethecountrateoverrecoilenergyaboveagiven(detector)backgroundlevel.Theexperimentalsituationhasbeenimprovingrapidlyoverthepastyears,withlarge-scalecollaborationssuchasDAMA,EdelweissandCDMS[13,14,15]startingtoprobeinterestingregionsofparameterspace(foranex-tensivediscussionsee[16]).Themainproblemcurrentlyliesinthehighlevelofbackgroundnoise,eitherfromambientradioactivityorcosmic-rayinducedactivity.In-formationonthedirectionoftherecoilscouldpotentiallyalsobeusefulandyieldalargeimprovementinsensitivity[17].

Inalltheseexperiments,thecountratestronglyde-pendsonthevelocitydistributionoftheincidentparti-

∗Electronic

address:ahelmi,swhite,volker@mpa-garching.mpg.decles,andamodulationeffectduetotheorbitalmotionoftheEartharoundtheSunisexpected[18].Inmostcases,anisotropicMaxwelliandistributionhasbeenas-sumed(e.g.[19,20]),althoughthereareotherexamplesintherecentliterature,discussingmultivariateGaussiandistributions[21,22,23].Attemptsatunderstandingtheeffectofsubstructureinthevelocitydistributionofdark-matterparticleshavealsobeenmade[24,25,26].ThissubstructurewouldhaveitsorigininthedifferentmergerandaccretioneventstheGalaxyshouldhaveexperiencedoveritslifetime[27].

Theprogressivebuildupofdarkhalosthroughmerg-ersandaccretionofsmallersubunitsimpliesthatthelatterwillleavesubstructureinthephase-spaceofthefinalobject.Thisisbecausethephase-spacevolumeofthefinalobjectismuchlargerthanthatinitiallyavailableforeachoneoftheobjectsindependently.Forexample,forasmallsatellitegalaxytheinitialphase-spacevolumeoccupiedbyitsparticlesisproportionalto(RsatVisthesizeofthesatellite,andVcsat)3,whereRsatvelocity.Thevolumeavailabletothesatellitecsatitscircularparticlesafterthemergingisdeterminedbytheirorbit,andisafactor(Rgal/Rsat)3×(Vcgal/Vcsat)3larger,whereRgalisthesizeofthefinalobjectandVinthecaseofamajorcgalitscircularvelocity.Notethatevenmerger,wherethemassisdoubled,thephase-spacevolumeavailableisal-ready4timeslarger[45].Thekeyquestioniswhetherthissubstructurewillbedirectlyorindirectlyobserv-able.Forexample,ifthereisaboundsatellitegoingthroughtheSolarneighbourhoodatthepresentday,itwilldominatethefluxofdark-matterparticlesonEarth.Theenergyspectrumoftheseparticleswillbestronglypeakedaroundtheorbitalenergyoftheclump,perhapsgivingasignalsimilartoadeltafunction.AsweshallshowinSec.IIIB2thefractionofmassinsatelliteswhichcouldhavesurvivedthetidal2fieldoftheGalaxybythepresentdayislessthan10−oftotalmassoftheGalaxy,implyingthatsuchascenarioisrelativelyunlikely.

MorerealisticistoassumethatthesatellitehalosthatcontributewithmasstotheSolarneighbourhoodwillbecompletelydisrupted.Theparticlesfreedfromsuchsatelliteswilltendtofollowtheinitialorbitoftheirpro-genitor,andeventuallywillfillavolumecomparabletothesizeoftheorbit.Becauseoftheconservationofphase-spacedensity(Liouville’stheorem),thisimpliesthatlocallytheyshouldhaveverysimilarvelocities[46].ThusonemayexpecttoseestreamsofparticlesgoingthroughtheSolarneighbourhood,whichhadtheirorigininthedifferentmergingevents.Suchstreamshaveal-readybeenobservedinthemotionsofnearbyhalostarsandintheouterregionsoftheGalactichalo[28,29].Streamsmanifestthemselvesaspeaksinthevelocitydis-tributionfunction.Clearlyitisimportanttodetermineforthedark-matterparticlesinthevicinityoftheSunwhetherthisdistributionfunctionwillbedominatedbyafewofthesepeaks,orwhethertheirnumberissolarge,thatitwillbeclosetoGaussian.

Thebestwaytounderstandtheexpectedproper-2

tiesoftheGalactichalointheSolarneighbourhoodisthroughhigh-resolutionsimulationsstartingfromappro-priatecosmologicalinitialconditions.Analyticmodellingcanprovideinsightsintotheprocessesthatdrivethebuild-upofstructuresuchasphase-mixing,ortidalstrip-ping.Neverthelessitneedstobecomplementedbycos-mologicalsimulations,thatprovidethemassspectrumoftheaccretedhalos,theirorbitalparameters,theirchar-acteristicmergingtimes,andthedetailedmixingofthematerialtheydeposit.Thehighlynon-linearcharacterofthehierarchicalbuildupofagalaxyliketheMilkyWay,forcesustoresorttonumericalsimulationstomakere-alisticpredictionsforitsproperties.Veryhighresolutionsimulationsarerequiredtobeabletoresolvethesub-structuresleftoverfrommergingevents,sincetheirden-sitycontrasts3areexpectedtofaderatherquicklywithtime(astforsufficientlylongtimescales[27]).

Themaingoalofthepresentpaperistounderstandthephase-spacestructureofadark-matterhalo.Wewishtoquantifytheexpectedamountofsubstructureandun-derstanditseffectondirectdark-matterdetectionexper-iments.Particularemphasiswillbeputondeterminingthepropertiesofthedark-matterdistributionintheSo-larneighbourhood:itsmassgrowthhistory,thespatialdistributionandthekinematicsofparticlesinthisregionoftheGalactichalo.Weaddresstheseissuesbyscalingdownahigh-resolutionsimulationoftheformationofaclusterofgalaxiesinaΛCDMcosmologytoagalacticsizehalo[30].

II.METHODOLOGY

Thesimulationsweanalyseherewerecarriedoutus-ingaparalleltree-code[31]ontheCrayT3EattheGarchingComputingCentreoftheMaxPlanckSociety.Thesesimulationsweregeneratedbyzoominginandre-simulatingwithhigherresolutionaparticularclusteranditssurroundingsformedinacosmologicalsimulation(asin[32]).TheΛCDMcosmologicalsimulationhasparam-etersΩ0=0.3,ΩΛ=0.7,h=0.7andσ8=0.9.Theclusterselectedisthesecondmostmassiveclusterinthesimulationandhasavirialmassof8.4×1014h−1M⊙.Theparticlesthatendupinthefinalclusterofthecos-mologicalsimulationandinitsimmediatesurroundings(definedbyacomovingsphereof70h−1Mpcradius)weretracedbacktotheirLagrangianregionintheinitialcon-ditionsforre-simulation.Theinitialmassdistributionbetween21and70h−1Mpcwasrepresentedby3×106particles.Intheinnerregion,wheretheoriginalsimula-tionhad2.2×105particles,newinitialconditionswerecreatedfor4.5×105,2×106,1.3×107and6.6×107particles,andsmallscalepowerwasalsoaddedontothisvolume.Theoriginalforcesofteningwasalsodecreasedtoobtainbetterspatialresolution.Allsimulationswererunfromveryhighredshiftuntilz=0.

Wecanscalethesimulatedclustertoa“MilkyWay”halobyscalingthecircularvelocitysothatatitsmax-

imumitisequalto220kms−1.Thescalingobtainedinthiscaseisγ=vcl

c/vMWfactor

c∼9.18.Thevirialradiusofoursimulated“MilkyWay”dark-matterhaloisrvir=228kpc.Thejustificationforthissimplescalingreliesbothontheoreticalandnumericalresults[34,35,36].Forexample,high-resolutionnumericalsim-ulationshaveshownthattheoverallpropertiesofgalax-iesandclustersofgalaxies,suchastheirdensityprofiles,numberofsatellitesandformationpathshaveoverlap-pingstatisticaldistributionsforthetwotypesofobjects.

III.

THEPHASE-SPACESTRUCTUREOFTHE

GALAXY

Inthissectionweshallfocusonthepropertiesofthedark-matterdistribution,withparticularemphasisonthevicinityofthe“Sun”.WeareinterestedinwhichhaloscontributemattertothisregionoftheGalactichalo,whatweretheirinitialproperties,andwhenweretheyaccreted.Wealsoinvestigatewhatisthepresent-dayspatialandvelocitydistributionofmaterialinthe“Solarneighbourhood”,andhowthedirectdetectionex-perimentsmaybefinetunedtodeterminethenatureofdark-matter.

AseriesofsnapshotsofthegrowthoftheclusterareshowninFigure1.Thesimulationsstartfromverysmalldensityfluctuations,assumedtohavebeenproduceddur-ingtheinflationaryexpansionoftheUniverse[33].Mat-teristhenaccretedontotheseinitialdensityfluctuationsthroughtheactionofgravity.AdarkhaloformswhenanoverdenseregiondecouplesfromtheexpandingUni-verse,turnsaroundandcollapsesontoitself.Thispro-cessrepeatsitselfonprogressivelylargerscales,andbighalosareformedthroughthemergingandaccretionofsmallerunitsasshowninFig.1.Thesesubunitswillor-bitthelargerhaloassatellitesforsometime,asshowninthebottompanelsinFig.1,untiltheyarecompletelydisrupted.Wewillfrequentlyrefertothemassubhalos.Theprogressivegrowthofmassoftheclusterisschemat-icallyshowninFigure2asa“mergertree”.

A.Massgrowthhistory1.

TheGalaxy

Letusfirststudythepropertiesofthedark-matterdis-tributionasafunctionofdistancefromthegalaxycentre.Weareinterestedindeterminingwhattypeofhalostyp-icallycontributetodifferentregionsofthegalaxyandtheirtimeofaccretion.Thisisrelevantbecausehalosaccretedatlatetimeswillbegenerallylessmixed,andcouldthusproducemoremassivestreamsdominatingthevelocitydistributionofparticlesneartheSun.WealsowanttoestimatetheprobabilitythatsuchhaloscouldcontributetotheSolarneighbourhoodmassbudget.

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Weproceedbydividingthehaloinsixsphericalshellsaroundthegalaxycentre.Theseshellsarelocatedat:r<10kpc,10≤r<25kpc,25≤r<50kpc,50≤r<75kpc,75≤r<100kpcand100≤r<200kpc.Foreachparticleinashell,wedeterminewhenitwasaccretedbythemainprogenitorofthegalaxystart-ingfromredshiftz=2.4or11Gyrago.Particlesmaycomefromaccretedsatellitesorfromthe“field”.“Field”particlesarethosewhichdidnotbelongtoanyboundstructurebeforebecomingpartofthegalaxy.Becauseofourresolutionlimit,fieldparticlesmayalsocomefromhaloswithlessthan10particles,i.e.withmasssmallerthan8.66×105M⊙.Theaccretiontimeforparticlesinasubhaloisdefinedtobethetimeofaccretionofthissub-halo.Inpractice,wesaythatasubhaloHsubidentifiedatredshiftzhasbeenaccretedbythemainprogenitorHmainatredshiftz′,ifatleasthalfoftheparticlesofHsubarecontainedwithinHmainatz′,aswellasthemostboundparticleofHsub.Foraparticlefromthefield,thetimeofaccretionissimplydefinedastheearli-esttimeatwhichthisparticlehasbecomeamemberofthemainprogenitorofthegalaxy,asdeterminedbyourFOFalgorithm.

InFigure3weshowthefractionofmassaccreted(nor-malisedtothepresentmass)foreachshellasafunctionoftheinitialmassoftheaccretedsatelliteandforthreedifferentredshiftbins.Wedividetheanalysisintothemassalreadypresentatredshiftz=2.4(shownindarkgrey);thataccretedbetweenz=2.4andz=0.83(lightgrey);andbetweenz=0.83andthepresentday(black).ThefirstpanelofFigure3showsthattheforma-tiontimeoftheinnergalaxyisstronglybiasedtowardshighredshifts,withmorethan60%ofthemassalreadypresentatz=2.4.Wealsonotethatlateaccretiondoesnotplayanyroleinbuildinguptheinnergalaxy.Sub-haloswithmassessmallerthan107M⊙accretedatlatetimesdonotmakeupmorethan10−3ofthetotalmassintheinnermostshell.Thishasimplicationsfordark-matterdetectionexperiments,sinceitimpliesthatre-centlyaccretedtinysubhaloswithveryhighphase-spacedensitydonotcontributeenoughtotheSolarneighbour-hoodtohaveasignificanteffectontheexpectedsignal,incontrasttothesuggestionin[37].TheonlywaysuchsmallsubhaloscouldmakeittothevicinityoftheSunwouldbebybeingaccretedfirstbyalargehalo,whichatsomelaterredshifthasamajormergerwiththemainprogenitoroftheGalaxy.Wewillquantifyhowlikelythismaybeinthenextsection.

Figure3alsoshowsthatsmallaccretedsubhalostendtodepositmostoftheirmassatlargedistances.Intheseouterregions,thecontributionofheavysubhalosisofthesameorderofmagnitudeasthatfromthesmallersatel-lites.Thedifferenceinthefinaldebrisdistributionfrommassivesubhalosandfromlighteronesisduetodynam-icalfriction;verymassivesatellitescansinktothecentreofthegalaxyinshorttimescales,whichenablesthemtodepositagoodfractionoftheirmassinthecentralre-gions.Alargefraction(about20%)ofthemassinthe

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FIG.1:Snapshotsofthegrowthofthedark-matterhaloinoursimulation.Eachpanelshowstheprojectedmassdensityinaboxofsidelength5.0Mpc/hintheoriginalclusterunits,whichcorrespondto778kpcinthescaledunitsusedthroughoutthepaper.Thepanelsarecentredonthemainprogenitorofthedark-matterhaloatthattime.Thefirstpanelcorrespondsto12.7Gyrago,andthelastpaneltothepresenttime.Notehowthehalogrowsthroughthemergingandaccretionofsmallerunits.

emitTodayFIG.2:Schematicrepresentationofa“mergertree”showingthegrowthofahalointime.Timeincreasesfromtoptobottom,andthewidthsofthebranchesareproportionaltothemassesoftheprogenitorhalos(basedon[34]).

outskirtscomesfromfieldparticles(asshowninthelastpanelofFig.3),stronglycontrastingwiththe0.7%seenfortheinnermostshell.

Finallyweremarkthatwhereastheformationofthein-nergalaxyisstronglyskewedtohighredshifts,theouterregionsgrowmuchmoregraduallyintime,withaccretionstillbeingimportantatlatetimes.

2.TheSolarneighbourhood

Wenowfocusonthe“Solarneighbourhood”,andanal-ysetheregion:7kpcInthetoppanelofFigure4weshowthemassaccretedatagiventime(tobemoreprecisebetweentwoconsec-utiveoutputs)fm(t)normalisedtothetotalpresent-daymassinthesphericalshell7kpcParticlesaccretedinthepastGyr,donotaccountfor

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FIG.3:Thepanelsshowtheaccretedmassfraction(nor-malisedtothepresent-daymass)asafunctionoftheinitialmassofthesatellitehaloforsphericalshellsaroundthegalaxycentre.Thedifferentcolourscorrespondtothefractionofmassaccretedatdifferentredshifts.Darkgreycorrespondstothemassalreadyinplaceatz=2.4;lightgreytomassaccretedbetweenz=2.4andz=0.83;andblacktothataccretedbetweenz=0.83andthepresentday.

morethanabout10−3ofthetotalpresentnumberofparticlesneartheSun.Theinfluenceofstreamsfromsuchrecentlyaccretedmaterialonthevelocitydistribu-tionfunctionneartheSunwillthusberelativelysmall,andmaydominateonlythehighenergytailofthedis-tributionfunction.ThelowerinFigure4showsthegrowth(Fm(t)=󰀅tpanelofmass0fm(t′

)dt′)asafunctionoftime.Weseethatmorethan50%ofthemassthatendsupinthe“Solarneighbourhood”todaywasalreadyinplace11Gyrago,andabout90%10Gyrago.Ofcourse,thisdependsonthespecificmergerhistoryofthishalo,sincethelargeincreaseofmassobservedatt=4Gyrisduetoamajormergertakingplaceatthetime.However,wealsonoticethataftert=7Gyr,thereisalmostnoincreaseofmassinthevicinityofthe“Sun”.Thelargecontrastwiththegradualmassgrowthofthehaloasawholecanbeclearlyperceivedfromthisfigure.

B.

SpatialdistributionintheSolarneighbourhood

Oneofthecriticalissuesinunderstandingtheoutcomeofthevariousdark-matterexperimentsconsistsinchar-acterisingtheexpectedsignal.Asdiscussedintheintro-duction,offundamentalimportanceistoknowwhetherthedistributionofparticlesinthevicinityoftheSunissmoothormightbedominatedbyajustafewstreams

FIG.4:Filledcirclesinthetoppanelshowthegrowthofmassinthe“Solarneighbourhood”normalisedtothemasspresentatz=0.Opencirclescorrespondtothegrowthofmassofthewholegalaxyhalo.Thebottompanelshowstothecumulativegrowthofmass.Notethat85%ofthemassintheSolarneighbourhoodwasalreadyinplace10Gyrago.Forthewholegalaxyhalo(dashedcurve),thisdidnothappenuntil6Gyrago.

orevenboundlumps(e.g.[37]).Belowweshalldescribethevelocitydistributionofparticlesthatendupinthisregionofthegalaxyhalo.Herewefocusontheirspatialproperties.

InFigure5weplotthepositionsofallparticlesinsideacubicvolumeof2kpconaside,located8kpcfromthe“MilkyWay”centre,whichweassumeisthedistancebetweentheSunandtheGalacticcentre.Becausethedarkhaloistriaxial,althoughalmostprolate(theaxesratiosareI1:I2:I3=0.65:0.71:1),weassumetheGalacticdisktobeperpendiculartothemajoraxisofthehalo.Thespatialdistributionofparticlesinsidethisrepresentativevolumeisextremelysmooth,asshowninFigure5.Thisismostlyduetothefactthatthematerialthatendsupintheinnergalaxymostlycomesfromafewverymassivehalos.Theveryshortdynamicaltimescalesinthisregionofthegalaxyarealsoresponsiblefortheveryrapidandefficientmixing,afterwhichthereremainsverylittleornospatialinformationontheirorigin.

1.

ThepropertiesofthehalosthatcontributetotheSolar

neighbourhood

TodeterminethecharacteristicsofthehalosthathavecontributedmattertotheSolarneighbourhood,wefo-cusonthesphericalshell:7kpc6

FIG.5:Thetoppanelsshowthespatialdistributionofpar-ticlesinsidea2kpconasidevolumelocatedat8kpcfromthegalaxycentre,i.e.thisvolumeiscentredonthe“Sun”.Thereare474particlesinthisbox.Thebottompanelshowstheirdistributiononthesky.

Sec.IIIA2,weidentifytheoriginoftheparticlesthatarelocatedinthisshellatthepresenttime.Wegrouptheparentsatellitesaccordingtotheirinitialmassinlog-arithmically.5spacedbinsofwidthdlogM=0.5startingfrom105M⊙to1012M⊙.Weestimatethecontributionfromhalosidentifiedatthreedifferentredshifts.Eithersuchhalosaredirectlyaccretedbythegalaxy,ortheyareaccretedbyanothermoremassivesubhalo,whicheven-tuallymergeswiththegalaxy.

IntheleftpanelofFigure6weshowthecontributionofmattertothesphericalshellnormalisedtothepresentmassintheshell.Thusweseethatalargefractionofthemassinthisshellcomesfromthemostmassivehalosidentifiedatz=2.4,andverylittlefromfieldparticles(aswasalsoshowninFigure3).Forhigherredshiftsthelargestcontributiontendstocomefromsmallersatel-lites,whichisjustaconsequenceofthefactthattheheaviesthaloshavenotyetcollapsedattheseredshifts.Forsufficientlyhighredshift,thefieldparticles(i.e.fromunresolvedhalos)arethelargestcontributortothemasspresenttodayintheSolarneighbourhood.

Justafewindividualhaloscontributetothelargestmassbins.However,forthesmallmassbins,thenumberofhaloscontributingwithinagivenbinactuallyincreasesdramatically,asshownintherightpanelofFigure6.Thelargefractionofthemassinthesphericalshell7kpc7

FIG.6:Theleftpanelshowsthefractionofmassinashellaroundthe“solar”radiuscomingfromhalosidentifiedatthreedifferentredshifts,normalisedtothepresentmassintheshell.Thefirstbinshowsthecontributionoffieldparticlesatthedifferentredshifts,whichendupinthe“Solarneighbourhood”.Byz=10.4notmanymassivehaloshaveyetcollapsed,andalmostallparticlescomefromthefield.However,byz=2.4,mostofthesefieldparticlesarefoundinhalos.ThelargestcontributiontomassinthevicinityoftheSuncomesfromhalosheavierthan1010M⊙.Therightpanelshowsthenumberofcontributinghalosinthedifferentmassbins.

2.

ArethereanyboundsubhalosintheSolar

neighbourhood?

TherehavebeenrecentsuggestionsintheliteraturethattherecouldbeapopulationofverytinysubhalosorbitingintheSolarneighbourhood[37].Thesesubhaloswhichcollapsedatveryhighredshift,mighthavesuffi-cientlylargedensitiestosurvivealmostintactuntilthepresentday.Ifthispicturewerecorrect,theirpresencewouldproduceasignalondark-matterdetectorsthatwouldbeverydifferentfromthatofaGaussiandistribu-tioncomingfromasmoothhalo.

Figure6showsthatthecontributionofmassfromha-losof105.5−106M⊙isnotnegligible,rangingfrom0.9%forhalosidentifiedatz=2.4to4%forthoseidentifiedatz=10.4.However,noneofthesehaloshasmanagedtosurviveboundinoursimulations,andsothemasstheyhavecontributedisinasmoothcomponentatthepresenttime.

Howeverwealsoneedtoquantifytheprobabilitythatsomeofthemassactuallycomesfromboundsubhalosbelowourresolutionlimit.Totacklethisproblemweneedtoestimatethefractionofthemassinsuchsubhaloswhichremainedbounduntilthepresentdaywithrespecttothetotalmassofthegalaxy.Weshalldosousingallsubhalosorbitingwithinthevirialradiusofthegalaxytoday.Thesesubhalosaremostlyfoundintheoutskirtsofthegalaxy,wheretheirsurvivaltimesarelongerduetothesmallergalactictidalforces.

ThesubhalomassfunctiondN/dMgivesthenumber

ofsatellitesofthegalaxyhalowithmassin[M,M+dM].WithasophisticatedsubhalofinderitispossibletodeterminedN/dMforoursimulatedhaloatthepresenttime[30].Figure7showsthatthenumberofsubhalosinagivenmassbincanbewellfitbyapowerlaw:

󰀆

dNM

M⊙

dM′

Thus

MT(󰀆

M′dM′.

M

107M⊙h−1

󰀇0.27

.(3)

Thusforexample,forthewholegalaxyhalothefractionofmassinboundsatellitessmallerthan107M⊙h−1at

FIG.7:DifferentialsatellitemassfunctionM−1

hostdN/dMatredshiftz=0normalisedtothemassofthehosthalo.Thesolidcirclescorrespondtothewholepopulationofsatel-lites,whiletheasterisksonlytothoselocatedintheinner30kpc.InthelattercaseMhostisthemasswithinthisspher-icalregion.ThestraightlinecorrespondstologdN/dM=a+blogM,showingthatthedifferentialmassfunctionisverywellfitbyapowerlaw.

thepresenttimeis4.4×10−2

.Thisoverestimatesthefractionofmassinsubhalosorbitingtheinnergalaxy,sincethespatialdistributionofsubhalosisskewedto-wardslargedistancesfromthegalaxycentre,wheretidalforcesareweaker.Forexample,ifweestimatep(C.KinematicsintheSolarneighbourhood

InFigure8weplotthevelocitiesofparticlesinsideacubicvolumeof2kpconaside,locatedat8kpcfromthe“MilkyWay”centre.(ThesamevolumeasinFigure5.)Weidentifywithdifferentcoloursandsymbolsparticlesthatbelongedtothesamehaloatz=2.4.Atthisred-shift,whichcorrespondsto11Gyrago,wefind252003haloswithatleast10particlesinoursimulation.Intheboxshowninthisfigure,thereare474particleswhichcomefrom39differenthalos;only3contributewithmorethantenparticles.Thesethreehaloscomprisethemainprogenitorofthegalaxy(i.e.thetrunkinFigure2)andthetwomostmassivehalosthatmergedwiththegalaxy

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(seeFigure4.Wedonotexpectallparticlesofthesamecolourtobeclusteredinasinglemassivestream,sinceeachindividualhaloispredictedtohavegivenrisetomanystreamsintheSolarneighbourhood[27,41].Forexample,thesixteenparticlesoriginatinginthethirdmostmassivehalo(3%oftheparticlesinthisvolume)aredistributedintendifferentstreams.Therefore,itisnotsurprisingthatitisdifficulttodistinguishstreamsinthisfigure.Thetotalnumberofparticlesinsidethisboxistoosmalltopopulateeachexpectedstreamwithmorethanoneortwoparticles.

1.Thevelocitydistributionfunction

Turningtoalargervolumeof4kpconaside,allowsustoincreasethenumberofparticlesbyroughlyafactorof8.InFigure9weplotthevelocitydistributionfunctioninsuchaboxinthevicinityofthe“Sun”.Theleftandrightpanelsshowthedifferentialandcumulativevelocitydistributions,respectively.Wealsoshowhowthesedis-tributionscomparetoamultivariateGaussianwiththesamevelocityellipsoidasthedata.Althoughslightdif-ferencesarevisible,itishardtodistinguishthetwodis-tributionsfromoneanotherwithoutmakingadetailedstatisticalcomparison.

2.Thefastestmovingparticles

InFigure10weshowthevelocitiesofparticleslocatedinthesame4kpconasideboxofFig.9.Wealsonoteherethattheirvelocitydistributionisrelativelysmooth.However,ifwefocusonthehighestenergyparticlesthisseemsnolongertobethecase,asshownbytheparticleshighlightedingrey.The1%fastestmovingparticlesarestronglyclumped.Mostofthissignalcomesfromahaloofmass1.94×1010M⊙thatmergedwiththegalaxyatz∼1.Figure11showsthedirectionsofmotionofallparticlesinthebox,whereweagainhighlightthefastestmovingones.Theirdistributionisclearlyanisotropic.

3.Thevelocitycorrelationfunction

ToquantifythedeviationsfromasmoothGaussiandistributionduetothevelocitysubstructurepresentinavolumeintheSolarneighbourhood,wecomputethecorrelationfunctioninvelocityspace.Wedefinethe(av-eraged)velocitycorrelationfunction󰀆ξ󰀇as

󰀆ξ󰀇=

󰀆DD󰀇

9

FIG.8:Principalaxesprojectionsofthevelocitiesofparticleslocatedinaboxof2kpconasideonthe“Solar”circle,whereallquantitieshavebeenscaledtothe“MilkyWayhalo”asdescribedinthetext.Thereare474particlesinthisbox.Thedifferentcoloursandsymbolsareusedheretoindicateparticlesoriginatinginthesamehaloidentified11Gyrago.Opencirclescorrespondtoparticlesfromhaloswhichdonotcontributesubstantiallytothisvolume.Blackfilledcirclesareparticlesfromthemainprogenitoridentifiedatthisredshift(226particles,i.e.47%).Opentrianglescorrespondto“field”particles,i.e.notassociatedtoanyhalo11Gyrago(42particles,i.e.9%).Thesquares(129particles,i.e.27%)correspondtothesecondmostmassivehaloidentifiedatthistime,whichmergedwiththegalaxyabout10Gyrago.Thelightgreycircles(16particles,3%)areforthethirdmostmassivehalo,whichmergedwiththegalaxyabout7Gyrago.TheseeventsareclearlyvisibleinFigure4,ashavingcontributedmostofthemassinthe“Solarneighbourhood”inthehistoryofthegalaxy.

value,i.e.󰀆DD󰀇=

󰀁

pairsofparticlesi,jwith|vi−vj|<∆.

(5)

󰀆RR󰀇isdefinedanalogouslyforthesamenumberofran-dompoints.Weestimatetheerrorin󰀆ξ󰀇as

∆󰀅ξ󰀆=

1+󰀆ξ󰀇

.󰀆DD󰀇

(6)

WecomparethemotionsofparticleslocatedintheboxofFigures9,10and11withthoseexpectedfromasmoothGaussiandistribution.WegenerateNreal=10differentMonteCarlosimulationswiththesamevelocitydispersiontensorandnumberofpointsasobservedinthisboxlocatedinthevicinityoftheSun.Wecompute󰀆ξ󰀇asinEq.(4),with

󰀆RR󰀇=

󰀄Nreal

i=1

󰀆RR󰀇i

FIG.9:Forparticleslocatedinaboxof4kpconasideatthe“Solar”radius,weplotthedifferential(left)andthecu-mulative(right)velocitydistributions(solidhistograms).ThedottedhistogramscorrespondtotheexpecteddistributionforamultivariateGaussianwiththesamevelocityellipsoidandnumberofpointsasthedata.ThemaindifferencesarethattheactualdistributionofvelocitiesappearstobebroaderandwithasharpercutoffthantheGaussian,anditisslightlylesspeaked.

10

FIG.10:PrincipalaxesprojectionsofthevelocitiesofparticleslocatedinthesameboxofFig.9.Ofthe4362particlespresentinthisvolume,wehighlightedthe1%fastest.Thevelocitydispersionsalongtheprincipalaxesareσ1=111.2kms−1,σ2=120.1kms−1,andσ3=141.4kms−1.Thelumpwithv1∼−185kms−1,v2∼−140kms−1andv3∼−370kms−1correspondstoahaloof1.94×1010M⊙identifiedatz=2.4,andaccretedatz=1,or8.2Gyrago.

FIG.11:ThisplotshowsthedirectionsofmotionofthesameparticlesdiscussedinFigs.9and10.Wehighlightingreythe1%fastestmovingparticles.Thepositionofaparticleintheplotisgivenbythesphericalangularcoordinatesofitsvelocityvector,e.g.v1=vcosφcosθ,whereθisthelatitudeandφthelongitude.

10%fastestmovingparticles,thedeviationisaslargeasthatobservedforthefulldataset,andwouldthusonlybevisiblewithalargenumberofdetectionevents,i.e.ofatleastafewthousanddark-matterparticles.

Althoughtheresultspresentedherecorrespondtotheanalysisofjustoneboxinthe“Solarneighbourhood”thefeaturesobservedherearerepresentativeofwhatisseenforothersimilarvolumesinthisregionofthegalaxy.

IV.DISCUSSION

Thebuildupofdark-matterhalosinahierarchicaluni-verseisaverynonlinearprocess,whichhappensthroughthemergingandaccretionofsmallersubunits.Thedom-inantdynamicalprocessesatworkaretidalstripping,bywhichasatellitehaloprogressivelylosesitsmass,and

phase-mixingbywhichthismassisprogressivelystrungoutalongstreams.Asanexamplewebrieflydiscussahaloof4.3×1010M⊙accretedatz=1.8(t=3.62Gyr),andthemasslostbetweenz=0.13andz=0.06(t=12.08andt=12.91Gyr).InFigure13weshowthedistributionofparticleslostinthisredshiftrange,ina2–dimensionalprojection(r,vr)ofphase-space.Intheleftpanelweshowtheirdistributionjustaftertheywerereleasedfromtheirparentsatellite.Therightpanelshowsthatwhatwasinitiallyarelativelycoherentsetofparticleswithsimilarmotions,hasbythepresenttimespreadoutintofourdifferentstreamswhichareseentooverlapinspace.Notethatthishappenedinlessthan1.7Gyr,whichshowshowshortthemixingtimescalesareintheinnergalaxy.

Thedensityinastreamdecreasesintime,asthepar-ticlesspreadoutalongtheorbitoftheirprogenitorsys-tem.Becausetheorbitdefinesa3–dimensionalvolume(adoughnutshapedvolumedefinedbytheorbitalturningpoints),thedensityinastreamwilldecreaseas(t/torb)−3[27].Thus,extrapolatingtheresultobtainedforthema-terialshowninFig.13–theformationof4streamsin1.7Gyr–,andnotingtheveryearlybuildupofthegalaxyhalo(asdemonstratedinSec.IIIA2),weestimatethatatleast4×(0.7Gyr/0.35Gyr)3×(10Gyr/1.7Gyr)3∼6500darkstreamsshouldbepresentintheSolarneighbour-hood(adetailedcarefulanalysis,takingalsointoaccountotherhalos,showsthatthisnumbershouldbeevenlarger[41]).HereweusedthattheorbitalperiodsofparticlesinFig.13areoftheorderof0.7Gyr,whilethemedianforparticlesneartheSuniscloserto0.35Gyr.

Wenoteherethatsomeauthors[24,26]haveassumedthatthedensitydecreaseinastreamisslowerintime(linearorquadratic),fromwhichthey(incorrectly)de-ducethatthestreamsshouldhavemuchhigherdensities.Theseworksthenarguethatstreamscanstronglyaffectthesignaldetectedforthefluxofdark-matterparticlesgoingthroughtheEarth.Wehavejustshownthisisnotthecase.

FIG.12:ForthesameboxasinFigures9,10and11,weplotthe“(averaged)velocitycorrelationfunction”󰀅ξ󰀆.Itisdefinedasthenumberofneighbourswithvelocitydifferencelessthanagivenvaluecomparedtowhatisexpectedforran-domdeviates.InthiscasetherandomdeviatesaredrawnfromamultivariateGaussianwiththesamenumberofpar-ticlesandvelocitydispersiontensorasthedata.Asteriskscorrespondto󰀅ξ󰀆overalltheparticlesinthebox,whereasdiamondstothe1%fastestmovingparticles.Bothforthefulldatasetandforthefastestmovingparticlesthereisasignalatsmallvelocitydifferences,indicativeofthepresenceofstreams.Thissignalisclearlymuchstrongerforthesub-setofmostenergeticparticles.TheerrorbarsarebasedonPoissoniancounts.

FIG.13:Hereweshowthephase-spaceprojection(r,vr)forparticleslostfromahalobetweenz=0.13andz=0.06(be-tween12.1and12.9Gyr).Theleftpanelshowstheirdistribu-tionclosetowhentheywerereleased.Therightpanelshowstheirpresentdaydistribution.Themultiplestreamswereproducedinthelapseofatmost1.7Gyr.Thegreycurveshavebeenaddedtohighlightthelocationofthesestreamsinthediagram.

11

Theseresultsalsoapplytothedensitycontrastincaus-ticsurfacesorrings,expectedtoformattheorbitalturn-ingpoints.Wenoteherethattheproposedlargeeffectassociatedwithsheetsofparticlesfallinginforthefirst,secondorthirdtimeisnotobservedinoursimulations.Thisisnotduetoourfiniteresolution(assuggestedby[42]),butduetothefactthattheformationtimeoftheinnerhaloissostronglybiasedtohighredshifts.SinceparticleswhichorbittheSolarneighbourhoodhaveor-bitalperiodsoftheorderof0.35Gyr,andtheinnergalaxywasalreadyinplace10Gyrago,thisimpliesthatthedensityofthesecausticstructuresshouldbeoftheorderof(10Gyr/0.35Gyr)−3∼4×10−5timessmallertoday.Forthesereasonsstructuressuchascausticringsorsurfaceswilllikelyhavenoeffectonthesignalthatdark-matterexperimentswilldetect.

Recentanalyticwork[25]hasfocusedontheeffectofrecentlyaccretedmaterialonthevelocitydistributionfunctionneartheSun.Astheseauthorsdiscuss,suchmaterialwillmostlydominatethehighenergytailofthevelocitydistribution,andprovideanon-GaussianandperhapsmoreeasilydetectablefeatureinthespectrumofparticlesthatgothroughtheEarth.AlthoughwereachasimilarconclusioninSec.C.2,wedonotagreeonthemagnitudeofthiseffect.Inparticular,wefindthatthedensityofstreamsfromsuchrecentlyaccretedmaterialisafactoroftensmallerthanhasbeenestimatedin[25].Probablecausesforthisdiscrepancycanbeperhapsbeattributedtothedifferentorbitaldistributionsoftheac-cretedlumps(providedabinitioinoursimulations,andassumedtohaveaparticularformin[25]);andinthedif-ferentmethodsusedtomeasurethedensityofastream(totaldensityofmaterialfromanaccretedhaloversusdensityofindividualstreams).

Althoughoursimulationrepresentstheformationofaclusterhalo,ratherthanthatofagalaxy,anditisonlyonepossiblerealizationoftheformationofadark-matterhalo,webelievetheconclusionswehavereachedarero-bust.Forexample,themassgrowthhistoryofthispartic-ularhaloisconsistentwiththatoftheMilkyWay:mostofthemasswasalreadyinplaceintheclusterhalo10Gyrago,ingoodagreementwiththeageoftheoldeststarsintheMilkyWaydisk.Highresolutionsimulationsbyotherauthorshaveshownthatgalaxyandclustersizedhalosgrowinstatisticallysimilarwaysandthatourparticularhaloisnotunusual[36](seealso[44]).Also,theprop-ertiesofthevelocitydistributionofdark-matterintheSolarNeighbourhoodareingoodagreementwiththeob-servedkinematicsofhalostars[43].Moreover,thechar-acteristicsofthestreamsinoursimulation,suchastheirlowdensitiesandlargenumber,areconsistentwiththesimpleanalyticestimatesdiscussedabove.Workalongthelinesof[25]canprovidefurtherinsightonhowmuchvariationonemayexpecttofindonahalo-to-halobasis.

12

V.

CONCLUSIONS

Weanalysedahighresolutionsimulationofthefor-mationofaclusterinaΛCDMcosmology.Byscalingitdowntoagalaxysizehalo(bytheratioofthemaximumcircularvelocities)wewereabletomakepredictionsfortheexpecteddark-matterdistributionneartheSun.Ourresultsindicatethatdirectdetectionexperimentsmayquitesafelyassumethatthedistributionofdark-matterparticlesintheSolarneighbourhoodiswellrep-resentedbyamultivariateGaussian.WefindthatnoneofthestreamspresentinanyofthevolumesattheSun’sdistancefromtheGalacticcentredominatetheirlocaldistribution.Themeandensityofanindividualstreamistypically0.3%thatofthelocaldark-matterdistribu-tion(deducedfromthenumberofparticlesintheratherdensestreamshowninFigure10).Thesesmallvaluesareduetothefactthatmostofthestreamsintheinnergalaxycomefromafewmassivehalosthatmergedathighredshifttobuilduptheobjectweseetoday.Theselargehalosmixextremelyquicklyandthereforegiverisetolowdensitystructures.Strongdensityenhancementssuchasthosepredictedin[24,42]areextremelyunlikelyintheinnerGalaxy.Oursimulationalsoshowsthatwe

shouldnotexpecttofinddense,recentlyformedstreamsneartheSun,sincethelastaccretioncontributingmattertotheSolarneighbourhoodtypicallytookplaceabout1Gyrago,andprovidesonly∼10−4ofthetotalmassinthisregion.Moreover,wefindthatfewerthan1%ofthelocaldark-matterparticlescouldbepartofsmalldensesubhaloswhichhavesurvivedintactwithinthelargerhalooftheMilkyWay.Itisthereforeunlikelythatanindividualhalowiththesecharacteristicswilldominatethesignalindirectdetectionexperiments.

Directdetectionexperimentswhicharesensitivetothedirectionofmotionofthefastestmovingdark-matterparticlesmaydiscoveradirectindicationofthehierar-chicalgrowthofourGalaxy’shalo.Theexpectedsignalforthesefastestmovingparticlesishighlyanisotropic,andcouldbeeventuallybeused,notjusttodeterminethenatureofthedark-matter,butalsotorecover,atleastpartially,therecentmerginghistoryoftheMilkyWay.WethankBenMooreandLeoStodolskyforusefuldiscussionswhichtriggeredmanyoftheideasdiscussedhere;UrosSeljakandAnneGreenforhelpingusimprovethemanuscript.AHwishestothankJokeforbeingacontinuoussourceofinspiration.

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[45]ThisisbecauseM∝R3∝V3,andthusifMf=2Mi

3333

thenRfVf∝4RiVi

[46]Ifinitially∆x∆visthephase-spacevolumeoccupiedby

′

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