Probing MACHOs Toward the Galactic Bulge

If the massive compact halo object (MACHO) fraction of the Galactic dark halo is f ~ 20% as suggested by some microlensing experiments, then about 1.2% of lensing events toward the Galactic bulge are due to MACHOs. For the 40% of these that lie nearby (D_l

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aProbingMACHOsTowardtheGalacticBulgeAndrewGouldDepartmentofAstronomy,TheOhioStateUniversity,140W.18thAve.,Columbus,OH43210gould@astronomy.ohio-state.eduABSTRACTIfthemassivecompacthaloobject(MACHO)fractionoftheGalacticdarkhaloisf～20%assuggestedbysomemicrolensingexperiments,thenabout1.2%oflensingeventstowardtheGalacticbulgeareduetoMACHOs.Forthe40%ofthesethatlienearby(Dl<4kpc),measurementoftheirdistanceDlwoulddistinguishthemfrombulgelenses,whilemeasurementoftheirtransversevelocityvlwoulddistinguishthemfromdisklenses.Hence,itwouldbepossibletoidentifyabout0.5%(f/20%)ofalleventsasduetoMACHOs.IshowthataplannedexperimentusingtheSpaceInterferometryMission(SIMPlanetQuest)couldtherebydetect1or2suchevents.Thisisatthemarginofwhatisrequiredbecauseofasmall,butnon-negligiblebackgroundfromspheroidstars.Subjectheadings:darkmatter–galaxies:stellarcontent–gravitationallensing–instrumentation:interferometers1.Introduction

FollowingthesuggestionofPaczy´nski(1986),theMACHO(Alcocketal.1993)andEROS(Aubourgetal.1993)collaborationsbegansearchingfordarkmatterintheformofmassivecompacthaloobjects(MACHOs)bymicrolensingobservationstowardtheLargeMagellanicCloud(LMC).ThistargetseemedidealbecauseofthesmallcolumnofknownpopulationsofstarscomparedtothehugevolumeofspacethatwouldbehometotheputativeMACHOs.Themicrolensingopticaldepthduetoknownstarswasestimatedtobeτ 9MWLMC=8×10fortheMilkyWaydisk(Gouldetal.1997)andτLMCLMC=1×10 8fortheLMCitself(Gould1995b).Bycontrast,ifthedarkhalowerecompletelycomposedofMACHOs,theiropticaldepthwouldbeoforder,

τLMCv2rothalo～

If the massive compact halo object (MACHO) fraction of the Galactic dark halo is f ~ 20% as suggested by some microlensing experiments, then about 1.2% of lensing events toward the Galactic bulge are due to MACHOs. For the 40% of these that lie nearby (D_l

roughly25timeshigher.Here,vrot=220kms 1istheMilkyWayrotationspeed.Hence,whentheexperimentsbegan,itseemedasthoughevenacrudemeasurementofτwouldunambiguouslydeterminewhetherthehalowascomposedofMACHOs.

Adecadelater,thesituationisfarlessclearthanwasanticipated.MACHO(Alcocketal.2000)foundτ～1×10 7,roughlytheroot-mean-squareoftheresultsexpectedfromMACHOsandstars.Theyinterpretedthistomeanthatthehalowas20%composedofMACHOsandestimatedthetypicalmasstobeM～0.4M⊙.Ontheotherhand,theEROScollaboration(Afonsoetal.2003a;Tisserand&Milsztajn2005)foundanupperlimitfortheopticaldepthduetoMACHOsof5%ofthefull-halovalue.

Oneoptionforresolvingthiscon ictistoexploreotherlinesofsight.Crotts(1992)andBaillonetal.(1993)advocatedM31,andseveralcollaborations,includingAGAPE(Ansarietal.1999),Columbia-VATT(Uglesichetal.2004),MEGA(deJongetal.2004),NainiTal(Joshietal.2005)POINT-AGAPE(Auri`ereetal.2001),SLOTT-AGAPE(CalchiNovati2003),andWeCAPP(Ri eser2003),havepursuedthissuggestion.InmanywaysthisissubstantiallymorechallengingthantheobservationstowardtheLMC,simplybecauseM31is15timesfartherawayandhencethesourcesaresubstantiallyfainter.Eventsarenowbeingreportedfromtheseexperiments,andtheirimplicationsfordarkmattershouldbeavailablesoon.

Themicrolensingtarget eldthathasbeenmonitoredthemostintensivelyistheGalac-ticbulge.OriginallyproposedbyPaczy´nski(1991)andGriestetal.(1991),majorsurveyshavebeencarriedoutbytheOGLE(Udalskietal.1993;Udalski2003),DUO(Alardetal.1995),MACHO(Popowski2005),EROS(Afonsoetal.2003b),andMOA(Abeetal.2004)collaborations.TheprimarymotivationofbothproposalswastoprobefordiskdarkmatterandotherexoticobjectssuchasalargepopulationofJupiters.Griestetal.(1991)doesmentionthatifthehaloiscomposedofMACHOs,thenthesewillgiverisetoanoptical

bulgedepthτhalo=1.3×10 7,butsincethisis4timessmallerthanthepredictedopticaldepth

bulgeduetodiskstarsτdisk=5.1×10 7,theredidnotappeartobeanywaytoisolatethe

MACHOevents.

Bulgemicrolensingobservationshavebeenenormouslyfruitful.Kiraga&Paczy´nski(1994)showedthattheopticaldepthduetobulgeself-lensingwasevengreaterthanthatduetodiskstars.Thehigheventrateencouragedsearchesforlensinganomaliesduetoplanetarycompanionsofthelenses(Mao&Paczy´nski1991;Gould&Loeb1992;Rhieetal.2000;Albrowetal.2001b;Gaudietal.2002;Abeetal.2004),whichhasnowyieldedthe rst rmmicrolensingplanetdetection(Bond2004).Bulgemicrolensinghasenabledthe rstmicrolensmassmeasurement(Anetal.2003)andtheprobingofbulge-staratmosphereswithµasresolutionbothphotometrically(Alcocketal.1997;Albrowetal.1999,2000;Fieldset

If the massive compact halo object (MACHO) fraction of the Galactic dark halo is f ~ 20% as suggested by some microlensing experiments, then about 1.2% of lensing events toward the Galactic bulge are due to MACHOs. For the 40% of these that lie nearby (D_l

al.2003)andspectroscopically(Castroetal.2001;Albrowetal.2001a;Cassanetal.2004).

HereIshowthatbulgemicrolensingcanalsobeusedtoprobeforhalodarkmatter(MACHOs)intheinnerGalaxy.Thisseemsabsurdat rstsightbecausetheobserved

obsopticaldepth,τbulge～2×10 6,isabout15timeshigherthantheratepredictedbyGriestet

haloal.(1991),τbulge～1.3×10 7,evenassumingthatthedarkhalowerecompletelycomposed

ofMACHOs.However,themicrolensingexperimentstowardtheLMCseemtoimplythatthisfractionisnolargerthan20%,whichmeansthatonlyabout1%ofGalacticbulgemicrolensingwouldbeduetohaloobjects.Howwouldoneidentifythesehalomicrolensingeventswithinthebarrageofmicrolensingbyordinarybulgeanddiskstars?

2.NeedleinHaystack

HalolensesaredistinguishedfromdisklensesbythetheirtransversevelocityvlrelativetotheSun,andfrombulgelensesbytheirdistancefromtheSun,Dl(orequivalently,theirabsoluteparallaxπl).Hence,toreliablyidentifythenearby,fastMACHOs,onemustreliablymeasurevlandπl.SincetheMACHOsarebyde nition“dark”matter,directobservationsofthelenscannotbeemployedinmakingthesedeterminations,astheywereforexampleforMACHO-LMC-5(Alcocketal.2001;Drakeetal.2004;Gould2004a;Gouldetal.2004).Instead,thesequantitiesmustbederivedentirelyfromobservationsofthesourceduringandafterthemicrolensingevents.

2.1.Observables

Thesetwoquantitiescanbeexpressedintermsofmicrolensingobservablesby(e.g.,Gould2000),

πl=πrel+πs,πrel=πEθE(2)

and

vl=.(3)tE

Here,πl,πsµl,µsaretheabsoluteparallaxesandpropermotionsofthelensandsource,πrel=πl πsandµrel=µl µsarethelens-sourcerelativeparallaxandpropermotion,θEistheangularEinsteinradius,tEistheEinsteintimescale,andπEisthemicrolensparallax(i.e.,theinverseoftheprojectedEinsteinradius,πE=AU/r E).ThedirectionofθEisthatofthelens-sourcerelativepropermotion.

Inbrief,todetermineπlandvl,onemustmeasure veobservables,two2-vectors(µsµrel+µs

If the massive compact halo object (MACHO) fraction of the Galactic dark halo is f ~ 20% as suggested by some microlensing experiments, then about 1.2% of lensing events toward the Galactic bulge are due to MACHOs. For the 40% of these that lie nearby (D_l

andθE)andthreescalars(πs,πE,andtE).

2.2.ParameterMeasurement

Twoofthese veparameters(πsandµs)arerelatedsolelytothesource,whiletheremainingthree(πE,tE,andθE)aremicrolensing-eventparameters.Ofthesethree,onlyone(tE)isroutinelymeasuredduringmicrolensingevents.Theothertwoarehigherorderparameters.WhilethereareavarietyofmethodstomeasureπEandθE(seeGould2001),thesegenerallyapplytoonlyasmallfractionofevents.Thereareonlytwoevents(outofalmost3000discovered)forwhichbothparametershavebeenmeasuredfrommicrolensingdataalone(Anetal.2003;Kubasetal.2005),andbothofthesewerebinarylenses.

TheonlyknownwaytoroutinelydetermineθEisbyhigh-precisionastrometricmea-surementsofthemicrolensingevent(Høgetal.1995;Miyamoto&Yoshii1995;Walker1995;Paczy´nski1998;Bodenetal.1998).ThecentroidofthemicrolensedimagesdeviatesfromthesourcepositionbyanamountanddirectionthatyieldsbothcomponentsofθE.

TheonlyknownwaytoroutinelydetermineπEistomakephotometricmeasurementsoftheeventfromtwolocationsseparatedbyoforderr E(Refsdal1966;Gould1994).Thedi erenceintheeventparametersthenyieldsboththesizeofr Eandthedirectionofmotion(thelatterpotentiallycon rmingthedirectionextractedfromθE).Sincer E～O(AU),inpracticethismeansplacingasatelliteinsolarorbit.Althoughthereisafour-foldambiguityinthedeterminationofπE,thiscanberesolvedbyhigher-ordere ects(Gould1995a).Moreover,measurementofthedirectionofθEalsohelpsresolvethisdegeneracy.

2.3.SIMPlanetQuestMeasurements

Gould&Salim(1999)showedthattheSpaceInterferometryMission(SIMPlanetQuest)combinedwithground-basedphotometry,coulddeterminebothoftheseparameterswithgood～3%precisionwithabout5hourstotalobsrvationtimeforbright(I～15)eventshavingtypicallensparameters.Moreover,theyshowedthatthesameobservationswouldalsoyieldgoodmeasurementsofπsandµs.Hence,SIM(combinedwithground-basedphotometry)couldmeasurealltherequiredquantitiesforabout200eventswithabout1000hoursofobservingtime.Indeed,aSIMKeyProjecthasbeenawarded1200hoursofobservationtimetocarryoutsuchobservations.Themainobjectiveofthisprojectistomeasurethebulgemassfunctionbutthesameobservationscouldculloutthehandfulofhaloeventsthatcouldbepresentinthesamesample.

If the massive compact halo object (MACHO) fraction of the Galactic dark halo is f ~ 20% as suggested by some microlensing experiments, then about 1.2% of lensing events toward the Galactic bulge are due to MACHOs. For the 40% of these that lie nearby (D_l

SIMhasbeendescopedsinceGould&Salim(1999)madetheiranalysis.Thenewperformanceisnotpreciselyknownbutitislikelythattheprecisionwilldegradetosomethinglike～5%forπEand～10%forθEforthecanonicaleventsconsideredbyGould&Salim(1999).Moreover,itisunlikelythat200I=15eventswillbefoundduringthe5-yearprimarySIMmission,andusingfaintersources(e.g.,I=16.5)wouldfurtherdegradetheprecisionbyafactor2.Nevertheless,asIshowbelow,thisprecisionwouldbequiteadequatefordistinguishinghalolenses.

3.BackgroundfromSpheroid/BulgeStars

HalolensescouldproduceeventsanywherealongthelineofsightfromtheSuntothebulgeand,assuminganisothermalhalomodelwithcoreradiusa=5kpc,thedensity

ρhalo=2vc

dDl=ρhaloc2f(a/R0)2+(1 x)2.(5)

Here,rE=(4GMDlDs/c2Dos)1/2istheEinsteinradius,DlandDsarethesourceandlensdistances,Dls=Ds Dl,fisthefractionofthehalointheformofMACHOs,R0=8kpcistheSolarGalactocentricdistance,x≡Dl/R0,andM(whichcancelsout)isthemassofthelens.

However,intheinnerGalaxy,thesehalolensesarecompletelysubmergedintheback-groundofbulgelenses,andsincetheyhavesimilarkinematics,thereisnowaytoreliablydistinguishthem.ItisonlyoutclosertotheSun,wherethespheroidalpopulation(hereusuallycalled“spheroid”or“stellarhalo”),thinsoutthatonemayhopetoseparatethetwopopulations.Evenhere,thereissomepossibilityofcontamination.Thelocalspheroiddensityisonlyabout1%ofthedarkhalo,butiff～20%asAlcocketal.(2000)suggest,thenMACHOsareonly20timesmorecommonthanspheroidstarslocally.Moreover,asoneapproachestheGalacticcenter,thespheroiddensitygrowssubstantiallymorerapidlythandoesthedarkhalo.Tomakeaquantitativecomparison,Iadopt

ρspheroid=1×10 4M⊙

R0 3.2.(6)

If the massive compact halo object (MACHO) fraction of the Galactic dark halo is f ~ 20% as suggested by some microlensing experiments, then about 1.2% of lensing events toward the Galactic bulge are due to MACHOs. For the 40% of these that lie nearby (D_l

Afteraccountingforobservedstarsandextrapolatingdowntobrowndwarfsanduptotheprogenitorsofremnants,Gouldetal.(1998)estimate6.4×10 5M⊙pc 3.However,bothDahnetal.(1995)andGould(2003) ndsubstantiallymorelow-luminosity(MV>8)starsthandidGouldetal.(1998)intheirmorelocalsample(seeFig.2fromGould2004b),soIhaveadjustedtheirestimateupward.Thepower-lawslopeismeasuredbyseveraltechniques(Gouldetal.1998andreferencestherein).

Figure1showstheopticaldepthperunitdistanceduetospheroidstarsandtoputativeMACHOsundertheassumptionthatf=20%.ItshowsthatevenwithaMACHOfractionof20%,thehalodominatesthespheroidfromR=R0toR=4kpc,whichlatterisaboutthelimittowhichthelocalspheroiddensitypro lecanbereliablyextrapolated.However,thisdominationisnotoverwhelming:atR=4kpcitisonlyafactorof5andevenatR=7kpc(wherethehaloopticaldepthhasfallenbyafactor5)thehaloonlydominatesbyafactor

10.Thismeansthat2or3halolenseswouldhavetobeidenti edtoconstituteareliable“MACHOdetection”.Otherwise,therewouldbeasigni cantpossibilitythatspheroidlenseswereresponsible.

SinceonemustrestrictattentiontoDl<4kpc,thetotalavailablehaloopticaldepthisreducedbyafactor0.4relativetothe1.3×10 7calculatedbyGriestetal.(1991).Ifwefurtherassumef=20%,theavailablehaloopticaldepthisfurtherreducedto10 8,about0.5%oftheobservedopticaldepthofτ～2×10 6(Afonsoetal.2003b;Popowski2005;Sumietal.2005).Hence,assumingforthemomentthattheeventratesareinproportiontotheopticaldepths,roughly200measurementswouldberequiredtoidentifyasinglehalolens.Thus,iftheSIMmissionwereextendedfrom5to10years(asiscurrentlyenvisioned)thenonemightexpectto ndabout2halolenses.Asnotedabovethisisjustatthemarginofaviabledetection.

4.PracticalConsiderations

4.1.EventTimescales

TheEinsteintimescalesofthesehaloeventsaregivenby,

tE=15day M

0.25 1/2 v⊥

If the massive compact halo object (MACHO) fraction of the Galactic dark halo is f ~ 20% as suggested by some microlensing experiments, then about 1.2% of lensing events toward the Galactic bulge are due to MACHOs. For the 40% of these that lie nearby (D_l

wellbeforepeakinordertomeasureπE(Gould&Salim1999).Hence,afairlyaggressivepostureisrequiredtokeepthehaloeventsinthesample.

However,thefactthatthesehaloeventsaresomewhatshorterthantypicalbulgeeventsmeansthattheyarealsomorefrequentthanwouldbeindicatedbytheiropticaldepthalone.Thatis,theeventrateΓ∝τ/tE,sotherateisinverselyproportionaltothetimescale.Hence,theshortertimescalesenhancestheviabilityofagivenexperimentrelativetowhatwasdiscussedin§3,providedthatnottoomanyhaloeventsarelostbecausetheyaretooshort.

4.2.Signal-to-NoiseRatios

Thetwomicrolensingparametersbeingmeasuredarerelatedtotheunderlyingphysicalparametersby, πE=,θE=κM

If the massive compact halo object (MACHO) fraction of the Galactic dark halo is f ~ 20% as suggested by some microlensing experiments, then about 1.2% of lensing events toward the Galactic bulge are due to MACHOs. For the 40% of these that lie nearby (D_l

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If the massive compact halo object (MACHO) fraction of the Galactic dark halo is f ~ 20% as suggested by some microlensing experiments, then about 1.2% of lensing events toward the Galactic bulge are due to MACHOs. For the 40% of these that lie nearby (D_l

8log(dτ/dDl[kpc]) 9 10

1134567

Galactocentric Distance (kpc)8

Fig.1.—OpticalDepthperunitpathlengthdτ/dDlasafunctionofdistancefromtheGalacticcenterforasourceneartheGalacticcenter.Thehalo(assumingaf=20%MACHOfraction)andthespheroidareshownbysolidanddashedcurves,respectively.Forf=20%,spheroidstarsarea20%backgroundatR=4kpcanda10%backgroundatR=7kpc,whichimpliesthat2or3halolensesmustbeidenti edatR>4kpcforareliablehalo“detection”.InsideR<4kpcthespheroidcontinuestogrow(andalsotransformsintothebulge),makingtheidenti cationofhalolenseslesssecure.Hence,theexperimentshouldberestrictedtoR>4kpc,wherethetotalopticaldepthisτ=5×10 8f.

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