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DraftversionJanuary25,2011 PreprinttypesetusingLATEXstyleemulateapjv.11/10/09 ASTROMETRICMICROLENSINGBYLOCALDARKMATTERSUBHALOS AdrienneL.Erickcek1,2andNicholasM.Law3 DraftversionJanuary25,2011 ABSTRACT High-resolution N-body simulations of dark matter halos indicate that the Milky Way contains numerous subhalos. When a dark matter subhalo passes in frontof a star, the light from that star will be deflected by 1 gravitationallensing,leadingtoasmallchangeinthestar’sapparentposition. Thisastrometricmicrolensing 1 signal depends on the inner density profile of the subhalo and can be greater than a few microarcseconds 0 for an intermediate-mass subhalo (M > 104M ) passing within arcseconds of a star. Current and near- 2 vir futureinstrumentscoulddetectthissignal,∼andwee⊙valuatetheSpaceInterferometryMission’s(SIM’s),Gaia’s, n and ground-basedtelescopes’ potential as subhalo detectors. We developa generalformalism to calculate a a subhalo’sastrometriclensingcrosssectionoverawiderangeofmassesanddensityprofiles,andwecalculate J thelensingeventratebyextrapolatingthesubhalomassfunctionpredictedbysimulationsdowntothesubhalo 3 massespotentiallydetectablewiththistechnique.Wefindthat,althoughthedetectableeventratesarepredicted 2 tobelowonthebasisofcurrentsimulations,lensingeventsmaybeobservedifthecentralregionsofdarkmatter subhalosaremoredensethancurrentmodelspredict(>1M within0.1pcofthesubhalocenter).Furthermore, ] targetedastrometricobservationscanbeusedtoconfir∼mthe⊙presenceofanearbysubhalodetectedbygamma- O ray emission. We show that, for sufficientlysteep density profiles, ground-basedadaptiveoptics astrometric C techniquescouldbecapableofdetectingintermediate-masssubhalosatdistancesofhundredsofparsecs,while . SIMcoulddetectsmallerandmoredistantsubhalos. h Subjectheadings:astrometry–darkmatter–Galaxy:halo–gravitationallensing:micro p - o r 1. INTRODUCTION halos’innermostregions. Itisthereforepossiblethatnumer- t oussubhalosarelocatedwithinafewkiloparsecsoftheSolar s Numerical simulations of dark matter halos have re- a System,withprofoundimplicationsforbothdirectdetection vealed the presence of numerous subhalos over a wide [ of the dark matter particle (Kamionkowski&Koushiappas range of masses extending down to the simulations’ res- 2008) and indirect detection through its annihilation signa- 2 olution limits (e.g., Ghignaetal. 1998, 2000; Klypinetal. ture (e.g., Bergstro¨metal. 1999; Calcańeo-Roldań&Moore v 1999a; Mooreetal. 1999; Kravtsovetal. 2004; Gaoetal. 2000; Stoehretal. 2003; Diemandetal. 2007a; Ando 2009; 8 2004;Reedetal. 2005;Diemandetal. 2007b;Springeletal. Kamionkowskietal.2010). 2 2008; Diemandetal. 2008). This substructure is the rem- 2 nant of hierarchical structure formation; as halos merge to Unfortunately, both indirect and direct detection of dark 4 formlargerstructures,theinnerportionsoftheancestorhalos matter continues to be elusive, and gravity still provides 7. become subhalos. If all halos leave subhalo remnants, then the only uncontested evidence for dark matter. Gravita- tional lensing is an especially powerful tool in the study of 0 the subhalo mass functionmay extendto masses far smaller dark matter substructure; subhalos within our galaxy could 0 than the resolution limit of any simulation (Huetal. 2000; be detected through their effects on signals from millisec- 1 Chenetal.2001;Profumoetal.2006;Diemandetal.2005). ond pulsars (Siegeletal. 2007), and substructure in lens- : Although subhalos may be destroyed by gravitational in- v ing galaxies has several observational signatures. Subha- teractions with the host halo, other subhalos, and stars, i los have been proposed as the origin of observed flux-ratio X thereareindicationsthattheirdenseinnerregionssurviveto anomalies between the multiple images of strongly lensed thepresentday(Hayashietal.2003;Kazantzidisetal.2004; r quasars (Mao&Schneider 1998; Metcalf&Madau 2001; a vandenBoschetal.2005;Readetal.2006;Berezinskyetal. Chiba 2002; Dalal&Kochanek 2002; Kochanek&Dalal 2006;Zhaoetal.2007;Green&Goodwin2007;Goerdtetal. 2004); theycan also alter thetime delaysbetweenthese im- 2007; Schneideretal. 2010; Ishiyamaetal. 2010). More- ages (Keeton&Moustakas 2009; Congdonetal. 2010) and over,high-resolutionsimulationsofhalossimilartotheMilky their separations (Koopmansetal. 2002; Chenetal. 2007; Way’s host suggest that subhalos with masses greater than Williamsetal. 2008; Moreetal. 2009). If the source is 106M are present at the Solar radius (Springeletal. 2008; extended, then subhalos can distort the image’s shape and Diema⊙ndetal. 2008). Although simulations indicate that surface brightness (Metcalf 2002; Inoue&Chiba 2005a,b; theselargesubhalosaresignificantlydisruptedbytheGalac- Koopmans 2005; Vegetti&Koopmans 2009a,b). Intrigu- ticdisk(D’Onghiaetal.2010;Romano-D´ıazetal.2010),the ingly, the observedlensing anomaliescan only be explained resolution is not sufficient to determine the fate of the sub- if the central regions of the lensing galaxies contain signifi- 1Canadian Institute for Theoretical Astrophysics, University of cantlymoresubstructurethanispredictedbyn-bodysimula- Toronto, 60 St. George Street, Toronto, Ontario M5S 3H8, Canada; tions(Maoetal.2004;Amaraetal.2006;Maccio` etal.2006; [email protected] Xuetal. 2009, 2010), although it has been suggested that 2PerimeterInstituteforTheoreticalPhysics,31CarolineSt. N,Water- thesestudiesuseatypicallensinggalaxies(Bryanetal.2008; loo,OntarioN2L2Y5,Canada 3Dunlap Fellow, Dunlap Institute for Astronomy and Astrophysics, Jacksonetal.2010). University of Toronto, 50 St. George Street, Toronto ON M5S 3H4, Individual subhalos within a lensing galaxy could be de- Canada 2 Erickcek&Law tected if they strongly lens one quasar image, splitting it explore the possibility of using astrometric lensing to con- into a closely spaced pair of images (Yoneharaetal. 2003; firmthepresenceofasubhalodetectedthroughitsgamma-ray Inoue&Chiba 2005a). Unfortunately, the diffuse nature of emission. We findthatground-basedtelescopescoulddetect dark matter subhalos implies that they have small Einstein lensing by a nearby ( 50 pc) subhalo with a post-stripping ∼ radii(Zackrissonetal.2008). Consequently,thesplitimages massgreaterthan1000M ,whileSIMcouldprobethesesub- areresolvableonlyforthelargestsubhalos,makingitunlikely halosatgreaterdistances⊙( 100pc)anddetectnearbysubha- to find a large enough subhalo with a sufficiently small im- loswithone-tenthofthism∼ass. pactparametertodetectablysplitaquasarimage(Riehmetal. This article is organized as follows. In Section 2, we de- 2009). scribetheimagemotioninducedbysubhalolenseswithava- Inthispaper, we considera differentwayto find subhalos riety ofdifferentdensityprofiles, namely, a singularisother- throughgravitationallensing: insteadoflookingforsplitim- mal profile, the NFW profile, and a generalized power-law ages,weinvestigatehowtheastrometricdeflectionofanim- density profile. In Section 3, we develop an astrometric ob- agechangesasasubhalomoves. Thisastrometricmicrolens- serving strategy designed to reliably detect subhalo lensing ing approach has two advantagesover strong lensing. First, while rejectingpossible false-positivedetections. In Section it is much easier to measure a change in the position of the 4, we calculate the areas of sky over which particular sub- centroid of an image than it is to resolve an image pair into halo lenses are detectable and determine the all-sky subhalo distinctsources. Theminimalseparationrequiredto resolve lensingeventratesforseveralmodelsofdarkmattersubstruc- apairofimagesislimitedtoapproximatelytheresolutionof ture. InSection5,weevaluatecurrentandplannedastromet- thetelescope(>25mas),whereasathighsignal-to-noiseratio ricsurveycapabilitiesinthecontextofdetectinglensingfrom (S/N), the pos∼ition of the centroidof the image can be mea- a subhaloin bothall-skyandtargetedsearches. We summa- suredwithhundredsoftimeshigherprecision.Second,strong rize our findingsand concludein Section 6. To evaluatethe lensingonlyoccurswhentheangularseparationbetweenthe lensing signatures and event rates, we developed models of sourceand the lens is smaller than the lens’s Einstein angle, theconcentration-massrelationandthemassfunctionforlo- whileastrometricdeflectionisdetectableforfarlargerimpact cal subhalos based on the findings of the Aquarius simula- parameters.Thedisadvantageassociatedwithastrometricmi- tions(Springeletal.2008);thesemodelsarepresentedinthe crolensingisthatitmustbeadynamicaleventbecausewedo appendix. notknowthe true position of the source. As the lens moves 2. ASTROMETRICSIGNATURESOFLENSINGBYSUBHALOS relative to a background star, the star’s image will move as well, and that is the detectable signature of astrometric mi- Theshapesoftheintermediate-masssubhalosthatwewill crolensing. We are therefore constrained to local subhalos beconsideringcannotbe probedbycurrentnumericalsimu- withsignificantpropermotions. lations;tosimplifyouranalysiswewillassumethatthesub- Astrometriclensing signals fromdarkmatter subhalosare halosare spherically symmetric. Numericalsimulations can necessarilysmallbecausesubhalosare diffuse, andastrome- only probe the shapes of the largest subhalos with masses tryhasonlyveryrecentlyprogressedtothepointthatanastro- greater than 108M (Kuhlenetal. 2007; Knebeetal. 2010). metricdarkmattersearchisfeasible. Therapiddevelopment Whilethesesubhal⊙osaretriaxial,theytendtobemorespher- of1-100µasastrometricprecisionhasbeendrivenbyawide icalthantheirhosthalos. Inasimulationthatincludesbary- varietyoffields–forexample,followingorbitsinthegalactic onicphysics,Knebeetal.(2010)foundthatsubhaloslocated center (e.g., Luetal. 2009; Gillessenetal. 2009); astromet- within half of the virial radius of the host halo have nearly ric detection of planets (e.g., Unwinetal. 2008; Lawetal. spherical mass distributions in their innermost resolved re- 2009; Malbetetal. 2010); accurate parallax determination gions, with a median minor-to-majoraxis ratio of 0.83. We (e.g., Henryetal. 2009; Subasavageetal. 2009); and the willseethatastrometricmicrolensingisonlysensitivetothe determination of stellar orbits (e.g., Hełminiaketal. 2009; density profile near the center of the subhalo (r < 0.1 pc); Pravdoetal. 2006; Konopackyetal. 2010; Irelandetal. aslongastheinnerregionofthesubhaloisnearlyspherically 2008;Dupuyetal.2009). symmetric,thesubhalo’striaxialityisirrelevantforouranaly- New space-based astrometric missions such as Gaia sis,andwedonotexpectdeviationsfromsphericalsymmetry (Lindegrenetal. 2008) and SIM (Unwinetal. 2008) are tosignificantlyaffectourresults. opening the possibility of ultra-high-precision all-sky and We will also assume that the darkmatter subhalois trans- targeted searches. Targeted ground-based astrometry is al- parent and that it contains no stars. The presence of a star ready capable of 100µas precision in arcminute-sizedfields, in the center of the subhalo would change its lensing signa- andnewlargertelescopeswillsignificantlyimprovethatpre- tureif the star’s massis comparableto thedarkmatter mass cision (Cameronetal. 2009), while even higher precisions in the central 0.1 pc of the subhalo, but stars further from are possible on bright stars (e.g., Muterspaughetal. 2006; the center would not have an effect. Furthermore, the dis- vanBelleetal.2008). crepancy between the number of intermediate-mass subha- Thesetechnicaladvanceshaveinspiredconsiderableinter- losseen in simulationsand the numberofdwarf spheroidals estinastrometricmicrolensingbystarsanddarkcompactob- observed in the Milky Way, known as the “missing satellite jects (Walker 1995; Paczynski 1995, 1998; Miralda-Escude problem” (Klypinetal. 1999b; Mooreetal. 1999), indicates 1996; Gould 2000; Gaudi&Bloom 2005), and by baryonic that stars are rare in subhalos with masses less than 106M clouds (Takahashi 2003; Leeetal. 2010). In this paper, we (Madauetal. 2008). Finally, we will assume that the sub⊙- investigateiftheseinstrumentscanalsobeusedtosearchfor halo’sdiameterissmallcomparedtoboththedistanced be- L darkmattersubhalos. We findthat, forstandarddarkmatter tweenthelensandtheobserverandthedistanced between LS models,observingasubhalolensingeventrateduringablind thelensandthestar. astrometric survey is highly unlikely. If the central regions Figure1showsaschematicviewoflensingbyanextended of dark matter subhalos are denser than expected, however, transparentobjectlikeadarkmattersubhalo.Whenalightray thelensingeventratecanbesignificantlyenhanced. Wealso passesthroughasphericallysymmetricthinlens,theimageof LocalDMSubhaloLensing 3 I a) b) I S α β S α ξ Obs θ β V /d y T L d d L LS d x S Fig.1.—a)Diagramshowingthepositionofthesourcestar(inblack),itsimage(ingray),andthelens(grayellipse). Wewillgenerallyassumethatα βso thattheray’simpactparameterinthelensplane(ξ dLθ)maybeapproximatedasξ dLβ. b)Thesamelensingsystemviewedasprojectedonthesk≪y. The centerofthelensismovingwithvelocityvTalongth≡ex-axis. ≃ thestarisshiftedfromitstruepositionbyanangle 2.1. Singularisothermalsphere Thedensityprofileforasingularisothermalsphere(SIS)is d 4GM (ξ) ~α= LS 2D ξˆ, (1) d ξ σ2 S " # ρ(r)= v , (3) 2πGr2 whered isthedistancebetweentheobserverandthestar,ξ S is the distance between the center of the lens and the star’s where σv is the velocity dispersion of the halo. Although image in the lens plane (~ξ d ~θ), and ξˆ ~ξ/ξ points from numericalsimulationsindicate that large dark matter subha- L thelenstothestar. Through≡outthiswork,w≡esetthespeedof los without baryons do not have this steep a density profile lightc = 1. Themass M (ξ)inEq.(1)isthemassenclosed (Springeletal. 2008; Diemandetal. 2008), we consider the 2D bythecylinderinteriortoξandisobtainedbyintegratingthe SIS case in detail because it simply illustrates key features projected surface mass density Σ over the area of the circle thataresharedbytheastrometriclensingsignaturesfromdark withradiusξ. matterhaloswithshallowerprofiles. As the subhalo moves relative to the backgroundstar, the The two-dimensional enclosed mass for an infinite SIS is angle ~β that extends from the lens to the star will change, M2D(ξ) = πσ2vξ/G. Since M2D dependslinearlyonξ,Eq.(1) reveals that α is independent of the separation between the andthe positionof thestar’s imagewillchangeaccordingly. lensandthestar. ThedeflectionangleisalwaystheEinstein We take the star to be fixed at the origin of an xy coordi- angleoftheSIS: natesystemonthesky,andwedefinethex-axistobeparallel tothe subhalo’stransversevelocityv , as shownin theright T d panelofFig.1. Theverticalcomponentof~βisthereforefixed θESIS= 1− dL 4πσ2v, (4) [β (t)=β ],and S! y y,0 σ 2 d =10µas v 1 L . (5) v 50pc t βx(t)=βx,0−4.2′′ 200kTm/s d 5yr , (2) 0.6km/s! − dS! ! L ! ! There are two images, with ~α = θSISβˆ, only if β < θSIS. where β is the value of β at t = 0. We see that a nearby Wewillonlyconsiderlargerimpact±paErameters,inwhichcEase x,0 x subhalowillmoveseveralarcsecondsduringamulti-yearob- thereisonlyoneimage,withα~ =θSISβˆ.AstheSISmovesrel- E servationalperiod. ativetothestar,thedirectionofthedeflectionanglechanges. For the subhalos we consider, the deflection angle α will ForaninfiniteSISmovingfromthedistantlefttothedistant beontheorderofmicroarcseconds. Sinceβchangesbysev- right,theimagestartsθSIS totherightofthestar’strueposi- E eralarcsecondsoverthecourseoftheobservation,β αfor tionandthentracesahalf-circlewithradiusθSIS untilitends mostoftheobservationalperiod.Wewillfurtherassu≫methat E θSIS totheleftofthestar’strueposition. βy,0 ≫ αsothatwearealwaysconsideringtheweak-lensing EFor an SIS, the mass enclosed in a sphere of radius R is regimewithβ α. We verifyin AppendixA thatthiscon- ≫ proportionaltoR;iftheSIShasinfiniteextent,thenitsmass dition is satisfied for all subhalo lensing scenarios, provided isinfinite. ItiscustomarytocharacterizeanSISbyitsvirial thatd 1000kpc. Thisconfirmsthatwearefirmlyinthe L ≪ mass:themasscontainedinaspherewithmeandensityequal weaklensingregimeaslongasweonlyconsidersubhalosin to the virial density ρ¯ . Bryan&Norman (1998) providea our local group. In this case, there is only one image of the vir fittingformulaforthevirialdensityinaflatΛCDMuniverse, star, and it is always located on the line connecting the lens positiontothestar’sposition,withthestarbetweentheimage ρ¯ 18π2+82[Ω (z) 1] 39[Ω (z) 1]2 ρ ,(6) andthelens. We willuse theβ αassumptiontosimplify vir≡ M − − M − crit btheealpenpsroexqiumaatitoendbays ~ξap≃prodxLi~βm,aatni≫ndgE~βq≃. (~θ1.)Ibnetchoimsceassae,s~ξimmpalye ΩM(z)=(cid:16)ΩM0(1Ω+M0z()13++1z)−3 ΩM0, (cid:17) (7) equationforthedeflectionangleα~ intermsoftheimpactpa- M rameter~β. Inthefollowingsubsections,wewilluseEq.(1)to ρcrit(z)= 0.0924 pc⊕3 h2 ΩM0(1+z)3+1−ΩM0 , (8) showhowthepathtakenbythestar’simageduringasubhalo ! h i transitdependsonthesubhalo’sdensityprofile. where H 100hkms 1Mpc 1 and Ω is the present-day 0 − − M0 ≡ 4 Erickcek&Law matter density divided by the critical density. We will use standard cosmologicalparameters: h = 0.7 and ΩM0 = 0.3. 14 βy = 1 arcsec The virial density is 4.6 M pc 3 at redshift zero and it in- 12 − 1 yr creasesmonotonicallywith⊕redshift. Thevelocitydispersion 10 σv intermsofthevirialmassis αµ [as]y 68 σ2 =G πρ¯vir(zv)Mv2ir 1/3, (9) 4 v 6 2   0 wherezvistheredshiftatwhichthehalo’svirialmassiseval- 12 βy = 50 arcsec uated.TofacilitatecomparisonswithN-bodysimulations,we 10 10 yrs take z = 0 in our calculations, but we note that increasing lzevnwsionuvgldsigmnaaklse.tIhnesesrutbinhgaltohsisdeexnpsreersasinodnwinotuolEdqe.n(h4a)ngcievethseir αµ [as]y 68 4 d M 2/3 ρ¯ (z ) 1/3 2 θSIS =(1.1µas) 1 L vir vir v . E − d 104M 4.6M pc 3 0 S! ⊙! ⊕ − ! (10) -15 -10 -5 0 5 10 15 α [µas] These properties describe an SIS in isolation. Once a x subhalo is accreted by a larger halo, the outer tails of its Fig.2.—Thedeflectionangleforastarat5kpcasanSISsubhalo50pc density profile are stripped of mass. Numerical simula- awaymovesatavelocityof200km/sfromthefarlefttothefarright. This tions indicate that a subhalo in our Galaxy may lose be- subhalo’svelocitydispersionis0.72km/s,whichcorrespondstoavirialmass tween 99% and 99.9% of its initial mass due to tidal strip- of5 105 M ,butithasbeenstrippedtoaradiusof0.02pcandcontains only×5 M .In⊙thetopplot,thesubhalocenterpasses1arcsecondbelowthe ping from the smooth componentof the halo (Hayashietal. star,andt⊙hecirclesshowtheimage’spositioneveryyearforthe100years 2003; Kazantzidisetal. 2004; vandenBoschetal. 2005; surrounding thetimeofclosest approach. Inthebottom plot, thesubhalo Readetal. 2006), and stars will further strip the outer por- centerpasses50arcsecondsbelowthestar,andthecirclesshowtheimage’s tions of subhalos (Berezinskyetal. 2006; Zhaoetal. 2007; position every 10 years for the 300 years surrounding the time of closest approach. Green&Goodwin 2007; Goerdtetal. 2007; Schneideretal. 2010; Ishiyamaetal. 2010). We will deal with this trunca- 1 1 1 tion by defininga truncationradiusRt and setting ρ = 0 for (x) tan−1 1+ 1. R > R. ThemasscontainedwithinR isthemassofthesur- F ≡ x2 − x − x2 − t t r r vivingsubhalo M. We will describethe tidal strippingwith t Figure 2 shows how the image of a fixed star moves as a theparameterm M/M ,whereM istheoriginalvirial bd t vir vir truncatedSISsubhalopassesbelowthestar’struepositionon ≡ massof the subhalo,evaluatedatredshiftzv. Thetruncation thesky. Thelens’sEinsteinangleisθSIS =15µas,whichcor- radiusisthengivenby E respondstoσ =0.72km/sandM =5 105M .Theimage v vir Rt = 0.56pc 0m.0b0d1 10M4vMir 1/3 4.6ρ¯M⊕(zp)c−3 1/3. (11) dmeotetiromninisehsihgohwly~βsecnhsiatnivgeestowthitehrtaitmioev[Ts/e×deLEbqe.c(a2⊙u)s]e. tIhfisvTra/tdioL (cid:18) (cid:19) ⊙! vir v ! is decreased, then the change in ~β during a set time interval (cid:0) (cid:1) Theangularsizeofthetruncatedhalois is decreased, and the image motion slows down. Through- m 50pc M 1/3 4.6M pc 3 1/3 out this work, we adopt vT = 200 km/s because that is the θt =0.64◦(cid:18)0.0b0d1(cid:19) dL ! 104vMir⊙! ρ¯vir⊕(zv)− ! (1.2) tDwyriplulikcsaieelervtehetlaaotlc.ait1dy9e8toe6fc;taaXbdulaeersekutbmahl.aat2ltoe0r0m8pu)as.rttWihcalivetheidnthits<hev1eh0loa0lcopitcy(,,ea.wngd.e, L ThusweseethatθESIS ≪θt forallsubhalosofinterest. we adoptdL = 50pc asourfiduciallensdistan∼ce. Provided ForatruncatedSIS,thetwo-dimensionalenclosedmassis that d d , the distance to the source has a minimal im- L S ≪ pactontheimagemotionbecaused onlyentersthroughthe S 2σ2 R2 factor(1 d /d ) inθSIS [see Eq.(10)]. We used = 5kpc M2D(ξ <Rt)= Gv ξtan−1 sξ2t −1+Rt− R2t −ξ2 asourfid−uciaLlvaSluebeEcausethisisareasonabledisStancetoa andM = M forξ R. Ifξ R,thenthedeflqectionan(1g3le) htaarvgeetnsotanr;octihcaenagbilnegeffdSectot oannythvealiumeaggreeamteortitohnan. 1Tokpilcluwstorualtde 2D t ≥ t ≥ t theeffectsofsubhalotruncationinFig.2,weassumethatthe isthesameasforapointmasswithEinsteinangle subhalohasbeenextremelystrippedbycloseencounterswith stars so that its radius is 0.02 pc (θ = 85 ), which implies 4GMd 2 t ′′ θPEM ≡ r dLdtSLS = rπθESISθt. (14) tthhraeteMsttag=es5:Mas⊙t.heWeedgseeeofthtahtetshuebihmalaogeapmprootiaocnhecsofnrsoimststhoef ThusweseethatθSIS θ impliesthatθPM θ. Therefore farleft,theimageveryslowlymovesrightwarduntilthestar E ≪ t E ≪ t is behind the subhalo, then the image rapidly traces an arc we may approximate the position of the brightest image as α = (θPM)2/βforβ > θ. Whenweinsert M fromEq.(13) as the subhalo center passes by the star, and finally the star E t 2D slowlyreturnstoitstruepositionasthesubhalomovesoffto into Eq. (1) for α and assume that α β, we find that the ≪ theright. deflectionangleforlensingbyatruncatedSISis The impact parameter β determines how quickly the im- y 2 β for θSIS <β<θ agemovesasthesubhalopassesbythestar. Inthetophalfof α~= θSISβˆ F θt E t , (15) Fig. 2thelensimpactparameteris1 ,andtheimagerapidly π E θt for β>θ ′′ β (cid:16) (cid:17) t   LocalDMSubhaloLensing 5 1 cosh 1 1for x<1 15 x √1 x2 − x 20µas motion (x)=ln + 1 − for x=1 10 G 2  √x12−1cos−1 1x for x>1 (Bartelmann1996;Golse&Kneib2002). 5 From Eq. (1), we see that the magnitudeof the deflection angleαisproportionaltoM /ξ. FortheNFWprofile,α ξ 2D arcsec 0 hifarlo≪aprpsr,oaancdheαsa∝sξta−r1,itfherd≫eflresc.tiTohnearnefgolerew,ailslainncNreFaWseus∝unbti-l Y/ thestarcrossesthescale radius(ξ r ),andthenitwillde- s ≃ creaseuntilthestarcrossesthesubhalocenter,atwhichpoint -5 itwillbegintoincrease againuntilthestar crossesr onthe s othersideofthesubhalo. Inthissense,thescaleradiusofan -10 NFWprofileplaysthesame roleas thetruncationradiusfor a truncatedSIS. Ifthe impactparameteris closeto the scale radius(β r /d ),thentheimagetrajectoryisroughlycircu- -15 y ≃ s L lar,anditresemblesthebottomhalfofFig.2. Unfortunately, -5 0 5 10 15 thesubhalosthataremassiveenoughtodeflectthestar’sim- X/arcsec age by several microarcseconds(M > 104M ) have large vir Fig.3.—Thedeflectioninducedover4yearsbyamovingSISlenswith scaleradii(r > 2pcforc < 100);β ∼ r /d i⊙sa largeim- s y s L thesamepropertiesasinFig.2. Thepathofthelensisdepictedbyadotted pact parameter∼, and the ima∼ge position≃changes very slowly arrow. Toshow theimagetrajectories, theimagemotionis exaggerated a asthesubhalomoves. Moreover,justaswithatruncatedSIS factorof106relativetothestar’spositions;ascalebarcorrespondingto20µas lens,thereversalintheimage’smotionasthestarcrossesthe motionisshown.Twentyequallyspacedmeasurementpointsoverthe4-year observationalperiodareshownforeachcurve. Notethatthestarsclosestto scaleradius(β rs/dL)isveryslow,regardlessoftheimpact thelenspositionundergomuchmorerapidpositionchanges. parameterβ . ≃ y AsintheSIScase,themostpromisinglensingscenariooc- traces out a semi-circle of radius θSIS during the few years curs when the center of the NFW subhalo passes very close E surrounding the time of closest approach, just as if the lens tothesource. Thekeydifferenceisthatαisnearlyconstant hadinfiniteextent. TheeffectoftheSIS’struncationismore forξ RtifthelensisanSIS,whichleadstothesemi-circle ≪ apparentinthebottomhalfofFig. 2,whereβ =50 ;theim- image trajectory displayed in the top portion of Fig. 2. For y ′′ age’strajectoryisclosertoacircle,anditwillbecomemore anNFWlenswithξ rs,thedeflectionangleisverysmall, ≪ andmorecircularasβ increases. Theimagetransversesthis as shown in the bottom panel of Fig. 4. The NFW density y circleveryslowly,taking10yearstomove2µas,incontrast profileleadstoano-winsituation: ifyoudecreasetheimpact to the image in the top panel, which movesnearly30µas in parameterβyinordertoenhancethechangeintheimage’spo- only5years. Thusweseethattheonlydetectableportionof sitionovera settimeperiod,themagnitudeofthedeflection theimage’spathintheskyistheperiodsurroundingthemo- decreases. Weareforcedtoconcludethatastrometriclensing mentofclosestapproachbetweenthestarandthelens,anda by subhalos is only detectable if the inner density profile is small impactparameteris requiredto make the image move steeperthanρ r−1. ∝ significantlyoverthecourseofafewyears. Figure3further illustratesthenecessityofasmallimpactparameterbyshow- 2.3. Generalizeddensityprofile ing how the images of stars at different positions relative to Wehaveseenthatastrometricgravitationallensingbysub- thelensmoveoverthecourseof4years;onlythestarsalong thelens’spathwithβ <2 haveimagesthataresignificantly halos is only detectable if the center of the subhalo passes y ′′ close to the star’s position during the observational period, movedduringtheobse∼rvationalperiod. Forstarsthatarethis duringwhichthesubhalomovesabout0.001pc(fora5-year closetothecenterofthesubhalo,withβ θ,thetruncation of the density profile does not affect the≪imatge trajectories, observational period). Therefore, only the innermost portion of the subhalo is responsible for the astrometric lens- asseeninthe toppanelofFig. 2. We willthereforeassume ing signature. Unfortunately, very little is known about thatβ θ forallinterestinglensingscenariosandignorethe ≪ t the intermediate-mass (10M < M < 106M ) subhalos subhalo’struncationwhenconsideringotherdensityprofiles. t that are capable of producing⊙de∼tectable∼astrome⊙tric lensing events. High-resolution N-body simulations can probe the 2.2. NFWdensityprofile density profiles of only the largest (M > 108M ) subha- t TheNFWprofile, los, and even these profiles are unresolv∼ed at r <⊙ 350 pc ρ (Springeletal. 2008; Diemandetal. 2008). For th∼ese large ρ(r)= s , (16) subhalos, Diemandetal. (2008) find that ρ r 1.2 in the 2 − r 1+ r innermost resolved regions, while Springelet∝al. (2008) see rs rs ρ r (1.2 1.7) attheirresolutionlimitforninelargesubhalos, wasfoundto be a universa(cid:16)lly(cid:17)g(cid:16)oodfit(cid:17)to the densityprofiles ∝ − − with no indicationthat the slope had reached a fixed central of galaxy and cluster halos in early numerical simulations value. Meanwhile,attheoppositeendofthemassspectrum, (Navarroetal. 1996, 1997). The two-dimensional enclosed Diemandetal.(2005)findthatthefirstEarth-massdarkmat- massforasubhalowithvirialmassMvirandanNFWdensity termicrohaloshavesteeperdensityprofileswithρ r (1.5 2.0) − − profilewithconcentrationc Rvir/rsis atredshiftz = 26,andhigher-resolutionsimulatio∝nsindicate ≡ thatthis steep profile extendsto within 20 AU of the micro- M ξ M (ξ)= vir (17) halocenter(Ishiyamaetal.2010). 2D ln(1+c) c G r − 1+c s! 6 Erickcek&Law timated, especiallyifγ < 1.2. As shownin Fig. 4, however, 15 detectingasubhalowith∼γ< 1.2ischallenging,andwecon- 10 1 yr cludethatEq.(20) isaccur∼ateto within 20%forsubhalos γ=2.0 (SIS) ofinterest. ∼ 5 If a dark matter subhalo with a density profile given by 0 Eq.(18)passesinfrontofastar,Eq.(1)tellsusthat 05 1 yr γ=1.8 ~α=θ ξ 2−γξˆ, (21) µas] 5 γ=1.5 α r0! α [y 0 5 yrs wherewehavedefined 05 50 yrs γ=1.2 θα ≡0.88µas 2(Γ30.5γ()γΓ−01.5)γ 1− ddL prc ρM0r03 . 05 100 yrs γ=1 (NFW) Like θSIS, θ dep end(cid:2)s−on th(cid:2)e d(cid:3)is(cid:3)t!a nces toS!th e0le!ns an⊙d(2th2e) E α -20 -10 0 10 20 sourceonlythroughthefactor(1 d /d ). Wealsonotethat L S αx [µas] θαisrelatedtotheEinsteinangle−θE: adnLFdig=i.ts45.0c—opncTcehannetdrdaevtfliToen=ctiis2o0nR0avinkr/gmrle/s2g.e=Tnhe9er9av.teiTdrihableymiananmsesorovdfientnghseilteylnesnpsrwoiifisthl5ed×oSf1=t0h55ekMlepn⊙cs,, θα =θEγ−1 dr0 2−γ. (23) γis.gTivheenibmypρac∝tpra−rγa,manedtetrhiesd1i−ffaercresnectopnadn,elasncdorornelsypothnedptoordtiioffneroenftthvealuimesagoef L! pathcorrespondingtothetimesurroundingthemomentofclosestapproach Wewillcontinuetoassumethatα βsothatξ (seeFig.1) ≪ betweentheimageandlensisshown.Notethattheimagepathbecomesmore isapproximatelyequaltod β. L linearandtheimagemotionslowsdownconsiderablyasγisdecreased. Equation (22) gives the magnitude of the deflection angle In light of this uncertainty, we consider a generic density intermsoftheparametersofthedensityprofiler0andρ0,but profile thisisnotthemostusefuldescriptionofthesubhalo. Instead r −γ wecharacterizethesubhalobyeitheritsmassaftertidalstrip- ρ(r)=ρ (18) ping (M m M ) or the mass contained within a radius 0 t bd vir r ≡ 0! of0.1pcfromthesubhalocenter(M0.1pc). Although Mt isa with 1 < γ 2. We assume that a constant-density core, morestandardand intuitivedescriptionof the subhalomass, ifpresent,iss≤ignificantlysmallerthanourtypicalimpactpa- using M0.1pc offers two advantages. First, M0.1pc completely rametersof0.001pc,andweassumethatthesubhalodoesnot determinesthedeflectionangle;withoutlossofgenerality,we contain a black hole. Largercores would decrease the lens- cansetr0 =0.1pc,inwhichcase ingsignalwhilethepresenceofablackholewouldenhance it by adding a point mass and steepening the density profile θ =8.8µas Γ 0.5(γ−1) 1 dL 3−γ M0.1pc . (Bertoneetal. 2005; Ricotti&Gould 2009). If we take this α 2(3 γ)Γ 0.5γ − d 4π M densityprofileasinfinitewhencalculatingtheprojectedsur- (cid:2)− (cid:3) ! S! ! ⊙(2!4) facedensityΣ,wefindthat Second,M istheport(cid:2)iono(cid:3)fthesubhalo’smassthatisac- 0.1pc tuallyprobedbyastrometricmicrolensingbecausetruncating Γ 0.5(γ 1) ξ 1−γ the subhalo’s density profile at R = 0.1 pc does not affect Σ(ξ)= √πρ r − , (19) t 0 0 (cid:2)Γ 0.5γ (cid:3) r0! icthsaarsatcrtoemrizeetritchelesnusbinhgalosi’gsnmatausrsea.llTohwesreufsortoe,cuosninsigdeMr0s.u1pbchtao- M (ξ)=2π3/2 ρ r3 (cid:2)Γ 0.5(cid:3)(γ−1) ξ 3−γ, (20) los that are more compact than standard virialized subhalos 2D 0 0 (3 γ)Γ 0.5γ r andmakesiteasytoapplyourresultstomoreexoticformsof (cid:2)− (cid:3) 0! (cid:16) (cid:17) darkmattersubstructure. whereΓ[x]istheEulergammafunction. (cid:2) (cid:3) To relate θ to the virial mass of the subhalo, we have to Of course, this density profile does not extend to infinity; α specify the full density profile. If γ = 2.0, we will assume the subhalo’s density profile will be truncatedby tidal strip- thatthesubhaloisatruncatedSISso thatEq.(18)holdsout ping,anditmayalso transitionto a steeperpowerlaw, asin tothetruncationradiusofthesubhalo. Inthiscase,Eq.(23) thecaseofanNFWprofile. Ifthedensityprofileistruncated tellsusthatθ =θSIS,andwecanuseEq.(10)toevaluateθ . atr=R,thenthesurfacedensitydivergesfromEq.(19)asξ α E α t Ifγ,2,wewillassumethatthesubhalo’sfulldensityprofile approachesR,butforξ R,Eqs.(19)and(20)arestillgood approximatiotns. For ins≪tancte, if γ = 1.5(1.2), M (ξ) for a priortoanytidalstrippingwas 2D sguivbehnalboytrEuqn.ca(2te0d)aift Rξt is0g.r1eRat.erWtheanw8il0l%sho(5w0%in)Athpepevnadluixe ρ(r)= ρ0 , (25) t γ 3 γ A that detectible astrom≤etric signaturesare only producedif r 1+ r − r0 r0 ξ < 0.03pc, andEq.(20)isaccuratetowithin20%forsubhalos with γ 1.5, M < 108M , and R > 0.1 pc. Fur- which reduces to Eq. (1(cid:16)8) (cid:17)fo(cid:16)r r (cid:17)r . In this case, the vir t 0 thermore, the≥lower bound on R is⊙significan∼tly smaller for virial mass does not uniquely det≪ermine θ , and we also t α subhaloswithM 108M . WewillthereforeuseEq.(20) have to specify the subhalo’s concentration. We define the vir andtakeR > 0.1pc≪asacon⊙servativelowerbound,although concentration as c R /r , where r is the radius at t vir 2 2 wenotethat∼theresultingdeflectionsmaybeslightlyoveres- which dlnρ/dlnr =≡ 2. Fo−r the profile−given by Eq. (25), − LocalDMSubhaloLensing 7 r =(2 γ)r . Itfollowsthat −2 − 0 15.0 Predicted PM r =27pc 1 94 Mvir 1/3 ρ¯vir(zv) −1/3. 0 2 γ c 106M 4.6M pc 3 10.0 Detection run − ! ! ⊙! ⊕ − ! (26) Weseethatr islargerthan5pcforthesubhalosweconsider 0 (Mvir 104M andγ 1.2),andweexpectthatlocalsubha- 5.0 loswi≥llbestri⊙ppedto≥muchsmallerradiibyencounterswith s stars. a µ Wecannowderivehowθα dependsonthesubhalo’svirial Y/ 0.0 massandconcentration. RecallfromEq.(22)thatθ ρ r2. α ∝ 0 0 Calibration period FortheprofilegivenbyEq. (25),thisfactorisrelatedtothe subhalo’sconcentrationandvirialmassthrough -5.0 pc ρ0r03 =810 c Mvir 2/3 ρ¯vir(zv) 1/3 PM subtracted trajectory r M 94 106M 4.6M pc 3 -10.0 0! ⊙ (cid:18) (cid:19) ⊙! ⊕ − !   3.57(−1)γ(γ−2) , (27) -15.0 -10.0 -5.0 0.0 5.0 10.0 × B[c(γ 2);3 γ,γ 2] " − − − # X/µas where B[z;a,b] is the incomplete Beta function. In Ap- Fig.5.—Thesubhalolensdetection scheme. Thedashedlineshowsthe pendix B.2, we use the findings of the Aquarius simulation trajectoryofanimageinducedover6yearsbyasubhalolenswiththesame (Springeletal.2008)toderivearelationshipbetweenthecon- propertiesasinFig.4andγ=1.8.Theimagemotionduringthecalibration centrationoflocalsubhalosandtheirvirialmass: periodandthedetectionrunarelabeled. Thedottedlineshowsthedirection ofthepropermotionfittedduringthecalibration period,andthedifference c=94 Mvir −0.067, (28) bbeytwtheeesnoltihdecmurevaes.uFreodrctrlaajreitcyt,orthyeainndtritnhseicprporpoeprermmotoitoinonparenddicptaioranlliasxsohfowthne 106M sourcearenotshown. ⊙! andwe use this relationto determinethe subhaloconcentra- The timescale of the calibration period is important, as it tionthroughoutthisinvestigation. needstobelongenoughbothtodetectbinarysystemsandto Figure 4 shows the paths taken by the star’s image as the obtainarobustparallaxandpropermotionmeasurement. In- center of a subhalopasses 1 arcsec below the star’s position creasingthecalibrationperiodlengthimprovesthepredicted- for several values of γ. In this figure, d = 50 pc, d = 5 position accuracy, but it also takes time away from the de- L S kpc,v =200km/s,andthesubhalolenshasavirialmassof tection run, reducing the probability of observing a lensing T 5 105M . Weseethattheimagepathdependsverystrongly event. The calibration periodmust be at least one year long on×γ. If γ⊙ 1, the motion is nearly linear, and the image to obtain a secure parallax, and we suggest two years as a ≃ sensiblelengthtoensureanaccurateparallaxandpropermo- movesverylittleandveryslowly. Asγincreases,anarcap- pearsintheimagepath;thereisnowsufficientmassenclosed tionmeasurement. Thelengthofthedetectionrunisthenset by the duration of the high-precision astrometric campaign. intheinnerarcsecondtocauseanobservableverticaldeflec- Inthisanalysis,weassumeafour-yeardetectionrun,imply- tion when the subhalo passes beneath the star. Increasing γ inga totalof six yearsofobservations,whichisclose tothe alsoincreasesthemotionoftheimageinagiventimeperiod, expectedmission lengthsof Gaia and SIM and a reasonable and the acceleration of the image as the subhalo approaches length for long-term ground-basedsurveys. We leave a full thepointofclosestapproachbecomesapparent. Asthesub- discussion of the optimal observing strategy to future work, halocenterpassesfromthe leftto therightofstar, thestar’s as the details of the observing scenario will depend on both image will jump from right to left; since the image moves theparticularastrometrictechniqueandthetimeavailablefor very slowly in the years before and after the crossing of the subhalo,thisshiftintheimage’spositionoffersthebesthope theobservations. Forthepurposesofthispaper,weadoptasimplemeasure- fordetection. mentoftheastrometricsignalfromalensingevent: 3. OBSERVINGSTRATEGY Nepochs TheimagemotionsshowninFig.4suggestasimpledetec- S = (Xm Xp)2+(Ym Yp)2, (29) tciaolnhsigtrha-tpegreycfisoirosnuabshtraolomleetnriscinsge,airlclhusptrraotgerdaminoFpiger.a5t.esAfotyrpuip- vut Xi=1 i− i i− i to 10years. We start with aninitialfew-yearcalibrationpe- where Xm and Ym give the 2D measured position of the i i riod, during which we assume that the star is relatively far star at each epoch, and Xp and Yp give the 2D position of i i from the lens. Duringthis calibrationperiod we 1) measure the star predicted from the proper motion, parallax and po- the star’s intrinsic proper motion, parallax, and starting position determined during the calibration period. This calcu- sition and 2) search for binary stars or other false positives. lation(followingGaudi&Bloom2005)essentiallymeasures Starsthatshowsignificantaccelerationduringthecalibration the total displacement of the star from its expected position periodprobablyhavebinarycompanions,andwerejectthem overthecourseofthemeasurements. Thesignal-to-noisera- fromtherestofthesearch. We followthecalibrationperiod tio (S/N) is simply S divided by the astrometric uncertainty with a several-year detection run, when we essentially wait per 1D datapoint (σ); σ includes contributions from the in- forasubhalolenstocomeclosetooneofourtargetstarsand strument’s intrinsic uncertainty per datapoint as well as un- inducesignificantlensing. certainty in the star’s proper motion, parallax and position. 8 Erickcek&Law Theuncertaintyinthestar’spredictedpositiongrowsintime Detectable dark matter subhalos are therefore likely to have duetotheproper-motionuncertainty,andtheparallaxuncer- proper motions greater than half an arcsecond per year, and tainty’seffectonthe2Dmeasurementsvariesindirectionand theywillastrometricallyaffectallstarswithinseveralarcsec- magnitudeacrossthesky. ondsofthesubhalocenter. Ifahaloisdetected,itspathcan ThiscalculationgivesaconservativeestimateoftheS/Nof be predicted (albeit initially at low precision), and a catalog a possible lensing detection. The displacements induced by preparedofstarsthatarelikelytobeaffectedbylensingover lensingareallinapproximatelythesamedirection,however, thenextfewyears. Intensiveastrometricmonitoringoffaint and the effective S/N would likely be improved by fitting a starsinthefieldcouldthenprovideafairlyrapidconfirmation model to the data that accounts for these correlations. We oftheexistenceofthesubhalolens. leavesuchenhancementsforfuturework. 4. CROSSSECTIONSFORASTROMETRICLENSINGBYSUBHALOS 3.1. Falsepositiveremoval 4.1. SignalCalculations Inadditiontosimplydetectingasubhalolensingsignal,we As described in Section 3, we calculate the lensing signal mustalso distinguishit fromotherastrometricsignals. Sub- by taking the square root of the sum of the squared differ- halolensingsignalstakeplaceovermonths,donotrepeat,oc- encesbetweenthestar’spositionontheskyandtheposition cur withouta visible lensing source, andhave a near-unique predictedbythepropermotionandparallaxmeasuredduring trajectory. The relatively short event timescales ensure that the calibration period [Eq. (29)]. At each epoch, the differ- ourmeasurementsareonlysensitivematterstructuresonthe encebetweenthemeasuredimagepositionandthepredicted spatialscaleswe considerhere. Theevents’otherproperties positionisproportionaltothedeflectionangleα~. Thislinear- canbeusedtoremovefalsepositives,suchasthosegenerated ityimpliesthatwemayuse anyvector~ηthatisproportional bystellarmicrolensingandmotioninbinarysystems. to α~ to calculate the signal; we just have to multiply the re- Astrometric microlensing by a passing point-like lens sultingsignalbyα/ηtoobtainthephysicalsignalthatwould moves the image centroid along an elliptical path that be- bemeasuredduringalensingevent. Weusethistechniqueto comes more circular as the impact parameter increases separatethesignal’sdependenceonthegeometryofthelens- (Walker 1995;Paczynski1995, 1998;Miralda-Escude1996; ingscenariofromitsdependenceonthepropertiesofthelens. Gould2000). Theimagepathcanbecomemorecomplicated Aswedescribeindetailbelow,wecanthencalculatethegeo- ifthe lenshasa smallfinite extentandisopaque(Takahashi metricalsignalonceandthenusethatresulttodeterminethe 2003;Leeetal.2010). Inallcases,however,theimagecom- signalforanylens. pletesitsorbitonobservabletimescales,quicklyapproaching Followingourconvention,weworkinacoordinatesystem itstruepositionasthecompactobjectmovesfurtheraway.We where the subhalo’s transverse velocity~vT lies along the x- sawintheprevioussectionthatsubhalosproducearadically axis. In this case, βy is the fixed impact parameter, and βx,0 different lensing signature; after the passage of the subhalo specifiestheinitialpositionofthelens. Itisusefultodefine center,theimagemovesveryslowlyanddoesnotapproachits d β d β true positionuntilthe edge of the subhalopasses by the star ϕ L x,0 and β˜ L y , (30) manydecadeslater. Lensingbysubhalosisthereforereadily ≡ vTtobs ≡ vTtobs distinguishablefromlensingbycompactobjects. whereβ and β are in radians, andt is the lengthofthe x,0 y obs The astrometric signal of a stellar binary is easily distin- observation(not includingthe calibrationperiod). If we de- guished from subhalo lensing signals in most cases simply fine∆βtobetheangulardistance,inradians,traversedbythe because the binary system produces a repeating signal. Al- lensduringtheobservationalperiod,wecaneasilyinterpretϕ most all long-period systems that do not produce repeating andβ˜.Thenormalizedimpactparameterβ˜ =β /∆β,whilethe y signalsinourdatasetwillberemovedbyourrequirementthat phaseϕ = β /∆βspecifiesthelocationofthepointofclos- x,0 thestardoesnotaccelerateduringthecalibrationperiod.Fur- estapproachtothestaralongthesubhalo’spath.Sinceweare thermore,roughlycircularbinarysystemsinducea verydif- onlyinterestedincaseswherethesubhalocenterpassesbythe ferentastrometricsignalfromsubhalolenses. starduringtheobservationalperiod,weconstrain0 < ϕ < 1. Rare highly eccentric systems with periods much longer WiththesedefinitionsEq.(21)mayberewrittenas thanourobservationlength(orevenverycloseunboundstel- larencounters)canproduceashort-timescaleastrometricsig- α~ =θ vTtobs 2−γ~η(ϕ,β˜,t/t ,γ), (31) nal during periastron, with little signal throughout much of α r obs rest of the orbit. The trajectory is similar to a lensing sig- 0 ! nature (with an additional very large radial velocity signal). whichallowsustoseparatethegeometryofthelens-starsys- However, simple simulations of such systems show that no tem (i.e. ϕ and β˜) from all of the lens characteristics apart Keplerianorbit(ofanyeccentricity< 0.999)thatproducesa fromγ. FromEq.(21),weseethat lensing-like signal can avoid producing detectable acceleration both duringthe calibrationperiod and after the putative ~η=βˆ ξ 2−γ (32) lensingevent. v t T obs! 1 γ 3.2. Finalsubhalolensconfirmation t 2 − t = ϕ +β˜2 ϕ xˆ+β˜yˆ . (33) lmenaTtsthienergus.ulWtbimheaawltoeilslteassrhteoowpfroianbcatahbnelydniedwxaitttehsiesncutb1ioh0na0ltophcaistodafepoterucertdaliboclcteiaotdnioaornkf; We use~ηsto ca−lctuoblast!e the signa"l of−atloebnss!ing eve#nt. This whileitispossibletodetectamoredistantsubhalo,thesub- is advantageous because the resulting signal is independent halowouldhaveto be massive (Mvir > 106M ), andwe ex- ofθα,dL,andvT;wecallthissignalSg (for“geometricalsig- pectsuch objectstorarelypass betwe∼enusan⊙da targetstar. nal”)becauseitdependsonlyonϕ,β˜,andγ. Thecalculation LocalDMSubhaloLensing 9 γ=1.8 10 8 6 Sg 4 γ=1.5 Ag 1 2 0.0 0.1 γ=2.0 0 0.2 γ=1.8 0.8 P0h.a6se 0.4 0.2 0.8 0.6 0.4 β~ 0.01 0.01γγ==11..52 0.1 1 10 0.0 1.0 S g Fig.6.—Left:ThesignalSgasafunctionofphase(ϕ)andimpactparameterβ˜ βy/∆βforatwo-yearcalibrationperiodandafour-yeardetectionrun. Note thedecrease insignaltowardlargerphases, wheretheimageonlypartially trave≡rsesitslensingpathduringtheobservational period. ThedecreaseinSg at thesmallestphasesandimpactparameterscorrespondstolensesthatstarttoproducelargeimagemotionduringthecalibrationperiod,whichisthenpartially subtractedoutbythepropermotionremovalduringthedetectionrun.Atlargervaluesofβ˜theapparentmotionsarestillrelativelylarge(seeFig.3,forinstance), butthesemotionsaredifficulttodistinguishfromthestar’spropermotion,leadingtothedecreaseinSg. Right: TheareaAgintheϕ β˜planethatproducesa geometricalsignalthatislargerthanagivenvalueforSgforavarietyofγvalues. Notethatthenormalizedareagoestozeroatavalu−eofSgthatisdependent onγ.ForSgvaluesmuchsmallerthanthiscut-off,Agisproportionalto(Sg)p,wheretheindexpdependsonγ. ofS takesintoaccountthesubtractionofthepropermotion infinityastheimpactparameterβ˜goestozero.Consequently, g and parallax measured duringthe calibration period, includ- thereisamaximalvalueofS thatisobtainableforeachvalue g ing any apparent motion generated by lensing, as shown in ofγ. AtthisvalueofS ,theareaA goestozero,asshownin g g Fig.5. Figure6showsS (ϕ,β˜)forγ = 1.5andγ = 1.8. We therightpanelofFig.6. ThismaximalobtainablevalueofS g g seethatS decreaseswithincreasingimpactparameterβ˜ and impliesthatthereisaminimumsubhalomassthatiscapable g increaseswithincreasingγ,whichisnotsurprisinggiventhe of generating a detectable signal, as we will see in the next image paths shown in Fig. 4. We also see a preference for subsection. ForvaluesofSg thataresmallerthanhalfofthe geometriesin whichthe subhalopassesby thestar earlierin maximalpossibleSg value,Ag (Sg)p,asshownintheright ∝ theobservationalperiod;thesignalisenhancedbecausethere panelofFig.6. Forthefourγvalueswetested,wefoundthat aremoreepochsaftertheshiftinthestar’spositionwhenthe p= 1/γ. Thissimplepower-lawbehaviorwillbesharedby − subhalocenterpassesthestar. thephysicallensingcrosssections. TorelatethephysicalsignalS tothegeometricalsignalS g weuse 4.2. PropertiesoftheLensingCrossSections S =θ vTtobs 2−γS . (34) The basic shape of the lensing cross sections can be de- α r g duced from the left panel of Fig. 6. This figure shows that 0 ! S does not depend strongly on the phase ϕ when S is g g Thisrelationcompletestheprocedurefordeterminingifastar smallerthanthemaximumobtainablegeometricalsignal. For isdetectablylensedbyagivensubhalo. Givenaspecificlens S values that correspond to these values of S , the lens- min g and a minimal detectable value for the signal Smin, Eq. (34) ingcrosssectionsarerectangular. Thewidthoftherectangle may be inverted to obtain the corresponding minimal value is given by the changein the lens position duringthe obser- for Sg. We then determine the area Ag of the region in the vational period (∆β) because we only consider lensing sce- ϕ β˜ plane (with 0 < ϕ < 1 and β˜ > 0) that produces a narios with 0<ϕ<1. The length of the rectangle is two − geometricalsignalthatexceedsthisminimalvalueforS . Fi- times the maximum impact parameter that a star may have g nally, we convert A to a physicalarea on the sky that gives and still be detectably lensed. The impact parameters that g S > S , and we call this area the cross-section A(S ); in lie within the lensing cross section are therefore bounded min min squareradianswehave by β < A(S )/(2∆β). For a lens at a distance of 50 pc y min with| v| ∼= 200 km/s, Eq. (2) tells us that ∆β 3 arcsec- T v t 2 onds for a four-year observational period, so th≃e maximum A(S )=2 T obs A (S ). (35) min dL ! g g iβmpacAt(pSara)m/(e6tearrcthseact)p.roducesa lensing signal S > Smin is y min ≃ Thefactorof2accountsforthefactthatthestarsbothabove Figure 7 shows how the lensing cross section depends on (β˜ > 0)andbelow(β˜ < 0)thelensareequallydeflected,and themassofthesubhaloforthreevaluesofS :5µas,20µas, min thetwofactorsof(v t /d )followfromthedefinitionsofϕ and 50 µas. We characterize the mass of the subhaloin two T obs L andβ˜ [seeEq.(30)]. ways,asdiscussedinSection2.3. Inthetoprow,weshowthe The geometrical area functions A (S ) are shown in the crosssection as a functionof the mass enclosed in the inner g g rightpanel of Fig. 6 for severalvalue of γ. As indicated by 0.1pcofthesubhalo(M ). Recallthattruncatingthesub- 0.1pc theS (ϕ,β˜)surfacesshownintheleftpanelofFig.6,thearea halodensityprofileataradiusof0.1pcdoesnotsignificantly g A that produces a geometrical signal larger than a specific alterthelensingsignal,whichimpliesthatM directlyde- g 0.1pc S valuedependsstronglyonγanddecreasesrapidlywithin- termines the lensing signature. In the bottom row, we show g creasingS . TheleftpanelalsoshowsthatS doesnotgoto howthelensingcrosssectiondependssubhalo’svirialmass. g g 10 Erickcek&Law 400 S = 5 µas S = 20 µas S = 50 µas γ=2.0 min min min γ=1.8 γ=1.5 300 γ=1.2 2] c e s c ar 200 a [ e Ar 100 0 1 10 100 1 10 100 1 10 100 M < 0.1 pc [M] M < 0.1 pc [M] M < 0.1 pc [M] O• O• O• 400 S = 5 µas S = 20 µas S = 50 µas γ=2.0 min min min γ=1.8 γ=1.5 300 2] c e s c ar 200 a [ e Ar 100 0 10 100 1000 10000 10 100 1000 10000 10 100 1000 10000 0.001 × M [M] 0.001 × M [M] 0.001 × M [M] vir O• vir O• vir O• Fig.7.—TheareaaroundasubhalothatwillproduceanastrometricsignalgreaterthanSminasafunctionofsubhalomass,withdL =50pc,dS =5kpc,and vT=200km/s.Top:Themassofthesubhaloisdefinedasthemasscontainedwithinaradiusof0.1pcfromitscenter.Truncatingthesubhalodensityprofileat aradiusof0.1pcdoesnotaffectitslensingsignature.Bottom:Themassofthesubhaloisdefinedas0.1%ofitsvirialmass.Theareacurvesforγ=1.2arenot shownbecausethesubhalomusthaveMvir>108M togenerateasignalofatleast5µasifγ=1.2. ⊙ We assumethat99.9%of thesubhalo’svirialmass hasbeen rectly inherited from the geometrical area functions A (S ) g g lost due to tidal strippingand take 0.001M as the present- discussedintheprevioussectionandshownintherightpanel vir daymassofthesubhalo. of Fig. 6. Consequently, the power index p is independent From Fig. 7, we can see how the inner slope of the den- of d ,d ,v , and subhalo mass, and it is even independent L S T sity profile determines the strength of the lens signature; in ofwhether M or M isusedto characterizethe subhalo 0.1pc vir allcases,thelensingcrosssectiondecreasessharplywithde- mass. As S increasestoward the maximalpossible value, min creasingγ. Thedependenceonγ is strongerwhenthe virial thecrosssectiondecreasesfasterthan(S )pandrapidlygoes min mass is used to define the mass of subhalo because a shal- tozerowhenthemaximalpossiblevalueforS isreached. min lower density profile requires a larger virial mass to get the 4.3. LensingEventRates same mass within a given radius. The bottom row of Fig. 7 indicatesthatsubhaloswithγ<1.5arenotcapableofgener- The lensing cross sections computed in the previous sec- atingeasilydetectibleastrometriclensingsignatures,andsub- tionsmaybecombinedwithasubhalonumberdensitytoyield haloswithγ = 1.5areonlydetectibleif astar passeswithin aprobabilitythatanygivenstarontheskywillbedetectably acoupleofarcsecondsofthesubhalocenter. Thesituationis lensed during the observational period. In this section, we farmorepromisingforsubhaloswithγ>1.8;inthiscase,an will compute these probabilities using three candidate sub- intermediate-masssubhalocouldproduc∼easignalofupto50 halonumberdensities. Thefirsttwocalculationswillassume µasif astar passeswithin10arcsecondsofthesubhalocen- that all the subhalos have the same mass; we will assume ter.Moreover,weseethatsmallsubhalos(M <1000M )are that a fraction f of the local dark matter halo is composed t soconcentratedthatdecreasingtheinnerslope∼ofthede⊙nsity of subhalos with mass M and radii of R = 0.1 pc, and then profile from γ = 2 to γ = 1.8 does not significantly change we will assume that the local dark matter halo was once in the astrometric lensingsignal. Finally, the top row of Fig. 7 subhaloswithvirialmass M . Wetakethelocaldarkmatter vir showsthatiftheinnerregionsofthesubhalosaredenserthan densityρ to be 0.4GeV cm 3, which impliesthat thereis dm − predictedbytheirvirialmasses,subhaloswithshallowerden- 3.5 108M ofdarkmatterwithin2kpc.Finally,wewilluse sityprofilesarecapableofproducingdetectiblesignals. a su×bhalo m⊙ass function derived from the Aquarius simula- ForS valuesthatcorrespondtogeometricalsignalswell tions(Springeletal.2008). Throughoutthissection,wewill min below the maximal possible value for S (the value of S at usev =200km/sandt =4years. g g T obs which A = 0), the cross section has a simple dependence Wecomputethelensingprobabilitiesbysummingtheindi- g on S : A is proportionalto (S )p, with p = 1/γ for the vidualcrosssectionsforallsubhaloswith d < d forsome min min L S − four values of γ we consider. This simple power law is di- fixedvalueof d . We choosed to be smallenoughthatwe S S mayneglectthespatialvariationinthesubhalonumberden-