Microlensing constraints on $10^{-10}M_\odot$-scale primordial black holes from high-cadence observation of M31 with Hyper Suprime-Cam

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by  Hiroko Niikura

TobesubmittedtotheAstrophysicalJournal
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
MICROLENSINGCONSTRAINTSON10−10M -SCALEPRIMORDIALBLACKHOLESFROMHIGH-CADENCE
(cid:12)
OBSERVATIONOFM31WITHHYPERSUPRIME-CAM
HirokoNiikura1,2,MasahiroTakada1,NaokiYasuda1,RobertH.Lupton3,TakahiroSumi4,SurhudMore1,AnupreetaMore1,
MasamuneOguri1,2,5,MasashiChiba6
TobesubmittedtotheAstrophysicalJournal
ABSTRACT
We use the Subaru Hyper Suprime-Cam (HSC) to conduct a high-cadence (2 min sampling) 7 hour long
observation of the Andromeda galaxy (M31) to search for the microlensing magnification of stars in M31
7 duetointerveningprimordialblackholes(PBHs)inthehaloregionsoftheMilkyWay(MW)andM31. The
1 combinationofanapertureof8.2m,afield-of-viewof1.5degreediameter,andexcellentimagequality(∼0.6(cid:48)(cid:48))
0 yieldsanidealdatasetforthemicrolensingsearch. IfPBHsinthemassrange M = [10−13,10−6]M make
2 PBH (cid:12)
up a dominant contribution to dark matter (DM), the microlensing optical depth for a single star in M31 is
n τ ∼ 10−4–10−7,owingtotheenormousvolumeandlargemasscontentbetweenM31andtheEarth. TheHSC
a observation allows us to monitor more than tens of millions of stars in M31 and in this scenario we should
J find many microlensing events. To test this hypothesis, we extensively use an image subtraction method to
9 efficientlyidentifycandidatevariableobjects,andthenmonitorthelightcurveofeachcandidatewiththehigh
cadencedata.Althoughwesuccessfullyidentifyanumberofrealvariablestarssuchaseclipse/contactbinaries

] andstellarflares,wefindonlyonepossiblecandidateofPBHmicrolensingwhosegenuinenatureisyettobe
O
confirmed. WethenusethisresulttoderivethemoststringentupperboundsontheabundanceofPBHsinthe
C massrange. Whencombinedwithotherobservationalconstraints,ourconstraintrulesoutalmostallthemass
. scalesforthePBHDMscenariowhereallPBHsshareasinglemassscale.
h
Subjectheadings:darkmatter–gravitationallensing: micro–stars: blackholes
p
-
o
r 1. INTRODUCTION gravitational wave sources, recently detected by the LIGO-
t Virgo collaboration (Abbott et al. 2016). The abundance of
s Thenatureofdarkmatter(DM)remainshotlystudiedand
a PBHs at different mass scales is constrained by various ob-
debated both theoretically and observationally, and is one of
[ servations except for a mass window of 1021–1024 g corre-
1 tihesehmavoestsiumggpeosrtteadntthpartobDlMemissinnonco-bsamryoolongicy,.nPorne-vreiolautsivsistutidc-, spondingto10−12–10−9M(cid:12) (seeCarretal.2016,forarecent
review). Thus it is of critical importance to further explore
v and interacts with ordinary matter only via gravity (Zwicky
observationalconstraintsonthePBHabundanceforthismass
1 1937;Rubinetal.1978;Davisetal.1985;Cloweetal.2006;
window.
5 Dodelson & Liguori 2006). Currently, unknown stable par-
Gravitational microlensing has been used as a powerful
1 ticle(s) beyond the Standard Model of Particle Physics, so-
2 called Weakly Interacting Massive Particles (WIMPs), are methodtoprobeDMintheMilkyWay(MW)sincethethe-
0 considered as viable candidates (e.g. Jungman et al. 1996). oretical proposal in Paczynski (1986) (also see Griest et al.
1. However they have yet to be detected in either elastic scat- 1991; Roulet & Mollerach 1997; Wambsganss 2006). Mi-
crolensing can be detected through the time-variable mag-
0 tering experiments or in collider experiments (e.g., Klasen
nification of a background star when the lensing object and
7 et al. 2015). Primordial black holes (PBH), first proposed
the background star move relative to each other on the sky.
1 by Zel’dovich & Novikov (1967) (also see Hawking 1971;
Although massive compact halo objects (MACHOs) such as
: Carr & Hawking 1974; Carr 1975), are also viable candi-
v brown dwarfs are some of the most secure baryonic candi-
dates for DM. PBHs can be formed by the gravitational col-
i dates that behave like DM, the MACHO and EROS experi-
X lapse of the Hubble patch in the radiation dominated era of
ments ruled out the scenario that MACHOs of [10−9,10]M
the Universe if a large overdensity of δ ∼ O(0.1) is injected (cid:12)
r mass scales are a majority component of DM, because these
a into the patch. It is also advocated that PBHs can be pro-
experimentsdidnotfindasufficientnumberoftheMACHO
genitorsofbinaryblackholes(Sasakietal.2016;Birdetal.
microlensing events of stars in the Large Magellanic Cloud
2016; Kawasaki et al. 2016b; Inomata et al. 2016) that are
(LMC) (Alcock et al. 2000; Bennett 2005; Tisserand et al.
2007;Wyrzykowskietal.2010). RecentlyGriestetal.(2014)
1KavliInstituteforthePhysicsandMathematicsoftheUniverse(WPI), (also see Griest et al. 2011) used the public 2-years Kepler
TheUniversityofTokyoInstitutesforAdvancedStudy(UTIAS),TheUni-
versityofTokyo,Chiba,277-8583,Japan data to search for the microlensing events from ∼ 105 stars
2PhysicsDepartment,TheUniversityofTokyo,Bunkyo,Tokyo113- in an open cluster at a distance of about 4 kpc and derived
0031,Japan newupperlimitsontheabundanceofPBHsof10−8M mass
(cid:12)
3DepartmentofAstrophysicalSciences,PrincetonUniversity,Peyton scale. However, the Kepler data primarily has a cadence of
Hall,PrincetonNJ08544USA
4DepartmentofEarthandSpaceScience,GraduateSchoolofScience, 30 min and the number of source stars (∼ 105) is relatively
OsakaUniversity,Toyonaka,Osaka560-0043,Japan smallcomparedtothemicrolensingopticaldepth(τ∼10−7).
5ResearchCenterfortheEarlyUniverse,UniversityofTokyo,Tokyo With the aim of constraining the abundance of PBH with
113-0033,Japan massscales10−10M ,wehavecarriedoutahighcadenceob-
(cid:12)
6Astronomical Institute, Tohoku University, Aoba-ku, Sendai 980- servation of the Andromeda galaxy (M31), with the Subaru
8578,Japan
2 Niikuraetal.
HyperSuprime-Cam(HSC),inordertosearchformicrolens- inSection6andpresentourconclusionsandsummaryinSec-
ingeventsofM31starsbyinterveningPBHsinboththehalo tion7.
regions of MW and M31. M31 is the largest spiral galaxy,
nearesttotheMW,atadistanceofabout770kpc(thedistance 2. EVENTRATEOFPBHMICROLENSINGFORM31
modulusµ (cid:39) 24.4mag). EvenasinglenightofHSC/Subaru STARS
datayieldsanidealdatasettosearchforthePBHmicrolensing
InthissectionweestimateeventratesofPBHmicrolensing
events. First, the 1.5 degree diameter field-of-view of HSC
forastarinM31.Weextendtheformulationinpreviousstud-
(Miyazaki et al. 2015) allows us to cover the entire region
ies(Griestetal.1991;Alcocketal.1996;Kerinsetal.2001;
of M31 covering both the bulge, disk and halo regions. The Riffeseretal.2006)tomicrolensingcasesduetoPBHsinthe
8.2mlargeapertureanditssuperbimagequality(0.6(cid:48)(cid:48)seeing)
haloregionsofMWandM31forasourcestarinM31.
allow us to detect fluxes from main sequence stars in M31,
even with short exposure, here 90 sec exposure, because we
2.1. Microlensingbasics
can reach to about 26 mag depth. This allows us to monitor
a sufficiently large number of stars in M31 simultaneously. IfastarinM317andaforegroundPBHarealmostperfectly
Secondly,the90secshortexposureandashort∼35secread- alignedalongtheline-of-sighttoanobserver,thestarismul-
tiplyimagedduetostronggravitationallensing. Incasethese
out time enable us to take data at an unprecedented cadence
multipleimagesareunresolved,thefluxfromthestarappears
of2min,whichallowsustosearchforPBHsofsmallermass
magnified. When the source star and the lensing PBH are
scales than done in Griest et al. (2014). Thirdly, there is a
separated by an angle β on the sky, the total lensing magni-
huge volume between M31 and the Earth, leading to a large
fication,i.e. thesumofthemagnificationofthetwoimages,
opticaldepthofPBHmicrolensingtoeachstarinM31aswe
will show; τ ∼ 10−7–10−4 for PBHs in the mass range of is
DMMPBHin(cid:39)the[1M0−W13,a1n0d−M6]M31(cid:12),haifloPs.BHHesnccoen,sbtyitumteonaitmoraijnogrimtyooref A= A1+A2 = √u2+2 , (1)
u u2+4
thantenmillionstarsinM31,weshouldbeabletofindmany
eventsofthePBHmicrolensingifPBHsexist. whereu≡(d×β)/R ,anddisthedistancetoalensingPBH.
E
However,theanalysisoftheobservationsisquitechalleng- TheEinsteinradiusR isdefinedas
E
ing. Since M31 is a dense star field and the counts in each
4GM D
CCD pixel can be from multiple stars, we are in the regime R2 = PBH , (2)
of having to deal with the microlensing effect on unresolved E c2
stars, i.e., pixel lensing (Crotts 1992; Baillon et al. 1993; where M isthePBHmass. Disthelensingweighteddis-
PBH
Gould1996)(alsoseeCalchiNovati2010,forareview).This tance,D≡d(1−d/d ),whered isthedistancetoasourcestar
s s
is one of the main reasons that the previous observational in M31, and d is the distance to the PBH. By plugging typi-
efforts for the microlensing search towards M31 have been calvaluesoftheparameters,wecanfindthetypicalEinstein
unsuccessful (Crotts & Tomaney 1996; Calchi Novati et al. radius:
2005;deJongetal.2006;Riffeseretal.2008;CalchiNovati
etal.2009;Leeetal.2012;CalchiNovatietal.2014,andalso θ ≡ RE (cid:39)3×10−8arcsec(cid:32) MPBH (cid:33)1/2(cid:32) d (cid:33)−1/2 (3)
see references therein). Another reason for the unsuccessful E d 10−8M 100kpc
(cid:12)
resultisthatthepreviousworksuseddatafromsmalleraper-
where we assumed d = 770 kpc for distance to a star in
ture telescopes, and the quality of the data is not as good as s
our HSC/Subaru data. To tackle the pixel lensing problem, M31 and we assumed D ∼ d for simplicity, and employed
M = 10−8M asaworkingexampleforthesakeofcom-
wewillheavilyusetheimagesubtractiontechniquedescribed PBH (cid:12)
parison with Griest et al. (2014). In the following analysis
in Alard & Lupton (1998), which is fully integrated in the
we will consider a wide range of PBH mass scales. The
HSC pipeline (Bosch et al. in preparation). By subtracting
thereferenceimagefromthetargetimage,wecanefficiently PBHlensingphenomenawesearchforareinthemicrolens-
identifyvariablestarcandidatesthatpopupinthedifference ingregime;wecannotresolvetwolensedimageswithangular
image. Thenbyrepeatingtheimagesubtractionfordifferent resolution of an optical telescope, and we can measure only
thetotalmagnification. AsizeofastarinM31isviewedas
images for multiple visits (188 visits for our case), we can
measure the light curve of each candidate, which allows us R
to efficiently search for microlensing candidates. In fact our θs (cid:39) ds (cid:39)5.8×10−9arcsec, (4)
analysis finds many secure candidates of variable stars such s
asstellarflaresandbinarysystems. if the source star has a similar size to the solar radius (R (cid:39)
(cid:12)
Thestructureofthispaperisasfollows. InSection2,after 6.96 × 1010 cm). Comparing with Eq. (3) we find that the
abriefreviewofthemicrolensingphenomena,wefirstderive EinsteinradiusbecomessmallerthanthesourcesizeifPBH
aneventrateformicrolensingduetotheinterveningPBHsfor mass MPBH ∼< 10−10M(cid:12) correspondingto MPBH ∼< 1023 g. We
asinglestarinM31,byemployingahalomodelfortheMW discuss such lighter PBHs in Section 6, where we will take
andM31.InSection3wedescribethedetailsofourdataanal- intoaccounttheeffectoffinitesourcesizeonthemicrolensing
ysisincludingtheimagesubtractiontechnique,anddefinethe (Witt&Mao1994;Cieplak&Griest2013;Griestetal.2014).
mastercatalogofvariablestarcandidates.InSection4wede- Since the PBH and the source star move relative to each
scribe the selection criteria for microlensing events from the other on the sky, the lensing magnification varies with time,
catalogofvariablestarcandidates.InSection5,weusethere- allowingustoidentifythestarasavariablesourceinadiffer-
sulttoderiveanexperimentalupperboundontheabundance enceimagefromthecadenceobservation. Themicrolensing
of PBHs as a function of PBH mass. We then discuss how
different assumptions in our analysis affect the upper bound 7ThroughoutthispaperweassumethatasourcestarisinM31,notinthe
MWhaloregion,becauseofthehighernumberdensityonthesky.
MicrolensingConstraintsonPrimordialBlackHoleswithHSCObservationofM31 3
lightcurvehasacharacteristictimescalethatisneededfora 10 5
−
lensingPBHtomoveacrosstheEinsteinradius:
R
tE ≡ vE, (5) 10−6
wherevistherelativevelocity. Assumingfiducialvaluesfor
τ
theseparameters,wecanestimatethetypicaltimescaleas
10 7
−
(cid:32) M (cid:33)1/2(cid:32) d (cid:33)−1/2(cid:32) v (cid:33)−1
t (cid:39)34min PBH ,
E 10−8M 100kpc 200km/s
(cid:12)
(6) 10 8
whereweassumed v = 200km/sforthetypicalrelativeve- − 101 102 103
locity. Thusthemicrolensinglightcurveisexpectedtovary d[kpc]
overseveraltensofminutes, andshouldbewellsampledby
10 7
our HSC observation. It should also be noted that a PBH −
closer to the Earth gives a longer timescale light curve for
afixedvelocity. Sincewecansafelyassumethattherelative 10 8
−
velocitystaysconstantduringtheEinsteinradiuscrossing,the
light curve should have a symmetric shape around the peak,
whichwewillusetoeliminatefakecandidates. dτdd 10−9
2.2. Microlensingeventrate 10 10
−
Here we estimate expected microlensing event rates from
PBHs assuming that they consist of a significant fraction of 10 11
−
DMintheMWandM31haloregions. 0 100 200 300 400 500 600 700 800
We first need to assume a model for the spatial distribu- d[kpc]
tionofDM(thereforePBHs)betweenM31andus(theEarth).
HerewesimplyassumethattheDMdistributionineachhalo Figure1. Upper:TheopticaldepthofPBHmicrolensingeffectonasingle
starinM31asafunctionofthedistancetoPBH,d,whichcanbeobtained
regionofMWorM31followstheNFWprofile(Navarroetal.
byintegratingtheintegrandinEq.(10)over[0,d],ratherthan[0,ds]. The
1997): opticaldepthisindependentofPBHmass,andweassumedNFWparameters
ρ (r)= ρc , (7) tomodeltheDMdistributionineachoftheMWandM31haloregions,where
NFW (r/r )(1+r/r )2 wedeterminedtheNFWparameterssoastoreproducetheirrotationcurves
s s (seetextfordetails). Lower: Similarplot,butthefractionalcontributionof
where r is the radius from the MW center or the M31 cen- PBHsatthedistance,d,totheopticaldepth. Notethatdinthex-axisisin
linearscale.Theareaunderthiscurveuptodgivestheopticaldepthtodin
ter, r is the scale radius and ρ is the central density pa-
s c theupperplot.
rameter. In this paper we adopt the halo model in Klypin
et al. (2002): Mvir = 1012M(cid:12), ρc = 4.88 × 106 M(cid:12)/kpc3, whichweassumetobeequaltods =770kpcthroughoutthis
and r = 21.5 kpc for MW, taken from Table 2 in the paper, paper.
s
while M = 1.6×1012M , ρ = 4.96×106 M /kpc3, and ByusingEqs.(7)-(9),wecancomputetheDMdensity,con-
vir (cid:12) c (cid:12)
r = 25 kpc for M31, taken from Table 3. Thus we assume tributed from both the MW and M31 halos, as a function of
s
a slightly larger DM content for the M31 halo than the MW thedistancetoPBH,d.
halo. Dark matter profiles with these parameters have been AssumingthatPBHsmakeustheDMcontentbyaafrac-
shown to fairly well reproduce the observed rotation curves tion,Ω /Ω ,wecancomputetheopticaldepthτforthe
PBH DM
forMWandM31,respectively. TheremightbeanextraDM microlensing of PBHs with mass M for a single star in
PBH
contributionintheinterveningspacebetweenMWandM31, M31. The optical depth is defined as the probability for a
e.g. due to a filamentary structure bridging MW and M31. source star to be inside the Einstein radius of a foreground
However,wedonotconsidersuchanunknowncontribution. PBH on the sky or equivalently the probability for the mag-
ConsideraPBHatadistanced(kpc)fromtheEarthandin nificationofsourcefluxtobegreaterthanthatattheEinstein
theangulardirectiontoM31, (l,b) = (121.2◦,−21.6◦)inthe radius,A≥1.34(Paczynski1986):
GatadlaiscttainccceoRor⊕di=na8t.e5skypstcefmro.mAsthsuemMinWgtcheantttehre,wEearcthanisepxlparceesds τ= ΩPBH (cid:90) dsdd ρDM(d)πR2(d,M ). (10)
the separationto thePBH fromthe MWcenter, r , in Ω M E PBH
MW−PBH DM 0 PBH
termsofthedistancefromtheEarth,d,as
Here the mass density field of DM is given by the sum
(cid:113) of NFW profiles for the MW and M31 halos: ρ (d) =
rMW−PBH(d)= R2⊕−2R⊕dcos(l)cos(b)+d2. (8) ρ (d)+ρ (d). Notethat,becauseofR2D∝MM ,
NFW,MW NFW,M31 E PBH
theopticaldepthisindependentofPBHmass.
If we ignore the angular extent of M31 on the sky (which is
In Fig. 1, we show the optical depth of PBH microlensing
restrictedto1.5degreeindiameterforourstudy),thedistance
forasinglestarinM31,calculatedusingtheaboveequation.
tothePBHfromtheM31center,r ,isapproximately
M31−PBH Here we have assumed that all the DM in the halo regions
givenby,
of MW and M31 is composed of PBHs, i.e., Ω /Ω =
r (d)(cid:39)d −d, (9) PBH DM
M31−PBH s 1. The optical depth for microlensing, τ ∼ 10−6, is larger
whereweapproximatedthedistancetoasourcestarinM31to comparedtothattoLMCorastarclusterinMW(τ ∼ 10−7)
bethesameasthedistancetothecenterofM31, D (cid:39) d , by an order of magnitude, due to the enormous volume and
M31 s
4 Niikuraetal.
abovegeometryisgivenbytˆ=2R cosθu /v. Thusthedif-
E T r
ferential rate of microlensing events, occurring per unit time
scaletˆ,isgivenby
dΓ Ω (cid:90) ds (cid:90) ∞ (cid:90) π/2 (cid:90) 2π ρ (d)
= PBH dd dv dθ dα DM
dtˆ Ω r M
DM 0 0 −π/2 0 PBH
(cid:34) (cid:35) (cid:32) (cid:33)
u R v2 2R u cosθ
× T E exp − r v2cosθ δ tˆ− E T .(14)
πv2 v2 r D v
c c r
UsingtheDirac-deltafunctionidentity,
(cid:32) (cid:33) (cid:32) (cid:33)
2R u cosθ 2R u cosθ v2
δ tˆ− E T =δ v − E T r ,
D v D r tˆ 2R u cosθ
Figure2. AschematicillustrationofconfigurationsofalensingPBHanda r E T
(15)
sourcestarinM31inthelensplane,followingFig.4ofGriestetal.(1991).
TheorbitofalensingPBH,aroundasourcestarinM31(placedattheorigin andintegratingoverαandvr,weobtain
inthisfigure),isparameterizedasinthefigure,whichisusedtoderivethe
microlensingeventrate(seetextfordetails). dΓ Ω (cid:90) ds (cid:90) π/2 ρ (d) (cid:34) v2(cid:35)
= PBH dd dθ DM v4 exp − r , (16)
dtˆ Ω M v2 r v2
large mass content between the Earth and M31. The PBHs DM 0 −π/2 PBH c c
in each of the MW and M31 halos result in a roughly equal with v = 2R u cosθ/tˆ. One can rewrite this equation by
r E T
contribution to the optical depth to an M31 star. Although changing variable θ to the minimum impact u = u sinθ,
(cid:113) min T
there is an uncertainty in the DM density in the inner region
suchthat,dθ=du / u2 −u2 . Thisresultsin
ofMWorM31(atradii ∼< 10kpc)duetopoorly-understood min T min
bgaivNreyenoxnttiimcweeesffceeascltteism,fotahrteeitcstholiengthrriatbtceuutifroovner.imsFiincrosrottlwleanergsmien.ogdeelvtehnetsvewloitchitya ddΓtˆ =2ΩΩPBH (cid:90) dsdd(cid:90) uT(cid:113)dumin MρDM(dv)2v4r exp(cid:34)−vv2r2(cid:35) ,
DM 0 0 u2 −u2 PBH c c
distributionofDMinthehaloregions. Wesimplyassumean T min
isotropic Maxwellian velocity distribution for DM particles (17)
(cid:113)
(e.g.Jungmanetal.1996): wherev = 2R u2 −u2 /tˆ. Tocomputetheeventratedue
r E T min
(cid:34) (cid:35)
1 |v|2 toPBHsinboththehaloregionsofMWandM31,wesumthe
f(v;r)d3v= exp − d3v (11) contributions, dΓ = dΓ +dΓ . Aswedescribedabove,
π3/2v (r)3 v (r)2 MW M31
c c wecanexpressthecentricradiusofeachhalo,r,enteringinto
where V (r) is the velocity dispersion at radius r from the v (r),intermsofthedistancetothelensingPBH,d;r=r(d).
halo c
MWorM31center. ForV (r),weassumethatitisgivenas Unlessexplicitlystated,wewillemployu =1asourdefault
halo T
(cid:114) choice.
GM (<r) Fig.3showstheexpectedeventrateforthePBHmicrolens-
v (r)= NFW , (12)
c r ing, computed using Eq. (17). Here we show the event
rate as a function of the full-width-half-maximum (FWHM)
where M (< r) is the interior mass within radius
NFW timescaleofthelightcurve,whichmatchesoursearchofmi-
r from the halo center, defined as M (< r) =
NFW crolensing events from the real HSC data. If a PBH is in
4πρsr3s[ln(1+c)−c/(1+c)],wherec = r/rs foreachofthe the mass range M = [10−12,10−7]M (cid:39) 2×[1021,1026] g,
MWandM31halos. (cid:12)
it causes the microlensing event that has a typical timescale
WestartfromthegeometryandvariablesshowninFig.4of
in the range of [10−1,1] hour. The lighter or heavier PBHs
Griestetal.(1991)andtheirEq.(10)(seeFig.2),whichgives
tend to cause a shorter or longer timescale event. The event
theratedΓofPBHsenteringavolumeelementalongtheline-
rateisquitehighupto10−4foramicrolensingtimescalewith
of-sightwheretheycancausemicrolensingforasinglestarin
[0.1,1]hours. Thatis,ifwetakeabout10hoursobservation
M31:
and observe 108 stars at once for each exposure, we expect
Ω ρ (d)u R (cid:34) v2(cid:35) many events up to 104 events (because 10−4 × 10 [hour] ×
dΓ= ΩPBH MDM πTv2E exp −vr2 v2r cosθdvrdθdddα. 0.1[hour] (cid:39) 104),assumingthatsuchPBHsconstituteama-
DM PBH c c jorityofDMintheinterveningspacebridgingMWandM31.
(13)
TherightfigureshowsthatthePBHsintheM31haloregion
Here n (d) is the number density of PBHs at the distance
PBH
give a slightly larger contribution to the event rate, because
d from the Earth, v is the velocity of the PBH in the lens
r
we assumed a larger halo mass for M31 than that of MW.
plane, θ is the angle at which the PBH enters the volume
Thus the high-cadence HSC observation of M31 is suitable
element, and α is an angle with respect to an arbitrary di-
forsearchingformicrolensingeventsofPBHs.
rection in the lens plane, as shown in Fig. 2. Microlensing
events are identified if they have a given threshold magni-
2.3. LightCurvecharacterizationinpixellensingregime
fication A at peak. This threshold magnification defines a
T
thresholdimpactparameterwithrespecttotheEinsteinradius Aswedescribedabove,thetimescaleforthePBHandM31
ofaPBH,u = R /R . ComparedtoGriestetal.(1991),we star microlensing system is typically several tens of minutes
T T E
havefurtherignoredmotionsofsourcestarsforsimplicity,i.e. for a PBH with 10−8M . However, there is an observational
(cid:12)
v = 0. Theparametersvaryintherangeofθ ∈ [−π/2,π/2], challenge. Since the M31 region is such a dense star field,
t
α∈[0,2π],v =[0,∞). fluxes from multiple stars are overlapped in each CCD pixel
r
Thetimescaleforthemicrolensingeventdescribedbythe (0.17(cid:48)(cid:48)pixelscaleforHSC/Subaru).Inotherwordsindividual
MicrolensingConstraintsonPrimordialBlackHoleswithHSCObservationofM31 5
r] r]
u u
ho10−2 ho10−5
hour/10−3 10−121M0−(cid:12)11M hour/
ar/10−4 10−(cid:12)10M ar/10−6 10−8M(cid:12)
ts/st10−5 (cid:12)10−9M(cid:12) ts/st
even10−6 10−8M(cid:12) even10−7 MW M31
[ [
rate10−7 10−7M rate
vent10−180−2 10−1 100 101(cid:12) vent10−180−2 10−1 100 101
E t [hours] E t [hours]
FWHM FWHM
Figure3. ThedifferentialeventrateofPBHmicrolensingforasingleM31star(Eq.17);therateperunitobservationtime(hour),perasinglesourcestarin
M31,andperunittimescaleofthemicrolensinglightcurve(hour)forPBHsofagivenmassscale.HereweassumedthatalltheDMintheMWandM31halo
regionsismadeofPBHs;ΩPBH/ΩDM =1. Thex-axisisthefull-width-half-maximum(FWHM)timescaleofmicrolensinglightcurve. Thelighterorheavier
PBHhasashorterorlongertimescaleofmicrolensinglightcurve.TherightpanelshowstherelativecontributiontothemicrolensingeventrateduetoPBHsin
eitherMWorM31haloregion,forthecaseofMPBH=10−8M(cid:12).
starsarenotresolvedevenwiththeSubaruangularresolution toasourcestaronthesky,whereu istheimpactparameter
min
(about 0.6(cid:48)(cid:48) for the seeing size). Hence we cannot identify relativetotheEinsteinradiusR (u isdimension-less).The
E min
which individual star in M31 is strongly lensed by a PBH, secondoneisthetimescaleofthelightcurve,whichdepends
even if it occurs. Such a microlensing of unresolved stars on the Einstein radius as well as the transverse velocity of
falls in the “pixel microlensing” regime (Gould 1996) (also thePBHmovingacrossthesky. Forthetimescaleparameter
seeCalchiNovati2010,forareview). weusetheFWHMtimescaleofthemicrolensinglightcurve,
To identify microlensing events in the pixel microlensing t ,insteadoft ,definedas
FWHM E
regime requires elaborate data reduction techniques. In this
paper,weusetheimagesubtractionorimagedifferencetech- A(cid:18)tFWHM(cid:19)−1≡ A0−1. (19)
nique first described in Alard & Lupton (1998). The image 2 2
difference technique allows us to search for variable objects Thusthelightcurveofmicrolensingcanbefullymodeledby
includingcandidatestarsthatundergomicrolensingbyPBHs. thethreeparameters,F ,u andt . Inthefollowingwe
0 min FWHM
Inbrief,startingwiththetimesequencedNexpimagesofM31, willusethethreeparameterswhenperformingafittingofthe
theanalysisproceedsasfollows. (i)Wegenerateareference microlensingmodeltotheobservedlightcurveofmicrolens-
imagebyco-addingsomeofthebest-seeingimagesinorderto ing candidate in the image difference. Note that the use of
gainahighersignal-to-noise. Nextwesubtractthisreference t ,insteadoft ,givesslightlylessdegenerateconstraints
FWHM E
imagefromeachofthe Nexp imagesaftercarefullymatching ontheparameters(Gondolo1999).
their point spread functions (PSFs) as described in Alard &
Lupton (1998). (ii) We search for candidate variable objects
3. DATAANALYSISANDOBJECTSELECTION
thatshowupinthedifferenceimage. Inreality, iftheimage
subtraction is imperfect, the difference image would contain 3.1. Observations
many fake candidates, as we will discuss further. (iii) Once The HSC camera has 104 science detectors with a pixel
secure variable objects are detected, we determine the posi- scaleof0.17(cid:48)(cid:48) (Miyazakietal.2015). The1.5degreediame-
tion (RA and dec) of each variable object in the difference terFoVofHSCenablesustocovertheentireregionofM31,
image. WeperformPSFphotometryforeachvariablecandi- fromtheinnerbulgetotheouterdiskandhaloregionswitha
dateusingthePSFcentertobeatthepositionofthecandidate singlepointing. Thepointingiscenteredatthecoordinatesof
inthedifferenceimage. ByrepeatingthePSFphotometryin theM31centralregion: (RA,dec)=(00h42m44.420s,+41d
eachdifferenceimageoftheNexpimages,wecanmeasurethe 16m10.1s). Wedonotperformanyditheringbetweendiffer-
light curve of the candidate as a function of the observation entexposuresinordertocomparestarsinthesameCCDchip,
time. whichmakestheimagedifferencesomewhateasier.However,
Thelightcurveofamicrolensingeventobtainedusingthe in reality the HSC/Subaru system has some subtle inaccura-
PSF flux in the difference image at time t, obtained as de- ciesinitsauto-guidanceand/orpointingsystem. Thisresults
scribedabove,canbeexpressedas in variations in the pointings of different exposures, typical
∆F(t)= F [A(t)−A(t )], (18) variationsrangefromfewtoafewtensofpixels.
0 ref Fig.4showstheconfigurationofthe104CCDchipsrela-
where ∆F(t) is the differential flux of the star at time t rela- tive to the image of M31 on the sky. The white-color boxes
tivetothereferenceimage,F istheintrinsicflux,A(t)isthe denotelocationsofHSC“patches”,whichareconvenienttes-
0
lensing magnification at t and A(t ) is the magnification at sellations of the HSC FoV. The image subtraction and the
ref
the time of the reference image, t . In the above equation, search of microlensing events will be done on a patch-by-
ref
∆F(t) is a direct observable, and others (F ,A(t),A(t )) are patchbasis. Thepatcheslabeled“patch-D1”,“patch-D2”and
0 ref
parametersthathavetobemodeled. “patch-H” denote the regions that represent inner and outer
As can be seen from Eq. (1), the light curve for the mi- disk regions (-D1 and -D2) and a halo (-H) region, respec-
crolensingofapointsourcebyapointmasscanbecharacter- tively. Theserepresentativeregionswillbeusedtoshowhow
izedbytwoparameters. Thefirstparameteristhemaximum theresultsvaryinthedifferentregions.
amplification A = A(u ) when the lensing PBH is closest Our observations were conducted on November 23, 2014
0 min
6 Niikuraetal.
patch-D2
patch-H
patch-D1
Figure4. ThebackgroundimageofM31showsconfigurationof104CCDchipsoftheSubaru/HSCcamera.Thewhite-colorgridsaretheHSC“patch”regions.
Thepatcheslabeledas“patch-D1”,“patch-D2”and“patch-H”aretakenfromrepresentativeregionsofthediskregionclosertothecentralbulge,theouterdisk
regionandthehaloregion,respectively,whichareoftenusedtoshowexampleresultsofourdataprocessinginthemaintext. Thedark-blueregionsarethe
patchesweexcludefromourdataanalysisduetotoodensestarfields,wherefluxesfromstarsaresaturatedandthedataarenotproperlyanalyzed.
1.6 light curve for each variable object. The total exposure time
was 90 seconds on source and about 35 seconds were spent
1.4
for readout on average. The weather was excellent for most
1.2 of our observation as can be seen from Fig. 5, which shows
eeing1.0 hoouwrotbhseersveaetiinogn.FTWheHsMeeicnhgasnigzeedwwasitbhettitmerethfaronm0.7th(cid:48)(cid:48)efostramrtoosft
s
0.8 oftheobservationperiod,withabestseeingFWHMofabout
0.4(cid:48)(cid:48) at t ∼ 10,000 sec (2.8hours). However, theseeing got
0.6
worsethan1(cid:48)(cid:48)towardstheendofourobservation.Weexclude
0.4 6 exposures which had seeing FWHM worse than 1.2(cid:48)(cid:48) and
0 5000 10000 15000 20000 25000
time[sec] usetheremaining188exposuresforourscienceanalysis.
Wealsousetheg-andr-banddata,whichweretakendur-
Figure5. ThePSFFWHM(seeingsize)ofeachexposure(90secexposure ingthecommissioningrunonJune16and17in2013,respec-
each)asafunctionoftimet[sec]fromthestartofourobservation.Wetook
tively,inordertoobtaincolorinformationofstarsaswellasto
theimagesofM31regionevery2min(90secexposureplusabout35sec
forreadout),andhave188exposuresintotal. Theredpointsshowthe10 testavariabilityofcandidatesatdifferentepochs. Theg-band
best-seeingimages(∼ 0.45(cid:48)(cid:48))fromwhichthereferenceimage,usedforthe data consist of 5×120 sec exposures and 5×30 sec expo-
imagedifference,wasconstructed. suresintotal, whilether-banddataconsistsof10×120sec
exposures.
whichwasadarknight,adayafterthenewmoon.Intotal,we
acquired194exposuresofM31withtheHSCr-bandfilter8, 3.2. DatareductionandSampleselection
fortheperiodofabout7hours,untiltheelevationofM31fell
3.2.1. Standarddataprocessing
belowabout30degrees. Wecarriedouttheobservationswith
acadenceof2minutes,whichallowsustodenselysamplethe We performed basic standard data reduction with the ded-
icated software package for HSC, hscPipe (version 3.8.6;
8 See http://www.naoj.org/Projects/HSC/forobservers.html alsoseeBoschetal.inpreparation),whichisbeingdeveloped
fortheHSCfiltersystem basedontheLargeSynopticSurveyTelescopesoftwarepack-
MicrolensingConstraintsonPrimordialBlackHoleswithHSCObservationofM31 7
age(Ivezicetal.2008;Axelrodetal.2010;Juric´ etal.2015) (Eq.18).
9. Thispipelineperformsanumberofcommontaskssuchas Inordertomakeamastercatalogofvariableobjectcandi-
biassubtraction,flatfieldingwithdomeflats,coadding,astro- dates, we constructed 63 target images by co-adding 3 time-
metricandphotometriccalibrations,aswellassourcedetec- consecutive images from the original 188 exposure images.
tionandmeasurements. A typical limiting magnitude is about 26 mag (5σ for point
After these basic data processing steps, we subtract the sources),andevenbetterforimageswhereseeingisgood(see
backgroundcontaminationfromlightdiffusionofatmosphere below). Whensubtractingthereferenceimagefromeachtar-
and/or unknown scattered light. However the background getimage,theAlard&Luptonalgorithmusesaspace-varying
subtraction is quite challenging for the M31 region, because convolutionkerneltomatchthePSFsoftwoimages. Theop-
thereisnoblankregionandeveryCCDchipistosomeextent timalconvolutionkernelisderivedbyminimizingthediffer-
contaminated by unresolved, diffuse stellar light. To tackle encebetweenconvolvedPSFsoftwoimages. Avariableob-
this problem, we first divide each CCD chip into different ject,whichhasafluxchangebetweenthetwoimages,shows
meshes(thedefaultsubdivisionisdoneinto64meshesineach upinthedifferenceimage.
CCDchip).Wethenemployahigher-orderpolynomialfitting Fig.6showstheresultoftheimagesubtractionperformed
to estimate a smooth background over different meshes. We by the pipeline. Even for a dense star region in M31, the
employeda10-thorderpolynomialfittingfortheCCDchips pipeline properly subtracts the reference from the target im-
aroundthebulgeregion,whichareparticularlydensestarre- age, by matching the PSFs and astrometry. A point source
gions. ForotherCCDchips,weusea6-thorderpolynomial which undergoes a change in its flux shows up in the differ-
fitting scheme. However, we found residual systematic ef- enceimage,asseenintherightpanel. Inthiscase,thecandi-
fects in the background subtraction, so we will further use dateappearsasablack-colorpointsourcemeaninganegative
additionalcorrectionforphotometryofthedifferenceimage, flux, because it has a fainter flux in the target image than in
aswewilldiscusslater. thereferenceimage.
Forourstudy,accuratePSFmeasurementsandaccurateas- Wedetectobjectsinthedifferenceimageeachofwhichis
trometric solutions are crucial, because those allow for an definedfromalocalminimumormaximuminthedifference
accurate subtraction of different images. The pipeline first image,whereweused5σforthePSFmagnitudeasdetection
identifiesbrighteststarobjects(S/N ∼> 50)tocharacterizethe threshold. Thepipelinealsomeasuresthecenterofeachob-
PSFanddoaninitialastrometricandphotometriccalibration. jectandthesizeandellipticityfromthesecondmoments. In
From this initial bright object catalog, we select star candi- thisprocesswediscardedobjectsthathaveill-definedcenter,
datesinthesizeandmagnitudeplaneforPSFestimation(see asaturatedpixel(s)inthedifferenceand/ororiginalimageor
Bosch et al. for details). The selected stars are fed into the if the objects are placed at a position within 50 pixels from
PSFExpackage(Bertin2011)todeterminethePSFasafunc- theCCDedge.
tion of positions in each CCD chip. The functional form of
thePSFmodelisthenativepixelbasisandweuseasecond- 3.2.3. PSFphotometryandmastercatalogofvariablestar
candidates
orderpolynomialperCCDchipforinterpolation. Forthede-
termination of the astrometry, we used a 30 sec calibration Foreachvariablestarcandidate,weobtainPSFphotometry
imagethatwetookatthebeginningofourobservation,where inthedifferenceimagetoquantifythechangeofflux. Weal-
brightstarsarelesssaturated. Weobtainanastrometrysolu- lownegativePSFfluxesforcandidatesthathavefainterfluxin
tion after every 11 images, 30 sec calibration frame plus 10 thetargetimagethaninthereferenceimage.Sincethephoton
time-consecutivescienceexposures,bymatchingthecatalog countsineachCCDpixelisgenerallycontaminatedbymulti-
of stars to the Pan-STARRS1 system (Schlafly et al. 2012; plestarsinmostoftheM31regions,weoftenfindaresidual
Tonry et al. 2012; Magnier et al. 2013). The HSC pipeline coherent background (large-scale modulated background) in
provides us with a useful feature, the so-called “hscMap”, eachdifferenceimage,duetoimperfectbackgroundsubtrac-
which defines a conversion of the celestial sphere to the flat tionintheoriginalimage. Toavoidcontaminationfromsuch
coordinatesystem,“hscMapcoordinate”,basedonatessella- a residual background, we first measure the spatially con-
tion of the sky. In Fig. 4 the white-color regions denote the stantbackgroundfromthemedianofcountsin41×41pixels
hscMap“patch”regions. Weperformimagedifferencesepa- aroundeachobjectinthepostage-stampimage,andthensub-
rately on each patch. Due to too many saturated stars in the tract this background from the image. Then we perform the
bulge region and M101, we exclude the patches, marked by PSF-photometry counts in ADU units taking the PSF center
darkbluecolor,fromthefollowinganalysis. tobeatthecandidatecenter. Hereafterwesometimesreferto
PSFmagnitudeinthedifferenceimageas“PSFcounts”. The
3.2.2. ImagesubtractionandObjectdetection pipelinealsoestimatesnoiseineachpixelassumingtheback-
Inordertofindvariableobjects, weemploythedifference ground limit (Poisson noise), and gives an estimation of the
image technique developed in Alard & Lupton (1998) (also noiseforthePSFphotometry(seeequations14and15forthe
seeAlard2000),whichisintegratedintotheHSCpipeline.To similardefinitioninMandelbaumetal.2014). However, the
dothis,wefirstgeneratedthe“reference”imagebyco-adding noiseestimationinvolvesanon-trivialpropagationofPoisson
10best-seeingimagesamongthe188exposureimages,where noise in the image difference procedures, so we will use an-
the10imagesarenottime-consecutive(mostofthe10images otherestimateforthePSFphotometryerrorineachpatch,as
arefromimagesaroundabout3hoursfromthebeginningof describedbelow.
the observation, as shown in Fig. 5). We use the mean of In the following we focus on the PSF photometry counts
the10imagesastheobservationtimeofthereferenceimage, in ADU units in the difference image, rather than the mag-
t , which is needed to model the microlensing light curve nitude, becauseitisthedirectobservable. However, wewill
ref
alsoneedtoinferthemagnitudeofeachcandidate;forexam-
9 Also see http://www.astro.princeton.edu/˜rhl/photo-lite. ple,toestimatetheluminosityfunctionofsourcestarsineach
pdffordetailsofthealgorithmusedinthepipeline. magnitudebinortoplotthelightcurveofvariablestarcandi-
8 Niikuraetal.
Figure6. Anexampleoftheimagesubtractiontechniqueweusefortheanalysisinthispaper.Theleft-panelimageisthereferenceimagewhichwasconstructed
byco-addingthe10best-seeingdata,withtypicalseeingof0.45(cid:48)(cid:48)(seeFig.5).Thesizeoftheimageis222×356pixels(correspondingtoabout0.63sq.arcmin),
whichistakenfrom“patch-D2”inFig.4.Themiddlepanelisthetargetimage(coaddedimageof3exposures)whoseseeingsizeis0.8(cid:48)(cid:48).Therightpanelshows
thedifferenceimage,showingthatthepipelineproperlysubtractsthetwoimagesevenforsuchadensestarregionandavariablestarcandidateshowsupatthe
center. Inthiscase,thecandidateobjectappearsasanegativefluxinthedifferenceimage,becausetheobjecthasafainterfluxinthetargetimagethaninthe
referenceimage.
dates in units of the magnitude. In this case we estimate the 3.2.4. Lightcurvemeasurement
magnitudeofanobjectinthei-thtargetimage,m,basedon
i Once each candidate is identified, we measure the PSF
mi =−2.5log(cid:32)CdiffF,i+Cref(cid:33), (20) ctooumnetsasiunreeathcehloigfhtthceu6rv3edwiffitehrean6cemiimnargeesos.lutTiohnis,aaslloawfusnucs-
0,i tionoftimefromthebeginningtotheendofour7hourlong
where Cdiff,i is the PSF flux for the object in the difference observations. In order to restore the highest time resolution
imageofthei-thtargetimage,C isthePSFfluxoftheref- of our data, we then used each of 188 exposures and mea-
erenceimageattheobjectpositiorenf,andF isthezero-point suredthePSFcountsineachofthe188differenceimagethat
0,i
flux in the i-th image. Note that the counts of the reference was made by subtracting the reference image (the coadded
image C can be contaminated by fluxes from neighboring image of 10 best-seeing exposures) from every single expo-
ref
stars,sotheabovemagnitudemightnotbeaccurate. sure. Hereweusedthesamepositionofcandidateasusedin
Fromtheinitialcatalogconstructedfromthe5σcandidates the63images. Inthiswaywemeasurethelightcurveofthe
from the 63 coadded images, we prune it down to a master objectwith2mintimeresolution.
catalog of “secure” variable star candidates by applying the Fig. 8 shows the light curves for examples of real vari-
followingcriteria: able stars. Note that we converted the PSF counts of each
candidate in the difference image to the magnitude based on
• PSF magnitude threshold – A candidate should have Eq.(20). However,themagnitudemightbecontaminatedby
a PSF magnitude, with a detection significance of 5σ fluxesfromblendedstarssurroundingthecandidatestar. This
or higher (including a negative flux), in any of the 63 demonstrates our ability to properly sample the light curves
differenceimages. withhightimeresolution. Thusthefigureshowsthatthedif-
ferenceimagetechniqueworkswellandcanidentifyvariable
• Minimum size – The size of the candidate should be starcandidatesaswellasmeasuretheirlightcurves.
greaterthan0.75timesthePSFsizeofeachdifference Fig. 9 shows the distribution of secure variable star candi-
image. datesdetectedinouranalysisovertheHSCfield-of-view,for
candidateswithmagnitudesm ≤ 24 and25magintheleft
r
• Maximum size – The size of the candidate should be and right panel, respectively. To estimate the magnitude of
smallerthan1.25timesthePSFsize. eachcandidate,weusedthePSFmagnitudeofthecandidate
inthereferenceimage. Basedontheshapeofthelightcurve
• Roundness–Thecandidateshouldhavearoundshape. for each candidate, we visually classified the candidates in
Werequireourcandidatestohaveanaxisratiogreater different types of variable stars; i) stellar flares, ii) eclipsing
than0.75,asthePSFdoesnotshowextremeaxisratios. or contact binary systems, iii) asteroids (moving object), iv)
Cepheidvariablesifthecandidatesappeartohavealongerpe-
• PSF shape – We impose that the shape of an ob- riod than our observation duration (7 hours), and v) “fakes”.
ject should be consistent with the PSF shape. The Here fakes are those candidates which show time variability
residual image, obtained by subtracting a scaled PSF onlywhentheseeingconditionsareasgoodas ∼< 0.6(cid:48)(cid:48).Since
model from the candidate image in the difference im- such good-seeing data is deeper as found from Figs. 5 and
age, should be within 3σ for the cumulative deviation 10,weseemtofindRR-Lyraetypevariableswhoseapparent
overpixelsinsidethePSFaperture. magnitudes would be around r ∼ 25 mag. When the seeing
getsworse,thesestarscannotbereliablyseeninthedifference
Fig. 7 shows examples of objects that pass or fail the above image. SinceRR-LyraestarsshouldexistintheM31region,
criteria. Note that the above conditions are broad enough in we think the “fake” stars are good candidates for RR-Lyrae
orderforusnottomissarealcandidateofmicrolensingifit stars. Thefigureshowsthatouranalysissuccessfullyenables
exists. Wemakeamastercatalogofvariablestarcandidates to find variable stars across the disk and halo regions. The
fromobjectsthatpassalltheaboveconditionsaswellasare totalnumberofcandidatesare1,334and2,740form ≤ 24
detected in the image difference at least twice in the 63 dif- and25mag,respectively. r
ference images at the same position within 2 pixels. These
criteriaresultin15,571candidatesofvariableobjects,which
4. STATISTICSANDSELECTIONCRITERIA
isourmastercatalogofvariablestarcandidates.
MicrolensingConstraintsonPrimordialBlackHoleswithHSCObservationofM31 9
Figure7. Examplesofdetectedobjectsinthedifferenceimage, whichpassordonotpasstheselectioncriteriatodefineamastercatalogofvariablestar
candidates(seetextfordetails). Eachpanelshows4postage-stampimages: theleftmostimageisthereferenceimage(thecoaddedimageof10best-seeing
exposures),the2ndleftisthetargetimage(thecoaddedimageof3time-consecutiveexposures),the3rdimageisthedifferenceimagebetweenthereferenceand
targetimages,andtherightmostimageistheresidualimageaftersubtractingthebest-fitPSFimagefromthedifferenceimageattheobjectposition. Thetwo
objectsintoprawaresuccessfulcandidatesthatpassedalltheselectioncriteria:theleft-panelobjecthasabrighterfluxinthetargetimagethaninthereference
image,whiletheright-panelobjecthasafainterflux(thereforeappearasablack-colorimagewithnegativeflux). Thelower-rowobjectsareremovedfromthe
catalogaftertheselectioncriteria.TheobjectsinthemiddlerowareexcludedbecausetheobjectiseithersmallerorlargerthanthePSFsize.Theleftobjectin
thebottomrowisexcludedbecauseithasatoolargeellipticitythanPSF.Therightobjectisexcludedbecauseoftoolargeresidualimage.
20.5 21.0
21.1
21.0 21.2
21.3
ag21.5 ag21.4
m m
21.5
22.0 21.6
21.7
22.5 21.8
0 5000 10000 15000 20000 25000 0 5000 10000 15000 20000 25000
time[sec] time[sec]
21 21.7
22 21.8
23 21.9
ag24 ag22.0
m m
25 22.1
26 22.2
27 22.3
0 5000 10000 15000 20000 25000 0 5000 10000 15000 20000 25000
time[sec] time[sec]
Figure8. Examplesoflightcurvesforrealvariablestarsidentifiedinourmethod. Thegreen-circledatapointsshowthelightcurvesampledbyouroriginal
dataof2minsamplingrate,whilethered-trianglepointsarethelightcurvemeasuredfromthecoaddeddataof3time-consecutiveexposures(therefore6min
cadence)(seetextfordetails). Upperleft: candidatestellarflare. Whenconvertingthemagnitudefromthecountsinthedifferenceimageateachobservation
time,weusedEq.(20).Notethattheestimatedmagnitudemightbecontaminatedbyfluxesofneighboringstarsinthereferenceimage.Upperright:candidate
contactbinarystars. Lowerleft: theeclipsebinarysystem,whichisprobablyasystemofwhitedwarfandbrowndwarf,becauseonestar(whitedwarf)hasa
totaleclipseoverabout10minduration,andthentheeclipsehasabout3hoursperiod. Lowerright:candidatevariablestar,whichhasalongerperiodthanour
observationduration(7hours).
Giventhecatalogofvariablestarcandidateseachofwhich shapes,itiscrucialtoproperlyestimatethephotometryerror
has its measured light curve, we now search for secure can- inthelightcurvemeasurement. However,accuratephotome-
didatesofPBHmicrolensing. Inthissectionwedescribeour tryfordensestarregionsinM31ischallenging. Toovercome
selectioncriteriatodiscriminatethemicrolensingeventfrom thisdifficulty,weusethefollowingapproachtoobtainacon-
othervariables. servative estimate of the error. The pipeline performs image
subtraction on each patch basis (as denoted by white-color
4.1. Photometricerrorsofthelightcurvemeasurement square regions in Fig. 4). For a given difference image, we
randomly select 1,000 points in each patch region, and then
Ourprimarytooltosearchforvariableobjectsinthedense
star regions of M31 is the use of the image difference tech- perform PSF photometry at each random point in the same
manner as that for the variable star candidates. In selecting
nique,aswehaveshown.Torobustlysearchforsecurecandi-
randompoints,weavoidedregionscorrespondingtobadCCD
datesofPBHmicrolensingthathavetheexpectedlightcurve
10 Niikuraetal.
fakes(inc. RR-Lyraecand.) fakes(inc. RR-Lyraecand.)
Cepheidvariable Cepheidvariable
asteroid asteroid
stellarflare stellarflare
eclipsingbinary eclipsingbinary
contactbinary contactbinary
Figure9. Distributionofsecurevariablestarcandidates,detectedfromouranalysisusingtheimagedifferencetechnique.Thedifferentsymbolsdenotedifferent
typesofcandidatesclassifiedbasedontheshapesoftheirlightcurves.Hereweexcludeothernon-securecandidatesthatareCCDartifactsandfakeeventsnear
totheCCDedgeorbrightstars.Theleftpanelshowsthedistributionforthecandidateswithmagnitudesmr≤24mag,whiletherightpanelshowsthecandidates
atmr≤25mag.Thenumberofcandidatesare1,334and2,740,respectively.
23.5 surement in the difference image, estimated based on the
above method. The shape of the photometric error appears
24.0 to correlate with the seeing conditions in Fig. 5. The figure
de24.5 showsthatmostofourdatareachesadepthof26magorso
u thankstothe8.2mlargeapertureofSubaru.
t
ni
g25.0
ma 4.2. Microlensingmodelfittothelightcurvedata
old25.5 Here we describe our selection procedure for PBH mi-
h
s crolensing events from the candidates. The unique part of
er26.0
h our study is the the high cadence for the light curve of each
t 5 sigma
candidate,sampledbyevery2minoverabout7hours. How-
26.5 3 sigma
everthemonitoringofeachlightcurveislimitedbyaduration
27.0 of7hours. Ifamicrolensingeventhasalongertimeduration
0 5000 10000 15000 20000 25000
than7hours,wecannotidentifysuchacandidate.Weusethe
time[sec]
statisticsinTable1toquantifythecharacteristicsofeachlight
Figure10. Thephotometricerrorusedforthelightcurvemeasurementin curve. Ourselectionprocedureforthecandidatesaresumma-
thedifferenceimage;werandomlyselect1,000pointsinthedifferenceim- rizedinTable2. Wewilldescribeeachoftheselectionsteps
ageofagivenpatch(hereshownforthepatch-D2inFig.4),measurethe indetail.
PSFphotometryateachrandompoint,andthenestimatethevarianceofthe
As we described, we start with the master catalog of vari-
PSFphotometries(seetextfordetails). Thesquaresymbolsshowthe3-or
5-sigmaphotometricerrorsestimatedfromthevariancewhenusingthedif- able star candidates, which contains 15,571 candidates, to
ferenceimagesconstructedfromthecoaddedimagesof3exposures, asa search for microlensing events. Our level 1 requirement is
functionofobservationtime. Thecirclesymbols,connectedbytheline,are thatacandidateeventshouldhavea“bump”initslightcurve,
theresultsforeachexposure. Althoughweusethephotometricerrorinthe
definedas3time-consecutivefluxchangeseachofwhichhas
ADUcountsforafittingofthemicrolensingmodeltothelightcurve,wehere
convertthecountstothemagnitudeforillustrativeconvenience. asignal-to-noiseratiogreaterthan5σinthedifferenceimage;
∆C ≥ 5σ, where the subscript i denotes the i-th difference
i i
pixelsorneartheCCDchipedges. Wethenestimatethevari- image(attheobservationtimeti). Thiscriterialeavesuswith
ance from those 1,000 PSF magnitudes, repeat the variance 11,703candidatesoverallthepatches.
estimationinthedifferenceimageforeveryobservationtime, Nextwefittheobservedlightcurvesofeachcandidatewith
and use the variance as a 1σ photometry error in the light amodeldescribingtheexpectedmicrolensinglightcurve. As
curve measurement at the observation time. The photomet- wedescribedinSection2.3,thelightcurveofamicrolensing
ricerrorestimatedinthiswaywouldincludeacontamination inthedifferenceimageisgivenas
fromvariouseffectssuchasalarge-scaleresidualbackground ∆C(t)=C [A(t)−A(t )], (21)
duetoanimperfectbackgroundsubtraction. Wefindthatthe i 0 i ref
photometric error is larger than the error estimated from the whereC isthePSF-photometrycountsofanunlensedimage
0
pipelineatthecandidateposition, whichislocallyestimated inthedifferenceimage,correspondingto F inEq.(18),and
0
by propagating the Poisson noise of the counts through the A(t)and A(t )arelensingmagnificationsattheobservation
i ref
imagesubtractionprocesses. timet andthetimeofthereferenceimaget . Asdescribed
i ref
Fig.10showsthephotometricerroronthelightcurvemea- inSection2.3,thelightcurveinthedifferenceimageischar-