Searching for intermediate-mass black holes in globular clusters with gravitational microlensing
aa r X i v : . [ a s t r o - ph . GA ] M a y Mon. Not. R. Astron. Soc. , 1–13 (2015) Printed September 13, 2018 (MN L A TEX style file v2.2)
Searching for intermediate-mass black holes in globularclusters with gravitational microlensing
N. Kains , D. M. Bramich , K. C. Sahu , A. Calamida Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, United States of America ⋆ Qatar Environment and Energy Research Institute (QEERI), HBKU, Qatar Foundation, Doha, Qatar National Optical Astronomy Observatory, 950 N Cherry Ave, Tucson, AZ 85719, United States of America
Received ... ; accepted ...
ABSTRACT
We discuss the potential of the gravitational microlensing method as a uniquetool to detect unambiguous signals caused by intermediate-mass black holes in glob-ular clusters. We select clusters near the line of sight to the Galactic Bulge and theSmall Magellanic Cloud, estimate the density of background stars for each of them,and carry out simulations in order to estimate the probabilities of detecting the as-trometric signatures caused by black hole lensing. We find that for several clusters,the probability of detecting such an event is significant with available archival datafrom the
Hubble Space Telescope . Specifically, we find that M 22 is the cluster withthe best chances of yielding an IMBH detection via astrometric microlensing. If M 22hosts an IMBH of mass 10 M ⊙ , then the probability that at least one star will yielda detectable signal over an observational baseline of 20 years is ∼ M ⊙ , the detectionprobability rises to > Key words: globular clusters – black holes – intermediate-mass black holes – gravi-tational lensing – microlensing
After formation, a stellar-mass black hole may grow via ac-cretion of surrounding material, or by merging with otherblack holes; eventually, supermassive black holes (SMBHs)may form with masses ranging upwards of ∼ M ⊙ . Thedetection of SMBHs at large redshifts indicates that someof them formed quickly and were already present only a fewhundred million years after the Big Bang (Fan 2006). Ex-plaining how these objects formed so rapidly is a challenge,because a stellar-mass seed black hole cannot reach a massof ∼ M ⊙ within 1 Gyr even by accreting material at thehighest possible rate, the Eddington rate, although mech-anisms have been put forward that could enable accretionat super-Eddington rates (Alexander & Natarajan 2014). Inthis context, one of the preferred scenarios for such rapidinitial growth is through the merger of smaller seed blackholes of intermediate mass (10 -10 M ⊙ , e.g. Ebisuzaki et al.2001), which serve as the missing link to understanding thegrowth of SMBHs. ⋆ [email protected] Globular clusters provide dense enough stellar en-vironments for intermediate-mass black holes (IMBHs)to form through runaway mergers of stars (e.g.Portegies Zwart & McMillan 2002, Miller & Hamilton2002), and they are approximately the same age as theirhost galaxy, suggesting that the IMBHs required forthe growth of SMBHs in the early Universe might havebeen delivered to galaxy centres by globular clusters(e.g. Capuzzo Dolcetta et al. 2001; L¨utzgendorf et al.2012). Further motivation for searching for IMBHs inclusters comes from the well-known M − σ relation (e.g.Ferrarese & Merritt 2000) for galaxies, which hints at a fun-damental connection between the formation and evolutionof central black holes and the central kinematics of galaxies.Extrapolating this relation to lower masses implies thatIMBHs should be found in systems with central dispersionsof ∼ c (cid:13) N. Kains et al. yielded upper mass limits that depend on the assump-tions made about the accretion process and the den-sity of the surrounding material (e.g. Grindlay et al. 2001;Maccarone, Fender & Tzioumis 2005; Haggard et al. 2013).In spite of this, some promising IMBH candidates (e.g.Farrell et al. 2012; Soria, Hau & Pakull 2013; Mezcua et al.2015) have been identified through observations of ultra-luminous X-ray sources (ULXs). These are extra-nuclearpoint sources with X-ray luminosities greater than 10 ergss − , corresponding to the Eddington limit of a 10 M ⊙ blackhole (Roberts 2007).Surface brightness profiles of globular clusters hostingcentral IMBHs are expected, from both theoretical predic-tions and N -body simulations, to have weak central cusps(e.g Bahcall & Wolf 1976; Baumgardt, Makino & Hut 2005)as opposed to core-collapsed clusters with steep profilesand pre-core collapsed systems with no cusp. However, ithas also been shown from numerical simulations that aphotometric profile with a shallow cusp might also be asign of ongoing core collapse (Trenti, Vesperini & Pasquato2010). Therefore a weak central cusp is not a unique sig-nature of a central IMBH, making claims of IMBH de-tections using this method contentious (e.g. Lanzoni et al.2007; Vesperini & Trenti 2010). Furthermore, the presenceof anisotropic orbits could mimic the signature of a centralIMBH in kinematic profiles, making the interpretation ofcusp data ambiguous (Ibata et al. 2009). Mass segregation ofstellar remnants can also replicate such signatures, as shownby Illingworth & King (1977) and then by Baumgardt et al.(2003) in their analysis of the kinematic profile of M 15. Be-cause remnants are natural products of stellar evolution, itis then difficult to favour an IMBH scenario.Combining photometric and spectroscopic observationsto yield kinematic data that can be compared to dynamicalmodels has recently led to a number of claims of IMBH de-tections, thanks to improvements in instrumental resolutionand the use of integral field units (L¨utzgendorf et al. 2013).This method has also yielded a mass estimate for an IMBHin ω Cen of (4 . ± . × M ⊙ (Noyola et al. 2010). How-ever, other authors (e.g. Anderson & van der Marel 2010)have found less compelling evidence for a central black holein ω Cen, which is in any case suspected to be the strippednucleus of a dwarf galaxy rather than a true globular cluster(e.g. Noyola, Gebhardt & Bergmann 2008). Different mea-surements using this technique have also led to conflictingpredictions as to the presence of an IMBH in a few clusters(e.g. Kamann et al. 2014).Further recent detection claims include the work ofPasham, Strohmayer & Mushotzky (2014), who reportedquasi-periodic oscillations in the X-ray emission of ULXM 82 - X-1, which they then used to estimate a blackhole mass of ∼ M ⊙ . However, the reliability of thistype of oscillations to constrain black hole masses has beenquestioned by other authors (e.g. Middleton et al. 2011).Baldassare et al. (2015) estimated a mass of 5 × M ⊙ forthe black hole in the centre of the dwarf galaxy RGG 118,using virial black hole mass estimate techniques, the limita-tions and caveats of which are discussed in detail by Shen(2013). Oka et al. (2016) concluded that the velocity disper-sion in the molecular cloud CO-0.40-0.22 is best modelledby the gravitational effect of a 10 M ⊙ black hole. The prox-imity of this molecular cloud to the Milky Way’s central SMBH, Sgr A ∗ , is particularly interesting within the con-text of IMBHs being potential seeds for SMBH formation.Despite this wealth of indirect observational evidence,there has not yet been an unambiguous detection of anIMBH. In this paper, we discuss how gravitational mi-crolensing would allow us to detect an astrometric signalthat could be unambiguously attributed to the presence ofan IMBH. While Safonova & Stalin (2010) have already pro-posed using microlensing as a technique to detect IMBHs incluster cores, they only considered the detection of photo-metric signals of microlensing of cluster stars by the IMBH,which have extremely low detection probabilities. Here, wewill show that astrometric microlensing is a far more promis-ing method to achieve a detection. We conduct a brief reviewof astrometric and photometric microlensing in Sec. 2, andhow it can be used to measure the mass of single objects(Sec. 3). The feasibility of such a detection is discussed inSec. 4, and we describe simulations to estimate expectedevent rates and the probabilities of detecting at least oneevent in several chosen globular clusters in Sec. 4.5. We dis-cuss our findings in Sec. 5, and draw conclusions for potentialfuture detections in Sec. 6. Astrometric microlensing has been discussed in detail byDominik & Sahu (2000). The interested reader is referredto that publication for a full discussion. Here we recall onlythe essential details.A microlensing event occurs when the observer, a sourceat distance D S , and a lens of mass M at a smaller distance D L become aligned. The time-dependent angular separation φ of the lens and source is usually expressed in units of theEinstein ring radius as u = φ/θ E , where θ E = r GMc ( D − − D − ) . (1)The photometric microlensing event then consists in theapparent magnification of the source, due to gravitationaldeflection of its light rays by the lens. A point source ismagnified by a factor (e.g. Paczy´nski 1986) µ ( u ) = u + 2 u √ u + 4 , (2)where u , and therefore µ ( u ), is time-dependent. Due to theasymmetric nature of the images of the source produced bythe gravitational lens, the apparent position of the sourcealso appears to change with time as the event unfolds follow-ing a characteristic pattern; this constitutes the astromet-ric microlensing. The apparent displacement of the centroidof a point source by an amount δ ( u ) can be expressed as(Hog, Novikov & Polnarev 1995) δ ( u ) = uu + 2 θ E , (3)with the displacement pointing away from the lens fromthe observer’s standpoint. The displacement has compo-nents parallel to the source-lens relative motion, δ k , and c (cid:13) , 1–13 earching for intermediate-mass black holes in globular clusters with gravitational microlensing perpendicular to it, δ ⊥ , which can be expressed (e.g.Dominik & Sahu 2000) as δ k = pu + p + 2 θ E δ ⊥ = u u + p + 2 θ E , (4)where u is the impact parameter, or minimum source-lensangular separation, in units of θ E . This occurs at time t ,and p ≡ p ( t ) = t − t t E , (5)where t is the time, and t E is the Einstein timescale, whichis the time taken by the source to cross the Einstein ringradius, such that t E = θ E /µ LS , where µ LS is the source-lens relative motion. Eq. (4) assumes a rectilinear uniformsource-lens relative motion, and is independent of the ob-servational point spread function (PSF). For a detailed dis-cussion of the behaviour of the expressions in Eq. (4), seeDominik & Sahu (2000). As the source moves relative to thelens, the components of the astrometric shift lead to a char-acteristic 1-dimensional (Fig. 1) pattern, or, in 2 dimensions,to an elliptical motion of the source’s centroid, as shown inFig. 2. These ellipses have eccentricity ǫ = [2 / ( u + 2)] / (Dominik & Sahu 2000).The photometric and astrometric effects behave differ-ently at small and large separations. From Eqns. (2) and (3),we see that for small values of u , the magnification becomesvery large, while the astrometric signal decreases linearlywith u . For large values of u , the photometric signal goesas u − , whereas the astrometric shift only decreases as u − .This means that the cross-section for astrometric events issignificantly larger than for photometric ones, making theman interesting channel to detect events for which lenses aremassive enough to cause a detectable signal. Microlensing has been used to measure the mass of sin-gle stars, which is made possible when subtle second-order effects are detectable in the light curves (e.g.Gould, Bennett & Alves 2004). For instance, the lens-sourceparallax π LS = D − − D − can be measured through lightcurve distortions, which means that the Einstein radius canthen be constrained using Eq. (1). If the size of the sourcecan also be constrained via additional second-order (“finitesource size”) effects (e.g. Gould 1992), then we can measure θ E and obtain a mass estimate for the lens (e.g. Kains et al.2013). Recently, observations from space telescopes havebeen used to constrain the parallax in microlensing events(e.g. Street et al. 2016; Zhu et al. 2015), with plans to dothis more routinely for events of interest with the SpitzerSpace Telescope (e.g. Udalski et al. 2015; Yee et al. 2015).In the case of lensing by an IMBH, however, finite sourcesize and parallax effects will not usually be detected, becausemost events will only be detectable through astrometry, andnot photometry, due to the much larger cross-section for
Figure 2. , , , , and 10 M ⊙ , with the ellipse sizeincreasing with mass, and the two lower masses shown in theinset), with the same IMBH parameters as Fig. 1, for u = 0 . u = 1 . t = t . astrometric events. Furthermore, for the kind of deep ob-serving campaigns towards the Galactic Bulge that wouldbe optimal for IMBH searches, the overwhelming majorityof source stars will be main-sequence stars, which are toosmall to produce significant source-size effects.Because the Einstein radius for an object scales with √ M (Eq. 1), observations over many years are needed todetect signals from IMBH lensing that allow us to constrainthe properties of the lens, whereas observations spanningmonths to a couple of years are usually sufficient for stellar-mass lenses. However, despite the extreme event timescalesproduced by IMBH lenses, they also lead to a much largerastrometric signal, making them easier to detect than forlow-mass lenses.If an astrometric signal is detected, the elliptical mo-tion of the source’s centroid can be used to measure θ E , viaEq. (4). In the case of field lens objects, only analysis ofsecond-order effects in the photometric event’s light curvecan then yield a constraint on D L , in order to combine itwith θ E to obtain a lens mass measurement. However, when D L is known, as is the case when considering IMBH lensesin the cores of globular clusters, the detection of the pho-tometric event is not necessary. We can derive or assume avalue D S (e.g. the distance to the Galactic Bulge for Bulgesources), so that the lens mass can be obtained from an as-trometric detection only, through Eq. (1). To do this fromthe time-series astrometric measurements, we fit the ellipti-cal trajectory due to lensing simultaneously with the sourceproper motion parameters. The lensing event can be char-acterised with the parameters t , t E , u , θ E , as well as aninclination angle α of the lens-source motion, while four pa-rameters are needed for the source proper motion: motionsalong the x and y axes, µ x and µ y , as well as arbitrary ref-erence points x and y . c (cid:13) , 1–13 N. Kains et al.
Figure 1.
The time-dependent centroid shift due to lensing by an IMBH of masses 10 , , , , and 10 M ⊙ plotted (with thesignal amplitude increasing with mass, and the lower two masses in the inset) for D L = 3 . D S = 8 . µ LS = 12 . u = 0 . left ) and 1.5 ( right ). The upper panels have a time axis in t E , while thelower panels show the signal for the various masses over a range of 20 years, with a time axis in years. The red segments in the upperpanels indicate the part of the astrometric curve that would be covered by a 20 year campaign centred on t = t , i.e. the same part thatis shown in the corresponding lower panels. Usually, campaigns focusing on stellar microlensing towardsthe Galactic Bulge (or other crowded regions) require care-ful estimates of the optical depth for both photometric andastrometric microlensing. In order to do this, one has to con-sider the entire populations of potential lens and source starsbetween the observer and the Galactic Bulge. When search-ing for IMBH in globular clusters, however, this is greatlysimplified because the location of the IMBH is known, in sofar as the distance to the cluster is known. Normally thisis the case to within a precision of 0.5 kpc or better forGalactic clusters.In theory, an IMBH in a cluster can lens both starswithin the cluster itself and background stars. In prac-tice, however, it is clear from Eq. (1) that the Einstein ra-dius, and therefore the lensing cross-section, tends to 0 for D L ∼ D S . Therefore the overwhelming probability for lens-ing comes from cases in which the source is a backgroundstar. For this reason, detections are only likely for clustersthat lie in front of the Galactic Bulge, the Small Magel-lanic Cloud (SMC), or the Large Magellanic Cloud (LMC), where background star number densities are high enoughthat a lensing event is reasonably likely to occur. This lim-its the sample of clusters to be considered. Furthermore,simulations have shown that IMBHs are highly unlikely toexist in core-collapsed clusters (Baumgardt, Makino & Hut2005), which excludes a significant number of targets. Wealso rejected clusters in high-extinction areas, only selectingclusters with a horizontal branch (HB) brighter than 19 magin V . Clusters with a fainter HB suffer from high extinction,meaning that background stars would also be highly extin-guished, and detecting sufficient numbers of them with agood enough signal-to-noise ratio (SNR) would require verylong exposure times. We excluded low-mass clusters such asAl 3 (BH 261), Djorg 2 (ESO456-SC38), and NGC 6540,and we rejected NGC 6809 because it is far away from theBulge, and has a low density of background stars, domi-nated by inner halo stars. The final list of clusters fulfillingall criteria is given in Table 1. We note that among these,NGC 362 is possibly currently undergoing core collapse (e.g.Dalessandro et al. 2013), but we include it because there isstill some debate as to the dynamical status of this clus-ter (e.g. McLaughlin & van der Marel 2005). We also notethat Leigh et al. (2014) used N -body simulations to showthat clusters containing stellar-mass black hole in binary c (cid:13) , 1–13 earching for intermediate-mass black holes in globular clusters with gravitational microlensing systems were less likely to host IMBHs with masses higherthan ∼ M ⊙ . M 22 contains two known stellar-mass blackholes (Strader et al. 2012), possibly meaning that any IMBHin this cluster is likely to have a mass lower than ∼ M ⊙ .However, the half-mass relaxation time of M 22 is ∼ x = D L /D S , theoptical depth for astrometric microlensing goes as (1 − x ) ,while for photometric microlensing the dependence goes as x (1 − x ). This means that while for photometric events, thelensing probability peaks for x = 0 .
5, i.e. a lens locatedhalf-way between the observer and the source, for astromet-ric events, the probability is highest for lenses that are muchcloser to the observer than the source, i.e. x ∼ For a stellar-mass lens event detected in an observing cam-paign lasting T obs , generally, t E ≪ T obs , so that the photo-metric event can be observed from baseline to peak and backto baseline as long as the magnification of the source reachesabove a threshold. On the other hand, the astrometric eventunfolds much more slowly, with t ast > T obs , and can usuallynot be observed in its entirety, except for low-mass lenses.Instead it is detected as long as the variation in centroidposition over the time T obs , δ obs , is above a threshold δ T ,as discussed by Dominik & Sahu (2000). Therefore the cen-troid shift itself, for the full event, might be larger than δ T ,but a short observing campaign may not be able to detectthe event because variations are slow, resulting in δ obs < δ T .For high-mass lenses such as IMBHs, t ast ≫ T obs for typical observing campaigns. Furthermore, because thecross-section for detection of astrometric events is muchlarger, and an astrometric detection is sufficient to constrainthe IMBH mass when D L is known, we will not consider thedetection of the photometric event.In addition to this, over large portions of the astromet-ric event, the centroid shift of the source caused by IMBHlensing will be linear and uniform in time, making it indis-tinguishable from the centroid shift due to proper motionof the source. Therefore only observations covering certainparts of the astrometric curve allow us to disentangle thereal lensing effect from the proper motion. This correspondsto parts of the astrometric curve where the change in totaldisplacement is not uniformly changing over T obs , as shownin Fig. 1, or for the 2-dimensional astrometric motion, asshown in Fig. 2. This 2-dimensional motion can allow usto detect curvature in the astrometric change even whenthe 1-dimensional displacement appears uniform. The 2-Dastrometric motion (Fig. 2) unfolds faster close to t = t (Fig. 3), so for a given T obs , the fraction of the 2-D astro-metric curve that is covered is larger when the position ofthe source is closer to u . Figure 3. t − t (in units of t E ), shown here for u = 1. Theblack filled circles are spaced equally by 1 t E . The source positionchanges more rapidly as it nears its closest approach to the lensat t = t . We carried out simulations to determine the probability ofan IMBH being detected unambiguously from an astromet-ric microlensing event in each cluster that passed our se-lection criteria. We considered 9 different IMBH masses, M BH /M ⊙ = 10 , 5 × , , × , , × , 10 , 5 × ,and 10 . Although we considered masses up to 10 M ⊙ in or-der to investigate the full mass range of IMBHs, an IMBHwith a mass larger than ∼ M ⊙ has a typical sphere ofinfluence (Peebles 1972) of ∼ . − ′ , which is compara-ble to, or larger than the core radius of globular clusters.Therefore, such IMBHs could also be detected by a numberof other techniques, since their surface brightness profileswould not be well fitted by King models, for example.For each cluster, D L is fixed by the distance to the clus-ter, and we assume D S is the distance to the Bulge or theSMC. The cluster distances we used are given in Table 1,and we assume distances to the Bulge and SMC of 8.5 kpc,which is a value within the range of different estimates inthe literature (e.g. Eisenhauer et al. 2003; Gillessen et al.2009; Vanhollebeke, Groenewegen & Girardi 2009) and 61kpc (Hilditch, Howarth & Harries 2005), respectively.Our ability to detect a lensing event unambiguously de-pends on a number of factors, and the expected number ofdetections can be expressed as h N det i = Z ∞−∞ Z ∞−∞ s ( x, y ) P det ( x, y ) dx dy , (6)where s ( x, y ) is the number density of background stars atcoordinates ( x, y ), and P det ( x, y ) is the probability for a sin-gle star that the event will be detected unambiguously, usingcriteria described in Sec. 4.6. The ( x, y ) coordinate systemis fixed relative to the IMBH, which may be assumed to beat the origin. Although necessary, the detection of curvature in the astro-metric curve of a background star is not sufficient to guar-antee the detection of a lensing event. Indeed, with moststars being members of binary, or multiple systems, and thewide range of orbital separations and eccentricities of suchsystems (e.g. Raghavan et al. 2010), many stars will exhibit c (cid:13) , 1–13 N. Kains et al.
ID RA DEC m V, HB r c Dist s µ LS [J2000.0] [J2000.0] [mag] [arcmin] [kpc] [arcsec − ] [mas/yr]BulgeNGC 6121 (M 4) 16:23:35 -26:31:33 13.45 1.16 2.2 0.10 a . b NGC 6304 17:14:32 -29:27:43 16.25 0.21 5.9 0.35 c . d, ∗ NGC 6528 18:04:50 -30:03:23 16.95 0.13 7.9 3.2 e . e NGC 6553 18:09:16 -25:54:28 16.60 0.53 6.0 1.6 f . f NGC 6626 (M 28) 18:24:33 -24:52:11 15.55 0.24 5.5 1.5 ‡ . g NGC 6656 (M 22) 18:36:24 -23:54:17 14.15 1.33 3.2 1.3 h . h, † SMCNGC 104 (47 Tuc) 00:24:06 -72:04:53 14.06 0.36 4.0 0.02 h . i NGC 362 01:03:14 -70:50:56 15.44 0.18 8.6 0.09 h . h, † Table 1.
Selected cluster targets. HB magnitudes, core radii, and distances are taken from the catalogue of Harris (1996), exceptfor the distance to NGC 104, which is from McLaughlin et al. (2006). References for the number densities of background stars s andproper motions relative to background Bulge/ SMC stars µ LS are (a)Bedin et al. (2013), (b)Dinescu et al. (1999) (c)Sarajedini et al.(2007), (d)Dinescu et al. (2003), (e)Lagioia et al. (2014), (f)Zoccali et al. (2001), (g)Casetti-Dinescu et al. (2013), (h)Bellini et al. (2014),(i)Anderson & King (2003). ∗ denotes an absolute proper motion measurement. † denotes a value of µ LS calculated from a proper motioncatalogue rather than taken from a reference paper. ‡ For NGC 6626, we estimated the stellar density based on its location along astraight line between NGC 6656 and NGC 6553. astrometric signatures due to orbital motion around the bi-nary system’s centre of mass. Although some binaries areeasily distinguished from single stars via their position ona colour-magnitude diagram (CMD), this is complicated inthe Bulge by a number of factors such as the metallicityspread of stars, differential reddening, the range of distancesdue to the size of the Bulge, and some contamination fromDisk stars. It is therefore useful to investigate the extent towhich signals caused by orbital motion in a binary mightmimic signals caused by the lensing of source stars by anIMBH along the line of sight.Little is known about the frequency of binaries in theGalactic Bulge or the SMC, and the distribution of theirorbital separations and eccentricities. In order to quantifyhow much of a confounding factor astrometric binaries canbe in this study, we use the distribution for Disk stars ofRaghavan et al. (2010), who found a Normal period distri-bution with h log P i = 5 .
03 and σ log P = 2 .
28, for P givenin days. They also found an approximately uniform distri-bution of eccentricities between e = 0 and ∼ .
9, except forsystems with
P < e = 0). Weassumed the scenario for Bulge stars that would producethe largest astrometric signal, which corresponds to equal-mass binaries and a total mass of 1 . M ⊙ , the largest pos-sible total binary mass in the Bulge (Calamida et al. 2015;Duquennoy & Mayor 1991). We also used uniform distribu-tions for the orientation of the system with respect to the x , y , and z axes.For Bulge stars, we find that shifts with a peak-to-peakamplitude of up to ∼ ∼ ∼ Figure 4.
Histogram of the peak-to-peak astrometric shift de-tected from astrometric binaries in the Bulge, for equal-mass bi-naries of total mass 1 . M ⊙ . The vertical dashed line indicatesthe highest shift that can be caused by a binary companion inthe Bulge. to-peak astrometric shift above 0.4 mas. Since this is thebest astrometric precision that can be achieved for brightsources (see Eq. 7 below), we can therefore safely assumethat any event that shows peak-to-peak shifts larger thanthat is caused by lensing. In theory, stars that are cluster members could also havenon-uniform astrometric curves, due to their orbits insideof the cluster. These could then potentially be confused forbackground stars being lensed by an IMBH in the cluster.However, we find that the number of such stars is negligible,due to the extremely long orbits involved. Furthermore, inthe vast majority of cases, the combination of proper mo-tions and the CMD allows us to determine cluster member-ship, even in cases where the cluster’s bulk motion relative to c (cid:13) , 1–13 earching for intermediate-mass black holes in globular clusters with gravitational microlensing the background stars is small (e.g. Lagioia et al. 2014). Fi-nally, we have conducted simulations to estimate the numberof astrometric lensing curves that can be mimicked by cir-cular orbits of cluster members, and found this to be withinthe error bars of our results (see Sec. 5). µ LS In order to estimate h N det i for each cluster, we must first es-timate the number density s of background stars. The num-ber density we calculate here is for stars with m F814W ≤ Hubble Space Telescope (HST)
Advanced Cam-era for Surveys (ACS) Survey of Galactic Globular Clus-ters (Sarajedini et al. 2007). We then compared the brightend (19 < m
F814W < .
5) of the resulting number den-sity distribution of background stars to the distribution ofBulge stars of Calamida et al. (2015) in the Sagittarius Win-dow Eclipsing Extrasolar Planet Search (SWEEPS) field( l = 1 . ◦ , b = − . ◦ ). We used the bright end of the distri-bution because this is where the photometry is not affectedby completeness issues. We calculated the number densityin the SWEEPS field by using the mass function foundby Calamida et al. (2015) to evaluate star counts down to m F814W ∼
26 mag, and found an average SWEEPS fielddensity of 6.5 stars/ arcsec . Finally, we used this to de-rive a scaling factor for number densities along the line ofsight to the globular clusters in our sample. For the clusterstoward the SMC, we used the same process, but with theSMC luminosity function of Kalirai et al. (2013) instead ofthe Bulge distribution, and found an average SMC numberdensity of 0.04 stars/ arcsec .For the source-lens relative motion µ LS , required as aningredient of our simulations, we used values from the lit-erature from proper motion studies, taking the bulk rela-tive proper motion of the cluster as a proxy for µ LS . ForNGC 362, we derived a value of µ LS from the proper motioncatalogue of Bellini et al. (2014) by calculating the medianproper motion of cluster and Bulge stars; we also did thiscalculation for NGC 6656 and NGC 104, to check that weobtained values consistent with those we adopted from theliterature. The resulting values of µ LS are listed, and rele-vant references are given, in Table 1.We adopt a scatter in the value of µ LS for the back-ground stars, using the dispersion in proper motion forBulge stars of 2.6 mas/yr along the directions of bothvelocity components. This value is in line with that ofClarkson et al. (2008), but we adopt the same value in bothdirections for simplicity. For SMC stars we use a scatter of0.3 mas/yr for both directions, in agreement with the find-ings of Vieira et al. (2010). We adopt a Monte Carlo approach to evaluating the de-tection probabilities P det ( x, y ) over a grid in ( x, y ) for eachIMBH mass. To keep the number of simulations reasonableas we evaluate P det ( x, y ) further away from the IMBH, wechose to perform a constant number of simulations, 1000,in rings of equal widths around the origin. We chose a ringwidth of 0.2 θ E .For each simulation, we draw the position of a back-ground star from a uniform distribution over the area of thecurrent grid element and assume that this is the source po-sition at t = 0. Without any loss of generality, we assumethat our ( x, y ) coordinate system is aligned such that thebulk cluster proper motion relative to the background starsmakes the background stars appear to move along the x-axis in the positive direction (since our IMBH is fixed atthe origin). Considering also that the background stars ex-hibit a velocity dispersion, we therefore draw the source-lensrelative proper motion from a two-dimensional Gaussian dis-tribution with means in the x − and y − directions of µ LS andzero, respectively, and σ in both directions equal to the dis-persion in the Bulge or SMC proper motions as appropriate.In the HST data archive, there are typically of the orderof 50 images for a cluster, spaced out over 20 years. Hence,for each simulation we adopt a time baseline of T obs = 20years with the first and last images obtained at t = 0 and t = 20 years, respectively. The epochs of the remaining 48images are drawn from a uniform distribution on the range[0, T obs ]. The astrometric motion curve of the backgroundstar is then generated using these epochs taking into accountthe source-lens relative proper motion, its position at t = 0,and the lensing effect of the IMBH.To simulate measurement noise in the astrometric mo-tion curve of the background star, we assign the star a ran-dom magnitude, drawn from the Bulge or SMC luminosityfunctions of Calamida et al. (2015) or Kalirai et al. (2013),respectively. We calculated the SNR in a 15-minute expo-sure with WFC3/UVIS for different magnitudes between m F814W = 18 and 26 mag, with a bin size of 0.2 mag, usingthe HST exposure time calculator. This allowed us to esti-mate the SNR for each background star, and thereby, theastrometric measurement precision z , through the expres-sion (Kuijken & Rich 2002), z = 0 . × √ N e , (7)where FWHM is the full-width half-maximum of the star’sPSF, and N e is the number of images per epoch. Thislevel of precision has been routinely achieved by severalprojects using HST observations for high-precision astro-metric measurements (e.g. Bedin et al. 2013; Bellini et al.2014; van der Marel et al. 2014). For simplicity, we use thepixel scale of WFC3/UVIS of 40 mas/ pixel for all simulatedobservations, and we use a conservative estimate of N e = 4.Each astrometric measurement of the background star isperturbed by a random number drawn from a Gaussian dis-tribution with zero mean and σ = z , thereby generating oursimulated noisy astrometric motion curve from HST obser-vations. c (cid:13) , 1–13 N. Kains et al.
For our detection criteria, we need a statistic that will al-low us to discriminate between astrometric motion modelswith different numbers of parameters (i.e. rectilinear uni-form motion, astrometric microlensing, and orbital motion).We use the Bayesian Information Criterion (BIC; Schwarz1978) derived by approximating the posterior probability ofeach model. It is valid for model parameters estimated bymaximum likelihood. For model parameters with a uniformprior, it is given by the expressionBIC = − L ) + N P ln( N D ) − N P ln(2 π ) (8)where L is the maximum-likelihood statistic, N D is the num-ber of data points, and N P is the number of model parame-ters. The use of an information criterion for discriminatingbetween models in our simulations is particularly appropri-ate since we have generated the astrometric measurementsfor each background star from independent Gaussian distri-butions for which the sigma values are known exactly. Inthis case, the BIC further reduces toBIC = χ + N P ln( N D ) − N P ln(2 π ) + K (9)where χ is the chi-squared statistic, and K is a constantterm that can be ignored for model selection purposes. Theratio of the posterior probabilities P ( A ) and P ( B ) of twomodels may be calculated from the BIC via: P ( A ) P ( B ) = exp(0 . BIC )) , (10)where ∆ BIC = BIC B − BIC A . We choose to adopt the thresh-old corresponding to a relative probability P ( A ) /P ( B ) =100, i.e. ∆ BIC ,T =9.21.For each noisy astrometric motion curve that we gen-erated in the simulations, we first fit a rectilinear uniformproper motion model (4 parameters) to the data and we cal-culate a corresponding BIC value which we denote as BIC lin .We then perform a slew of tests which must be satisfied inorder for a successful detection of astrometric microlensingevent to be declared:(i) We compute the peak-to-peak amplitude δ obs of theresiduals to the fit. If this is above 2z, then we proceed tostep (ii).(ii) We fit the 9-parameter astrometric lensing model tothe data (see Section 3) and calculate the correspondingBIC lens .(iii) We check that the mass of the IMBH lens is recoveredcorrectly to within a factor of ten, and if so, then we proceedto step (iv).(iv) We verify that the astrometric lensing model isfavoured over the rectilinear uniform proper motion modelabove our threshold, i.e. BIC lin − BIC lens ≥ ∆ BIC ,T . If so,then we proceed to step (v).(v) If δ obs ≥ Figure 5.
Cumulative expected number of detections of IMBHlensing events, as a function of the distance from the lens, foreach of the nine IMBH masses considered, with h N det i increasingwith mass. proper motion, and seven parameters for binary orbitalmotion, see Section 4.3.1) and calculate the correspondingBIC bin .(vii) We verify that the astrometric lensing model isfavoured over the orbital motion model above our thresh-old, i.e. BIC bin − BIC lens ≥ ∆ BIC ,T . If so, then we declaresuccessful detection of astrometric microlensing. Otherwise,we finish.The computation of P det ( x, y ) for each grid element isthen trivial as the ratio of the number of successful detec-tions to the number of simulations performed. The detectionprobabilities tend to zero as the distance from the IMBH in-creases due to lack of curvature and decreasing peak-to-peaksignal in the astrometric motion curves over the observa-tional baseline. In Fig. 5, we plot h N det i evaluated via equa-tion 6 as a function of integration radius from the IMBH. Wecan stop the integration when asymptotic limits for h N det i are reached, for example at ∼ θ E for an IMBH in M 22and T obs = 20 years. Results from our simulations are given in Table 3.From this, we see that h N det i is significant for mostof the selected Bulge clusters for IMBH masses above ∼ M ⊙ , and for some, down to masses of ∼ M ⊙ . Unsur-prisingly, the most promising clusters for such detections arefour of the five clusters in our sample closest to the SolarSystem, with M 22, NGC 6553, NGC 6121, and NGC 6626having the largest number of expected events. For clusterstoward the SMC, low background star densities make lensingprobabilities, and h N det i , very low.We plot h N det i as a function of lens mass for baselinesof 20, 25, and 30 years for these four clusters in Fig. 6.With a time baseline of 20 years, h N det i in M 22 is large for M > × M ⊙ , and declines with mass down to h N det i =0 . M = ∼ M ⊙ . This makes it the best candidatefor further analysis. The next best candidate is NGC 6553, c (cid:13) , 1–13 earching for intermediate-mass black holes in globular clusters with gravitational microlensing Figure 6.
The expected number of detected events as a function of black hole mass. The detection rates from simulations are plottedas triangles, diamonds, and open circles, along with a power-law fit, for baselines of 20, 25, and 30 years, respectively. with h N det i = 0 .
93, 0.70, and 0.34 for M = 10 , 5 × and 10 M ⊙ respectively, and h N det i = 0 .
11 at M = 10 M ⊙ .Thanks to the fast motion of M 4 relative to the Bulge, anddespite low stellar densities, the expected numbers of eventsare slightly higher than NGC 6553 for a high-mass IMBH,with h N det i = 1 .
03 and 0.75 for M = 10 and 5 × M ⊙ respectively.For the Bulge clusters, we fit a power law for the massdependence of h N det i , of the form h N det i = a (cid:18) MM f (cid:19) b , (11)where a and b are the fitted power-law parameters, and M f is an arbitrary fiducial mass. We find that the mass depen-dence is best fitted with two mass regimes, and therefore wefit two power laws for masses below and above 10 M ⊙ . Thecoefficients a and b for both regimes are given in Table 2.It is also useful to turn values of h N det i into probabili-ties that are easier to interpret. Under the assumption thatan IMBH exists in the cluster, then the discrete Poisson dis-tribution is appropriate for the number of stars observed tofeature a detectable astrometric signal caused by an IMBH.We know that the detectable astrometric events occur withan expected average rate of h N det i calculated from our sim-ulations, and we can therefore express the probability of n events being detected as P ( n ) = h N det i n n ! e −h N det i , (12)from which we can express the probability of at least oneevent being detected as P ( n >
0) = 1 − e −h N det i . (13)In Table 4, we give P ( n >
0) for each IMBH mass, clus-ter, and time baseline, calculated from the values of h N det i listed in Table 3.In order to assess how to best exploit the availablearchival data, or how to devise the most efficient future long-term observing strategy to maximise chances of IMBH de-tections, we looked at how the probabilities given in Table 4change for different values of T obs . We considered values of25 to 30 years and scaled the number of observations linearlywith T obs . The effect of T obs on the expected detection rateis shown in Fig. 6. The expected number of detections risesapproximately linearly, and at a rate that is faster for largerIMBH masses. For the lower mass ( < M ⊙ ) IMBHs, theincrease is small: since t E is smaller, even a relatively shortobserving campaign is sufficient to cover a significant por-tion of the astrometric curve, and the returns of extendingthe observing baseline are modest. On the other hand, forhigh-mass IMBHs, even a small increase in observing base-line improves event numbers significantly. For the four most c (cid:13) , 1–13 N. Kains et al. T obs = 20 years M/M ⊙ × × × × BulgeNGC 6121 (M 4) 1.03(2) 0.75(2) 0.34(1) 0.24(0) 0.10(0) 0.06(0) 0.02(0) 0.01(0) 0.004(0)NGC 6304 0.07(0) 0.06(0) 0.03(0) 0.03(0) 0.01(0) 0.008(0) 0.003(0) 0.002(0) 0NGC 6528 0.26(1) 0.18(1) 0.09(0) 0.06(0) 0.03(0) 0.02(0) 0.005(0) 0.003(0) 0NGC 6553 0.93(3) 0.70(2) 0.34(1) 0.24(0) 0.11(0) 0.07(0) 0.03(0) 0.02(0) 0.003(0)NGC 6626 (M 28) 0.69(3) 0.55(2) 0.29(1) 0.21(0) 0.09(0) 0.06(0) 0.02(0) 0.01(0) 0.004(0)NGC 6656 (M 22) 5.66(15) 4.20(10) 1.99(3) 1.42(2) 0.58(1) 0.37(0) 0.13(0) 0.08(0) 0.02(0)SMCNGC 104 (47 Tuc) 0 0 0 0 0 0 0 0 0NGC 362 0 0 0.002(0) 0.002(0) 0.002(0) 0.002(0) 0.001(0) 0 0 T obs = 25 years M/M ⊙ × × × × BulgeNGC 6121 (M 4) 1.69(3) 1.20(2) 0.52(1) 0.36(0) 0.14(0) 0.09(0) 0.03(0) 0.02(0) 0.006(0)NGC 6304 0.14(1) 0.11(0) 0.06(0) 0.04(0) 0.02(0) 0.01(0) 0.004(0) 0.003(0) 0.001(0)NGC 6528 0.43(1) 0.31(1) 0.14(0) 0.10(0) 0.04(0) 0.02(0) 0.008(0) 0.004(0) 0NGC 6553 1.64(5) 1.22(3) 0.56(1) 0.39(1) 0.16(0) 0.10(0) 0.04(0) 0.02(0) 0.005(0)NGC 6626 (M 28) 1.24(4) 0.92(3) 0.45(1) 0.33(1) 0.14(0) 0.09(0) 0.03(0) 0.02(0) 0.005(0)NGC 6656 (M 22) 9.60(21) 7.05(13) 3.03(4) 2.12(3) 0.83(1) 0.54(1) 0.19(0) 0.12(0) 0.03(0)SMCNGC 104 (47 Tuc) 0 0 0 0.001(0) 0.001(0) 0.001(0) 0 0 0NGC 362 0 0.002(0) 0.006(0) 0.006(0) 0.005(0) 0.004(0) 0.001(0) 0.001(0) 0 T obs = 30 years M/M ⊙ × × × × BulgeNGC 6121 (M 4) 2.51(4) 1.73(3) 0.73(1) 0.50(1) 0.18(0) 0.12(0) 0.04(0) 0.03(0) 0.007(0)NGC 6304 0.25(1) 0.19(1) 0.09(0) 0.06(0) 0.02(0) 0.02(0) 0.006(0) 0.004(0) 0.001(0)NGC 6528 0.65(2) 0.48(1) 0.20(0) 0.14(0) 0.05(0) 0.03(0) 0.01(0) 0.005(0) 0NGC 6553 2.43(6) 1.79(4) 0.79(1) 0.53(1) 0.22(0) 0.14(0) 0.05(0) 0.03(0) 0.006(0)NGC 6626 (M 28) 1.91(6) 1.42(3) 0.67(1) 0.46(1) 0.19(0) 0.12(0) 0.04(0) 0.03(0) 0.006(0)NGC 6656 (M 22) 14.55(27) 10.28(17) 4.42(6) 2.95(3) 1.13(1) 0.72(1) 0.25(0) 0.15(0) 0.04(0)SMCNGC 104 (47 Tuc) 0 0 0.001(0) 0.002(0) 0.001(0) 0.001(0) 0 0 0NGC 362 0.004(0) 0.009(0) 0.01(0) 0.01(0) 0.008(0) 0.005(0) 0.002(0) 0.001(0) 0
Table 3.
Expected number of detected events for each of the 9 IMBH masses considered, for T obs = 20, 25, and 30 years. 1- σ error barson the last decimal place are given in parentheses. An error bar of 0 indicates that the Poisson error is smaller than the precision quoted. promising clusters we have identified, adding even just fiveyears to the baseline increases the probability of detectingan event significantly for the largest IMBH masses. For ex-ample, for M 22, compared to a 20-year campaign, for a10 M ⊙ IMBH, a baseline of T obs = 25 years increases h N det i by ∼ T obs = 30 years yields approximately double theexpected detection rate, bringing the probability to 0.68.We also looked at the impact of varying the number ofobservations between 1 and 5 epochs per year, and foundthat the improvement is not significant. In fact observingcadences down to 1 observation every ∼ − There are many existing observations from various scienceprograms that can already be used to search for the astro-metric gravitational lensing signals caused by the presence ofIMBHs in globular clusters. In addition to these, future ob-servations that will be obtained for many clusters for a widerange of science objectives, in particular stellar population c (cid:13) , 1–13 earching for intermediate-mass black holes in globular clusters with gravitational microlensing T obs = 20 years M/M ⊙ × × × × BulgeNGC 6121 (M 4) 0.64(1) 0.53(1) 0.29(0) 0.21(0) 0.09(0) 0.06(0) 0.02(0) 0.01(0) 0.004(0)NGC 6304 0.07(0) 0.06(0) 0.03(0) 0.02(0) 0.01(0) 0.008(0) 0.003(0) 0.002(0) 0NGC 6528 0.23(1) 0.16(1) 0.09(0) 0.06(0) 0.03(0) 0.02(0) 0.005(0) 0.003(0) 0NGC 6553 0.60(1) 0.51(1) 0.29(1) 0.22(0) 0.10(0) 0.07(0) 0.02(0) 0.02(0) 0.003(0)NGC 6626 (M 28) 0.50(2) 0.42(1) 0.25(1) 0.19(0) 0.09(0) 0.06(0) 0.02(0) 0.01(0) 0.004(0)NGC 6656 (M 22) 1.00(0) 0.98(0) 0.86(0) 0.76(1) 0.44(0) 0.31(0) 0.12(0) 0.08(0) 0.02(0)SMCNGC 104 (47 Tuc) 0 0 0 0 0 0 0 0 0NGC 362 0 0 0.002(0) 0.002(0) 0.002(0) 0.002(0) 0.001(0) 0 0 T obs = 25 years M/M ⊙ × × × × BulgeNGC 6121 (M 4) 0.82(1) 0.70(1) 0.41(0) 0.30(0) 0.13(0) 0.08(0) 0.03(0) 0.02(0) 0.006(0)NGC 6304 0.13(1) 0.10(0) 0.06(0) 0.04(0) 0.02(0) 0.01(0) 0.004(0) 0.003(0) 0.001(0)NGC 6528 0.35(1) 0.27(1) 0.13(0) 0.10(0) 0.04(0) 0.02(0) 0.008(0) 0.004(0) 0NGC 6553 0.81(1) 0.70(1) 0.43(1) 0.33(0) 0.14(0) 0.10(0) 0.03(0) 0.02(0) 0.005(0)NGC 6626 (M 28) 0.71(1) 0.60(1) 0.36(1) 0.28(0) 0.13(0) 0.09(0) 0.03(0) 0.02(0) 0.005(0)NGC 6656 (M 22) 1.00(0) 1.00(0) 0.95(0) 0.88(0) 0.56(0) 0.42(0) 0.17(0) 0.11(0) 0.03(0)SMCNGC 104 (47 Tuc) 0 0 0 0.001(0) 0.001(0) 0.001(0) 0 0 0NGC 362 0 0.002(0) 0.006(0) 0.006(0) 0.005(0) 0.004(0) 0.001(0) 0.001(0) 0 T obs = 30 years M/M ⊙ × × × × BulgeNGC 6121 (M 4) 0.92(0) 0.82(0) 0.52(0) 0.39(0) 0.17(0) 0.11(0) 0.04(0) 0.02(0) 0.007(0)NGC 6304 0.22(1) 0.17(0) 0.08(0) 0.06(0) 0.02(0) 0.02(0) 0.006(0) 0.004(0) 0.001(0)NGC 6528 0.48(1) 0.38(1) 0.18(0) 0.13(0) 0.05(0) 0.03(0) 0.01(0) 0.005(0) 0NGC 6553 0.91(1) 0.83(1) 0.54(1) 0.41(0) 0.19(0) 0.13(0) 0.05(0) 0.03(0) 0.006(0)NGC 6626 (M 28) 0.85(1) 0.76(1) 0.49(1) 0.37(0) 0.17(0) 0.12(0) 0.04(0) 0.03(0) 0.006(0)NGC 6656 (M 22) 1.00(0) 1.00(0) 0.99(0) 0.95(0) 0.68(0) 0.51(0) 0.22(0) 0.14(0) 0.04(0)SMCNGC 104 (47 Tuc) 0 0 0.001(0) 0.002(0) 0.001(0) 0.001(0) 0 0 0NGC 362 0.004(1) 0.009(1) 0.01(0) 0.01(0) 0.008(0) 0.005(0) 0.002(0) 0.001(0) 0
Table 4.
Probability of detecting at least one astrometric lensing event for each of the 9 IMBH masses considered, for T obs = 20, 25, and30 years. 1- σ error bars on the last decimal place are given in parentheses. An error bar of 0 indicates that the Poisson error is smallerthan the precision quoted. studies, will extend the time baseline of astrometric data setsfor the clusters in our sample. The current available baselinefor M 22 in the HST archive is 22 years, meaning that theexpected number of detectable lensing events in the existingdata set for a 10 M ⊙ IMBH is around h N det i = 0 . James Webb Space Tele- scope (JWST) and the
Wide-field Infrared Survey Telescope (WFIRST). These telescopes will be able to make astromet-ric measurements with precisions similar to or better thanwhat can be achieved with HST, further extending our as-trometric baseline throughout their mission lifetimes. Thiswould allow for the detection of IMBHs in several globu-lar clusters if they exist, and to obtain constraints on thedemographics of these elusive objects. c (cid:13) , 1–13 N. Kains et al. a LM b LM a HM b HM T obs = 20 yearsNGC 6121 (M 4) 0.10(1) 0.64(7) 0.33(1) 0.50(1)NGC 6304 0.01(1) 0.59(53) 0.03(1) 0.37(4)NGC 6528 0.03(1) 0.69(28) 0.09(1) 0.47(2)NGC 6553 0.11(1) 0.65(6) 0.33(1) 0.47(1)NGC 6626 (M 28) 0.09(1) 0.61(7) 0.27(1) 0.44(1)NGC 6656 (M 22) 0.58(1) 0.64(1) 1.90(1) 0.51(1) T obs = 25 yearsNGC 6121 (M 4) 0.14(1) 0.66(5) 0.51(1) 0.53(1)NGC 6304 0.02(1) 0.63(36) 0.05(1) 0.42(3)NGC 6528 0.04(1) 0.72(20) 0.14(1) 0.50(1)NGC 6553 0.16(1) 0.65(4) 0.54(1) 0.51(1)NGC 6626 (M 28) 0.14(1) 0.65(5) 0.44(1) 0.47(1)NGC 6656 (M 22) 0.84(1) 0.65(1) 2.94(2) 0.54(1) T obs = 30 yearsNGC 6121 (M 4) 0.18(1) 0.66(4) 0.72(1) 0.55(1)NGC 6304 0.03(1) 0.64(27) 0.08(1) 0.48(2)NGC 6528 0.06(1) 0.75(15) 0.20(1) 0.53(1)NGC 6553 0.22(1) 0.67(3) 0.76(1) 0.53(1)NGC 6626 (M 28) 0.19(1) 0.64(4) 0.64(1) 0.50(1)NGC 6656 (M 22) 1.13(1) 0.67(1) 4.22(2) 0.56(1) Table 2.
Power-law coefficients (see Eq. 11), for the two massregimes (subscripts LM and HM denote the low- and high-massregimes, respectively, with the limiting mass between the tworegimes set at 10 M ⊙ ), for the Bulge clusters, and T obs = 20years. We used M f = 10 M ⊙ and 10 M ⊙ for the low- and high-mass regimes, respectively. ACKNOWLEDGEMENTS
We thank Andrea Bellini for sharing with us his cataloguesof proper motions for NGC 104, NGC 362, and NGC 6656,Manuela Zoccali for sending us her proper motion catalogueof NGC 6553, Luigi Bedin for sharing his proper motionand photometry catalogue of M 4, and Cristina Pallancafor her photometric catalogue of M 28. NK thanks MurrayBrightman for useful discussions.
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