Gravitational Lensing and Microlensing in Clusters: Clusters as Dark Matter Telescopes

The methodology starts with identification of a sample of clusters at intermediate redshifts, whose mass distributions are well-constrained by arcs and multiple images of known redshifts. Next is the search along the critical curves of the cluster. Sources that lie on or near the caustics (one-dimensional loci of formally infinite magnification) in the source plane are imaged on or in the vicinity of the critical curves (lens plane images of the caustics) with very high factor. Location of critical lines depends on cluster’s mass distribution, angular diameter distances between lens, source and observer, and $z$ of the source, thus is calculable once the mass distribution of the cluster lens is constrained. For sources occupying a specific redshift range, e.g. $2<z<7$, particularly high magnification can be expected ($\sim$ 40-fold), and they are detectable in optical bands. At $z>7$, the most relevant signatures are expected in the near-IR $\lambda\geq 1\,\mu$m (due to shift of Lyman-$\alpha$ emission). By only searching say $z>5$ critical line, we minimize contamination from lower redshift sources.

Large field of view combined with a good angular resolution ($0.^{\prime\prime}4$) provides an advantage in efficient imaging the regions in and around the image plane critical curves (which are typically $\sim$ tens of arcseconds in extent). Multi-waveband observational capabilities will provide an aid to identifying images on the basis of colour, facilitating separation of foreground objects (including the lens cluster members) from the background objects. Generating deep exposure by stacking successive observations will enable us to go down to faint image magnitudes of around 25 – 26, accessing source magnitudes that are several times fainter. Spectroscopic follow-up will allow accurate determination of the redshifts of the images, and aid in identifying and separating possible blends.

Globular clusters are hypothesized to harbour intermediate-mass ($10^{3}-10^{4}$ $M_{\odot}$ ) black holes in their centres. Gravitational ML of a background star by the central IMBH is the only currently existing tool that can resolve the nature of the central dark mass, as there is a significant difference in the lensing signatures of a single point-mass lens (a single black hole) and an ensemble of point-mass lenses (which would be if the central mass consisted of many low-mass objects). In 2010, our group initiated monitoring of a set of selected GCs looking for ML signature of possible central IMBH, which would act as a lens, amplifying light of stars passing behind (Safonova & Stalin, 2010). Since the time scale of such event is of the order of $\sim$yrs, the monitoring cadence is once a month. We use differential photometry to search for variability in our data using the package originally developed by Wozniak (2000) and modified by W. Pych (2012). Differential photometry is unaffected by blending and, more importantly, is sensitive to ML events due to stars that are too faint to be detected at baseline, which is usual for the cores of globular clusters. The reference frame (RF) is matched to the seeing of each science image and subtracted to obtain a series of difference images. The GC centre is examined for variability and candidates transient/variable are further studied after constructing the light curves. In Fig. 1 we present the sample data from the globular cluster M13, demonstrating the technique (Joseph et al. 2015).

It is now believed that stars with planets are a rule rather than exception, with estimates of the actual number of planets exceeding the number of stars in our Galaxy alone by orders of magnitude (Cassan et al. 2012). However, most of the planets $\sim 4000$ detected till now are in the field of the Galaxy. By a simple estimate, Milky Way’s $\sim 200$ globular clusters having $\sim$ million stars each, have a total $\sim 200$ million stars. If there are 10 planets for each star, there shall be $\sim 2$ billion planets in galactic GCs. Actually, there is only 1! – in a globular cluster M4, discovered accidentally by a pulsar timing. Interesting question arises: where are the GCs planets? It was suggested that in dense GC environment most of the planets will get liberated – turn ‘rogue’. The current mass fraction of FFPs may reach $\sim 10\%$ in the entire cluster, and the central density in planets may be $10^{5}$ pc^-3 at current epoch, since a significant population of FFPs is retained in relaxed systems (Soker et al. 2001). Their numbers in top 20 dense clusters may even exceed the number of stars by a factor of $\sim 100$ (Fregeau et al. 2002, Hurley & Shara 2002). FFPs are undetectable by transit or radial velocity methods, and their direct imaging is also not possible. The only way to detect the population of FFPs in GCs is by gravitational microlensing. We have observed GC M4 (chosen due to its proximity, location and the actual existence of a planet) in 2011 in search for ML signatures of FFPs (Safonova et al. 2016). The number of ML events in a time of observation $t^{\rm obs}$ can be calculated considering a lens (1 Jupiter-mass FFP) with Einstein radius $\theta_{\rm E}$ moving with angular velocity $v$,

$$\frac{{\Delta N}_{\rm FFP}}{t^{\rm obs}}=\pi\theta_{\rm E}^{2}v\cdot 2\theta_{\rm GC}\cdot\frac{N_{*}}{4/3\pi\theta^{3}_{\rm GC}}\times N_{\rm FFP}\,,$$

(1)

where $\pi\theta^{2}_{\rm E}$ is event cross-section, $\theta_{\rm GC}$ is the radius within which we estimate the event rate (e.g. tidal or half-mass), $N_{*}$ is number of stars and $N_{\rm FFP}$ is number of FFPs inside the radius. Velocity dispersion in GCs was shown to depend weakly on mass due to the lack of equipartition (Trenti & van der Marel 2013), such that $\sigma\propto m^{-0.08}$. In our case, for a mean stellar mass of 1/3 $M_{\odot}$ , we estimate for FFPs $\sigma_{\rm FFP}\approx 1.61\sigma_{*}$. We assume that velocity vectors have random but isotropic distribution, sources are not moving, and all events have the same characteristic time scale $t_{\rm E}=R_{\rm E}/\sigma_{\rm FFP}$. For an optimistic stellar density $N_{*}$ of $10^{3}$ pc^-3, the rate of ML events by FFP in 4 months of observations is

	$\displaystyle N^{\rm low}_{\rm FFP}=5.8\times 10^{-3}\,,\quad\text{where low means }N_{\rm FFP}=N_{*}\,,$
	$\displaystyle N^{\rm high}_{\rm FFP}=0.58\,,\quad\text{where high means }N_{\rm FFP}=100\,N_{*}\,.$		(2)

Einstein radius of an FFP is about $100$ times smaller than that of central IMBH, and so we expect the duration of ML events to be $\lesssim$ days: e.g. in M4 it is $\sim 19$ hrs for a Jupiter-mass planet.