Joint Strong and Weak Lensing Analysis of the Massive Cluster Field J0850+3604

We select background source galaxies for our weak lensing analysis via the ${{BRz}}^{\prime }$ color–color selection developed by Medezinski et al. (2010). Selection of the background sources is key to accurate weak lensing measurements, as contamination by foreground sources or cluster members will dilute the lensing signal when unaccounted for, leading to a bias in the inferred cluster parameters (e.g., Medezinski et al. 2010; Okabe et al. 2010).

We start by placing all galaxies in the field on a B − R versus $R\mbox{--}{z}^{\prime }$ color–color diagram. We divide this color–color space into discrete cells and calculate the mean projected distance of all galaxies from the cluster center (as reported by Ammons et al. 2014) within each cell (Figure 1, left panel). Figure 1 shows a clear region where the mean projected distance of galaxies is smaller than the rest of the color–color space. This region corresponds to colors of the cluster members themselves. The number density of galaxies in the same color–color space (Figure 1, right panel) shows an overdensity in this same region, i.e., the massive cluster. We want to exclude these galaxies from our source population.

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We define photometric selection criteria to select red background galaxies, blue background galaxies, cluster galaxies, and foreground galaxies. Our criteria are listed in Table 2. The left panel of Figure 2 shows the individual galaxies in the field, color-coded by subsample (cluster, foreground, blue background, or red background).

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Table 2.  Color–Color Selected Galaxies

Sample	Selection Criteria	N	$\langle z\rangle$ a	$\langle {\beta }_{\mathrm{wl}}\rangle$ a
Cluster	${z}^{\prime }\leqslant 26.5$	1147	0.44	...
	$B-R\leqslant 3.3$
	$0.4\leqslant R-{z}^{\prime }\leqslant 0.8$
	$(B-R)-(R-{z}^{\prime })\geqslant 2.0$
Background (blue)	$22\leqslant {z}^{\prime }\leqslant 26$	7220	1.53	0.56
	{$R-{z}^{\prime }\leqslant 0.35$ AND $[(B-R)+0.3\times (R-{z}^{\prime })]\leqslant 1.3$} OR {$R-{z}^{\prime }\leqslant -0.1$}
Background (red)	$21\leqslant {z}^{\prime }\leqslant 26$	13593	1.11	0.56
	$R-{z}^{\prime }\geqslant 0.35$
	$(B-R)-(R-{z}^{\prime })\leqslant 0.9$
	$(B-R)+(R-{z}^{\prime })\leqslant 2.5$

Note.

^aMean photometric redshift and distance ratio, measured from COSMOS galaxies with the same selection criteria applied.

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As a check of the robustness of our selection criteria, we plot the number density of the different galaxy subsamples as a function of distance from the cluster center (Figure 2, right panel). The cluster galaxy sample is concentrated near the cluster center and drops off with increasing radius, as expected. The foreground galaxy subsample remains flat across the field, as those galaxies are not associated with the cluster or affected by lensing. Both background source galaxy samples are flat at large radii, but show a decline within ∼2' of the cluster center. This is indicative of depletion of the galaxy number counts, due to lensing magnification by the cluster (e.g., Broadhurst et al. 1995). When no source selection is applied, the cumulative galaxy number counts as a function of magnitude tend to show a power law behavior with a slope that is close to s = 0.4, where no magnification bias due to lensing is expected. However, owing to our color–color source selection, the count slope as a function of magnitude cut progressively decreases as we go to fainter magnitudes (e.g., Chiu et al. 2016). Because of the depth of the imaging data, the effective count slope at our limiting magnitude (${R}_{c}=27.5$) reaches s ≲ 0.2, for which depletion of source counts is expected.

This depletion cannot be explained by the masking of background sources by cluster members, foreground objects, and defects (e.g., saturated stars and stellar trails), which is only a ∼10% effect in the central regions (e.g., Umetsu et al. 2016). This masking effect is estimated and accounted for in the right panel of Figure 2, following the procedure given in (Umetsu et al. 2011b, see their Appendix A.2) and Umetsu et al. (2016).

The weak lensing signal scales with the angular diameter distance ratio ${\beta }_{\mathrm{wl}}\equiv {D}_{\mathrm{LS}}/{D}_{{\rm{S}}}$, with ${\beta }_{\mathrm{wl}}\equiv 0$ for foreground objects with $z\lt {z}_{{\rm{L}}}$. For statistical weak lensing measurements, the mean distance ratio is

$$\begin{eqnarray}&&\langle {\beta }_{\mathrm{wl}}\rangle =\displaystyle \frac{{\int }_{0}^{\infty }N(z){\beta }_{\mathrm{wl}}(z){dz}}{{\int }_{0}^{\infty }N(z){dz}},\end{eqnarray} \tag{ 3 }$$

where N(z) is the redshift distribution function of the background population. We estimate and correct the depths of the different subsamples by applying similar selection criteria to the Cosmic Evolution Survey (COSMOS; Capak et al. 2007), for which accurate photometric redshifts have been derived from 30-band photometry (Ilbert et al. 2009). Because the COSMOS filters do not include the R_c band, we calculate fluxes from the best-fit templates with EAZY (Brammer et al. 2008). We then calculate the mean redshifts and distance ratios for the background subsamples (Table 2). The redshift histograms of the various galaxy subsamples are shown in Figure 3. The mean redshifts of the red and blue background subsamples are $\langle {z}_{\mathrm{red}}\rangle =1.11$ and $\langle {z}_{\mathrm{blue}}\rangle =1.53$, respectively. Despite this difference, these two subsamples coincidentally have the same mean distance ratio of $\langle {\beta }_{\mathrm{wl}}\rangle =0.56$, due to the fact that the blue population has a larger fraction of low redshift (${\beta }_{\mathrm{wl}}=0$) interlopers. We use the value $\langle {\beta }_{\mathrm{wl}}\rangle =0.56\pm 0.03$ for our weak lensing analysis, which accounts for a typical 5% uncertainty (e.g., Umetsu et al. 2014) that is marginalized over in the modeling. Updated photometric redshifts from Laigle et al. (2016) are available in the COSMOS field, but using them only changes the mean distance ratio to $\langle {\beta }_{\mathrm{wl}}\rangle =0.60$, which is still consistent with our previous estimate.

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We exclude background galaxies in a 1' radius region around the cluster centroid determined by Ammons et al. (2014) when performing the weak lensing analysis, as sources there may be in either the strong lensing or highly nonlinear regime. This central region also has a higher density of cluster member galaxies, where residual contamination is more likely.

We use a weak-lensing analysis pipeline, based on a modified version of the IMCAT software package (Kaiser et al. 1995), to perform shape measurements of the background galaxies. Full details of the formalism, including the shear calibration method, are described in Umetsu et al. (2010, 2012, 2014). Its implementation has been applied extensively to cluster weak-lensing studies with Subaru/Suprime-Cam observations (e.g., Umetsu & Broadhurst 2008; Umetsu et al. 2009, 2010, 2011a, 2011b, 2012, 2014, 2015; Medezinski et al. 2010, 2011, 2013, 2016; Zitrin et al. 2011; Coe et al. 2012; Wegner et al. 2017). In this work, we follow the analysis procedures that Umetsu et al. (2014) employed for the CLASH survey (Postman et al. 2012). Briefly summarizing, the procedures include (see also Section 3 of Umetsu et al. 2016): (1) object detection using the IMCAT peak finder, hfindpeaks, (2) conservative close-pair rejection to reduce the crowding and deblending effects, and (3) shear calibration developed by Umetsu et al. (2010) to minimize the inherent noise bias.

Using simulated Subaru/Suprime-Cam images (Massey et al. 2007; Oguri et al. 2012), Umetsu et al. (2010) found that the shear signal can be recovered with $| m| \sim 0.05$ of the multiplicative calibration bias and $c\sim {10}^{-3}$ of the residual shear offset (as defined by Heymans et al. 2006; Massey et al. 2007). Accordingly, we include for each galaxy a shear calibration factor of $g\to g/0.95$ (see Equation (2)) to account for residual calibration. As discussed by Umetsu et al. (2012), m depends modestly on the width and quality of the point-spread function (PSF). This variation with the PSF properties limits the accuracy of shear calibration to $\delta m\sim 0.05$ (Umetsu et al. 2012).

We use the R_c band for the weak lensing analysis, as it has the best combination of depth and seeing conditions. We exclude frames where the seeing is $\gt 0\buildrel{\prime\prime}\over{.} 9$, and do not match the PSF across frames before coadding to the final stacked frame. The median seeing of the final data frame is 081.

The mass centroid of the cluster is allowed to vary. It is given a Gaussian prior centered on the position of the main cluster halo, as determined by Ammons et al. (2014) on the basis of the mean position of the spectroscopically confirmed cluster members. The 1σ width of the prior is 12'', which is the uncertainty on the Ammons et al. (2014) centroid position estimated from a bootstrap resampling of the cluster galaxies. The halo is assumed to be a Navarro–Frenk–White (NFW; Navarro et al. 1996) profile with log-uniform priors on its mass and concentration, which are appropriate for positive-definite quantities (e.g., Feroz et al. 2008; Sereno & Covone 2013; Umetsu et al. 2014). The halo ellipticity (defined as $\epsilon \equiv 1-b/a$, where b/a is the minor-to-major projected axis ratio) and orientation ${\theta }_{\epsilon }$ are given uniform priors. We account for the foreground halo at $z=0.2713$ by fixing its centroid to the coordinates determined from Ammons et al. (2014) and assuming that it is a spherical NFW profile. Its mass is allowed to vary and is given a log-uniform prior. The foreground halo's concentration is assumed to vary monotonically with mass, according to the mass–concentration relation of Dutton & Macciò (2014).

Using the posterior distribution of the cluster properties from our weak lensing analysis, we generate 100,000 Monte Carlo realizations of the mass distribution in the field, including the main cluster halo, the foreground halo, and the cluster and LOS galaxies. The main cluster is given a random triaxiality and 3D orientation, such that its projected ellipticity and orientation matches that of the weak lensing results for a particular model. Our analysis accounts for LOS effects using the full multi-plane lens equation (e.g., Blandford & Narayan 1986; Kovner 1987; Schneider et al. 1992; Petters et al. 2001; Collett & Auger 2014; McCully et al. 2014), as ignoring LOS structure can lead to biases in the inferred model parameters and resulting magnification maps (e.g., Bayliss et al. 2014).

The procedure for constructing the full LOS mass distribution in this manner is described in Ammons et al. (2014), with some small modifications detailed here. The virial mass of the main cluster is divided among a common NFW dark matter halo and the spectroscopically confirmed member galaxies, which are assumed to be truncated singular isothermal spheres (see Wong et al. 2011). The foreground group's mass is also apportioned between a common dark matter halo and confirmed member galaxies in a similar way. In addition to the spectroscopically observed galaxies, we add galaxies with photometric redshifts from SDSS to our mass model by selecting those galaxies brighter than $i\lt 21.1$ and within 15 of the field center. Galaxies with $| z-{z}_{{\rm{L}}}| /(1+{z}_{{\rm{L}}})\leqslant 0.1$ are treated as members of the main cluster, and their redshifts are fixed to the cluster redshift. For the remaining galaxies, we assign redshifts from a Gaussian distribution centered on their SDSS photometric redshift, with a 1σ width equal to their photo-z uncertainty. Redshifts are drawn from these distributions for each Monte Carlo realization, such that photo-z errors are accounted for in our modeling. We also include two galaxies in close proximity to the lensed arcs that are further from the cluster center and not in the spectroscopic catalog. These galaxies appear blended in the Suprime-Cam imaging data. We use GALFIT (Peng et al. 2002) to deblend them to determine their relative fluxes, then scale their magnitudes proportionally, such that the sum of their fluxes matches the i-band photometry from SDSS. The two galaxies are treated as cluster members because the color inferred from the blended photometry is consistent with that of the spectroscopically confirmed cluster galaxies.

We use the two images of the lensed background galaxy as constraints for a strong lensing analysis. The source is assumed to be at a fixed redshift of z = 5.03. The uncertainty in the photometric redshift estimate is unlikely to matter, because the lensing properties of a cluster at low and intermediate redshifts are fairly insensitive to source redshift beyond ${z}_{{\rm{S}}}\sim 4$. We use the pixsrc software (Tagore & Keeton 2014), an extension to lensmodel (Keeton 2001) that performs a pixelated reconstruction of the arcs using the full surface brightness distribution in the R_c band. Due to a lack of discernible structure within the arcs, we assume the source follows a Sérsic profile with n = 1, although our results are robust to different choices of n. The position, brightness, scale radius, ellipticity, and orientation of the source are optimized so that the lensed images, convolved with a PSF that mimics seeing conditions during observations, most closely match the data (as measured by the ${\chi }^{2}$ sum over residuals). The strong lensing analysis is depicted in Figure 5.

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Using the full pixel information differs from standard modeling methods that use only the centroid positions of images as constraints. This technique is advantageous in this case, where the lensed images are near critical curves (implying that the source is close to a caustic) and information from the extended images can be used. Figure 6 shows our best-fit model in detail. This model predicts the core of the source galaxy to lie outside the caustic, so that it is seen only in the southern arc and does not produce multiple images. A second arc is visible because part of the source overlaps the caustic; that part is stretched into two additional images that merge together to create the northern arc. In other words, most parts of the source yield just one image, but the western outskirts actually create three images, two of which meet at the critical curve to become the northern arc. A standard modeling approach that treats the arcs as two point-like images would not capture the full complexity of the lensing.

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To compute ${\chi }^{2}$ values from image residuals, we measure a noise level of ${\sigma }_{\mathrm{noise}}=9$ counts in a galaxy-subtracted image near—but not overlapping—the observed arcs. Applying this noise level to all 100,000 Monte Carlo realizations of the mass distribution yields the χ² histogram in Figure 7. The best model has ${\chi }^{2}=1502$ for ${N}_{\mathrm{pix}}=1011$ pixels. The residuals do not have obvious structure (see the top row of Figure 5), and the pixel distribution is roughly Gaussian, with a mean of −2.0 and a standard deviation of 10.8. We conjecture that the noise properties vary slightly between the region where we measure the noise and the location of the arcs, due to statistical variations and/or imperfect galaxy subtraction. We keep this in mind when interpreting χ² values, but do not attempt to rescale the noise.

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The vast majority of models cannot reproduce two arcs because the critical curves are not in the right place, as seen in the bottom row of Figure 5, as well as in Figure 7 (the large jump at χ² ≈ 1855). Only 107 of the models lie below this threshold and are able to reproduce the two arcs. We consider these models successful according to the strong lensing analysis. When deriving parameter constraints (Section 4.2), we treat all of the successful strong lensing models with equal weight. Because our assumption of a single Sérsic source with fixed index is probably too simplistic (limited by current data), and also due to questions about the noise level, we do not weight by likelihood ${ \mathcal L }\propto {e}^{-{\chi }^{2}/2}$. We emphasize that these issues do not affect whether models can produce two arcs, nor does changing the χ² threshold significantly affect the derived parameter constraints.