MC²: CONSTRAINING THE DARK MATTER DISTRIBUTION OF THE VIOLENT MERGING GALAXY CLUSTER CIZA J2242.8+5301 BY PIERCING THROUGH THE MILKY WAY

Merging galaxy clusters are receiving growing attention because of their potential to provide constraints on properties of dark matter, which is gravitationally the most dominant constituent of galaxy clusters. The capability of weak lensing in constraining the mass distribution of a foreground lens based on its gravitational impact on the shapes of background galaxies provides a unique opportunity to learn from the difference in distribution between dark matter and baryonic components (galaxies and gas). If collisionless, dark matter particles will distribute nearly in the same way as galaxies do. The famous example supporting this standard paradigm is the "Bullet Cluster" 1E0657-56 (Clowe et al. 2006), which exhibits no significant difference between the locations of the cluster galaxies and mass peaks. However, there have been reports on some exceptional cases, where weak-lensing reveals mass distributions apparently at odds with the conventional paradigm of collisionless dark matter. For example, in A520 (Mahdavi et al. 2007; Jee et al. 2012, 2014a) and A2744 (Merten et al. 2011), significant (≳ 6σ in case of A520) mass peaks without any strong concentration of luminous cluster galaxies are detected. In addition, in the case of the Musket Ball cluster (Dawson et al. 2012), a marginal detection of a weak-lensing mass peak trailing galaxies is claimed (Dawson 2013). Kahlhoefer et al. (2014) have shown that such observational anomalies may be produced by self-interacting dark matter (SIDM). Thus they open up interesting research opportunities to study a collection of these rare systems and put meaningful constraints on the properties of dark matter.

We recently launched a Merging Cluster Collaboration⁸ (MC²; PI: W. Dawson) project with aims to enhance our understanding of cluster physics and constrain properties of dark matter from systematically analyzing a number of prominent merging clusters. Combining data from different instruments (e.g., X-ray, spectroscopy, weak-lensing, radio relic observation, etc.) will greatly reduce the parameter space that we need to explore in setting up initial conditions. The simulation of the MC² collaboration will include various models of dark matter (e.g., Rocha et al. 2013). Iterative comparisons between the simulations and observations will enable us to constrain properties of dark matter with unprecedented precision.

This paper is part of a upcoming series of the MC² publications, presenting our first weak-lensing analysis of CIZA J2242.8+5301 at z = 0.19. The cluster is one of the most remarkable clusters in our sample, possessing a prominent giant radio arc (van Weeren et al. 2010) stretched over ∼2 Mpc. Radio relics are discovered at the edge of merging galaxy clusters elongated perpendicular to the merger axis and are believed to trace shock fronts (Ensslin et al. 1998). Therefore, the observed morphology of the radio relics of CIZA J2242.8+5301 unambiguously indicates that we are observing a post-merger system colliding along the north–south axis. The north–south elongation of the X-ray emission from XMM-Newton and Chandra also supports this merger scenario (Ogrean et al. 2013, 2014). In Figure 1, we display the pseudo-color composite image of CIZA J2242.8+5301 with illustration of the radio relics and X-ray emission.

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These rich data sets offer opportunities for detailed simulations of the system in order to understand the exact physical mechanism leading to the observed features of CIZA J2242.8+5301. Our weak-lensing study will provide reliable mass properties of the cluster, which are among the key input parameters of the simulation, but are currently missing. Because of the apparent disruption of the system, we need to derive the mass without relying on any dynamical assumption, which is an important merit of weak lensing probes.

We structure our paper in the following way. In Section 2, we briefly review the discovery and development of our understanding of CIZA J2242.8+5301. Section 3 describes the data and reduction method. A basic description of weak lensing theory is presented in Section 4. We present our mass reconstruction in Section 5. The cluster mass estimation is described in Section 6. Our results are discussed in Section 7 before we conclude in Section 8.

We assume a flat Λ cold dark matter (ΛCDM) cosmology with H₀ = 70 km s⁻¹ Mpc⁻¹, Ω_M = 0.3, and Ω_Λ = 0.7. At the redshift of CIZA J2242.8+5301 z  ∼  0.19, the plate scale is ∼3.17 kpc arcsec⁻¹.

The galaxy cluster CIZA J2242.8+5301 at $(\alpha ,\delta)=(22^{{\rm h}}42^{{\rm m}}49\buildrel{\mathrm{s}}\over{.}1, 53{^\circ}00{^\prime}51{^{\prime \prime }})$ or (l, b) = (104191, −5111) was discovered in the Clusters In the Zone of Avoidance (CIZA) survey as one of the 57 clusters located at low Galactic latitude (Kocevski et al. 2007). The clusters in the CIZA survey are in general difficult to identify based on galaxy overdensity because of the severe extinction and stellar obscuration. Kocevski et al. (2007) report that the X-ray luminosity from the original ROSAT All Sky Survey data is L_X = 6.8 × 10⁴⁴ erg s⁻¹, which is converted to M₅₀₀ ∼ 5.5 × 10¹⁴ M_☉ according to the scaling relation of Pratt et al. (2009).

The spectacular radio relics of the cluster were discovered by van Weeren et al. (2010) from Westerbork Synthesis Radio Telescope (Katgert et al. 1973) and Giant Metrewave Radio Telescope (GMRT; Swarup et al. 1991) data with a clear detection of a spectral index gradient toward the cluster center, which led the authors to conclude that the radio signal is arising from electrons accelerated at large-scale shocks due to a head-on binary merger of roughly equal masses. They also argued that the fraction of the polarization constrains the angle between the plane of the sky and the merger axis to be less than ∼30°.

Based on the constraints imposed by the radio data, van Weeren et al. (2011) carried out hydrodynamical simulations to determine the merger geometry. They concluded that CIZA J2242.8+5301 might be undergoing a merger with a mass ratio of 2:1 and an impact parameter less than 400 kpc nearly in the plane of the sky (<10°). The estimated core passthrough time is about 1 Gyr ago. As we will discuss in Section 7, we believe that the simulation set-up of van Weeren et al. (2011) should be revised because the total mass in the two systems there is assumed to be only 5.5 × 10¹⁴ M_☉, which is based on the estimation of Kocevski et al. (2007), who converted the ROSAT X-ray luminosity to the mass with the scaling relation of Pratt et al. (2009). Our weak-lensing analysis shows that the total mass of CIZA J2242.8+5301 well exceeds ∼2 × 10¹⁵ M_☉ (Section 6).

Additional evidence supporting the north–south merger scenario is the presence of tailed radio sources. Stroe et al. (2013) found that the tailed radio sources in the CIZA J2242.8+5301 field are pointing either north or south, consistent with the direction of the merger hypothesis.

The ROSAT Position Sensitive Proportional Counter image already showed that the intracluster medium (ICM) of CIZA J2242.8+5301 is severely disturbed (van Weeren et al. 2010). A more detailed study of the ICM was presented by Ogrean et al. (2013) using XMM-Newton observations. The X-ray emission clearly shows a north–south elongation of the cluster ICM, which is consistent with the merger axis inferred from the radio relics and numerical simulations (van Weeren et al. 2010, 2011). However, no evidence for shock compression is found near the northern relic in the XMM-Newton data. A weak indication of shock is present near the southern relic, although the feature cannot be exclusively attributed to a shock. Ogrean et al. (2013) concluded that many features in their X-ray data including temperature gradient, surface brightness distribution, etc. suggest that perhaps the actual merger is much more complicated than a simple binary merger, which was the initial hypothesis of van Weeren et al. (2010, 2011).

With the Suzaku X-Ray Telescope, Akamatsu & Kawahara (2013) found significant jumps in temperature at the position of the radio relics in CIZA J2242.8+5301, which is the first confirmation that the radio relics in the cluster indeed traces the location of the shocks. The implied Mach number from the temperature drop (∼8 keV to ∼2 kev) is $\mathcal {M}_X=3.15\pm 0.52$, which is slightly lower than, but statistically consistent with the radio measurement $\mathcal {M}_{{\rm radio}}=4.6_{-0.9}^{+1.3}$ derived from the spectral index (van Weeren et al. 2010).

Higher resolution X-ray data from Chandra were analyzed by Ogrean et al. (2014). Their investigation of the surface brightness profile does not provide any evidence of a shock near the northern relic in agreement with their previous study with XMM-Newton (Ogrean et al. 2013). Instead, Ogrean et al. (2014) report that multiple density discontinuities are present in other regions both on and off the hypothesized merger axis, speculating that the X-ray shock features arise, if real, from violent relaxation of the dark matter cores of the clusters. The presence of the large temperature discontinuity of Akamatsu & Kawahara (2013) was confirmed by the reanalysis of the Suzaku data by Ogrean et al. (2014).

Stroe et al. (2014a) report that the relic features of CIZA J2242.8+5301 are detected even at high-frequency (∼16 GHz) using the Arcminute Microkelvin Imager (Zwart et al. 2008). Interestingly, the high-frequency data detects a clear trend of spectral steepening toward 16 GHz. However, this finding is somewhat inconsistent with the diffusive shock acceleration model, which is believed to be the main physical mechanism responsible for most radio relics in merging clusters. Another noteworthy feature in the high-frequency data is a flat-spectrum diffuse extension of the southern relic, which was not present in the lower frequency maps of van Weeren et al. (2010) and Stroe et al. (2013).

The Hα survey of the CIZA J2242.8+5301 field (Stroe et al. 2014b, 2014c) shows a significant excess of Hα sources with respect to the blank field. Possibly, this excess is linked to the enhanced star formation activities triggered by the passage of the shock wave.

3.1. Subaru/SuprimeCam Data Reduction

3.2. CFHT/MegaCam Data Reduction

3.3. Spectroscopic Data

3.4. Object Detection, Photometry, and Extinction Correction

3.5. Point-spread Function Modeling

A point-spread function (PSF) distorts galaxy shapes, and thus it is important to carefully model it and remove its effect when we measure weak lensing signals. Our PSF modeling is based on the principal component analysis (PCA) described in-depth in Jee et al. (2007a) and Jee & Tyson (2011). The method has been successfully applied to a wide range of data from ground (e.g., Jee et al. 2013; Dawson et al. 2012) to space (e.g., Jee et al. 2014a, 2014b). In the most recent competition called the 3rd GRavitational lEnsing Accuracy Testing (Mandelbaum et al. 2014), the method (team name: sFIT) won the data challenge. The application to ground-based data was described in detail in Jee et al. (2013). Below, we briefly summarize the key steps.

The SWARP software allows us to save the intermediate files (RESAMP) to be used for the final co-add. These files are already resampled to match the final co-add exactly (only integer pixel shifts are required). Our PSF model starts with identifying stars from these intermediate resampled images. In the CIZA J2242.8+5301 field, the typical number of stars per CCD usable for PSF modeling is ∼400 or higher. This is more than enough to obtain reliable principal components (we kept the most significant 20 principal components) and describe the coefficients along these principal components using a polynomial interpolation method. Although recent studies suggest that a more sophisticated PSF interpolation method is required for future cosmic shear surveys,¹² we find that a simple third order polynomial interpolation method provides sufficient accuracy in the current cluster weak-lensing analysis. Because we choose to measure shapes from final co-adds, we model the PSF in the co-adds by propagating the PSF models of individual CCD frames.

The fidelity of the result obtained from this procedure can be examined by comparing the PSF statistics directly measured from stars in the co-add with those inferred from our PSF model propagated from individual exposure data. In Figure 2, we present such comparisons for both Subaru and CFHT images. A few features are noteworthy here. First, the PSF pattern in the coadd is complicated and shows a large spatial variation. Second, conspicuous discontinuities in the PSF variation is present (often these features are found across the CCD boundaries). Third, the PSF pattern predicted by our PCA-based model closely matches the observed PSF pattern. This agreement verifies that the PSF model obtained after stacking PSFs from individual exposures is indeed valid in representing the PSF in the co-add. In addition, it also warrants that there is no major systematic error in our image registration. For example, if there is any frame misaligned by as few as ∼0.5 pixels in the x-axis, the (observed) stars in that region would possess large ellipticities along the x-axis in the coadd image, which in turn would cause large discrepancies between observation and model.

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3.6. Shape Measurement

We use forward-modeling to determine galaxy ellipticity. The details are described in our past publications (e.g., Jee et al. 2013). In brief, we first convolve elliptical Gaussian with a PSF profile and then fit the result to galaxy images. We define the ellipticity with

$\begin{equation} e=\frac{a-b}{a+b}, \end{equation} \tag{ 1 }$

where a and b are the semi-major and -minor, axes, respectively, that are determined from the fit. Because real galaxy profiles are not Gaussian, the result is biased. Also, photon noise causes non-negligible bias in the shape parameters, in particular, for faint objects. In Jee et al. (2013), we determined the correction factor by carrying out image simulations with real galaxy images and found that our ellipticity should by multiplied by ∼1.1 to recover input shears. This procedure calibrates out the isotropic part of the shear bias. Typically, shear measurement is also subject to anisotropic bias often correlated with PSF elongation. We corrected for this additive bias in our cosmic shear study presented in Jee et al. (2013). However, this additive bias is several orders of magnitude smaller than the statistical error and thus can be safely ignored in the current cluster lensing study.

As the ellipticity has both direction and magnitude, it is convenient to represent it using the following two components:

$\begin{equation} e_+= e \cos (2 \theta) \end{equation} \tag{ 2 }$

$\begin{equation} e_\times = e \sin (2 \theta), \end{equation} \tag{ 3 }$

where θ is the angle between the x-axis and the major axis of the ellipse.

3.7. Multiplicative Bias by Stellar Contamination

3.8. Source Selection

We base our source selection mainly on the color–magnitude relation with some additional steps to minimize the contamination from stars misidentified as galaxies. Figure 3 shows the relation between g − i and i. The color–magnitude relation of the spectroscopically confirmed red members is tight. The objects in the yellow polygon were explicitly rejected because a significant fraction of the objects there is either stars or cluster members. We choose the "red" edge to be 0.5 mag redder than the mean red-sequence color. We let the "blue" edge have a larger margin(2 mag) to minimize stellar contamination. This color cut is applied to sources with i ∼ 22 because beyond this limit the photometric scatter will make the color–magnitude relation-based source selection much less reliable. The blue dots represent the sources that we select for weak-lensing analysis. We do not use the r-band photometry in this selection because photometric redshift estimation with the three g, r, and i bands does not provide significant advantage in isolating the cluster members. In addition, it is difficult to obtain reliable r-band photometry for many sources because of the charge bleeding.

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In addition to the above color–magnitude relation, we require the object half light radius (in the i band) to be greater than 044, which is the upper limit of the stellar locus in the current data. Also, the pre-seeing semi-minor axis (determined after PSF-convolved elliptical Gaussian fitting) should be significantly greater than zero. This last measure is needed to remove remaining stars and avoid the aliasing effect pertinent to small objects. We also applied S/N cuts by ensuring that the ellipticity error per component is less than 0.3 and the detection significance is above 5σ. From the Subaru i-band image, we obtain ∼16 galaxies per sq. arcmin, which is roughly a factor of two lower than the typical number that we achieve with Subaru in other fields (e.g., Dawson et al. 2012). We repeat that this low source density is due to both high stellar density and high extinction. Because of both relatively shallow depth (the 5σ limiting mag is r_limit ∼ 24.1) and obscuration by bleeding trails, the source number density is a factor of two lower (∼8 galaxies per sq. arcmin) in the CFHT r-band data. When objects appear both in Subaru and CFHT, we combine their shapes by weight-averaging their ellipticities. Weights are computed using ellipticity errors, which are in turn derived from a Hessian matrix. Although we merge the two shape catalogs from Subaru and CFHT to perform our final weak-lensing analysis, the result is consistent with the case when only Subaru shapes are used. This is not surprising because most of the objects detected in the CFHT image are also visible in the Subaru image. However, it is important to note that the results based on the CFHT shapes are also statistically consistent with the results obtained with the Subaru shapes. In particular, we emphasize that the two versions of mass maps from both Subaru and CFHT are in excellent agreement, which will be demonstrated in Section 5.

3.9. Cluster Member Selection

Interested readers are referred to Bartelmann & Schneider (2001) and Hoekstra (2013) for more details on weak lensing and its application to galaxy clusters. Here we only provide a brief review.

In a weak-lensing regime, we can assume that averaging the ellipticity components defined in Equations (2) and (3) become reduced shears g₁ and g₂, respectively. That is,

$\begin{equation} \left< e_+ \right> \simeq g_1 \end{equation} \tag{ 4 }$

and

$\begin{equation} \left< e_{\times } \right> \simeq g_2. \end{equation} \tag{ 5 }$

The reduced shears comprise the matrix elements of the following weak-lensing transformation:

$\begin{equation} {\bf A}=(1-\kappa) \left(\begin{array}{@{}c c@{}}1 - g _1 & -g _2 \\ -g_2 & 1+g _1 \end{array} \right), \end{equation} \tag{ 6 }$

where κ is the surface mass density in units of the critical density:

$\begin{equation} \Sigma _c = \frac{c^2}{4 \pi G D_l \beta }. \end{equation} \tag{ 7 }$

In Equation (7), c is the speed of light, G is the gravitational constant, and D_l is the angular diameter distance to the lens. β is defined below.

The lensing signal strength depends on the distance ratios between lens, source, and the observer. In order to interpret and quantify the observed signal properly, it is necessary to estimate the source redshift. Often, a lensing efficiency is expressed in terms of β defined as

$\begin{equation} \beta ={\rm max }\left[ D_{\rm ls}/D_s,0 \right], \end{equation} \tag{ 8 }$

where D_ls and D_s are the angular diameter distances between the lens and the source, and between the observer and the source, respectively. Because sources in front of the lens do not contribute to the lensing signal, we assign zero to β in those cases (D_ls/D_s becomes negative). Therefore, we need to obtain <β > to characterize the effective lensing efficiency of the source population. We refer to the corresponding redshift as effective redshift z_eff. However, because the lensing efficiency is nonlinear, using <β > alone to represent a broad distribution biases the result. A first order correction is given when we update the reduced shear g with the following equation (Seitz & Schneider 1997):

$\begin{equation} g^{\prime } = [1 + (\langle\beta ^2 \rangle/ \langle \beta \rangle^2-1) \kappa ]g. \end{equation} \tag{ 9 }$

Since we base our selection on the two-filter broadband photometry as described in Section 3.8, we estimate the source redshift indirectly utilizing a publicly available reference catalog. We choose the Cosmic Evolution Survey (COSMOS) photometric redshift catalog¹⁴ (Ilbert et al. 2009) as our reference because (1) the magnitude-limit I < 25 is well-suited to the current analysis, (2) the result is obtained based on 30 filters, and (3) the survey area is reasonably large (2 sq. degs) so that the result does not greatly suffer from the cosmic variance. In addition, the Ilbert et al. (2009) photometry is also derived from the Subaru Telescope, which obviates the need for any transformation between the current CIZA J2242.8+5301 and the COSMOS photometric systems.

We apply the source selection criteria described in Section 3.8 to the COSMOS galaxy catalog. We obtain 〈β〉 = 0.656 and the corresponding effective redshift of this subset is z_eff = 0.626. Because the depth of our CIZA J2242.8+5301 image is shallower than the COSMOS images, it is necessary to correct for the difference. We compute this correction factor by constructing magnitude histograms for the two catalogs and weight galaxies according to the ratio of our source number density per magnitude bin to the COSMOS one. The stellar contamination discussed in Section 3.7 is also taken into account in this step. Both <β > and z_eff decrease to 0.616 and 0.549, respectively. We obtain <β² > = 0.438, which is needed to account for the width of the redshift distribution (Equation (9)).

The reduced shear defined in Equations (4) and (5) is directly observable (up to shear calibration) by averaging object ellipticities. However, in order to obtain the surface mass density κ, we need to know shears γ, which are related to reduced shears via

$\begin{equation} g=\frac{\gamma }{1-\kappa }. \end{equation} \tag{ 10 }$

Often, g = γ is assumed in the very weak-lensing regime where κ ≪ 1. Under this assumption, we can obtain the two-dimensional convergence field by performing the following integral:

$\begin{equation} \kappa ({\bf x}) = \frac{1}{\pi } \int D^*({\bf x}-{\bf x}^\prime) \gamma ({\bf x}^\prime) d^2 {\bf x}, \end{equation} \tag{ 11 }$

where D* is the convolution kernel defined as D(x) = −1/(x₁ − ix₂)² and x is the coordinate. This direct inversion first used in Kaiser & Squires (1993) is still a popular method to reconstruct a two-dimensional mass distribution.

However, near cluster centers, κ is non-negligible, and thus we need to include the nonlinear relation between g and γ. In the current paper, we used the maximum-entropy inversion code of Jee et al. (2007b), which utilizes the entropy of the mass pixels as prior to regularize the result.

We show our mass reconstruction results in Figure 4 from both Subaru and CFHT. Clearly, the results demonstrate that the mass distribution is elongated along the merger axis inferred from the radio relics. Note that the substructures seen in both instruments are consistent. We regard this consistency as verification that the observed substructures are not due to any residual systematics in weak-lensing measurements. As shown in Figure 2, the PSF ellipticity patterns of Subaru and CFHT are different. Thus, if our PSF correction error were significant, we would not observe this level of consistency between the two instruments. Also, remember that the source number density in the CFHT weak-lensing catalog is a factor of two lower. This implies that the CFHT result can be regarded as one of the bootstrapping realizations of the Subaru result (in the absence of systematic errors). Most of the sources used for the CFHT weak-lensing analysis are present in the Subaru source catalog. Therefore, little difference from the Subaru result is observed when we combine the two source catalogs. Nevertheless, we note that our weak-lensing analysis hereafter is based on the union of both Subaru and CFHT shape catalogs.

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Figure 5 compares the mass reconstruction from the combined catalog (CFHT+Subaru) with the cluster galaxies. The cluster member selection is described in Section 3.9. We smooth the galaxy distribution using a Gaussian kernel with a FWHM ≃ 34. Both number and luminosity maps indicate that the cluster galaxies in CIZA J2242.8+5301 have a bimodal distribution, and our mass reconstruction reveals two dominant mass clumps that can be associated with the two peaks in both luminosity and number density maps. It is apparent that the mass centroids do not perfectly align with the cluster galaxies. The mean offset between the mass and luminosity/number density peaks is ∼1'. We present our analysis on the statistical significance of the mass–galaxy offset in Section 7.3.

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Together with the radio relics, galaxy distribution, and X-ray morphology, our weak-lensing mass reconstruction shows that CIZA J2242.8+5301 is comprised of at least two massive halos separated by ∼1 Mpc. This is analogous to the situation where we need to perform accurate photometry for two blended objects that are of comparable luminosity. Ignoring the presence of the companion will lead to significant overestimation of each object's luminosity. Of course, this companion bias is avoided when we fit the two objects simultaneously and estimate the luminosity based on the resulting best-fit parameters. Similarly, if we assume a single halo and attempt to estimate the mass using the azimuthally averaged shear profile, the resulting mass will become non-negligibly biased. Hence, for the mass estimation of CIZA J2242.8+5301 , we fit the two halos simultaneously. One caveat in this approach is that we still have to assume that both halos can be represented by a Navarro–Frenk–White (NFW; Navarro et al. 1997) profile. In general, the difference between our model and actual cluster profiles is an important topic for cluster cosmology, where one desires to know the mass with bias smaller than 1% accuracy. In this study, we do not suggest a solution to this cluster model bias. Instead, we perform a separate mass estimation using aperture mass densitometry (Fahlman et al. 1994; Clowe et al. 2000), which provide non-parametric masses within an aperture, and compare these projected masses with the results derived from our parameterization. We demonstrate that the discrepancy is negligible within the current statistical error.

Although we emphasize above that the traditional method of fitting an analytic profile to one-dimensional (1D) tangential shear profile causes bias, we still present the results from this approach in Figure 6 because the results give rough estimates and provide opportunity to examine a possible presence of systematics (so-called B-mode test). The B-mode signal is consistent with zero, which shows that no significant systematic errors are present in our analysis. The singular isothermal sphere (SIS) model is used to predict velocity dispersions whereas the NFW model is assumed for mass estimation. The vertical dotted line in Figure 6 is to denote the minimal radius inside which we exclude the data for fitting. This need for excluding the tangential shear at small radii is shared by the weak-lensing community because of the following reasons. First, crowding makes reliable shape measurements difficult. Second, the cluster galaxy contamination rate into the source catalog is higher. Third, mis-centering distorts the shear profile. Fourth, weak-lensing assumption becomes invalid when the cluster possesses strong-lensing features. We summarize the fitting results in Table 1, which are to be compared to the results from our simultaneous fitting explained below.

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Table 1. Mass Estimates of CIZA J2242.8+5301 Based on 1D Fit

Subclusters	σv	c200c	R200c	M200c
	(km s−1)		($h_{70}^{-1}$ Mpc)	($h_{70}^{-1}10^{14}M_{\odot }$)
North	1046 ± 65	3.19 ± 0.06	2.0 ± 0.1	11.1 ± 2.2
South	1182 ± 58	3.10 ± 0.04	2.3 ± 0.1	15.7 ± 2.5

Note. The σ_v value is derived from the SIS fit whereas the rest are from the NFW fit.

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The complicated shape of the observed tangential shear profiles indicate that CIZA J2242.8+5301 cannot be described by a single halo without significant bias in cluster mass estimation. At relatively small radii (r ≲ 10'), it seems that the substructures and halo mis-centering suppress the signal amplitude whereas at r ≳ 10' the tangential shear profile rises because of the contribution from both halos. In Figure 6 the two-component models (blue) described below possess these features and better-fit the observed reduced shear profile.

Because we desire to determine parameter uncertainties and examine their correlations robustly, we perform Markov Chain Monte Carlo (MCMC) analysis. In fact, this approach was also used in the case of ACT-CL J0102−4915 (Jee et al. 2014b), which closely resembles CIZA J2242.8+5301 in terms of their structure and mass. ACT-CL J0102−4915, nicknamed "El Gordo" (Menanteau et al. 2012) meaning "the Fat One" in Spanish, is a merging system with two comparably massive (∼10¹⁵M_☉ each) clumps. Similarly to the case of ACT-CL J0102−4915, we place two NFW halos over the northern and southern sub-clusters. Throughout the paper, the Duffy et al. (2008) mass–concentration relation is assumed. We choose the galaxy luminosity center as an initial centroid and allow it to vary as we run our MCMC. We use a flat prior with a bounding box 200'' × 200'' centered on the luminosity peak. For the concentration parameter, we use a flat prior ranging from c = 2 to 4, which corresponds to the range M₂₀₀ = 7 × 10¹³–3 × 10¹⁷ M_☉ according to the Duffy et al. (2008) relation. The 1D fitting results above are used as initial values. We exclude source galaxies when they are located within r_cut = 200'' circle. The mass varies ∼11% when r_cut varies between 150'' < r_cut < 250''.

The resulting parameter constraints are displayed in Figure 7. The most probable masses are M_200c ∼ 10¹⁵M_☉ for both clusters. We summarize the fitting results in Table 2. The mass of the southern halo based on the 1D analysis is higher by ∼60%, although the results are in good agreement for the northern clump. The large difference in the southern subcluster indicates that cluster mass estimation with weak-lensing requires care when non-negligible substructures are present.

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Table 2. Mass Estimates of CIZA J2242.8+5301 Based on Simultaneous 2D Fit

Subclusters	σv	c200c	R200c	M200c
	(km s−1)		($h_{70}^{-1}$ Mpc)	($h_{70}^{-1}10^{14}M_{\odot }$)
North	$967_{-128}^{+113}$	$3.20_{-0.08}^{+0.09}$	$2.0_{-0.2}^{+0.2}$	$11.0_{-3.2}^{+3.7}$
South	$1137_{-101}^{+93}$	$3.23_{-0.08}^{+0.08}$	$1.9_{-0.2}^{+0.1}$	$9.8_{-2.5}^{+3.8}$

Note. The σ_v value is derived from the SIS fit whereas the rest are from the NFW fit.

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The aperture mass densitometry can be performed by evaluating the following integral.

$\begin{eqnarray} &&\zeta _c (r_1, r_2,r_{\rm max}) = \bar{\kappa }(r \le r_1) - \bar{\kappa }(r_2 < r \le r_{\rm max}) \nonumber \\ &&= 2 \int _{r_1} ^{r_2} \frac{ \left< \gamma _T \right> }{r}d r + \frac{2}{1-r_2^2/r_{\rm max}^2} \int _{r_2}^{r_{\rm max}} \frac{ \left<\gamma _T \right>}{r} d r, \end{eqnarray} \tag{ 12 }$

where 〈γ_T〉 is the azimuthally averaged tangential shear, r₁ is the aperture radius, and r₂ and r_max are the inner- and the outer-radii of the annulus. ζ_c(r₁, r₂, r_max) provides a density contrast of the region inside r < r₁ with respect to the control annulus (r₂, r_max). We choose r₂ = 600''(∼ 1.9 Mpc) and r_max = 800''(∼ 2.5 Mpc) for the control annulus and estimate the density within this region by projecting our NFW fitting results (Table 2). The input to the equation of the densitometry is a shear, not a reduced shear. Therefore, we determine the aperture mass iteratively by updating κ. The result is displayed in Figure 8. Also, displayed in Figure 8 is the aperture mass estimated with the NFW fitting results above. In order to obtain this estimation, we first projected each NFW profile along the line-of-sight direction and sum the two cluster results. The projected masses from aperture mass densitometry and NFW fitting are in good agreement at large radii, which indicates that despite the apparent violent merger, parameterizing the halos with NFW profiles still provides a reasonable global mass in CIZA J2242.8+5301. At smaller radii (r ≲ 1 Mpc), we observe some disagreements, which may arise not only from a genuine difference at the core, but also from centroid mis-alignments and/or data binning artifacts.

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Given that CIZA J2242.8+5301 is comprised of two halos of approximately equal mass, we can estimate the total mass of the system in the following way. We adopt the mean position of the two halos as the cluster center. Namely, the coordinate of this location is $(\alpha ,\delta) \simeq (22^{{\rm h}}42^{{\rm m}}47\buildrel{\mathrm{s}}\over{.}3, 53{^\circ}01{^\prime}56{^{\prime \prime }})$. We set up a three-dimensional box (3D) with a side of 6 Mpc containing the two halos of CIZA J2242.8+5301. A density is assigned to each volume element based on the parameters in Table 2. Then, it is straightforward to compute the total spherical mass as a function of radius. We estimate that M_200c = (2.51 ± 0.53) × 10¹⁵ M_☉ is reached at r_200c = 2.63 Mpc. This mass is comparable to the value M_200c = (2.76  ±  0.51) × 10¹⁵ M_☉ of "El Gordo" similarly comprised of two ∼10¹⁵M_☉ halos. We remind readers that the total mass becomes greater than a mere sum of the two halos' masses because r_200c should increase. Although this total mass estimation assumes that the two subclusters are at the same distance from us, the result does not vary significantly with viewing angles (see Jee et al. 2014b for the case of El Gordo) as long as the angle between the merger axis and the plane of the sky is less than ∼70°. The polarization of the radio relic indicates that the angle is less than ∼30° (van Weeren et al. 2010), which is consistent with the result from our detailed dynamical analysis (W. Dawson et al., in preparation).

7.1. Comparison with Dynamical and X-Ray Studies

7.2. Implication for Merging Scenario

The giant radio relic of CIZA J2242.8+5301 provides direct evidence for the on-going merger, which is also supported by the gas and galaxy distribution. The numerical simulation of van Weeren et al. (2011) successfully reproduces the observed features of the radio relics in CIZA J2242.8+5301. Here we discuss the implication of the current weak-lensing results on the initial condition by noting the following two apparent differences between our findings and the assumptions made for the simulation of van Weeren et al. (2011).

First, the most striking difference is the total mass of the system. Our weak-lensing analysis finds that the total mass of CIZA J2242.8+5301 should exceed M_200c  ∼  2 × 10¹⁵ M_☉ (Section 6). On the other hand, in van Weeren et al. (2011) the total mass was assumed to be 5.5 × 10¹⁴ M_☉, which is a factor of four lower.

Second, the mass ratio is different. van Weeren et al. (2011) found that the mass ratio of the northern cluster to the southern cluster should be close to 2:1 in order to reproduce the observed asymmetry of the two relics. However, this 2:1 mass ratio is not favored by our weak-lensing analysis, which shows that the best-fit masses of the two sub-clusters are in fact of nearly equal mass.

The first difference (i.e., total mass) comes from the fact that van Weeren et al. (2011) assume that the total mass of CIZA J2242.8+5301 is 5.5 × 10¹⁴ M_☉, which is based on the luminosity-mass (L_X − M) relation of Pratt et al. (2009). One of the most important merger parameters that will be affected by updating the mass would be the impact velocity between the two halos. Using the simplified assumption made in Sarazin (2002), we can estimate the initial separation d₀ using Kepler's 3rd law

$\begin{equation} d_0 \sim 4.5 \left(\frac{M_1 + M_2}{10^{15}\, M_{\odot } } \right) ^{1/3} \left(\frac{t_{{\rm impact}}}{10^{10}\, {\rm yr} } \right) ^{2/3}\, {\rm Mpc}, \end{equation} \tag{ 13 }$

where t_impact is the time elapsed at the impact since the unknown epoch t₀ when the two halos detached from the Hubble flow and started to accelerate toward each other. It is important to remember that the impact velocity that we discuss below is not sensitive to the exact value of d₀ (t₀).

It is impractical to set up a numerical simulation to begin at t = t₀ and run it until t = t_impact. Therefore, most N-body simulations choose a reasonable initial distance d as a starting point. The relative velocity between the two halos when they are separated by d is given by (Sarazin 2002)

$\begin{eqnarray} v&\sim& 2930 \left(\frac{M_1 + M_2}{ 10^{15}\, M_{\odot } } \right)^{1/2} \left(\frac{1-d/d_0}{1-{ (b/d_0)^2} } \right)^{1/2}\nonumber \\ &&\times \left(\frac{d}{1\, {\rm Mpc} } \right)^{-1/2}\, {\rm km}\,{\rm s}^{-1}, \end{eqnarray} \tag{ 14 }$

where b is the impact parameter. When we assume t_impact = t_age (∼11 Gyr), b = 0, and M₁ + M₂ = 5.5 × 10¹⁴, the relative velocity of the two halos is ∼1000 km s⁻¹ at d = 2 Mpc. In fact, this relative velocity is used as an initial condition in the simulation of van Weeren et al. (2011). Instead, if we substitute our weak-lensing masses into M₁ + M₂, the relative velocity increases to ∼2500 km s⁻¹ at the same separation d = 2 Mpc. This difference is non-negligible and will greatly amplify at the core pass-through. The estimation of the exact relative speed at the core impact requires N-body simulations, and we will defer the estimation to our future studies.

The second difference may appear puzzling because the simulation clearly shows that a mass ratio of 2:1 (the northern clump should be more massive than the southern clump by a factor of two) is required to reproduce the observed asymmetry in the two radio relics. Nevertheless, given the size of the mass uncertainties in the current study, our result does not rule out the case of the 2:1 mass ratio. As shown in Figure 7, the tension is only slightly greater than 1σ. In addition, even in the case that the global mass ratio is 1:1, the asymmetry in the relics of CIZA J2242.8+5301 can be accommodated in the scenario, wherein the two subcluster's concentration values are different. Since a halo with a higher concentration packs more mass within the same volume (at a small radius), it may be possible to cause the observed disparity in radio relics if the northern cluster has a denser core.

7.3. Mass–Galaxy Offset

The comparison of the mass and galaxy centroids in CIZA J2242.8+5301 shows that on average they are offset by ∼1'. The offset may be regarded as interesting in the context of a SIDM theory with a non-negligible collisional cross-section, which predicts that in general mass and galaxy distributions will differ after collision in merging clusters. In CIZA J2242.8+5301 the northern galaxy centroid appears to lead the corresponding mass centroid whereas the mass–galaxy offset is rather lateral for the southern clump. Also, the axis defined by the line connecting the two mass centroids is rotated by ∼10° with respect to the merger axis defined by the cluster mass, X-ray gas, and radio relics.

In order to investigate the significance of the mass centroid, we re-sample source galaxies while allowing duplication. Similarly, we bootstrap the cluster galaxies as well to estimate the uncertainty of their centroid in both luminosity and number density. The resulting centroid distributions are displayed in Figure 9. We note that the luminosity centroid uncertainty shown here should be considered an upper limit because we assign equal probabilities of elimination/selection to all galaxies regardless of their luminosity; a tighter centroid distribution would be obtained if brighter galaxies were made harder to drop in the re-sampled catalog.

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Before we discuss the mass–galaxy offset, it is instructive to examine individual contours in detail. First, we note that the mass centroid contours closely resemble the shapes of the mass peaks (Figure 5). Second, the northern mass peak, which is more peaked, has a tighter centroid distribution. Third, the galaxy centroid distributions also resemble the corresponding smoothed galaxy (both luminosity and number density) distribution. Fourth, the size of the galaxy centroid uncertainties are similar between the northern and southern clumps.

The significance of the mass–galaxy offset is not easily deduced from Figure 9. One question is whether number density or luminosity is a better tracer of the galaxy population. Luminosity is a better tracer of the mass in stars in principle, but its location can be more highly disturbed by interloping foreground galaxies. We therefore tabulate both types of offset in both north and south subclusters, yielding four potential offsets. In Figure 10, we plot the spatial distribution of these four types of offsets extracted from the bootstrap resampling. The blue contours show the 68% and 95% confidence limits. The origin in each plot corresponds to the null hypothesis of no offset, and the red contour shows the confidence limit that just touches this point in parameter space. The corresponding p-values are listed in Table 3. The lowest p-value is seen in the mass–luminosity offset in the north, meaning that the probability of detecting the observed offset by a random chance is ∼8%.

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Table 3. Significance of the Mass–Galaxy Offsets in CIZA J2242.8+5301

Subclusters	Mass–Luminosity	Mass–Number
North	0.082	0.310
South	0.288	0.431

Note. We quote p-values for the null hypothesis that the mass and galaxy centroids coincide.

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Can we use the current mass–galaxy (luminosity) offset for the argument of SIDM? We believe that it is still premature to conclude that our observation supports SIDM because of the following caveats. First, we do not know the ICM mass exactly. As seen in Figure 1, the X-ray emission is more centrally concentrated than the cluster galaxies, which may imply that a significant fraction of ICM may be present between the two halos of CIZA J2242.8+5301. Hence, one can suspect that our mass centroids are somewhat pulled inward (toward the cluster center) if the fraction of the mass locked up in the ICM is significant. Dawson (2013) showed that for the Musket Ball Cluster (Dawson et al. 2012), this systematic offset can be of the same order of magnitude as the weak-lensing and galaxy centroid uncertainties. Second, we cannot yet exclude the possibility that any line-of-sight interlopers are present. This issue may be resolved in the future when we reach more completeness in our spectroscopic survey of the field. Currently, our existing spectroscopic catalog does not hint at the possibility. Third, our weak-lensing source catalog may be "asymmetrically" contaminated by the cluster galaxies. With our dependence on the color–magnitude relation in source selection, we expect that our source catalog contains some blue cluster galaxies inevitably. Typically, we assume that the contamination is azimuthally symmetric in isolated clusters. However, in clusters with elongated morphology like CIZA J2242.8+5301, the assumption may turn out to be incorrect, especially, when the merger triggers star-formation activity preferentially along the merger axis in the outskirts (Stroe et al. 2014c). Fourth, interpretation of galaxy ellipticity is ambiguous in the strong lensing regime. That is, we do not know whether a galaxy ellipticity e becomes g or 1/g* a priori. The two cases can be reliably distinguished only when strong lensing data with known redshifts are used. Because CIZA J2242.8+5301 is a merging cluster, a probable shape of the critical curve is a long ellipse elongated in the north–south direction similar to the one modeled by Zitrin et al. (2013) for "El Gordo." Given the weak-lensing mass of CIZA J2242.8+5301, the Einstein radius of each halo is likely to be r = 05 ∼ 1', which is comparable to the size of the observed mass–galaxy offset. Therefore, it is worth investigating in the future how weak-lensing mass centroids are affected in a similar mass configuration.

We have presented a detailed weak gravitational lensing analysis of CIZA J2242.8+5301, which is a rare target boasting a textbook example of a giant radio relic. Our mass reconstruction shows that the cluster consists of two halos of nearly equal mass, consistent with the cluster galaxy distribution. In order to quantify the mass while minimizing the contamination bias, we perform MCMC by simultaneously fitting the two halos with NFW profiles. Although we adopt the luminosity centers as our initial values, the mass centroids are not fixed, but allowed to move during our MCMC run. We obtain $M_{200c}=11.0_{-3.2}^{+3.7}\times 10^{14}\, M_{\odot }$ and $9.8_{-2.5}^{+3.8}\times 10^{14}\, M_{\odot }$ for the northern and southern halos, respectively. The total mass of the system is M_200c = (2.51 ± 0.53) × 10¹⁵ M_☉ at r_200c = 2.63 Mpc, which makes CIZA J2242.8+5301 among the most massive clusters detected so far. Because of the low redshift of the cluster, this mass alone does not cause any tension with the ΛCDM paradigm. This large cluster mass suggests that the previous N-body simulation assuming the total mass ∼5.5 × 10¹⁴M_☉ should be revised.

We detect somewhat large offsets (∼1') between galaxy and mass centroids. To quantify the significance, we run bootstrapping for both mass and galaxy distributions and find that the offsets are significant at least at the 2σ level for the northern clump. Although the direction of the offset is in favor of SIDM at face value, we identify some caveats that may lead to the current observation. They are (1) the gas mass that can pull the total mass (galaxy+gas+dark matter) centroids toward the cluster center with respect to the galaxies, (2) a possible line-of-sight structure that is uncorrelated with the cluster, but may bias our mass centroids, (3) the cluster galaxy contamination in our source catalog, whose contamination rate may differ azimuthally and thus shift the two weak-lensing mass centroids, and (4) the ambiguity of shear interpretation between strong and weak lensing regimes. We believe that all these four issues can be addressed in the future as more concerted efforts are made to enhance our understanding of the system.

We thank the Merging Cluster Collaboration for useful discussions and comments. M.J.J., D.W., and W.D. acknowledge support from HST-GO-13343.01-A. Part of this work was performed under the auspices of the U.S. DOE by LLNL under Contract DE-AC52-07NA27344. A.S. acknowledges financial support from NWO. M.B. acknowledges funding from the Deutsche Forschungsgemeinschaft under SFB 676. D.S. acknowledges financial support from the Netherlands Organisation for Scientific research (NWO) through a Veni fellowship, from FCT through a FCT Investigator Starting Grant and Start-up Grant (IF/01154/2012/CP0189/CT0010) and from FCT grant PEst-OE/FIS/UI2751/2014. R.W. is supported by NASA through the Einstein Postdoctoral grant number PF2-130104 awarded by the Chandra X-Ray Center, which is operated by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-03060.