Precise Mass Determination of SPT-CL J2106-5844, the Most Massive Cluster at z > 1

Our WL sources are selected based on the following shape criteria. First of all, ellipticities of sources (after PSF deconvolution) should have converging values. We fulfill this condition by selecting sources whose MPFIT STATUS parameter (Markwardt 2009) is unity. Second, objects should have their semiminor axis from our elliptical Gaussian fitting (after PSF correction) greater than 0.4 pixels, which efficiently excludes point sources and small galaxies whose shapes are uncertain and subject to residual PSF systematics. Third, errors on ellipticity should be less than 0.25. The objects that have ellipticity error larger than this tend to have bright neighbors, very low surface brightness, and/or very small FWHM values. Moreover, their noise bias is large, which makes their shear calibration highly uncertain. Lastly, half-light radii (before PSF deconvolution) of the objects should be larger than those of stars (Figure 2). This requirement also prevents us from including spurious features such as cosmic rays, hot pixels, etc. In spite of the above efforts, some spurious objects (e.g., diffraction spikes around bright stars, cosmic rays, fragmented parts of foreground galaxies, clipped sources at the field boundaries, etc.) still remain. We manually identify and remove these objects by visual inspection.

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Source selection with ACS photometry. The redshifted rest-frame 4000 Å break feature (∼8500 Å at the cluster redshift) at z = 1.132 is bracketed by F606W and F814W. The early-type members occupy a narrow locus in the color–magnitude diagram shown in the top panel of Figure 3. We select objects bluer and fainter than the red sequence (F606W − F814W < 1.0 and F606W > 25) as our background sources (blue dots). This enables us to remove only red cluster galaxies and thus the resulting catalog may include blue cluster members. In the peripheral region where only F606W is available, we select sources purely based on their magnitudes 25.0 < F606W < 28.0. Therefore, we need to correct for the contamination for both red and blue members.

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In order to assess the cluster member contamination, we utilize two control fields: the HUDF (Rafelski et al. 2015) and the Great Observatories Origins Deep Survey (GOODS; Giavalisco et al. 2004). After applying the same selection criteria to the control fields, we compare the resulting magnitude distributions with those from our sources as shown in the top panel of Figure 4. If the contamination is significant in our source selection, it should appear as a significant number density excess beyond the sample variance.

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In the central region (blue box in Figure 1) where the F606W − F814W color is available, the magnitude distribution (dark gray) is consistent with those from the control fields in the F606W ≲ 26 regime. The small difference between GOODS-N and GOODS-S suggests that the sample variance might be much smaller than the Poissonian scatter, shown with error bars. For fainter sources, the densities in the control field greatly outnumber those in our cluster field because the images of the control field are deeper. These tendencies are also observed in our previous high-z cluster WL studies (e.g., Jee et al. 2011). Therefore, in the central region where both F606W and F814W are available, we decide to proceed with our WL study without any correction for blue cluster contamination.

In the peripheral region, we only apply the magnitude cut (25.0 < F606W < 28.0), and the resulting distribution is shown in light gray. Obviously, this magnitude-only selection includes red galaxies that would have been discarded if F814W had been available. The excess due to the inclusion of these red sources is clearly seen (dark gray versus light gray in the top panel of Figure 4). However, one should not incorrectly attribute the difference exclusively to the red cluster members because many non-cluster members can also have their color satisfying the condition F606W − F814W > 1.0. In Section 3.4, we find that in fact the second issue dominates the excess and the red cluster member contamination is not significant. This is not surprising because the fraction of the early-type galaxies is expected to be low in the faint magnitude regime and also decreases rapidly with clustocentric radius.

Source selection with WFC3 NIR photometry. Since all objects observed with WFC3 are located inside the central region and have color, we select the sources using both color and magnitude criteria: F105W − F140W < 0.5 and F140W > 23.5. Similarly to the ACS-based selection, these thresholds in color and magnitude are chosen to select sources bluer and fainter than the locus of the red sequence (bottom panel of Figure 3). Again, we check for the presence of a possible contamination by blue cluster members by comparing the resulting magnitude distributions with control field statistics. The bottom panel of Figure 4 indicates that the source number density is consistent with that from HUDF at F140W ≲ 26. At F140W ≳ 26, the source density becomes gradually lower because of the shallower depth.

Our high-resolution mass reconstruction enables a detailed comparison with the cluster galaxy distributions. Both luminosity and number density maps show that the galaxy distribution is not symmetric and can be described as consisting of the main clump and the western extension. Our WL mass map clearly detects these two structures. As observed in both luminosity and number density maps, the mass clump corresponding to the western galaxy concentration is also much weaker than the main mass clump. Our bootstrapping analysis shows that the significance of the western mass clump is $\sim 3.4\sigma$.

In Figure 6, the distribution of the western galaxies (both luminosity and number density) is offset from that of the mass contours by ∼30'' (distance between the two centroids). We investigate the centroid uncertainties of mass and galaxies in the western substructure by using bootstrapping analysis. The centroid is determined by 2D circular Gaussian fitting. Among the parameters that we described in Section 3.2, only centroid (x, y) is set free; we set zero for background, the highest pixel value for amplitude, and the effective smoothing scales for the semimajor and minor axes. The bottom panel of Figure 7 shows that the 1σ contours for mass and galaxy touch each other.

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The mass map further resolves two substructures within the main mass clump. Our bootstrapping analysis shows that the northwestern (southeastern) peak is detected with a significance of ∼5.7σ (∼6σ). The orientation of the vector connecting the two substructures is similar to the direction (northwest to southeast) of the elongation seen in the galaxy distributions. The distance between the two substructure centroids is ∼150 kpc (∼20''). The effective smoothing scale of our mass reconstruction is FWHM ∼ 17''. If we increase the effective smoothing scale above ∼20'', the two peaks merge into a single peak elongated in the northwest–southeast direction.

Interestingly, the centroid of the northwestern substructure lies near the luminosity peak whereas that of the southeastern substructure is closer to the peak in number density. The difference between the luminosity and number density distributions arises from the BCG being much more luminous than the other member candidates by at least ∼2 mag, although more galaxies are concentrated near the southeastern WL peak. Figure 7 shows our significance test of the mass peak centroids based on bootstrapping. An important caveat to remember in comparison with cluster galaxies is that our galaxy map is based on photometric selection targeted for the red sequence and subject to non-member contamination and omission of blue members. Currently, the number of spectroscopically confirmed members is 18 (F11).

The asymmetric X-ray surface brightness distribution of SPT2106 was considered as an indication of merger activity in F11. Comparison of the X-ray image with our mass reconstruction and galaxy distributions also supports this merger hypothesis. As seen in Figure 6(b), the Chandra image shows elongated X-ray emission whose peak location is consistent with the centroid of the northwestern mass substructure. No strong X-ray emission is found in the other substructure. In some cluster mergers with small impact parameters, a bright cool core is often found near the less massive system while there is relatively very weak X-ray emission around the more massive (thus hot) component (e.g., Jee et al. 2014; Golovich et al. 2017; van Weeren et al. 2017). Also, the elongation of the X-ray emission is consistent with the orientation of the vector connecting the two substructures seen in both mass and galaxy distributions. Thus, the absence of any significant X-ray peak near the southeastern substructure may indicate that the two substructures might have passed through each other.

We use the open-source Python package emcee (Foreman-Mackey et al. 2013) when performing our MCMC analysis. We use flat priors for both scale radius r_s and concentration c. The prior interval of c is set to 0 < c < 10. We verify that the upper limit is sufficiently high and most of our MCMC samples stay below this upper limit. The lower limit is set to zero because the concentration in the NFW halo, by definition, should have a positive value. We limit the range of the scale radius to 10'' < r_s < 110''. Together with the interval of the concentration parameter, this r_s range results in cluster masses bound within ∼10¹¹ M_☉ to ∼10¹⁷ M_☉.

Figure 8 shows the posterior distributions from our 500,000 MCMC samples in M_200c and c_200c for the four choices of the cluster center. All four results are consistent and show that SPT2106 is indeed an extremely massive system. For example, when we select our WL mass peak as the cluster center, we obtain ${M}_{200c}={10.4}_{-3.0}^{+3.3}\,\times$ ${10}^{14}\,{M}_{\odot }$. The marginalized masses are summarized in Table 1.

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Although we claim that our two-parameter result presented in Section 4.2.1 is the least biased mass, here we present our mass estimates based on the one-parameter fitting method to enable fair comparisons with previous studies. We consider the three M−c relations from Duffy et al. (2008, hereafter D08), Dutton & Macciò (2014, hereafter DM14), and Diemer & Kravtsov (2015, hereafter DK15). In Figure 9 we display the radial tangential shear profile for the four different centers. The resulting masses are summarized in Table 2 for the three M−c relations and also for the singular isothermal sphere (SIS) model. Our results show that the cluster mass is insensitive to the choice among the four centers considered in this paper whereas the profile assumption is an important systematic factor. We obtain the highest masses for the D08 relation and the lowest for the DK15 relation, although their error bars marginally overlap.

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Table 2.  Mass Estimates Based on One-parameter 2D Fitting for Various Choices of Centroids and M−c Relations

	2D SIS		2D NFW
Centroids	${\sigma }_{v}$	${M}_{200c,\mathrm{SIS}}$	${M}_{200c,{\rm{D}}08}$	${M}_{200c,\mathrm{DM}14}$	${M}_{200c,\mathrm{DK}15}$
	(${\rm{km}}\,{{\rm{s}}}^{-1}$)	(${10}^{14}\,{M}_{\odot }$)	(${10}^{14}\,{M}_{\odot }$)	(${10}^{14}\,{M}_{\odot }$)	(${10}^{14}\,{M}_{\odot }$)
WL mass	1152 ± 104	${7.7}_{-1.9}^{+2.3}$	${10.6}_{-2.6}^{+3.5}$	${8.8}_{-2.1}^{+2.8}$	${7.3}_{-1.9}^{+1.9}$
Galaxy Number	1227 ± 93	${9.3}_{-2.0}^{+2.3}$	${12.5}_{-2.7}^{+3.6}$	${10.5}_{-2.2}^{+2.9}$	${8.7}_{-1.9}^{+1.9}$
X-ray	1134 ± 105	${7.4}_{-1.9}^{+2.3}$	${10.2}_{-2.5}^{+3.4}$	${8.4}_{-2.0}^{+2.8}$	${7.0}_{-1.9}^{+1.9}$
SZ	1202 ± 95	${8.8}_{-1.9}^{+2.3}$	${11.9}_{-2.7}^{+3.5}$	${9.9}_{-2.2}^{+2.8}$	${8.2}_{-1.9}^{+1.9}$

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S18 use a moment-based approach to shape measurement while our shape measurement is based on a model-fitting method. The moment-based approach was pioneered by Kaiser et al. (1995, hereafter KSB) and has been modified by a number of authors (e.g., Hoekstra et al. 1998; Erben et al. 2001; Schrabback et al. 2007). The specific KSB branch used by S18 is described in detail by Schrabback et al. (2010), where the authors apply the method to measure cosmic shears. One important difference made in S18 from the version of Schrabback et al. (2010) is the use of the pixel-based correction for CTE degradation of Massey et al. (2014). In Schrabback et al. (2010), the correction is applied at the catalog level. The pixel-based correction for CTE degradation is similar to our method, although our WL pipeline employs the STScI pipeline correction, which implements the algorithm of Ubeda & Anderson (2012). The fidelity test of this STScI correction in the context of WL is presented in Jee et al. (2014).

Since the galaxy shape catalog of S18 is not public, it is impossible to compare shape measurements for individual galaxies. Nevertheless, it is possible to compare aggregate statistics when we closely mimic the S18 procedure for tangential shear measurement. We match their source selection, source redshift estimation ($\langle \beta \rangle =0.282$), shape measurement filter (ACS/F606W), applied cosmology (Ω_M = 0.3, Ω_Λ = 0.7, and h = 0.7), reference point (X-ray peak from Chiu et al. 2016), and M−c relation (Diemer & Kravtsov 2015) between the two analyses. We read off the S18 tangential shears from Figure G8 of their paper. Because we compare the shears only in the region where both F606W and F814W colors are available, we choose the maximum distance to be 100''. Although S18 exclude the shear at θ < 60'' in their mass estimation, we use all shear profiles at θ < 100'' to increase the statistical significance. This is justified because we are only interested in comparing the amplitude of the shears (shear calibration) between the two studies. The resulting mass difference becomes ∼10% (our mass is lower). This discrepancy is ∼16% of the statistical error.

S18 estimate the lensing efficiency $\langle \beta \rangle$ using the photometric redshift catalog from the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS, Skelton et al. 2014). They apply the same source selection criteria to the CANDELS photo-z catalog and obtain $\langle \beta \rangle =0.282$ for their source population. When we apply the S18 color and magnitude selection to the CANDELS photo-z catalog, we obtain $\langle \beta \rangle =0.260$, which would increase the cluster mass by ∼8%. We believe that the discrepancy of $\langle \beta \rangle \sim 0.02$ is due to the difference in the shape criteria. A larger difference ($\langle \beta \rangle =0.323$) is found when we replace the CANDELS catalog with the HUDF one. This higher $\langle \beta \rangle$ value would lower the cluster mass by ∼15%. The difference in $\langle \beta \rangle$ between CANDELS and HUDF may be attributed to the systematics in the photo-z catalogs or the sample variance. Note that since we use the HUDF catalog for our analysis and since our final mass (central value) is higher than the S18 one by ∼18%, the mass shift due to the increase in $\langle \beta \rangle$ is in the opposite direction.

S18 perform their 1D tangential shear fitting utilizing only the DK15 M−c relation, whereas we present results using three M−c relations and also the one without the M−c relation. As shown in Section 4.2.2 and Table 2, our mass estimates become lowest when the DK15 relation is used. The D08 and DM14 relations give mass estimates larger than the DK15 relation by ∼21% and ∼10%, respectively. We confirm the same trend also with the S18 tangential shears (read off from Figure G8 of their paper). S18 reports ${8.8}_{-4.6}^{+5.0}\,\times$ ${10}^{14}\,{M}_{\odot }$ using the DK15 relation and X-ray center. For the same choices, our mass estimate is (7.0 ± 1.9) × ${10}^{14}\,{M}_{\odot }$. Our central value is lower than the S18 value by ∼20%. Since in Sections 5.2.1 and 5.2.2 we show that the differences in shear calibration and $\langle \beta \rangle$ estimation make our mass lower than the S18 one by ∼10% and ∼15%, respectively, it is plausible that this ∼20% difference in mass might be attributed to the combined effect of the two.

As mentioned in Section 4.2.2, we regard the result obtained without any M−c relation as the main mass estimate because we believe that it is the most unbiased value. When we use the X-ray center, the central value of this result (${10.1}_{-3.0}^{+3.2}\times {10}^{14}\,{M}_{\odot }$) is ∼45% higher than the one obtained from the DK15 relation ((7.0 ± 1.9) × ${10}^{14}\,{M}_{\odot }$). Consequently, the difference in the M−c relation is the leading cause of the discrepancy between our results and those of S18.

The two WL studies use different source selection (especially color cut) criteria; S18 select bluer (F606W − F814W < 0.2) and brighter (24.0 < F606W < 26.5) sources while we include redder (F606W − F814W < 1.0) and fainter (25.0 < F606W < 28.0) objects. S18 claim that their color cut maximizes the background galaxy fraction and that the expected (non-background) contamination rate is ∼2.4%.

When the color–redshift relation is tight and the photometric scatter is negligible, we agree with S18 that the S18 color cut is optimal for maximizing the background fraction. As shown in the top panel of Figure 11, these conditions are satisfied for the HUDF galaxies. Most sources bluer than F606W − F814W = 0.2 are above the cluster redshift z = 1.132 (dark gray). However, this color–redshift relation is substantially diluted if the field becomes shallower. Even for the GOODS-S field, which is deeper than our cluster field, the scatter is severe (light gray).

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Given the large increase in scatter for the color–redshift relation, we do not think that the F606W − F814W < 0.2 cut is still optimal for the cluster field. To investigate the issue quantitatively, we define "border" objects as sources whose 1σ photometric error bars touch the color thresholds. A "border" source can be included or excluded depending on the direction of the scatter. The fraction of "border" objects increases as the image depth becomes shallower because of the increase in photometric error. The middle panel of Figure 11 shows this trend for several choices of the F606W − F814W color constraint. In the case of the bluest selection (F606W − F814W < 0.2), the border fraction is nearly ∼50% while the fraction is negligible for our case (F606W − F814W < 1). Consequently, we believe that the purity (background fraction) is nonnegligibly compromised by the increased "border" fraction when one uses a very tight color–redshift relation only observed in extremely deep fields, such as HUDF.

Apart from the decrease in the border fraction mentioned above, our selection has two additional advantages: a reduction in shot noise and improved accuracy in $\langle \beta \rangle$ estimation. Our source density is more than a factor of 9 higher than the S18 one, which results in a decrease in shot noise by more than a factor of 3. Also, the increase in source galaxy number leads to a significant reduction in the uncertainty of the $\langle \beta \rangle$ value because of the decrease in sample variance between the cluster and control fields.

Therefore, including red galaxies (F606W − F814W < 1) in our source selection enables us to increase the overall S/N in cluster mass estimation compared to the case where only blue sources are selected (F606W − F814W < 0.2). To illustrate this point, we display the S/N value of the mass estimation as a function of the color cut in the bottom panel of Figure 11. Although the purity of the background galaxies (i.e., numerator in S/N) decreases as the selection includes redder galaxies, the overall S/N value increases because of the reduction in the denominator, which comprises the shot noise and the $\langle \beta \rangle$ uncertainty.

If found to be large, the sample variance weakens the reliability of our determination of source redshift. In Section 3.4, we obtain the information on source redshift using the HUDF photo-z catalog by assuming that the color–magnitude–redshift relation is identical between the cluster field and HUDF. Because the two fields are of similar size, we are interested in quantifying how much the source redshift varies when a different sky patch is chosen. Our previous studies (e.g., Jee et al. 2014, 2017) have shown that in fact the sample variance is subdominant compared to the statistical errors. Here we repeat the same test to quantify the level of the sample variance on the SPT2106 field by comparing $\langle \beta \rangle$ measured from the HUDF, GOODS-S, and GOODS-N control fields.

We examine the sample variance on two angular scales. The first angular scale is the size of the GOODS fields; the second is the distance between GOODS-N and GOODS-S. For the sample variance estimation on the first scale, we divide each GOODS field into 9–11 subregions and measure the fluctuation of the $\langle \beta \rangle$ values. For ACS sources (F606W − F814W), the standard deviation of the distribution in $\langle \beta \rangle$ is ∼0.01 for each GOODS field. For WFC3 sources (F105W − F140W), the distribution width decreases by a factor of two (${\sigma }_{\langle \beta \rangle }\sim 0.005$). We attribute this reduction to the greater depth of the WFC3/IR data, which gives a higher total volume because of its long line-of-sight (LOS) baseline. These $\langle \beta \rangle$ fluctuations are translated to less than ∼7% in mass, which shows that the sample variance is negligible on the first angular scale (corresponding to the size of the GOODS field). For both ACS and WFC3 sources, the difference in the average $\langle \beta \rangle$ value between GOODS-N and GOODS-S is ∼0.01, similar to the level of the fluctuations observed within each GOODS field. Although future studies still need to investigate the sample variance on the above and other angular scales with more than just these two fields, we have not found any strong evidence that the sample variance is a significant source of systematics when WL sources are selected from deep imaging data (sufficient volume along the LOS direction) and their redshift estimation is based on deep control fields.

The use of M−c relations can be a source of systematic uncertainties. In Section 4.2 we measure the mass estimates of SPT2106 with and without the M−c relations. Figure 8 shows the distributions of posteriors on the M−c plane and their marginalized values. Also displayed are the mass estimates for three M−c relations. The masses obtained from the three M−c relations are consistent with the results derived from our two-parameter model. On the other hand, the two concentration values favored by the DM14 and DK15 M−c relations are outside the 1σ contours.

Although applying an M−c relation is an efficient and popular method to determine a cluster mass, we argue that the mass obtained without the assumption is least biased when the relation is highly uncertain as in the current case. Because massive high-z clusters are rare even in the current large state-of-the-art N-body simulations, statistical properties such as mass function, M−c relations, etc. are only loosely determined. For example, the M−c relations for massive, high-z clusters come from the power laws extrapolated from the results for less massive clusters; note that the shapes of the two power laws in DM14 and DK15 are different (e.g., flattening, upturn, etc.) in the high-redshift regime. In addition, the M−c relations are valid for the "mean" (or "median") properties of simulated clusters. Thus there is no guarantee that SPT2106 follows the mean M−c relation even if the extrapolated relation holds.

In Section 4.2, we assume a spherical NFW profile for measuring the mass of SPT2106. This assumption, however, is only applicable to a cluster with average properties. It is possible that SPT2106 might deviate greatly from this mean cluster profile, which if true is an additional source of uncertainties in our mass determination. According to Becker & Kravtsov (2011), this cluster model bias amounts to ∼20% for massive clusters. Some impacts of the assumed profile on our cluster mass estimation can be seen in Table 2, where we compare mass estimates using SIS and NFW profiles with various M−c relations. For example, the SIS mass estimate is ∼30% lower than our two-parameter NFW result; we already discussed the impacts of the various M−c relations in Section 5.4.2.

As an attempt to assess the amount of systematics due to the cluster model bias, we perform aperture mass densitometry (hereafter AMD), which minimizes the impact from the assumption of any particular halo model. AMD uses tangential shears of sources (Equation (11)) and computes a projected overdensity of the region inside a specific aperture radius with respect to the control annulus. For the estimation of the mean surface density in the control annulus, we inevitably have to assume a particular halo model. Nevertheless, the assumption plays a much less significant role than in the pure model-fitting method.

The projected overdensity within the r = r₁ aperture is given by

$$\begin{eqnarray}&&\begin{array}{l}{\zeta }_{c}({r}_{1},{r}_{2},{r}_{\max })=\bar{\kappa }(r\leqslant {r}_{1})-\bar{\kappa }({r}_{2}\lt r\leqslant {r}_{\max })\\ =2{\displaystyle \int }_{{r}_{1}}^{{r}_{2}}\displaystyle \frac{\left\langle {\gamma }_{T}\right\rangle }{r}{dr}+\displaystyle \frac{2}{1-{r}_{2}^{2}/{r}_{\max }^{2}}{\displaystyle \int }_{{r}_{2}}^{{r}_{\max }}\displaystyle \frac{\left\langle {\gamma }_{T}\right\rangle }{r}{dr},\end{array}\end{eqnarray} \tag{ 13 }$$

where $\left\langle {\gamma }_{T}\right\rangle$ is the azimuthal average of tangential shears, r₁ is the aperture radius, and r₂ and r_max are the inner and outer radii of the annulus, respectively. The aperture radius should be sufficiently separated from the inner radius of the annulus because the AMD uses tangential shears between the aperture radius r₁ and the inner radius r₂ for evaluation of the integral as expressed in Equation (13). The statistics ${\zeta }_{c}({r}_{1},{r}_{2},{r}_{\max })$ is then converted into the aperture mass within the radius r₁ through the equation below:

$$\begin{eqnarray}&&{M}_{\mathrm{proj}}(r\lt {r}_{1})=\pi {r}_{1}^{2}{\zeta }_{c}({r}_{1}){{\rm{\Sigma }}}_{\mathrm{crit}},\end{eqnarray} \tag{ 14 }$$

where Σ_crit is the critical surface density obtained in Equation (2).

We choose the inner and outer radii of the control annulus to be 155'' (∼1.27 Mpc) and 175'' (∼1.44 Mpc), respectively. The average surface density of the annulus is estimated to be $\bar{\kappa }=0.009$ from our two-parameter NFW fitting result (Table 1). We iteratively update the tangential shear using γ = (1 − κ)g because what we measure is not a strict shear γ but a reduced shear g = γ/(1 − κ). The process stops when the density is converged. Readers are referred to Clowe et al. (2000) and Jee et al. (2005) for details of the AMD implementation.

Figure 12 shows the resulting cumulative projected mass profile. Also displayed are the profiles derived from the best-fit NFW models for comparison. At r = 840 kpc, the AMD mass agrees with our NFW result obtained with two-parameter fitting within ∼6%, which we adopt as the systematic error due to the model bias. The shaded region represents the 1σ uncertainties computed from bootstrapping runs. The statistical error at the radius is ∼22%. We find a ∼21% difference at the same radius when we compare the AMD result with the NFW result based on the DK15 M−c relation. These comparisons demonstrate that an NFW profile assumption might not be a significant source of systematics in our determination of the SPT2106 mass. And the excellent agreement with the AMD profile shows that the intrinsic mass profile of SPT2106 might be better described with the NFW parameters derived from our two-parameter fitting, although the result from the DK15 M−c relation is still marginally consistent with the AMD profile.

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The good agreement of the AMD profile with our parametric (NFW) result also suggests that the mass profile of SPT2106 can be well approximated by a single halo. In general, treating well-separated merging clusters as a single halo is an important source of bias in estimating the total mass of the systems (e.g., Jee et al. 2014). However, since in SPT2106 the two substructures are very close (∼150 kpc), we believe that this single-halo approximation is not a significant source of systematic error.