First Results on the Cluster Galaxy Population from the Subaru Hyper Suprime-Cam Survey. III. Brightest Cluster Galaxies, Stellar Mass Distribution, and Active Galaxies

The HSC-SSP survey (Aihara et al. 2017b) is one of the Subaru Strategic Programs, which are designed to enable large-scale projects to be conducted with the facility instruments of the Subaru Telescope. The 300 night survey is carried out using the wide-field Hyper Suprime-Cam (Miyazaki et al. 2017) by a collaboration of astronomers from Japan, Taiwan, and Princeton University in the United States, and consists of a wide, a deep, and an ultradeep layer. Each layer is observed in the grizy broadband filters. There are also narrowband observations in the deep and ultradeep layers. The wide layer reaches to a depth of $r\sim 26$ mag over $1400\,$ deg². In the deep and ultradeep layers, the target depth and area are $r\sim 27$ mag, $27\,$ deg² (over four separate fields) and $r\sim 28$ mag, $3.5\,$ deg² (two fields: Subaru-XMM Deep Field and COSMOS), respectively. The primary science goals of the survey are to constrain the properties of dark matter and dark energy via the cosmic structure growth and expansion history derived from weak-lensing tomography and Type Ia supernovae, and to trace galaxy and active galactic nucleus (AGN) evolution from the local universe all the way to the epoch of reionization.

The survey data is reduced by a pipeline (Bosch et al. 2017) derived from that developed for the Large Synoptic Survey Telescope (Ivezic et al. 2008). Astrometric and photometric calibrations are carried out by comparison with data from the PanSTARRS1 survey (Chambers et al. 2016). The first public release of the HSC-SSP survey data is presented in Aihara et al. (2017a).

The camira algorithm (Oguri 2014) is run on the "S16A" internal data release of the HSC-SSP survey (Aihara et al. 2017a), covering roughly 230 deg² observed in all five filters. The resulting 1921 clusters above the richness limit $\hat{N}\gt 15$ span the redshift range of $z=0.1\mbox{--}1.1$ (Oguri et al. 2017). The richness is defined to be the number of red member galaxies with stellar mass ${M}_{\mathrm{star}}\geqslant {10}^{10.2}\,{M}_{\odot }$ lying within a physical radius of $\approx 1.4\,$ Mpc. Based on the abundance of clusters, the richness limit $\hat{N}=15$ roughly corresponds to ${M}_{200}\approx 1.3\times {10}^{14}\,{M}_{\odot }$. Comparisons with spectroscopic catalogs and existing X-ray clusters indicate that the photometric redshifts ${z}_{\mathrm{ph},c}$ of the clusters are quite accurate (with bias and scatter of −0.0013 and 0.0081, respectively), and the richness is a good proxy of cluster mass. Although camira produces the member catalogs for each of the clusters, they only contain red member galaxies. Since we would like to understand the evolution of blue populations as well, we do not make use of these member catalogs directly; rather, we adopt a statistical background subtraction approach that can be applied to both populations.

The photometric catalog we use to measure the cluster galaxy properties is the same one used in the cluster detection. The catalog is limited to cmodel magnitude ${z}_{\mathrm{cmodel}}\lt 24$. Additional magnitude limits of ${r}_{\mathrm{cmodel}}\lt 26.5$ and ${i}_{\mathrm{cmodel}}\lt 26$, as well as various flags, are applied to ensure clean photometry (please refer to Oguri et al. 2017 for more details). All magnitudes are corrected for Galactic extinction. Star–galaxy separation is performed using i-band measurements as this band is of the best imaging quality. Unless noted explicitly, cmodel grizy measurements are used throughout our analysis.

As described in Aihara et al. (2017a) and Bosch et al. (2017), the version of the HSC pipeline used to produce the S16A release occasionally generates problematic photometry in crowded fields such as cluster centers. As the pipeline has a hard time deblending galaxies in such regions, the magnitudes of the deblended objects could be estimated erroneously. As a remedy, galaxy colors are measured on the objects in the undeblended images after the point-spread function (PSF) sizes are matched for all five broad bands; the PSF sizes are degraded to $1\buildrel{\prime\prime}\over{.} 1$, and the aperture size of $1\buildrel{\prime\prime}\over{.} 1$ in diameter is used for the color measurement. We still use the cmodel magnitudes in the z-band for the flux measurement.

The stellar mass and luminosity of galaxies are estimated via the machine-learning algorithm demp (Direct Empirical Photometric method; Hsieh & Yee 2014). The training set is produced by cross-matching the HSC ultradeep layer data in the COSMOS field with the COSMOS2015 catalog (Laigle et al. 2016), which we take as the "truth" table, as the stellar mass and photometric redshift therein are derived robustly using 30 photometric bands ranging from near-UV to 24 μm, including deep data from UltraVISTA (Muzzin et al. 2013) and SPLASH (Steinhardt et al. 2014). The exquisite data quality of the HSC ultradeep layer (reaching to $5\sigma$ depth of $i\sim 27.2$ for point sources; Aihara et al. 2017a) ensures an accurate mapping between the observed grizy magnitudes and the stellar mass (or luminosity) as a function of redshift. To validate our training, we compare the demp-derived stellar mass (${M}_{\mathrm{star}}^{\mathrm{hsc}}$), based on the wide-layer-depth data in the COSMOS field,¹⁰ with that from the Laigle et al. (2016) catalog (${M}_{\mathrm{star}}^{\mathrm{cosmos}}$). The result for galaxies at $z=0.7\mbox{--}1$ is shown in the top panel of Figure 1. It is clearly shown that our stellar mass is unbiased with respect to the COSMOS masses (mean of ${\rm{\Delta }}\mathrm{log}{M}_{\mathrm{star}}\equiv \mathrm{log}{M}_{\mathrm{star}}^{\mathrm{hsc}}-{M}_{\mathrm{star}}^{\mathrm{cosmos}}$ is −0.02), with a scatter of 0.2 dex. The comparisons done for lower-redshift galaxies show similar results. We note in passing that for galaxies at $z\gt 0.8$ or so, the grizy photometry from the HSC does not sample much of the galaxy spectral energy distribution (SED) in the rest-frame optical, and therefore SED-fitting-based methods using solely the HSC photometry are prone to larger uncertainties and/or biases for high-z galaxies compared to the empirical approach adopted here.

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In the lower panel of Figure 1, we examine the completeness of our stellar mass estimates. The differential completeness is defined for galaxies within a given mass range ($\mathrm{log}M-{\rm{\Delta }}/2$ to $\mathrm{log}M+{\rm{\Delta }}/2$) to be

$$\begin{eqnarray}&&C(\tilde{M})\\ &&=\,\displaystyle \frac{N({\tilde{M}}_{\mathrm{star}}^{\mathrm{cosmos}}\in \{\tilde{M}-{\rm{\Delta }}/2,\tilde{M}+{\rm{\Delta }}/2\},{\tilde{M}}_{\mathrm{star}}^{\mathrm{hsc}}\in \{6,14\})}{N({\tilde{M}}_{\mathrm{star}}^{\mathrm{cosmos}}\in \{\tilde{M}-{\rm{\Delta }}/2,\tilde{M}+{\rm{\Delta }}/2\})},\end{eqnarray} \tag{ 1 }$$

where a tilde denotes a quantity in logarithm, the denominator is the number of all galaxies detected in the COSMOS2015 catalog with ${M}_{\mathrm{star}}^{\mathrm{cosmos}}$ in that mass range, while the numerator is the same, except we also require the demp-derived stellar mass to be in the range ${10}^{6}\mbox{--}{10}^{14}\,{M}_{\odot }$ (a condition that removes about 6% of the galaxies; restricting the demp-derived stellar mass to be within 2 dex of the COSMOS2015 value results in essentially the same completeness curves). In practice, we choose ${\rm{\Delta }}=0.25\,$ dex to generate the figure. The four curves in the panel show the completeness for galaxies in the four redshift bins used in our analysis. Generally speaking, our completeness is $\gt 95 \%$ at all redshifts for ${M}_{\mathrm{star}}^{\mathrm{cosmos}}\geqslant 3\times {10}^{9}\,{M}_{\odot }$.

Now that we have shown that demp could recover stellar masses in an unbiased and accurate fashion using the HSC wide-layer photometry, for galaxies likely associated with a cluster (see Section 2.3 below), we derive the stellar mass using demp. We note that, as Laigle et al. (2016) adopt the Chabrier (2003) initial mass function (IMF), our demp-based masses inherit the same assumption on the IMF. Finally, following essentially the same procedure, demp is trained with the COSMOS2015 catalog as well as the HSC ultradeep data to derive the (rest-frame) i-band absolute magnitude.

A part of our analysis of cluster galaxy evolution concerns the radio luminosity distribution (RLD) and the fraction of galaxies that are active in the radio wavelength, where the radio-active galaxies are simply defined to be above a certain threshold in radio luminosity. The identification of radio-active galaxies is carried out by matching our galaxy catalog with the source catalog from the FIRST (Faint Images of the Radio Sky at Twenty-Centimeters; Becker et al. 1995) survey, which has a $5\sigma$ flux limit of 1 mJy. We note that the HSC-SSP survey area is entirely within the FIRST footprint. We consider a match if an HSC galaxy has a counterpart in the FIRST catalog within $1^{\prime\prime}$. For BCGs, we have visually inspected the matching results and take proper flux measurements when the radio counterpart is an obvious multicomponent source.

To construct the composite SMD, we proceed as follows. For a given cluster, we assume all galaxies lying within a projected physical distance of ${r}_{\max }$ to be at the cluster redshift, and estimate the stellar mass using demp. The "apparent" SMD is obtained by simply counting the number of galaxies as a function of stellar mass, for all galaxies within a projected distance ${r}_{\mathrm{cl}}$. This obviously has contributions from both cluster members and foreground/background galaxies. The latter is estimated by an annulus with inner and outer radii ${r}_{\mathrm{an},\mathrm{in}}$ and ${r}_{\max }$. For all of the clusters in a given redshift bin, we sum over both the apparent stellar mass distribution and the distribution in the background annulus and then subtract the latter from the former (accounting for differences in the area). After further normalizing by the number of clusters in the redshift bin, we obtain the SMD.

In our analysis, we adopt ${r}_{\mathrm{an},\mathrm{in}}=5\,$ Mpc and ${r}_{\max }=7$ Mpc, and ${r}_{\mathrm{cl}}={r}_{200}$, our best estimate of the virial radius of the cluster (see Section 2.4). It is found that as long as ${r}_{\mathrm{an},\mathrm{in}}$ is sufficiently large, our results do not sensitively depend on the exact choice of values (e.g., Lin & Mohr 2007). Some clusters are close to the boundary of the survey and thus the annulus region is not completely covered by the survey data; others have big holes in the annulus region due to bright stars or other data reduction issues. These clusters are excluded when we construct the composite SMDs (see further discussion in Section 3.2).

To test if the above procedure can unbiasedly uncover the true SMD, we use a set of mock cluster and galaxy catalogs, which are based on the MICE mock catalog (Carretero et al. 2015). The MICE mock catalog is produced by populating halos in a light-cone simulation with galaxies using a combination of halo occupation distribution and abundance-matching techniques. The algorithm is tuned to reproduce the observed luminosity function, color–magnitude relation, and galaxy clustering properties in the local universe. The mock catalog provides apparent magnitudes in the Dark Energy Survey (DES) grizY and VISTA JHK filters for galaxies out to z = 1.4 over one octant of the sky, and is complete to r-band absolute magnitude ${M}_{r}\leqslant -18.9$.

For our purpose, we extracted catalogs of galaxies projected within ${r}_{\max }$ around the top 100 most massive halos at $z=0.9\mbox{--}1.02$ over a $200\,$ deg² area from the full MICE mock catalog, and treated the mock galaxies the same way as we do real galaxies, that is, assuming they all lie at the redshift of their respective halos, and estimating the stellar mass with the SED-fitting technique, using the software package NewHyperz¹¹ with the Bruzual & Charlot (2003, hereafter BC03) templates.¹² We also apply survey masks and artificially create boundary effects to mimic observational defects at the catalog level.

Figure 2 shows the performance of our method in recovering the SMD. While the dashed histogram shows the true SMD in the mocks (constructed from galaxies lying within the projected virial radius, and with redshifts within ${\rm{\Delta }}z=0.005$ from the halos, chosen to approximate what is attainable with spectroscopic redshifts¹³ ), the solid one is obtained using the background subtraction scheme. We see that over the mass range of interest (e.g., $\gt {10}^{9}\,{M}_{\odot }$), our method works fairly well. We repeated this exercise in another redshift range ($z=0.6\mbox{--}0.77$), also finding excellent agreement between the underlying and recovered SMDs.

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We would also like to examine the spatial distribution of stellar mass in clusters, again by stacking clusters and employing the statistical background subtraction scheme. This is tested with the MICE mocks as well. Figure 3 (top panel) shows, with the dashed curve, the true stellar mass surface density profile, and the solid curve is the derived profile, for galaxies more massive than ${10}^{10}\,{M}_{\odot }$, for mock halos in the $z=0.6\mbox{--}0.77$ range. The bottom panels shows a similar comparison, but for projected number density profiles. The number density profile can be recovered better than the stellar mass profile, which could be due to the higher degree of background contamination in the stellar mass field. To better quantify our ability to infer the true profile shape, we rescale the radial distance by the mean virial radius of the mock clusters, and find that the Navarro et al. (1997, hereafter NFW) profile can describe both the number density and stellar mass density spatial distribution well (see the dotted curves in Figure 3). Assuming that both the mean mass of the clusters and the cluster center can be accurately known, it is found that we can recover the underlying radial profiles well, in the sense that the concentration of the NFW profiles can be estimated to within 20%.

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The redshift bins used in our analysis are at $z=0.3\mbox{--}0.6$, $0.6\mbox{--}0.77$, $0.77\mbox{--}0.9$, and $0.9\mbox{--}1.02$, each occupying a comoving volume of about ${(423.6{h}^{-1}\mathrm{Mpc})}^{3}$. Extending our previous work in Inagaki et al. (2015), we employ the top N cluster selection in each of these redshift bins, and posit that these clusters could be regarded to represent a progenitor–descendant relationship. Underpinning the usefulness of the top N selection the assumptions that (1) the rate of mass assembly is similar in massive galaxy clusters and (2) the merging rates among massive clusters are negligible. As such, the top N most massive clusters at a given redshift would remain the top N most massive clusters at a later epoch, thus naturally providing a progenitor–descendant relationship. While extensive tests have been presented in Inagaki et al. (2015), here we use data from the Millennium Run simulation (Springel et al. 2005) to further test the validity and limits of this approach, concluding that even though these assumptions are not strictly true, the top N selection remains a very useful approach for studying the evolution of clusters and their associated galaxy populations.

We use the version of the simulation run with the WMAP7 cosmology for our tests, and only consider a cubical volume with $424\,{h}^{-1}\,\mathrm{Mpc}$ on a side. Four snapshots, at $z\,=0.45,0.68,0.83$, and 0.98, are considered, and we use merger trees to figure out the progenitor–descendants of the halos in these snapshots. In the perfect case where the halos are selected by their mass, we find that between z = 0.98 and 0.83, 86% of the top N = 100 halos selected at the higher redshift remains among the top 100 at the later epoch. The corresponding fractions for the $z=0.83\to 0.68$ and $z=0.68\to 0.45$ snapshots are 86% and 79%, respectively. We find that, even if we consider snapshots that are further separated in time, the remaining fraction does not degrade too much: 76% (66%) of the top 100 halos selected at z = 0.98 remains even at z = 0.68 (0.45), which is likely due to the rarity of major mergers. These results are summarized in Table 1 (second to fourth columns).

In reality, one cannot select clusters by their halo mass. An observable serving as the mass proxy, such as richness or X-ray luminosity, is often used instead. After perturbing the true mass by a log-normal random variate with ${\sigma }_{\mathrm{log}M}=0.1$ (which is equivalent to using a proxy that exhibits a ∼25% fractional scatter in mass, a reasonable assumption for our richness $\hat{N};$ Oguri 2014; Oguri et al. 2017), we find that the remaining fractions all approach 55%–70%, irrespective of the time interval between the snapshots. The results are shown in Table 1 (fifth to seventh columns).

As is shown in Inagaki et al. (2015), the remaining fraction is not a strong function of N. Given our survey volume, choosing N = 50 or 200 only changes the results as shown in Table 1 by a few percent. We adopt N = 100 in this study primarily because of the desire to focus on reasonably massive clusters (e.g., $\gt {10}^{14}\,{M}_{\odot };$ see Section 2.5) while still having a sufficient number of clusters to work with.

For the purpose of our study, it is more interesting to examine whether the top N selection can recover the evolution of the SMD, that is, whether the composite SMD built from the top N clusters at a later epoch is representative of that of the descendants of top N clusters selected at an earlier cosmic time. For this purpose, we use model galaxies generated by the semi-analytic model of Guo et al. (2013), which is tuned to reproduce the galaxy stellar mass function and clustering in the local universe. From Table 1, we see that the remaining fraction ranges from 58% to 67% by going from z = 0.98 to lower redshifts when a 0.1 dex mass scatter is introduced. In Figure 4 (lower three panels on the left), we show pairwise comparisons of the SMDs between that of the top 100 halos at z = 0.45, z = 0.68, and z = 0.83 (solid histogram) and that of the descendants of the top 100 halos at z = 0.98 observed at these lower redshifts (dashed histogram), with this level of scatter in mass–observable relations. For example, the solid histogram in the bottom panel is the SMD of the top 100 richest halos found at z = 0.45, while the dashed histogram represents the SMD of the descendants at z = 0.45 of the top 100 richest halos selected at z = 0.98. The dotted histogram, on the other hand, represents the SMD of the top 200 richest halos identified at z = 0.45. For ease of comparing different SMDs, in the lower-right panel we show the ratios of the dashed and dotted histograms to the solid histogram. The difference between the dotted histogram and the other two histograms in the bottom panels, together with the good agreement between the solid and dashed histograms in the lower three rows of the figure (the right-hand panels show the ratio of non-solid histograms to the solid histogram in the corresponding left panels), suggest that, even though the remaining fraction is not very high, the SMDs in clusters are universal enough (at least in the semi-analytic model) that the top N selection can still allow us to study the evolution of SMDs across cosmic time. We emphasize that we have effectively assumed that the SMD depends primarily on the halo mass and only weakly on the exact assembly history of the halos.

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We can also repeat the same exercise, but focusing on BCGs in these simulated clusters. Among the BCGs in the z = 0.45 descendant halos of the top 100 halos at z = 0.98, the median of the logarithm of the stellar mass is 11.64 ± 0.02. The same number for the top 100 halos selected at z = 0.45 is 11.60 ± 0.03, which can be clearly distinguished from that for the top 200 halos at the same redshift (11.54 ± 0.02). Thus, it is also reasonable to expect the top N selection to provide a useful way to disentangle the stellar mass assembly history of BCGs.

Finally, we estimate the mean masses of our cluster samples. As a rigorous determination of the richness–halo mass relation for camira via weak lensing is not yet completed (although see Murata et al. 2017), we here adopt two different approaches, one from the stacked lensing and the other from an abundance consideration.

In our lensing analysis, the shape of each galaxy is measured on the coadded i-band image using the re-Gaussianization method, which measures moments of the galaxy image taking account of the non-Gaussianity of the PSF perturbatively (Hirata & Seljak 2003). Shape measurements are carefully calibrated using image simulations. We include multiplicative and additive bias derived from the image simulations in our weak-lensing analysis (see Mandelbaum et al. 2017 for more details). Using the shear catalog as presented in Mandelbaum et al. (2017), we derive the average differential surface mass density profile following Medezinski et al. (2017) in the range of comoving radii between $0.27\,{h}^{-1}$ Mpc to 3 h⁻¹ Mpc with the bin width of ${\rm{\Delta }}(\mathrm{log}r)=0.15$. For the photometric redshift of each source galaxy, we adopt the mlz photometric redshift (see Tanaka et al. 2017). As discussed in Medezinski et al. (2017), the secure background galaxy selection is important for cluster weak-lensing analyses. We adopt the P(z) cut method in which the integrated probability density function of the photometric redshift of each galaxy is used to select galaxies behind clusters (see Medezinski et al. 2017 for details). The derived differential surface mass density profile is fitted with the NFW density profile predictions to infer the average mass (M₂₀₀) and concentration parameter (${c}_{200}$). We find that, from our highest to lowest-redshift bins, the typical masses are ${M}_{200}=2.0,1.9,3.0,$ and $4.4\times {10}^{14}\,{M}_{\odot }$. The best-fit concentration parameters are ${c}_{200}\sim 2$, which is smaller than the expected value of ${c}_{200}\sim 5$ (e.g., Zhao et al. 2009). This is most likely due to the miscentering effect, which reduces the weak-lensing signals near the center (although removing the innermost regions essentially has no effect on the resulting mass; see Section 4 for more discussion regarding miscentering). If we force the concentration parameter to ${c}_{200}=5$, the best-fit masses are decreased by ∼20%.

As the second method, we make use of halo samples from the ${(424{h}^{-1}\mathrm{Mpc})}^{3}$ subvolume of the Millennium simulation (see Section 2.4). Since the exact scatter between the camira richness and mass is still to be measured from lensing, we perturb the true mass of the halos by a set of values of ${\sigma }_{\mathrm{log}M}$ (from 0.06 to 0.16, which spans the reasonable range of possible values) and take the average of the median mass derived from the top 100 halos for each of the ${\sigma }_{\mathrm{log}M}$ value. From our highest to lowest-redshift bins, the typical masses are found to be ${M}_{200}=2.1,2.6,3.0,$ and $3.7\times {10}^{14}\,{M}_{\odot }$, in reasonable agreement with the weak-lensing estimates.

In the following analysis, we adopt the weak-lensing-based masses as the default, but note that our results do not change if we use the abundance-based estimates. In Table 2, we provide some basic information of our cluster sample, including the redshift ranges of each of the four redshift bins, the estimated masses and radii of the subsamples, and the limiting richness ${\hat{N}}_{\mathrm{lim}}$ for the top 100 selection.

Table 2.  Basic Cluster Properties

			Stacked Lensing		Abundance
Bin	Redshift Range	Mean z	M200	r200	M200	r200	${\hat{N}}_{\mathrm{lim}}$
			(${10}^{14}\,{M}_{\odot }$)	(Mpc)	(${10}^{14}\,{M}_{\odot }$)	(Mpc)
1	0.30–0.60	0.45	4.4 ± 0.2	1.33	3.7	1.27	30.0
2	0.60–0.77	0.69	3.0 ± 0.3	1.07	3.0	1.09	22.7
3	0.77–0.90	0.84	1.9 ± 0.4	0.86	2.6	0.98	21.6
4	0.90–1.02	0.96	2.0 ± 0.4	0.84	2.1	0.87	18.0

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We close by estimating the probable descendant cluster mass range at $z\approx 0$ for our cluster sample, making use of the mass growth history of the top 100 richest halos from (the subvolume of) the Millennium simulation at z = 0 (selected by assuming ${\sigma }_{\mathrm{log}M}=0.1$). In Figure 5, we show with the solid curve and shaded region the median and interquartile range of the main progenitor mass for these 100 halos. The curve is consistent with the masses of our sample, for both lensing-based (solid symbols) and abundance-based (open points) estimates. The median mass at z = 0 is about $6.5\times {10}^{14}\,{M}_{\odot }$. As our lensing-based mass mainly lies in the upper part of the shaded region, we consider an independent way of estimating the descendant mass, by utilizing the mean mass growth history following the prescription of Zhao et al. (2009). The dashed curve in the figure shows the prediction from that analytic model, which passes through the locus of the weak-lensing-based mass estimates, and reaches ${M}_{200c}\approx {10}^{15}\,{M}_{\odot }$ by z = 0. It is reasonable to assume that the typical descendant mass lies in the range bracketed by the two methods employed here.

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Strictly speaking, with our cluster sample, we can only study the evolution of red BCGs, as camira considers solely red-sequence galaxies in the cluster detection and characterization process. Once a candidate cluster is found, a BCG is chosen to maximize the likelihood that takes into account the centrality of BCGs in the spatial distribution of member galaxies, as well as the stellar mass distribution in the massive end of the stellar mass–halo mass relation. The cluster parameters, such as the center location and cluster redshift, are then refined iteratively, taking into consideration the properties of the candidate BCG. The above procedure is repeated until a converged solution is found (Oguri 2014).

Our main goal here is to infer the average degree of stellar mass growth in BCGs from $z\approx 0.96$ to $z\approx 0.45$. As BCGs are among the largest galaxies in the universe, defining and measuring their total luminosity and stellar mass contents have always been a challenging task (e.g., Gonzalez et al. 2005; Kravtsov et al. 2014). As an exploratory study seeking the first-look result on BCG mass growth, we thus perform simplistic photometric measurements that should at least allow us to capture the luminosity in the main/inner part of the galaxies (i.e., ignoring any very extended components such as cD envelope or intracluster light (ICL)). To really trace the light profile out to large scales (e.g., $\gt 100\,$ kpc), one needs to carefully mask out (or model) all detectable sources in, close to, and around BCGs, and pay special attention to the sky level and scattered light, which is left for a dedicated study in the near future. We refer to Huang et al. (2017) for a detailed analysis of relatively nearby BCGs (at $z=0.3\mbox{--}0.5$) using the HSC-SSP survey data.

For our analysis, three measurements of the BCG flux are used. The first one is the cmodel magnitudes from the HSC pipeline, which are the results of a linear combination of an exponential disk model and a de Vaucouleur profile. The cmodel flux is the flux of the best-fit model galaxy, obtained by adding all the profile-weighted flux in the object, so there is no well-defined aperture (see Bosch et al. 2017 for more details). As mentioned above, there are still problems with the cmodel magnitudes from the pipeline in crowded regions. Furthermore, the cmodel magnitudes may be systematically underestimated for relatively bright objects ($i\lesssim 20$ mag; see Aihara et al. 2017a). We therefore run SExtractor (Bertin & Arnouts 1996) on the final-stacked i-band images and obtain the "total" magnitude (mag_auto) and aperture magnitude. The aperture photometry is interpolated to derive the magnitude at a fixed physical (circular) aperture, which we choose to be 50 kpc in diameter. demp is used to derive the stellar mass of BCGs from the grizy cmodel magnitudes (${M}_{\mathrm{star},\mathrm{bcg}}^{\mathrm{cmodel}}$). Stellar masses corresponding to mag_auto and the aperture magnitude are obtained by scaling from ${M}_{\mathrm{star},\mathrm{bcg}}^{\mathrm{cmodel}}$ with the differences between ${i}_{\mathrm{mag}\_\mathrm{auto}}$ and ${i}_{\mathrm{cmodel}}$, and ${i}_{\mathrm{aper}}$ and ${i}_{\mathrm{cmodel}}$, respectively.

Our measurements are presented in Figure 6. The top, middle, and bottom panels show the results using cmodel, mag_auto, and the 50 kpc diameter aperture, respectively. In each of the panels, the blue points represent the median of the BCG stellar masses (derived with demp) of the top 100 clusters in each of the redshift bins. As an independent check, we also show as red triangles the stellar mass estimated by camira. In a nutshell, camira uses the calibrated stellar population synthesis model from BC03 with a single burst formed at z_f = 3 and the Salpeter (1955) IMF to derive the stellar mass of a particular galaxy by maximizing its likelihood of being on the red sequence at a given redshift. After adjusting for the difference between the adopted IMFs (recall that the demp-based masses inherit the choice of Laigle et al. 2016, that is, the Chabrier IMF), we see that the redshift trends indicated by the two mass estimates generally agree with each other.

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We see a gentle increase in stellar mass with both cmodel and mag_auto (although the increase is mostly at $z\geqslant 0.68$). The growth based on the median of the former from $z\approx 0.96$ to $z\approx 0.45$ is about 25%, while the latter suggests a ∼40%–45% increase. On the other hand, the lack of change in the 50 kpc aperture-based stellar mass implies the mass growth must happen primarily over larger scales, which is consistent with the findings of Huang et al. (2017). We note that the median Kron radii characterizing the mag_auto photometry are 22, 28, 29, and 36 kpc, respectively, from the highest to lowest-redshift bins. The differences in stellar mass based on mag_auto and the 50 kpc aperture can thus be understood from the relative sizes between the Kron radii and the fixed 25 kpc radius.

The degree of stellar mass growth we find here is consistent with that in Lin et al. (2013), where we use a 4.5 μm selected cluster sample over a ∼8 deg² area to infer a ∼50% increase between z = 1 and z = 0.5. It is also consistent with the amount of growth seen by Lidman et al. (2012) and Zhang et al. (2016), both of which are based on clusters detected in the X-ray. We show in Figure 7 the correlation between the BCG stellar mass (based on mag_auto) and cluster mass (using the abundance-based estimates). The color of the data points indicates the redshift of the clusters (from high to low: blue, green, orange, and red). The two dashed lines show the relative growth between $z\approx 0.96$ (cyan) and 0.45 (magenta) from the best-fit relation of Zhang et al. (2016): ${M}_{\mathrm{bcg}}\propto {M}_{200}^{0.24}{(1+z)}^{\gamma }$ with $\gamma =-0.19$. As our cluster subsamples are constructed to represent clusters along an evolutionary sequence, we can clearly see how our BCGs evolve from the lower-left portion toward the upper-right part in this parameter space. Although the redshift exponent γ in Zhang et al. (2016) is only weakly constrained (−0.19 ± 0.34), the consistency between our data and their relation argues for a non-negligible evolution in the BCG stellar mass–cluster mass correlation, and the stellar mass growth in BCGs.

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Some of the earlier studies do not find evidence of stellar mass growth in BCGs (e.g., Whiley et al. 2008; Collins et al. 2009; Stott et al. 2010). Some of these results may be due to the use of a fixed metric aperture (e.g., Whiley et al. 2008) that is reminiscent of our results using the 50 kpc aperture, while others may be attributed to the use of samples of limited sizes. With small samples of clusters, it might be difficult to control the cluster mass distributions within the sample so that progenitor–descendant relations can be facilitated. If the clusters at higher redshifts are selected such that they are of masses comparable to the lower-redshift counterparts, then, given the seemingly weak redshift evolution of the ${M}_{\mathrm{bcg}}$–M₂₀₀ relation (e.g., Zhang et al. 2016; also see Brough et al. 2008) and the large intrinsic scatter about that relation (e.g., Lin & Mohr 2004; Lidman et al. 2012), it is plausible that subtle stellar mass growths in BCGs would be missed by previous studies that only employ small samples of clusters.

For halos of masses similar to our sample (see Table 2), the semi-analytic model of Guo et al. (2013) predicts an ∼80% increase between z = 0.98 and z = 0.45, which is higher compared to the value we find. However, given the large scatter in the mass distributions of our BCGs, this discrepancy is only at the $1\sigma$ level, and is thus not significant.

We next examine the evolution of the general galaxy population in clusters, in terms of the SMD. Following the methodology presented in Section 2.3, we construct the composite SMD from galaxies within r₂₀₀ from the cluster center (using the mean r₂₀₀ for all clusters in a redshift bin), for each of the redshift bins. Seventy-six out of the 400 clusters are excluded due to large holes in the annulus or central regions, as the background estimates for these clusters would be problematic.¹⁴

The composite SMDs for all galaxies but the BCGs are shown in Figure 8. The blue, green, orange, and red histograms are the SMDs from our highest- to lowest-redshift bins. Using the completeness curves derived in Section 2.2 (see Figure 1), we corrected for incompleteness down to $3\times {10}^{9}\,{M}_{\odot }$. We see that overall, the mean number of galaxies increases with time at all mass scales, although the effect appears to be most prominent at the low-mass (${M}_{\mathrm{star}}\lt {10}^{10}\,{M}_{\odot }$) regime, which is reminiscent of the findings by Vulcani et al. (2011).

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To understand the buildup of the cluster galaxies through time, we consider the blue and red members separately. We use the locus on the color–magnitude diagrams of the red sequence as a function of redshift as provided by camira to classify galaxies into red and blue populations. More specifically, as a function of magnitude, we model the observed color distribution of the red-sequence galaxies by a Gaussian, and consider as red galaxies those that lie within $2.5\sigma$ from the mean of the Gaussian. Those that are bluer are regarded as blue galaxies. Depending on the redshift and magnitude, the typical width lies in the range $\sigma \sim 0.04\mbox{--}0.09$ mag. The resulting SMDs for the red and blue galaxies are shown in Figure 9. Each of the panels shows comparisons of SMDs in two redshift bins; from top to bottom, the bins considered are 0.96 versus 0.84, 0.84 versus 0.69, 0.69 versus 0.45, and 0.96 versus 0.45. The solid (dashed) histograms represent the SMDs in the lower (higher) redshift bin, while the red and blue histograms represent the red and blue populations. Generally speaking, for both red and blue populations, the number of galaxies increase with time at all mass scales. At the very low-mass scales (e.g., ${M}_{\mathrm{star}}\,\lt 5\times {10}^{9}\,{M}_{\odot }$), blue galaxies always dominate over red ones, whereas the opposite holds at the massive end (a few times ${10}^{10}\,{M}_{\odot }$), where the contrast becomes larger with time. The changes are most apparent when we contrast the $z\approx 0.96$ bin with the $z\approx 0.45$ bin (bottom panel).

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For these SMD measurements, we derived the completeness curves for the blue and red galaxies separately in each of the redshift bin (again using the data in the COSMOS field with the same criteria for red–blue demarcation as for the clusters), and applied completeness corrections down to ${M}_{\mathrm{star}}=3\times {10}^{9}\,{M}_{\odot }$.

The abundance of red galaxies at $z\approx 0.45$ is twice as large as the number at $z\approx 0.96$ when we integrate the SMD above ${10}^{10}\,{M}_{\odot }$. Given that the total number of galaxies N scales with cluster mass as $N\propto {M}_{200}^{\beta }$, where $\beta \approx 0.8$, and that the scaling does not vary much with redshift (e.g., Lin et al. 2004, 2006, 2012; Chiu et al. 2016b; Hennig et al. 2017), a factor of 2 growth in the number of galaxies is consistent with the expectation from the factor of 2.2 increase in cluster mass between these two epochs. This also implies that the change in the (red) galaxy population is mostly dominated by processes associated with cluster growth (e.g., accretion and merging with smaller galactic systems or from the field), rather than in situ star formation.

For low-mass red galaxies with ${M}_{\mathrm{star}}=3\times {10}^{9}\mbox{--}{10}^{10}\,{M}_{\odot }$, we find that the abundance has grown by a factor of 7.3 from $z\approx 0.96$ to $z\approx 0.45$. Comparing to the factor of 2 growth for the more massive red galaxies, we see a clear manifestation of "downsizing." As for the blue galaxies, the relative growths over the same period for massive (${M}_{\mathrm{star}}\gt {10}^{10}\,{M}_{\odot }$) and low-mass ($3\times {10}^{9}\mbox{--}{10}^{10}\,{M}_{\odot }$) ones are 1.5 and 3, respectively.

We next investigate the correlation between the total stellar mass content (${M}_{\mathrm{gal}}$) and the cluster mass. Here, ${M}_{\mathrm{gal}}$ is obtained by integrating the observed SMD down to ${10}^{10}\,{M}_{\odot }$, including the contribution from BCGs. We find that the stellar mass content in clusters is totally dominated by red galaxies. From our highest-redshift bin to the lowest, the red galaxies account for 82%, 82%, 86%, and 91% of the stellar mass in cluster galaxies (the corresponding numbers when BCGs are excluded are 78%, 78%, 82%, and 88%). In the top panel of Figure 10, the ${M}_{\mathrm{gal}}$–M₂₀₀ correlation is shown, while the stellar-to-total mass ratio (${M}_{\mathrm{gal}}/{M}_{200}$) is presented in the bottom panel. As we have two ways of estimating cluster mass, we present results from both methods. In the figure, the solid points are derived by using the weak-lensing-based mass, while the open ones are calculated using the abundance-based mass (Table 2). Both methods give consistent results. As in Figure 7, the color indicates the redshift of the clusters (from high to low: blue, green, orange, and red). In Figure 10, we can clearly see the direction that clusters move toward with time; the more massive clusters become, the smaller the stellar-to-total mass ratio gets. The solid line in the figure is the ${M}_{\mathrm{gal}}\propto {M}_{500}^{0.71\pm 0.04}$ scaling obtained by (Lin et al. 2012; note that both their stellar mass and total mass are measured within r_500c), adjusted only in normalization by the differences in the IMFs adopted. It is a bit surprising to see that the line matches very well with the locus of our data points, given the differences in the two analyses (different cluster samples, redshift ranges, cluster mass definition, and calibration). Our result confirms the finding of Lin et al. (2012) that the stellar mass–cluster mass relation shows no evidence for redshift evolution (see also Chiu et al. 2016b).

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We try to understand the lack of redshift evolution of the ${M}_{\mathrm{gal}}$–M₂₀₀ relation by considering the following simplistic model, which may shed light on the hierarchical buildup of stellar mass content in clusters. We start by constructing the complete merging history of massive halos (e.g., ${M}_{200}\geqslant {10}^{14}\,{M}_{\odot }$) selected from the Millennium simulation at z = 0, including progenitor halos down to galactic-scale halos (${M}_{200}\geqslant {10}^{11}\,{M}_{\odot }$). Whenever a halo is formed, a stellar mass is "assigned" to it, following a certain ${M}_{\mathrm{gal},\mathrm{ini}}({M}_{200})$ function that is assumed to be invariant in time. When a halo merges with a more massive halo, a fraction ${f}_{\mathrm{loss}}$ of the stellar mass is assumed to be lost to the intrahalo or interhalo space (thus becoming unaccounted for with our SMD measurements). Our goal is to see if the model can be tuned to reproduce the observed behavior of the ${M}_{\mathrm{gal}}$–M₂₀₀ relation, that is, with no noticeable evolution in both amplitude and slope.

For simplicity, we start tracking the stellar mass buildup at z = 3 and assume the ${M}_{\mathrm{gal},\mathrm{ini}}({M}_{200})$ function to be a power law (${M}_{\mathrm{gal},\mathrm{ini}}=A\,{M}_{200}^{\theta }$). With such a setting, there are two relevant parameters in our model, namely, the stellar mass loss fraction ${f}_{\mathrm{loss}}$ and the power-law index θ. Without stellar mass loss (i.e., ${f}_{\mathrm{loss}}=0$), the slope of the resulting ${M}_{\mathrm{gal}}$–M₂₀₀ relation is steeper than the observed value of $\approx 0.7$, irrespective of the value of θ. Some stellar mass loss is thus required to balance the accumulation of stellar mass in massive halos. A set of combinations of these parameters is found to reproduce the value of the slope of the ${M}_{\mathrm{gal}}-{M}_{200}$ relation and its lack of redshift evolution, although the model still results in weak evolution in the amplitude (which may be due to mass accretion below the mass limit in our merger tree treatment and/or mergers taking place in between simulation snapshots). Without an exhaustive exploration of the parameter space, we find that a model with $({f}_{\mathrm{loss}},\theta )=(0.4,0.5)$ appears to work well (see the dashed lines in Figure 10). The best model also suggests that about 10%–15% of the final stellar mass is from the initial assignment to the main progenitor, about equal amounts (∼23%) from major and minor mergers (mass ratios of ≲3 and ≲20, respectively), and the rest (∼40%) from the accretion of smaller systems; a large contribution from small galactic systems is consistent with the conclusions of Chiu et al. (2017).

Admittedly, the model is rather crude, but it does show that a quasi-steady state could be obtained (in terms of the slope of the stellar mass–cluster mass relation), and it allows us to estimate the relative contribution to the stellar mass content from progenitors of different masses. With more detailed treatments in both the stellar mass loss process and the ${M}_{\mathrm{gal},\mathrm{ini}}({M}_{200})$ function, such as the variation with time and halo mass dependence, the model might be tuned to generate the non-evolving ${M}_{\mathrm{gal}}-{M}_{200}$ relation as observed, but it is entering the regime of semi-analytic modeling and is left for a future study.

Finally, it is instructive to compare our measurements with the stellar mass function in the field, taken from the latest study using data from the COSMOS survey (Davidzon et al. 2017). As the COSMOS study employs the rest-frame UVJ color–color diagram to distinguish between red and blue galaxies, while we use the red-sequence color in optical bands, we do not expect the comparison to be exact (for example, we cannot distinguish quiescent galaxies from very dusty ones). Since the redshift bins used by Davidzon et al. (2017) are different from what we adopted ($z=0.2\mbox{--}0.5$, 0.5–0.8, $0.8\mbox{--}1.1$), we have regrouped our clusters into three redshift bins ($z=0.3\mbox{--}0.5$, $0.5\mbox{--}0.8$, $0.8\mbox{--}1.02$) and remeasured the SMDs. In Figure 11, we show the results of this comparison. The histograms show our cluster SMDs; the mean redshift is shown in each of the panels. The dashed curves are the Schechter (1976) function fits to the observed stellar mass functions from COSMOS, scaled in amplitude by a factor of $(4\pi /3){r}_{200}^{3}\,\times \,200/{{\rm{\Omega }}}_{M}(z)$ to account for differences in the mean densities of the two environments. We find a generally good agreement in the overall shape for both red and blue populations. We see that the cluster red galaxy abundance is always much higher compared to the (scaled) field value, indicating that the red cluster galaxy population is not simply produced by the agglomeration of the quenched field population; the cluster environment (or any "pre-processing" accompanying the hierarchical cluster growth) must have much enhanced the quenching processes. On the other hand, the good agreement in amplitude of the blue SMDs (except for the lowest-z bin) suggests that clusters at $z\gt 0.5$ still have a "fair" share of the blue galaxy population.

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The difference in amplitude between the cluster and scaled field SMDs of blue galaxies in the $z=0.3\mbox{--}0.5$ bin, together with the overabundance of massive, red galaxies with respect to the field (particularly at the highest-redshift bin), may be regarded as a manifestation of the environmental dependence of the downsizing phenomenon (Tanaka et al. 2005); while the most massive galaxies (${M}_{\mathrm{star}}\sim {10}^{11}\,{M}_{\odot }$) in clusters have already been quenched and become red at $z\gtrsim 0.9$, quenching has finally progressed to lower-mass cluster galaxies (a few times ${10}^{10}\,{M}_{\odot }$) by $z\sim 0.4$, thus lowering the abundance of blue cluster galaxies with respect to the expectations from the field (see also van der Burg et al. 2013; Vulcani et al. 2013).

Another fundamental statistic characterizing the galaxy population is the LD. The composite LDs are constructed following the same methodology as for the SMDs, and are shown on the left panels of Figure 12. Completeness corrections, derived in an analogous way to those used for the SMDs, have been applied. Consistent with the finding from the previous section, the red galaxies dominate over blue ones among the luminous members. For the low-luminosity galaxies, the increase of the red population is very dramatic, which is consistent with the finding of De Lucia et al. (2007, see below).

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One can see that the LD of higher-redshift clusters often shows a (slightly) higher amplitude in the bright end compared to that of the lower-redshift LDs. This is likely due to the passive evolution of stellar populations. To correct for this effect and reveal the true level of evolution, we seek simple stellar population models that best describe our data. We start by constructing composite LDs of red galaxies in the apparent magnitude space in fine redshift bins, following the methodology presented in Lin et al. (2012). The best-fit characteristic magnitude (${m}_{i}^{\star }$), derived by fitting a Schechter (1976) function to the observed LDs, as a function of redshift is shown in Figure 13. After comparing the predictions from single-burst models constructed from combinations of various stellar population synthesis models (BC03; Maraston 2005; Conroy et al. 2009) and different formation time z_f and IMFs, we find that the Conroy et al. (2009) model with z_f = 3.5 and Chabrier IMF fits our observations best. This model is shown as the curve in Figure 13.

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We then construct composite LDs in the absolute magnitude space again (for red galaxies only), but this time subtracting all the magnitudes by the evolving absolute magnitude of the single-burst model, effectively taking out passive evolution. The results are shown in the right panels of Figure 12. After the passive evolution is removed, we can see the true level of growth with time, which is similar to that found from the SMDs.

We show as black points in two of the panels on the left side of Figure 12 the LDs measured by the EDisCS survey (Rudnick et al. 2009). The points on the second panel from the top (bottom) are the LD for their clusters at z = 0.6–0.8 ($0.4\mbox{--}0.6$), which can be compared to the red solid histogram in the same panel. We see that while at $z\approx 0.45$ the two measurements agree, their $z\approx 0.7$ LD shows a much more dramatic downturn at the faint magnitudes than ours, and is actually in better agreement with our $z\approx 0.84$ LD (dotted histogram in the second panel from the top). It is unclear what the cause of the discrepancy is, but our measurement suggests that the emergence of faint red galaxies occurs over a longer period of time compared to what EDisCS has shown.

We finish by providing a look-up table that allows one to check the correspondence between the i-band absolute magnitude (M_i) and stellar mass, based on the Laigle et al. (2016) catalog (and with our definition of red and blue galaxies). In Table 3, we show, at four redshifts, the median M_i corresponding to ${M}_{\mathrm{star}}={10}^{10}\,{M}_{\odot }$ and ${10}^{11}\,{M}_{\odot }$, for blue and red galaxies separately and, in the last column, the absolute magnitude of the evolving characteristic magnitude (${M}_{i}^{\star }$) from the best-fit passive evolution model (corresponding to the curve shown in Figure 13). Down to ${M}_{\mathrm{star}}={10}^{10}\,{M}_{\odot }$, we are probing to ${M}_{i}^{\star }+1.5$ (${M}_{i}^{\star }+1.2$) for red galaxies at our highest (lowest) redshift bin.

Table 3.  i-band Absolute Magnitude vs. Stellar Mass

z	${10}^{11}\,{M}_{\odot }$		${10}^{10}\,{M}_{\odot }$		${M}_{i}^{\star }$
	red	blue	red	blue
0.45	−22.43	−22.76	−20.68	−21.07	−21.91
0.68	−22.57	−23.02	−20.75	−21.19	−22.16
0.83	−22.77	−23.04	−20.81	−21.56	−22.29
0.96	−22.91	−23.04	−20.85	−21.50	−22.39

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It is interesting to investigate how the stellar mass content is built up spatially with time. We thus study the evolution of the spatial distribution of stellar mass in clusters. The averaged profiles, produced by our statistical background subtraction method as outlined in Section 2.3, are shown in Figure 14. We include only galaxies more massive than ${10}^{10}\,{M}_{\odot }$. In the left panels, the projected distance from the cluster center is in units of Mpc, while that in the right panels is scaled by the mean r₂₀₀. In the top two panels, the blue, green, orange, and red curves are the profiles from our highest- to lowest-redshift bins. For clarity, data points are only shown for the profiles of clusters in the highest and lowest bins in the top-left panel. The contribution from the BCGs is included in the left panels, but not in the right-hand ones; the difference is only in the innermost radial bin. In the bottom two panels, only the results for the highest- and lowest-redshift bins are shown (dashed curve for the highest, solid one for the lowest), now split by color (red curves are for red galaxies, while blue curves are for the blue ones).

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It is apparent from the top-left panel that the contribution from BCGs makes the profile rather peaky, uncharacteristic of the NFW profile that is found to describe the spatial distribution of (non-BCG member) galaxies in clusters (e.g., Lin et al. 2004; Hennig et al. 2017). From the bottom-left panel, we see that blue galaxies do not play a significant role in the stellar mass density at any redshift, and thus the growth of the surface mass density is mainly driven by the red galaxy population, which is consistent with the finding of van der Burg et al. (2015). By comparing the mean profile of $z\sim 1$ clusters with that of $z\sim 0.15$ clusters, van der Burg et al. (2015) observe an "inside-out" growth of stellar mass density, in the sense that high-z clusters show an excess (deficit) of stellar mass in the inner (outer) parts compared to the low-z ones. This is not found in our comparison of $z\approx 0.96$ and $z\approx 0.45$ clusters (dashed and solid red curves in the lower-left panel), from which we see that the profile of the lower-z clusters is always above that of the higher-z one. We note that the $z\sim 1$ clusters studied by van der Burg et al. (2015) are of masses comparable to ours at $z\approx 0.96$, and that both studies employ schemes to ensure clusters along evolutionary sequences are traced. A late buildup of ICL at $z\lesssim 0.4$ may be able to reconcile the two results. Another possibility is a potentially higher fraction of our clusters with a misidentified center at higher redshifts, although this does not seem likely based on our tests using camira clusters with X-ray counterparts (see Section 4).

From the top-right panel, it can be seen that the stellar mass density increases with time at all radii, when the growth in cluster size is taken into account (i.e., when the radius is scaled by the mean r₂₀₀). This is true also when BCGs are included (not shown). Excluding the BCGs allows us to compare the stellar mass profiles with the NFW one. We find that the $z\approx 0.45$ profile can be well fit by an NFW profile with $c={2.7}_{-0.2}^{+0.3}$, while that at $z\approx 0.96$ can be described by one with $c={3.1}_{-0.2}^{+0.3}$. These profiles are shown as dotted curves in the top-right panel. Given the uncertainties in the fitting, there is no strong evidence suggesting an evolution of the profile shape between $z\approx 0.96$ and $z\approx 0.45$.

We conclude this section by commenting on the shape of the stellar mass surface density profiles for the red and blue galaxies. From the measurements shown in the lower-right panel, we find that at $z\approx 0.96$, the red and blue galaxies can be described by an NFW profile with c = 3.1 and $c\approx 1$, respectively. At $z\approx 0.45$, while the red galaxies follow an NFW profile with c = 2.7, the blue profile is extremely noisy and, if anything, would be described by a flattened/cored profile. Thus, we would conclude that the fraction of red galaxies always increases toward the cluster center (see also Goto et al. 2004; Hansen et al. 2009 and references therein), and this trend becomes stronger toward lower redshift.

Finally, we study the population of radio-active galaxies in clusters. We first construct the 1.4 GHz RLD by cross-correlating our cluster sample with the radio source catalog from the FIRST survey, then study the fraction of galaxies that are radio active by cross-matching the FIRST sources with the HSC photometric objects in and around the clusters. To enable comparison with previous studies, we measure the RLD within the estimated mean virial radius r₂₀₀.

The RLDs are constructed in a similar fashion to the SMDs. For a given cluster, we regard all radio sources with a flux greater than 1 mJy around it to be at the cluster redshift; counts within ${r}_{\mathrm{cl}}={r}_{200}$ and in a large background region (of about $1500\,$ deg²) are recorded as a function of radio luminosity ${P}_{1.4}$ (a power-law index of −0.8 is assumed in the conversion from flux to luminosity). The contributions from all of the clusters are summed and then the expected background level is subtracted. The resulting RLDs are shown in Figure 15. Because of the fixed flux limit, the RLDs at different redshifts are probed to different luminosity limits. Given the large error bars, it is hard to determine the exact shape of the RLD and its redshift evolution, although there is some hint of higher abundance of high-power sources in the higher-redshift bins (e.g., $z\gt 0.77$). The black points in the figure are from Lin & Mohr (2007), representing the RLD of $z\lt 0.2$ clusters, scaled by the typical volume of clusters used in that study. The amplitude of this local RLD is similar to, if slightly lower than, our $z\approx 0.45$ RLD, hinting at some evolution between $z\lt 0.2$ and $z\approx 0.45$.

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Although there have been several studies of the redshift evolution of RLDs (or radio luminosity functions) in clusters (Sommer et al. 2011; Ma et al. 2013; Gupta et al. 2017), we note that our measurements are the first one probing beyond z = 0.7. However, given the rarity of the radio galaxies, a much larger cluster sample is needed to tightly constrain the shape and possible evolution of the RLD.

Although it would be informative to compare with radio luminosity function in the field, we note that most of such studies are based on pencil beam surveys, and thus cannot probe the most powerful radio sources. An interesting comparison we can make is regarding the "overdensity" of radio galaxies in clusters with respect to the field. The solid curve in Figure 15 is the field radio luminosity function from Smolčić et al. (2009) at $z=0.9\mbox{--}1.3$, scaled in amplitude by $(4\pi /3){r}_{200}^{3}\,\times \,200/{{\rm{\Omega }}}_{M}(z)$ to facilitate the comparison with cluster RLDs, as has been done for a similar comparison on SMDs in Section 3.2 (we used ${r}_{200}=0.8\,$ Mpc for this rough estimate). To match the curve to the RLD of our $z=0.77\mbox{--}0.9$ or $z=0.9\mbox{--}1.02$ bin, we need to further scale it up by a factor of 10. This exercise shows that even at high z, when the whole universe is expected to be more active, the cluster environment further galvanizes radio activities. We note that for $z\lt 0.2$, Lin & Mohr (2007) estimate that the enhancement of radio activity in clusters is a factor of ∼7, thus hinting at an even stronger promotion of dense environment on radio activity at $z\gtrsim 1$.

Next, we investigate the fraction of galaxies that have ${P}_{1.4}$ above a threshold ${P}_{\mathrm{lim}}$ chosen to be high enough that the origin of the radio activity is certain to be due to the central supermassive black hole. We measure the "radio-active fraction" (RAF) again using a statistical background correction method; in both the cluster region ($r\leqslant {r}_{\mathrm{cl}}$) and the annulus region, we measure the numbers of galaxies in a given stellar mass bin and with ${P}_{1.4}\geqslant {P}_{\mathrm{lim}}$. The RAF is then the ratio of the number of radio galaxies to the number of galaxies in the stellar mass bin, both corrected for the background value estimated from the annulus region. We accumulate the numbers of both radio-active and quiet galaxies from all clusters, and subtract the total background contribution before calculating the averaged RAF. In practice, we choose $\mathrm{log}{P}_{\mathrm{lim}}=24.7$, ${r}_{\mathrm{cl}}={r}_{200}$.

The resulting RAF as a function of stellar mass is shown in Figure 16. We see that it is a strong function of stellar mass, which is consistent with the findings at low z (e.g., Best et al. 2005). There is a nontrivial dependence on redshift, however. Considering all galaxies more massive than ${10}^{10}\,{M}_{\odot }$, the RAF in our two highest-redshift bins is 1.5–2 times higher than that of the two lower-redshift bins. It is worth noting that at $z\gt 0.77$, the high RAF for galaxies more massive than ${10}^{11}\,{M}_{\odot }$ is mainly due to blue galaxies. Although such a massive blue population disappears at lower z (see Figure 9), the appearance of massive red galaxies increases the RAF in the two lower-redshift bins. We may thus have witnessed a likely change in the dominant accretion mode powering radio galaxies in clusters at $z\sim 0.8$ or so (e.g., from cold gas-powered high Eddington ratio mode to low Eddington ratio mode).

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Finally, we note that the RAF of BCGs is found to be about 7%, which is also consistent with the value found by Lin & Mohr (2007, c.f. Table 5 therein). Interestingly, it does not show much redshift dependence, which is consistent with the finding of Gralla et al. (2011).