# First results on the cluster galaxy population from the Subaru Hyper Suprime-Cam survey. II. Faint end color–magnitude diagrams and radial profiles of red and blue galaxies at 0.1 < z < 1.1

First results on the cluster galaxy population from the Subaru Hyper Suprime-Cam survey. II.... Abstract We present a statistical study of the redshift evolution of the cluster galaxy population over a wide redshift range from 0.1 to 1.1, using ∼1900 optically-selected CAMIRA clusters from ∼232 deg2 of the Hyper Suprime-Cam (HSC) Wide S16A data. Our stacking technique with a statistical background subtraction reveals color–magnitude diagrams of red-sequence and blue cluster galaxies down to faint magnitudes of mz ∼ 24. We find that the linear relation of red-sequence galaxies in the color–magnitude diagram extends down to the faintest magnitudes we explore with a small intrinsic scatter σint(g − r) < 0.1. The scatter does not evolve significantly with redshift. The stacked color–magnitude diagrams are used to define red and blue galaxies in clusters in order to study their radial number density profiles without resorting to photometric redshifts of individual galaxies. We find that red galaxies are significantly more concentrated toward cluster centers and blue galaxies dominate the outskirts of clusters. We explore the fraction of red galaxies in clusters as a function of redshift, and find that the red fraction decreases with increasing distances from cluster centers. The red fraction exhibits a moderate decrease with increasing redshift. The radial number density profiles of cluster member galaxies are also used to infer the location of the steepest slope in the three-dimensional galaxy density profiles. For a fixed threshold in richness, we find little redshift evolution in this location. 1 Introduction Clusters of galaxies are the largest gravitationally bound objects in the Universe. Because their dynamics are determined mostly by gravity and the spatial distribution of clusters also follows the large-scale structure, clusters of galaxies are considered a good probe of cosmological structure. Therefore, understanding the formation history of clusters is key for both studying the structure formation and for cosmological applications of clusters (e.g., Rosati et al. 2002; Planelles et al. 2015; Castorina et al. 2014). One of the most prominent features of clusters of galaxies is the presence of a large number of red galaxies, which exhibit a tight relation in the color–magnitude diagram (CMD) (e.g., Stanford et al. 1998). This tight relation in the CMD, often referred to as the red-sequence, has been observed out to relatively high redshift, z ≃ 2 (Tanaka et al. 2010; Andreon et al. 2014; Cerulo et al. 2016; Romeo et al. 2016). The tilt of the color in the red-sequence can be explained mainly by the decrease of metallicity for low-mass galaxies (Faber 1973; Kodama & Arimoto 1997; De Lucia et al. 2007), and the age of the galaxies (Ferreras et al. 1999; Gallazziet al. 2006). Another notable property of clusters is that they also contain blue galaxies. Thus the color distribution of cluster member galaxies has a bimodal distribution (Gilbank et al.; Loh et al. 2008; Li et al. 2012). Historically, it is well known that there are more blue galaxies observed at high redshifts than at low redshifts compared to the red galaxies (Butcher & Oemler 1978). Also, it has been found that the abundance of morphologically different types of galaxies in clusters correlates with local density (Dressler 1980; Whitmore & Gilmore 1991) and distance from the cluster center (Whitmore et al. 1993; George et al. 2013). These relations seen in the local universe can also be seen in z ∼ 1 clusters (Postman et al. 2005). De Propris et al. (2004) showed that the fraction of blue galaxies in a blank field is a strong function of the local density, with a sharp decline near the density of the cluster environment toward ∼10%. This also suggests that member galaxies of clusters are dominated by the red population, although there are still a non-negligible amount of blue galaxies in clusters. In Hennig et al. (2017), Sunyaev–Zel’dovich (SZ) detected clusters are stacked to derive the number density profile for red and blue galaxies to find a clear difference in the concentration between the two populations. Because clusters of galaxies are rare objects, it has been difficult to construct a large sample of clusters at high redshifts, which requires both wide area and sufficient survey depth. Therefore previous galaxy population studies of massive high-redshift (z ∼ 1) clusters have been made using only a handful of clusters, either from deep optical/infrared (IR) imaging surveys over small areas or from follow-up imaging observations of X-ray and SZ-selected high-redshift clusters. For instance, the IRAC Shallow Cluster Survey (ISCS) includes 13 clusters at 1.01 < z < 1.49 (Snyder et al. 2012), the Gemini Cluster Astrophysics Spectroscopic Survey (GCLASS) has 10 spectroscopically confirmed rich clusters at 0.85 < z < 1.34 (Muzzin et al. 2012), and the HAWK-I Cluster Survey (HCS) contains only nine clusters at 0.84 < z < 1.46 (Lidman et al. 2013). Conversely, there are large cluster samples at lower redshifts for different data and different cluster-finding methods. Gladders and Yee (2000, 2005) used two optical and near-IR filters to define the red sequence on Red-Sequence Cluster Survey (RCS), Koester et al. (2007) applied a red-sequence based finding method called maxBCG to Sloan Digital Sky Survey (SDSS) galaxies, and Rykoff et al. (2014) developed a red-sequence based method with a more sophisticated algorithm and applied it to SDSS galaxies. In this paper, we explore the average picture of the cluster galaxy population using a large, homogeneous sample of clusters with a uniform selection over the wide redshift range of 0.1 < z < 1.1, which is selected from the Subaru Hyper Suprime-Cam (HSC) survey. The sample contains more than 1900 clusters optically identified by the CAMIRA (Cluster finding Algorithm based on Multi-band Identification of Red-sequence gAlaxies: Oguri et al. 2018) algorithm applied to the S16A internal data release of the HSC survey. The unique combination of the area and depth of the HSC survey is crucial for accurate galaxy population studies for clusters at z ∼ 1, as well as galaxy population studies down to very faint magnitudes for low-redshift clusters. An advantage of using a large homogeneous sample of clusters is that it enables a statistical subtraction of projected foreground and background galaxies, which is important for mitigating projection effects in cluster galaxy studies. With this unprecedented data set, we first examine the width of the red-sequence of galaxies down to mz = 24 mag. Then we trace the redshift evolution of the radial profiles of red and blue galaxies in clusters. This enables us to understand the formation history and dynamical evolution of the clusters from z = 1.1 to the present. We refer the interested reader to a series of complementary papers describing different aspects of the properties of these cluster galaxies (Jian et al. 2018; Lin et al. 2017). This paper is organized as follows. In section 2, we describe the HSC photometric data and briefly overview the CAMIRA cluster catalog. Our selection criteria for the photometric sample is also presented. In section 3, we describe our method for identifying cluster member galaxies without using photometric redshifts. Section 4 presents the intrinsic scatter of red galaxies down to the magnitude limit. The radial profile and fraction of red member galaxies in clusters and their evolution over redshifts are discussed in section 5. The comparison with semi-analytical model results is also presented there. We summarize our results in section 6. Unless otherwise stated, we assume cosmological parameters as $$\Omega _m=0.31, \Omega _\Lambda =0.69$$, and h = 0.7 that are consistent with Planck Collaboration (2014) results throughout the paper. 2 HSC data 2.1 HSC photometric sample The HSC survey started observing in 2014 March and is continuously collecting photometric data over a wide area under good photometric conditions (Aihara et al. 2018b). The HSC survey consists of three different layers: Wide, Deep and UltraDeep. In this paper, we use the photometric data from the Wide layers of the internal S16A data release, which covers more than 200 deg2 of the sky with five broadband filters (grizy). In the S16A data, the Wide layer reaches the limiting magnitude of 26.4 in the i band. During the survey, i- and r-band filters were replaced by newer ones which have better uniformity over the entire field of view (Miyazaki et al. 2018; Kawanomoto et al. 2017) They are denoted as the i2 and r2 filters, respectively. In this paper we do not discriminate between the old and new filters and simply denote them as i and r. For each pointing, we divide the exposure into 4 for the g and r bands, and 6 for the i, z, and y bands, with a large dithering step of ∼0.6 deg, to recover images at gaps between CCDs and to obtain a uniform depth over the entire field. The total exposure times for each pointing are 150 × 4 = 600 s for g and r bands, and 200 × 6 = 1200 s for i, z, and y bands. The expected 5σ limiting magnitudes for the 2″ diameter aperture are 26.5, 26.1, 25.9, 25.1, and 24.4 for g, r, i, z, and y bands, respectively. In point spread function (PSF) magnitudes, which is an aperture magnitude convolved with the measured and modeled PSF function, we reach down to i = 26.4 with a 5σ level. The median seeing of i-band images is 0$${^{\prime\prime}_{.}}$$61 (Aihara et al. 2018a). The images are reduced by the processing pipeline hscpipe (Bosch et al. 2018), which was developed as a part of the LSST (Large Synoptic Survey Telescope) pipeline (Ivezic et al. 2008; Axelrod et al. 2010; Jurić et al. 2015). The photometry and astrometry are calibrated in comparison with the Pan-STARRS1 3π catalog (Schlafly et al. 2012; Tonry et al. 2012; Magnier et al. 2013), which fully covers the HSC survey footprint and has a set of similar filter response functions. The photometry of the current version of hscpipe in crowded regions such as cluster centers is not accurate due to the complexity of object separation on the image, (deblending: Bosch et al. 2018). For this reason, we combine two different methods of measuring the photometry as described in detail in subsection 2.3. 2.2 CAMIRA clusters In this section, we briefly overview the CAMIRA cluster-finding algorithm and properties of the catalog (Oguri et al. 2018). CAMIRA is a cluster-finding algorithm based on the red-sequence galaxies (Oguri 2014). By applying the algorithm to the HSC S16A data, Oguri et al. (2018) constructed a catalog of ∼1900 clusters from ∼230 deg2 of the sky over the wide redshift range 0.1 < z < 1.1 with almost uniform completeness and purity. Oguri et al. (2018) first applied specific color cuts to a spectroscopic redshift-matched catalog to remove the obvious blue galaxies. Those galaxies are used only for calibrating the color of the red-sequence and this color cut is not applied in cluster-finding itself. CAMIRA computes the likelihood of a galaxy being on the red-sequence as a function of redshift, using a stellar population synthesis (SPS) model of Bruzual and Charlot (2003) with accurate calibration of colors using the spectroscopic galaxies mentioned above. In contrast with the complex color degeneracy among different parameters of SPS, CAMIRA simplifies the parametrization to quantify the red-sequence galaxies. CAMIRA applies a single instantaneous burst at the formation redshift zf = 3 and assumes no dust attenuation, since these prescription is good enough to represent the red-sequence color. The metallicity of galaxies is modeled as a logarithmic linear function of total mass at z = zf; i.e., log ZSPS = log Z11 + aZlog [M*(zf)/1011 M⊙], with log Z11 = −2, aZ = 0.15. CAMIRA additionally introduces the scatter of metallicity to model the intrinsic scatter of the red-sequence with σlog Z = 0.14 (Oguri 2014). For individual galaxies, the likelihood is calculated by finding the best-fitting parameters, but color calibration is simultaneously applied by using the spectroscopic sample, which recovers the imperfect prediction of the SPS model [see equation (2) of Oguri (2014). These likelihoods are used to compute the richness parameter with which cluster candidates are identified by searching for peaks of the richness. For each cluster candidate, the brightest cluster galaxies (BCG) is identified by searching a bright galaxy near the richness peak. The photometric redshift and richness are iteratively updated until the result converges. As a result, the accuracy of the photometric redshift reaches 1% out to z ∼ 1, which is better than other cluster-finding algorithms for different data sets (e.g., Rykoff et al. 2016; Mints et al. 2017). In this paper, we use 1902 CAMIRA clusters selected from ∼230 deg2 HSC Wide fields with richness $$\hat{N}_{\rm mem}>15$$. We note that the number of CAMIRA clusters is slightly smaller than in Oguri et al. (2018), because we further impose the condition that all galaxies have well-measured y-band photometry. The difference is less than 1% and this does not change our results. Although there is a large scatter in the relation between richness and mass, Oguri et al. (2018) argued that the richness limit of $$\hat{N}_{\rm mem}=15$$ roughly corresponds to a constant mass limit of M200m = 1014 h−1 M⊙. Due to the large scatter of the mass–richness relation and lack of the mass calibration from weak lensing, in this paper we do not divide the sample into richness bins. In contrast, in order to study the redshift evolution of the cluster galaxy property, we divide these clusters into different redshift bins with a bin size of Δz = 0.05 at z < 0.4 and Δz = 0.1 at z ≥ 0.4. Table 1 summarizes the mean redshift of clusters and the number of clusters with richness $$\hat{N}_{\rm mem}>15$$ in each redshift bin. The effect of cluster evolution on sample selection will be addressed in a future paper once the halo mass estimates from weak lensing is available. The different sample selection can be found in our complementary paper (Lin et al. 2017). Table 1. Definition of redshift bins used in this paper.* Redshift bin  Mean redshift  Color  Number of clusters  0.10–0.15  0.13  g − r  43  0.15–0.20  0.18  g − r  91  0.20–0.25  0.22  g − r  74  0.25–0.30  0.28  g − r  145  0.30–0.35  0.32  g − r  130  0.35–0.40  0.38  g − r  70  0.40–0.50  0.45  r − i  192  0.50–0.60  0.55  r − i  220  0.60–0.70  0.65  r − i  179  0.70–0.80  0.75  i − y  231  0.80–0.90  0.85  i − y  217  0.90–1.00  0.95  i − y  137  1.00–1.10  1.05  i − y  173  Redshift bin  Mean redshift  Color  Number of clusters  0.10–0.15  0.13  g − r  43  0.15–0.20  0.18  g − r  91  0.20–0.25  0.22  g − r  74  0.25–0.30  0.28  g − r  145  0.30–0.35  0.32  g − r  130  0.35–0.40  0.38  g − r  70  0.40–0.50  0.45  r − i  192  0.50–0.60  0.55  r − i  220  0.60–0.70  0.65  r − i  179  0.70–0.80  0.75  i − y  231  0.80–0.90  0.85  i − y  217  0.90–1.00  0.95  i − y  137  1.00–1.10  1.05  i − y  173  *For each redshift bin, the mean redshift of clusters, the color combination used to define red and blue galaxies, and number of clusters with $$\hat{N}_{\rm mem}>15$$ are shown. View Large 2.3 HSC sample selection In this section, we describe our selection of the HSC photometric galaxy sample, which is used for our analysis of the cluster galaxy population. In this paper, we use magnitudes for each object that is derived by combining CModel magnitudes (Abazajian et al. 2004; Bosch et al. 2018) with PSF-matched aperture magnitudes, the so-called afterburner photometry (Bosch et al. 2018). The CModel magnitude is obtained by fitting the object’s light profile with the sum of a de Vaucouleurs bulge and an exponential disk convolved with the PSF. PSFs are measured at the positions of stars and then modeled to interpolate over the entire field of view (Bosch et al. 2018). The PSF-matched aperture magnitude in the afterburner photometry is obtained by stacking the image after blurring each exposure toward the target PSF size and measuring the photometry at a given aperture size. All the magnitudes are corrected for the Galactic extinction (Schlegel et al. 1998). First we define the total magnitude of each galaxy with the z-band CModel magnitude.1 Then we derive the magnitudes in other bands as   $$m_{x} = m_{z}^{\rm CM} + \left( {m}_{x}^{\rm ab} - {m}_{z}^{\rm ab} \right),$$ (1)where $${m}_{z}^{\rm CM}$$ is the CModel magnitude in the z-band measured with forced photometry on the PSF-unmatched coadd image (cmodel_mag), and $${m}_{x}^{\rm ab}$$ ($${m}_{z}^{\rm ab}$$) is the PSF-matched aperture magnitude in the x band (z band) measured in the 0$${^{\prime\prime}_{.}}$$55 aperture in radius on the stacked image, where the PSF is convolved to homogenize the target PSF size of 1$${^{\prime\prime}_{.}}$$1 (parent_mag_convolved_2_0). As the error of the afterburner photometry is significantly underestimated because the neighboring pixels are highly correlated due to the blurring, we use the photometric error associated with the PSF-unmatched aperture photometry with a corresponding aperture instead of the afterburner photometry error. In the following we describe flags applied to select galaxies with high-quality photometry (Aihara et al. 2018b). Flags in forced photometry table [grizy]flags_pixel_edge is not True [grizy]flags_pixel_interpolated_center is not True [grizy]flags_pixel_cr_center is not True We apply the above three constraints to avoid objects geometrically overlapping with the masked region [EDGE or NO DATA], or object centers close to interpolated pixels or suspected cosmic rays. [grizy]cmodel_flux_flags is not True These flags ensure that the CModel flux is successfully measured. [grizy]centroid_sdss_flags is not True This excludes objects for which measurements of centroids failed, using the same method as in the Sloan Digital Sky Survey (Bosch et al. 2018). [gr]countinputs >1 [izy]countinputs >3 In the HSC-Wide, we divide every pointing into 4 for the g and r bands, and 6 for the i, z, and y bands. Here we use objects taken twice or more for g and r, and four times or more for i, z and y bands. detect_is_primary is True We also remove blended objects to avoid ambiguous photometric measurements. zcmodel_mag - a_z <24.0 We limit our sample in these magnitude ranges so that all the objects have a high signal-to-noise ratio (S/N). rcmodel_mag - a_r <28.0 icmodel_mag - a_i <28.0 These two flags are not for the primary object selection but for removing objects that are too faint. zcmodel_mag_err <0.1 We also directly impose the S/N cut corresponding to S/N ≳ 10. iclassification_extendedness =1 Stars are excluded. Flags in afterburner table [grizy]parent_flux_convolved_2_0_flags is not True In addition to the above selection, we use only galaxies brighter than the limits shown in figure 1 depending on the redshift of clusters. Fig. 1. View largeDownload slide Redshift and magnitude distribution of all CAMIRA member galaxies. The dashed line represents the observer-frame z-band magnitude of an SPS with constant absolute magnitude Mz = −18.5 as seen at different redshifts. The thick solid line is the same constant absolute magnitude but after applying K-correction and taking into account passive evolution. Shaded rectangles are regions where the galaxy sample is complete and suitable for exploring the redshift evolution of clusters in various aspects over different redshifts. The horizontal dotted line represents the mz = 24.0 mag cut that we apply to remove the galaxies affected by large photometric errors especially in the highest redshift bins of the cluster sample (see subsection 2.3 for details). (Color online) Fig. 1. View largeDownload slide Redshift and magnitude distribution of all CAMIRA member galaxies. The dashed line represents the observer-frame z-band magnitude of an SPS with constant absolute magnitude Mz = −18.5 as seen at different redshifts. The thick solid line is the same constant absolute magnitude but after applying K-correction and taking into account passive evolution. Shaded rectangles are regions where the galaxy sample is complete and suitable for exploring the redshift evolution of clusters in various aspects over different redshifts. The horizontal dotted line represents the mz = 24.0 mag cut that we apply to remove the galaxies affected by large photometric errors especially in the highest redshift bins of the cluster sample (see subsection 2.3 for details). (Color online) As a sense check, we compare colors of CAMIRA cluster member galaxies with model colors from the SPS model used in CAMIRA cluster finding (Oguri et al. 2018), which is based on the SPS model of Bruzual and Charlot (2003) with the calibration of colors from spectroscopic redshifts as described above. Figure 2 shows the redshift evolution of colors of CAMIRA cluster member galaxies, where we derive colors of cluster member galaxies by matching the photometric galaxy sample constructed above with a catalog of CAMIRA member galaxies from Oguri et al. (2018). We find that model colors and median colors of the member galaxies agree well, as expected. As shown in figure 2, in the redshift ranges of 0.1 < z < 0.4, 0.4 < z < 0.7, and 0.7 < z < 1.1, g − r, r − i, and i − y respectively show rapid color changes, because these colors cover the 4000 Å break at these redshifts. Moreover, we find that the above combination of filters shows the tightest scatter around the theoretical prediction in each redshift range. Therefore, we use these colors to construct the CMD to see the red-sequence galaxies at fainter magnitude with a stacking analysis (see table 1). Fig. 2. View largeDownload slide Redshift–color relation of red-sequence galaxies. Red points are cluster member galaxies identified by CAMIRA, and their median and 1σ region are denoted by filled circles and thin solid lines. The thick solid lines show the model colors from the SPS model (Bruzual & Charlot 2003) calibrated with spectroscopic redshifts. (Color online) Fig. 2. View largeDownload slide Redshift–color relation of red-sequence galaxies. Red points are cluster member galaxies identified by CAMIRA, and their median and 1σ region are denoted by filled circles and thin solid lines. The thick solid lines show the model colors from the SPS model (Bruzual & Charlot 2003) calibrated with spectroscopic redshifts. (Color online) 3 Statistical identification of cluster member galaxies 3.1 Stacking analysis In this paper, we study distributions of cluster member galaxies statistically, without resorting to spectroscopic or photometric redshifts. Since we do not use both photometric redshift and spectroscopic redshift for each galaxy, physical quantities of individual galaxies such as stellar masses or star formation rates are not available; however, an advantage of our approach is that we can exclude any uncertainties associated with photometric redshift measurements, which may also introduce uncertainties in the identification of cluster member galaxies. This statistical method has been used previously in the literature (e.g., Lin et al. 2004; Hansen et al. 2005; Loh & Strauss 2006). Here we describe our specific procedure. First we divide the CAMIRA cluster catalog into subsamples at different redshift bins, as shown in table 1. In each redshift bin, we have roughly ∼50–200 clusters. The galaxy distribution associated with the ith cluster and jth annulus can be written as   \begin{eqnarray} N_{ij}^{\rm in}(z_{\rm cl}) =\sum _k \Theta \left(|\Delta {\boldsymbol{\theta }}_{ik}|\chi _{{\rm cl},i}-r_{P,j}\right) \Theta \left(r_{P,j+1}-|\Delta {\boldsymbol{\theta }}_{ik}|\chi _{{\rm cl},i}\right),\nonumber\\ \end{eqnarray} (2)where the summation k runs over galaxies, χcl is the comoving distance to the cluster redshift zcl, $$\Delta {\boldsymbol{\theta }}_{ik}$$ is the sky separation of the kth galaxy with respect to the ith cluster center, and Θ is a Heaviside step function. We consider the comoving radial distance from the cluster center rP in the range from 0.1 to 5 h−1 Mpc (comoving), which is divided into 15 logarithmically uniform bins. After stacking over all clusters within each redshift bin, we have   $$N_j^{\rm in}(r_P) = \sum _{i} N_{ij}^{\rm in}.$$ (3)This number $$N_j^{\rm in}$$ includes not only cluster member galaxies but also foreground and background galaxies along the line of sight. In order to remove these foreground and background galaxies, we assume that the galaxy number distribution outside the cluster region defined by $$r=|\Delta {\boldsymbol{\theta }}_{ik}|\chi _{{\rm cl},i}>5\, h^{-1}\:{\rm Mpc}$$ represents the distribution of the foreground and background galaxy population well. Although the distribution may differ field-by-field due to both inhomogeneous observing conditions and large-scale structure of the Universe, such local variation of the galaxy number distribution is expected to be averaged out after stacking many clusters at different positions on the sky, as long as the sky coverage of the survey is sufficiently large (e.g., Goto et al. 2003). The galaxy number outside the cluster region Nout for each redshift bin is estimated by using all galaxies that are located at r > 5 h−1 Mpc for all the clusters in the redshift bin. The number of galaxies is rescaled by the area before subtraction. We then subtract the contamination by foreground and background galaxies using Nout estimated above. We measure the areas occupied by the galaxies outside the cluster regions by counting the number of randoms in the random catalog (Coupon et al. 2018). The random catalog is created based on the pixel-based information and inherits most of the photometric flags on the object images. One can find the corresponding version of the random catalog to the object catalog in the same data release site. The random catalog takes account of both the selection criteria described in subsection 2.3 and masks. Now the foreground and background subtracted number of galaxies can be written as   $$N_j = N^{\rm in}_j - N^{\rm out} \frac{R^{\rm in}_j}{R^{\rm out}},$$ (4)where R is the number of randoms which is defined in exactly the same manner as in equations (2) and (3). 3.2 Color correction As shown in figure 2, red-sequence galaxies distribute in a narrow range of color which evolves with redshift. This means that galaxy colors evolve with redshift, even within the same redshift bin. In order to obtain accurate stacking results, including accurate separation of red and blue galaxies based on their colors, we apply a correction for the color evolution as a function of redshift before stacking many clusters to study the population of red and blue galaxies within each redshift bin. For clusters at z = z1, we derive the corrected color as $$C_{\rm corr}=C_{\rm obs}-C^{\rm th}(z_1)+C^{\rm th}_{\star }$$, where Cth denotes the theoretically derived galaxy color based on the stellar population synthesis model of Bruzual and Charlot (2003), with the calibration of colors using spectroscopic galaxies in the HSC survey (Oguri et al. 2018). $$C_{\star }^{\rm th}$$ is Cth at the median redshift z⋆ within the redshift bin. Furthermore, we correct the color gradient as a function of magnitude using z-band magnitudes. We fit the color–magnitude relation along the red sequence using the linear function as g(mz) = g⋆ + α(mz − mz,⋆), and correct the colors of all galaxies to the red-sequence zero-point (intercept of the linear relationship) estimated at the median magnitude of the cluster member galaxies, g⋆ ≡ g(mz, ⋆). To summarize, the corrected color of a galaxy with magnitude mz for a cluster at zcl stands for the color difference from that of the red sequence and is derived from the observed raw color Cobs as   $$C(m_z | z_{\rm cl}, m_{z,\star }) = C_{\rm obs} -C^{\rm th}(z_{\rm cl})+C^{\rm th}_{\star } - g(m_z) + g_{\star }.$$ (5)We show the color gradient in terms of redshift in figure 2 for CAMIRA member galaxies and the color gradient in terms of z-band magnitude in figure 3. Fig. 3. View largeDownload slide Color–magnitude diagrams (CMDs) that are derived by stacking photometric galaxies over all CAMIRA clusters. Color level stands for the number of galaxies in each cell after foreground and background galaxies are statistically subtracted (see the text for details). From left to right, we show CMDs for the cluster centric radii rP < 0.1, 0.5, and 1.0 h−1 Mpc. From top to bottom, the mean redshifts of the clusters are zcl = 0.18, 0.33, 0.55, and 0.85. Overlaid contours in each panel are the distribution of cluster member galaxies identified by CAMIRA. Solid lines show the slopes that minimize the scatter of CAMIRA member galaxies around the line, i.e., g(mz) correction. We define red and blue galaxies for each redshift bin by those above and below the dashed line (which is defined by a line 2σ below the solid line), respectively. Vertical dotted lines are the apparent magnitude cut corresponding to the rest frame Mz < −18.5. (Color online) Fig. 3. View largeDownload slide Color–magnitude diagrams (CMDs) that are derived by stacking photometric galaxies over all CAMIRA clusters. Color level stands for the number of galaxies in each cell after foreground and background galaxies are statistically subtracted (see the text for details). From left to right, we show CMDs for the cluster centric radii rP < 0.1, 0.5, and 1.0 h−1 Mpc. From top to bottom, the mean redshifts of the clusters are zcl = 0.18, 0.33, 0.55, and 0.85. Overlaid contours in each panel are the distribution of cluster member galaxies identified by CAMIRA. Solid lines show the slopes that minimize the scatter of CAMIRA member galaxies around the line, i.e., g(mz) correction. We define red and blue galaxies for each redshift bin by those above and below the dashed line (which is defined by a line 2σ below the solid line), respectively. Vertical dotted lines are the apparent magnitude cut corresponding to the rest frame Mz < −18.5. (Color online) 3.3 Definition of red and blue galaxies While there are a variety of definitions of red and blue galaxies in the literature, we introduce an empirical definition based on the observed data. Since CAMIRA cluster member galaxies represent the population of quiescent galaxies, the location of the CAMIRA cluster member galaxies in the CMD is well localized. This means that, at a given redshift, galaxies that have different star formation activities have different colors. We define blue galaxies as those that are located in the CMD 2σ away (on the bluer side) from the linear relation of the red-sequence obtained in subsection 3.2. Figure 3 shows a CMD after foreground and background subtraction in different redshifts (low-z to high-z from top to bottom) and different cluster centric radii (inner to outer from left to right). The color tilts are not corrected (but see figure 4 for a color-corrected diagram for z = 0.55 and rP < 0.5 h−1 Mpc). Horizontal dashed lines are the locations dividing the sample into red and blue galaxies. It is clearly seen that there are few blue galaxies at the inner regions of clusters and the number of blue galaxies increases with the cluster centric radius. We will see this more in detail in subsection 5. We note that if we carefully focus on the faint end of the CMD, the linear function obtained by CAMIRA member galaxies are slightly off from the peak of the red galaxy distribution. Unlike the CAMIRA member galaxies which have evolved to the red sequence, faint galaxies near the red-sequence track are still in the stage of star-forming and are in the transition phase from star-forming galaxies to quiescent galaxies. For the thorough investigation, we need to divide the cluster sample into finer mass bins, which our future paper will be devoted to. Fig. 4. View largeDownload slide Same as figure 3 for z = 0.55 and rP < 0.5 h−1 Mpc, but color is corrected according to equation (5). (Color online) Fig. 4. View largeDownload slide Same as figure 3 for z = 0.55 and rP < 0.5 h−1 Mpc, but color is corrected according to equation (5). (Color online) 4 Red-sequence at the faint end In this section, we study the red-sequence galaxies within cluster centric radius rP < 0.5 h−1 Mpc at the very faint end down to mz ∼ 24, which is enabled by our careful statistical subtraction of foreground and background galaxies. Specifically, we study how the scatter of the red-sequence changes as a function of magnitude. We model the color-corrected, foreground- and background-subtracted CMD distribution with the following double Gaussian (e.g., Hao et al. 2009)   \begin{eqnarray} n(C|m_z) & =& \frac{A_R(m_z) }{\sqrt{2\pi \sigma _R^2(m_z)}} \exp \left[ -\frac{(C-C_R)^2}{2\sigma _R^2(m_z)} \right] \nonumber \\ && +\, \frac{A_B(m_z)}{\sqrt{2\pi \sigma _B^2(m_z)}} \exp \left[ -\frac{(C-C_B)^2}{2\sigma _B^2(m_z)} \right] \end{eqnarray} (6)where the parameters Ax, σx, and Cx, with x being either R (red) or B (blue), are treated as free parameters. As we already corrected for the color tilt against the magnitude in subsection 3.2, the mean of the red component, CR, can well be described by a constant. For simplicity, we also assume that the blue component (CB) has constant mean, which is equivalent to assuming that the tilt of the color–magnitude relation for blue galaxies is the same as that for red galaxies. This assumption is reasonable because the blue galaxies do not have a tight relation with the mz but are rather broadly distributed and thus insensitive to the choice of color correction; as far as the tilt correction is linear, the color correction simply changes the CB and σB at each mz bin but it does not affect the estimate of σR that we are interested in. We divide the CMD into several magnitude bins, and estimate the values of σR for each magnitude bin with the Markov Chain Monte Carlo method by keeping other parameters free but fixing the CR to its corrected value obtained in subsection 3.2. Figure 5 shows the best-fitting scatter parameter σR as a function of magnitude, for different cluster redshifts. As discussed above, we use different colors for clusters at different redshift, such that these colors refer to approximately the same color in the cluster rest frame. At the very faint end, we need to take account of the scatter associated with the photometric error, which has a significant contribution to the observed scatter at mz ∼ 24. As the intrinsic scatter is not correlated with the photometric error, we can separate their contributions as   $$\sigma _{\rm obs}^2 = \sigma _{\rm photo}^2 + \sigma _{\rm int}^2,$$ (7)where σobs, σphoto, and σint are the observed scatter, the scatter due to the photometric error, and the intrinsic scatter of the red-sequence, respectively. We find that the intrinsic scatter of the red-sequence galaxies after subtracting the photometric error is almost constant over a wide range of magnitudes. We also find that there is no significant redshift evolution of the scatter, which is consistent with previous work which used smaller samples of clusters (e.g., Cerulo et al. 2016; Hennig et al. 2017). The figure suggests a slight decrease of the scatter at the faint end, but being the photometric error large at faint magnitudes, the intrinsic scatter is likely to be underestimated. Over most of the magnitude range, however, the photometric error is much smaller than the intrinsic scatter, which indicates that our result is robust against the photometric error. Fig. 5. View largeDownload slide Scatter of colors of red-sequence galaxies as a function of z-band magnitude. Results are shown for different redshift bins. Shaded regions show observed scatter, whereas dashed lines show the estimated intrinsic scatter after subtracting the scatter due to the photometric error. which is shown by filled circles with error bars. The circles and error bars are the median and one-sigma scatter of the photometric error in each magnitude bin. (Color online) Fig. 5. View largeDownload slide Scatter of colors of red-sequence galaxies as a function of z-band magnitude. Results are shown for different redshift bins. Shaded regions show observed scatter, whereas dashed lines show the estimated intrinsic scatter after subtracting the scatter due to the photometric error. which is shown by filled circles with error bars. The circles and error bars are the median and one-sigma scatter of the photometric error in each magnitude bin. (Color online) 5 Cluster profile and fraction of red galaxies 5.1 Cluster profile Given the timescale for the evolution of galaxies, tracking the redshift evolution of the number-density profiles for red and blue components can help us understand the dynamical history of the formation of galaxy clusters. The radial mass density profile of dark matter halos has long been thought to have a long tail that goes as ρ ∝ r−3 (Navarro et al. 1996). With such a profile, the total enclosed mass of a cluster diverges logarithmically, and the total mass associated with the halo depends upon the arbitrary boundary imposed on the halo. The splashback radius, marked by the apocenter of the recently infalling material, provides a clear physical boundary for the halo and can be used to identify the edges of dark matter halos (Diemer & Kravtsov 2014; More et al. 2015). The splashback radius manifests itself as a sharp drop in the matter density at its location (Diemer & Kravtsov 2014; Adhikari et al. 2014). A dark matter profile with such a density jump can be modeled with an inner universal profile multiplied by a transition function and an outer profile which represents the so-called two-halo contribution (Diemer & Kravtsov 2014). This can be explicitly written as   \begin{eqnarray} \rho (r) = \rho ^{\rm in}(r) \left[ 1+\left( \frac{r}{r_t} \right)^\beta \right]^{-\gamma /\beta } + \rho ^{\rm out}(r), \end{eqnarray} (8)  \begin{eqnarray} \rho ^{\rm out}(r) = \rho _m \left[ b_e \left( \frac{r}{5R_{200}}\right)^{-s_e} + 1 \right], \end{eqnarray} (9)where rt, γ, and β denote the location of the dip in the profile, the steepness of the dip, and how rapidly the slope changes, respectively. They all characterize the transition between inner and outer profiles. For the outer profile, ρm, be, and se represent the overall normalization, the relative normalization of the power-law profile, and the index of the power law, respectively. Diemer and Kravtsov (2014) express rt/R200m as a function of the accretion rate Γ, rt/R200m = [0.62 + 1.18 exp (−2Γ/3)], but in this paper we keep rt as a free parameter since we do not have a reliable estimate of the either the R200m or the accretion rate of our optically-selected clusters. We use the NFW (Navarro–Frenk–White; Navarro et al. 1996) profile to describe the inner profile,   $$\rho ^{\rm NFW}(r) = \frac{\rho _{\rm s}}{(r/r_{\rm s}) (1+r/r_{\rm s})^2},$$ (10)where rs and ρs denote the transition scale of slope from −1 to −3 and overall normalization, respectively. While we use rt for fitting observed radial profiles, following More, Diemer, and Kravtsov (2015) we define the splashback radius, Rsp, as the radius where the radial profile attains its steepest slope. As in More et al. (2016), we allow β and γ to take the values with log β = log 6 ± 0.2 and log γ = log 4 ± 0.2. The profiles of equations (8), (9), and (10) are numerically integrated along the line of sight to project to the two-dimensional sky. Figure 6 shows the radial number density profiles for red and blue components at redshifts z = 0.18, 0.33, 0.55, and 0.85. The covariance matrices are estimated from the jackknife resampling as   \begin{eqnarray} \widehat{{\rm Cov}_{ij}} &=& \frac{N_{\rm a}-1}{N_{\rm a}}\sum _{\rm a} \left[w_{\rm a}^{\rm cg}\left(r_{{\rm p}, i}\right) - \overline{w^{\rm cg}}\left(r_{{\rm p}, i}\right)\right] \nonumber \\ &&\times \left[w_{\rm a}^{\rm cg}\left(r_{{\rm p}, j}\right) - \overline{w^{\rm cg}} \left(r_{{\rm p}, j}\right)\right], \end{eqnarray} (11)where $$\overline{w^{\rm cg}}$$ is the arithmetic mean of $$w^{\rm cg}_{\rm a}$$. We divide the entire area into 35 rectangular regions with a side of 5.°0. That scale corresponds to the comoving angular separation of 25 h−1 Mpc at z = 0.1 which is sufficiently larger than the scale of our interest and includes more than one cluster in all sub-divided regions. We first fit to an NFW profile by using data below 2 h−1 Mpc. Best-fitting scale radii are summarized in table 2. We find notable differences of concentrations between two populations. Red galaxies are more concentrated toward the cluster center (i.e., smaller rs) and blue galaxies are less concentrated (i.e., larger rs). The difference in the concentration can be accounted for by the merger of clusters as discussed in e.g., Okamoto and Nagashima (2003, 2001). Another possible explanation is that red galaxies at the same luminosity live in more massive halos than their blue counterparts (Mandelbaum et al. 2006; More et al. 2011) and have experienced more dynamical friction so that they are concentrated toward the cluster center as we see in the discussion below. If we focus on the redshift evolution, it is seen that the overall profile tends to be more concentrated for lower redshifts. While the evolution of the concentration of red galaxies is subtle, blue galaxies evolve rapidly from z = 0.5 to 0.3. Fig. 6. View largeDownload slide Radial number density profiles of galaxies around clusters. From left to right, the mean redshifts of clusters are 0.18, 0.33, 0.55, and 0.85. Red and blue symbols show profiles for red and blue member galaxies. Gray symbols show profiles for all galaxies. Dashed lines are best-fitting models of the projected NFW profile. Solid lines show the best-fitting Diemer and Kravtsov (2014) model. Vertical solid and dashed lines indicate the best-fitting scale radii rs for red and blue galaxies, respectively. Bottom panels show the slope of the best-fitting profiles with vertical lines being the best-fitting value of rt, which can be compared with the splashback radius, Rsp. (Color online) Fig. 6. View largeDownload slide Radial number density profiles of galaxies around clusters. From left to right, the mean redshifts of clusters are 0.18, 0.33, 0.55, and 0.85. Red and blue symbols show profiles for red and blue member galaxies. Gray symbols show profiles for all galaxies. Dashed lines are best-fitting models of the projected NFW profile. Solid lines show the best-fitting Diemer and Kravtsov (2014) model. Vertical solid and dashed lines indicate the best-fitting scale radii rs for red and blue galaxies, respectively. Bottom panels show the slope of the best-fitting profiles with vertical lines being the best-fitting value of rt, which can be compared with the splashback radius, Rsp. (Color online) Table 2. Best-fitting values of cluster profile parameters.* z  Sample  rs  rt  Rsp  ΔAIC  ΔBIC    Red  0.50 ± 0.01  1.20 ± 0.13  1.21 ± 0.70  −13.9  −7.98  0.18  Blue  1.14 ± 0.03  3.63 ± 1.07  2.66 ± 1.29  −18.0  −12.1    All  0.57 ± 0.01  1.11 ± 0.13  1.21 ± 0.66  −16.1  −10.2    Red  0.51 ± 0.01  1.34 ± 0.16  1.26 ± 0.66  −14.7  −8.8  0.33  Blue  8.61<  1.43 ± 0.18  1.52 ± 0.58  −16.6  −10.7    All  0.91 ± 0.02  1.53 ± 0.22  1.15 ± 0.64  −16.2  −10.4    Red  0.54 ± 0.01  1.20 ± 0.10  1.26 ± 0.73  −14.9  −9.0  0.55  Blue  8.42<  1.20 ± 0.18  1.26 ± 0.54  −16.4  −10.5    All  1.11 ± 0.03  1.12 ± 0.12  1.10 ± 0.52  −14.4  −8.5    Red  0.77 ± 0.03  1.40 ± 0.13  1.32 ± 0.79  −15.1  −9.2  0.85  Blue  7.73<  1.75 ± 0.47  1.67 ± 0.54  −17.4  −11.5    All  1.72 ± 0.12  1.29 ± 0.27  1.21 ± 0.39  −17.1  −11.2  z  Sample  rs  rt  Rsp  ΔAIC  ΔBIC    Red  0.50 ± 0.01  1.20 ± 0.13  1.21 ± 0.70  −13.9  −7.98  0.18  Blue  1.14 ± 0.03  3.63 ± 1.07  2.66 ± 1.29  −18.0  −12.1    All  0.57 ± 0.01  1.11 ± 0.13  1.21 ± 0.66  −16.1  −10.2    Red  0.51 ± 0.01  1.34 ± 0.16  1.26 ± 0.66  −14.7  −8.8  0.33  Blue  8.61<  1.43 ± 0.18  1.52 ± 0.58  −16.6  −10.7    All  0.91 ± 0.02  1.53 ± 0.22  1.15 ± 0.64  −16.2  −10.4    Red  0.54 ± 0.01  1.20 ± 0.10  1.26 ± 0.73  −14.9  −9.0  0.55  Blue  8.42<  1.20 ± 0.18  1.26 ± 0.54  −16.4  −10.5    All  1.11 ± 0.03  1.12 ± 0.12  1.10 ± 0.52  −14.4  −8.5    Red  0.77 ± 0.03  1.40 ± 0.13  1.32 ± 0.79  −15.1  −9.2  0.85  Blue  7.73<  1.75 ± 0.47  1.67 ± 0.54  −17.4  −11.5    All  1.72 ± 0.12  1.29 ± 0.27  1.21 ± 0.39  −17.1  −11.2  *For each redshift range, the top, middle and bottom rows are for red galaxy, blue galaxy and all galaxy samples, respectively. All the values are in comoving h−1 Mpc. Also shown are difference of information criteria. A minus value means that the NFW profile is favored over the density jump model. View Large Next we fit all the data to the full profile of equation (8), keeping rs and ρs fixed to their best-fitting values from the simple NFW profile fitting to simplify the degeneracies inherent in the fitting procedure. The best-fitting curves are presented with solid lines in figure 6. We also show the logarithmic slope of the profile, which can be used to define the splashback radius Rsp as a local minimum of the slope. We compare the splashback radii Rsp, rt, and rs in table 2. As shown in More, Diemer, and Kravtsov (2015), there is a tight relation between the mass accretion rate of clusters and the normalized splashback radius Rsp/R200m obtained by stacking clusters at each redshift. Given the fact that the richness limit of our cluster sample approximately corresponds to a constant mass limit of M200m > 1014 h−1 M⊙ over the whole redshift range (Oguri et al. 2018), we find that the splashback radii from our fits roughly correspond to Rsp ∼ R200m. This is broadly consistent with More et al. (2016) in which splashback radii were derived for SDSS clusters to argue that the observed splashback radii are smaller than the standard cold dark matter model prediction, implying a faster mass accretion than the standard model. However, more careful estimates of the mass of these clusters using the weak gravitational lensing signal ought to be performed before doing a more quantitative comparison to More et al. (2016), as well as quantifying the redshift evolution. We will explore this in the near future. We compare the goodness-of-fit between NFW and full profile of Diemer and Kravtsov (2014) by computing two different criteria; the Akaike Information Criteria (AIC) corrected for the finite data size and the Bayesian Information Criteria (BIC). They are defined as   \begin{eqnarray} \displaystyle {\rm AIC} &=& -2\ln ({\mathcal {L}}) + 2p + \frac{2p(p+1)}{N-p-1} \end{eqnarray} (12)  \begin{eqnarray} {\rm BIC} &=& -2\ln ({\mathcal {L}}) + \ln (N)p, \end{eqnarray} (13)where $${\mathcal {L}}$$ is the likelihood and p and N are number of parameters and data, respectively; (N, p) for NFW is (12, 2) and (12, 6) for the density jump model. We compare the information criteria for the two profile fittings, and find that the density jump model is disfavored as summarized in table 2. As a cautionary note, we also mention that selection effects in optical cluster finding can significantly complicate the inference of the splashback radius from observations. Optical clusters are more likely to have their major axes oriented along the line-of-sight, which breaks the spherical symmetry assumption involved in the inference of the splashback radius (see e.g., Busch & White 2017). In addition, degeneracies related to cluster mis-centering can also reduce the significance of the detection of the splashback radius (Baxter et al. 2017). The values of the splashback radii inferred from optical clusters should therefore be carefully compared to expectations, a topic we will focus on in the near future. For clusters at z > 0.5, we find a slight decline of radial profiles in the central region, r < 0.2 h−1 Mpc. This might reflect the mis-centering of the optically-selected clusters, which have been inferred in comparison with the X-ray profile (Oguri et al. 2018). It may be more difficult to identify the center of clusters correctly at higher redshift simply because in crowded regions, like cluster centers, angular separations of neighboring galaxies are smaller at higher redshifts, given the same physical scale. This is mainly due to the fact that the pipeline fails to deblend galaxies in crowded regions. We leave the effect of mis-centering for future work, after the weak lensing measurement of the CAMIRA cluster sample becomes available. 5.2 Red fraction as a function of redshift We derive the red galaxy fraction by summing up the number of red and blue galaxies in each radial bin out to the maximum comoving distances, for which we adopt 0.2, 0.5, and 1.0 h−1 Mpc. Figure 7 shows the evolution of fractions of red galaxies as a function of redshift, for three different maximum distances. We can clearly see the evolution of the red fraction over the cosmological timescale, from z = 1.1 to 0.1, such that the red fraction increases at lower redshift. This qualitative trend is consistent with Hennig et al. (2017) and Jian et al. (2018), while Loh et al. (2008) reported steeper evolution. The red fraction significantly increases with decreasing maximum distance from the cluster center, which is due to the different radial number density profiles between red and blue galaxies. We note that there are gaps in the fraction of red galaxies at z = 0.4 and z = 0.75 because we have used the different combination of filters to define red and blue galaxies. This implies that the global evolution of the red fraction over the redshifts 0.1 < z < 1.1 is subject to the choice of colors; however, within the redshift range in which we use the same filter combination, we observe apparent decreases of the red fraction. However, we note that the red fraction at higher redshift bins, 0.7 < z, are almost flat or slightly increasing. As we will discuss it at the end of subsection 5.3, this trend is partly due to our bad photometry at crowded regions like cluster centers. Fig. 7. View largeDownload slide Fraction of the red galaxies as a function of redshift and the maximum distance from the cluster center for counting up the galaxies. The x-axis for different rP is slightly shifted for visual purposes. (Color online) Fig. 7. View largeDownload slide Fraction of the red galaxies as a function of redshift and the maximum distance from the cluster center for counting up the galaxies. The x-axis for different rP is slightly shifted for visual purposes. (Color online) Figure 8 shows the fraction of red galaxies as a function of projected cluster-centric radius rP. Symbols are observed points and solid lines are predictions from the semi-analytical model described in subsection 5.3. The fractions of red galaxies are high, ∼0.6–1.0 in the inner regions (rP < 0.3 h−1 Mpc), decreasing to ∼0.2–0.4 in the outer regions (rP > 1 h−1 Mpc). On the intermediate scale in between the inner and outer regions, the fraction of red galaxies is gradually decreasing. Although the absolute value of the observed fraction is slightly higher than predicted by the semi-analytical model, the declining slopes are in good agreement with the model. Fig. 8. View largeDownload slide Radial profile of the fraction of red galaxies as a function of projected cluster-centric radius at four different redshifts. Symbols are the observed data points and solid lines are the prediction from the semi-analytical model described in subsection 5.3. (Color online) Fig. 8. View largeDownload slide Radial profile of the fraction of red galaxies as a function of projected cluster-centric radius at four different redshifts. Symbols are the observed data points and solid lines are the prediction from the semi-analytical model described in subsection 5.3. (Color online) 5.3 Comparison with a semi-analytical model It is important to check the consistency of our results with theoretical models of galaxy formation. We compare our results with a semi-analytical model of galaxy formation, ν2GC (Makiya et al. 2016). Semi-analytical models have an advantage over mock galaxy catalogs based on the halo occupation distribution technique in that semi-analytical models are constructed from physically motivated prescriptions of several astrophysical processes which, in comparison with observations, will lead to better understanding of the build-up of cluster galaxies. They also contain physical properties of galaxies, such as galaxy stellar and gas masses and star formation rates. This information may also help reveal physical processes affecting the evolution of cluster galaxies. We examine the spatial distribution of galaxies in ν2GC. In ν2GC, we construct merger trees of dark matter halos using cosmological N-body simulations (Ishiyama et al. 2015) with the Planck cosmology (Planck Collaboration 2014). The simulation box is 280 h−1 Mpc on a side containing 20483 particles, corresponding to a particle mass of 2.2 × 108 h−1 M⊙. The semi-analytical model includes the main physical processes involved in galaxy formation: formation and evolution of dark matter halos; radiative gas cooling and disc formation in dark matter halos; star formation, supernova feedback and chemical enrichment; galaxy mergers; and feedback from active galactic nuclei. The model is tuned to fit the luminosity functions of local galaxies (Driver et al. 2012) and the mass function of neutral hydrogen (Martin et al. 2010). The model well reproduces observational local scaling relations such as the Tully–Fisher relation and the size–magnitude relation of spiral galaxies (Courteau et al. 2007). We use ν2GC results to create mock galaxy catalogs. Rest-frame and apparent magnitudes of galaxies are estimated in the same filter as used in the HSC survey (Kawanomoto et al. 2017). We apply the same magnitude cut of the HSC (Mz < −18.5 in rest frame) to the simulated galaxies. In the analysis in this paper, we extract galaxy samples at different redshifts, residing in dark matter halos with masses greater than 1014 M⊙. We regard those galaxies as cluster galaxies. We have ∼175 simulated clusters at z = 1.1 and ∼1000 clusters at z = 0.13 which are defined in the same realization. Since the mock catalogs do not perfectly reproduce the colors of galaxies, we cannot apply the selection condition that is applied to the HSC data to the mock catalogs. We therefore re-define the selection criterion for red and blue galaxies in the mock galaxy catalogs. To define red and blue galaxy populations, we determine the red-sequence using a subsample of the mock galaxies as follows. We perform linear regression to fit the slope α and zero-point β of the following equation to describe the red sequence   $$C = \alpha \ m_z + \beta ,$$ (14)where C is the color corresponding to the cluster redshift, e.g., g − r at redshift [0.1–0.2]. To reduce contaminations by blue galaxies, we only fit to galaxies with specific star formation rates (sSFRs) log10(sSFR/Gyr−1) ≤ −1 and with distances from the cluster center d ≤ 0.1 Mpc h−1. These conditions are reasonable for extracting red-sequence galaxies which correspond to the CAMIRA-identified red-sequence. Finally, in exactly the same manner as in the observation, we define galaxies redder than the C − 2σ line on the CMD as red galaxies, where σ is the standard deviation of the distribution of the galaxies used in the fitting. Figure 9 shows radial profiles of cluster member galaxies for different redshift bins and different populations identified in the simulation. We see clearly that blue galaxies are more diffuse and red galaxies are more concentrated, which is consistent with our observational results. Fig. 9. View largeDownload slide Same as figure 6 but obtained from simulations with a semi-analytical model. (Color online) Fig. 9. View largeDownload slide Same as figure 6 but obtained from simulations with a semi-analytical model. (Color online) Figure 10 shows the redshift evolution of the red fraction in the simulation with the same binning of distance from the cluster center. We see a clear decrease of the red fraction with redshift and the result shows a reasonable agreement with the observation except for the two highest-redshift bins. The simulated results show a monotonic decrease of the red fraction at the highest redshift ranges, but our observational results show a slight increase. The monotonic decrease of the red fraction to 0.5 at z = 1 is consistent with the results of Hennig et al. (2017), and therefore the slight increase seen in our data may not be accounted for by the difference in the depth of the sample, (our sample is ∼1 mag deeper in the z band) or the different filter combination used to define the red and blue populations. We do not make strong conclusions about the source of this discrepancy in this paper, but it may be partly due to the bad photometry in crowded regions, which significantly affects the color of galaxies in clusters. We will revisit the issue and address it in the future work once the photometry of HSC in the crowded regions is improved. Fig. 10. View largeDownload slide Same as figure 7 but obtained from simulations with a semi-analytical model. (Color online) Fig. 10. View largeDownload slide Same as figure 7 but obtained from simulations with a semi-analytical model. (Color online) 6 Summary In this paper, we have used the HSC S16A internal data release galaxy sample over ∼230 deg2 to explore the properties of cluster galaxies over wide redshift and magnitude ranges. Clusters are identified by the red-sequence cluster finding method CAMIRA (Oguri 2014; Oguri et al. 2018). Thanks to the powerful capability of the Subaru telescope to collect light and the good sensitivity of the HSC detector, we can study faint cluster galaxies down to the 24th magnitude in the z band. This sample is ∼1 mag deeper than the cluster sample of Hennig et al. (2017) which reaches m⋆ + 1.2 ∼ 23.2 in the Dark Energy Survey (DES) z band at z = 1. Together with a reliable CAMIRA cluster catalog out to z = 1.1, the excellent HSC data allows us to continuously track the evolution history of cluster galaxies from z = 1.1 to the present. We have used the stacked CMD to divide red and blue galaxies into clusters. We have statistically subtracted background and foreground galaxies after area corrections using the well-defined random catalog, which is also available from the HSC data base. After subtracting the foreground and background galaxies, color–magnitude relations for red galaxies (red-sequence) and blue clouds are clearly detected over wide ranges in redshift and magnitude. We have used these CMDs for defining red and blue galaxies, studying the tightness of the red-sequence down to very faint magnitudes, and the radial number density profiles of red and blue galaxies. Our results are summarized as follows. Red galaxies in clusters follow a clear linear relation in the CMD down to the HSC completeness limit for all redshifts. However, we observe a slight offset of the red populations in the cluster from the linear relation determined using the CAMIRA member galaxies. This may be partly due to our sample selection, i.e., a constant mass cut over all redshift ranges, and we will revisit it once the cluster mass is measured well with the weak lensing. We have measured the intrinsic scatter of the red-sequence as a function of the observed z-band magnitude and cluster redshift. We have found that the intrinsic scatter is almost constant over wide range of magnitudes. The intrinsic scatter shows little evolution with redshift. Red galaxies are more concentrated toward the cluster center compared with blue galaxies. We fitted the cluster member radial profile at r < 1.0 h−1 Mpc to an NFW profile, and find the transition scale rs is significantly smaller for red galaxies than for blue galaxies. Given that the cluster sample has approximately constant mean mass over different redshifts (Oguri et al. 2018), the mildly decreasing rs with redshift implies that the galaxy profiles in clusters become less concentrated at higher redshift. We note, however, that it is important to derive the virial mass of the clusters independently; this will soon be provided by the HSC weak lensing analysis (Mandelbaum et al. 2018). Special care is required that the mass profile measured by weak lensing is for dark matter and this should be different from the profile of member galaxies. We fitted the radial number density profiles with the density jump model of Diemer and Kravtsov (2014), and find that the splashback radius Rsp defined by the minimum of the logarithmic slope is almost constant over the redshift range. However, given the large statistical uncertainties, we do not detect the splashback radius for our current data set. The fraction of red galaxies is not only a strong function of the distance from the cluster center, but also exhibits a moderate decrease with increasing redshift. We note that the estimated red fraction shows a slight discontinuity at the redshift where the red and blue galaxies are defined in different combinations of colors, i.e., z ∼ 0.4 and z ∼ 0.7. This discontinuity might reflect that our definition of red and blue galaxies are not optimal near the transition redshifts because the redshift of the 4000 Å break mismatches with the filter response functions of a given combination of the color. We note that the same discontinuity is also seen in the simulations. We also compared our results with semi-analytical model predictions. We find that the observed cluster profiles and the redshift evolution of the red fraction are broadly consistent with the semi-analytical model prediction. Further studies for more quantitative comparisons are important. The total mass and mass profile of the CAMIRA clusters can be measured by stacked weak lensing. With the help of the mass–richness relation by the forward modeling (R. Murata et al. in preparation), this will allows us to explore cluster physical quantities, such as virial radius, mass accretion rate, and the mass dependence of those quantities, in more detail. We will revisit this in our future work. Acknowledgements We thank the anonymous referee for providing useful comments. AN is supported in part by MEXT KAKENHI Grant Number 16H01096. This work was supported in part by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, MEXT as “Priority Issue on Post-K computer” (Elucidation of the Fundamental Laws and Evolution of the Universe) and JICFuS, and JSPS KAKENHI Grant Number 26800093 and 15H05892. SM is supported by the Japan Society for Promotion of Science grants JP15K17600 and JP16H01089. This work was supported in part by MEXT KAKENHI Grant Number 17K14273 (TN). HM is supported by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. The Hyper Suprime-Cam (HSC) collaboration includes the astronomical communities of Japan and Taiwan, and Princeton University. The HSC instrumentation and software were developed by the National Astronomical Observatory of Japan (NAOJ), the Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), the University of Tokyo, the High Energy Accelerator Research Organization (KEK), the Academia Sinica Institute for Astronomy and Astrophysics in Taiwan (ASIAA), and Princeton University. Funding was contributed by the FIRST program from Japanese Cabinet Office, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), the Japan Society for the Promotion of Science (JSPS), Japan Science and Technology Agency (JST), the Toray Science Foundation, NAOJ, Kavli IPMU, KEK, ASIAA, and Princeton University. The Pan-STARRS1 Surveys (PS1) have been made possible through contributions of the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, Queen’s University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation under Grant No. AST-1238877, the University of Maryland, and Eotvos Lorand University (ELTE). 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M., Jones C. 1993, ApJ , 407, 489 CrossRef Search ADS   © The Author 2017. Published by Oxford University Press on behalf of the Astronomical Society of Japan. All rights reserved. For Permissions, please email: journals.permissions@oup.com http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications of the Astronomical Society of Japan Oxford University Press

# First results on the cluster galaxy population from the Subaru Hyper Suprime-Cam survey. II. Faint end color–magnitude diagrams and radial profiles of red and blue galaxies at 0.1 < z < 1.1

17 pages

/lp/ou_press/first-results-on-the-cluster-galaxy-population-from-the-subaru-hyper-BJNHUMJbQg
Publisher
Oxford University Press
ISSN
0004-6264
eISSN
2053-051X
D.O.I.
10.1093/pasj/psx106
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### Abstract

Abstract We present a statistical study of the redshift evolution of the cluster galaxy population over a wide redshift range from 0.1 to 1.1, using ∼1900 optically-selected CAMIRA clusters from ∼232 deg2 of the Hyper Suprime-Cam (HSC) Wide S16A data. Our stacking technique with a statistical background subtraction reveals color–magnitude diagrams of red-sequence and blue cluster galaxies down to faint magnitudes of mz ∼ 24. We find that the linear relation of red-sequence galaxies in the color–magnitude diagram extends down to the faintest magnitudes we explore with a small intrinsic scatter σint(g − r) < 0.1. The scatter does not evolve significantly with redshift. The stacked color–magnitude diagrams are used to define red and blue galaxies in clusters in order to study their radial number density profiles without resorting to photometric redshifts of individual galaxies. We find that red galaxies are significantly more concentrated toward cluster centers and blue galaxies dominate the outskirts of clusters. We explore the fraction of red galaxies in clusters as a function of redshift, and find that the red fraction decreases with increasing distances from cluster centers. The red fraction exhibits a moderate decrease with increasing redshift. The radial number density profiles of cluster member galaxies are also used to infer the location of the steepest slope in the three-dimensional galaxy density profiles. For a fixed threshold in richness, we find little redshift evolution in this location. 1 Introduction Clusters of galaxies are the largest gravitationally bound objects in the Universe. Because their dynamics are determined mostly by gravity and the spatial distribution of clusters also follows the large-scale structure, clusters of galaxies are considered a good probe of cosmological structure. Therefore, understanding the formation history of clusters is key for both studying the structure formation and for cosmological applications of clusters (e.g., Rosati et al. 2002; Planelles et al. 2015; Castorina et al. 2014). One of the most prominent features of clusters of galaxies is the presence of a large number of red galaxies, which exhibit a tight relation in the color–magnitude diagram (CMD) (e.g., Stanford et al. 1998). This tight relation in the CMD, often referred to as the red-sequence, has been observed out to relatively high redshift, z ≃ 2 (Tanaka et al. 2010; Andreon et al. 2014; Cerulo et al. 2016; Romeo et al. 2016). The tilt of the color in the red-sequence can be explained mainly by the decrease of metallicity for low-mass galaxies (Faber 1973; Kodama & Arimoto 1997; De Lucia et al. 2007), and the age of the galaxies (Ferreras et al. 1999; Gallazziet al. 2006). Another notable property of clusters is that they also contain blue galaxies. Thus the color distribution of cluster member galaxies has a bimodal distribution (Gilbank et al.; Loh et al. 2008; Li et al. 2012). Historically, it is well known that there are more blue galaxies observed at high redshifts than at low redshifts compared to the red galaxies (Butcher & Oemler 1978). Also, it has been found that the abundance of morphologically different types of galaxies in clusters correlates with local density (Dressler 1980; Whitmore & Gilmore 1991) and distance from the cluster center (Whitmore et al. 1993; George et al. 2013). These relations seen in the local universe can also be seen in z ∼ 1 clusters (Postman et al. 2005). De Propris et al. (2004) showed that the fraction of blue galaxies in a blank field is a strong function of the local density, with a sharp decline near the density of the cluster environment toward ∼10%. This also suggests that member galaxies of clusters are dominated by the red population, although there are still a non-negligible amount of blue galaxies in clusters. In Hennig et al. (2017), Sunyaev–Zel’dovich (SZ) detected clusters are stacked to derive the number density profile for red and blue galaxies to find a clear difference in the concentration between the two populations. Because clusters of galaxies are rare objects, it has been difficult to construct a large sample of clusters at high redshifts, which requires both wide area and sufficient survey depth. Therefore previous galaxy population studies of massive high-redshift (z ∼ 1) clusters have been made using only a handful of clusters, either from deep optical/infrared (IR) imaging surveys over small areas or from follow-up imaging observations of X-ray and SZ-selected high-redshift clusters. For instance, the IRAC Shallow Cluster Survey (ISCS) includes 13 clusters at 1.01 < z < 1.49 (Snyder et al. 2012), the Gemini Cluster Astrophysics Spectroscopic Survey (GCLASS) has 10 spectroscopically confirmed rich clusters at 0.85 < z < 1.34 (Muzzin et al. 2012), and the HAWK-I Cluster Survey (HCS) contains only nine clusters at 0.84 < z < 1.46 (Lidman et al. 2013). Conversely, there are large cluster samples at lower redshifts for different data and different cluster-finding methods. Gladders and Yee (2000, 2005) used two optical and near-IR filters to define the red sequence on Red-Sequence Cluster Survey (RCS), Koester et al. (2007) applied a red-sequence based finding method called maxBCG to Sloan Digital Sky Survey (SDSS) galaxies, and Rykoff et al. (2014) developed a red-sequence based method with a more sophisticated algorithm and applied it to SDSS galaxies. In this paper, we explore the average picture of the cluster galaxy population using a large, homogeneous sample of clusters with a uniform selection over the wide redshift range of 0.1 < z < 1.1, which is selected from the Subaru Hyper Suprime-Cam (HSC) survey. The sample contains more than 1900 clusters optically identified by the CAMIRA (Cluster finding Algorithm based on Multi-band Identification of Red-sequence gAlaxies: Oguri et al. 2018) algorithm applied to the S16A internal data release of the HSC survey. The unique combination of the area and depth of the HSC survey is crucial for accurate galaxy population studies for clusters at z ∼ 1, as well as galaxy population studies down to very faint magnitudes for low-redshift clusters. An advantage of using a large homogeneous sample of clusters is that it enables a statistical subtraction of projected foreground and background galaxies, which is important for mitigating projection effects in cluster galaxy studies. With this unprecedented data set, we first examine the width of the red-sequence of galaxies down to mz = 24 mag. Then we trace the redshift evolution of the radial profiles of red and blue galaxies in clusters. This enables us to understand the formation history and dynamical evolution of the clusters from z = 1.1 to the present. We refer the interested reader to a series of complementary papers describing different aspects of the properties of these cluster galaxies (Jian et al. 2018; Lin et al. 2017). This paper is organized as follows. In section 2, we describe the HSC photometric data and briefly overview the CAMIRA cluster catalog. Our selection criteria for the photometric sample is also presented. In section 3, we describe our method for identifying cluster member galaxies without using photometric redshifts. Section 4 presents the intrinsic scatter of red galaxies down to the magnitude limit. The radial profile and fraction of red member galaxies in clusters and their evolution over redshifts are discussed in section 5. The comparison with semi-analytical model results is also presented there. We summarize our results in section 6. Unless otherwise stated, we assume cosmological parameters as $$\Omega _m=0.31, \Omega _\Lambda =0.69$$, and h = 0.7 that are consistent with Planck Collaboration (2014) results throughout the paper. 2 HSC data 2.1 HSC photometric sample The HSC survey started observing in 2014 March and is continuously collecting photometric data over a wide area under good photometric conditions (Aihara et al. 2018b). The HSC survey consists of three different layers: Wide, Deep and UltraDeep. In this paper, we use the photometric data from the Wide layers of the internal S16A data release, which covers more than 200 deg2 of the sky with five broadband filters (grizy). In the S16A data, the Wide layer reaches the limiting magnitude of 26.4 in the i band. During the survey, i- and r-band filters were replaced by newer ones which have better uniformity over the entire field of view (Miyazaki et al. 2018; Kawanomoto et al. 2017) They are denoted as the i2 and r2 filters, respectively. In this paper we do not discriminate between the old and new filters and simply denote them as i and r. For each pointing, we divide the exposure into 4 for the g and r bands, and 6 for the i, z, and y bands, with a large dithering step of ∼0.6 deg, to recover images at gaps between CCDs and to obtain a uniform depth over the entire field. The total exposure times for each pointing are 150 × 4 = 600 s for g and r bands, and 200 × 6 = 1200 s for i, z, and y bands. The expected 5σ limiting magnitudes for the 2″ diameter aperture are 26.5, 26.1, 25.9, 25.1, and 24.4 for g, r, i, z, and y bands, respectively. In point spread function (PSF) magnitudes, which is an aperture magnitude convolved with the measured and modeled PSF function, we reach down to i = 26.4 with a 5σ level. The median seeing of i-band images is 0$${^{\prime\prime}_{.}}$$61 (Aihara et al. 2018a). The images are reduced by the processing pipeline hscpipe (Bosch et al. 2018), which was developed as a part of the LSST (Large Synoptic Survey Telescope) pipeline (Ivezic et al. 2008; Axelrod et al. 2010; Jurić et al. 2015). The photometry and astrometry are calibrated in comparison with the Pan-STARRS1 3π catalog (Schlafly et al. 2012; Tonry et al. 2012; Magnier et al. 2013), which fully covers the HSC survey footprint and has a set of similar filter response functions. The photometry of the current version of hscpipe in crowded regions such as cluster centers is not accurate due to the complexity of object separation on the image, (deblending: Bosch et al. 2018). For this reason, we combine two different methods of measuring the photometry as described in detail in subsection 2.3. 2.2 CAMIRA clusters In this section, we briefly overview the CAMIRA cluster-finding algorithm and properties of the catalog (Oguri et al. 2018). CAMIRA is a cluster-finding algorithm based on the red-sequence galaxies (Oguri 2014). By applying the algorithm to the HSC S16A data, Oguri et al. (2018) constructed a catalog of ∼1900 clusters from ∼230 deg2 of the sky over the wide redshift range 0.1 < z < 1.1 with almost uniform completeness and purity. Oguri et al. (2018) first applied specific color cuts to a spectroscopic redshift-matched catalog to remove the obvious blue galaxies. Those galaxies are used only for calibrating the color of the red-sequence and this color cut is not applied in cluster-finding itself. CAMIRA computes the likelihood of a galaxy being on the red-sequence as a function of redshift, using a stellar population synthesis (SPS) model of Bruzual and Charlot (2003) with accurate calibration of colors using the spectroscopic galaxies mentioned above. In contrast with the complex color degeneracy among different parameters of SPS, CAMIRA simplifies the parametrization to quantify the red-sequence galaxies. CAMIRA applies a single instantaneous burst at the formation redshift zf = 3 and assumes no dust attenuation, since these prescription is good enough to represent the red-sequence color. The metallicity of galaxies is modeled as a logarithmic linear function of total mass at z = zf; i.e., log ZSPS = log Z11 + aZlog [M*(zf)/1011 M⊙], with log Z11 = −2, aZ = 0.15. CAMIRA additionally introduces the scatter of metallicity to model the intrinsic scatter of the red-sequence with σlog Z = 0.14 (Oguri 2014). For individual galaxies, the likelihood is calculated by finding the best-fitting parameters, but color calibration is simultaneously applied by using the spectroscopic sample, which recovers the imperfect prediction of the SPS model [see equation (2) of Oguri (2014). These likelihoods are used to compute the richness parameter with which cluster candidates are identified by searching for peaks of the richness. For each cluster candidate, the brightest cluster galaxies (BCG) is identified by searching a bright galaxy near the richness peak. The photometric redshift and richness are iteratively updated until the result converges. As a result, the accuracy of the photometric redshift reaches 1% out to z ∼ 1, which is better than other cluster-finding algorithms for different data sets (e.g., Rykoff et al. 2016; Mints et al. 2017). In this paper, we use 1902 CAMIRA clusters selected from ∼230 deg2 HSC Wide fields with richness $$\hat{N}_{\rm mem}>15$$. We note that the number of CAMIRA clusters is slightly smaller than in Oguri et al. (2018), because we further impose the condition that all galaxies have well-measured y-band photometry. The difference is less than 1% and this does not change our results. Although there is a large scatter in the relation between richness and mass, Oguri et al. (2018) argued that the richness limit of $$\hat{N}_{\rm mem}=15$$ roughly corresponds to a constant mass limit of M200m = 1014 h−1 M⊙. Due to the large scatter of the mass–richness relation and lack of the mass calibration from weak lensing, in this paper we do not divide the sample into richness bins. In contrast, in order to study the redshift evolution of the cluster galaxy property, we divide these clusters into different redshift bins with a bin size of Δz = 0.05 at z < 0.4 and Δz = 0.1 at z ≥ 0.4. Table 1 summarizes the mean redshift of clusters and the number of clusters with richness $$\hat{N}_{\rm mem}>15$$ in each redshift bin. The effect of cluster evolution on sample selection will be addressed in a future paper once the halo mass estimates from weak lensing is available. The different sample selection can be found in our complementary paper (Lin et al. 2017). Table 1. Definition of redshift bins used in this paper.* Redshift bin  Mean redshift  Color  Number of clusters  0.10–0.15  0.13  g − r  43  0.15–0.20  0.18  g − r  91  0.20–0.25  0.22  g − r  74  0.25–0.30  0.28  g − r  145  0.30–0.35  0.32  g − r  130  0.35–0.40  0.38  g − r  70  0.40–0.50  0.45  r − i  192  0.50–0.60  0.55  r − i  220  0.60–0.70  0.65  r − i  179  0.70–0.80  0.75  i − y  231  0.80–0.90  0.85  i − y  217  0.90–1.00  0.95  i − y  137  1.00–1.10  1.05  i − y  173  Redshift bin  Mean redshift  Color  Number of clusters  0.10–0.15  0.13  g − r  43  0.15–0.20  0.18  g − r  91  0.20–0.25  0.22  g − r  74  0.25–0.30  0.28  g − r  145  0.30–0.35  0.32  g − r  130  0.35–0.40  0.38  g − r  70  0.40–0.50  0.45  r − i  192  0.50–0.60  0.55  r − i  220  0.60–0.70  0.65  r − i  179  0.70–0.80  0.75  i − y  231  0.80–0.90  0.85  i − y  217  0.90–1.00  0.95  i − y  137  1.00–1.10  1.05  i − y  173  *For each redshift bin, the mean redshift of clusters, the color combination used to define red and blue galaxies, and number of clusters with $$\hat{N}_{\rm mem}>15$$ are shown. View Large 2.3 HSC sample selection In this section, we describe our selection of the HSC photometric galaxy sample, which is used for our analysis of the cluster galaxy population. In this paper, we use magnitudes for each object that is derived by combining CModel magnitudes (Abazajian et al. 2004; Bosch et al. 2018) with PSF-matched aperture magnitudes, the so-called afterburner photometry (Bosch et al. 2018). The CModel magnitude is obtained by fitting the object’s light profile with the sum of a de Vaucouleurs bulge and an exponential disk convolved with the PSF. PSFs are measured at the positions of stars and then modeled to interpolate over the entire field of view (Bosch et al. 2018). The PSF-matched aperture magnitude in the afterburner photometry is obtained by stacking the image after blurring each exposure toward the target PSF size and measuring the photometry at a given aperture size. All the magnitudes are corrected for the Galactic extinction (Schlegel et al. 1998). First we define the total magnitude of each galaxy with the z-band CModel magnitude.1 Then we derive the magnitudes in other bands as   $$m_{x} = m_{z}^{\rm CM} + \left( {m}_{x}^{\rm ab} - {m}_{z}^{\rm ab} \right),$$ (1)where $${m}_{z}^{\rm CM}$$ is the CModel magnitude in the z-band measured with forced photometry on the PSF-unmatched coadd image (cmodel_mag), and $${m}_{x}^{\rm ab}$$ ($${m}_{z}^{\rm ab}$$) is the PSF-matched aperture magnitude in the x band (z band) measured in the 0$${^{\prime\prime}_{.}}$$55 aperture in radius on the stacked image, where the PSF is convolved to homogenize the target PSF size of 1$${^{\prime\prime}_{.}}$$1 (parent_mag_convolved_2_0). As the error of the afterburner photometry is significantly underestimated because the neighboring pixels are highly correlated due to the blurring, we use the photometric error associated with the PSF-unmatched aperture photometry with a corresponding aperture instead of the afterburner photometry error. In the following we describe flags applied to select galaxies with high-quality photometry (Aihara et al. 2018b). Flags in forced photometry table [grizy]flags_pixel_edge is not True [grizy]flags_pixel_interpolated_center is not True [grizy]flags_pixel_cr_center is not True We apply the above three constraints to avoid objects geometrically overlapping with the masked region [EDGE or NO DATA], or object centers close to interpolated pixels or suspected cosmic rays. [grizy]cmodel_flux_flags is not True These flags ensure that the CModel flux is successfully measured. [grizy]centroid_sdss_flags is not True This excludes objects for which measurements of centroids failed, using the same method as in the Sloan Digital Sky Survey (Bosch et al. 2018). [gr]countinputs >1 [izy]countinputs >3 In the HSC-Wide, we divide every pointing into 4 for the g and r bands, and 6 for the i, z, and y bands. Here we use objects taken twice or more for g and r, and four times or more for i, z and y bands. detect_is_primary is True We also remove blended objects to avoid ambiguous photometric measurements. zcmodel_mag - a_z <24.0 We limit our sample in these magnitude ranges so that all the objects have a high signal-to-noise ratio (S/N). rcmodel_mag - a_r <28.0 icmodel_mag - a_i <28.0 These two flags are not for the primary object selection but for removing objects that are too faint. zcmodel_mag_err <0.1 We also directly impose the S/N cut corresponding to S/N ≳ 10. iclassification_extendedness =1 Stars are excluded. Flags in afterburner table [grizy]parent_flux_convolved_2_0_flags is not True In addition to the above selection, we use only galaxies brighter than the limits shown in figure 1 depending on the redshift of clusters. Fig. 1. View largeDownload slide Redshift and magnitude distribution of all CAMIRA member galaxies. The dashed line represents the observer-frame z-band magnitude of an SPS with constant absolute magnitude Mz = −18.5 as seen at different redshifts. The thick solid line is the same constant absolute magnitude but after applying K-correction and taking into account passive evolution. Shaded rectangles are regions where the galaxy sample is complete and suitable for exploring the redshift evolution of clusters in various aspects over different redshifts. The horizontal dotted line represents the mz = 24.0 mag cut that we apply to remove the galaxies affected by large photometric errors especially in the highest redshift bins of the cluster sample (see subsection 2.3 for details). (Color online) Fig. 1. View largeDownload slide Redshift and magnitude distribution of all CAMIRA member galaxies. The dashed line represents the observer-frame z-band magnitude of an SPS with constant absolute magnitude Mz = −18.5 as seen at different redshifts. The thick solid line is the same constant absolute magnitude but after applying K-correction and taking into account passive evolution. Shaded rectangles are regions where the galaxy sample is complete and suitable for exploring the redshift evolution of clusters in various aspects over different redshifts. The horizontal dotted line represents the mz = 24.0 mag cut that we apply to remove the galaxies affected by large photometric errors especially in the highest redshift bins of the cluster sample (see subsection 2.3 for details). (Color online) As a sense check, we compare colors of CAMIRA cluster member galaxies with model colors from the SPS model used in CAMIRA cluster finding (Oguri et al. 2018), which is based on the SPS model of Bruzual and Charlot (2003) with the calibration of colors from spectroscopic redshifts as described above. Figure 2 shows the redshift evolution of colors of CAMIRA cluster member galaxies, where we derive colors of cluster member galaxies by matching the photometric galaxy sample constructed above with a catalog of CAMIRA member galaxies from Oguri et al. (2018). We find that model colors and median colors of the member galaxies agree well, as expected. As shown in figure 2, in the redshift ranges of 0.1 < z < 0.4, 0.4 < z < 0.7, and 0.7 < z < 1.1, g − r, r − i, and i − y respectively show rapid color changes, because these colors cover the 4000 Å break at these redshifts. Moreover, we find that the above combination of filters shows the tightest scatter around the theoretical prediction in each redshift range. Therefore, we use these colors to construct the CMD to see the red-sequence galaxies at fainter magnitude with a stacking analysis (see table 1). Fig. 2. View largeDownload slide Redshift–color relation of red-sequence galaxies. Red points are cluster member galaxies identified by CAMIRA, and their median and 1σ region are denoted by filled circles and thin solid lines. The thick solid lines show the model colors from the SPS model (Bruzual & Charlot 2003) calibrated with spectroscopic redshifts. (Color online) Fig. 2. View largeDownload slide Redshift–color relation of red-sequence galaxies. Red points are cluster member galaxies identified by CAMIRA, and their median and 1σ region are denoted by filled circles and thin solid lines. The thick solid lines show the model colors from the SPS model (Bruzual & Charlot 2003) calibrated with spectroscopic redshifts. (Color online) 3 Statistical identification of cluster member galaxies 3.1 Stacking analysis In this paper, we study distributions of cluster member galaxies statistically, without resorting to spectroscopic or photometric redshifts. Since we do not use both photometric redshift and spectroscopic redshift for each galaxy, physical quantities of individual galaxies such as stellar masses or star formation rates are not available; however, an advantage of our approach is that we can exclude any uncertainties associated with photometric redshift measurements, which may also introduce uncertainties in the identification of cluster member galaxies. This statistical method has been used previously in the literature (e.g., Lin et al. 2004; Hansen et al. 2005; Loh & Strauss 2006). Here we describe our specific procedure. First we divide the CAMIRA cluster catalog into subsamples at different redshift bins, as shown in table 1. In each redshift bin, we have roughly ∼50–200 clusters. The galaxy distribution associated with the ith cluster and jth annulus can be written as   \begin{eqnarray} N_{ij}^{\rm in}(z_{\rm cl}) =\sum _k \Theta \left(|\Delta {\boldsymbol{\theta }}_{ik}|\chi _{{\rm cl},i}-r_{P,j}\right) \Theta \left(r_{P,j+1}-|\Delta {\boldsymbol{\theta }}_{ik}|\chi _{{\rm cl},i}\right),\nonumber\\ \end{eqnarray} (2)where the summation k runs over galaxies, χcl is the comoving distance to the cluster redshift zcl, $$\Delta {\boldsymbol{\theta }}_{ik}$$ is the sky separation of the kth galaxy with respect to the ith cluster center, and Θ is a Heaviside step function. We consider the comoving radial distance from the cluster center rP in the range from 0.1 to 5 h−1 Mpc (comoving), which is divided into 15 logarithmically uniform bins. After stacking over all clusters within each redshift bin, we have   $$N_j^{\rm in}(r_P) = \sum _{i} N_{ij}^{\rm in}.$$ (3)This number $$N_j^{\rm in}$$ includes not only cluster member galaxies but also foreground and background galaxies along the line of sight. In order to remove these foreground and background galaxies, we assume that the galaxy number distribution outside the cluster region defined by $$r=|\Delta {\boldsymbol{\theta }}_{ik}|\chi _{{\rm cl},i}>5\, h^{-1}\:{\rm Mpc}$$ represents the distribution of the foreground and background galaxy population well. Although the distribution may differ field-by-field due to both inhomogeneous observing conditions and large-scale structure of the Universe, such local variation of the galaxy number distribution is expected to be averaged out after stacking many clusters at different positions on the sky, as long as the sky coverage of the survey is sufficiently large (e.g., Goto et al. 2003). The galaxy number outside the cluster region Nout for each redshift bin is estimated by using all galaxies that are located at r > 5 h−1 Mpc for all the clusters in the redshift bin. The number of galaxies is rescaled by the area before subtraction. We then subtract the contamination by foreground and background galaxies using Nout estimated above. We measure the areas occupied by the galaxies outside the cluster regions by counting the number of randoms in the random catalog (Coupon et al. 2018). The random catalog is created based on the pixel-based information and inherits most of the photometric flags on the object images. One can find the corresponding version of the random catalog to the object catalog in the same data release site. The random catalog takes account of both the selection criteria described in subsection 2.3 and masks. Now the foreground and background subtracted number of galaxies can be written as   $$N_j = N^{\rm in}_j - N^{\rm out} \frac{R^{\rm in}_j}{R^{\rm out}},$$ (4)where R is the number of randoms which is defined in exactly the same manner as in equations (2) and (3). 3.2 Color correction As shown in figure 2, red-sequence galaxies distribute in a narrow range of color which evolves with redshift. This means that galaxy colors evolve with redshift, even within the same redshift bin. In order to obtain accurate stacking results, including accurate separation of red and blue galaxies based on their colors, we apply a correction for the color evolution as a function of redshift before stacking many clusters to study the population of red and blue galaxies within each redshift bin. For clusters at z = z1, we derive the corrected color as $$C_{\rm corr}=C_{\rm obs}-C^{\rm th}(z_1)+C^{\rm th}_{\star }$$, where Cth denotes the theoretically derived galaxy color based on the stellar population synthesis model of Bruzual and Charlot (2003), with the calibration of colors using spectroscopic galaxies in the HSC survey (Oguri et al. 2018). $$C_{\star }^{\rm th}$$ is Cth at the median redshift z⋆ within the redshift bin. Furthermore, we correct the color gradient as a function of magnitude using z-band magnitudes. We fit the color–magnitude relation along the red sequence using the linear function as g(mz) = g⋆ + α(mz − mz,⋆), and correct the colors of all galaxies to the red-sequence zero-point (intercept of the linear relationship) estimated at the median magnitude of the cluster member galaxies, g⋆ ≡ g(mz, ⋆). To summarize, the corrected color of a galaxy with magnitude mz for a cluster at zcl stands for the color difference from that of the red sequence and is derived from the observed raw color Cobs as   $$C(m_z | z_{\rm cl}, m_{z,\star }) = C_{\rm obs} -C^{\rm th}(z_{\rm cl})+C^{\rm th}_{\star } - g(m_z) + g_{\star }.$$ (5)We show the color gradient in terms of redshift in figure 2 for CAMIRA member galaxies and the color gradient in terms of z-band magnitude in figure 3. Fig. 3. View largeDownload slide Color–magnitude diagrams (CMDs) that are derived by stacking photometric galaxies over all CAMIRA clusters. Color level stands for the number of galaxies in each cell after foreground and background galaxies are statistically subtracted (see the text for details). From left to right, we show CMDs for the cluster centric radii rP < 0.1, 0.5, and 1.0 h−1 Mpc. From top to bottom, the mean redshifts of the clusters are zcl = 0.18, 0.33, 0.55, and 0.85. Overlaid contours in each panel are the distribution of cluster member galaxies identified by CAMIRA. Solid lines show the slopes that minimize the scatter of CAMIRA member galaxies around the line, i.e., g(mz) correction. We define red and blue galaxies for each redshift bin by those above and below the dashed line (which is defined by a line 2σ below the solid line), respectively. Vertical dotted lines are the apparent magnitude cut corresponding to the rest frame Mz < −18.5. (Color online) Fig. 3. View largeDownload slide Color–magnitude diagrams (CMDs) that are derived by stacking photometric galaxies over all CAMIRA clusters. Color level stands for the number of galaxies in each cell after foreground and background galaxies are statistically subtracted (see the text for details). From left to right, we show CMDs for the cluster centric radii rP < 0.1, 0.5, and 1.0 h−1 Mpc. From top to bottom, the mean redshifts of the clusters are zcl = 0.18, 0.33, 0.55, and 0.85. Overlaid contours in each panel are the distribution of cluster member galaxies identified by CAMIRA. Solid lines show the slopes that minimize the scatter of CAMIRA member galaxies around the line, i.e., g(mz) correction. We define red and blue galaxies for each redshift bin by those above and below the dashed line (which is defined by a line 2σ below the solid line), respectively. Vertical dotted lines are the apparent magnitude cut corresponding to the rest frame Mz < −18.5. (Color online) 3.3 Definition of red and blue galaxies While there are a variety of definitions of red and blue galaxies in the literature, we introduce an empirical definition based on the observed data. Since CAMIRA cluster member galaxies represent the population of quiescent galaxies, the location of the CAMIRA cluster member galaxies in the CMD is well localized. This means that, at a given redshift, galaxies that have different star formation activities have different colors. We define blue galaxies as those that are located in the CMD 2σ away (on the bluer side) from the linear relation of the red-sequence obtained in subsection 3.2. Figure 3 shows a CMD after foreground and background subtraction in different redshifts (low-z to high-z from top to bottom) and different cluster centric radii (inner to outer from left to right). The color tilts are not corrected (but see figure 4 for a color-corrected diagram for z = 0.55 and rP < 0.5 h−1 Mpc). Horizontal dashed lines are the locations dividing the sample into red and blue galaxies. It is clearly seen that there are few blue galaxies at the inner regions of clusters and the number of blue galaxies increases with the cluster centric radius. We will see this more in detail in subsection 5. We note that if we carefully focus on the faint end of the CMD, the linear function obtained by CAMIRA member galaxies are slightly off from the peak of the red galaxy distribution. Unlike the CAMIRA member galaxies which have evolved to the red sequence, faint galaxies near the red-sequence track are still in the stage of star-forming and are in the transition phase from star-forming galaxies to quiescent galaxies. For the thorough investigation, we need to divide the cluster sample into finer mass bins, which our future paper will be devoted to. Fig. 4. View largeDownload slide Same as figure 3 for z = 0.55 and rP < 0.5 h−1 Mpc, but color is corrected according to equation (5). (Color online) Fig. 4. View largeDownload slide Same as figure 3 for z = 0.55 and rP < 0.5 h−1 Mpc, but color is corrected according to equation (5). (Color online) 4 Red-sequence at the faint end In this section, we study the red-sequence galaxies within cluster centric radius rP < 0.5 h−1 Mpc at the very faint end down to mz ∼ 24, which is enabled by our careful statistical subtraction of foreground and background galaxies. Specifically, we study how the scatter of the red-sequence changes as a function of magnitude. We model the color-corrected, foreground- and background-subtracted CMD distribution with the following double Gaussian (e.g., Hao et al. 2009)   \begin{eqnarray} n(C|m_z) & =& \frac{A_R(m_z) }{\sqrt{2\pi \sigma _R^2(m_z)}} \exp \left[ -\frac{(C-C_R)^2}{2\sigma _R^2(m_z)} \right] \nonumber \\ && +\, \frac{A_B(m_z)}{\sqrt{2\pi \sigma _B^2(m_z)}} \exp \left[ -\frac{(C-C_B)^2}{2\sigma _B^2(m_z)} \right] \end{eqnarray} (6)where the parameters Ax, σx, and Cx, with x being either R (red) or B (blue), are treated as free parameters. As we already corrected for the color tilt against the magnitude in subsection 3.2, the mean of the red component, CR, can well be described by a constant. For simplicity, we also assume that the blue component (CB) has constant mean, which is equivalent to assuming that the tilt of the color–magnitude relation for blue galaxies is the same as that for red galaxies. This assumption is reasonable because the blue galaxies do not have a tight relation with the mz but are rather broadly distributed and thus insensitive to the choice of color correction; as far as the tilt correction is linear, the color correction simply changes the CB and σB at each mz bin but it does not affect the estimate of σR that we are interested in. We divide the CMD into several magnitude bins, and estimate the values of σR for each magnitude bin with the Markov Chain Monte Carlo method by keeping other parameters free but fixing the CR to its corrected value obtained in subsection 3.2. Figure 5 shows the best-fitting scatter parameter σR as a function of magnitude, for different cluster redshifts. As discussed above, we use different colors for clusters at different redshift, such that these colors refer to approximately the same color in the cluster rest frame. At the very faint end, we need to take account of the scatter associated with the photometric error, which has a significant contribution to the observed scatter at mz ∼ 24. As the intrinsic scatter is not correlated with the photometric error, we can separate their contributions as   $$\sigma _{\rm obs}^2 = \sigma _{\rm photo}^2 + \sigma _{\rm int}^2,$$ (7)where σobs, σphoto, and σint are the observed scatter, the scatter due to the photometric error, and the intrinsic scatter of the red-sequence, respectively. We find that the intrinsic scatter of the red-sequence galaxies after subtracting the photometric error is almost constant over a wide range of magnitudes. We also find that there is no significant redshift evolution of the scatter, which is consistent with previous work which used smaller samples of clusters (e.g., Cerulo et al. 2016; Hennig et al. 2017). The figure suggests a slight decrease of the scatter at the faint end, but being the photometric error large at faint magnitudes, the intrinsic scatter is likely to be underestimated. Over most of the magnitude range, however, the photometric error is much smaller than the intrinsic scatter, which indicates that our result is robust against the photometric error. Fig. 5. View largeDownload slide Scatter of colors of red-sequence galaxies as a function of z-band magnitude. Results are shown for different redshift bins. Shaded regions show observed scatter, whereas dashed lines show the estimated intrinsic scatter after subtracting the scatter due to the photometric error. which is shown by filled circles with error bars. The circles and error bars are the median and one-sigma scatter of the photometric error in each magnitude bin. (Color online) Fig. 5. View largeDownload slide Scatter of colors of red-sequence galaxies as a function of z-band magnitude. Results are shown for different redshift bins. Shaded regions show observed scatter, whereas dashed lines show the estimated intrinsic scatter after subtracting the scatter due to the photometric error. which is shown by filled circles with error bars. The circles and error bars are the median and one-sigma scatter of the photometric error in each magnitude bin. (Color online) 5 Cluster profile and fraction of red galaxies 5.1 Cluster profile Given the timescale for the evolution of galaxies, tracking the redshift evolution of the number-density profiles for red and blue components can help us understand the dynamical history of the formation of galaxy clusters. The radial mass density profile of dark matter halos has long been thought to have a long tail that goes as ρ ∝ r−3 (Navarro et al. 1996). With such a profile, the total enclosed mass of a cluster diverges logarithmically, and the total mass associated with the halo depends upon the arbitrary boundary imposed on the halo. The splashback radius, marked by the apocenter of the recently infalling material, provides a clear physical boundary for the halo and can be used to identify the edges of dark matter halos (Diemer & Kravtsov 2014; More et al. 2015). The splashback radius manifests itself as a sharp drop in the matter density at its location (Diemer & Kravtsov 2014; Adhikari et al. 2014). A dark matter profile with such a density jump can be modeled with an inner universal profile multiplied by a transition function and an outer profile which represents the so-called two-halo contribution (Diemer & Kravtsov 2014). This can be explicitly written as   \begin{eqnarray} \rho (r) = \rho ^{\rm in}(r) \left[ 1+\left( \frac{r}{r_t} \right)^\beta \right]^{-\gamma /\beta } + \rho ^{\rm out}(r), \end{eqnarray} (8)  \begin{eqnarray} \rho ^{\rm out}(r) = \rho _m \left[ b_e \left( \frac{r}{5R_{200}}\right)^{-s_e} + 1 \right], \end{eqnarray} (9)where rt, γ, and β denote the location of the dip in the profile, the steepness of the dip, and how rapidly the slope changes, respectively. They all characterize the transition between inner and outer profiles. For the outer profile, ρm, be, and se represent the overall normalization, the relative normalization of the power-law profile, and the index of the power law, respectively. Diemer and Kravtsov (2014) express rt/R200m as a function of the accretion rate Γ, rt/R200m = [0.62 + 1.18 exp (−2Γ/3)], but in this paper we keep rt as a free parameter since we do not have a reliable estimate of the either the R200m or the accretion rate of our optically-selected clusters. We use the NFW (Navarro–Frenk–White; Navarro et al. 1996) profile to describe the inner profile,   $$\rho ^{\rm NFW}(r) = \frac{\rho _{\rm s}}{(r/r_{\rm s}) (1+r/r_{\rm s})^2},$$ (10)where rs and ρs denote the transition scale of slope from −1 to −3 and overall normalization, respectively. While we use rt for fitting observed radial profiles, following More, Diemer, and Kravtsov (2015) we define the splashback radius, Rsp, as the radius where the radial profile attains its steepest slope. As in More et al. (2016), we allow β and γ to take the values with log β = log 6 ± 0.2 and log γ = log 4 ± 0.2. The profiles of equations (8), (9), and (10) are numerically integrated along the line of sight to project to the two-dimensional sky. Figure 6 shows the radial number density profiles for red and blue components at redshifts z = 0.18, 0.33, 0.55, and 0.85. The covariance matrices are estimated from the jackknife resampling as   \begin{eqnarray} \widehat{{\rm Cov}_{ij}} &=& \frac{N_{\rm a}-1}{N_{\rm a}}\sum _{\rm a} \left[w_{\rm a}^{\rm cg}\left(r_{{\rm p}, i}\right) - \overline{w^{\rm cg}}\left(r_{{\rm p}, i}\right)\right] \nonumber \\ &&\times \left[w_{\rm a}^{\rm cg}\left(r_{{\rm p}, j}\right) - \overline{w^{\rm cg}} \left(r_{{\rm p}, j}\right)\right], \end{eqnarray} (11)where $$\overline{w^{\rm cg}}$$ is the arithmetic mean of $$w^{\rm cg}_{\rm a}$$. We divide the entire area into 35 rectangular regions with a side of 5.°0. That scale corresponds to the comoving angular separation of 25 h−1 Mpc at z = 0.1 which is sufficiently larger than the scale of our interest and includes more than one cluster in all sub-divided regions. We first fit to an NFW profile by using data below 2 h−1 Mpc. Best-fitting scale radii are summarized in table 2. We find notable differences of concentrations between two populations. Red galaxies are more concentrated toward the cluster center (i.e., smaller rs) and blue galaxies are less concentrated (i.e., larger rs). The difference in the concentration can be accounted for by the merger of clusters as discussed in e.g., Okamoto and Nagashima (2003, 2001). Another possible explanation is that red galaxies at the same luminosity live in more massive halos than their blue counterparts (Mandelbaum et al. 2006; More et al. 2011) and have experienced more dynamical friction so that they are concentrated toward the cluster center as we see in the discussion below. If we focus on the redshift evolution, it is seen that the overall profile tends to be more concentrated for lower redshifts. While the evolution of the concentration of red galaxies is subtle, blue galaxies evolve rapidly from z = 0.5 to 0.3. Fig. 6. View largeDownload slide Radial number density profiles of galaxies around clusters. From left to right, the mean redshifts of clusters are 0.18, 0.33, 0.55, and 0.85. Red and blue symbols show profiles for red and blue member galaxies. Gray symbols show profiles for all galaxies. Dashed lines are best-fitting models of the projected NFW profile. Solid lines show the best-fitting Diemer and Kravtsov (2014) model. Vertical solid and dashed lines indicate the best-fitting scale radii rs for red and blue galaxies, respectively. Bottom panels show the slope of the best-fitting profiles with vertical lines being the best-fitting value of rt, which can be compared with the splashback radius, Rsp. (Color online) Fig. 6. View largeDownload slide Radial number density profiles of galaxies around clusters. From left to right, the mean redshifts of clusters are 0.18, 0.33, 0.55, and 0.85. Red and blue symbols show profiles for red and blue member galaxies. Gray symbols show profiles for all galaxies. Dashed lines are best-fitting models of the projected NFW profile. Solid lines show the best-fitting Diemer and Kravtsov (2014) model. Vertical solid and dashed lines indicate the best-fitting scale radii rs for red and blue galaxies, respectively. Bottom panels show the slope of the best-fitting profiles with vertical lines being the best-fitting value of rt, which can be compared with the splashback radius, Rsp. (Color online) Table 2. Best-fitting values of cluster profile parameters.* z  Sample  rs  rt  Rsp  ΔAIC  ΔBIC    Red  0.50 ± 0.01  1.20 ± 0.13  1.21 ± 0.70  −13.9  −7.98  0.18  Blue  1.14 ± 0.03  3.63 ± 1.07  2.66 ± 1.29  −18.0  −12.1    All  0.57 ± 0.01  1.11 ± 0.13  1.21 ± 0.66  −16.1  −10.2    Red  0.51 ± 0.01  1.34 ± 0.16  1.26 ± 0.66  −14.7  −8.8  0.33  Blue  8.61<  1.43 ± 0.18  1.52 ± 0.58  −16.6  −10.7    All  0.91 ± 0.02  1.53 ± 0.22  1.15 ± 0.64  −16.2  −10.4    Red  0.54 ± 0.01  1.20 ± 0.10  1.26 ± 0.73  −14.9  −9.0  0.55  Blue  8.42<  1.20 ± 0.18  1.26 ± 0.54  −16.4  −10.5    All  1.11 ± 0.03  1.12 ± 0.12  1.10 ± 0.52  −14.4  −8.5    Red  0.77 ± 0.03  1.40 ± 0.13  1.32 ± 0.79  −15.1  −9.2  0.85  Blue  7.73<  1.75 ± 0.47  1.67 ± 0.54  −17.4  −11.5    All  1.72 ± 0.12  1.29 ± 0.27  1.21 ± 0.39  −17.1  −11.2  z  Sample  rs  rt  Rsp  ΔAIC  ΔBIC    Red  0.50 ± 0.01  1.20 ± 0.13  1.21 ± 0.70  −13.9  −7.98  0.18  Blue  1.14 ± 0.03  3.63 ± 1.07  2.66 ± 1.29  −18.0  −12.1    All  0.57 ± 0.01  1.11 ± 0.13  1.21 ± 0.66  −16.1  −10.2    Red  0.51 ± 0.01  1.34 ± 0.16  1.26 ± 0.66  −14.7  −8.8  0.33  Blue  8.61<  1.43 ± 0.18  1.52 ± 0.58  −16.6  −10.7    All  0.91 ± 0.02  1.53 ± 0.22  1.15 ± 0.64  −16.2  −10.4    Red  0.54 ± 0.01  1.20 ± 0.10  1.26 ± 0.73  −14.9  −9.0  0.55  Blue  8.42<  1.20 ± 0.18  1.26 ± 0.54  −16.4  −10.5    All  1.11 ± 0.03  1.12 ± 0.12  1.10 ± 0.52  −14.4  −8.5    Red  0.77 ± 0.03  1.40 ± 0.13  1.32 ± 0.79  −15.1  −9.2  0.85  Blue  7.73<  1.75 ± 0.47  1.67 ± 0.54  −17.4  −11.5    All  1.72 ± 0.12  1.29 ± 0.27  1.21 ± 0.39  −17.1  −11.2  *For each redshift range, the top, middle and bottom rows are for red galaxy, blue galaxy and all galaxy samples, respectively. All the values are in comoving h−1 Mpc. Also shown are difference of information criteria. A minus value means that the NFW profile is favored over the density jump model. View Large Next we fit all the data to the full profile of equation (8), keeping rs and ρs fixed to their best-fitting values from the simple NFW profile fitting to simplify the degeneracies inherent in the fitting procedure. The best-fitting curves are presented with solid lines in figure 6. We also show the logarithmic slope of the profile, which can be used to define the splashback radius Rsp as a local minimum of the slope. We compare the splashback radii Rsp, rt, and rs in table 2. As shown in More, Diemer, and Kravtsov (2015), there is a tight relation between the mass accretion rate of clusters and the normalized splashback radius Rsp/R200m obtained by stacking clusters at each redshift. Given the fact that the richness limit of our cluster sample approximately corresponds to a constant mass limit of M200m > 1014 h−1 M⊙ over the whole redshift range (Oguri et al. 2018), we find that the splashback radii from our fits roughly correspond to Rsp ∼ R200m. This is broadly consistent with More et al. (2016) in which splashback radii were derived for SDSS clusters to argue that the observed splashback radii are smaller than the standard cold dark matter model prediction, implying a faster mass accretion than the standard model. However, more careful estimates of the mass of these clusters using the weak gravitational lensing signal ought to be performed before doing a more quantitative comparison to More et al. (2016), as well as quantifying the redshift evolution. We will explore this in the near future. We compare the goodness-of-fit between NFW and full profile of Diemer and Kravtsov (2014) by computing two different criteria; the Akaike Information Criteria (AIC) corrected for the finite data size and the Bayesian Information Criteria (BIC). They are defined as   \begin{eqnarray} \displaystyle {\rm AIC} &=& -2\ln ({\mathcal {L}}) + 2p + \frac{2p(p+1)}{N-p-1} \end{eqnarray} (12)  \begin{eqnarray} {\rm BIC} &=& -2\ln ({\mathcal {L}}) + \ln (N)p, \end{eqnarray} (13)where $${\mathcal {L}}$$ is the likelihood and p and N are number of parameters and data, respectively; (N, p) for NFW is (12, 2) and (12, 6) for the density jump model. We compare the information criteria for the two profile fittings, and find that the density jump model is disfavored as summarized in table 2. As a cautionary note, we also mention that selection effects in optical cluster finding can significantly complicate the inference of the splashback radius from observations. Optical clusters are more likely to have their major axes oriented along the line-of-sight, which breaks the spherical symmetry assumption involved in the inference of the splashback radius (see e.g., Busch & White 2017). In addition, degeneracies related to cluster mis-centering can also reduce the significance of the detection of the splashback radius (Baxter et al. 2017). The values of the splashback radii inferred from optical clusters should therefore be carefully compared to expectations, a topic we will focus on in the near future. For clusters at z > 0.5, we find a slight decline of radial profiles in the central region, r < 0.2 h−1 Mpc. This might reflect the mis-centering of the optically-selected clusters, which have been inferred in comparison with the X-ray profile (Oguri et al. 2018). It may be more difficult to identify the center of clusters correctly at higher redshift simply because in crowded regions, like cluster centers, angular separations of neighboring galaxies are smaller at higher redshifts, given the same physical scale. This is mainly due to the fact that the pipeline fails to deblend galaxies in crowded regions. We leave the effect of mis-centering for future work, after the weak lensing measurement of the CAMIRA cluster sample becomes available. 5.2 Red fraction as a function of redshift We derive the red galaxy fraction by summing up the number of red and blue galaxies in each radial bin out to the maximum comoving distances, for which we adopt 0.2, 0.5, and 1.0 h−1 Mpc. Figure 7 shows the evolution of fractions of red galaxies as a function of redshift, for three different maximum distances. We can clearly see the evolution of the red fraction over the cosmological timescale, from z = 1.1 to 0.1, such that the red fraction increases at lower redshift. This qualitative trend is consistent with Hennig et al. (2017) and Jian et al. (2018), while Loh et al. (2008) reported steeper evolution. The red fraction significantly increases with decreasing maximum distance from the cluster center, which is due to the different radial number density profiles between red and blue galaxies. We note that there are gaps in the fraction of red galaxies at z = 0.4 and z = 0.75 because we have used the different combination of filters to define red and blue galaxies. This implies that the global evolution of the red fraction over the redshifts 0.1 < z < 1.1 is subject to the choice of colors; however, within the redshift range in which we use the same filter combination, we observe apparent decreases of the red fraction. However, we note that the red fraction at higher redshift bins, 0.7 < z, are almost flat or slightly increasing. As we will discuss it at the end of subsection 5.3, this trend is partly due to our bad photometry at crowded regions like cluster centers. Fig. 7. View largeDownload slide Fraction of the red galaxies as a function of redshift and the maximum distance from the cluster center for counting up the galaxies. The x-axis for different rP is slightly shifted for visual purposes. (Color online) Fig. 7. View largeDownload slide Fraction of the red galaxies as a function of redshift and the maximum distance from the cluster center for counting up the galaxies. The x-axis for different rP is slightly shifted for visual purposes. (Color online) Figure 8 shows the fraction of red galaxies as a function of projected cluster-centric radius rP. Symbols are observed points and solid lines are predictions from the semi-analytical model described in subsection 5.3. The fractions of red galaxies are high, ∼0.6–1.0 in the inner regions (rP < 0.3 h−1 Mpc), decreasing to ∼0.2–0.4 in the outer regions (rP > 1 h−1 Mpc). On the intermediate scale in between the inner and outer regions, the fraction of red galaxies is gradually decreasing. Although the absolute value of the observed fraction is slightly higher than predicted by the semi-analytical model, the declining slopes are in good agreement with the model. Fig. 8. View largeDownload slide Radial profile of the fraction of red galaxies as a function of projected cluster-centric radius at four different redshifts. Symbols are the observed data points and solid lines are the prediction from the semi-analytical model described in subsection 5.3. (Color online) Fig. 8. View largeDownload slide Radial profile of the fraction of red galaxies as a function of projected cluster-centric radius at four different redshifts. Symbols are the observed data points and solid lines are the prediction from the semi-analytical model described in subsection 5.3. (Color online) 5.3 Comparison with a semi-analytical model It is important to check the consistency of our results with theoretical models of galaxy formation. We compare our results with a semi-analytical model of galaxy formation, ν2GC (Makiya et al. 2016). Semi-analytical models have an advantage over mock galaxy catalogs based on the halo occupation distribution technique in that semi-analytical models are constructed from physically motivated prescriptions of several astrophysical processes which, in comparison with observations, will lead to better understanding of the build-up of cluster galaxies. They also contain physical properties of galaxies, such as galaxy stellar and gas masses and star formation rates. This information may also help reveal physical processes affecting the evolution of cluster galaxies. We examine the spatial distribution of galaxies in ν2GC. In ν2GC, we construct merger trees of dark matter halos using cosmological N-body simulations (Ishiyama et al. 2015) with the Planck cosmology (Planck Collaboration 2014). The simulation box is 280 h−1 Mpc on a side containing 20483 particles, corresponding to a particle mass of 2.2 × 108 h−1 M⊙. The semi-analytical model includes the main physical processes involved in galaxy formation: formation and evolution of dark matter halos; radiative gas cooling and disc formation in dark matter halos; star formation, supernova feedback and chemical enrichment; galaxy mergers; and feedback from active galactic nuclei. The model is tuned to fit the luminosity functions of local galaxies (Driver et al. 2012) and the mass function of neutral hydrogen (Martin et al. 2010). The model well reproduces observational local scaling relations such as the Tully–Fisher relation and the size–magnitude relation of spiral galaxies (Courteau et al. 2007). We use ν2GC results to create mock galaxy catalogs. Rest-frame and apparent magnitudes of galaxies are estimated in the same filter as used in the HSC survey (Kawanomoto et al. 2017). We apply the same magnitude cut of the HSC (Mz < −18.5 in rest frame) to the simulated galaxies. In the analysis in this paper, we extract galaxy samples at different redshifts, residing in dark matter halos with masses greater than 1014 M⊙. We regard those galaxies as cluster galaxies. We have ∼175 simulated clusters at z = 1.1 and ∼1000 clusters at z = 0.13 which are defined in the same realization. Since the mock catalogs do not perfectly reproduce the colors of galaxies, we cannot apply the selection condition that is applied to the HSC data to the mock catalogs. We therefore re-define the selection criterion for red and blue galaxies in the mock galaxy catalogs. To define red and blue galaxy populations, we determine the red-sequence using a subsample of the mock galaxies as follows. We perform linear regression to fit the slope α and zero-point β of the following equation to describe the red sequence   $$C = \alpha \ m_z + \beta ,$$ (14)where C is the color corresponding to the cluster redshift, e.g., g − r at redshift [0.1–0.2]. To reduce contaminations by blue galaxies, we only fit to galaxies with specific star formation rates (sSFRs) log10(sSFR/Gyr−1) ≤ −1 and with distances from the cluster center d ≤ 0.1 Mpc h−1. These conditions are reasonable for extracting red-sequence galaxies which correspond to the CAMIRA-identified red-sequence. Finally, in exactly the same manner as in the observation, we define galaxies redder than the C − 2σ line on the CMD as red galaxies, where σ is the standard deviation of the distribution of the galaxies used in the fitting. Figure 9 shows radial profiles of cluster member galaxies for different redshift bins and different populations identified in the simulation. We see clearly that blue galaxies are more diffuse and red galaxies are more concentrated, which is consistent with our observational results. Fig. 9. View largeDownload slide Same as figure 6 but obtained from simulations with a semi-analytical model. (Color online) Fig. 9. View largeDownload slide Same as figure 6 but obtained from simulations with a semi-analytical model. (Color online) Figure 10 shows the redshift evolution of the red fraction in the simulation with the same binning of distance from the cluster center. We see a clear decrease of the red fraction with redshift and the result shows a reasonable agreement with the observation except for the two highest-redshift bins. The simulated results show a monotonic decrease of the red fraction at the highest redshift ranges, but our observational results show a slight increase. The monotonic decrease of the red fraction to 0.5 at z = 1 is consistent with the results of Hennig et al. (2017), and therefore the slight increase seen in our data may not be accounted for by the difference in the depth of the sample, (our sample is ∼1 mag deeper in the z band) or the different filter combination used to define the red and blue populations. We do not make strong conclusions about the source of this discrepancy in this paper, but it may be partly due to the bad photometry in crowded regions, which significantly affects the color of galaxies in clusters. We will revisit the issue and address it in the future work once the photometry of HSC in the crowded regions is improved. Fig. 10. View largeDownload slide Same as figure 7 but obtained from simulations with a semi-analytical model. (Color online) Fig. 10. View largeDownload slide Same as figure 7 but obtained from simulations with a semi-analytical model. (Color online) 6 Summary In this paper, we have used the HSC S16A internal data release galaxy sample over ∼230 deg2 to explore the properties of cluster galaxies over wide redshift and magnitude ranges. Clusters are identified by the red-sequence cluster finding method CAMIRA (Oguri 2014; Oguri et al. 2018). Thanks to the powerful capability of the Subaru telescope to collect light and the good sensitivity of the HSC detector, we can study faint cluster galaxies down to the 24th magnitude in the z band. This sample is ∼1 mag deeper than the cluster sample of Hennig et al. (2017) which reaches m⋆ + 1.2 ∼ 23.2 in the Dark Energy Survey (DES) z band at z = 1. Together with a reliable CAMIRA cluster catalog out to z = 1.1, the excellent HSC data allows us to continuously track the evolution history of cluster galaxies from z = 1.1 to the present. We have used the stacked CMD to divide red and blue galaxies into clusters. We have statistically subtracted background and foreground galaxies after area corrections using the well-defined random catalog, which is also available from the HSC data base. After subtracting the foreground and background galaxies, color–magnitude relations for red galaxies (red-sequence) and blue clouds are clearly detected over wide ranges in redshift and magnitude. We have used these CMDs for defining red and blue galaxies, studying the tightness of the red-sequence down to very faint magnitudes, and the radial number density profiles of red and blue galaxies. Our results are summarized as follows. Red galaxies in clusters follow a clear linear relation in the CMD down to the HSC completeness limit for all redshifts. However, we observe a slight offset of the red populations in the cluster from the linear relation determined using the CAMIRA member galaxies. This may be partly due to our sample selection, i.e., a constant mass cut over all redshift ranges, and we will revisit it once the cluster mass is measured well with the weak lensing. We have measured the intrinsic scatter of the red-sequence as a function of the observed z-band magnitude and cluster redshift. We have found that the intrinsic scatter is almost constant over wide range of magnitudes. The intrinsic scatter shows little evolution with redshift. Red galaxies are more concentrated toward the cluster center compared with blue galaxies. We fitted the cluster member radial profile at r < 1.0 h−1 Mpc to an NFW profile, and find the transition scale rs is significantly smaller for red galaxies than for blue galaxies. Given that the cluster sample has approximately constant mean mass over different redshifts (Oguri et al. 2018), the mildly decreasing rs with redshift implies that the galaxy profiles in clusters become less concentrated at higher redshift. We note, however, that it is important to derive the virial mass of the clusters independently; this will soon be provided by the HSC weak lensing analysis (Mandelbaum et al. 2018). Special care is required that the mass profile measured by weak lensing is for dark matter and this should be different from the profile of member galaxies. We fitted the radial number density profiles with the density jump model of Diemer and Kravtsov (2014), and find that the splashback radius Rsp defined by the minimum of the logarithmic slope is almost constant over the redshift range. However, given the large statistical uncertainties, we do not detect the splashback radius for our current data set. The fraction of red galaxies is not only a strong function of the distance from the cluster center, but also exhibits a moderate decrease with increasing redshift. We note that the estimated red fraction shows a slight discontinuity at the redshift where the red and blue galaxies are defined in different combinations of colors, i.e., z ∼ 0.4 and z ∼ 0.7. This discontinuity might reflect that our definition of red and blue galaxies are not optimal near the transition redshifts because the redshift of the 4000 Å break mismatches with the filter response functions of a given combination of the color. We note that the same discontinuity is also seen in the simulations. We also compared our results with semi-analytical model predictions. We find that the observed cluster profiles and the redshift evolution of the red fraction are broadly consistent with the semi-analytical model prediction. Further studies for more quantitative comparisons are important. The total mass and mass profile of the CAMIRA clusters can be measured by stacked weak lensing. With the help of the mass–richness relation by the forward modeling (R. Murata et al. in preparation), this will allows us to explore cluster physical quantities, such as virial radius, mass accretion rate, and the mass dependence of those quantities, in more detail. We will revisit this in our future work. Acknowledgements We thank the anonymous referee for providing useful comments. AN is supported in part by MEXT KAKENHI Grant Number 16H01096. This work was supported in part by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan, MEXT as “Priority Issue on Post-K computer” (Elucidation of the Fundamental Laws and Evolution of the Universe) and JICFuS, and JSPS KAKENHI Grant Number 26800093 and 15H05892. SM is supported by the Japan Society for Promotion of Science grants JP15K17600 and JP16H01089. This work was supported in part by MEXT KAKENHI Grant Number 17K14273 (TN). HM is supported by the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. The Hyper Suprime-Cam (HSC) collaboration includes the astronomical communities of Japan and Taiwan, and Princeton University. The HSC instrumentation and software were developed by the National Astronomical Observatory of Japan (NAOJ), the Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), the University of Tokyo, the High Energy Accelerator Research Organization (KEK), the Academia Sinica Institute for Astronomy and Astrophysics in Taiwan (ASIAA), and Princeton University. Funding was contributed by the FIRST program from Japanese Cabinet Office, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), the Japan Society for the Promotion of Science (JSPS), Japan Science and Technology Agency (JST), the Toray Science Foundation, NAOJ, Kavli IPMU, KEK, ASIAA, and Princeton University. The Pan-STARRS1 Surveys (PS1) have been made possible through contributions of the Institute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, Queen’s University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation under Grant No. AST-1238877, the University of Maryland, and Eotvos Lorand University (ELTE). 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