GALAXIES IN X-RAY SELECTED CLUSTERS AND GROUPS IN DARK ENERGY SURVEY DATA. I. STELLAR MASS GROWTH OF BRIGHT CENTRAL GALAXIES SINCE z ∼ 1.2

The DES is a ground-based optical survey that uses the wide-field DECam camera (Flaugher et al. 2015) mounted on the 4 m Blanco telescope to image 5000 deg² of the southern hemisphere sky (Sánchez 2010). The paper is based on 200 deg² DES Science Verification (SV) data. This data set was taken during the 2012B observing season before the main survey (Diehl et al. 2014) began. A large fraction of the SV data have full DES imaging depth (Lin et al. 2013a) and are processed with the official DES data processing pipeline (Mohr et al. 2012). A more detailed review can be found in Sánchez et al. (2014).

The XCS serendipitously searches for galaxy cluster (and group) candidates in the XMM-Newton archive (Lloyd-Davies et al. 2011; Mehrtens et al. 2012; Viana et al. 2013). The cluster candidates are then verified with optical/infrared imaging data, which confirm the existence of red sequence galaxies. Photometric redshifts of the confirmed clusters are also subsequently derived with the red sequence locus. Using DES SV data, C. J. Miller et al. (2015, in preparation; referred to as M15 in the rest of the paper) have identified ∼170 X-ray selected clusters and groups from XCS. M15 also measures their photometric redshifts and verifies the measurements against archival spectroscopic redshifts.⁴⁶ In this paper, we use a sub-sample from M15 that consists of 106 clusters and groups with masses above 3.0 × 10¹³ M_⊙. These clusters and groups are all referred to as "clusters" in the rest of the paper. In Figure 1, we show their masses and redshift distributions. For comparison, we also show the mass and redshift distribution of the cluster sample used in a similar study (Lidman et al. 2012). Our sample covers a lower mass range, and appears to be more evenly distributed in the redshift-mass space.

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The cluster mass (M₂₀₀, the mass inside a three-dimensional (3D) aperture within which the averaged matter density is 200 times the critical density) is either derived with X-ray temperature or X-ray luminosity, using a lensing calibrated M − T relation (Kettula et al. 2013). Because XCS is a serendipitous survey, not all the clusters have high-quality X-ray temperature measurements. For these clusters, we derive their masses from X-ray luminosity. Further details about this procedure and about the mass uncertainties can be found in Appendix A.

We note that a handful of the clusters do not seem to have significant galaxy overdensity associated with them. It is possible that our sample contains spurious clusters that originate from foreground/background X-ray contaminations. We have reanalyzed our analysis after removing eight clusters that are not associated with significant galaxy overdensity. The results are consistent with those presented in this paper within 0.5σ. Given that these eight clusters are in the low-mass range (generally below 10¹⁴ M_⊙), removing them may introduce an artificial mass selection effect. We therefore do not attempt to do so in this paper. We also note that other factors, including BCG photometry measurement and cluster mass scaling relations at the low-mass end (see discussion in Sections 4.2 and 5.2), have a bigger effect on our results than the possible spurious clusters in the sample.

The BCGs are selected through visually examining the DES optical images, the X-ray emission contours, and the galaxy color–magnitude diagram. In this procedure, we aim to select a bright, extended, elliptical galaxy close to the X-ray emission center, which also roughly lies on the cluster red sequence. If there exist several red, equally bright and extended ellipticals close to the X-ray center, we select the nearest one. We did not notice a proper BCG candidate with a blue color.

We check our visual BCG selection against the central galaxy choice of a preliminary version of the DES SV RedMaPPer cluster catalog (see the algorithm in Rykoff et al. 2014). Out of the 106 XCS clusters and groups, 64 are matched to RedMaPPer clusters and the majority (61) identify the same BCG. In the cases where we disagree with the BCG, we choose the brighter, more extended galaxy closest to the X-ray center while RedMaPPer selects a galaxy further away. The other 42 non-matches are caused by the different data coverage, redshift limit, and mass selection of the two catalogs: the RedMaPPer catalog employs only a subset of the SV data to achieve relatively uniform depth for selecting rich clusters below redshift 0.9.

In Figure 2, we show the distance distribution between the selected BCGs and the X-ray emission centers. Half of the BCGs are separated by less than 0.07 Mpc (comparable to Lin & Mohr 2004) from the X-ray emission centers, regardless of the redshifts of the clusters.

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Measuring BCG photometry is among the most controversial topics in BCG studies. In Appendix B, we discuss complications and possible biases associated with Petrosian magnitude, Kron magnitude, profile fitting magnitude, and aperture magnitude, with extended details. We use magnitude measured with circular apertures of 15, 32, 50, and 60 kpc radii. The main results are derived with the 32 kpc radius apertures, considering the BCG half light radius measurements in Stott et al. (2011). Detailed rationalization about this choice and description about our measurement procedure can also be found in Appendix B.

We correct for galactic extinction using the stellar locus regression method (High et al. 2009; Kelly et al. 2014; E. Rykoff et al. 2015, in preparation), and compute BCG luminosities and stellar masses using the stellar population modeling technique. We employ a Chabrier (2003) Initial Mass Function (IMF) and the Conroy et al. (2009) and Conroy & Gunn (2010) simple stellar population (SSP) models to construct stellar population templates, and select templates according to BCG DES g, r, i, z photometry. We use the best-fit model to compute the K-correction factor and the mass-to-light ratio. We evaluate uncertainties associated with BCG apparent magnitude, redshift, and BCG mass-to-light ratio. Further details about these procedures can be found in Appendix C.

Since BCG luminosity and stellar mass are known to be correlated with cluster mass, the comparison between the observations and simulations needs to be made between clusters of similar masses. For each BCG in our sample, we compare it to a simulation subsample of 100 BCGs hosted by clusters of similar masses and redshifts. The simulation data are selected with the following procedure.

• 1.  
  Identify simulation clusters with redshifts closest to that of the XCS cluster. Ideally, we would have identified a cluster sub-sample with a redshift distribution matching the redshift uncertainty of the XCS cluster, but this is not possible since simulations are stored at discrete redshifts.
• 2.  
  Select from the redshift sub-sample of 100 clusters with a posterior mass distribution (log-normal) matching the mass uncertainty of the XCS cluster. Note that we are not using the cluster mass function as a prior. Application of this prior leads to sampling clusters that are ∼0.1 dex less massive, but leave the conclusions unchanged.

Note that in the above procedure, we are not considering additional cluster properties beyond M₂₀₀ and redshift. There is emerging evidence that X-ray selected clusters may be biased in terms of cluster concentration distribution (Rasia et al. 2013), but it is unclear how the bias would affect a BCG formation study. We also do not consider the Eddington bias associated with L_X. The M₂₀₀ of the lowest L_X/T_X systems are derived with T_X. Future studies yielding higher precision on BCG growth may wish to take these selection effects into consideration.

In Figure 3, we show the redshift and the mass distribution of the XCS clusters together with the resampled DL07 simulation clusters. The above procedure produces a simulation sub-sample that resembles the probability distribution of the XCS sample.

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We directly compute the relative luminosity and stellar mass difference between the observed and simulated BCGs, as shown in Figure 4.⁴⁷

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Note that the differences between the observed and simulated BCGs change with redshift. The effect suggests that the observed BCGs do not grow as rapidly as in DL07—a different redshift evolution history in the observation. We fit the differences with a linear dependence on lookback time: if the redshift evolution of the observed BCGs is consistent with that in DL07, the slope of the linear fit shall be 0. This null hypothesis is not favored.

In Figure 4, we show the linear fitting result with blue bands that encompass the 1σ uncertainties. The luminosity redshift evolution in the observation is different from the simulation with a 2.5σ significance (0.028 ± 0.011). The significance from the stellar mass comparison is lower at 1.3σ (0.015 ± 0.012), but BCG stellar mass is less certain (recall that it requires a choice for the mass-to-light ratio) and therefore the result is noisier.

The redshift evolution difference shows that the observed BCGs become increasingly under-massive/under-luminous at decreasing redshift compared to DL07 (compare the result to Lidman et al. 2012; Lin et al. 2013b; Oliva-Altamirano et al. 2014). At the lowest redshift bin (z ∼ 0.1) in Figure 4, the observed BCGs appear to be 0.1–0.2 dex⁴⁸ under-massive/under-luminous as a result of a different redshift evolution history.

Arguably, the above statement relies on a fitting function connecting the difference between the observed and simulated BCG properties to redshift. The significance level of this statement depends on the exact form of the fitting function. In Sections 4 and 5, we present stronger evidence for this statement through modeling the BCG redshift evolution for both observational data and simulation data, testing the model, and eventually showing the model constraints being different in the observation and in the simulation.

In addition, we are not considering BCG luminosity and stellar mass uncertainties in this section (they are not included in the linear fitting procedure). We also address this in Sections 4 and 5.

At z > 0.9, we notice that two of the four BCGs in our sample appear to be massive/luminous outliers by ∼0.5 dex, which matches previous findings about massive BCGs at z > 1.0. In Collins et al. (2009), five 1.2 < z < 1.5 BCGs are identified to be 0.5 ∼ 0.7 dex more massive than the DL07 simulation BCGs, and in Liu et al. (2013), a massive z = 1.096 cD-type galaxy is discovered in a 5 arcmin² Hubble Deep Field. However, after considering cluster mass uncertainty, and the BCG luminosity and stellar mass uncertainties, we can only detect the over-massive/over-luminous BCG effect with ∼1σ significance.

We model the BCG-cluster mass relation as redshift-dependent using the following equation,

$$\begin{eqnarray}&&\mathrm{log}{m}_{*}=\mathrm{log}{m}_{0}+\alpha \mathrm{log}\left(\displaystyle \frac{{M}_{200,z}}{{M}_{\mathrm{piv}}}\right)+\beta \mathrm{log}(1+z).\end{eqnarray} \tag{ 1 }$$

This equation adopts a power-law dependence on cluster mass (Brough et al. 2008; Moster et al. 2010, 2013; Kravtsov et al. 2014; Oliva-Altamirano et al. 2014) as well as a power-law dependence on redshift. We choose M_piv to be 1.5 × 10¹⁴ M_⊙, about the median mass of the XCS clusters. We also assume that there exists an intrinsic scatter, , between the observed BCG stellar mass and this relation, as $\mathrm{log}{m}_{*,\mathrm{obs}}\sim { \mathcal N }(\mathrm{log}{m}_{*},{\epsilon }^{2})$. Hence, the relation contains four free parameters: log m₀, α, β, and .

We perform a Markov Chain Monte Carlo (MCMC) analysis to sample from the following posterior likelihood:

$$\begin{eqnarray}&&\mathrm{log}{ \mathcal L }=-\displaystyle \frac{1}{2}\mathrm{log}| {\boldsymbol{C}}| -\displaystyle \frac{1}{2}{{\boldsymbol{Y}}}^{T}{C}^{-1}{\boldsymbol{Y}}+\mathrm{log}p({\boldsymbol{Q}}).\end{eqnarray} \tag{ 2 }$$

In this function, ${\boldsymbol{Y}}$ is a 106 dimension vector (y₁, y₂ ..., y₁₀₆ ), with the kth element being the difference between the modeled and the observed BCG stellar masses, as:

$$\begin{eqnarray}&&{y}_{k}={y}_{\mathrm{model},k}-{y}_{\mathrm{obs},k}.\end{eqnarray} \tag{ 3 }$$

The covariance matrix, ${\boldsymbol{C}}$, in Equation (2) is the combination of the covariance matrices for cluster mass measurements, BCG stellar mass measurements, redshift measurements, and the intrinsic scatter. It has the following form:

$$\begin{eqnarray}{\boldsymbol{C}} & = & \mathrm{Cov}({{\boldsymbol{m}}}_{*})+{\alpha }^{2}\mathrm{Cov}(\mathrm{log}\;{{\boldsymbol{M}}}_{200})\\ & & +{\beta }^{2}\mathrm{Cov}(\mathrm{log}({\rm{1}}+{\boldsymbol{z}}))+{\epsilon }^{2}{\boldsymbol{I}}.\end{eqnarray} \tag{ 4 }$$

Additionally, we implement an outlier pruning procedure as we "fit" (or sampling the posterior distribution in Bayesian statistics) for the BCG-cluster mass relation, as described in Hogg et al. (2010). To summarize this procedure, we adopt a set of binary integers ${\boldsymbol{Q}}$ = (q₁, q₂,..., q₁₀₆) as flags of outliers. q_k = 0 indicates an outlier and y_k is correspondingly modified as,

$$\begin{eqnarray}&&{y}_{k}=\mathrm{log}{m}_{*,k}-\mathrm{log}{m}_{\mathrm{outlier}},\end{eqnarray} \tag{ 5 }$$

where m_outlier is treated as a 5th free parameter. To penalize data pruning, we assume a Bernoulli prior distribution for ${\boldsymbol{Q}}$, characterized by another free parameter p as,

$$\begin{eqnarray}&&p({\boldsymbol{Q}})=\displaystyle \prod _{k}{p}^{{q}_{k}}{(1-p)}^{1-{q}_{k}}.\end{eqnarray} \tag{ 6 }$$

Eventually, the parameters to be sampled from Equation (2) are log m₀, α, β, , ${\boldsymbol{Q}}$, p, and log m_outlier. More details about deriving the posterior likelihood (Equation (2)) as well as choosing the covariance matrix can be found in Appendix D. We assume uniform truncated priors for all the free parameters except ${\boldsymbol{Q}}$, and the final result appears to be insensitive to this choice. We perform the fitting procedure for both the observed BCGs from the XCS sample and the simulation BCGs sampled from the DL07 simulation (Section 3.1).

In Figure 5, we plot the posterior distribution of log m₀, α, β, in Equation (1). We also list their marginalized means and standard deviations in Table 1.

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Table 1.  Constraints on log m₀, α, β, and σ of the BCG-cluster Mass Relation (Equation (1)) and p from the Outlier Pruning Procedure

	Prior	15 kpc	32 kpc	50 kpc	60 kpc	32 kpc (log M200 > 13.85)	32 kpc (log M200 < 14.70)	DL07
log m0	[10, 13]	11.37 ± 0.08	11.52 ± 0.08	11.60 ± 0.09	11.61 ± 0.09	11.58 ± 0.08	11.49 ± 0.08	11.698 ± 0.004
α	[−0.5, 0.8]	0.20 ± 0.08	0.24 ± 0.08	0.30 ± 0.08	0.32 ± 0.09	0.37 ± 0.10	0.19 ± 0.11	0.452 ± 0.004
β	[−2, 2]	−0.15 ± 0.31	−0.19 ± 0.34	−0.24 ± 0.39	−0.19 ± 0.40	−0.62 ± 0.34	−0.06 ± 0.37	−0.912 ± 0.026
	[0.001,1]	0.172 ± 0.015	0.180 ± 0.018	0.192 ± 0.019	0.198 ± 0.020	0.169 ± 0.019	0.186 ± 0.020	0.1628 ± 0.0012
p	[0.5, 1.0]	0.970 ± 0.017	0.970 ± 0.018	0.971 ± 0.017	0.970 ± 0.018	0.965 ± 0.019	0.967 ± 0.017	N.A.

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The constraint we derive on α agrees well with the reported values from the literature (Brough et al. 2008; Kravtsov et al. 2014; Oliva-Altamirano et al. 2014). We also notice that α increases with bigger BCG apertures, indicating a stronger correlation with cluster mass in the BCG outskirts (also see Stott et al. 2012). This effect seems to be justifiable, considering an inside-out growth scenario for BCGs (van Dokkum et al. 2010; Patel et al. 2013; Bai et al. 2014). Further analysis with large apertures is limited by the increasing amount of background noise at the BCG outskirts, but a larger BCG sample may help quantify the effect. This effect also illustrates the importance of understanding BCG photometry measurement when deriving BCG-cluster mass relations.

Our estimation of log m₀, the normalization of Equation (1), appears to be lower than the corresponding value in DL07 by 0.1–0.2. As log m₀ is mainly constrained by low-redshift BCGs, this result is completely consistent with BCGs being under-massive at low redshift, as discussed in Section 3.2.

Our estimation of β, the index of the redshift component in Equation (1), also disagrees with the corresponding value in DL07. The constraint on β derived from the whole cluster sample is different from the simulation value at a significance level of 2.3σ. The constraint from our data is closer to 0, suggesting less change of BCG stellar mass with redshift. Note that a further, quantitative conclusion should not be drawn. Although β is the dominant parameter that describes BCG redshift evolution in Equation (1), it is not the only one. The mass term in Equation (1) also contains information about BCG redshift evolution as cluster M₂₀₀ evolves with time. A quantitative analysis of BCG redshift evolution is presented in Section 5.

Our constraint on β is highly covariant with log m_* (recall the bivariate normal distribution), but the covariance shall not be interpreted as "degeneracy": a reasonable m_* sampled from its marginalized posterior distribution does not make β consistent with the simulation. We also notice that different conventions for BCG magnitude measurement can bias the constraint on β. For example, using the Kron magnitude from the popular SExtractor software (Bertin & Arnouts 1996), which tends to underestimate BCG Kron Radius and therefore BCG total magnitude (see the discussion in Appendix B.2. This effect happens frequently for our intermediate redshift BCGs), shifts β downward by ∼1σ.

We detect hints that the constraints on α and β may depend on cluster mass (see Table 1). For clusters with log M₂₀₀ above 13.85, we notice a stronger correlation between BCG and cluster masses (larger α, compare it to Chiu et al. 2014; van der Burg et al. 2014) and steeper redshift evolution (smaller β) at ∼1.0σ significance level. However, BCGs in low-mass clusters (log M₂₀₀ < 13.85) are possibly over-massive compared to our simulation-calibrated BCG-cluster mass relation. Evaluating the masses of low-mass clusters and groups through their X-ray observables needs to be handled with care. In this paper, we use the lensing calibrated M − T relation of galaxy groups and clusters to derive M₂₀₀ for most low-mass clusters (see Figure 10 and Appendix A). Arguably, the accuracy of X-ray inferred masses of low-mass clusters is less well characterized than the higher-mass end. Thus, in our growth rate determinations we show the difference after excluding the lowest-mass systems (log M₂₀₀ < 13.85, about 10% of the sample). For consistency, we also examine the effect of excluding the highest-mass systems (log M₂₀₀ > 14.7, about 10% of the sample).

We need to know how the cluster mass evolves with redshift in our method. To acquire this information, we select a sample of halos with z ∼ 1.0, log M₂₀₀ ∼ 13.8 from the Millennium simulation, and extract their evolution histories by identifying descendants of these halos all the way to z = 0 (using the descendantid keyword). We then compute the mean M₂₀₀ evolution of these halos, shown in Figure 6.

The second step is to use the BCG-cluster mass relation to derive the average stellar mass of the BCGs hosted by these halos at different redshifts. From Equation (1), the average BCG stellar mass relative to some normalization epoch, z₀, can be expressed as:

$$\begin{eqnarray}&&\mathrm{log}\displaystyle \frac{{m}_{*,z}}{{m}_{*,{z}_{0}}}=\alpha \mathrm{log}\displaystyle \frac{{M}_{200,z}}{{M}_{200,{z}_{0}}}+\beta \mathrm{log}\displaystyle \frac{1+z}{1+{z}_{0}}.\end{eqnarray} \tag{ 7 }$$

We take $\mathrm{log}\frac{{m}_{*,z}}{{m}_{*,{z}_{0}}}$ from the above equation to describe the average BCG stellar mass growth. The $\frac{{M}_{200,z}}{{M}_{200,{z}_{0}}}$ component in the equation is the average cluster mass growth extracted from the simulation.

The result of applying Equation (7) to the average M₂₀₀ growth with the simulation data is also shown in Figure 6. We estimate the uncertainties on the BCG stellar mass growth rate through sampling the joint constraint on α and β. We do not consider the uncertainties of cluster mass evolution, as it is marginal and is cosmology-dependent.

We test our method by applying it to the DL07 simulation BCGs. We first derive the BCG-cluster mass relation in DL07 using the procedure from Section 4.1 for the sample drawn from Section 3.1. We compare the computed BCG growth rate to the values obtained through directly tracking cluster descendants. The latter is acquired through recording the central galaxy stellar mass of the halo descendants since redshift 1.0. We consider the result from this second approach as the true growth of simulation BCGs.

In the bottom panel of Figure 6, we show the BCG growth rate derived with Equation (1), and the true growth rate encompassed by uncertainty from bootstrapping. Overall, for low-mass clusters, our approach reproduces the average BCG growth rate from z = 1.0 to z = 0 within 1σ. Bias associated with this method (like progenitor bias: see Shankar et al. 2015), if there is any, appears to be negligible.

We compute the BCG stellar mass growth rate using Equation (7) and compare it to the simulation value obtained with the same method. We discuss the observational result based on BCG 32 kpc aperture stellar masses in this paper–the result derived with other apertures looks similar. From z = 1.0 to z = 0, we estimate the BCG growth rate to be 0.13 ± 0.11 dex, compared to 0.40 ± 0.05 dex in simulations (uncertainty estimated for the BCG sample in Section 3.1), as shown in Figure 7. This result is in agreement with our conclusion from the simulation matching analysis (Section 3.2) and also in agreement with previous studies (Lidman et al. 2012; Lin et al. 2013b,). Even after considering all the uncertainties, biases, and covariances associated with BCG luminosity and stellar mass measurements, we still confirm that the observed BCG growth is slower than the prediction from DL07 at a significance level of ∼2.5σ.

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Like our constraint on the BCG-cluster mass relation, our result here shifts by ∼1σ (0.29 ± 0.11 dex) when we exclude the lowest-mass systems (log M₂₀₀ < 13.85, about 10% of the sample). Note that the shift may be caused by the inaccuracy of X-ray cluster mass scaling relations at the low-mass end (see the discussion in Section 4.2). For consistency, we also show the result (0.07 ± 0.12 dex) after excluding the highest-mass systems (log M₂₀₀ > 14.7, about 10% of the sample). Our result is also susceptible to improper BCG magnitude measurements. Using the Kron magnitude from SExtractor, the result will be biased toward more rapid BCG growth by ∼1σ. We also considered applying our method with stellar-to-halo mass relations from the literature, but as many previous studies are based on magnitude conventions with various problems for BCGs (see the discussion in Appendix B), we opt not to use them in this paper.

In this paper, we have shown, from two different perspectives, that the BCG stellar mass growth rate in clusters with log M₂₀₀ = 13.8 at z = 1.0 is slower than the prediction naively expected in a hierarchical formation scenario (De Lucia & Blaizot 2007). This effect is not that surprising considering the processes that contribute to (or counter-act) BCG formation.

A hierarchical structure formation scenario predicts that galaxy mergers add stars to BCGs. The BCG stellar build-up can be further augmented by in situ star formation, but a reduction in stellar mass is possible from mergers that eject stars into the intracluster space. The competition between these mechanisms remains a subject for large modeling uncertainties in simulations. If we assume that BCGs experience the rapid build-up events (mostly merging events) as prescribed in the DL07 simulation, there must be a mechanism that offsets BCG growth to mimic the slower evolution we observe in this paper.

In Figure 8, we experiment with incorporating extra stellar mass gain or loss into the DL07 simulation. Stellar mass gain tends to steepen BCG growth over time, while stellar mass loss tends to slow down BCG build-up and flatten the BCG growth curve. In order to explain the observed BCG growth rate in our data, the BCGs in DL07 would need to go through extra stellar mass loss at 20–40 M_⊙ yr⁻¹, ending up with (2.0–3.5) × 10¹¹ M_⊙ at z = 0, which agrees with our data.

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Such stellar content stripping from mergers would produce ICL. Our result indicates that ICL accumulates at 20–40 M_⊙ yr⁻¹ after z = 1.0, totaling (1.5–3) × 10¹¹ M_⊙ ICL in the present epoch. This amount corresponds to about 30%–60% of the total of BCG and ICL stellar masses, consistent with the observed ICL fraction in low and medium redshift clusters (z < 0.5: Zibetti et al. 2005; Krick et al. 2006; Gonzalez et al. 2007; Krick & Bernstein 2007; Toledo et al. 2011; Giallongo et al. 2014; Montes & Trujillo 2014; Presotto et al. 2014).

In fact, ICL production has already been suggested as an explanation for the seemingly mild evolution of massive galaxies (Monaco et al. 2006; Conroy et al. 2007; Burke et al. 2012; Behroozi et al. 2013; Oliva-Altamirano et al. 2014). Although not completely settled, recent studies indicate that ICL possibly forms late, mostly after z = 1.0 (Conroy et al. 2007; Contini et al. 2014). Specifically, the Contini et al. (2014) study updates the DL07 simulation with more realistic ICL production processes and predicts a slower BCG growth rate (Figure 8), in excellent agreement with our measurement. Hence, the slow BCG stellar mass growth since z = 1.0 observed throughout this paper is completely justifiable if ICL forms late after z = 1.0.

Admittedly, the DL07 simulation also includes stellar stripping that would produce ICL. Unfortunately the amount of ICL from this simulation is not retrievable, and we are not able to analyze if it meets our expectation. The Guo et al. (2011) SAM simulation has explicitly included ICL production and predicts very similar BCG growth with DL07, but much of the ICL is already in place before z = 1.0, which is not favored in our interpretation.

We use a lensing calibrated M − T scaling relation from Kettula et al. (2013) to derive cluster mass from X-ray temperature (T_X, core not excised). In Kettula et al. (2013), weak lensing mass measurements are obtained for 10 galaxy groups in the mass range (0.3–6.0) × ${10}^{14}{h}_{70}^{-1}{M}_{\odot }$. Together with 55 galaxy clusters (many above $2\times {10}^{14}{h}_{70}^{-1}{M}_{\odot }$) from Hoekstra et al. (2011, 2012), Mahdavi et al. (2013), and Kettula et al. (2013) derive weak lensing calibrated M − T relation across the group to cluster range.⁴⁹

We check the Kettula et al. (2013) M − T scaling relation against a few other studies based on the gas content (Figure 9). The hydrostatic equilibrium (HSE, Sun et al. 2009; Vikhlinin et al. 2009; Eckmiller et al. 2011) and gas mass fraction (Mantz et al. 2010) calibrated relations agree with the Kettula et al. (2013) M − T relation at the cluster scale, but have trouble matching to it at the group scale (Sun et al. 2009; Eckmiller et al. 2011). As known from simulations (Nagai et al. 2007; Rasia et al. 2012; see Kettula et al. 2013 for a discussion), the disagreement is not surprising, as HSE masses are biased low at the group scale and the gas mass fraction relation is only derived with the most massive clusters.

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We use the Kettula et al. (2013) scaling relation to derive cluster M₅₀₀ from X-ray temperature, and then the Hu & Kravtsov (2003) relation to derive M₂₀₀ from M₅₀₀. We assume the cluster concentration parameter to be 5 in this procedure. Using a different concentration parameter in the [3, 5] range only changes M₂₀₀ at the percent level. We also assume the intrinsic scatter of M₅₀₀ to be 0.1 dex, as typically found in simulation studies (Kravtsov et al. 2006; Henry et al. 2009).

We resort to X-ray luminosity (L_X, core not excised) to estimate masses for clusters/groups that do not have high-quality temperature measurements. We first derive X-ray temperature using a L − T scaling relation, and then derive M₂₀₀ using the procedure above.

We use a self-similar, redshift-dependent L − T scaling relation from Hilton et al. (2012), but also experimented with a few other self-similar L − T relations (see Figure 10) from Stott et al. (2012), Maughan et al. (2012), and Pratt et al. (2009). The Hilton et al. (2012) relation, which is also based on an XCS sample (Mehrtens et al. 2012), provides the best fit to our data. We assume a 0.1 dex intrinsic scatter for the derived temperatures, as it is constrained in Hilton et al. (2012).

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We decide between L_X- and T_X-based masses through comparing their uncertainties. To estimate the mass uncertainties associated with each method, we produce 200 "pseudo-measurements" for each cluster, sampling through temperature/luminosity measurement uncertainty, the scaling relation uncertainty, and the intrinsic scatter of the relations. We derive mass uncertainties for L_X- or T_X-based masses assuming a log-normal distribution for the 200 "pseudo-measurements," If the uncertainty of the T_X mass is larger than the uncertainty of the L_X mass by 0.05 dex (we prefer T_X mass since L_X mass is more susceptible to biases), we use L_X mass in lieu of T_X mass. In the end, about half of the cluster masses are derived with L_X, and most of the clusters' masses have <0.25 dex uncertainty.

The 200 "pseudo-measurements" are also used to derive the M₂₀₀ covariance between the cluster sample. Because we are including scaling relation uncertainty, the M₂₀₀ covariance matrix is not diagonal.

Petrosian magnitude measures the flux enclosed within a scaled aperture known as the "Petrosian radius," which is calculated considering background noise level and object light profile (Petrosian 1976; Blanton et al. 2001; Yasuda et al. 2001). It is extraordinarily robust under exposure-to-exposure variations, but not appropriate for extended galaxies. Although Petrosian magnitude accounts for most of the flux of a disk-like (Sérsic index = 1) galaxy, it will only recover 80% of the flux for a bulge-like galaxy with a de Vaucouleurs (Sérsic index = 4) profile (Blanton et al. 2001).

Indeed, a series of studies have found that using Petrosian magnitude (see Bernardi et al. 2013; He et al. 2013, for relevant discussion), the brightness of luminous red galaxies (LRGs) is underestimated by about 0.3 mag. Moreover, the missing flux problem is sensitive to the profiles of extended galaxies, and worsens quickly with higher Sérsic index. Graham & Driver (2005) estimate that Petrosian magnitude at its most popular configuration (one that is adopted by SDSS) underestimates the luminosity of Sérsic index = 10 galaxies by 44.7% (0.643 mag)! For this reason, we are not exploiting Petrosian magnitude in this paper.

Kron magnitude is another scaled aperture magnitude, measuring the flux enclosed within a few "Kron radius" (usually 2.5 Kron radius), and the Kron radius is decided from the light profile (Kron 1980). Kron magnitude is not as robust as Petrosian magnitude under exposure-to-exposure variations, but does appear to be more proper for extended galaxies.

Like Petrosian magnitude, Kron magnitude recovers most of the flux of a disk-like galaxy, but misses 10% of the flux for a bulge-like galaxy (Sérsic index ∼4, Andreon 2002; Graham & Driver 2005). Unlike Petrosian magnitude, the flux missing ratio is insensitive to the galaxy Sérsic index. Graham & Driver (2005) estimate that the missing flux varies only at the percent level when Sérsic index changes from 2 to 10. Indeed, tests with simulated skies (see Andreon 2002, or our test in Figure 11) show that Kron magnitude only underestimates the brightness of bulge-like galaxies by about 0.2 mag. It also appears to be indifferent to the presence of ICL: when we apply the measurement to simulated BCGs enclosed by ICL (we use the model in Giallongo et al. 2014), the measurement changes by only <∼0.1 mag.

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As proper as the design of Kron magnitude seems to be, the real problem comes from observationally deriving the Kron radius. As pointed out in Graham & Driver (2005), correctly estimating Kron radius requires integration over the light profile to a very large radius, usually many times the half light radius for extended galaxies. If the integration is improperly truncated, the measured Kron radius will be much smaller, and Kron magnitude turns out to be catastrophically wrong—it may underestimate the flux of an extended galaxy by as much as 50% (Bernstein et al. 2002)!

We find this to be a frequent problem for BCG measurements from the widely used SExtractor software (i.e., mag_auto), as demonstrated in Figures 12 (a) and (b). The Kron radius from SExtractor is two times smaller than it should be for one of the BCGs, and the BCG light intensity at 2.5 SExtractor Kron radius is still high. As a result, SExtractor underestimates the Kron flux of this BCG by ∼0.5 mag. This problem seems purely algorithmic though. Using the galaxy intensity profile to recalculate Kron radius until it converges, we are able to correct this measurement error. Comparing the corrected measurements to the magnitude measurements from profile fitting (see Appendix B.3), we recover the 0.2 mag accuracy of Kron magnitude as discussed above.

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For this paper, we have redone our analysis using Kron magnitude. We recompute the Kron radius instead of using SExtractor output, but the result remains qualitatively similar.

We have also experimented with BCG profile fitting magnitude from the GALFIT software (Peng et al. 2002, 2010). We fit the BCGs with a model consisting of two Sérsic profiles, one with a Sérsic index = 1 (i.e., a disk profile) and one with a flexible Sérsic index as suggested by Bernardi et al. (2014) and Meert et al. (2015). We convolve these models to point-spread functions (PSF) derived with the PSFex software (Bertin 2011), and carefully mask all neighboring objects, including blended objects identified with the GAIN deblender (Zhang et al. 2014). Overall, the design of this procedure is similar to the Galapagos fitting software (Barden et al. 2012).

For this paper, we only use the profile fitting magnitude for testing purpose (see Appendices B.2 and B.4). We hesitate about using it for scientific purposes, as we realize that the measurement needs to be extensively tested with sky simulations as in Häussler et al. (2007), Bernardi et al. (2014), and Meert et al. (2015). Upon evaluating the profile fitting magnitude uncertainties (see Figure 13), we do not find it to improve BCG measurement accuracy and therefore do not consider the testing efforts to be worthwhile for this paper. We nevertheless have redone our analysis using this magnitude, but the result remains qualitatively similar.

In this paper, we measure BCG magnitude with circular apertures of 15, 32, 50, and 60 kpc radii. The main results in this paper are derived with the 32 kpc measurements, considering the BCG half light radius measurements in Stott et al. (2011). The 32 kpc aperture choice is also comparable to the popular Kron magnitude aperture (2.5 Kron radius) measurements from the SExtractor software. We carefully mask BCG neighbors (including blended objects identified with the GAIN deblender, Zhang et al. 2014) and interpolate for the BCG intensity in the masked area. To realistically evaluate BCG magnitude uncertainty, we perform the procedure on processed single exposure images, use the median as the measurement, and evaluate the uncertainty through bootstrapping. We find our typical measurement uncertainty to be ∼0.4 mag, significantly larger than the SExtractor estimation from co-added images (but not larger when we bootstrap the SExtractor measurements from single exposure images). Since we perform the measurements independently on different exposures, our uncertainty is more comprehensive than the SExtractor uncertainty from co-added images (also see the magnitude measurement scatter test in Figure 11). Our measurement becomes uncertain when we have few exposures to work with (see Figure 13), which will be improved as DES assembles more exposures in the coming years.

To evaluate the sky background level around BCGs, we use background check maps generated with the SExtractor software from DESDM, configured with the "Global evaluation" process. We sample the values in a ring with inner and outer radius of ∼13 and 18 arcsec from the BCG. We have investigated how sky background estimation affects our measurement, as it was considered a difficult task for BCGs. We find it to have only marginal influence. Indeed, even by using a "Global" setting, SExtractor still overestimates the background around some extremely bright sources, known as the dark halo problem within DES (after background subtraction, the light intensity of a bright object falls slightly below 0 at the outskirt). However, changing the background sampling location only marginally shift our final measurements. In fact, other details of the measurement procedure, like incomplete masking of neighboring sources, may cause bigger problems.

We test for measurement bias associated with aperture magnitude and Kron magnitude with simulated DES images, using the UFIG simulation (Bergé et al. 2013; Chang et al. 2014). This sky simulation is based on an N-body dark matter simulation populated with galaxies using the Adding Density Determined GAlaxies to Lightcone Simulations (ADDGALS) algorithm (Dietrich et al. 2014; M. Busha 2015, in preparation; R. Wechsler 2015, in preparation, for a review). We find that both aperture magnitude and Kron magnitude tend to underestimate the brightness of fainter sources, but the effect is negligible for even the furthest BCGs (z-band apparent magnitude is about 22). In addition, the bias would only have suppressed the significance of our result, as further objects are evaluated to be less massive/luminous. We also perform the test with sky simulations based on adding simulated galaxies into real DES coadd images, known as the Balrog simulation (Suchyta et al. 2015), and come to the same conclusion.

We estimate BCG luminosity uncertainty by combining BCG magnitude and redshift (see M15) uncertainties. To simplify subsequent analyses, we assume the redshift uncertainty to be statistical (systematic uncertainty is about ∼0.001, comparing to ∼0.05 statistical uncertainty; see M15). The redshift uncertainty is taken as 0.001 if archival spectroscopic redshift is available. We ignore K-correction uncertainty, as it is decided.

We estimate BCG stellar mass uncertainty by combining BCG luminosity uncertainty and BCG mass-to-light ratio (MLR) uncertainty. In this section, we pay special attention to estimating the MLR uncertainty from modeling star formation histories, which is the uncertainty from fitting a τ-model to BCGs that formed through merging with galaxies of various star formation histories. We evaluate the uncertainty by applying the stellar population fitting procedure to DL07 BCGs (selected in Section 3.1). We compare the derived BCG MLRs to their true values in the simulation.

In the left panel of Figure 14, we show the difference between the derived and true MLR plotted against redshift. We notice a systematic uncertainty of ∼0.05 dex, likely caused by the mismatch of IMF in our procedure and in DL07 (a Chabrier IMF produces a mass-to-light ratio that is 0.05 dex higher than that of a Kroupa IMF; see Papovich et al. 2011).

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We also notice a statistical scatter with the derived values, ranging from 0.05 dex to 0.1 dex with weak dependence on redshift (Figure 14), but no dependence on cluster mass or BCG stellar mass. We evaluate the uncertainty and covariance for our BCG sample, taking the corresponding values in simulation. To elaborate, for each BCG in our sample, we assume its MLR to have been measured 100 times (each BCG is matched to 100 simulation BCGs in Section 3.1), and the error of each measurement is the offset between the derived and true MLR for one simulation BCG. As a result of this set-up, the MLR uncertainty for each BCG contains about 0.05 dex systematic uncertainty and 0.05 to 0.1 dex statistical uncertainty depending on its redshift.

Admittedly, it is more than likely that we are underestimating the BCG MLR uncertainty. In our simulation test, the systematic uncertainty originates from using slightly different SSP models and IMFs (Conroy et al. 2009 and Conroy & Gunn 2010 SSP models, and Chabrier 2003 IMF in our procedure VS the Bruzual & Charlot 2003 SSP models and Kroupa 2001 IMF in DL07). The statistical uncertainty originates from matching τ star formation history and fixed metallicity to DL07 BCGs. We have not considered uncertainties associated with SSP models, dust distributions, and possible IMF variations

Estimating the uncertainties from these so-called "known unknowns" is difficult. Conroy et al. (2009) shows that one may at best recover the MLR of bright red galaxies with 0.15 dex uncertainty at z = 0, or 0.3 dex at z = 2.0. According to this result, we would have underestimated BCG MLR uncertainty by ∼0.1 dex. We also experimented with the SSP models from Maraston (2005) and Bruzual & Charlot (2003), but the derived MLR differences are lower than 0.1 dex.

Since the redshift dependence of our estimation is qualitatively similar to that presented in Conroy et al. (2009), it is unlikely that we are affected by our conclusion about BCG redshift evolution. We therefore do not attempt to include additional uncertainties from the "known unknowns." Eventually, the BCG stellar mass uncertainty is dominated by the uncertainty from magnitude measurement or redshift (See Figure 14(b)), rather than from MLR.