THE STELLAR MASS GROWTH OF BRIGHTEST CLUSTER GALAXIES IN THE IRAC SHALLOW CLUSTER SURVEY

In a universe dominated by cold dark matter, structures are expected to grow hierarchically (Springel et al. 2005). Taken at face value, such a structure formation scenario suggests that the most massive galaxies should form late. Indeed, in the semi-analytic model (SAM) of De Lucia & Blaizot (2007), the mass assembly of brightest cluster galaxies (BCGs)—the most massive galaxies in the universe—occur relatively late, in the sense that typical BCGs acquire 50% of their final mass at z < 0.5 through galactic mergers. Although there is ample evidence of mergers involving BCGs at low redshifts (e.g., Lauer 1988; Rines et al. 2007; Tran et al. 2008; Lin et al. 2010), the importance of stellar mass growth at late times remains unclear. Using deep near-IR data to infer the stellar mass of BCGs across wide redshift ranges, it was suggested that BCGs in massive clusters have attained high stellar mass and exhibited little change in mass since z ∼ 1 (Collins et al. 2009; Stott et al. 2010). Using the correlation between the BCG stellar mass and cluster mass, Lidman et al. (2012) found that the BCGs have grown by a factor of 1.8 between z = 0.9 and z = 0.2, from a large sample of X-ray luminous clusters.

Some of the contradicting results may arise from inconsistent cluster sample selection (e.g., drawing different cluster samples at different redshifts that do not have any evolutionary links), and some may be due to incompatible comparisons between observations and theories (e.g., while theoretical models predict "total" magnitudes for galaxies, it is particularly difficult observationally to measure such a quantity for BCGs; Whiley et al. 2008). Ideally, one would like to identify a cluster sample that forms an evolutionary sequence, that is, the higher-z clusters are expected to be the progenitors of lower-z clusters in the same sample. With such a sample, one could then meaningfully follow the evolution of the galaxy populations, including the mass assembly of BCGs. This approach also facilitates more direct comparisons with theoretical models.

In this paper we attempt to construct such a cluster sample and study the evolution of the stellar mass content of the BCGs, using a subset of a complete cluster sample drawn from the Spitzer IRAC Shallow Cluster Survey (ISCS; Eisenhardt et al. 2008, hereafter E08). We will show that the observed growth is similar to that predicted by the Millennium Simulation (Springel et al. 2005; Guo et al. 2011) at z = 0.5–1.5, but the two disagree at z < 0.5 at the 2σ level.

In Section 2 we describe our cluster sample and the numerical simulations used in this analysis. For the construction of a cluster sample that represents an evolutionary sequence, knowledge of cluster mass is critical. We have developed a method to infer cluster mass from the total 4.5 μm luminosity of the clusters (Section 3). We then proceed to use two methods that rely on the dark matter halo merger history to infer the BCG mass growth in progenitors of present-day clusters with a mass of (2.5–4.5) × 10¹⁴ M_☉, and compare the results to SAMs based on the Millennium Simulation (Section 4). We conclude in Section 5.

Throughout this paper we adopt a Wilkinson Microwave Anisotropy Probe 5 (WMAP5; Komatsu et al. 2009) ΛCDM cosmological model where $\Omega _M=1-\Omega _\Lambda =0.26$, H₀ = 71 km s⁻¹ Mpc⁻¹, and the normalization of the matter power spectrum σ₈ = 0.8.

2.1. Cluster Sample

2.2. Numerical Simulations

Absent traditional cluster mass proxies such as X-ray observables, weak gravitational lensing, or velocity dispersion, the simplest way to estimate a cluster's mass is via its luminosity/stellar content or N_gal (e.g., Yang et al. 2009; Rozo et al. 2009). Assuming a monotonic relationship between the mass and luminosity is unrealistic, however, because of the non-negligible scatter in the luminosity–mass correlation (Lin et al. 2004, hereafter L04). However, the median mass of the N most luminous clusters (or "top N" hereafter) should have considerably lower scatter. Here we make use of mock cluster catalogs to derive a "lookup table" that tells us the median mass of a sample of clusters that is rank ordered by L_tot.

Our basic procedure is as follows: (1) extract a patch of the sky whose area is the same as the Boötes field from the lightcone simulation, (2) assign a luminosity to each of the dark matter halos, and (3) rank the halos by luminosity and produce the lookup table. In practice, however, we need to take several complications into account, such as the uncertainties of the slope s, scatter σ_s, and redshift evolution of the luminosity–mass correlation (hereafter L–M relation). Another consideration is that the mass M₂₀₀ for the halos is measured within r₂₀₀ (the radius within which the mean overdensity is 200 times the critical density of the universe at the cluster redshift), while for our Boötes cluster sample L_tot is measured within a metric radius of 0.8 Mpc. We thus need a model for the spatial distribution of galaxies within clusters, and we assume that the galaxies follow the Navarro et al. (1997) profile with concentration of c and scatter of σ_c (see, e.g., L04).

For s and σ_s, we assume the possible values follow a Gaussian distribution with the mean corresponding to the observed values based on a sample of 93 clusters at z ∼ 0 (e.g., L04; s = 0.85, σ_s = 0.15), and the standard deviation of 0.05. It was found that once the luminosity is measured with respect to the evolving L*, the L–M relation does not show strong hints of redshift evolution (Lin et al. 2006). We therefore assume the redshift evolution of these parameters to be negligible.¹⁰ As for the concentration and its scatter, we note that the absolute value of c does not matter for the luminosity ranking; rather, it is the ratio of σ_c/c that perturbs the luminosity ranking. As this ratio has not been observationally determined, we simply assume σ_c/c = 0.1, 0.2, and 0.3, with equal probability. Our results are robust against different choices of this ratio, however.

We have extracted 16 Boötes-like patches from the lightcone. For each patch and combination of s, σ_s, and σ_c/c, in six redshift bins (z = 0.2–0.4, 0.4–0.6, 0.6–0.8, 0.8–1.0, 1.0–1.2, and 1.2–1.5), we generate 500 Monte Carlo realizations of the mock cluster catalog, and combine the results to produce a lookup table that marginalizes over the parameter uncertainties. Figure 1 is an illustration of the lookup tables for the 16 patches at z = 0.8–1.0. It is clear that cosmic variance is large: at a fixed N, the inferred M₂₀₀ could differ by a factor of 1.6.

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Our primary goal is to follow the evolution of BCGs in a sample of clusters that is believed to form an evolutionary sequence.¹¹ We first consider a simple approach that combines the lookup tables discussed above with the knowledge of average dark matter halo growth (Section 4.1). We then take into account the stochastic nature of halo merger history, and show that the two methods produce very similar results (Section 4.2). We further compare the results with theoretical models (Section 4.3).

4.1. Average Dark Matter Halo Growth History

Using the hi-res run, we can extract the full merger history for the dark matter halos. In Figure 2 (top panel) we show the mass growth history of halos whose present-day mass is M₂₀₀ = (2.5–4.5) × 10¹⁴ M_☉ (hereafter the target mass range). The mass range is chosen to have enough z ∼ 0 BCGs (see below), while keeping the spread in mass of high-z progenitors small. The solid curve and the shaded region represent the median and the 68% range spanned by the most massive progenitors, respectively. Therefore, after specifying the mass of the z ∼ 0 clusters, using such curves we could know the typical mass of their progenitors at any redshift. We can then use the lookup tables of Section 3 to select the top N most luminous clusters at the target redshift whose median mass matches the expected progenitor mass, and study the properties of BCGs in these clusters.

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In Figure 2 (bottom panel) we show as solid blue points the mean BCG luminosity L_bcg (scaled by the evolving L*) within 32 kpc diameter for progenitors of present day clusters whose mass is in the target mass range. As we have scaled out the stellar aging by normalizing the luminosity to L*, we could attribute the change of L_bcg/L* as due to merger, accretion, and star formation, and will regard this quantity as a measure of stellar mass growth of BCGs. The error bars include the cosmic variance (estimated by the scatter in L_bcg which resulted from using lookup tables from the 16 mock Boötes patches) and a conservative 20% systematic uncertainty to account for the impact of photometric redshift selection.

At z < 0.1, the volume probed by ISCS is not large enough to contain any cluster in the target mass range. We thus use an enlarged version of the low-z cluster sample presented in Vikhlinin et al. (2009). These 76 clusters are selected by the same criteria as described in Vikhlinin et al. (2009), with the exception of a lower X-ray flux limit (7.5 × 10⁻¹² erg s⁻¹ cm⁻²). All of them have high quality Chandra observations, allowing us to estimate their mass accurately via the Y_X–M₅₀₀ scaling relation, where Y_X is the product of X-ray temperature and the intracluster medium mass (Kravtsov et al. 2006), and M₅₀₀ is defined analogously as M₂₀₀. After converting M₅₀₀ to M₂₀₀ assuming a Navarro et al. (1997) profile with c = 5, there are 22 clusters within the target mass range at z < 0.1.

The BCG luminosity for these nearby clusters within the 32 kpc diameter aperture is measured from the Wide-field Infrared Survey Explorer (WISE; Wright et al. 2010) all-sky data release, with a redshift-dependent correction factor applied, as described in the Appendix. We use the 3.4 μm data from WISE, as it is closer to the rest frame wavelength probed by the ISCS 4.5 μm data. The small black points in the figure represent the individual BCGs, while the magenta star is their mean luminosity (scaled by L*). Taking these results together, the stellar mass content of BCGs has increased by a factor of 2.7 or so since z ≈ 1.5.

4.2. Taking Detailed Halo Merger History into Account

The approach employed in Section 4.1 ignores variations in the merger history of clusters. In Figure 3 we show a comparison of descendant and progenitor halo masses at z = 0 and 0.5 from the hi-res run. The horizontal lines delineate the target mass range of the present day clusters used in Section 4.1. It can be seen that the halos at z = 0.5 that grow into the target mass range at z = 0 span a wide range in mass [i.e., (0.4–4) × 10¹⁴ M_☉]. Here we try to take this varied degree of growth into account.

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For every redshift bin considered in Section 3, we need to determine two normalized probability distributions: (1) the distribution in progenitor mass p₁(M_p|M_d) of those halos that will grow into the target mass range M_d at z = 0, and (2) the likelihood of a progenitor having the probable mass [i.e., p₁(M_p|M_d) > 0] that actually becomes a halo within the target mass range at z = 0, p₂(M_p, M_d). In other words, for p₁(M_p|M_d), we measure the mass distribution of those halos that lie in the horizontal band in Figure 3, and for p₂(M_p, M_d) we would like to know, for all progenitors that have the same masses as those lying in the horizontal band in Figure 3, the likelihood that their descendant will end up in the narrow mass range bracketed by the horizontal lines.

We measure p₁(M_p|M_d) and p₂(M_p, M_d) at the six redshift bins using the hi-res run. The mean BCG luminosity is determined as

$$\begin{eqnarray} &&\overline{L_{\rm bcg}} = \int \int L_{\rm bcg}(M_p) p_1(M_p|M_d) p_2(M_p,M_d) dM_p dM_d, \nonumber\\ \end{eqnarray} \tag{ 1 }$$

where a cluster's mass is again inferred by the lookup table and the luminosity ranking. The BCG mass growth derived from this method is shown as open black squares in Figure 2 (bottom panel), which are plotted alongside those points deduced by the method presented in Section 4.1 (solid blue points). It is reassuring to see that the two methods give similar results. For simplicity, hereafter we only present the results based on the "simpler" approach of Section 4.1.

4.3. Comparison with Millennium Simulation

Our results show that, since z ∼ 1.5, the BCGs in progenitors of present day intermediate mass clusters have grown by a factor of ∼2.7. Here we compare these measurements with the predictions from the Millennium Simulation, based on the SAM of Guo et al. (2011).

We have queried the Millennium Simulation database and found all the most massive progenitors of z = 0 halos in the target mass range. In each of the six redshift bins, we identify the central galaxies and measure the mean of their stellar mass.¹² Using the most massive galaxies instead of the central ones does not change our results. At z ∼ 0, the aperture that encloses half of the mass in model BCGs is ≈27 kpc, comparable to our chosen aperture of 32 kpc.

In practice, to take account of cosmic variance and thus make a fairer comparison with observations, for each redshift bin, we divide the Millennium Simulation box into 8–27 sub-volumes comparable to the Boötes observations (or the local observations, for the z ∼ 0 bin), and adopt the weighted mean and error from all the sub-volumes for the model BCG stellar mass estimates.

The model predictions for the stellar mass growth are shown as green open triangles in Figure 4. For the stellar mass of observed BCGs, we simply multiply the (time-dependent) stellar mass of the passively evolving BC03 model with the observed L_bcg/L* ratio. We regard the uncertainties associated with the conversion from luminosity to stellar mass (typically ∼0.2 dex) as a systematic error, which is not included in the error bars in Figure 4. Both our BC03 model and the SAM of Guo et al. (2011) employ the Chabrier IMF, thereby reducing one potential concern of such a comparison. The model agrees with our measurements remarkably well, between z = 0.5 and 1.5, where a factor of 2.3 growth is found. However, the two disagree at the 2σ level at z < 0.5. While the observed BCGs show only a small increase in stellar mass content down to z ∼ 0, the model BCGs appear to exhibit an accelerated growth below z = 0.5; that is, 46% of the final mass of the model BCGs is acquired between z = 0.5 and 0. The corresponding fraction for the observed BCGs is <10%.

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We find quantitatively very similar results when we compare to the model of De Lucia & Blaizot (2007), which is also based on the Millennium Simulation.

One potential concern of our comparison with the model stems from uncertainties in the mass estimates for the ISCS clusters. Although our luminosity ranking method described in Section 3 should allow us to reliably select clusters of the desired mass, a definitive calibration of this method awaits a formal weak lensing analysis (Y.-T. Lin et al. 2013, in preparation). Here we evaluate the effect of uncertainties in the cluster mass by first selecting halos whose present-day mass lies in the range (1–9) × 10¹⁴ M_☉, and then perturbing their masses with a log-normal random variable with standard deviation of σ = 0.3 (except for z ∼ 0 halos, for which σ = 0.08 is chosen to reflect the much better accuracy of Chandra-based masses). We then study the BCG growth in halos whose perturbed mass lies in the appropriate halo mass range. The net effect is to lower the stellar mass of model BCGs, and slightly increase the discrepancy between the observations and model, but does not qualitatively alter our conclusions. We conclude that the different behavior of BCG mass assembly history between the model and observation at z  <  0.5 as found above is robust against cluster mass uncertainties.

Given its area and depth, the ISCS provides a unique data set to study cluster galaxy population evolution from z ∼ 2 to present day. We have developed an empirical approach that takes into account the effect of cosmic variance and overcomes the difficulty in inferring cluster masses from optical/IR cluster surveys, and have applied it to study the evolution of BCGs in progenitors of present day clusters with mass of (2.5–4.5) × 10¹⁴ M_☉. Our two methods to construct cluster samples that form an evolutionary sequence rely heavily on our knowledge of the merger history of dark matter halos (provided by numerical simulations), and both give consistent results (Sections 4.1 and 4.2).

Using a large but heterogeneous cluster sample, Lidman et al. (2012) have detected a factor of 1.8 growth in BCG stellar mass between z = 0.9 and 0.2, which is similar to our finding. They have taken into account of the expected cluster mass growth between different cosmic epochs (a bit similar to our approach in Section 4.1), and have focused on more massive clusters. It is encouraging to see consistent results emerging from two independent analyses. It is worth emphasizing that our method in Section 4.2 allows us to take the stochastic merger history into account, while follow the evolution of clusters. Our way of inferring the cluster mass via the luminosity ranking also enables us to probe a wide range in redshift, pushing the upper limit beyond the current capability of the X-ray surveys.

A comparison of our results with the SAM of Guo et al. (2011) shows good agreement between z = 1.5 and 0.5. At lower redshifts, there are suggestions of different behavior between the model and observation, however. While the growth of model BCGs is accelerating at late times, that of the observed BCGs is slowing down. Such a contrast suggests the period of z = 0–0.5 is potentially key in differentiating models of BCG assembly history.

Our method is designed to trace the evolution of galaxies in clusters. For the field galaxies, van Dokkum et al. (2010) have studied the stellar mass growth by selecting galaxies at a fixed number density. As advocated by these authors, such a selection provides a meaningful way to pick up galaxies that form an evolutionary sequence (although strictly not in the sense of dark matter halo growth), and is therefore complementary to our approach here.

Although in principle our method can be applied to study the BCG evolution in progenitors of present-day clusters of any mass range, extending much beyond the limited mass range presented here is beyond the capability of the ISCS cluster sample. Studying higher mass clusters requires progenitors too massive to be found in sufficient numbers in the Boötes field at intermediate-z. The depth of our photometry and the cluster sample size at high-z also prevents us from tracing lower mass present-day clusters. With the depth and large area coverage of the upcoming Subaru HyperSuprime Cam Survey (Takada 2010) and SPT-Spitzer Deep Field (M. L. N. Ashby et al. 2013, in preparation), we can apply this method and study the evolution of BCGs in much greater detail, especially for the z = 0–0.5 period.

We thank Laurie Shaw and Antonio Vale for constructing the merger trees used in this work. We are grateful to the anonymous referee for a report that improved the paper. Y.T.L. thanks Gabriella De Lucia, David Spergel, and Jerry Ostriker for helpful discussions, and I.H. for constant encouragement. Y.T.L. acknowledges supports from the National Science Council grant NSC 102-2112-M-001-001-MY3, as well as WPI Research Center Initiative, MEXT, Japan, during the course of this work. This work was supported by National Science Foundation grants AST-0707731 and AST-0908292. Computer simulations and analysis were supported by the NSF through resources provided by XSEDE and the Pittsburgh Supercomputing Center, under grant AST070015; computations were also performed at the TIGRESS high performance computer center at Princeton University, which is jointly supported by the Princeton Institute for Computational Science and Engineering and the Princeton University Office of Information Technology. This work is based in part on observations made with the Spitzer Space Telescope, which is operated by the JPL/Caltech under a contract with NASA. This publication makes use of data products from WISE, a joint project of UCLA and JPL/Caltech, funded by NASA. The Millennium Simulation databases were constructed as part of the activities of the GAVO.

As the resolution of WISE channel 1 (W1) is low (61) compared to IRAC, and the filter response function is somewhat different from IRAC channel 1 ([3.6]), we have compared the aperture photometry between W1 and [3.6] for a sample of 25 nearby BCGs and derived the difference in flux within the 32 kpc aperture. More specifically, we have searched the Spitzer Heritage Archive for well-known X-ray clusters whose BCG can be unambiguously identified, and the query returned 25 clusters. We have then measured the photometry within several apertures (up to 60'' diameter) on both the IRAC images and WISE atlas images using SExtractor (Bertin & Arnouts 1996), and interpolated the measurements to infer the magnitude within 32 kpc diameter (in Vega system). Figure 5 shows the difference in magnitude (W1−[3.6]) for the BCGs as a function of redshift. The trend is mainly driven by the difference in resolution between the two instruments. A least squares fit to the data points yields

$$\begin{equation} {\rm W1}-[3.6] = 4.5z-0.09. \end{equation} \tag{ A1 }$$

We have applied the correction factor thus inferred to the WISE photometry for the BCGs in the Vikhlinin et al. (2009) sample.

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