LoCuSS: Brightest Cluster Galaxy Dominance at z=0.2

LoCuSS: Connecting the Dominance and Shape of Brightest Cluster Galaxies with the Assembly History of Massive Clusters


We study the luminosity gap, , between the first and second ranked galaxies in a sample of 59 massive () galaxy clusters, using data from the Hale Telescope, the Hubble Space Telescope (HST), Chandra, and Spitzer. We find that distribution, , is a declining function of , to which we fitted a straight line: . The fraction of clusters with “large” luminosity gaps is , which represents a excess over that obtained from Monte Carlo simulations of a Schechter function that matches the mean cluster galaxy luminosity function. We also identify four clusters with “extreme” luminosity gaps, , giving a fraction of . More generally, large luminosity gap clusters are relatively homogeneous, with elliptical/disky brightest cluster galaxies (BCGs), cuspy gas density profiles (i.e. strong cool cores), high concentrations, and low substructure fractions. In contrast, small luminosity gap clusters are heterogeneous, spanning the full range of boxy/elliptical/disky BCG morphologies, the full range of cool core strengths and dark matter concentrations, and have large substructure fractions. Taken together, these results imply that the amplitude of the luminosity gap is a function of both the formation epoch, and the recent infall history of the cluster. “BCG dominance” is therefore a phase that a cluster may evolve through, and is not an evolutionary “cul-de-sac”. We also compare our results with semi-analytic model predictions based on the Millennium Simulation. None of the models are able to reproduce all of the observational results on , underlining the inability of the current generation of models to match the empirical properties of BCGs. We identify the strength of AGN feedback and the efficiency with which cluster galaxies are replenished after they merge with the BCG in each model as possible causes of these discrepancies.

galaxies:clusters:general – galaxies:elliptical – galaxies:halos – X-ray:galaxies – X-rays:galaxies:clusters – gravitational lensing

1 Introduction

Numerical simulations and large-scale redshift surveys both indicate that we live in a hierarchical universe, i.e. one in which the large-scale structure of the universe grows from the bottom up with smaller objects forming earlier than larger objects. This picture rests on the matter content of the universe being dominated by collisionless dark matter particles, smoothly distributed at early times, and seeded with small density perturbations. Exploring this picture observationally in the non-linear regime of gravitational collapse, i.e. within collapsed dark matter halos that host individual galaxies through to massive clusters of galaxies, complements the statistical analysis of the linear regime probed by galaxy redshift surveys. Indeed, a promising route to fleshing out our understanding of hierarchical structure formation is to measure observable quantities that are sensitive to the age and/or assembly history of dark matter halos, and thus in principle to test the hierarchical paradigm by comparing the observed and predicted distributions. Any discrepancies found between observation and theory may ultimately point to modifications to the theoretical model including, for example, the properties of the dark matter particle and the distribution of initial density fluctuations (e.g. Komatsu et al., 2009). Quantities discussed in the literature that may be useful probes of the age and assembly history of dark matter halos include the luminosity gap between the first and second ranked galaxies in a group or cluster (often expressed as the difference between their magnitudes, ; e.g. Dariush et al. 2007), the concentration of dark matter halos (e.g. Neto et al., 2007; Okabe et al., 2010), and the sub-halo population of dark matter halos (e.g. Taylor & Babul, 2004; Zentner et al., 2005).

The luminosities of the first and second ranked galaxies in clusters was first studied, as far as we are aware, by Sandage & Hardy (1973, see also ). More recently the luminosity gap, , was studied in the context of galaxy groups, the term “fossil” being coined to describe virialized systems with (Ponman et al., 1994; Jones et al., 2000, 2003). galaxies are absent from fossil groups, which were thus interpreted as having formed at early times, with dynamical friction then having sufficient time to cause the galaxy population to merge and form the brightest group galaxy (BGG). Fossil groups are expected to be more common than fossil clusters, at least in part because the probability of galaxy-galaxy merging is anti-correlated with galaxy velocity, and thus with cluster mass. Nevertheless, two clusters with masses of have been found with (Khosroshahi et al., 2006; Mendes de Oliveira et al., 2006; Cypriano et al.., 2006). These two objects (RX J1416.42315, and RX J15552.22013) raise the interesting question of whether the most massive () clusters might host similarly dominant BCGs. Theoretical studies suggest that this is the case, for example, Milosavljević et al. (2006) and Dariush et al. (2007) predict that and of clusters have respectively.

The concentration parameter of a dark matter halo describes the shape of its density profile following the so-called universal profile proposed by Navarro et al. (1997) and variants thereon. Halos with smaller concentrations have a flatter density profile, while larger concentrations imply a steeper density profile. Bullock et al. (2001) analyzed numerical simulations of CDM universes finding a weak dependence of concentration on halo mass: with (see also Dolag et al., 2004; Neto et al., 2007; Duffy et al., 2008). This relationship arises from the relative timing of the formation of dark matter halos as a function of mass. On average less massive halos form at earlier times than more massive halos in a hierarchical universe. At earlier times the universe was denser than at later times, and thus the central regions of less massive halos are relatively dense, leading to higher concentration parameters than for more massive halos. Some observational studies have reported very high concentration parameters in individual systems, for example, one of the fossil groups studied by Khosroshahi et al. (2004, 2006) was found to have , based on modeling of X-ray observations. Lensing studies of several individual strong-lensing clusters have also obtained very high concentrations of in contrast to the theoretical prediction of (Kneib et al., 2003; Gavazzi, 2003; Broadhurst et al., 2005; Limousin et al., 2007). More recently, observational studies have begun to study larger samples and thus to constrain the concentration-mass relation itself, and to probe the general population rather than a small number of potentially extreme objects (e.g. Buote et al., 2007; Okabe et al., 2010).

Theoretically, substructures within dark matter halos, i.e. the sub-halo population, are also sensitive to the assembly history of the host dark matter halo (e.g. Taylor & Babul, 2004; Zentner et al., 2005). Observationally substructures in galaxy clusters can be identified via detailed modeling of the observed gravitational lensing signal (Smith et al., 2005, 2009; Richard et al., 2010a, b). Specifically, group- and galaxy-scale perturbers are required to achieve statistically acceptable fits to the strong-lensing data. The contribution of these structures to the total cluster mass is quantified via the “substructure fraction”, , defined as the amount of mass within the adopted cluster-centric radius that is assigned to substructures divided by the total cluster mass within the same aperture. Smith & Taylor (2008) combined Smith et al.’s (2005) observational measurements of for 10 X-ray luminous galaxy clusters with Taylor & Babul’s (2004) semi-analytic model of structure formation to explore the interpretation of lensing-based measurements of . The main conclusion was that depends on both when the cluster formed, and on the level of recent mass assembly, each defined as the lookback time to when each cluster had acquired 50% and 90% of their observed mass respectively. For example: clusters at with formed on average at , and suffered mass growth since ; in contrast, clusters at with the highest substructure fractions () formed on average since and acquired of their mass between and , i.e. a time interval of just .

A complementary view of hierarchical merging within galaxy clusters is available from BCG morphology. Based on their isophotal shapes, elliptical galaxies have been classified as disky or boxy (Bender et al., 1989) – with positive and negative fourth-order Fourier coefficients respectively. The interpretation of boxy and disky isophotes in terms of the details of galaxy merger histories is a controversial subject (Faber et al., 1997; Naab & Burkert, 2003; Khochfar & Burkert, 2005). In this study we will side-step these difficulties, and concentrate simply on disky/boxy isophotes as an indicator of the presence of gas that has dissipated, and settled into a disk-like structure, either because the last massive galaxy to merge with the BCG was gas rich, or because gas has been accreted by the merger product from its environment, e.g. by a BCG in a cool core cluster. BCGs are the most massive early-type galaxies and are generally expected to have boxy isophotes consistent with formation via mergers of early-type (gas-poor) galaxies (Lin & Mohr, 2004; Khosroshahi, Ponman & Jones, 2006). However, BGGs in some fossil groups are as bright as BCGs and do not have boxy isophotes (Khosroshahi, Ponman & Jones, 2006), suggesting that (i) fossil BGGs may form early from the mergers between gas-rich spiral galaxies, and (ii) some fossil BGGs may have subsequently evolved into BCGs.

The main aim of this article is to combine measurements of the luminosity gap, cool core strength, concentration, substructure fraction, and BCG isophotal shape for a large sample of clusters to assemble an empirical picture of the hierarchical assembly of clusters and their BCGs. We present the first observational measurement of the distribution of the luminosity gap statistic of clusters, and compare this distribution with the other probes of hierarchical assembly discussed above. This allows us to build an empirical picture with which to assess the usefulness of the respective measurements for assessing the age of clusters. We also investigate how well the current generations of galaxy formation models can reproduce the observed luminosity gap distribution. The data used for this study are drawn from the Local Cluster Substructure Survey (LoCuSS; PI: Smith; A summary of LoCuSS is provided in §2, together with a description of the data used in this paper. The analysis and results are then presented in §3, and compared with theoretical predictions in §4. The main conclusions are summarized and discussed in §5. We assume , and throughout. In this cosmology corresponds to a physical scale of , at . All photometric measurements are relative to Vega.

2 Data

2.1 Sample Selection and Observing Strategy

LoCuSS is a morphologically-unbiased multi-wavelength survey of X-ray luminous galaxy clusters at . The overall aim of the survey is to measure the cluster-cluster scatter in key observables such as the X-ray temperature, and parameter (Smith et al., 2005; Zhang et al., 2008; Okabe et al., 2010), the Sunyaev-Zeldovich Effect parameter (Marrone et al., 2009), and the obscured and unobscured star formation activity (Haines et al., 2009a, b, 2010; Smith et al., 2010; Pereira et al., 2010), and to correlate this scatter with the structure and thus hierarchical assembly history of the clusters. The backbone of the survey is the gravitational lensing analysis of HST (Smith et al., 2005; Richard et al., 2010b, Hamilton-Morris et al. in prep.; May et al. in prep.) and Subaru (Okabe et al., 2010; Oguri et al., 2010) imaging data, because the lensing-based mass maps can be used to infer the likely assembly history of the clusters (Smith & Taylor, 2008).

The parent sample for this study comprises 115 clusters satisfying , , drawn from the ROSAT All-sky Survey catalogs (Ebeling et al., 1998, 2000; Böhringer et al., 2004). The cut at ensures that the clusters are observable from Palomar Observatory at elevations above (an airmass of ). The clusters span a decade in X-ray luminosity in the band: (Fig. 1), which corresponds to a mass range of (Reiprich & Böhringer, 2002) – i.e. well-matched to .

Figure 1: The distribution of the parent sample of 115 clusters in the redshift plane. Filled data points indicate clusters that we observed in acceptable conditions (see §2.2) with WIRC on the Hale 200-in telescope, and are colour-coded as follows: black – also observed with both HST and Chandra; blue – also observed with HST, but not with Chandra; green – also observed with Chandra, but not with HST; red – observed with neither HST nor Chandra. Open data points were not observed with WIRC at Palomar, and therefore do not form part of the sample studied in this article. The dashed lines delineate the volume limited samples against which the observed sample is compared statistically in §2.5. The absence of clusters from the parent sample to the lower right is caused by the flux-limit of the ROSAT All-sky Survey.

The radius at which the mean enclosed density of a dark matter halo at the median redshift of the cluster sample () is is . To ensure that our results are comparable with previous studies of the luminosity gap statistic, data that probe out to () are required. This requirement is met by the Wide-field Infrared Camera (WIRC) on the Hale 200in Telescope at Palomar Observatory (§2.2). Traditionally, the luminosity gap statistic has been studied at optical wavelengths. In contrast, working in the near-infrared permits the use of colours as a surrogate for a photometric redshift estimate of cluster galaxies (§3.1), taking advantage of the relative insensitivity of near-infrared colours to spectral type (e.g. Mannucci et al., 2001). This is vital, in the absence of exhaustive spectroscopic catalogs, to weed out non-cluster members when calculating the luminosity gap statistic.

2.2 Ground-based Near-infrared Data

The parent sample of 115 clusters were used as a back-up observing program during observing runs with WIRC (Wilson et al., 2003) on the Hale 200in Telescope3 during observing runs spanning April 2004 to July 2005. Data were acquired when the full width half maximum (FWHM) of point sources exceeded 1 arcsec. In total, data were obtained on 78 clusters, with no pre-selection on cluster properties other than the X-ray selection described above (§2.1). Each cluster was observed with a single WIRC pointing. BCGs at have a typical angular extent of ; the individual exposures were therefore dithered within a box of full-width to minimise inclusion of BCG flux in sky-flats constructed from the science data. Each cluster was observed for a total of 600 sec per filter, split into 5 dither positions.

Figure 2: The colour-magnitude relation for A 1763. The horizontal lines show the region within which likely cluster galaxies were selected. The filled circle denotes the BCG.
Cluster [J2000] Redshift HST PID Also known as
A 68 00 37 05.28 09 09 10.8 0.255 26.65 0.25 8249
A 115 00 55 50.65 26 24 38.7 0.197 26.29 0.39 11312
A 141 01 05 37.17 24 40 49.7 0.230 26.54 0.40 10881 RXC J0105.52439
ZwCl 0104.40048 01 06 49.50 01 03 22.1 0.255 26.37 0.52 11312 Z 348
A 209 01 31 53.00 13 36 34.0 0.206 26.76 0.87 8249 RXC J0131.81336
A 267 01 52 48.72 01 01 08.4 0.230 26.56 1.43 8249 RXCJ0152.70100
A 291 02 01 43.11 02 11 48.1 0.196 25.49 1.79 8301 RXC J0201.70212
A 383 02 48 02.00 03 32 15.0 0.188 26.24 1.76 8249 RXC J0248.00332
RXC J0331.12100 03 31 05.87 21 00 32.7 0.188 27.07 0.96 10881
A 521 04 54 06.88 10 13 24.6 0.247 26.47 0.00 11312 RXC J0454.11014
A 586 07 32 20.42 31 37 58.8 0.171 26.31 0.51 8301
ZwCl 0740.41740 07 43 23.16 17 33 40.0 0.189 26.26 1.62 11312 Z 1432
A 611 08 00 55.92 36 03 39.6 0.288 26.90 1.28 9270
A 665 08 30 57.36 65 51 14.4 0.182 25.81 0.66
ZwCl 0839.92937 08 42 56.06 29 27 25.7 0.194 26.45 0.67 11312 Z 1883
ZwCl 0857.92107 09 00 36.86 20 53 40.0 0.235 25.82 0.34 8301 Z 2089
A 750 09 09 12.74 10 58 29.1 0.163 26.52 0.95 11312
A 773 09 17 54.00 51 42 57.6 0.217 26.68 0.47 8249
ZwCl 0923.65340 09 27 10.69 53 27 30.9 0.205 25.86 0.55 Z 2379
ZwCl 0949.65207 09 52 47.52 51 53 27.6 0.214 26.45 1.24 8301 Z 2701
A 901 09 56 26.40 10 04 12.0 0.163 25.90 0.08 10395 RXC J0956.41004
RX J1000.54409 10 00 31.16 44 08 42.5 0.153 25.72 0.35 10881
A 963 10 17 01.20 39 01 44.4 0.205 26.42 1.73 8249
A 1201 11 12 54.61 13 26 08.2 0.169 26.25 1.37 8719
A 1204 11 13 20.55 17 35 39.1 0.171 25.80 0.95 8301
A 1246 11 23 58.83 21 28 45.4 0.190 25.92 0.42 8301 RXCJ1123.92129
A 1423 11 57 17.43 33 36 38.6 0.213 26.16 1.77 8719
A 1553 12 30 48.95 10 32 45.6 0.165 26.82 0.94
ZwCl 1231.41007 12 34 17.45 09 45 58.1 0.229 26.21 0.71 8719 Z 5247
A 1634 12 54 01.84 06 42 14.4 0.196 25.82 0.61 RXC J1254.00642
A 1682 13 06 47.89 46 33 32.5 0.226 26.96 0.09 8719
ZwCl 1309.12216 13 11 46.15 22 01 36.8 0.266 26.18 2.17 Z 5768
A 1704 13 14 24.38 64 34 31.0 0.220 26.34 1.41
A 1758 13 32 44.47 50 32 30.5 0.280 26.26 0.22
A 1763 13 35 16.32 40 59 45.6 0.228 26.84 1.59 8249
A 1835 14 01 02.40 02 52 55.2 0.253 27.32 2.44 8249
A 1914 14 25 59.78 37 49 29.1 0.171 26.62 1.33 8301
A 1961 14 44 31.85 31 13 34.3 0.234 26.47 0.31
A 1994 14 56 13.48 05 48 56.6 0.220 26.45 0.55 RXC J1456.30549
MS 1455.02232 14 57 15.23 22 20 34.0 0.258 26.06 0.11 8301 ZwCl 1454.82233, Z7160
A 2009 15 00 19.63 21 22 08.9 0.153 25.99 0.17 8301
ZwCl 1459.44240 15 01 23.13 42 20 39.6 0.290 26.28 0.07 Z 7215
A 2111 15 39 40.51 34 25 27.0 0.229 25.58 0.37
A 2146 15 56 09.05 66 21 33.1 0.234 26.07 0.41 8301
A 2163 16 15 34.10 06 07 26.0 0.169 25.46 0.49
A 2204 16 32 46.94 05 34 31.3 0.152 25.82 0.11 8301
A 2218 16 35 52.80 66 12 50.4 0.171 26.12 0.32 5701
A 2219 16 40 22.56 46 42 21.6 0.228 26.62 1.18 6488
A 2254 17 17 45.96 19 40 48.0 0.178 26.05 0.88 8301
RX J1720.12638 17 20 10.14 26 37 30.9 0.164 26.37 1.76 11312
A 2261 17 22 27.24 32 07 56.7 0.224 26.34 2.32 8301
RXC J2102.12431 21 02 09.98 24 32 01.8 0.188 26.82 2.04
A 2345 21 27 13.73 12 09 46.1 0.176 26.67 1.09 11312 RXC J2127.11209
RX J2129.60005 21 29 40.02 00 05 20.9 0.235 26.78 1.93 8301
A 2390 21 53 36.72 17 41 31.2 0.233 26.21 1.50 5352
RXC J2211.70350 22 11 45.95 03 49 45.3 0.270 26.35 1.71
A 2485 22 48 31.13 16 06 25.6 0.247 26.59 0.00 11312 RXC J2248.51606
A 2537 23 08 23.20 02 11 31.0 0.297 26.19 0.63 9270 RXC J2308.32011
A 2631 23 37 39.82 00 16 16.9 0.278 26.35 0.64 11312 RXC J2337.60016

 Uncertainties on and are dominated by the uncertainties on the photometric calibration, which is  mag in J- and K-bands.

Table 1: The observed sample of clusters.

The data were reduced in a uniform and standard manner using an automated pipeline of iraf tasks to dark subtract, flat-field, align, and co-add the individual frames at the telescope. Data acquired in conditions worse than suffered strongly variable transparency and/or non-uniform background, and were therefore excluded from the analysis, leaving a total of 59 clusters with good quality data (Table 1). Astrometric and photometric calibration were achieved by reference to the 2MASS catalogs, to root mean square (rms) precisions of 1 arcsec and  magnitudes respectively (Stott et al., 2008). The results described in this article are insensitive to the uncertainty on the photometric calibration. An example colour magnitude diagram is shown in Fig. 2. The typical depth reached by the data is ; an galaxy has and a typical BCG has at .

2.3 Hubble Space Telescope Observations

Hubble Space Telescope (HST)4 imaging data are available through a broad red filter (F606W, F702W, and/or F814W) for 45 of the 59 clusters (Table 1) of which 13 are drawn from new LoCuSS ACS (PID:10881) and WFPC2 (PID:11312) observations. The reduction of the data on 10 clusters observed under PID:5701, PID:6488 and PID:8249 is described by Smith et al. (2005). Of the remaining 36 clusters, the 18 with WFPC2 data (PIDs:5352, 8301, 8719, 11312) were all reduced onto a pixel scale using wfixup, wmosaic, imshift and crrej tasks within iraf to clean, register and combine the individual exposures. Details of the reduction of the remaining clusters observed with ACS are described by Hamilton-Morris et al. (2010, in preparation).

2.4 Chandra X-ray Observations

Chandra X-ray observations are available for 41 of the total sample of 59 clusters. The reduction and analysis of these data are described in detail by Sanderson et al. (2009). In brief, for each cluster, an annular spectral profile was extracted and used to deproject the X-ray emission to measure the gas density and temperature in spherical shells. The phenomenological cluster model of Ascasibar & Diego (2008) was then jointly fitted to the temperature and density profiles to determine the mass profile, assuming hydrostatic equilibrium, following the procedures described in Sanderson & Ponman (2010). The model is based on a Hernquist (1990) density profile, which yields larger scale radii (and correspondingly lower mass concentrations) than the commonly-used NFW profile. Following Sanderson et al. (2009), we also use the logarithmic slope of the gas density profile at 0.04 (; Vikhlinin et al., 2007) as an indicator of cool core strength, which has also been shown to correlate with the substructure fraction of cluster cores, based on strong lens models (Richard et al., 2010b). A more negative value of indicates a steeper central gas density profile, and thus a stronger cool core, and vice versa.

2.5 Statistical Comparison of Sub-samples

Incomplete coverage of the parent sample of 115 clusters with WIRC, and heterogeneous coverage of the WIRC-observed clusters with other facilities (Fig. 1) may introduce subtle biases into our results. We therefore compare statistically the various observed sub-samples, including for completeness the sub-sample for which Spitzer data are available (§3.2). Specifically, the cluster X-ray luminosities are compared, after correction for the modest redshift evolution within the sample due to the expansion of the universe: , where following Evrard et al. (2002).

The mean X-ray luminosity of the full sample of 59 clusters is statistically indistinguishable from the mean luminosity of the sub-samples observed at other wavelengths (Table 2). We also draw 100,000 samples of 59 clusters at random from the combined volume-limited samples defined by , , and , (see Fig. 1). The average X-ray luminosity of the observed sample of 59 clusters is well within one standard deviation of the average X-ray luminosity of these randomized samples (Table 2). We therefore conclude that the distributions of the full sample of 59 clusters, the sub-samples observed with other telescopes, and the volume limited-sample defined above, are all statistically indistinguishable from each other. We therefore expect any biases to be negligible, and that our results can be treated as comparable with those that would be achieved with a volume-limited sample. We also take care to double check that the -distributions of the various observational sub-samples are statistically indistinguishable from each other in §3.

All clusters observed with WIRC
Clusters observed with WIRC & HST
Clusters observed with WIRC & Chandra
Clusters observed with WIRC & Spitzer
Mean of 100,000 samples drawn randomly from volume-limited sample

 The uncertainties are errors on the mean X-ray luminosity of each sample, with the exception of the last row, in which we quote the standard deviation of the 100,000 samples around the mean luminosity of all of these randomly drawn samples.

Table 2: Statistical Comparison of Sub-samples

3 Analysis and Results

3.1 Source Detection and Photometry

The - and -band frames were analyzed with SExtractor (Bertin & Arnouts, 1996), extracting all objects subtending at . The resulting catalogs were matched using a search radius comparable with the seeing disk, and point sources were excluded based on the stellarity index calculated by SExtractor. In the absence of spectroscopic redshift information we rely on the red ridge line of galaxies seen in the colour-magnitude diagrams for each cluster (e.g. Fig. 2) to isolate likely cluster galaxies. A simple model based on redshifting local galaxy spectral templates (King & Ellis, 1985) confirms that the colour of galaxies varies by between E/S0 and Scd spectral types. We therefore selected galaxies within  magnitudes of the BCG colour in each cluster as likely cluster members (see horizontal lines in Fig. 2).

The extended envelope of the BCGs typically spans a diameter of in the HST frames. In contrast, BCGs typically span just in the near-infrared frames. The difference is due to the brighter sky in the near-infrared relative to the optical. We therefore use the deep F702W HST/WFPC2 data available for 10 clusters in our sample (Smith et al., 2005) to estimate the -band flux lost due to the bright -band sky, under the assumption that the colour of BCGs does not vary significantly with radius on large scales. This assumption introduces negligible systematic uncertainty into our results because colour gradients in elliptical galaxies are measured to be (La Barbera et al., 2004, 2010), which translates into a possible  magnitude systematic error on the factor of 2 radial corrections to the -band photometry estimated below.

After masking out other galaxies from the data, the BCG - and -band light distributions are modelled using ellipse in the stsdas package in iraf. The -band model is then used to extrapolate the -band light distribution out to above the mean local background. The same procedure was applied to a sample of galaxies detected in the WFPC2 frame of each of these ten clusters. This analysis revealed that reliance on solely -band data causes the the total flux of BCGs to be under-estimated by , with a median of . This effect is much less severe for non-BCG’s, with total flux being under-estimated by , with a median of . We fit a straight-line to these data: , obtaining and . The correction, , was then applied to all galaxies within our sample. The amplitude of this systematic correction to the luminosity gap statistic measurements is therefore and is typically in the range with an uncertainty of , both of which are smaller than the bin-width in our subsequent analysis. Our results are therefore not significantly affected by the uncertainties on this correction.

3.2 Luminosity Gap Statistic of Clusters

Figure 3: Distribution of the observed luminosity gap (black points and solid line). The gray filled histogram is the expected distribution if the galaxies are drawn at random from a Schechter function following Dariush et al. (2007) (see §3.2 for more details). The dashed, dot-dashed, and dotted histograms show the -distributions of the sub-samples of clusters for which HST, Chandra, and Spitzer data are available.

In the absence of models of the mass distribution, and thus measurements of for all clusters in the sample we adopt a fixed projected physical radius of within which to calculate for each cluster. This aperture fits comfortably within the observed field of view for all clusters, and corresponds to for a cluster at . The distribution of the luminosity gap statistic is shown in Fig. 3; is a declining function of . We therefore fit a straight-line to the data: , weighting the data-points by where is the Poisson uncertainty on in each bin. The best-fit parameter values are: and . We also measure the fraction of “fossil clusters”: a total of 4 clusters have , yielding a fraction of clusters satisfying this selection of , where the error bar is at using binomial statistics (Gehrels 1986).

Following Dariush et al. (2007) we also show in Fig. 3 the distribution derived from a Monte Carlo simulation in which galaxies were drawn at random from a Schechter function with and , adopted from a fit of the Schechter function to the -band galaxy luminosity function of the Millennium semi-analytic catalogue, and is also consistent with observed luminosity functions (e.g. Lin et al., 2004). This simulation allows us to identify whether the distribution presents any excess probability over random statistical sampling of a common underlying luminosity function. Excess probability over random is only found at . We measure the observed probability of a cluster to have a luminosity gap of to be , compared with the estimated probability based on the Monte Carlo simulation of . We therefore detect an excess probability over random sampling at of at significance, and conclude that the distribution at has a physical origin.

The distributions of the sub-samples of clusters for which HST, Chandra, and Spitzer data are available are statistically consistent with that of the full sample of 59 clusters (Fig. 3). Two sample Kolmogorov-Smirnov (KS) tests that compare the HST, Chandra, and Spitzer sub-samples in turn with the full sample confirm that the probability of the respective sub-samples being drawn from a different underlying distribution than the full sample is in all cases, with the largest difference between the cumulative distributions being , between the Chandra sub-sample and the full sample.

Figure 4: Absolute -band magnitude of the first (black circles) and second ranked (red triangles) galaxies as a function of luminosity gap. The solid black and red lines show the best-fit straight-line to the data – see §3.2 for more details. The horizontal dashed line is at , the absolute magnitude of an galaxy, taken from Lin et al. (2004).

We also look at how the absolute magnitude of the first and second ranked cluster galaxies vary with (Fig. 4). The luminosity of the first ranked galaxy increases very slowly with , remaining in the range across the full range of . In contrast, the luminosity of the second ranked galaxy declines from at to at . We characterize these trends by fitting the following relations to the data: and , where the numerical subscripts denote the first and second ranked galaxies respectively. The best-fit values are: , , and , . Empirically large luminosity gap statistics are therefore due to both an over-bright BCG, , and an under-bright second ranked galaxy, . The relative faintness of second ranked galaxies in large luminosity gap clusters supports the idea that the growth of dominant BCGs is driven by the merging of luminous cluster galaxies with the BCG. Indeed the current SFR of BCGs discussed above lends additional support – the BCG in a cluster with a luminosity gap of is more luminous and has a stellar mass of more than the second ranked galaxy. Just two of the four clusters with in Fig. 5 host an active BCG. The most active of these, A 1835, is forming stars at (Egami et al., 2006), and the other, RXJ 2129.60005, is forming stars at (Quillen et al., 2008). These two BCGs would therefore have to form stars continuously at this rate for and  years respectively for their large luminosity gap to be caused exclusively by gas cooling and consequent star formation.

Finally, we note that on average second ranked galaxies in clusters with have where the uncertainty is the rms scatter around the mean. Lin et al. (2004) measured for cluster galaxies at , in agreement with similar studies of field galaxies and of higher redshift clusters (De Propris et al., 1999; Cole et al., 2001). The distribution of luminosities of second ranked cluster galaxies in clusters with is therefore statistically consistent with them being galaxies. This contrasts with low mass systems, i.e. fossil groups, in that galaxies are absent from low mass systems. This difference is probably due, at least in part, to the relative inefficiency of galaxy merging in massive clusters.

3.3 Comparing Luminosity Gap with Cool Core Strength

Figure 5: The gradient of the logarithmic gas density profile at versus luminosity gap for 41 clusters that have also been observed with Chandra and studied by Sanderson et al. (2009). Blue stars correspond to clusters with an H emitting BCG (see Sanderson et al., 2009); blue stars with a black outline have also been identified as hosting a BCG that is forming stars at using Spiter/MIPS observations; filled red circles denote clusters with BCGs that are not H emitters and are forming stars at ; open black circles indicate clusters that have not been observed with Spitzer.

To explore further the physical origin of large luminosity gaps we plot versus , the slope of the logarithmic gas density profile at , for 41 clusters that have also been observed with Chandra in Fig. 5. The measurements of are based on Sanderson et al.’s (2009) analysis of the Chandra data (§2.4). At the clusters span the full range of cool core strengths: . This dynamic range shrinks to just at – the clusters with large luminosity gaps also host relatively strong cool cores. We also identify star-forming BCGs in Fig. 5. It has long been known that H emission from the BCG is closely associated with the presence of significant central cooling in the cluster core (e.g. Heckman, 1981; Crawford et al., 1999). More recently, Sanderson et al. (2009) found in their sample of 65 clusters that H emitting BCGs occur exclusively in those clusters with the most cuspy inner gas density profiles (), and where the projected offset between the X-ray centroid and the BCG is . The same is true of the five BCGs with star formation rates (SFR) of , based on mid-infrared observations with Spitzer/MIPS – this SFR corresponds to a flux of from a BCG at . These measurements are drawn from the literature (Egami et al., 2006; Quillen et al., 2008) and our own measurements using data from Cycle 4 (PID:40827, PI: Smith; PID: 41011, PI: Egami) the details of which will be published elsewhere (Egami et al., in prep.). Fig. 5 therefore confirms that cool core clusters tend to host actively star-forming BCGs (e.g. Edge et al., 1999; Egami et al., 2006; Quillen et al., 2008). However, cool core clusters () with active BCGs (, and/or H emission) are found across the full range of in Fig. 5.

Figure 6: Example isophotal shape profiles. From left to right the BCGs are classified as boxy, disky, pure ellipse and unclassified. The vertical line at the left is set to the FWHM of point sources and the one at the right indicates the half-light radius of the BCG. Note that to keep the analysis simple and conservative, no flux was masked out of the HST data. So, for example, the BCG in A 1201 was unclassified because of the impact of the gravitational arc at a BCG-centric radius of (Edge et al., 2003) on the isophotal analysis.

These results are consistent with the interpretation of large luminosity gap clusters as objects that formed relatively early, and subsequently developed a large luminosity gap through the merging of bright cluster galaxies with the BCG. A similarly long period of time – a few Gyr – is required to form a strong cool core following cluster formation. Conversely, if all clusters with smaller luminosity gaps formed more recently than those with larger gaps, and thus have had insufficient time to form a large luminosity gap and a cool core, then they should all host relatively weak cool cores. However this is not the case. This can be understood if the so-called “fossil” status of a large luminosity gap cluster is not the end-point of its evolution. If bright () galaxies fall into a cool core “fossil” cluster, then that cluster would move immediately leftward from the bottom right of Fig. 5. As the in-falling system (presumably a group) reaches the cluster core later, it may disrupt partially or fully the cooling of gas onto the BCG, and cause the cluster to move vertically in the plane. This scenario naturally explains the triangular distribution of points in Fig. 5, and is consistent with hierarchical infall (i.e. mergers) playing a role in regulating cooling in cluster cores.

To place this discussion on a more quantitative footing we adopt a strategy that we return to often in §3 – we split the sample into low- () and high- () sub-samples and perform a two sample KS test on the cumulative distribution of the other variable, in this case . The hypothesis that high- clusters are drawn from the same underlying distribution as low- clusters is rejected at just confidence, i.e. slightly over significance, based on a maximum difference between the cumulative distributions of . In the absence of a decisive test, we therefore divide the sample at , i.e. a more extreme value of , attempting to identify roughly the luminosity gap at which the distribution diverges from that of lower- clusters. This time the two sample KS test rejects the null hypothesis at confidence – i.e.  – based on a maximum difference between the respective cumulative distributions of .

3.4 Comparing the Luminosity Gap with BCG Morphology

Cluster Effective Extremum Method Mean Method
radius Classification Classification
A 68 Boxy Boxy
A 115 Unclassified Disky
A 141 Disky Disky
ZwCl 0104.40048 Unclassified Elliptical
A 209 Disky Disky
A 267 Disky Disky
A 291 Disky Disky
A 383 Unclassified Unclassified
RXC J0331.12100 Unclassified Unclassified
A 521 Boxy Elliptical
A 586 Unclassified Elliptical
ZwCl 07401740 Elliptical Boxy
A 611 Unclassified Boxy
ZwCl 0839.92937 Unclassified Disky
ZwCl 0857.92107 Boxy Boxy
A 750 Disky Unclassified
A 773 Boxy Boxy
ZwCl 0949.65207 Disky Elliptical
A 901 Disky Disky
RX J1000.54409 Unclassified Boxy
A 963 Unclassified Disky
A 1201 Unclassified Elliptical
A 1204 Elliptical Elliptical
A 1246 Unclassified Unclassified
A 1423 Unclassified Unclassified
ZwCl 1231.41007 Boxy Boxy
A 1682 Elliptical Boxy
A 1763 Elliptical Elliptical
A 1835 Unclassified Disky
A 1914 Elliptical Disky
MS 1455.02232 Unclassified Boxy
A 2009 Unclassified Elliptical
A 2146 Unclassified Elliptical
A 2204 Unclassified Unclassified
A 2218 Disky Disky
A 2219 Unclassified Disky
A 2254 Unclassified Elliptical
RXJ 1720.12638 Unclassified Unclassified
A 2261 Unclassified Unclassified
A 2345 Elliptical Elliptical
RX J2129.60005 Unclassified Elliptical
A 2390 Disky Disky
A 2485 Elliptical Elliptical
A 2537 Elliptical Elliptical
A 2631 Elliptical Boxy
Table 3: Results from Isophotal Analysis of Brightest Cluster Galaxies using HST data.

We use the high angular resolution HST imaging observations of the 45 clusters discussed in §2.3 to measure the isophotal shape of the BCGs in these clusters. The ellipse task in IRAF was used to measure the fourth Fourier coefficient () of the light distribution. This coefficient indicates whether the galaxy has a disky or boxy shape (Bender, 1988). In Fig. 6 we show the profile of four BCGs to illustrate the diversity within the sample. Following Bender et al. (1989) we tried to use the extremum value of (i.e.  in Table 3) to classify galaxies as either disky () or boxy (). If the profile passes through a stationary point, then the extremum is obtained by finding the maximum or minimum value of in the radial range enclosed by the FWHM of point sources and the effective radius derived from a de Vaucouleurs profile fit. In the absence of a stationary point, the extremum value of is the value at the effective radius, under the assumption that is a monotonic function of radius. However is in general not a monotonic function of radius for BCGs in our sample, even for those with isophotes that have, on average, boxy and disky isophotes (Fig. 6). For these reasons, the isophotal shapes of 22 out of 45 BCGs cannot be classified based on . We also find some clusters (e.g. A 1204 – see Fig. 6) in which is consistent with zero across the full radial range of the data.

We therefore implement a modified scheme, in which we calculate the error-weighted mean value of in the same radial range as above, with no weighting of the bins to account for the variation of the bin solid angle as a function of radius. BCGs with consistent with zero within the uncertainties were classified as elliptical, otherwise BCGs are classified as Boxy or Disky if or respectively. Finally, a BCG is “Unclassified” if the error on is comparable with the dynamic range of the data, i.e. . BCG morphologies derived under both Bender et al.’s “extremum” scheme and our own “mean” scheme are listed in Table 3 along with the Boxy/Disky/Elliptical/Unclassified classification based on each method. The respective methods agree on morphological classification for 16 of the 22 BCGs for which classification was possible under both methods. However, only three of the six discrepant BCGs have and values that formally disagree between the methods within the quoted uncertainties – A 521, A 750 and ZwCl 0949.65207. The important advantage of our method is that classification is possible for an additional 15 BCGs that were unclassifiable under the Bender et al. scheme. We therefore adopt as our measure of BCG morphology for all clusters with HST data for the reasons outlined above regarding the general absence of clearly defined stationary points and monotonic behaviour of the profiles.

In summary, out of 45 clusters, 10 are classified as Boxy, 13 as Disky, 14 as Elliptical, and 8 are Unclassified. In Fig 7 we plot versus , the most striking feature of which is the lack of clusters with large and negative , i.e. boxy BCGs appear not to live in large luminosity gap clusters. As in §3.3, we split the clusters into low- () and high- () samples and perform a two-sample KS test. The low- and high- samples contain and clusters respectively, with a maximum difference between their cumulative -distributions of . The hypothesis that the low- and high- samples are drawn from the same underlying distribution is therefore disfavoured at confidence, i.e. . Unlike the situation for the analysis of the distributions of high- and low- clusters in §3.3, the significance with which the null hypothesis is rejected does not increase if the sub-samples are re-defined by splitting the full sample at . This is obvious from a comparison of Figs. 5 & 7, and suggests that the relationship between and BCG morphology is stronger than between and cool core strength.

Figure 7: Luminosity gap statistic () versus error-weighted mean fourth Fourier component of the BCG light distribution (). Positive values of correspond to Disky BCGs; negative values correspond to Boxy BCGs; values consistent with zero are consistent with elliptical isophotes. Clusters with host BCGs with both Boxy and Disky isophotes. In contrast clusters with host only non-Boxy (i.e. Elliptical or Disky BCGs).

3.5 Comparing the Luminosity Gap with Cluster Concentration

Figure 8: Concentration versus luminosity gap for 41 clusters for which the X-ray-based mass profiles are available from Sanderson et al.’s (2009) analysis of archival Chandra data.

To investigate the possibility that high- clusters formed at earlier times than low- clusters, we explore the relationship between and the shape of the cluster dark matter halos via the concentration parameter. In Fig. 8 we plot versus concentration, for the 41 clusters with available Chandra data Sanderson et al. (2009)2.4). The distribution is similar to the distribution in that the lower-right of both plots is empty, and that clusters with span the full dynamic range in the vertical axis. To quantify this we again perform a two sample KS test, on the and sub-samples. In this case the low- and high- samples contain and clusters respectively, with a maximum difference between their cumulative -distributions of . Acceptance/rejection of the null hypothesis that low- and high- clusters are drawn from same underlying -distribution therefore have roughly equal probability. However if we modify the definition of the low- and high- sub-samples by splitting the full sample at we are able to reject the null hypothesis at . We therefore conclude that the plane qualitatively supports the interpretation of the plane, however statistically this is not decisive. Specifically, clusters with a large luminosity gap tend to have a relatively large concentration parameter, although there is a curious deficit of clusters with and . Clusters with lower luminosity gaps plausibly comprise both clusters that formed more recently than clusters with large gaps – and thus have lower concentration parameters – and clusters that used to have a large luminosity gap, and thus formed early, and have a higher concentration parameter, but that then suffered infall of bright () galaxies. Put another way, the existence of clusters in the top left corner of Fig. 8 is consistent with the timescale on which the concentration parameter of a cluster may be reset following a cluster-cluster merger being long compared with the infall timescale of .

3.6 Comparing the Luminosity Gap with Cluster Substructure

Figure 9: Luminosity gap versus substructure fraction measured within in 10 clusters from our sample by Smith et al. (2005).

Measurements of the substructure fraction (), i.e. the fraction of the total cluster mass that resides in substructures, are available for ten of the clusters (Smith et al., 2005) from our sample of 59. Smith et al.’s gravitational lens models include mass components that account explicitly for substructures required to reproduce the observed positions of multiply-imaged background galaxies – these substructures comprise both galaxy group and individual galaxy masses. We plot versus for these ten clusters in Fig. 9, revealing a relationship between these quantities in the sense that clusters with simpler gravitational potentials (low ) have more dominant BCGs (high ), and vice versa. To quantify this relationship, we fit a simple model to the data: , and obtain best-fit parameters of: and . This result is consistent with that found by Richard et al. (2010b), despite the smaller aperture of used in their study. This consistency arises because the typical projected separation of the first and second ranked galaxies in our sample is . We also double-check that the distribution of the 10 clusters in Fig. 9 is consistent with that of the full sample, finding a maximum difference between the cumulative distributions of , indicating roughly equal probability of rejection/acceptance of the hypothesis that the two samples are drawn from different underlying populations.

3.7 Summary

We now summarize the comparison of our luminosity gap measurements with other probes of the structure, and thus the age and assembly history of clusters, and discuss the interpretation of these results.

The clearest empirical relationship found is between the and in the sense that clusters with a dominant BCG () have a lower substructure fraction () and vice versa. The strong correlation between and is in stark contrast with the triangular distributions of clusters in the -, -, and - planes. A simple physical interpretation of the - relation is that both quantities are sensitive to the same thing. As galaxies and groups of galaxies fall into clusters the light emitted by the galaxies will either cause to decrease or stay the same, depending on how bright the infalling galaxies are. At the same time the total mass of these galaxies and the group-scale halos within which they may be embedded causes to increase. Early studies discussed the idea that galaxy groups with may have formed at earlier times than groups with . However, more recently, a variety of studies have shown that both and are correlated with both the formation epoch of the host dark matter halo, and the recent hierarchical assembly history of the halo (Dariush et al., 2007, 2010; Smith & Taylor, 2008). Therefore both theoretical and observational studies across a broad range of dark matter halo mass are converging on the view that “fossil” status is not an end-point in the evolution of galaxy systems that formed early. Rather it is a phase that a galaxy system can evolve through if it formed early and then suffered minimal hierarchical infall after the formation of a bright massive central galaxy. The triangular distribution of clusters in the -, -, and planes are all consistent with this interpretation, and inconsistent with the idea that fossil galaxy systems are evolutionary cul-de-sacs. Specifically, if a cluster forms early and then sufficient time elapses for a large luminosity gap to form via merging of gas-rich galaxies to form the BCG, and for a cool core to form, then this cluster will reside in the top-right corner of Figs. 7 & 8 and the bottom right of Fig. 5. If a galaxy then falls into the cluster, either on its own or in a group, then the cluster would move left-ward in all of Figs 5, 7, and 8 as soon as the infalling galaxy system crosses the aperture within which is measured (in our case ). Several Gyr later the infalling structure will reach the center of the cluster, and its merger with the cluster may be sufficiently energetic to modify the strength of the cluster cool core, the shape of the BCG, and the concentration of the cluster dark matter halo. In this way, clusters can move vertically in Figs. 5, 7, and 8, and produce the observed triangular distribution of clusters.

Figure 10: Strength of the cool core in each cluster, as measured by the slope of the logarithmic gas density profile at from Sanderson et al. (2009) versus the . The absence of a relationship between and suggests that disky BCG isophotes are more likely caused by such BCGs being formed from mergers between gas rich galaxies than by cooling of gas onto the BCG. The typical error bar on is .

The interpretation of non-boxy morphologies () of BCGs in clusters with large luminosity gaps is an important element of the discussion above. Khochfar & Burkert (2005) showed that the morphology of early-type galaxies is sensitive to the morphology (indicative of gas content) of their progenitors and subsequent gas infall. The straightforward interpretation of the observables is therefore that dominant BCGs formed from mergers of gas rich (presumably spiral) galaxies and/or have accreted gas since the last major merger in their assembly history. Formation of dominant BCGs from gas rich progenitors is consistent with the early formation of these BCGs as discussed above, because at earlier times the galaxies from which BCGs formed would have been more gas rich than at later times.

To disentangle the relative contribution of gas rich mergers and accretion of gas to the disky shape of some BCGs we plot in Fig. 10 , the slope of the logarithmic gas density profile at versus . If BCG morphology is strongly influenced by gas cooling onto the BCG then one would expect a relationship between and in the sense that disky BCGs () would live in clusters with a steep central () gas density profile. This is because clusters with steep central gas density profiles host a cool core – i.e. a central positive temperature gradient, absence of an entropy core, and a cooling timescale short compared with the age of the universe. However we do not find any strong relationship between and in Fig. 10. We divide the 41 clusters with Chandra data into those with the strongest cool cores – – and the rest. A two sample KS test on these two sub-samples yields a maximum difference between the cumulative -distributions of , indicating roughly equal probability of accepting/rejecting the hypothesis that the two distributions are drawn from the same underlying distributions.

In the absence of a strong relationship between cool core strength and BCG morphology, we therefore conclude that BCG morphology is more sensitive to the gas content of the galaxies that merged to form it, than to the subsequent gas accretion history of the BCG. This view is consistent with the comparison of , and BCG activity in Fig. 5 and the discussion of the dependence of on in §3.2. The key point being that the merging of luminous cluster galaxies to form the BCG appears to have a much stronger influence on luminosity gap than gas cooling and subsequent star formation within BCGs.

4 Comparison with Theoretical Predictions

Modern galaxy formation and evolution models contain physical prescriptions for many physical processes relevant to the formation and evolution of galaxies, including dynamical friction, conversion of cold gas into stars during galaxy mergers, and AGN feedback. These processes are particularly important in the centres of galaxy clusters where they regulate the cooling of gas onto the most massive galaxies in the universe – BCGs. However the models were not constrained by the luminosity gap distribution; our observational results can therefore provide a strong test of the models.

Figure 11: Distribution of the observed luminosity gap (black points – see also Fig. 3) compared with the same for clusters with , measured within a projected BCG-centric radius of using the the Millennium simulation-based semi-analytic galaxy formation models of Bower et al. (2006), Croton et al. (2006), and de Lucia & Blaizot (2007). The error bar on each bin in the theoretical histograms is comparable with the observational errors.

Figure 12: Absolute -band magnitude of the first (black circles) and second ranked (red triangles) galaxies as a function of luminosity gap from the three semi-analytic galaxy formation models discussed in §4. The solid black and red lines show the best straight-line fit to the observational data shown in Fig. 4. The horizontal dashed line in each panel is at , the absolute magnitude of an galaxy, taken from Lin et al. (2004).

We compare our observations with the Bower et al. (2006), Croton et al. (2006), and de Lucia & Blaizot (2007) semi-analytic models, all of which are based on the Millennium Simulation5 – a cosmological numerical simulation of dark matter in a volume spanning containing particles. An important difference between the models is that the Bower et al. model implements “quasar” mode AGN feedback, whereas the Croton et al. and de Lucia & Blaizot models implement “radio” mode AGN feedback. We also note that de Lucia & Blaizot compared their model predictions with the observed properties of BCGs, however they didn’t compare with observed luminosity gaps.

First we select dark matter halos from the Millennium dark matter friends of friends catalogue. Within the whole simulated volume, 209 hales were found with masses greater than , i.e. above the mass threshold of the observed sample. We then extracted galaxies in these 209 halos from the semi-analytic galaxy catalogues based on each of the three models. The -band luminosity gap was computed for each halo within a projected cluster-centric radius of . The predicted luminosity gap statistic distributions are over-plotted on the observed distribution in Fig 11.

The observed distribution is consistent, within the uncertainties with a monotonically declining function of 3.2). The Bower et al. model matches this observational result well, and the predicted fraction of clusters with the most extreme luminosity gaps is , just below the observed fraction of 3.2). In contrast, the Croton et al. model peaks at – i.e. it does not predict a monotonic decline of – however it predicts which is in excellent agreement with the observations. The de Lucia & Blaizot model predicts a yet more prominent peak at , and a yet higher fraction of clusters with extreme , , that disagrees with the observations at .

Following the same approach as in §3.2, we also decompose the predicted distributions into the predicted absolute magnitudes of the first () and second () ranked galaxies (Fig. 12). The most striking feature of this figure is that the slopes of versus and versus are much steeper and shallower than the observations respectively in the Bower et al. model. In contrast, the Croton et al. and de Lucia & Blaizot models succeed much better in reproducing the observed trends. Interestingly, the discrepant trends in and within the Bower et al. model conspire to produce a distribution of in Fig. 3 that is in good agreement with observations.

The absolute magnitudes of BCGs span in the Bower et al. model, in contrast to the observed range of . As BCGs grow, the largest increase in luminosity from purely ingesting another galaxy is a brightening by , i.e. a merger between the brightest two galaxies in a cluster with . The very large spread in for BCGs in the Bower et al. model therefore indicates that the conversion of cold gas into stars is too efficient in their model. In the model, most of the mergers that form BCGs are between gas poor galaxies. The main source of gas for formation of new stars is that which cools from the intracluster medium. The steep relationship between and therefore implies that AGN feedback in BCGs is too weak in the Bower et al. model. An important caveat on this interpretation is that we showed in §§3.4 & 3.7 that clusters with large luminosity gaps () have non-boxy isophotes and therefore likely formed from mergers of gas rich galaxies, i.e. probably at higher redshift than the BCGs in the model.

The shallow slope of the relationship between and in the Bower et al. model implies that the replenishment of the supply of cluster galaxies that are ingested into their respective BCGs is too efficient in this model. Specifically, the difference in slopes of between Bower et al. and the other two models could arise from differing treatments of the merging of galaxies in the respective models following the time at which individual galaxy halos lose their identity following ingestion into the parent cluster halo. We also comment, more generally, that the galaxies in Bower et al.’s model tend to be less luminous than the observed galaxies by , and those in Croton et al.’s model tend to be over-luminous by . This suggests that the strength of feedback in the general cluster population may be too strong in the former and too weak in the latter model.

For completeness, we also compare our measurement of the fraction of clusters that satisfy with predictions from Milosavljević et al.’s (2006) analytic model. Our measurement of is well within of Milosavljević et al.’s prediction of . The most obvious difference between their model and our observations is that the prediction is calculated within the cluster virial radii, in contrast to our calculation within a projected cluster-centric radius of . The larger volume within each cluster probed by Milosavljević et al. will reduce the probability of finding clusters with large luminosity gap statistics. The same authors also estimate the fraction of clusters with using data from the Sloan Digital Sky Survey (Miller et al., 2005), obtaining a similar fraction to their prediction. The possible disagreement between this estimate and our own is harder to understand because both use a similar physical aperture for the calculation of . We note, however, that the two observed cluster samples are selected in different ways; our sample is X-ray selected whilst SDSS is optically selected.

5 Conclusions

We have combined wide-field near-infrared imaging from the WIRC camera on the Hale 200in telescope, with HST, Chandra, and Spitzer observations of 59 massive galaxy clusters at to explore the connections between the formation histories of BCGs and the galaxy clusters that they inhabit. This large statistical sample is intended to be representative of the underlying population of massive X-ray luminous clusters. Extensive tests confirm that results based on this sample can be regarded as statistically compatible with those from a complete volume-limited sample. Our main empirical results are as follows:

(i) We have made the first observational measurement of the distribution of the luminosity gap statistic, , of massive clusters. The probability distribution of the luminosity gap statistic is a monotonically declining function of , well described by the relation with and .

(ii) Following Dariush et al. (2007) we used Monte Carlo simulations to quantify the fraction of clusters with large luminosity gaps expected from random sampling of a Schechter function. The observed distribution exceeds the statistical distribution derived from the Monte Carlo simulation at at significance, confirming that the most extreme luminosity gaps have a physical origin, and are not statistical flukes.

(iii) Four of our sample of 59 clusters have extreme luminosity gaps of – ZwCl 1309.12216, A 1835, A 2261, and RXC J2102.12431 – which equates to a fraction of clusters that have of .

(iv) The morphology of 45/59 BCGs was measured by analyzing the shape of the BCG isophotes in archival and new HST observations of the cluster cores. The split between boxy, elliptical and disky isophotes is: 22% boxy, 32% elliptical, 29% disky, with 17% unclassified.

(v) A strong correlation is found between and , the fraction of mass in the cluster cores associated with group- and galaxy scale dark matter halos, the latter coming from published gravitational lens models of the cluster cores (Smith et al., 2005). The relationship between and is parameterized thus: , with best fit parameters ad .

(vi) Clusters with large luminosity gaps, , have cuspy gas density profiles, and thus relatively strong strong cool cores (, where is the logarithmic gas density profiles at ), elliptical or disky BCGs (, where is the fourth-order Fourier co-efficient of the optical isophotes), concentrated dark matter density profiles (, where is based on a Hernquist 1990 model fit to the Chandra data), and small substructure fraction (, where is based on strong lens modeling of the mass distribution).

(vii) In contrast, clusters with small luminosity gaps, , span the full range of observed cool core strengths (), span the full range of boxy, elliptical, and disky BCG morphologies (), span the full range of concentrations (), and have large substructure fractions ().

Clusters with are therefore a more homogeneous population than clusters with . The stronger cool cores, more concentrated mass distribution, and non-boxy BCGs, all point towards high- clusters forming at early times. Such early formation is required to allow sufficient time to pass for the BCG to ingest (aided by dynamical friction) the bright cluster galaxy population in order to develop the large luminosity gap, and for the establishment of the cool core. The formation of more concentrated dark matter halos at earlier times than less-concentrated halos is a generic prediction of cold dark matter theory (e.g. Neto et al., 2007). The interpretation of disky BCGs is less straightforward, however such morphologies can plausibly be interpreted as evidence for the last major mergers in a BCG’s formation history comprising gas-rich galaxies – the presence of gas thus leading to the establishment of a disk-like structure in the BCG. This gas-rich merger scenario for BCG formation is consistent with the early formation of large- clusters.

How can the heterogeneous population of low- clusters, and more specifically, the fact that some low- clusters have strong cool cores, non-boxy BCGs, and high concentrations, be interpreted within the context of the early formation of high- clusters? The most natural explanation is that large- clusters can evolve into low- clusters when the supply of bright cluster galaxies is replenished by episodes of hierarchical infall of smaller galaxy systems, such as galaxy groups. Such infall would depress and increase immediately that the group entered the measurement aperture (in this case a clustercentric radius of ), and would modify other cluster properties such the cool core strength, BCG morphology, and concentration of the mass distribution on longer timescales of several Gyr. The observed heterogeneity of low- clusters can therefore be explained by these clusters comprising both (i) clusters that have formed more recently, and thus have a low concentration, haven’t had time to develop a large luminosity gap and cool core, and have a BCG formed from relatively gas-poor mergers, and (ii) clusters that formerly had a large luminosity gap, and have suffered hierarchical infall in the previous few Gyr. We therefore conclude that a large luminosity gap (and large substructure fraction) is a phase through which a cluster can evolve if sufficient time elapses between episodes of hierarchical merging of other galaxies and groups of galaxies with the cluster. The large scatter seen in the theoretical age- and age- relationships (Dariush et al., 2007, 2010; Smith & Taylor, 2008) lend further weight to the view that both the age and the recent merger history of a cluster contribute to the observed values of and .

We also compare our observational results with predictions from Millennium simulation-based semi-analytic models of galaxy evolution. We find that none of the models can successfully reproduce the observations in their entirety. Bower et al. (2006) succeeds best at reproducing the monotonically declining , however they predict a relationship between BCG luminosity and that is far too steep. In contrast, both Croton et al. (2006) and de Lucia & Blaizot (2007) predict that peaks at , in disagreement with the observations, with de Lucia & Blaizot predicting the more prominent peak. de Lucia & Blaizot also predict , in contrast to the observed value of . Nevertheless, both Croton et al. and de Lucia & Blaizot match the observed slope of the relationship between BCG luminosity and very well. We discuss the possible causes of these disagreements, and suggest that Bower et al.’s model may be too efficient at converting cold gas to stars in BCGs, and may also to be too efficient at replenishing the supply of galaxies in clusters.

We also note that semi-analytic galaxy evolution models also fail to reproduce observational results on high redshift BCGs (Collins et al., 2009; Stott et al., 2010). Our new results add to this picture of the inability of models to reproduce observations of BCGs. An important strength of our results is that we do not rely on calculations of the stellar mass of BCGs, and thus are insensitive to possible systematic uncertainties in stellar mass estimates for observed BCGs arising from alternative stellar population models.

Our future work on the hierarchical assembly of clusters at will take advantage of the wide-field multi-wavelength dataset that we are assembling, including mid/far-IR observations with Spitzer and Herschel, joint strong/weak-lens modeling of the cluster mass distributions, our spectroscopic redshift survey of cluster galaxies with MMT/Hectospec, and X-ray observations with XMM-Newton and Chandra.


We acknowledge helpful comments from the anonymous referee. We thank our LoCuSS collaborators, in particular Alastair Edge, Victoria Hamilton-Morris, Jean-Paul Kneib, Yuying Zhang, and Nobuhiro Okabe, for encouragement, assistance and many stimulating discussions. HGK, GPS, AJRS, TJP, and JPS acknowledge support from PPARC and latterly from STFC. GPS acknowledges support from the Royal Society. GPS thanks Andrew Benson, Richard Bower, Gabriella de Lucia, and Malcolm Bremer for helpful discussions and comments; Kevin Bundy, Brad Cenko, Chris Conselice, Richard Ellis, Avishay Gal-Yam, Sean Moran, David Sand and Keren Sharon for assistance with acquiring some of the near-infrared data presented in this article; and Rick Burruss and Jeff Hickey for their support at Palomar Observatory.


  1. pagerange: LoCuSS: Connecting the Dominance and Shape of Brightest Cluster Galaxies with the Assembly History of Massive ClustersLoCuSS: Connecting the Dominance and Shape of Brightest Cluster Galaxies with the Assembly History of Massive Clusters
  2. pubyear: 2009
  3. The Hale Telescope at Palomar Observatory is owned and operated by the California Institute of Technology.
  4. Based on observations with the NASA/ESA Hubble Space Telescope obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5–26555.
  5. The Millennium Simulation used in this paper was carried out by the Virgo Supercomputing Consortium at the Computing Centre of the Max-Planck Society in Garching. The semi-analytic galaxy catalogue is publicly available at


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