Mass–Richness Relations for X-Ray and SZE-selected Clusters at 0.4 < z < 2.0 as Seen by Spitzer at 4.5 μm

The starting point for this sample is the Meta-catalog of X-ray detected clusters of galaxies (MCXC), a catalog of compiled properties of X-ray detected clusters of galaxies (Piffaretti et al. 2011, and references therein). This catalog is based on the ROSAT All Sky Survey (Voges et al. 1999) data on 1743 clusters at 0.003 < z < 1.261 that have been homogeneously evaluated within the radius, R₅₀₀, corresponding to an overdensity of 500 times the critical density. For each cluster, the MCXC provides redshift,⁴ coordinates, R₅₀₀, and X-ray luminosity in the 0.1–2.4 keV band, L_{500,[0.1–2.4keV]}. Based on the values published in Piffaretti et al. (2011), we also derive the angular size, θ₅₀₀ = R₅₀₀/DA(z) for each cluster, where DA(z) is the angular diameter distance.

In order to define a richness parameter to be used as a proxy for cluster mass, M₅₀₀, we need to define the aperture radius in which galaxies should be counted and the redshift range for which this radius is still representative of the cluster R₅₀₀. To this aim, in Figure 1, we show the entire MCXC sample θ₅₀₀ versus redshift relation. The red horizontal line indicates the 2.5 arcmin radius of a single Spitzer/IRAC field of view. The red asterisks indicate the mean θ₅₀₀ per redshift bin of Δz = 0.1, the error bars are the standard deviation of the mean per redshift bin. We note that at z > 0.4 (dot–dashed line) the average θ₅₀₀ of the sample is included within the Spitzer/IRAC field of view. Therefore we adopt this lower redshift cut to the cluster samples considered in our study. There are 142 clusters in MCXC at z > 0.4.

Zoom In Zoom Out Reset image size

The most reliable X-ray masses are obtained by solving the equation of hydrostatic equilibrium, which requires measurements of the density and temperature gradients of the X-ray emitting gas (see discussion in Maughan 2007). This is only possible for nearby bright clusters, therefore it remains a challenge for the majority of clusters detected in X-ray surveys, especially at high redshifts where surface brightness dimming effects become significant. Thus, in most cases, cluster masses are estimated from simple properties such as X-ray luminosities (L_X) or from adopting a single global temperature (kT) via the calibration of scaling relations.

To derive an estimate of the total cluster mass, M₅₀₀, within R₅₀₀, we adopt the most recent calibrations, in particular in their redshift evolution, of the relations between X-ray global properties and cluster total mass, as presented in Reichert et al. (2011). Reichert et al. (2011, see also references therein) obtained these relations by homogenizing published estimates of X-ray luminosity and total mass. These values were rescaled at different radii and overdensities by using their dependence upon the gas density, which was described by a β-model (Cavaliere & Fusco-Femiano 1976).

Reichert et al. (2011) scaling relations, together with the MCXC luminosities, are then used here to run the following iterative process: (i) an input temperature is assumed; (ii) a conversion from the MCXC L_{500,[0.1–2.4keV]} luminosity to the pseudo-bolometric (0.01–100 keV) value, L_500,bol, is derived assuming the thermal apec model in XSPEC (Arnaud 1996), adopting the temperature assumed at step (i) and a metal abundance of 0.3 times the solar value; (iii) a value of the mass within an overdensity of 500 with respect to the critical density of the universe at the cluster's redshift is then calculated from Equation (26) in Reichert et al. (2011),

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{M}_{500,{\rm{X}}}}{{10}^{14}\,{M}_{\odot }} & = & (1.64\pm 0.07)\cdot {\left(\displaystyle \frac{{L}_{500,\mathrm{bol}}}{{10}^{44}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{0.52\pm 0.03}\\ & & \ \cdot \ {\left(\displaystyle \frac{H(z)}{{H}_{0}}\right)}^{\alpha },\end{array}\end{eqnarray} \tag{ 1 }$$

where $H(z)=\sqrt{{{\rm{\Omega }}}_{{\rm{\Lambda }}}+{{\rm{\Omega }}}_{m}^{* }{(1+z)}^{3}+{{\rm{\Omega }}}_{k}^{* }{(1+z)}^{2}}$, Ω_k = (1 − Ω_m − Ω_Λ), and $\alpha =-{0.90}_{-0.15}^{+0.35}$;

(iv) a new temperature is recovered from the M–T relation (Equation (23) in Reichert et al. 2011) and compared to the input value assumed at step (i); (v) the calculations are repeated if the relative difference between these two values is larger than 5%.

We consider also a correction on the given luminosity due to the change in the initial R₅₀₀. This correction, typically a few percent, is obtained as described in Piffaretti et al. (2011), by evaluating the relative change of the square of the gas density profile integrated over the cylinder with dimension of r = R₅₀₀ and height of 2 × 5 R₅₀₀.

As a consistency test, for a subsample of common clusters, we can also compare the bolometric luminosities we have obtained with those independently derived by Maughan et al. (2012). Maughan et al. (2012) used a sample of 115 galaxy clusters at 0.1 < z < 1.3 observed with Chandra to investigate the relation between X-ray bolometric luminosity and Y_X (the product of gas mass and temperature) and found a tight L_X–Y_X relation (Maughan 2007). They also demonstrate that cluster masses can be reliably estimated from simple luminosity measurements in low quality data where direct masses, or measurements of Y_X, are not possible.

There are 26 clusters in common between our ROSAT-based sample and their Chandra sample. In Figure 2, we compare the bolometric luminosities obtained independently and find the values to be in very good agreement.

Zoom In Zoom Out Reset image size

We then searched for Spitzer/IRAC archival observations homogeneously covering at least an area within a 2.5 arcmin radius from the cluster center coordinates and with a minimum exposure time of 90 s. This depth ensures that we reach at least a 5σ sensitivity limit of 21.46 AB mag (9.4 μJy) at 4.5 μm (see Section 3.1 for further discussion of required depth).

These requirements result in a final X-ray-selected sample comprised of 47 galaxy clusters at 0.4 < z < 1.27 (indicated by red circles in Figure 1). We note that a few large clusters (indicated by red circles above the red line in Figure 1) have still been considered throughout this work. This is because the mean θ₅₀₀ in those redshift bins is smaller than the IRAC field of view. It also ensures an adequate sample size and avoids biasing our derived richness–mass relation against large, less-concentrated clusters. The derived cluster mass and redshift distributions of our X-ray sample are illustrated in Figure 3 (blue circles and histograms).

Zoom In Zoom Out Reset image size

Recent years have seen rapid progress of both the quality and quantity of SZE measurements using a variety of instruments. Therefore, several programs have been launched in the past few years with the aim of measuring total masses through the SZ effect of large samples of clusters, both for cosmology and astrophysics studies. Spitzer/IRAC data coverage over some of the SZ survey fields have been requested, as well as targeted SZE observations of existing MIR-selected clusters have also been obtained by various investigators.

We have therefore mined the Spitzer/IRAC archive and drawn a heterogeneous sample of spectroscopically confirmed SZE-selected clusters based from a number of these programs. Applying the same redshift and photometric coverage selection criteria illustrated in Section 2.1, the final SZE-selected sample considered in our study is comprised of 69 galaxy clusters at 0.4 < z < 2.0.

In particular, our sample is comprised of 4 clusters from the Planck Cluster Catalog (Planck Collaboration et al. 2015), 4 clusters from the Massive Distant Clusters of WISE Survey (MADCoWS, Brodwin et al. 2015), 1 cluster from the IRAC Distant Cluster Survey (IDCS, Brodwin et al. 2012), 1 cluster from the XMM-Newton Large Scale Structure Survey (XLSSU, Pierre et al. 2011; Mantz et al. 2014), and 59 clusters from the SPT-SZ Cluster Survey (SPT-SZ, Bleem et al. 2015),.

Cluster masses, M_500,SZ, as reported in the aforementioned papers, are based on the spherically integrated Comptonization measurement, Y_500,SZ, obtained by either the Planck Space Telescope, the Combined Array for Research in Millimeter-wave Astronomy (CARMA⁵ ), or the South Pole Telescope (SPT; Carlstrom et al. 2011; Austermann et al. 2012; Story et al. 2013). Cluster mass and redshift distributions of the final SZE-selected subsample are illustrated in Figure 3 (red circles and histograms) and can be compared with the X-ray sample shown therein.

We also note that only five clusters in our sample, Clus ID 26, 355, 621, 1050, and OBJ8 have mass estimates in the literature, derived both from L_X and Y_500,SZ, and that with the exception of Clus ID 621, 1050, the majority have consistent mass estimates within 2σ of the associated errors.

The publicly accessible Spitzer Enhanced Imaging Products⁶ (SEIP) provide super mosaics (combining data from multiple programs where available) and a source list of photometry for sources observed during the cryogenic mission of Spitzer. The SEIP includes data from the four channels of IRAC (3.6, 4.5, 5.8, 8 μm) and the 24 μm channel of the Multi-Band Imaging Photometer for Spitzer (MIPS) where available. In addition to the Spitzer photometry, the source list also contains photometry for positional counterparts found in the AllWISE release of the Wide-Field Infrared Survey Explorer (WISE) and in the Two Micron All Sky Survey (2MASS). To ensure high reliability, strict cuts were placed on extracted sources and some legitimate sources may appear to be missing. These sources were removed by cuts in size, compactness, blending, shape, and SNR, along with multiband detection requirements. In most fields, the completeness of the source list is well matched to expectations for a SNR = 10 cutoff, as reliability is favored over completeness. However, the list may be incomplete in areas of high surface brightness and/or high source surface density. This is most relevant for this work for objects near bright sources or the centers of clusters, which may have a higher source density.

Following the recommendations in Surace et al. (2004), for our richness estimate we adopt the aperture corrected IRAC 4.5 μm flux density measured within an aperture of diameter 3.8 arcsec from the SEIP source list. The chosen aperture is twice the instrumental FWHM, which provides accurate photometry with an aperture correction for a point source already applied, which is customary for cluster studies with Spitzer in the literature (e.g., Bremer et al. 2006; Rettura et al. 2006). IRAC PSF has a FWHM ∼ 2 arcsec, thus we note that a star/galaxy separation in Spitzer data, especially at faint fluxes, is not straightforward. Therefore we will describe in Section 4 how we account and correct for foreground stars in our richness estimates.

For the Warm Mission data, a SEIP source list is not available in the Spitzer archive. However, we have adopted the same SEIP source extraction pipeline and applied it ourselves in exactly the same way as for the Cryo mission clusters.

As we deal with a heterogeneous sample that has been observed by Spitzer at varying depths, for consistency of our analysis, we aim to be able to calibrate our method to a depth that is reached by all our archival data.

For illustration purposes, we show in Figure 4 the number counts of four representative clusters in our samples along with the number counts derived from a reference deep Spitzer legacy program that we adopt as a control field. The Spitzer UKIDSS Ultra Deep Survey (SpUDS, PI: J. Dunlop) data used here come from a program covering ∼1 deg² in the UKIRT Infrared Deep Sky Survey, Ultra Deep Survey field (UKIDSS UDS Dye et al. 2006), centered at R.A. = 02^h:18^m:45^s, Decl. = −05°:00':00''. Note that we use the SEIP source list photometry available in the archive for our control (SpUDS) field as well. The SpUDS survey reaches greater sensitivities than the data on the majority of our clusters, in particular for the SZE sample, as shown by examples on the right column of the panel Figure 4.

Zoom In Zoom Out Reset image size

As shown in Figure 5 for the entire sample, the IRAC coverage of our samples is not uniform. The median depth of the SZE cluster observations reach [4.5] = 21 AB, for instance, while the median depth for the X-ray sample reaches [4.5] = 22.5 AB. For the sake of overall consistency of our analysis and to be able to calibrate our method to a depth that the vast majority of current and future Spitzer surveys can easily reach (with even 90 s exposure), we adopt [4.5]_cut = 21 AB as the magnitude cut for all subsequent analyses. We will also further investigate the dependence of richness estimates on image depth in Section 4.1.

Zoom In Zoom Out Reset image size

For galaxy stellar populations formed at high redshift, a negative k-correction provides a nearly constant 4.5 μm flux density over a wide redshift range. An ${L}_{[4.5]}^{* }$ galaxy formed at z_f = 3 will have [4.5] ∼ 21 (AB) at 0.4 ≲ z ≲ 2.0, which is sufficiently bright that it is robustly seen in even just 90 s integrations with Spitzer (e.g., Eisenhardt et al. 2008).

While we recognize that using the simple approach of a single apparent magnitude cut at 0.4 < z < 2.0 would introduce a bias for optical mass–richness relationships, we note here that an infrared relation is not significantly affected because of the k-correction. Adopting Mancone et al. (2010) results on the evolution with redshift of the characteristic absolute magnitude ${M}_{[4.5]}^{* }(z)$, we note that at 4.5 μm, in the redshift range spanned by our sample, the stellar population evolution and redshift evolution are roughly matched, thus sampling a similar rest-frame luminosity range of the cluster galaxy population as a function of redshift. Because of cluster galaxy population evolution with redshift seen through the 4.5 μm band filter at 0.4 ≲ z ≲ 2, our adopted apparent magnitude limit [4.5]_cut always corresponds to a roughly similar absolute magnitude ${M}_{{[4.5]}_{\mathrm{cut}}}\sim {M}_{[4.5]}^{* }(z)+1$ over this large redshift range. We find in fact that ${M}_{{[4.5]}_{\mathrm{cut}}}$ varies between ${M}_{[4.5]}^{* }(z)$ + 0.87 (at z ≳ 1.2) and ${M}_{[4.5]}^{* }(z)+1.17$ (at z ∼ 0.5), thus by 0.3 mag. This small variation in limit magnitude will not significantly increase the scatter in the mass–richness relation we will derive in the next Section 5. In Section 4.1, based on a subsample of clusters for which deeper data are available, we will study the dependence of richness estimates on survey depth and will parameterize a linear relation (Equation (3)) to account for these effects. Accordingly, a variation in magnitude cut by 0.3 mag will result in a variation in richness, ${\rm{\Delta }}R\sim 6\,\mathrm{gals}\times {\mathrm{Mpc}}^{-2}$, hence in logarithmic scale ΔLog R ∼ 0.05, which is very small and will not significantly increase the scatter in the derived mass–richness relation.

In order to investigate the effect of the chosen aperture radius and depth of the IRAC 4.5 μm data on richness estimates, we performed a series of tests on a subsample of clusters where the IRAC data is deep enough to allow us to measure richness values at different sensitivity levels. As shown in Figure 5, the X-ray sample contains a "deep" subsample of 36 clusters for which their depth is ≥22.5 mag AB. We measure the average (and standard deviation of the mean) richness of this sample down to various depths, [4.5]_cut, ranging 21 < [4.5] < 22.5, and with different aperture radii, $0\buildrel{\,\prime}\over{.} 5\lt r\lt 2^{\prime}$. As shown in Figure 8, richness increases with increasing magnitude cut adopted; this is not surprising since there are typically more galaxies at fainter luminosities for a canonical luminosity function. This test validates the importance of adopting a uniform magnitude cut while dealing with a heterogeneous, archival sample. We also note that the slope and standard deviation of richness is much smaller for the larger, adopted 2 arcmin radius aperture than for the smaller apertures. For the adopted 2 arcmin aperture radius, the dependence of the mean richness, $\langle {R}_{[4.5]}\rangle$, with the magnitude cut adopted, [4.5]_cut, is best fitted with the linear relation

$$\begin{eqnarray}&&\displaystyle \frac{\langle {R}_{[4.5]}\rangle }{\mathrm{galaxies}\,{\mathrm{Mpc}}^{-2}}=(-358.71)+(18.97)\times \displaystyle \frac{{[4.5]}_{\mathrm{cut}}}{\mathrm{AB}\,\mathrm{mag}}.\end{eqnarray} \tag{ 3 }$$

Zoom In Zoom Out Reset image size

This relation allows us to quantify the expected increase in average richness value at increasing depths of the observations due simply to an intrinsic photometric effect, not due to cluster-to-cluster variations or dynamical state. By means of extrapolation, this relation could be also used to predict the expected average richness value for samples of clusters at similar redshifts with upcoming, wide-area, infrared surveys (see discussion in Section 6.4).

In the top panel of Figure 11 we show the distributions of richness for our samples. The X-ray sample (blue histogram and blue dashed line) shows larger median richness values than the SZ sample (red histogram and red dashed line). This difference is statistically significant, as shown by the Kolmogorov–Smirnov (KS) test, which provides a probability P_KS ∼ 10⁻⁶ that the observed distributions of richness are extracted from the same parent population.

Zoom In Zoom Out Reset image size

As the SZE sample probes the redshift range z > 1 more extensively, we tested whether this difference in median richness could depend on the relative redshift distribution of the two samples. Hence we performed the KS test again only for clusters in the redshift range 0.4 < z < 0.8, which is well sampled by both methods. However, the latter test produces a similar outcome of the former, ruling out this possibility.

In the bottom panel of Figure 11 we show the dependence of richness on aperture radius for both cluster samples. We divide each sample in two equally populated bins of X-ray or SZE derived cluster masses. We calculate the richness profiles by measuring the background-subtracted projected surface density of galaxies with [4.5] < 21 AB within r = 30'', 60'', 120'' from the cluster center. We find the shapes of the richness profiles to be similar for both samples and in each mass bins and the richness values consistent within the large scatter. However, the low-mass X-ray sample appears to be as rich as the high-mass SZ sample. Hence at fixed cluster mass, X-ray clusters appear to have, on average, higher near-infrared richness than SZ-selected clusters.

One possibility is that this difference is due to systematics in derived masses for the X-ray and SZ samples. For example, if, despite the careful re-calibration in Section 2, the X-ray masses are underestimates, then the higher richness of the X-ray clusters would imply that they are more massive clusters. Given the extensive work on calibrating X-ray and SZ observables with mass, this seems unlikely. Yet, without having a statistically significant SZ, X-ray, and richness estimates sample of the same clusters in this redshift range, we cannot definitively rule it out.

Alternately, could this be due to a selection effect, where the X-ray surveys, at a fixed cluster mass, typically select richer systems where galaxy merging has been, on average, less effective than in SZE-selected clusters? X-ray surveys are indeed usually considered biased toward selecting more relaxed, more evolved systems (e.g., Eckert et al. 2011). This is because of the presence of a surface brightness peak in the so-called "cool core" clusters. The clear peak of X-ray emission is more easily detected in the wide shallow surveys of ROSAT, for instance, and it is considered to be associated with a decrease in the gas temperature, typical of relaxed structures.

On the other hand, most of the clusters newly discovered by Planck via SZE show clear indication of morphological disturbances in their X-ray images, suggesting a more active dynamical state (Planck Collaboration et al. 2011). Interestingly, Rossetti et al. (2016), measuring the offset between the X-ray peak and the BCG population, a known indicator of an active cluster dynamical state (e.g., Sanderson et al. 2009; Mann & Ebeling 2012), found evidence of the dynamical state of SZ-selected clusters to be significantly different from X-ray-selected samples, with a higher fraction of non-relaxed, merging systems.

In the hierarchical cluster formation scenario, clusters form by the infall of less massive groups along the filaments. Therefore while it is possible that a larger fraction of SZ-selected clusters in our sample are in a less developed stage of cluster formation, if the difference in richness between our samples was to be explained with the fact that X-ray clusters were more evolved and had accreted more galaxies within R₅₀₀, then they should also be the most massive. This has, however, not been found, as clearly shown in Figure 3, unless there are mass calibration uncertainties larger than those discussed.

An intriguing possibility is that there are differences in the intrinsic baryon fraction within R₅₀₀ relative to the cosmological baryon fraction between the cluster samples. If X-ray clusters at these redshifts harbor a larger baryon fraction per unit dark-matter halo mass compared to SZE cluster samples within the aperture radius, it could account for lower total masses, more efficient cooling of the intracluster medium by thermal Bremsstrahlung, and the resultant increased star-formation efficiency could result in a larger population of luminous galaxies translating to a higher richness. Indeed, simulations like the millennium simulations show a factor of two variation in the baryon fraction of >10¹⁴ M_☉ dark-matter halos, which could easily account for both the differences in derived masses and richness values. Even among the X-ray cluster sample, Vikhlinin et al. (2009) showed that the baryon fraction increases with increasing mass among which there is likely the origin of a richness–mass correlation that we derive. A comparison between the density profiles of SZ clusters and X-ray clusters in Planck Collaboration et al. (2011) shows that SZ clusters show shallower density profiles than X-ray-selected clusters, which may again argue that the baryons in SZE-selected clusters are predominantly at larger radii than in the X-ray sample. However, extracting effects such as these from the data would again require SZ, X-ray, and richness estimates of the same clusters in this redshift range while our study has rather heterogeneous samples with different origins that we have attempted to place on the same calibration scale. Apart from stating these possibilities, it is challenging for us to definitively claim one of them as an origin for the observed difference.

In an attempt to understand the source of the scatter in the mass–richness relation, we also measure the galaxy concentration of our cluster samples, defined as the ratio between the richness measured within r = 60'' and r = 120''. By definition, a higher value of galaxy concentration corresponds to systems with a steeper galaxy surface density profile. As shown in Figure 12, there is a hint that clusters that deviate the most from the fitted mass–richness relation are also the ones with the most centrally concentrated galaxy surface density profile. If we include a correction for galaxy concentration, the scatter of the mass–richness relation slightly decreases so that the associated error in mass at fixed richness is found to be ±0.22 dex, indicating that surface density concentration does play a significant role in the scatter.

Zoom In Zoom Out Reset image size

Possible origins for this are blending and source confusion in the IRAC image, beam dilution when the 2' aperture is much larger than the cluster overdensity, or the impact of galaxy merging on richness estimates. Images of galaxy clusters that are more centrally concentrated are more likely to be affected by the blending of some cluster galaxies in the core, given the Spitzer angular resolution. This could result in underestimating their richness. The amount of confusion is linearly proportional to the surface density of galaxies. A cluster that is four times as concentrated in surface density would have its richness underestimated by 0.6 dex, which is the amount of offset of the red points in Figure 12 from the best-fit line.

Alternatively, systems with higher central galaxy concentrations have their richness estimate, calculated within a radius r = 120'', biased by the fact that the aperture chosen is too large, hence their richness is relatively smaller compared to less centrally concentrated systems. However, if this was purely an observational bias, we should have found a correlation between galaxy concentration and θ₅₀₀ derived for the hot gas in the intracluster medium, a proxy of cluster size, which is not apparent.

If mergers were responsible for the scatter in the richness–mass relation, clusters of the same total mass that might have experienced more or less merger events in their cores with respect to their outskirts would result in lower or higher concentration measurements and would therefore be found to have lower or higher values of richness. In the most extreme cases, as shown in Figure 13, galaxy clusters of the same total mass (as probed by their gas) and at the same lookback time may have experienced very different evolutionary processes, resulting in large differences in richness and concentration inferred from the number and location of their cluster members.

Thus we conclude that the scatter we derive in the richness–mass relation is likely due to a combination of source confusion and differences in evolutionary history of the clusters.

The AllWISE⁸ program combined data from the cryogenic Wide-field Infrared Survey Explorer mission (WISE, Wright et al. 2010) and the NEOWISE (Mainzer et al. 2011) post-cryogenic survey to deliver a survey of the full MIR sky. Since all-sky catalogs in the W2 band at 4.6 μm are available, it could allow us to potentially extend our method outside the Spitzer footprint. Therefore in this section we test whether our proposed method of deriving MIR richness estimates can be applied robustly to the publicly available WISE data.

To this aim, we use archival WISE 4.6 μm data available at the location of all our clusters and apply the same method described in Section 4. The SEIP source list contains W2 photometry for positional counterparts found in the AllWISE release both for all clusters in our sample and for the control SpUDS field. In Figure 14 (left panel), we show the richness estimates based on data from Spitzer and WISE for our SZE sample. The dashed lines indicates the straight-line fit to be compared to the 1:1 (solid) line. We note that there are several catastrophic outliers and that Spitzer-based richness values are systematically higher than the WISE-based counterparts. We note that WISE is less sensitive than Spitzer and that its angular resolution is also poorer (64 versus 20).

Zoom In Zoom Out Reset image size

To test whether the catastrophic discrepancies in the Spitzer versus WISE richness estimates could be ascribed to confusion, we sum the flux densities of every object detected within r = 120'' from each cluster center by the two instruments. We also subtract a median flux density from the background field to get a flux overdensity at the location of the cluster. Source confusion makes groups of sources in a high resolution image appear as a single bright source in a low-resolution image. By adding the flux densities we take out the effect of confusion, which would bias low richness estimates. We then use the redshift of the cluster to translate the summed flux overdensity to a luminosity surface density. As shown in the right panel of Figure 14, we find that the total luminosity densities for each cluster appear to be conserved, as the two instruments provide matching measurements. Therefore we can ascribe the aforementioned discrepancies solely to the poorer angular resolution of WISE, with richness estimates highly depending on the particular projected cluster galaxy geometry. At the WISE image quality, we expect a higher number of sources to be blended, resulting in lower counts of galaxies per cluster, yielding lower richness values than those measured by Spitzer. For example, source overdensities of 30 galaxies in the 2' radius aperture would correspond to 10 gals Mpc⁻², which in turn would correspond to a ratio of 11 for the number of WISE beams per source. This is well below the classical confusion limit. In reality, the confusion is even higher since the average underlying foreground source density would also contribute to confusion noise.

To summarize, we deem WISE-based richness estimates to be poorer proxies for cluster mass preventing us from effectively extending our method beyond the Spitzer footprint. Calibrating a mass–richness relation for the WISE data set will require a different technique and is therefore beyond the scope of this paper.

The upcoming Euclid and WFIRST missions aim to survey large portions of the extragalactic sky in the near infrared (e.g., H band) to measure the effects of dark energy, but also have distant cluster studies as a key scientific goal. The Euclid wide-area survey, in particular, will observe 15,000 deg², almost the entire extragalactic celestial sphere, down to a 5σ point-source depth of H = 24 mag (AB). They plan to use photometric redshift overdensities to identify clusters but that requires ancillary ground-based optical data which is currently being taken. In this section, based on the results of our analysis, we try to provide a simple prediction of the expected richness values for Euclid-selected clusters and the range of masses that will be accessible.

Euclid is expected to detect ∼2 × 10⁶ clusters at all redshifts, with ∼4 × 10⁴ of them at 1 < z < 2 with cluster masses M₂₀₀ ≳ 8 × 10¹³ M_⊙ (Sartoris et al. 2016; Ascaso et al. 2017). The cluster sample in our study spans a similar mass and redshift range, hence we can attempt to predict the average richness expected for clusters at 0.4 < z < 2.0 as a function of mass in the Euclid survey.

In Figure 7, we show the evolution of the [3.6]–[4.5] (left panel) and H − [4.5] color (middle panel) with redshift for a set of Bruzual & Charlot (2003) stellar population models with exponentially declining star-formation rates. We show both the typical color evolution expected for an early-type galaxy (assuming τ = 0.1 Gyr) and a star-forming galaxy (τ = 1.0 Gyr) as described in Rettura et al. (2010, 2011). As noted by several authors (e.g., Papovich 2008; Muzzin et al. 2013; Wylezalek et al. 2013; Rettura et al. 2014), the [3.6]–[4.5] color is fairly insensitive to different modes of star formation out to z ∼ 3 and can be used as a good redshift indicator at z > 1.3. The H − [4.5] color, instead, is more sensitive to galaxy star-formation history, in particular between 1 < z < 2.

Zoom In Zoom Out Reset image size

According to the models shown in the middle panel of Figure 7, we expect the H − [4.5] color of a galaxy to vary between −0.7 and 0.75 AB (dashed lines) depending on its type at 0.4 < z < 2.0 (gray shaded area). This would imply that to match the Euclid H_cut = 24 AB depth, we need an equivalent Spitzer 4.5 μm survey to reach [4.5]_cut = 24.7 AB.

In Section 4.1, based on our "deep" subsample of 36 clusters for which the deepest IRAC coverage was available in the Spitzer archive, we derived a relation between the survey depth and the average richness of our cluster sample. As we have demonstrated, we are already below the knee of the galaxy luminosity function at these redshifts and we do not expect the slope of the relation to change as we go deeper. Using Equation (3), we can then predict the richness of clusters at an equivalent depth of [4.5]_cut = 24.7, i.e., down to H_cut = 24. We predict the following levels of richness (in galaxies Mpc⁻²) for Euclid-detected clusters as a function of cluster mass: log R_[H] = 1.78 ± 0.26 (log M₅₀₀ < 14.5), log R_[H] = 1.87 ± 0.21 (14.5 ≤ log M₅₀₀ ≤ 14.75), and log R_[H] = 1.99 ± 0.16 (log M₅₀₀ > 14.75). In the right panel of Figure 7 we show the expected mean richness (and standard deviation of the mean) in bins of cluster mass for the Euclid clusters at 0.4 < z < 2.0, propagating the uncertainty in the fit required to extrapolate to these faint flux densities. For immediate comparison with the future observational data, the figure reports the expected richness values in units of galaxies arcmin⁻². We conclude that typical Euclid clusters that are about 3 × 10¹⁴ M_☉ will show galaxy overdensities of ∼12 galaxies arcmin⁻².