ROBUST OPTICAL RICHNESS ESTIMATION WITH REDUCED SCATTER

Following Paper I, cluster locations, redshift estimates, and initial richness estimates are taken from the maxBCG cluster catalog (Koester et al. 2007a, 2007b), an optically selected cluster catalog. The maxBCG algorithm identifies galaxy clusters by relying on the observation that the galaxy population of massive halos clusters tightly in space and color, forming what is known as the E/S0 ridgeline or red sequence (e.g., Dressler 1984; Kormendy & Djorgovski 1989; Hansen et al. 2009). This feature allows for high-contrast detection of galaxy clusters with optical data, both locally and out to high redshift (e.g., Gladders & Yee 2000; Eisenhardt et al. 2008).

The maxBCG catalog is approximately volume limited in the redshift range of interest (0.1 ⩽ z ⩽ 0.3), with very accurate cluster photometric redshifts (δz ∼ 0.01). Studies with mock SDSS catalogs indicate that the completeness and purity are above 90% (Koester et al. 2007a; Rozo et al. 2007). The maxBCG catalog has been used to investigate the scaling of multiple cluster mass proxies with richness, including line-of-sight velocity dispersion (Becker et al. 2007), X-ray luminosity (Rykoff et al. 2008b), and weak-lensing shear (Sheldon et al. 2009a), as well as derive cosmological constraints from cluster counts (Rozo et al. 2007, 2010a).

The richness estimator used in the maxBCG catalog is N₂₀₀, defined as the number of galaxies with g − r colors within 2σ of the E/S0 ridgeline as defined by the brightest cluster galaxy (BCG) color, that are brighter than 0.4 L_* (in i band), and found within a scaled aperture r^gal₂₀₀ of the cluster center (Hansen et al. 2005). The full catalog comprises 13, 823 objects with a richness threshold N₂₀₀ ⩾ 10, corresponding to M ≳ 5 × 10¹³ h⁻¹ M_☉ (Johnston et al. 2007).

The scatter in L_X at fixed richness, $\sigma _{\ln L_X|N}$ (where N is an arbitrary richness), is estimated using the methods of Paper I and Rykoff et al. (2008b). In brief, we use the RASS photon maps to estimate the 0.5–2.0 keV X-ray flux in an aperture centered at the BCG of each cluster, which is in turn used to derive L_X (0.1–2.4 keV) given the cluster's photometric redshift (the conversion factors are similar to those used in Böhringer et al. 2004). The local background is calculated in a 20'–40' annulus using the sector rejection method of Böhringer et al. (2000). As detailed in Rykoff et al. (2008b), for the clusters in the NORAS catalog our L_X estimates are consistent within errors with ∼10% scatter and a systematic offset of <5%, primarily attributable to differences in apertures. Upper limits are estimated from the local noise level. We then perform a Bayesian linear regression to ln L_X as a function of ln N, where N is the richness parameter to be tested. The variance in ln L_X is included as a free parameter. The fit is done following the algorithm presented in Kelly (2007), which performs a full Bayesian modeling of the power-law distribution, and correctly takes into account errors on the independent variable as well as upper limits on L_X for those clusters without significant detection of X-ray emission. This method has several advantages, in that it takes into account all the available X-ray data, not only those for clusters in X-ray catalogs. With Monte Carlo tests of simulated cluster profiles placed at random locations in the RASS field, we have confirmed that this method is able to accurately recover the input relation (see Appendix A of Rozo et al. 2009a). We note that in our tests detailed in Section 4.1 approximately 70% of the 2000 richest clusters are detected at >1σ.

When estimating L_X, one must specify an aperture. The matched filter richness estimators described in this paper have the benefit of assigning a cluster radius R_c, to each individual cluster. As in Paper I, we estimate L_X using the aperture derived from the cluster richness. Alternatively, using a fixed 0.9 h⁻¹ Mpc aperture to estimate L_X does not have a significant effect on our results.

As discussed in Rykoff et al. (2008b, see Section 5.6), there is clear evidence that strong cool core clusters increase the scatter in X-ray cluster properties. High-resolution X-ray imaging of clusters allows the exclusion of cluster cores, reducing the scatter in observed X-ray properties (e.g., O'Hara et al. 2006; Chen et al. 2007; Maughan 2007). Unfortunately, the broad point spread function (PSF) of ROSAT means that it is impossible to do so in this work. In Paper I, we analyzed both the full sample of maxBCG clusters as well as a "clean" sample after removing all ten known strong cool core maxBCG clusters that may significantly bias our results (see Section 2.4 in Paper I). We concluded that although the absolute value of the scatter in L_X at fixed richness is reduced by using the "clean" sample, the same general trends were evident with the full and "clean" samples. In this work, we focus exclusively on the "clean" sample of Paper I.

The input galaxy catalog for this work is derived from SDSS DR7 photometric data (Abazajian et al. 2009). This data release includes nearly 10,000 deg² of drift-scan imaging in the Northern Galactic Cap. However, as the maxBCG cluster catalog was created using data from DR4 (Adelman-McCarthy et al. 2006), the relevant area covered is ∼7500 deg². Survey edges, regions of poor seeing, and bright stars are masked as previously described (Scranton et al. 2002; Koester et al. 2007a; Sheldon et al. 2009b). In this work we use CMODEL_COUNTS as our total magnitude, and MODEL_COUNTS when computing colors. All magnitudes are corrected for Galactic extinction. The input catalog is complete in both luminosity and g − r color down to an i-band limit of 21.3 mag, or ∼0.1 L_*, at z = 0.3, the redshift limit of the maxBCG catalog.

The careful selection of a clean input catalog is required for proper richness estimation. In Section 5.6, we discuss the effects of "catalog noise" on the richness–mass relation, by which we mean the inclusion of stars and/or artifacts as well as catastrophic photometric errors in the galaxy catalog employed when estimating richnesses. Our best input catalog was based on the same cuts used in Sheldon et al. (2009b). After selecting galaxies based on the default SDSS star/galaxy separator, we filter all objects with any of the following flags set in the g, r, or i bands: SATURATED, SATUR_CENTER, BRIGHT, NOPETRO, DEBLENDED_AS_MOVING. These cuts remove ∼30% of the objects brighter than i > 22, including a significant number of relatively bright stars that are erroneously tagged as galaxies in the SDSS pipeline.

For the radial filter, Paper I adopted an NFW profile (Navarro et al. 1995), which is a good description of the dark matter profile in N-body simulations, and is found to be a good descriptor of the distribution of cluster galaxies (Lin & Mohr 2004; Hansen et al. 2005; Popesso et al. 2007). The corresponding two-dimensional surface density profile is (Bartelmann 1996)

$$\begin{equation} \Sigma (R) \propto \frac{1}{(R/R_s)^2-1}f(R/R_s), \end{equation} \tag{ 7 }$$

where R_s is the characteristic scale radius, and

$$\begin{equation} f(x) = 1-\frac{2}{\sqrt{x^2-1}}\tan ^{-1}\sqrt{ \frac{x-1}{x+1} }. \end{equation} \tag{ 8 }$$

This formula assumes x > 1. For x < 1, one uses the identity tan ⁻¹(ix) = itanh (x).

Following Koester et al. (2007a), Paper I set R_s = 0.15 h⁻¹ Mpc. Also, in order to avoid the singularity at R = 0, they assumed that Σ was constant for R ⩽ R_core = 0.1 h⁻¹ Mpc. This core density is chosen so that the mass distribution Σ(R) is continuous. Finally, the profile Σ(R) is truncated at the cluster radius R_c(λ) and is normalized such that

$$\begin{equation} 1 = k_\mathrm{NFW}\int _0^{R_c(\lambda)} dR\ 2\pi R\Sigma (R). \end{equation} \tag{ 9 }$$

For the NFW profile with the given values of R_s and R_c, the normalization factor k_NFW can be parameterized as

$$\begin{eqnarray} k_\mathrm{NFW} & = & \exp (1.6517-0.5479\rho +0.1382\rho ^2-0.0719\rho ^3\nonumber \\ && -\, 0.01582\rho ^4-0.00085499\rho ^5), \end{eqnarray} \tag{ 10 }$$

where ρ = ln (R_c) and 0.001 < R_c < 3.

The luminosity distribution of maxBCG clusters is well represented by a Schechter function (e.g., Hansen et al. 2009) which we write as

$$\begin{equation} \phi (m) \propto 10^{-0.4(m-m_*)(\alpha +1)}\exp (-10^{-0.4(m-m_*)}). \end{equation} \tag{ 11 }$$

Paper I set α = 0.8 independent of redshift. The characteristic magnitude, m_*, is calculated for a k-corrected passively evolving stellar population (Koester et al. 2007b). We assume Mⁱ_* = −21.22 for red galaxies, corresponding to 2.25 × 10¹⁰ L_☉. A PEGASE.2 stellar population/galaxy formation model (e.g., Eisenstein et al. 2001) was used to calculate the k-corrected magnitude at each redshift. In the redshift range 0.05 < z < 0.35 appropriate for maxBCG, m_*(z) is well approximated by a fourth-order polynomial:

$$\begin{equation} m_*(z) = 12.27 \,{+}\, 62.36z \,{-}\, 289.79z^2\,{+}\,729.69z^3\,{-}\,709.42z^4. \end{equation} \tag{ 12 }$$

For each cluster, m_* is taken at the appropriate redshift and the filter is normalized by integrating down to the magnitude cutoff. In Paper I, this was chosen to be 0.4 L_* or m_* + 1 mag.

The old stellar populations in the red-sequence galaxies have a prominent 4000 Å break in their spectra. In the redshift range targeted by maxBCG, z ≲ 0.35, the 4000 Å break is located in the g band. Therefore, the g − r color of red-sequence galaxies correlates strongly with redshift, and results in tight E/S0 ridgelines. Consequently, Paper I relied on c = g − r for their color filter. They assume G(c) is Gaussian with a small intrinsic dispersion of σ_int = 0.05 mag. The corresponding color filter, G(c), is

$$\begin{equation} G(c|z) =\frac{1}{\sqrt{2\pi }\sigma }\exp \left[ \frac{(c-\left\langle c|z \right\rangle)^2}{2\sigma ^2} \right], \end{equation} \tag{ 13 }$$

where c = g − r is the color of interest, 〈c|z〉 is the mean of the Gaussian color distribution of early-type galaxies at redshift z, and the net dispersion σ is the sum in quadrature of the intrinsic color dispersion σ_int = 0.05 and the estimated color error σ_c. The mean color 〈c|z〉 = 0.625  +  3.149z was determined by matching maxBCG cluster members to the SDSS LRG (luminous red galaxy; Eisenstein et al. 2001) and "main" (Strauss et al. 2002) spectroscopic galaxy samples. In Section 4.3, we investigate modifications of this color model based on measurements of the red sequence of maxBCG clusters measured in Hao et al. (2009).

The last necessary ingredient for estimating λ is a background model. We assume that the background galaxy density is constant in space, so that $b(\mathbf {x})=2\pi R \bar{\Sigma }_g(m_i,c)$ where $\bar{\Sigma }_g(m_i,c)$ is the galaxy density as a function of galaxy i-band magnitude and g − r color. The mean galaxy density is obtained by binning the full galaxy catalog in color and magnitude using a cloud-in-cells (CIC) algorithm (e.g., Hockney & Eastwood 1981) and dividing by the survey area. For our cells, we use 40 evenly spaced bins in g − r ∈ [ − 1, 3] and 100 bins in i ∈ [12, 22]. The final galaxy number density is normalized by the width of each color and magnitude bin (0.1 mag each). We mark as "bad" all cells that have fewer than 5 galaxies deg⁻². This has the effect of masking out erroneous photometric artifacts that are called bright galaxies in the DR7 catalog. Although these artifacts are rare, they may nevertheless significantly bias the luminosity-weighted richness for a few clusters (see Section 4.6).¹⁹ We emphasize that because the background is measured per square degree, the average number of background galaxies is automatically accounted for as the angular size of the clusters changes with redshift.

The primary goal of this paper is to improve upon the Paper I richness estimator. To do that, however, we must first define the metric used to gauge improvement. As discussed in the introduction, our chosen figure-of-merit is the scatter in L_X at fixed richness. As in Paper I, for each individual richness estimator we rank order the full sample of maxBCG clusters by richness and take the top 2000. A constant number threshold yields a consistent number density of clusters (for maxBCG, this is 1.7 × 10⁻⁶ clusters h⁻³₇₀ Mpc⁻³), and therefore a roughly consistent mass threshold (see also Reyes et al. 2008). Because this ordering changes as the richness estimator is varied, the precise set of clusters that are used will vary from test to test. It is important to limit ourselves to a rich subsample of any given richness estimate so that our results are insensitive to the N₂₀₀ ⩾ 10 cut in the maxBCG catalog. For the original maxBCG richness estimator N₂₀₀, this is equivalent to N₂₀₀ ⩾ 20 or an equivalent mass of M₂₀₀ ≳ 1 × 10¹⁴ h⁻¹ Mpc (Johnston et al. 2007). In Paper I we confirmed that our results with the top 1000 or 3000 clusters are consistent with the top 2000.

We denote the original matched filter richness estimator described in Paper I as λ₀. Using the top 2000 richest clusters in the clean sample, and a fixed 0.9 h⁻¹ Mpc aperture, we find $\sigma _{\ln L_X|\lambda _0} = 0.69\pm 0.02$.²⁰ A representative plot of the L_X–λ₀ relation, including upper limits, is shown in Figure 4 of Paper I. We emphasize that the L_X–richness relations from this paper are of much the same form. As discussed in Section 5.3 of Paper I, comparing two richnesses is complicated by the fact that the errors are correlated. In all pairwise richness comparisons, we perform bootstrap resampling on the clean cluster catalog and calculate the scatter in the top 2000 clusters for both λ₀ and the new richness λ_new. For each resampling we calculate a bootstrap ratio $r_\mathrm{boot}=\sigma _{\ln L_X|\lambda _{\mathrm{new}}}/\sigma _{\ln L_X|\lambda _0}$. If the improvement in scatter is not significant, we will find that r is consistent with unity, whereas an improved (worse) scatter will result in an r_boot value that is significantly less than (greater than) 1. This is our primary diagnostic for improvement.

In addition to the scatter, we also monitor the redshift evolution in the L_X–richness relation of each of the richness estimators we consider to ensure that no strong redshift evolution is introduced by our alterations. We measure the evolution using a stacking analysis as described in Paper I and Rykoff et al. (2008b, see Section 5.3). In brief, for the stacking analysis we extract photons from the RASS map around each cluster BCG, as well as a background annulus as described in Section 2.2. Each source and background photon is weighted relative to the median luminosity distance to the clusters in the bin, and a weighted count-rate is estimated (e.g., Böhringer et al. 2000). For simplicity, we then use the conversion factors from Section 2.2 to convert from mean flux to luminosity. In practice, this gives the same results as the full spectral fit on the photons in each stack as implemented in Rykoff et al. (2008b), primarily because the soft ROSAT band is not very sensitive to cluster temperature.

We now measure 〈L_X|N〉 where N is the richness measure of interest in three different redshift bins (0.10 < z < 0.18; 0.18 < z < 0.26; and 0.26 < z < 0.30). As shown in Rykoff et al. (2008b), the stacking analysis allows us to go much further down the richness function than for the scatter analysis and still obtain significant detections. For this analysis we use the top ∼6500 clusters in seven richness bins, with as few as 13 clusters in the richest bins and as many as 1600 clusters in the poorest bins. While this introduces selection function effects near the N₂₀₀ ⩾ 10 threshold of the maxBCG cluster catalog, we expect these to be minor in the redshift evolution, which, as mentioned earlier, we only use as a sanity check. We fit the stacked data with a power-law evolution in redshift,

$$\begin{equation} \left\langle L_X|N \right\rangle = A \left(\frac{N}{40} \right)^\alpha \left(\frac{1+z}{1+\tilde{z}} \right)^\gamma, \end{equation} \tag{ 14 }$$

where $\tilde{z}$ is the median redshift of the cluster sample and N is the richness measure of interest. We find that γ = 0.7 ± 0.8 for λ₀, consistent with no evolution. A representative plot of the stacked L_X–λ₀ relation split by redshift is shown in Figure 5 of Paper I.

As noted in Paper I, even if the relation between N and cluster mass is redshift independent, we may expect to observe evolution in the L_X–N relation due to evolution in the L_X–M relation. The expectation for self-similar evolution in L_X at fixed mass is $L_X \propto \bar{\rho }_c$ for soft-band X-ray luminosities (Kaiser 1986), where ρ_c is the critical density of the universe at redshift z. In a ΛCDM universe with Ω_m = 0.25, L_X∝ρ_c is well approximated by L_X∝(1 + z)^1.10 or γ = 1.10. Thus, for λ₀, the evolution is consistent with both the no-evolution and the self-similar evolution models. As we do not know what the true evolution should be, we do not use the evolution parameter γ as a true comparison between richness estimators. That said, modest evolution in the richness–mass relation is a desirable property for richness estimators, so we do check that γ remains in the range ≈0–1.5 as we modify λ.

The red sequence is well suited to cluster finding due to its high contrast with the background and its strong redshift dependence (e.g., Bower et al. 1992; Gladders & Yee 2000; Koester et al. 2007a). However, not all cluster galaxies are red, only the majority, with typical red fractions being of order ≈80% (e.g., Hao et al. 2009). We now explore whether accounting for this blue galaxy population in our color filter improves our richness estimator.

We first empirically construct a new color filter that accounts for both red and blue galaxies. As an input, we use the 2000 richest clusters as measured by our original matched filter richness estimator, λ₀. We then bin these clusters in ten redshift bins of width 0.02, and select all galaxies within 0.9 h⁻¹ Mpc of the BCG brighter than 0.4 L_*. To estimate galaxy luminosity, we assume that all galaxies are at the cluster redshift. The empirical color distribution of these galaxies is then background subtracted and fit as a sum of two Gaussians using the error-corrected Gaussian mixture model (ECGMM) of Hao et al. (2009), which allows us to properly take into account photometric errors. The location, width, and relative amplitude of these two Gaussians define the appropriate color filter for the color distribution of all (red + blue) cluster galaxies within our chosen aperture. Additional tests have shown that the color distribution does not depend significantly on cluster richness λ₀.

Figure 1 shows our best-fit model (solid green line) for the lowest redshift bin (0.1 < z < 0.12) and the highest redshift bin (0.28 < z < 0.30). Note that the width of the histogram in the high redshift bin is dominated by photometric errors. Therefore, the fit model is narrower than the binned histogram, as the ECGMM model takes into account these photometric errors. Over the full redshift range we find that the width of the red sequence is consistent with no or mild evolution. For simplicity we assume that there is no evolution, while emphasizing that at high redshift the color model is dominated by photometric errors.

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Our empirically determined model filter takes the form

$$\begin{eqnarray} G(c|z) & = & \frac{\alpha _{\mathrm{red}}}{\sqrt{2\pi }\sigma _{\mathrm{red}}}\exp \left[ \frac{(c-\mu _{\mathrm{red}})^2}{2\sigma _{\mathrm{red}}^2} \right] \nonumber \\ & & \hspace{14.45377pt} + \frac{\alpha _{\mathrm{blue}}}{\sqrt{2\pi }\sigma _{\mathrm{blue}}}\exp \left[ \frac{(c-\mu _{\mathrm{blue}})^2}{2\sigma _{\mathrm{blue}}^2} \right], \end{eqnarray} \tag{ 15 }$$

where α_red and α_blue are the relative weights of the red and blue Gaussians; μ_red and μ_blue are the mean color of the red and blue galaxies at redshift z; and σ_{red, int} and σ_{blue, int} are the intrinsic scatter for the red and blue galaxies. Each of these parameters is fit as a function of redshift using a simple linear relation, with the exception of the Gaussian widths, which we find is consistent with no evolution. Our final model parameters are

$$\begin{eqnarray} \mu _{\mathrm{red}} = & 0.629+3.016z \end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray} \mu _{\mathrm{blue}} = & 0.590+2.115z \end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray} \alpha _{\mathrm{red}} = & 0.786-0.303z \end{eqnarray} \tag{ 18 }$$

$$\begin{eqnarray} \alpha _{\mathrm{blue}} = & 1.0 - \alpha _{\mathrm{red}} \end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray} \sigma _{\mathrm{red,int}} = & 0.05 \end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray} \sigma _{\mathrm{blue,int}} = & 0.25. \end{eqnarray} \tag{ 21 }$$

We can now replace the Gaussian color filter in Equation (13) with this new filter, and combine it with the original luminosity and radial filters to measure a new richness, λ_{red + blue}. The scatter in L_X at fixed richness is $\sigma _{\ln L_X|\lambda _{\mathrm{red}+\mathrm{blue}}} = 0.72\pm 0.02$, and the corresponding redshift evolution parameter is γ = 0.1 ± 0.6. We note that the width of the color filter for the blue cluster members derived here is slightly wider than that derived from spectroscopically confirmed subsamples (Hao et al. 2009), implying that there is some contamination from background galaxies. However, further tests have confirmed that our results are insensitive to the precise width of this component (see also Section 5.3).

To measure the significance of the change, we use the bootstrap resampling described in Section 4.1 and find r_boot = 1.04 ± 0.02. That is, including the blue galaxies in the filter increases the scatter at a significance of 2σ. This may be due to the fact that measuring blue cluster members is inherently noisier due to the smaller contrast with the background, or it may reflect that blue galaxies in clusters tend to have fallen in more recently (e.g., Abraham et al. 1996), thereby adding stochasticity to the richness measure. Adequately determining the astrophysical origin of this increased scatter would require us to repeat this analysis using full spectroscopic membership information, which we do not currently have available for this sample. We conclude that attempting to include blue galaxies in richness estimates in the common case when only photometric data are available will increase the scatter of the L_X–richness relation.

Thus far, we have treated the color of the red-sequence cluster members as a function of redshift only. However, in addition to a zero point (mean color) and scatter, the ridgeline has a slope in color–magnitude space: fainter (less massive) galaxies have bluer colors, possibly reflecting trends from the mass–metallicity relation (e.g., Kodama & Arimoto 1997; Bernardi et al. 2005; De Lucia et al. 2007). In this section, we investigate the effect of red-sequence tilt and its redshift evolution on the matched filter richness estimation.

Hao et al. (2009) used the maxBCG cluster catalog to measure the slope and intercept of the g − r versus i color–magnitude relation for red-sequence galaxies. They found that the ridgeline slope is nearly independent of cluster richness, consistent with findings that the slope is independent of environment (Hogg et al. 2004). The color filter that incorporates this tilt is described as follows:

$$\begin{equation} G(c,m|z) = \frac{1}{\sqrt{2\pi }\sigma }\exp \left[ -\, \frac{d(c,m|z)^2}{2\sigma ^2} \right], \end{equation} \tag{ 22 }$$

where

$$\begin{equation} d(c,i|z) = c - (m_{17}(i-17.0)+b_{17}). \end{equation} \tag{ 23 }$$

The slope and zero point at i = 17 are taken from Hao et al. (2009):

$$\begin{eqnarray} m_{17} & = & {-}0.0701z-0.008969 \end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray} b_{17} & = & 3.2982z+0.58907. \end{eqnarray} \tag{ 25 }$$

As before, c = g − r is the relevant color and m denotes the i-band magnitude. The net dispersion σ is taken as the sum in quadrature of the intrinsic color dispersion σ_int = 0.05 and the estimated color error σ_c.²¹

After replacing the Gaussian color filter in Equation (13) with the tilt-based filter in Equation (22), we measure the new richness λ_tilt. The scatter in L_X for the new richness is $\sigma _{\ln L_X|\lambda _{\mathrm{tilt}}} = 0.69\pm 0.02$, and the corresponding redshift evolution parameter γ = 1.3 ± 0.5. The change in scatter is insignificant, with r_boot = 0.996 ± 0.01. Therefore, incorporating the tilt of the red sequence does not make a significant difference in the scatter or redshift evolution.

This is not particularly surprising: not only is the tilt of the red sequence small compared to the intrinsic scatter, in Paper I we demonstrated that the adopted Gaussian color filter was very robust relative to small systematic offsets between the center of the filter and the true mean color of cluster galaxies. That said, the tilt of the red sequence evolves with redshift, becoming increasingly important at higher redshifts. Thus, it is possible—even likely—that our above conclusion will not hold at high redshifts, or when extending λ to fainter luminosities, thereby increasing the lever-arm over which the tilt of the red sequence can act. Therefore, despite the fact that we do not observe a significant improvement, we have opted to incorporate the tilt-based color filter from Equation (13) into our standard definition of λ for further tests.

When estimating richness, one must adopt a luminosity or magnitude cut. In Paper I, we adopted a luminosity cut L_cut = 0.4 L_*, which was chosen to allow consistent richness estimates across the entire survey volume (0.1 ⩽ z ⩽ 0.3) while maintaining high precision photometry for the faintest galaxies considered. However, this cut is not uniquely specified by these conditions. The full DR7 input catalog is complete to i ≈ 21.3, which corresponds to a luminosity of 0.1 L_* at a redshift of z = 0.3, so we can easily go deeper. We now explore whether doing so can reduce the scatter in the richness–mass relation.

We have calculated the matched filter richness λ for a set of luminosity cuts: L_cut/L_* = {0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40} for every maxBCG cluster.²² For this test, we use the color filter including the red-sequence tilt, as described in the previous section.

Figure 2 shows the comparison of the scatter in L_X at fixed richness for different luminosity cuts for the top 2000 clusters (black diamonds). We find that there does in fact appear to be an optimal luminosity cut L_cut = 0.2 L_* (m_* + 1.75 mag) below which λ fails to improve any further. We compare the richness for the different luminosity cuts relative to this value using the bootstrap resampling method and find that the proposed cut of 0.2 L_* is significantly (5σ) better than the original luminosity cutoff of 0.4 L_*. The resulting scatter value is $\sigma _{\ln L_X|\lambda } = 0.63\,{\pm}\, 0.02$, with a corresponding redshift evolution parameter of γ = 0.6 ± 0.5, consistent with both no evolution and self-similar evolution.

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One question that may arise from looking at Figure 2 is whether the flattening in the scatter as a function of luminosity cut is driven by the fact that by L_cut = 0.1 L_* one begins to approach the limiting magnitude of the SDSS galaxy catalog for galaxies at z = 0.3. To test this hypothesis, we have run the same scatter analysis on the lower redshift half of our cluster sample, i.e., clusters with redshift below the median redshift of the sample, z_med < 0.23. We maintain the same equivalent space-density cutoff as the full run (black diamonds) by limiting the scatter measurement to the top 1000 clusters. The resulting points are shown in Figure 2 as red squares, and clearly display the same behavior as the full cluster catalog. At z = 0.23, a luminosity of 0.2 L_* (0.1 L_*) corresponds to i = 19.9 (20.7), significantly brighter than the limiting magnitude of the catalog. Thus, we can only conclude that the flattening of the scatter below 0.2 L_* is real, and not due to photometric errors of faint galaxies. For the time being, we simply adopt this new optimal luminosity cut, postponing the discussion of the origin of this cut to Section 6. For reference, decreasing our luminosity cut from 0.4 L_* to 0.2 L_* increases the cluster richness by an average of ≈65%.

The NFW profile was originally introduced as a good fit for the dark matter distribution in N-body simulations (Navarro et al. 1995). Although galaxies will not necessarily follow the same distribution as that of dark matter, studies have shown that the number density of cluster galaxies can be described by an NFW function (e.g., Lin & Mohr 2004; Popesso et al. 2007; Hansen et al. 2005). Nevertheless, a filter based on an NFW profile might not necessarily be ideal. In this section, we investigate the effect of changing the radial filter function.

A projected NFW profile extends to infinity, so we are required to normalize the filter taking into account the cutoff radius. An alternative radial profile suggested by Postman et al. (1996) instead goes to zero at the cutoff radius:

$$\begin{equation} \Sigma _\mathrm{post}(R/R_\mathrm{core}) \propto \frac{1}{\sqrt{1+(R/R_\mathrm{core})^2}} - \frac{1}{\sqrt{1+(R_c/R_\mathrm{core})^2}}. \end{equation} \tag{ 26 }$$

The equation assumes R < R_c, where R_c is the cluster radius as defined in Equation (5); outside of this radius, P = 0. We follow Postman et al. (1996) in setting R_core = 100 h⁻¹ kpc, and we normalize the filter as in Equation (9). This profile gives more weight to the central galaxies and less weight to the peripheral galaxies than the NFW filter. We denote the resulting richness, λ_post. Finally, we also investigate a third possibility, that of replacing the radial filter 2πRΣ(R) with a flat top-hat function, i.e., Σ(R)∝1/R or isothermal, which defines λ_flat. This gives equal weight to cluster galaxies at the center and those at the periphery.

We perform a pairwise comparison of the scatter in L_X at fixed richness among our three richness estimators generated with three radial functions, λ_NFW, λ_post, and λ_flat. For these tests, we use the best luminosity cutoff (0.2 L_*), the color filter including tilt, and fix the cutoff radius at 0.9 h⁻¹ Mpc. In the comparison of the flat profile to the NFW profile, we find $r_\mathrm{boot}= \sigma _{L_X|\lambda _{\mathrm{flat}}}/\sigma _{L_X|\lambda _{\mathrm{NFW}}} = 1.03\pm 0.015$. Thus, using the NFW profile is better than using a flat radial profile at the 2σ level. Comparing the Postman profile to the NFW profile, we find $r_\mathrm{boot}= \sigma _{L_X|\lambda _{\mathrm{post}}}/\sigma _{L_X|\lambda _{\mathrm{NFW}}} = 1.016\pm 0.016$.

The NFW profile gives slightly more weight to peripheral cluster galaxies than the Postman filter, and less weight to the peripheral galaxies than the flat filter. The NFW filter, somewhere in the middle, narrowly outperforms the other two, which are closer to the extremes. That said, it is worth noting that up to ≈30% of BCGs in the maxBCG catalog may not be at the halo center (Johnston et al. 2007). Therefore, we cannot rule out the possibility that our results are driven at least in part by miscentering inherent to the maxBCG catalog. Nevertheless, our tests in this section show that the shape of the radial profile has a relatively weak effect on the fidelity of the richness estimator λ. Thus, we do not feel it is necessary to expend further energy in exploring a broad variety of possible radial filter functions.

Total optical luminosity of a cluster has been suggested as a superior mass tracer to simply galaxy counting (e.g., Popesso et al. 2005, and others), though any such improvement is likely to be small as the optical luminosity and total number of cluster galaxies are highly correlated (e.g., Popesso et al. 2005, 2007; Koester et al. 2007a). In this section, we explore this possibility by using the total optical luminosity of red-sequence galaxies in clusters as an X-ray luminosity tracer.

We have already seen that our λ formalism naturally produces a red-sequence-based cluster membership probability. Consequently, we can readily estimate the total red-sequence cluster luminosity L_λ via

$$\begin{equation} L_{\lambda } = \sum _j L_j p_j, \end{equation} \tag{ 27 }$$

where L_λ is the luminosity weighted lambda, p_j is the membership probability of galaxy j, and L_j is the luminosity of galaxy j. The luminosity is defined as the i-band luminosity of a galaxy at z = 0.25, and all galaxies are k-corrected assuming the galaxies are red galaxies at the redshift of the cluster. We calculated L_λ for our clusters using the best luminosity cutoff (0.2 L_*), the color filter including tilt, and with a fixed 0.9 h⁻¹ Mpc aperture. We find a scatter of $\sigma _{\ln L_X|L_\lambda } = 0.68\pm 0.02$, and a redshift evolution parameter γ = −1.0  ±  0.6. The bootstrap comparison to the corresponding λ estimate results in $r_\mathrm{boot}=\sigma _{\ln L_X|L_\lambda }/\sigma _{\ln L_X|\lambda } = 1.09\pm 0.02$. Thus, calculating the red-sequence luminosity of the clusters is a worse tracer of L_X than pure red-sequence counts.

One worry when interpreting this results is that the result may be systematics driven. Specifically, in estimating the cluster luminosity we assume all galaxies are at the cluster redshift, which can dramatically boost the luminosity assigned to any foreground galaxies. Even with low membership probabilities, such boosts might significantly bias the estimated cluster luminosity. To test this hypothesis, we have repeated our experiment while restricting the sum in Equation (27) to galaxies fainter than the BCG. We find that our results are robust to this change, which suggests that foreground interlopers are not the culprit. There is the additional possibility that the more numerous faint galaxies, with larger photometric errors, are increasing the noise. However, the same trend—that simple number counts are superior to luminosity weights—appears with richnesses using brighter luminosity cuts. Thus, we conclude that the total red-sequence cluster luminosity is a worse L_X tracer than red-sequence galaxy counts.

Another type of luminosity weighting that has been suggested in the past is weighting by the luminosity of the BCG (e.g., Reyes et al. 2008). In Paper I, we showed that the richness estimate of Reyes et al. (2008), while superior to N₂₀₀, has a significantly larger scatter than λ₀. Similarly, in the preset work we did not observe any significant difference when combining λ with the BCG luminosity. This may be simply a fact that our tests are primarily focused on the high richness end. It is clear that at sufficiently low richness—e.g., λ = 1, or a single red galaxy—the luminosity of the BCG must contain additional information about the mass of the host halo. We emphasize that these results do not contradict those of Reyes et al. (2008). The reason is that although significant improvements can be made relative to N₂₀₀, these are not as easily achieved relative to our optimized λ richness estimator.

In Table 1 we summarize the results of our various tests to improve the fidelity of the richness estimator λ. As discussed in Section 4.1, due to the fact that the errors in the scatter of two richness estimators are correlated, our primary diagnostic for improvement is the bootstrap ratio $r_\mathrm{boot}=\sigma _{\ln L_X|\lambda _\mathrm{new}}/\sigma _{\ln L_X|\lambda }$. The other values in the table, the absolute value of the scatter, and the evolution parameter γ, are presented only for reference.

Table 1. Summary of Richness Comparisons

Description	$\sigma _{\ln L_x|N}$	rboot	γ	Section
Benchmark λ0	0.69 ± 0.02	—	0.7 ± 0.8	4.1
With blue members	0.72 ± 0.02	1.04 ± 0.02a	0.1 ± 0.6	4.2
With tilt	0.69 ± 0.02	1.0 ± 0.1a	1.3 ± 0.5	4.3
Deeper (0.2 L*)	0.63 ± 0.02	0.90 ± 0.02a	0.6 ± 0.5	4.4
Flat radial profile	0.64 ± 0.02	1.03 ± 0.015b	0.3 ± 0.5	4.5
Postman radial profile	0.63 ± 0.02	1.016 ± 0.016b	0.7 ± 0.5	4.5
Luminosity weight	0.68 ± 0.02	1.09 ± 0.02b	−1.0 ± 0.6	4.6

Notes. All richnesses are measured in a fixed metric aperture of 0.9 h⁻¹ Mpc. ^aBootstrap scatter ratio r relative to benchmark λ₀. ^bBootstrap scatter ratio r relative to λ computed with "Deeper" luminosity cut.

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In our first suite of tests we compare to the benchmark richness λ₀ presented in Paper I, estimated with a luminosity cut of 0.4 L_*. We find that while counting the blue cluster members slightly increases the scatter, changing the luminosity cut decreases the scatter significantly. The second suite of tests is performed relative to the richness estimated with a luminosity cut of 0.2 L_*, described in Section 4.4. In this suite we find that the radial filter is not very important, although the NFW filter is close to ideal. Finally, we find that weighting all red galaxies by their luminosity significantly increases the scatter, by almost 5σ.

Having explored ways in which λ could be improved while relying on a fixed metric aperture, we now turn toward optimizing cluster radii. Throughout this section, we use our final set of filters, given by

$$\begin{equation} u(c,m,R|z,\lambda) = [2\pi R\Sigma (R)]\phi (m)G(c,m), \end{equation} \tag{ 28 }$$

where Σ(R) is given by Equation (7), ϕ(m) is given by Equation (11), and G(c, m) is given by Equation (22). In addition, our luminosity filter now extends to a luminosity cutoff L_cut = 0.2 L_*. For the rest of this paper, we denote the matched filter richness estimate from this filter simply as λ. We proceed now to optimize the radial scaling parameters R₀ and β from Equation (5).

We optimize the radial aperture using the procedure laid out in detail in Paper I. Briefly, we begin by defining a coarse grid in R₀ and β, and estimate the scatter $\sigma _{\ln L_X|\lambda }$ along this grid. Once we have a rough idea of where the minimum lies in parameter space, we repeat this search using a finer grid centered on the expected minimum. We then use bootstrap resampling to estimate the 1σ and 2σ confidence contours of the location of the scatter minimum in the R₀–β plane. For a more detailed description of this algorithm, we refer the reader to Section 4 of Paper I.

Figure 3 shows our final set of contours. The minimum on the fine grid is consistent with the coarse grid, with radial scaling parameters of R₀ = 1.0 h⁻¹ Mpc and β = 0.2 for use in the scaling relation R_c = R₀(λ/100)^β. These values are in good agreement with the trends seen in Paper I. Although the 1σ contour is closed, there is a broad degeneracy region that extends down to a fixed metric aperture with R₀ = 0.9 h⁻¹ Mpc and β = 0.0. In particular, for a cluster with λ = 45, the median richness of the richest 2000 clusters, the scaled aperture is ∼0.9 h⁻¹ Mpc. Therefore the reference fixed aperture of 0.9 h⁻¹ Mpc is consistent with the scaled aperture, and the richness comparisons in the previous sections are indeed valid for our final richness estimator. However, as discussed in Paper I, the existence of the degeneracy line shown in Figure 3 is largely driven by the limited richness range that we can probe using X-ray luminosity. We fully expect that the variable aperture approach is superior to a fixed metric aperture, particularly when probing lower richness systems, although those systems are out of reach for our present analysis.

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For the radial scaling parameters of R₀ = 1.0 h⁻¹ Mpc and β = 0.2 used in the final version of λ, the scatter in L_X for the top 2000 clusters is $\sigma _{\ln L_X|\lambda } = 0.63\pm 0.02$, and the evolution parameter is γ = 0.5 ± 0.5. These values are consistent with those estimated for the fixed 0.9 h⁻¹ Mpc aperture, and the relative improvement is not significant, with r_boot = 0.99  ±  0.02.

With our fully optimized richness estimator λ in hand, we find that the L_X–λ relation for the top 2000 clusters can be parameterized as

$$\begin{equation} \left< \ln L_X|\lambda \right> = (3.19\pm 0.03) + (1.69\pm 0.06) \ln (\lambda /40), \end{equation} \tag{ 29 }$$

with a scatter of $\sigma _{\ln L_X|\lambda } = 0.63\pm 0.02$, where L_X is calculated in units of 10⁴³ h⁻²₁₀₀ erg s⁻¹ in the 0.1–2.4 keV energy range, within an aperture of r_λ. This relation is plotted in Figure 4, where the dashed lines show the ±2σ_ln L|λ scatter constraints. We note that we show in Rozo et al. (2009a, see Appendix A) that in order to convert the aperture to a total L_X, or L_X within r₂₀₀, we must increase the normalization by ∼10%–20% due to cluster miscentering and RASS PSF effects. In addition, we can achieve smaller statistical errors on the slope and normalization by implementing the full stacking analysis from Rykoff et al. (2008b). Unfortunately, due to the selection effects of the N₂₀₀ threshold, this analysis is liable to be biased at the low richness end. Therefore, we defer a full calibration—including aperture corrections—to a full catalog selected on the λ richness.

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Up until now, we have focused our analysis on the color–magnitude relation as described by g − r versus i. As discussed above, the g − r color is well suited for cluster photometric redshift and richness estimation in the 0.1 < z < 0.3 range covered by the maxBCG catalog. However, this filter combination is not uniquely able to isolate the cluster red sequence and evolve smoothly with redshift. In order for the richness estimator λ to be generally useful, and to be extended to higher redshifts, it must be robust to changes in filters. In this section, we use additional SDSS data to investigate how λ changes with alternative filter choices.

We have chosen to focus our analysis on three alternative color combinations: g − i, r − i, and u − r. Three bands—g, r, and i—have high quantum efficiency in SDSS, and thus we do not need to worry about significant problems with photometry near the limiting magnitude. When using the u-band data we limit the redshift range to z < 0.25 (see below). The colors g − i and u − r have the advantage of containing the g band which changes strongly as the 4000 Å break moves with redshift. Although the r − i color evolves with redshift and allows us to pick out the red sequence, this evolution is not as strong as when combined with the g color, and thus we expect that λ may not perform as well.

To create the red-sequence model of the filter, we follow the method of Hao et al. (2009, see Section 5.1). For each cluster with λ₀ > 10 we take all galaxies within 0.9 h⁻¹ Mpc and brighter than 0.4 L_*. We use the ECGMM to decompose the red sequence and blue cloud/background for the color of interest. We then fit a linear model to all galaxies with ±2σ of the red sequence, yielding a slope and intercept at i = 17, m₁₇ and b₁₇ (as in Section 4.3). Next we bin the clusters in redshift bins of width 0.02 and calculate the mean <m₁₇ > and <b₁₇ >. Finally, we fit a linear model as a function of redshift to obtain a simple functional form of <m₁₇|z > and <b₁₇|z > akin to Equation (24). This model is used to calculate the given richness within a fixed metric aperture of 0.9 h⁻¹ Mpc.

Figure 5 shows the comparison of λ calculated with various color filters to the basic λ_{g − r}. In the top panel we compare λ_{g − i}, in the middle panel we compare λ_{r − i}, and in the bottom panel we show λ_{u − r}. The top two panels show all maxBCG clusters, and the bottom panel only those with z < 0.25 due to the smaller sensitivity in the u band. The inset histograms illustrate the distribution of differences between the given richness and λ_{g − r}.

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Table 2 shows the results of comparing the richness estimates to the baseline λ_{g − r}. First, we have estimated the width of the distribution, Δλ_rms, shown in the inset plots of Figure 5. We report Δλ rather than Δλ/λ because it is the former that is roughly constant with richness. Second, we have compared the scatter in $\sigma _{L_X|\lambda }$ using the bootstrap resampling ratio r described in Section 4.1. For the first two alternative versions of λ we use the top 2000 clusters in the full redshift range; for λ_{u − r} we use the top 1000 clusters for the lower redshift range 0.1 < z < 0.25.

It is readily apparent that the change from λ_{g − r} to λ_{g − i} is nearly insignificant. The median richness of the top 2000 clusters is ∼45, so the observed scatter is ≲ 4%. For reference, Poisson scatter at this richness would correspond to ≈15% scatter, while Rozo et al. (2010a) estimate the scatter in N₂₀₀ at fixed mass to be ≈35%. In short, g − i is effectively as efficient as g − r for the purposes of selecting red-sequence galaxies, reflecting the fact that the 4000 Å break falls within the g filter.

Focusing now on the r − i and u − r filter combinations, we see these choices exhibit a significantly larger scatter in Δλ, of order 10%, still less than Poisson scatter, and much less than the Rozo et al. (2010a) estimate for N₂₀₀. This increased scatter relative to g − r is also reflected as increased scatter in L_X at fixed richness. In the case of r − i, this can be understood by the fact that our filter combination does not straddle the 4000 Å break, and therefore the red sequence is not as prominent against the background. The u − r color does straddle the 4000 Å break, but suffers from the much lower SDSS sensitivity in the u band.

In summary, we see that an effective use of the λ richness does not require the specific g − r filter combination used in this work. Indeed, use of any filter combination that can effectively isolate the red sequence will result in nearly identical and unbiased values for λ, although the scatter is reduced when using high sensitivity filters that straddle the 4000 Å break. Even in the case of the less optimal r − i and u − r combinations we get a reasonable, although not ideal, richness estimator. Due to the stability of λ when using different filter combinations, we expect these methods to be easily generalizable for other telescopes and higher redshifts.

With the entire SDSS DR7 at our disposal, covering ∼7500 deg², we can make a very accurate estimation of the global background as a function of color and magnitude. In other instances, however, there may be greater uncertainties in the true mean background for a given filter combination. In this section we investigate the effect of varying the background level on λ.

In general, uncertainties in the background will be a function of color and magnitude. For simplicity, we model uncertainty in the background as a boosting or de-boosting of the background normalization. Our intention with this approximation is to give a rough estimate of the systematic uncertainty in richness estimates given the uncertainty in the background. We expect that a boost of the background normalization $b(\mathbf {x})$ will cause the richness λ to decrease by a fixed amount (independent of λ) and a decrease of the background normalization will cause λ to increase by a similar amount.

In Figure 6, we show the effect of changing the background normalization on the calculation of λ in a fixed 0.9 h⁻¹ Mpc aperture. When the background normalization is deboosted by a factor of 0.5, then λ changes by Δλ = 6.6 ± 2.9, and when it is boosted by a factor of 1.5, then Δλ = −4.2 ± 1.7. For a moderate richness cluster of λ ∼ 30, incorrectly estimating the background by ∼50% can bias the richness by ∼20%, approximately equal to the Poisson scatter. On the other hand, a ∼10% error in the background normalization has a negligible effect on the richness estimation of ∼3%. We emphasize these values are only meant to help in estimating how precisely the background must be modeled in order to ensure a negligible impact on richness estimates.

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In our adapted color filter from Section 4.3, and in all our tests, we fix the intrinsic scatter in the red sequence at σ_int = 0.05. We now investigate the effect on λ and $\sigma _{L_X|\lambda }$ on changes in the intrinsic scatter, similar to the tests of Section 4.3 in Paper II. For this test, we fixed the radial size at 0.9 h⁻¹ Mpc and set the red-sequence width to 0.03 mag and 0.07 mag, representing a narrow and wide extreme.

In each case, we find that the effect on both the scatter and the richness estimate is negligible. For σ_int = 0.05, then Δλ = −1.5 ± 1.0; and for σ_int = 0.07, then Δλ = 1.2 ± 0.9. Unsurprisingly, if we take a narrower (wider) red sequence then the richness estimate decreases (increase) as we lose (gain) galaxies at the margins. However, the total probability of these additional galaxies is quite small, and the overall bias in the richness estimate is negligible. Note that one reason that λ is not very sensitive to the red-sequence width is because the color filter in Equation (13) uses the sum in quadrature of the photometric error and intrinsic color dispersion. We conclude that our richness estimator is robust to changes in the red-sequence width, and thus is suitable for other filter combinations that span the 4000 Å break that may have slightly different ridgeline widths.

The calculation of λ requires measuring all the galaxies above a given luminosity threshold that evolves with redshift. In practice, uncertainty in the luminosity threshold is equivalent to uncertainty in the photometric zero point as well as systematic offsets in differing methods of calculating galaxy magnitudes. In this section, we explore the effect of an offset in the photometric zero point, which is equivalent to each of these effects.

As before, we run with a fixed 0.9 h⁻¹ Mpc aperture with a range of systematic offsets in the magnitude limit for the luminosity function filter (Equation (11)). We find that for small zero-point shifts—up to ±0.05 mag—the effect on λ is negligible (δλ  <  1) for the vast majority of clusters, consistent with the observation in Paper II that such errors do not significantly impact the scatter of the richness–mass relation. This is because only rarely is a red-sequence galaxy added or removed from consideration when making such a small shift. Of course, large photometric shifts in rich clusters can be significant. For instance, a ±0.1 mag shift in a rich cluster can change λ by ≈10%. Note, however, that such photometric errors are much larger than expected for upcoming photometric surveys.

In this section, we investigate the effect of uncertainty in the color–redshift relation for the Gaussian color filter. If we were to follow the strategy of the preceding sections, we would approach this issue by systematically shifting the slope and intercept of the color filter. However, this treatment would unrealistically ignore the data at hand. When observing an actual galaxy cluster, we can always measure the red sequence for each individual cluster. Therefore, we explore the effect on λ if we measure the red sequence intercept and slope for each cluster individually.

For each maxBCG cluster, we first take all the galaxies within 0.9 h⁻¹ Mpc and brighter than 0.2 L_*. To measure the intercept and slope of the red sequence, we follow the method of Sections 4.2 and 4.3. The color distribution of galaxies is decomposed into two Gaussians (the red-sequence and the blue/background galaxies) using the ECGMM of Hao et al. (2009). To limit the degrees of freedom to allow for accurate fitting of relatively poor clusters (λ ≲ 30), we fix the width of the red-sequence component at σ_int = 0.05 mag. One could in principle also attempt to fit this from the data, but we have already shown that the recovered richness is largely insensitive to this parameter, and the typical ridgeline width is ≈0.05. If the mixture model is unable to identify two distinct components then the cluster is flagged and λ is not measured. We then take all galaxies within 2σ_int and fit the red-sequence slope and intercept. These values are then substituted for Equation (24) to calculate the richness λ.

Figure 7 shows the comparison of λ_fit, where we fit for the red sequence on a cluster-by-cluster basis to λ in a fixed 0.9 h⁻¹ Mpc aperture. Results are nearly identical with a variable aperture with R₀ = 1.0 h⁻¹ Mpc and β = 0.2. The inset plot shows that for individual clusters λ shifts by a negligible 0.1 ± 0.6. Overall, the richness measurement is extremely robust to perturbations in the red-sequence location. This is due to a combination of the contrast of the red sequence with the background galaxies, and the fact that the smooth Gaussian color filter is more tolerant of color offsets than a top-hat filter (as explored in Section 6 of Paper I). It should be noted that our automated algorithm for red-sequence fitting succeeded in measuring the red sequence for all clusters of richness λ ≳ 50. At λ ⩽ 40, the failure rate was ≈10%. We are confident that this failure rate could be decreased if one incorporated even mild priors on the red-sequence parameters, for instance from population synthesis models. The main point here is not the automated red-sequence fitting, but rather the fact that so long as the red sequence can be properly fit from the data, one does not need an a priori model for the red sequence in order to be able to compute λ. Of course, having such a model will help in the limit of low signal-to-noise ratio, and is less computationally intensive.

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As discussed in Section 2.3, a clean input catalog is required for accurate cluster finding and richness estimation. The old mantra "garbage in, garbage out" is especially apt. We refer to the inclusion of any object that is not a correctly measured galaxy as "catalog noise." This includes stars, asteroids, image artifacts, and improperly measured photometric errors. In general, it also includes pipeline-to-pipeline variations in the reduction, detection, and photometry for the same object. In this section, we obtain a first-order estimate of the magnitude of the effect of catalog noise on our richness estimation.

To test the effect of catalog noise, we compare the scatter in L_X at fixed richness for two versions of the input catalog. The first is the "clean catalog" that has been filtered as described in Section 2.3. The second is the raw catalog, using all the objects marked as galaxies in DR7 without any additional filtering. Figure 8 shows the histogram of the number of objects as a function of the i-band magnitude for stripe 10 in SDSS. There are significantly more galaxies at the faint end in the uncleaned catalog (red dashed histogram), many of which are false detections.

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For each input galaxy catalog, we first compute the background as described in Section 3.4. We then compute both λ₀, the original matched filter richness, and λ, the optimized richness, both at the fiducial fixed 0.9 h⁻¹ Mpc scale. We expect that the effect of catalog noise will be greater on λ, as it uses a deeper luminosity cut where the catalog noise is greater. This is indeed what we find for the top 2000 clusters, where the scatter in the L_X at fixed richness increases from 0.70 ± 0.02 to 0.72 ± 0.02 for λ₀, and from 0.63 ± 0.02 to 0.66 ± 0.02 for λ. Using the bootstrap resampling ratio r described in Section 4.1, we find that using the noisy catalog increases the scatter by r_boot = 1.03 ± 0.02 for λ₀ and r_boot = 1.04 ± 0.02 for λ. In addition, we confirm that the scatter measured is consistent between the cleaned DR7 catalog and the cleaned DR4 catalog used in Paper I. Thus, at least for different versions of the SDSS pipeline, λ is robust to this level of pipeline-to-pipeline variations.

Overall, with our current catalog and our best richness estimator, we can detect the effect of catalog noise at the 2σ level in our chosen figure of merit. Therefore, catalog noise is a nuisance, though it is not a critical path item. This is partly due to the high quality of the raw SDSS DR7 catalog. However, we emphasize that we can detect this effect even though we are recomputing the background for each input catalog, and these false galaxies are presumably not correlated with the cluster positions. Thus, increased noise in the background does translate to increased scatter in the richness estimator and should be controlled as well as possible.

• 1.  
  All galaxies within ∼1.25 h⁻¹ Mpc of the cluster center. The richest maxBCG cluster has λ = 216 and R_c = 1.17 h⁻¹ Mpc, so this is the largest size that is needed.
• 2.  
  If the cluster redshift z is between 0.1 and 0.3, you will need the g − r color (c) of each galaxy. As described in Section 5.1, any pair of sensitive bands that straddle the 4000 Å break may be substituted. Similarly, if you work at higher or lower redshift, you will need to change your filter appropriately.
• 3.  
  The i-band magnitude of each galaxy or a suitable red filter.
• 4.  
  The k-corrected value of 0.2 L_* for the band in question. For SDSS i band and 0.05 < z < 0.35, m_*(z) is well described by Equation (12), and the luminosity cut is given as m_* + 1.75 mag.
• 5.  
  The mean galaxy background as a function of color and magnitude (see Sections 3.4 and 5.2).
• 6.  
  The ridgeline slope and intercept, as described in Equation (24). Alternatively, you can fit for the ridgeline slope and intercept with σ_int = 0.05 from the list of galaxies.

• 1.  
  Calculate the luminosity filter value for each galaxy, ϕ(m), as described in Equation (11) in Section 3.2. Note that the filter is normalized such that the integral from m = −∞ to m_cut is exactly one.
• 2.  
  Calculate the radial filter value for each galaxy, 2πRΣ(R), as described in Equation (7) in Section 3.1. The radial filter needs to be normalized by integrating R_c as per Equation (9). Thus, the amplitude of Σ(R) is a function of R_c, with the normalization constant for the particular implementation of the NFW profile adapted in this work given in Equation (10).
• 3.  
  Calculate the color filter value for each galaxy, G(c, m|z), as described in Equation (22) in Section 4.3.
• 4.  
  Calculate the background filter value for each galaxy, $b(m,c)=2\pi R \bar{\Sigma }_g(m,c)/C(z)^2$. We have defined $\bar{\Sigma }_g(m,c)$ as the mean galaxy background in N/sq.deg./mag/mag, and the conversion factor C(z) is given in degrees/Mpc at the redshift of the cluster.
• 5.  
  Using a zero finder (e.g., bisector or Newton's method), solve the following equation:

  $$\begin{equation} 0 = \lambda - \sum _{R<R_c(\lambda)} p_i \end{equation} \tag{ A1 }$$

  where

  $$\begin{equation} p_i= \frac{\lambda 2\pi R_i \Sigma (R_i)\phi (m_i)G(c_i)}{\lambda 2\pi R_i \Sigma (R_i)\phi (m_i)G(c_i) + b(m_i,c_i)} \end{equation} \tag{ A2 }$$

  and

  $$\begin{equation} R_c(\lambda) = R_0(\lambda /100.0)^\beta. \end{equation} \tag{ A3 }$$

  Note that once the root λ of Equation (A1) has been found, Equation (A2) can be used to estimate the membership probability of every galaxy.
• 6.  
  To estimate the statistical error in λ, use Equation (4). This measurement error is typically very small, of order Δλ/λ = 0.4λ^−1/2 or ≈4% (9%) for λ = 100 (15).

Sample IDL code that will calculate λ for SDSS data will also be released online.

The goal of this paper was to extensively test and optimize the richness measure proposed by Rozo et al. (2009b). A careful calibration of the mass–richness relation is beyond the scope of the current paper, though we do intend to address this problem in subsequent work. Nevertheless, we felt it was important to provide a rough calibration that may be used for comparison purposes, and to test the efficacy of our estimator. To this end, we have relied on abundance matching techniques. The primary drawback of this calibration is that it is cosmology dependent, so the output cannot be directly used to obtain independent cosmological constraints. Briefly, we compute the cumulative cluster abundance function N_clusters(> λ), defined as the number of maxBCG clusters of richness λ or higher. We then also compute the expected cumulative mass function N_halos(> m) as a function of mass by integrating the Tinker et al. (2008) mass function for our fiducial cosmology over the maxBCG survey volume.²⁵ When computing the mass function, we adopt as our fiducial mass definition M_200m, the mass contained with a 200 overdensity relative to the mean matter density. A cluster of richness λ is assigned a mass M by solving the equation N_clusters(> λ) = N_halos(> M) for M(λ). We call this mass estimate the density matching mass, denoted as M_dm(λ).

As detailed in Mortonson et al. (2011), mass estimates from density matching are expected to be biased high relative to the Bayesian posterior by an amount equal to (1/2)γσ² in the log. Here, γ is the slope of the halo mass function at mass M, and σ is the scatter in mass at fixed richness. Thus, in order to correct for this bias, we must first estimate the scatter in mass at fixed richness for λ. To do so, we rely on the scatter in L_X at fixed richness. Assuming ln L_X = a + αln M, it follows that a scatter in mass σ_M|λ corresponds to a scatter in L_X given by ασ_M|N. If L_X and λ are uncorrelated, then one simply needs to add in quadrature the intrinsic scatter in L_X at fixed mass, $\sigma _{L_X|M}$, to arrive at the total scatter in L_X at fixed richness:

$$\begin{equation} \sigma _{L_X|\lambda }^2 = \alpha _{L_X|M}^2 \cdot \big(\sigma _{L_X|M}^2 + \sigma _{M|\lambda }^2\big). \end{equation} \tag{ B1 }$$

With this equation, one can straightforwardly solve for σ_M|λ. In the slightly more general case when L_X and λ are correlated, one finds

$$\begin{equation} \sigma _{M|\lambda } = r\sigma _{M|L_X} + \left[ \frac{\sigma _{L_X|\lambda }^2}{\alpha _{L_X|M}^2} - \sigma _{M|L_X}^2 (1-r^2) \right]^{1/2}. \end{equation} \tag{ B2 }$$

In the above equation, r is the unknown normalized correlation coefficient between the richness λ and L_X at fixed mass (e.g., Allen et al. 2011), while $\alpha _{L_X|M}$ is the slope of the L_X–M relation. We adopt the fiducial values $\alpha _{L_X|M}=1.61$ and $\sigma _{M|L_X}=0.246$ as per Vikhlinin et al. (2009). As for the correlation coefficient r, because the scatter in L_X is dominated by emission from the core, we do not expect λ and L_X to be strongly correlated. Here, we simply consider three values for r, r = ±0.3, and r = 0. Setting the scatter $\sigma _{L_X|\lambda }=0.63$ as appropriate for our top 2000 clusters, we arrive at σ_M|λ = 0.31^+0.08_−0.07, where the error bars reflect the change in the scatter for r = ±0.3. Figure 11 (left panel) illustrates how our recovered scatter σ_M|λ changes as a function of the minimum richness λ of the sample under consideration.

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One interesting feature of the scatter in mass at fixed richness as a function of λ is that the scatter appears to increase slowly with decreasing richness for λ ≳ 60, but begins to climb much faster below λ ≲ 60. Remarkably, in Paper II we found that λ ≈ 60 is the richness at which the miscentering of maxBCG galaxy clusters is expected to become important. Moreover, as we argue in Paper II, cluster miscentering "turns-on" very quickly. Consequently, it is possible that the rapid rise of the scatter with decreasing richness below λ ≲ 60 does not reflect the true intrinsic scatter of the estimator λ, but is rather a reflection of the miscentering properties of maxBCG clusters. Above λ ≳ 60, however, we do not expect cluster miscentering to play as significant a role. Thus, it is likely that the observed scatter at λ ≳ 60 is close to the intrinsic scatter of our richness estimator.

Based on the results shown in Figure 11 (left panel), we adopt a fiducial scatter σ_M|λ = 0.25. With this scatter in hand, we can correct the density matched masses by the expected bias. If M_dm(λ) is the density matched mass of a cluster of richness λ, we set the cluster's final mass to

$$\begin{equation} M(\lambda) = \exp \left(-\frac{1}{2}\gamma (M_\mathrm{dm}(\lambda)) \sigma _{M|\lambda }^2 \right) M_\mathrm{dm}(\lambda), \end{equation} \tag{ B3 }$$

where γ(M_dm) is the slope of the halo mass function dN/dln M∝M^−γ evaluated at the density matching mass M_dm. Having assigned a mass to every cluster in this fashion, the resulting mass–richness relation is fit to a power law using all clusters of richness λ ⩾ 60. We restrict ourselves to these clusters given that we believe cluster miscentering might be starting to become important below this richness. We finally arrive at

$$\begin{equation} \ln \left(\frac{M_{200m}}{h_{70}^{-1} 10^{14}\ M_\odot } \right) = 1.72 + 1.08 \ln (\lambda /60). \end{equation} \tag{ B4 }$$

Note that we have scaled the masses relative to h = H₀/100 km s⁻¹ Mpc⁻¹ = 0.7, and we have made explicit that this scaling relation is appropriate for a mass overdensity of 200 relative to the mean matter density. Equation (B4) is our proposed scaling between mass and richness, while σ_M|λ = 0.25 is our fiducial value for the scatter in mass at fixed richness. This value my be somewhat overestimated at the high-mass end. Note, however, that the amplitude A of this relation can shift by ≈0.1 depending on the value of the correlation coefficient r, the choice of fiducial cosmology, etc. Adopting a 20% systematic uncertainty the overall amplitude, and adding in quadrature to the expected statistical uncertainty σ_M|λ = 0.25, we find that the total uncertainty in the mass of any given cluster is ≈0.33 at the 1σ level.

We can also compute the corresponding mass–richness relations for other mass definitions by rescaling all assigned cluster masses using the formulae in Hu & Kravtsov (2003) and refitting to a power law. We find

$$\begin{eqnarray} \ln \left(\frac{M_{200c}}{h_{70}^{-1}\,10^{14}\ M_\odot } \right) & = & 1.48 + 1.06 \ln (\lambda /60) \end{eqnarray} \tag{ B5 }$$

$$\begin{eqnarray} \ln \left(\frac{M_{500c}}{h_{70}^{-1}\,10^{14}\ M_\odot } \right) & = &1.14 + 1.04 \ln (\lambda /60). \end{eqnarray} \tag{ B6 }$$

We emphasize again, however, that these are not meant to be a rigorous mass calibration, a problem that we defer to future work.

As a check of our mass calibration, we have assembled two cluster samples that overlap the maxBCG footprint and redshift range. The first sample is drawn from the LoCuSS observations of high luminosity clusters from RASS cluster catalogs (Ebeling et al. 1998, 2000; Böhringer et al. 2004). These clusters have been observed with XMM-Newton to obtain hydrostatic mass estimates (Zhang et al. 2008, 2010) and Subaru to obtain weak-lensing observations to provide independent estimates of the masses of several clusters (Zhang et al. 2010). Here we concentrate on the hydrostatic mass estimates of Zhang et al. (2008, 2010). The ROSAT Brightest Cluster Sample (BCS; Ebeling et al. 1998) has been observed with Chandra, with hydrostatic mass estimates obtained by Mantz et al. (2010). The X-ray mass values for the BCS clusters have been reduced by 11% to account for the Chandra calibration update described in that paper (A. Mantz 2011, private communication). For each cluster, we have used the X-ray center and spectroscopic redshift to estimate λ from SDSS DR7 photometric data. We emphasize that we cannot use these data to provide a rigorous mass calibration due to the fact that these cluster samples do not constitute a random sampling of the maxBCG clusters. We only use these data for illustrative purposes and to test whether the mass calibration derived from density matching is reasonable.

Figure 11 (right panel) shows the mass as a function of λ obtained via density matching, scaled to M_500c (Equation (B6)). Dotted lines are the expected 90% confidence interval assuming an uncertainty Δln M = 0.33 as discussed above (25% intrinsic scatter, 20% systematic uncertainty). The LoCuSS sample is overplotted with red triangles, and the BCS clusters are shown with blue squares. It is clear that there is a normalization offset between the two data sets, which corresponds to a systematic offset Δln M = 0.45 in mass, which includes five clusters that overlap between the samples. A full accounting of this offset is beyond the scope of this paper, but there are several possibilities. First, the analyses were done with different data sets (XMM and Chandra); second, there is an additional 10% systematic uncertainty in the normalization of the BCS clusters due to the uncertainty in f_gas used to calculate the masses; third, because the masses are calculated in a scaled aperture (r_500c), any slight normalization offset is magnified in the full analysis; fourth, Zhang et al. (2008) note that if one changes their parameterization of the temperature profile below ≈0.5 R₅₀₀ so as to allow for a rapid drop in the cluster temperature, their hydrostatic masses can increase by as much as 25%.

Regardless of what the ultimate source of the difference in the normalization between the two data sets is, it is reassuring to see that our rough mass calibration fits between these two data sets. Moreover, the scatter in the mass–richness relation is clearly smaller than the overall difference in normalization between the two sets, and consistent with our 25% estimate for the scatter. This demonstrates both that the richness estimator λ is indeed tightly correlated with cluster mass with scatter at the ∼20%–30% level, and that our 33% estimate for the uncertainty in the mass of any one cluster is reasonable. Note that since the scatter in mass at fixed richness for the original maxBCG richness (N₂₀₀) was estimated to be σ_ln M|N = 0.45 (Rozo et al. 2009a), this means that even this rough mass calibration allows us to predict individual cluster masses with significantly higher precision than we could using the maxBCG richness.

Table 3 contains the improved richness, λ, for each cluster in the maxBCG cluster catalog. The positions (R.A., decl.), photometric redshifts (z), and original richness estimate (N₂₀₀) are taken directly from Table 1 in Koester et al. (2007a). The spectroscopic redshifts (z^BCG_spec) are obtained by cross-identifying the maxBCG BCG positions with the full spectroscopic catalog from SDSS DR8,²⁶ thus increasing the number of BCGs with spectra from 5413 to 9409. Finally, the improved richness and error estimates (λ, λ_e) are taken from this work. Note that the catalog is complete in N₂₀₀, but is not complete in λ. We set z^BCG_spec = −1.0 when a spectrum is not available, and set λ = −1.0 when no significant number of red galaxies above background is found by the richness estimator. In addition, we note that the DR7 galaxy catalog used to calculate λ covers a slightly different footprint than the original DR4 catalog used to construct the original maxBCG catalog. There are 80 maxBCG clusters that we cannot calculate reliable richness estimates as they are in newly masked regions. For these, we set λ = −2.0 in the catalog.