THE REDMAPPER GALAXY CLUSTER CATALOG FROM DES SCIENCE VERIFICATION DATA

DES is an ongoing five-band (grizY) photometric survey performed with the Dark Energy Camera (DECam, Flaugher et al. 2015) on the 4 m Blanco Telescope at Cerro Tololo Inter-American Observatory (CTIO). Prior to the beginning of the DES, from 2012 November to 2013 March, DES conducted a $\sim 250\;{\mathrm{deg}}^{2}$ "SV" survey. The largest contiguous region covers $\sim 160\;{\mathrm{deg}}^{2}$ of the eastern edge of the SPT survey ("SPT-E" hereafter). A smaller $\sim 35\;{\mathrm{deg}}^{2}$ region is in the western edge of the footprint ("SPT-W" hereafter). In addition, the DES surveys 10 Supernova fields ("SN fields" hereafter) every 5–7 days, each of which covers a single DECam 2.2-degree-wide field of view, for a total of $\sim 32\;{\mathrm{deg}}^{2}$ of deeper imaging (including extra offset pointings of SN fields taken during SV). Finally, there are smaller discontinguous regions targeting massive clusters (Melchior et al. 2015) and the COSMOS field (Scoville et al. 2007). We utilize this DES SV data set to construct the first DES redMaPPer cluster catalog. The redMaPPer footprint used in this paper is the same as that used for the associated redMaGiC (red-sequence matched-filter Galaxies Catalog) of red galaxies with well-behaved photo-z performance (Rozo et al. 2015c, henceforth RM15).

The DES SV data were processed by the DES Data Management (DESDM) infrastructure (R. Gruendl et al 2016, in preparation), which includes image detrending, astrometric registration, global calibration, image coaddition, and object catalog creation. Details of the DES single-epoch and coadd processing can be found in Sevilla et al. (2011) and Desai et al. (2012). We use SExtractor to create object catalogs from the single-epoch and coadded images (Bertin & Arnouts 1996; Bertin 2011).

After the initial production of these early data products, we detected several issues that were mitigated in post-processing, leading to the creation of the "SVA1 Gold" photometry catalog.⁵⁷ First, we masked previously unmasked satellite trails. Second, we use a modified version of the big-macs stellar-locus regression (SLR) fitting code (Kelly et al. 2014)⁵⁸ to recompute coadded zero-points over the full SVA1 footprint. Third, regions around bright stars ($J\lt 13$) from the Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006) were masked. Finally, we removed 4% of the area with a large concentration of centroid shifts between bandpasses in individual objects, indicating scattered light, ghosts, satellite trails, and other artifacts (Jarvis et al. 2015, Section 2.1). We utilize the SExtractor MAG_AUTO quantity derived from the coadded images for galaxy total magnitudes and colors. This choices reflects the fact that methods used to compute multi-epoch fitting photometric quantities are still under development. The added noise in the color results in a larger observed red-sequence width, which results in slightly poorer photometric redshifts, as shown in Section 4.1.2. For the present work, we have not made use of the DES Y-band imaging because of uncertain calibration and the minimal lever-arm gained at the redshifts probed in this paper. Finally, our fiducial star/galaxy separation is performed with the multi-band multi-epoch image processing code ngmix used for galaxy shape measurement in DES data (Jarvis et al. 2015), as detailed in Appendix A of RM15.

The footprint is initially defined by MANGLE (Swanson et al. 2008) maps generated by DESDM which describe the geometry of the coadded data in polygons of arbitrary resolution. For ease of use, these are then averaged over HEALPIX NSIDE=4096 pixels (Górski et al. 2005), where each pixel is approximately 07 on a side. The pixelized MANGLE maps are combined with maps of the survey observing properties (e.g., airmass, FWHM, etc.) compiled by Leistedt et al. (2015) using the method of Rykoff et al. (2015) to generate 10σ MAG_AUTO limiting magnitude maps. We first restrict the footprint to regions with deep MAG_AUTO in the z band (${m}_{z,\mathrm{lim}}$) such that ${m}_{z,\mathrm{lim}}\gt 22$, as shown in Figure 1 for $\sim 125\;{\mathrm{deg}}^{2}$ in the SPT-E region.⁵⁹ Only galaxies brighter than the local $10\sigma$ limiting magnitude are used in the input catalog.

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The ngmix runs used for star/galaxy separation in this paper and in RM15 were primarily used for galaxy shape estimation for DES cosmic shear (Becker et al. 2015) and cosmological constraints (The Dark Energy Survey Collaboration et al. 2015). Therefore, the runs were performed on regions with very tight tolerance for image quality and were restricted to the largest contiguous region (SPT-E) as well as to supplementary runs on the SN fields. These regions comprise our fiducial footprint for the input galaxy catalog of $148\;{\mathrm{deg}}^{2}$ (of which $125\;{\mathrm{deg}}^{2}$ is in SPT-E). However, mask boundaries and holes reduce the effective area for extended cluster sources to $\sim 100\;{\mathrm{deg}}^{2}$ (see Section 3.6 for details). In Section 5, we describe an expanded footprint where we relax some of these constraints, and include SPT-W and COSMOS, with less robust star/galaxy separation.

The spectroscopic data used in this paper comes from the Galaxy and Mass Assembly survey (GAMA Driver et al. 2011), the VIMOS VLT Deep Survey (VVDS Garilli et al. 2008), the 2dF Galaxy Redshift Survey (2dFGRS Colless et al. 2001), SDSS (Ahn et al. 2013), the VIMOS Public Extragalactic Survey (VIMOS Garilli et al. 2014), and the Arizona CDFS Environment Survey (ACES Cooper et al. 2012). In addition, we have a small sample of cluster redshifts from SPT used in the cluster validation of Bleem et al. (2015). These data sets have been further supplemented by galaxy spectra acquired as part of the OzDES spectroscopic survey, which is performing spectroscopic follow-up on the AAOmega instrument at the Anglo-Australian Telescope in the DES supernova fields (Yuan et al. 2015). In all, there are 36,607 photometric galaxies with spectroscopic redshifts in our input catalog, although only ∼2000 are red cluster members, and ∼1400 are used in the calibration of the red sequence in Section 3.3.

In addition to our new catalog of DES SVA1 data, we have updated the redMaPPer catalog for SDSS DR8 photometric data (Aihara et al. 2011), which remains the most recent photometric data release from SDSS. The DR8 galaxy catalog contains $\sim {\rm{14,000}}\;{\mathrm{deg}}^{2}$ of imaging, which we cut to the ${\rm{10,401}}\;{\mathrm{deg}}^{2}$ of contiguous high-quality observations using the mask from the Baryon Acoustic Oscillation Survey (BOSS, Dawson et al. 2013). The mask is further extended to exclude all of the stars in the Yale Bright Star Catalog (Hoffleit & Jaschek 1991), as well as the area around objects in the New General Catalog (Sinnott 1988). The resulting mask is that used by RM1 to generate the SDSS DR8 redMaPPer catalog. We refer the reader to that work for further discussion on the mask, as well as object and flag selection.

Total magnitudes are determined from i-band SDSS CMODEL_MAG, which we denote as ${m}_{i}$, and colors from ugriz SDSS MODEL_MAG. All of our spectroscopy is drawn from SDSS DR10 (Ahn et al. 2013). Finally, we make use of the $10\sigma$ limiting magnitude maps from Rykoff et al. (2015, see, e.g., Figure 4). As with SVA1 data, only galaxies brighter than the local $10\sigma$ limiting magnitude are used in the input catalog.

Variable survey depth can lead to galaxies being "masked out" from galaxy clusters. Specifically, if a member galaxy (with $L\geqslant 0.2{L}_{*}$) has a magnitude below our brightness threshold, then one needs to statistically account for this missing galaxy, as per the above formalism. To do so, however, one needs to know the survey depth over the full area coverage of the galaxy cluster.

The original redMaPPer application to SDSS DR8 in RM1 (redMaPPer v5.2) assumed that the survey had a uniform depth with ${m}_{i}\lt 21.0$. In the update described in RM4 (redMaPPer v5.10), we empirically computed the local survey depth averaged over the location of each cluster. This was superior to assuming a constant-depth survey, but ignored small-scale depth variations, as well as being somewhat noisy. In this updated version (redMaPPer v6.3), we have extended redMaPPer to incorporate variable survey limiting magnitude maps as detailed in Rykoff et al. (2015) and described in Section 2. Specifically, we utilize the local survey depth from these depth maps to estimate the fraction of cluster galaxies that are masked, as defined in Section 3 and detailed in Appendix B of RM4. In the present version of the algorithm, we assume that the red galaxy detection is complete (modulo masking) at magnitudes brighter than the local $10\sigma$ limiting magnitude used to select the input catalog. In future versions, we intend to track the full completeness function, as described in Section 5 of Rykoff et al. (2015).

As with the previous versions of the redMaPPer algorithm, our luminosity filter is based on a Schechter function (e.g., Hansen et al. 2009) of the form

$$\begin{eqnarray}&&\phi ({m}_{i})\propto {10}^{-0.4({m}_{i}-{m}_{*})(\alpha +1)}\mathrm{exp}(-{10}^{-0.4({m}_{i}-{m}_{*})}),\end{eqnarray} \tag{ 3 }$$

where we set the faint-end slope $\alpha \;=\;1.0$ independent of redshift. Previously, we set the characteristic magnitude, ${m}_{*}(z)$, using a k-corrected, passively evolving stellar population which we had derived from a PEGASE.2 stellar population/galaxy formation model (Fioc & Rocca-Volmerange 1997; Eisenstein et al. 2001; Koester et al. 2007b). As this was derived specifically for the SDSS filters at relatively low redshift, we have updated our reference ${m}_{*}(z)$ to more simply allow for different filter sets and a broader redshift range.

The new value of ${m}_{*}(z)$ is computed using a Bruzual & Charlot (2003, BC03) model to predict the magnitude evolution of a galaxy with a single star formation burst at z = 3 (with solar metallicity and Salpeter IMF) as implemented in the EzGal Python package (Mancone & Gonzalez 2012). We normalize m_* so that ${m}_{i,\mathrm{SDSS}}\;=\;17.85$ at z = 0.2 for an L_* galaxy. This was chosen to match the ${m}_{*}(z)$ relation from RM1 and Rykoff et al. (2012). We have additionally confirmed that the evolution of ${m}_{*}(z)$ is within 8% of that used in RM1 over the RM1 redshift domain ($0.1\lt z\lt 0.5$), with the largest deviations at $z\sim 0.5$. The normalization condition for ${m}_{z}$ for DES is then derived from the BC03 model using the DECam passbands (Flaugher et al. 2015).

As described in RM1, the initial calibration of the red sequence relies on spectroscopic "seed" galaxies. This may be comprised of a set of training clusters with spectroscopic redshifts (as in DES SVA1) or a large spectroscopic catalog with a sufficient number of red galaxies in clusters (as in SDSS DR8). In RM1 (see Section 6.2), we selected red galaxies by splitting the spectroscopic catalog into narrow redshift bins and using a Gaussian mixture model (Hao et al. 2009) to separate galaxies in each redshift bin into blue and red components. However, we have found that this method is only robust when we have a plethora of spectra, as is the case with SDSS.

In this paper, the initial red galaxy selection is performed by computing the color residuals of galaxies in a broad range of redshifts relative to the BC03-derived color models from Section 3.2. As we are only concerned with making an initial selection of red and blue galaxies, any color calibration offsets between the data and the BC03 model are irrelevant; we just need to get an initial sample of red galaxies. We again employ a Gaussian mixture model to obtain a first estimate for the mean color and intrinsic scatter of the red spectroscopic galaxies. To ensure a clean selection, we use the g − r color for ${z}_{\mathrm{spec}}\lt 0.35;\;$ r − i for $0.35\lt {z}_{\mathrm{spec}}\lt 0.7;$ and i − z for ${z}_{\mathrm{spec}}\gt 0.7$. At this point, we proceed as described in Step 3 of RM1, Section 6.2.

Ideally, a photometric survey would be deep enough to detect the faintest $0.2{L}_{*}$ galaxies that contribute to our richness estimator λ over the full redshift range and footprint of the catalog. In a roughly uniform survey such as the SDSS, this limitation translates into a maximum redshift, ${z}_{{\rm{max}}}$, below which the cluster catalog is volume limited; for SDSS, ${z}_{{\rm{max}}}\lt 0.33$. By contrast, the observing strategy of a multi-epoch survey such as the DES may yield much greater depth variations, as shown in Figure 1. Furthermore, the depth variations can be different in different bands. Consequently, the redshift range that can be successfully probed with redMaPPer will depend on the local survey depth, with deeper regions allowing us to detect galaxy clusters to higher redshifts.

We define a maximum redshift ${z}_{{\rm{max}}}$ at each position in the sky as follows. Given a point in the survey, our initial depth map for the main detection band (${m}_{z}$ in the case of SVA1), and a luminosity threshold (${L}_{\mathrm{thresh}}$), we calculate the maximum redshift to which a typical red galaxy (defined by our red-sequence model) of ${L}_{\mathrm{thresh}}$ is detectable at $\gt 10\sigma$ in the main detection band (z band for DES), and at $\gt 5\sigma$ in the remaining bands. Only clusters with $z\leqslant {z}_{{\rm{max}}}$ are accepted into our cluster catalog, with ${z}_{{\rm{max}}}$ defining the redshift component of our survey mask. In this way, we can simply (and conservatively) account for the regions that are extremely shallow in one or more bands. This happens in SVA1 primarily at the boundaries, and other regions that were observed in non-photometric conditions. The result is a map of ${z}_{{\rm{max}}}$ in HEALPIX format with NSIDE=4096, where each pixel is approximately 07 on a side, which is matched to the resolution of the input depth maps.

Given this procedure, we still have an arbitrary decision as to where to set our luminosity threshold ${L}_{\mathrm{thresh}}$. The most conservative option would be to demand that every cluster in the final catalog be at a redshift such that we can detect red galaxies to ${L}_{\mathrm{thresh}}=0.2{L}_{*}$. However, we have chosen to be somewhat more aggressive in the interest of increasing the number of galaxy clusters and redshift reach of the DES SV catalog, as the impact on the uncertainty estimate of λ (see Equation (2)) is modest for clusters that only require a small extrapolation. For SVA1, we have chosen the luminosity threshold to be ${L}_{\mathrm{thresh}}=0.4{L}_{*}$ for the construction of the ${z}_{{\rm{max}}}$ map. For DR8, we have chosen ${L}_{\mathrm{thresh}}=1.0{L}_{*}$, such that ${z}_{{\rm{max}}}\sim 0.6$ over $\gt 99\%$ of the DR8 footprint. Although this requires a large richness extrapolation at high redshift (and hence large richness errors), this cut maintains consistency with previous redMaPPer catalogs (versions 5.2 and 5.10) where we did not use a ${z}_{{\rm{max}}}$ map. However, if users wish to utilize a volume-limited subset of the DR8 redMaPPer catalog, a restriction of ${z}_{\lambda }\lt 0.33$ will ensure that the local depth at every cluster is deep enough to detect $0.2{L}_{*}$ galaxies.

Although the code used to run on SVA1 and DR8 is the same, there are a few key differences that we highlight here.

• 1.  
  For DR8, we use the i band for the detection magnitude; for SVA1, we use the z band, which is better suited to the broad redshift range and the excellent z-band performance of DECam.
• 2.  
  For DR8, we use ugriz for galaxy colors, while for SVA1 we only use griz. The lack of a u band has a negligible effect on the cluster detection and cluster photo-zs at $z\gt 0.2$ (see Section 8.1 of RM15).
• 3.  
  For DR8, reddening corrections are applied to catalog magnitudes. For SVA1, these are incorporated into the SLR zero-point calibration.
• 4.  
  For DR8, we train the red-sequence model over $2000\;{\mathrm{deg}}^{2}$ (∼20% of the full footprint), as in RM1, to ensure sufficient statistics of spectroscopic training while avoiding any possibility of over-training. For the much smaller SVA1 catalog, we use the full footprint and all available spectra. The impact of this is detailed in Section 4.1.2.

In RM1, we describe a method of estimating the purity and completeness of the cluster catalog using the data itself, by placing fake clusters into the data and recovering the richness. While this method (described in Section 11 of RM1) is useful for estimating the selection function and projection effects, it is not appropriate for generating a cluster random catalog for cross-correlation measurements, such as the cluster–shear cross-correlation used for stacked weak-lensing mass estimates (e.g., Johnston et al. 2007; Reyes et al. 2008), as existing large-scale structure is imprinted on the random catalog.

In this section, we describe a new way of generating cluster random points by making use of the ${z}_{{\rm{max}}}$ map from Section 3.4. A particular challenge is the fact that galaxy clusters are extended objects, and thus the detectability depends not just on the redshift, but also on the cluster size and the survey boundaries. We generate a random cluster catalog that has the same richness and redshift distribution of the data catalog by randomly sampling $\{\lambda ,{z}_{\lambda }\}$ pairs from the data catalog. To ensure that the random catalog correctly samples the survey volume, we utilize the redshift mask. Specifically, after sampling a cluster from the cluster catalog, we randomly sample a position ($\{\alpha ,\delta \}$) for the random point. If the cluster redshift ${z}_{\lambda }$ is larger than the maximum redshift at which the cluster can be detected, then we draw a new $\{\alpha ,\delta \}$, repeating the procedure until the cluster is assigned a position consistent with the cluster properties. In all, we sample each cluster ${n}_{\mathrm{samp}}\sim 1000$ times to ensure that any correlation measurements we make are not affected by noise in the random catalog.

Having assigned a position, we use the depth map and the footprint mask to estimate the local mask fraction ${f}_{\mathrm{mask}}$ and scale factor S, as defined in Section 3. This is the point at which the finite extent of the clusters is taken into account. Only random points that have ${f}_{\mathrm{mask}}\lt 0.2$ and $\lambda /S\gt 20$ are properly within the cluster detection footprint. These cuts will locally modify the richness and redshift distribution of the random points relative to the data. In particular, the random points will tend to undersample the regions from which we discard clusters, particularly for low-richness and high-redshift clusters.

We address this difficulty by using weighted randoms. Specifically, given all of the ${n}_{\mathrm{samp}}$ random points generated from a given $\{\lambda ,{z}_{\lambda }\}$ pair, we calculate the number of random points that pass our mask and threshold cuts, denoted as ${n}_{\mathrm{keep}}$. Each random point is then upweighted by a factor $w={n}_{\mathrm{samp}}/{n}_{\mathrm{keep}}$. This ensures that the weighted distribution of random points matches the cluster catalog as a function of both λ and ${z}_{\lambda }$, while taking into account all of the boundaries and depth variations. As we typically sample each cluster ∼1000 times, the weight w is sufficiently well measured that we neglect noise in w when making use of the weighted random points. We note that in this procedure, we neglect sample variance from large-scale structure that may be imprinted in the cluster catalog; while this may be a small issue for SVA1, this will be averaged out over large surveys such as DES and SDSS.

Finally, we compute the effective area of the survey for cluster detection. For any given redshift z, we compute the total area (${A}_{\mathrm{tot}}$) covered where we might have a chance of detecting a cluster, such that $z\lt {z}_{{\rm{max}}}$. Taking into account boundaries and the finite size of clusters, the effective area is simply ${A}_{\mathrm{tot}}\times {n}_{\mathrm{samp}}/{n}_{\mathrm{keep}}$, where ${n}_{\mathrm{samp}}/{n}_{\mathrm{keep}}$ is computed for all random points with $z\lt {z}_{{\rm{max}}}$. We then use a cubic spline to perform a smooth interpolation as a function of redshift. Due to the finite size of the clusters and the small footprint of SVA1 with a lot of boundaries, the effective area for $\lambda \gt 20$ cluster detection is reduced from $148\;{\mathrm{deg}}^{2}$ to $\sim 100\;{\mathrm{deg}}^{2}$ at $z\lt 0.6$.

4.1.1. SDSS DR8

In Figure 4, we compare the photometric redshift ${z}_{\lambda }$ to the spectroscopic redshift of the CG (where available) for all of the clusters in DR8 with $\lambda \gt 20$. The top panel shows a density map of the ${z}_{\mathrm{spec}}$–${z}_{\lambda }$ relation with $4\sigma$ outliers (such that $| ({z}_{\mathrm{spec}}\mbox{--}{z}_{\lambda })/{\sigma }_{{z}_{\lambda }}| \gt 4$), which make up $1.1\%$ of the population, marked as red points. The outlier clump at ${z}_{\lambda }\sim 0.4$ is due to cluster miscentering rather than photometric redshift failures. In RM1, we demonstrated that this clump of outliers is due to errors in cluster centering rather than photometric redshift estimation. Specifically, these outliers represent those clusters in which the photometrically assigned CG has a spectroscopic redshift that is inconsistent not only with the photometric redshift of the cluster, but also with the spectroscopic redshift of the remaining cluster members (see Figure 10 in RM1). This failure mode is particularly pronounced near filter transitions. The bottom panel shows the bias (magenta dotted–dashed line) and scatter (cyan dot–dot–dashed line) about the 1–1 line (blue dashes). The performance is equivalent to that from RM1, with ${\sigma }_{z}/(1+z)\lt 0.01$ over most of the redshift range.

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4.1.2. DES SVA1

Figure 5 is the analog to Figure 4 for DES SVA1. Because of the significantly smaller number of spectra, we show all clusters with $\lambda \gt 5$, despite the fact that this will increase the rate of $4\sigma$ outliers due to miscentering. Nevertheless, the performance is still very good with only 5% outliers. All of these outliers have $\lambda \lt 20;$ thus, there are no $4\sigma$ outliers in the set of 52 clusters with spectra in the fiducial $\lambda /S\gt 20$ catalog. The bias and scatter are all very good at $z\lesssim 0.7$, with an increase of ${\sigma }_{z}/(1+z)$ from ∼0.01 to ∼0.02 at high redshift. This increase is caused by both the variations in survey depth, as well as noise in the high-z red-sequence model that will be reduced as we obtain more cluster spectra and increase our footprint in full DES operations. At low redshift, we note that the scatter in ${z}_{\lambda }$ is larger in DES SVA1 than in SDSS DR8. This is primarily caused by the relatively noisy MAG_AUTO galaxy colors employed for our SVA1 catalog which increase the red-sequence width, and hence the noise, in ${z}_{\lambda }$.

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Because our analysis utilized all of the available spectroscopy for training redMaPPer, it is possible that our photo-z performance is artificially good due to over-training. To test for this, we have performed a second full training of the red-sequence model using only 50% of the cluster spectra, and reserving the second half for a validation test. This is not ideal, as we then fall below the required number of spectra for a good fit to the red-sequence model (see Appendix B of RM1). Nevertheless, the ${z}_{\lambda }$ statistics of the validation catalog are equivalent to those of the full fiducial run.⁶⁰

In Figure 6, we show the comoving density of redMaPPer clusters for DR8 (red) and SVA1 (blue). Densities are computed using our fiducial cosmology for clusters with $\lambda /S\gt 20$ by summing individual cluster P(z) functions. The widths of the lines are smoothed over a redshift range $\delta z=0.02$ and assume Poisson errors (which are consistent with jackknife errors). The black dashed line shows the predicted abundance for halos with ${M}_{500c}\gt 1\times {10}^{14}\;{h}_{70}^{-1}\;{M}_{\odot }$, with the dash–dotted lines showing the same with mass thresholds of $0.7\times {10}^{14}\;{h}_{70}^{-1}\;{M}_{\odot }$ and $1.3\times {10}^{14}\;{h}_{70}^{-1}\;{M}_{\odot }$ (Tinker et al. 2008).

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We note that the redMaPPer cluster is volume limited only out to $z\leqslant 0.33$. Above this redshift, the cluster density as a function of redshift reflects two competing trends: an increasing Eddington (1913) bias in the estimated cluster richness, which tends to increase the cluster density as a function of richness, and an increasing detection threshold due to the shallow survey depth of the SDSS. For $z\approx 0.4$, the number of galaxies lost due to the shallow survey depth is relatively small, and Eddington bias dominates, leading to an apparent increase in the cluster density. As one moves toward even higher redshifts, the increasing detection threshold quickly dominates and the density of clusters falls as an increasing function of redshift.

The SVA1 density is roughly consistent with DR8 at low redshift, although the volume probed is much smaller; the peak at $z\sim 0.6$ is caused by the same Eddington bias effects as in DR8 at lower redshift. The number density slowly declines with redshift in SVA1, which is consistent with a constant mass threshold at fixed richness. However, we caution that the possibility of a varying mass threshold (due to the build-up of the red sequence, for example) as well as Eddington bias and projection effects must be taken into account to compute a proper cluster abundance function $n(z,M)$ for cosmological studies.

The initial star/galaxy classifier in SVA1 data is the modest classifier based on the SExtractor SPREAD_MODEL quantity (Chang et al. 2015a; Jarvis et al. 2015, Section 2.2) which compares the fit of a point-spread function (PSF) model to that of a PSF convolved with a small circular exponential model for morphological classification. While the modest classifier works reasonably well at bright magnitudes, at $z\sim 0.7$ the stellar locus (in the DES optical bands griz) comes close to the galaxy red sequence. For accurate selection of individual red galaxies as in the redMaGiC catalog, this required our improved star/galaxy classification based on ngmix (Rozo et al. 2015c), which reduced stellar contamination from $\gtrsim 15\%$ at $z\sim 0.7$ to less than 5%.

In order to estimate the impact of star/galaxy separation, we have rerun the redMaPPer cluster finder on a slightly expanded footprint using the modest star/galaxy classifier, while leaving everything else (including the red-sequence calibration) the same. We then match clusters from this catalog to our fiducial catalog. The first thing we find is that a small number of clusters (∼1.4%) are now badly miscentered on bright, red, misclassified stars (as determined from our improved star/galaxy separation from ngmix). We also note that the global background is slightly increased at high redshift, thus slightly depressing the richness estimates. The richness bias is ∼3% at z = 0.8, with the bias decreasing linearly with redshift such that the cluster richnesses at z = 0.2 are unbiased. We calibrate this bias with a simple linear model, and correct for it in our final expanded catalog. The associated systematic uncertainty in richness due to the inefficient star/galaxy separation is ∼2%, which is smaller than the statistical uncertainty on λ. Thus, aside from mild miscentering problems, redMaPPer richness estimates are quite insensitive to stellar contamination in the galaxy catalog, as expected.

In addition to the overall geometric mask, our fiducial footprint includes masking for bright ($J\lt 13$) 2MASS stars and 4% of the area with a larger-than-typical concentration of object centroid shifts. However, we have found that several good cluster centers are masked in these regions, causing significant offsets from the X-ray and SZ centers (e.g., Section 2.3 of S15).

In order to estimate the impact of masking (in addition to star/galaxy separation), we have rerun redMaPPer on the expanded footprint using the modest classifier (as above) and including galaxies that had been rejected by both the 2MASS mask and the "4%" mask. We then match clusters from this expanded catalog to the fiducial catalog. Aside from those clusters which are now badly miscentered due to stellar contamination, two SPT clusters (SPT-CLJ0417–4748 and SPT-CL0456–5116; see S15) are now properly centered, as the central galaxies are no longer masked.

Figure 7 shows the comparison in cluster redshift ${z}_{\lambda }$ between the expanded (${z}_{{\lambda }^{\prime }}$) and fiducial (${z}_{\lambda }$) catalogs. The cluster redshifts are very consistent, with a few outliers at ${\rm{\Delta }}{z}_{\lambda }\gt 0.01$. The red curve in the right panel shows a Gaussian fit to the ${\rm{\Delta }}{z}_{\lambda }$ histogram, with mean 5 × 10⁻⁵ and rms 7 × 10⁻⁴. Thus, the worse star/galaxy separation and less conservative mask have no significant impact on the cluster redshift estimation.

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Figure 8 shows the richness bias as the ratio of $\lambda ^{\prime}$ (expanded catalog) to λ (fiducial catalog) in the SPT-E region. All values of ${\lambda }^{\prime }$ have been corrected for the star/galaxy separation bias model in Section 5.1. Again, the richness estimates are consistent, with a Gaussian fit showing $\lambda ^{\prime} /\lambda \;=\;0.99\pm 0.04$. We note that this ∼4% richness scatter is fully consistent with expectations based on the richness extrapolations in the fiducial catalog which made use of a more aggressive mask. However, we also find that for ∼7% of clusters, $\lambda ^{\prime} /\lambda$ differs from unity by more than 3σ. These apparent outliers are caused by clusters seen in projection. Changes in masking can change the way these projected clusters are deblended or merged by the redMaPPer algorithm, leading to these outliers. This result suggests a lower limit of ≈7% for the redMaPPer projection rate, and demonstrates the need for a full model of projection effects incorporated into a cluster abundance function.

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In Figure 9, we show the comoving density of clusters in the SPT-E region for our fiducial (blue) and expanded (magenta) catalogs. The number densities are clearly consistent at all redshifts. Therefore, in future versions of redMaPPer on DES data, our fiducial runs will be performed with a less aggressive mask (with more area) as it has no impact on the richness estimation, yet it does improve cluster centering in a small number of cases. While improved star/galaxy separation is helpful for many purposes, it is heartening to know that our richness estimates are not strongly biased by a less-than-ideal separator. For this version of the catalog, however, we recommend that the fiducial catalog should be used for all purposes except where the greater area can be made use of in cross-checks with X-ray catalogs, as in Section 6.

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A detailed comparison of the DES SVA1 redMaPPer and SPT SZ cluster catalogs has been published in S15. We briefly summarize their most important results. Using $129\;{\mathrm{deg}}^{2}$ of overlapping data, they find 25 clusters between $0.1\lt z\lt 0.8$, including 3 new clusters that did not have identified optical counterparts in Bleem et al. (2015). Every SZ cluster within the redMaPPer footprint and at $z\lt {z}_{\mathrm{max}}$ was detected in the redMaPPer catalog. Due to the high-mass threshold of the Bleem et al. (2015) sample, these are all high-mass and high-richness clusters, with a typical richness of $\lambda \sim 70$. Using the method of Bocquet et al. (2015), they implement a full likelihood formalism to constrain the λ–mass relation of SPT-selected clusters. By inverting the scaling relation from S15 using the methods of Evrard et al. (2014), they determine that the mass of a $\lambda \sim 20$ cluster is ${M}_{500c}\sim {10}^{14}\;{h}_{70}^{-1}\;{M}_{\odot }$, which is consistent with the density of clusters from Section 4.2. In addition, they find a mass scatter at fixed richness, ${\sigma }_{\mathrm{ln}M| \lambda }\;=\;{0.18}_{-0.05}^{+0.08}$, at a richness of $\lambda \;=\;70$. Thus, they confirm that the redMaPPer richness λ is a low-scatter mass proxy for DES data across a much broader range in redshift than was probed in Rozo & Rykoff (2014). Furthermore, the parameters of the λ–mass relation are consistent with what was derived from SDSS DR8 data using a rough abundance matching argument (Rykoff et al. 2012), thus providing further confirmation of the fact that redMaPPer is probing a similar cluster population in SDSS and DES data.

In S15, they further constrain the optical-SZE positional offsets. The offset distribution is characterized by a two-component Gaussian model. The central component describes "well-centered" clusters where the optical and SZ positions are coincident (given the SZ positional uncertainty from the finite beam size of SPT). There is also a less populated tail of central galaxies with large offsets. For this work, we have modified the model such that the central Gaussian component is a one-dimensional rather than a two-dimensional Gaussian, as we have that found this produces superior ${\chi }^{2}$ fits to the X-ray offsets in Section 6.2. The positional offsets, x, are now modeled as

$$\begin{eqnarray}&&p(x)=\displaystyle \frac{{\rho }_{0}}{{\sigma }_{0}\sqrt{2\pi }}{e}^{-\tfrac{{x}^{2}}{2{\sigma }_{0}^{2}}}+\displaystyle \frac{(1-{\rho }_{0})x}{{\sigma }_{1}^{2}}{e}^{-\tfrac{{x}^{2}}{2{\sigma }_{1}^{2}}},\end{eqnarray} \tag{ 4 }$$

where $x=r/{R}_{\lambda }$, ${\rho }_{0}$ is the fraction of the population with small offsets with variance ${\sigma }_{0}^{2}$, and the population with large offsets is characterized with variance ${\sigma }_{1}^{2}$. We have refit the offset model of SPT clusters from S15 $r/{R}_{\lambda }$ rather than $r/{R}_{500}$, in addition to using the redMaPPer positions from the expanded SVA1 catalog. This change allows us to better compare to the X-ray cluster samples described in Section 6.2. In all of the cases, we marginalize over the parameter ${\sigma }_{0}$ since it is not relevant to the overall fraction and distribution of incorrect central galaxies.

The optical-SZ positional offset distribution has a central component with ${\rho }_{0}={0.80}_{-0.37}^{+0.15}$ and a large-offset population with ${\sigma }_{1}={0.27}_{-0.08}^{+0.21}{R}_{\lambda }$. Given the matched clusters, the mean of the centering probability of the central galaxies of the clusters in the matched sample is $\langle {P}_{\mathrm{cen}}\rangle =0.82$. This is consistent with the constraints from the optical-SZ matching, although the 21 clusters in the sample do not have a lot of constraining power.

In this section, we make use of the overlap of the redMaPPer SVA1 expanded catalog with X-ray observations from Chandra and XMM to measure the T_X–λ relation as well as to further constrain the centering properties of the catalog. More extensive comparisons to X-ray observations, including a full analysis of the redMaPPer DR8 catalog, will be presented in D. Hollowood et al. (2016, in preparation) and A. Bermeo-Hernandez et al. (2016, in preparation).

6.2.1. Chandra Analysis

6.2.2. XCS Analysis

6.2.3. The T_X–λ Relation

For this study, we wished to determine the redMaPPer ${T}_{X,2500}$–λ relation using clusters with either XMM or Chandra observations. However, it is well known that X-ray cluster temperatures derived from XMM are systematically offset from Chandra observations (e.g., Schellenberger et al. 2015). Therefore, we have determined a correction factor to make the Chandra and XMM temperatures consistent. For this, we required access to more redMaPPer clusters with X-ray observations than are available in DES SVA1. We rely on recent compilations of T_X measurements of SDSS redMaPPer clusters using Chandra (D. Hollowood et al. 2016, in preparation) and XMM (A. Bermeo-Hernandez et al. 2016, in preparation). There are 41 DR8 redMaPPer clusters in common between these samples, allowing us to fit a correction factor of the form

$$\begin{eqnarray}&&{\mathrm{log}}_{10}\left(\displaystyle \frac{{T}_{X}^{{\text{}}{Chandra}}}{1\;{\rm{keV}}}\right)=1.0133{\mathrm{log}}_{10}\left(\displaystyle \frac{{T}_{X}^{{XMM}}}{1\;{\rm{keV}}}\right)+0.1008\end{eqnarray} \tag{ 5 }$$

using BCES orthogonal fitting (Akritas & Bershady 1996). We note that this relation is consistent with that found by Schellenberger et al. (2015). Of the 14 redMaPPer SVA1 clusters with ${T}_{X,2500}^{\mathrm{XMM}}$ values, 4 are in common with the Chandra sample. From these 4, we have used the ${T}_{X,2500}$ value with the lowest uncertainty (3 from XMM and 1 from Chandra).

Figure 10 shows the T_X–λ scaling relation derived from XCS and Chandra clusters. All Chandra temperatures have been corrected according to Equation (5). We use an MCMC to fit the full cluster sample to a power-law model:

$$\begin{eqnarray}&&\mathrm{ln}({T}_{X})=\alpha +\beta \mathrm{ln}(\lambda /50)+\gamma \mathrm{ln}[E(z)/E(0.4)],\end{eqnarray} \tag{ 6 }$$

with intrinsic scatter ${\sigma }_{\mathrm{ln}T| \lambda }$. Given the limited number of clusters in our sample, we fix the redshift evolution parameter $\gamma \;=\;-2/3$, assuming self-similar evolution. We find that $\alpha \;=\;1.31\pm .07$, $\beta \;=\;0.60\pm 0.09$, and $\sigma \;=\;{0.28}_{-0.05}^{+0.07}$. The best-fit scaling relation (including $1\sigma$ error) is shown with the gray bar in Figure 10, and the dashed lines show the $\pm 2{\sigma }_{\mathrm{ln}T| \lambda }$ constraints. We note that the slope is consistent with $\beta \;=\;2/3$, which is what we expect if clusters are self-similar and $\lambda \propto M$, as in S15.

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S15 used SZ-selected clusters to place a constraint on the scatter in mass at fixed richness ${\sigma }_{\mathrm{ln}M| \lambda }\;=\;{0.18}_{-0.05}^{+0.08}$. To compare against S15, we transform our constraints on the scatter in T_X at fixed mass to constraints on scatter in mass at fixed richness by assuming a self-similar slope ${T}_{X}\propto {M}^{2/3}$. We also require an estimate for the scatter in mass at fixed X-ray temperature when T_X is not core-excised. We rely on the results by Lieu et al. (2015), who find an intrinsic scatter in weak-lensing mass at a fixed temperature of ${\sigma }_{\mathrm{ln}{M}_{{WL}}| T}\;=\;0.41$. Our choice is motivated by the fact this study, like ours, measures cluster temperatures with no core-excision. Adopting a 25% intrinsic scatter in weak-lensing mass at fixed mass, we arrive at an intrinsic scatter in mass at fixed temperature ${\sigma }_{\mathrm{ln}M| T}\;=\;0.32$. Finally, given that the X-ray cluster sample extends to low-richness systems, which are expected to have a larger scatter, we adopt a richness-dependent scatter as a function of mass:

$$\begin{eqnarray}&&{\rm{Var}}(\mathrm{ln}\lambda | M)=\langle \lambda | M{\rangle }^{-1}+{\sigma }_{\mathrm{ln}\lambda | M}^{2}.\end{eqnarray} \tag{ 7 }$$

Following Evrard et al. (2014), we arrive at ${\sigma }_{\mathrm{ln}M| \lambda }\;=\;0.3\pm 0.15$. This result is higher than but consistent with that of S15. The large error bars reflect in part the large intrinsic scatter in the mass–T_X relation for non-core-excised temperatures.

6.2.4. Positional Offset Distribution

Using each of the SPT, Chandra, and XCS redMaPPer-matched samples, we have fit the positional offset model of Equation (4). For the SZ sample, the error on the position was given by Equation (11) from S15; for the X-ray samples, we used a fixed error of $10^{\prime\prime}$. However, we note that this does not fully account for systematic errors in X-ray centroids, especially for clusters with complex morphologies. For the X-ray samples, we do not require the cluster to be bright enough to obtain a temperature constraint in order for it to have a well-detected center.

The offset model results are summarized in Table 2 with errors quoted as 68% confidence intervals as derived from an Markov Chain Monte Carlo fit to the data, similar to Section 4.3 of S15. The constraints on ${\rho }_{0}$ and ${\sigma }_{1}$ (the large-offset "miscentered" component) are all consistent within errors for all three samples.

Table 2.  redMaPPerCentral Offset Fits

Sample	No.	$\langle {P}_{\mathrm{cen}}\rangle$	${\rho }_{0}$	${\sigma }_{1}(R/{R}_{\lambda })$	${\chi }^{2}/\mathrm{dof}$
SPT	21	0.83	${0.80}_{-0.37}^{+0.15}$	${0.27}_{-0.08}^{+0.21}$	6.0/10
Chandra	35	0.80	${0.68}_{-0.18}^{+0.22}$	${0.27}_{-0.05}^{+0.12}$	4.7/10
XCS	29	0.82	${0.85}_{-.11}^{+0.07}$	${0.22}_{-0.04}^{+0.08}$	9.1/10
Combined	74	0.81	${0.78}_{-0.11}^{+0.11}$	${0.31}_{-0.05}^{+0.09}$	9.9/10

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To better constrain the overall centering of the redMaPPer SVA1 expanded cluster sample, we have also performed a joint likelihood fit to all three cluster samples. In the cases where we have multiple observations of the same cluster, we first take the XCS position, followed by the Chandra position, followed by the SPT position. As our goal is to better constrain the well-centered fraction ${\rho }_{0}$ as well as the miscentering kernel ${\sigma }_{1}$, our joint likelihood constrains these two parameters for the full sample. However, to allow for differences in centering precision, we use a separate value of ${\sigma }_{0}$ for each individual sample. In all, we have five parameters, but we treat the set of $\{{\sigma }_{0}\}$ as nuisance parameters in our figures below.

The histogram of offsets for the combined sample is shown in Figure 11. The results of our joint fit are shown in Figure 12 and described in Table 2. The best-fit model has been binned to match the data and is overplotted with black points in Figure 11. Our final constraint on the fraction of clusters that are correctly centered is ${\rho }_{0}={0.78}_{-0.11}^{+0.11}$, compared to the redMaPPer predicted fraction of 0.82, which is in very good agreement. By comparison, redMaPPer clusters in SDSS are correctly centered ≈86% of the time (see Rozo & Rykoff 2014).

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