PaperlayerView exact revision on arXiv

Source-linked AI summary

redMaPPer I: Algorithm and SDSS DR8 Catalog

E. S. Rykoff, E. Rozo, M. T. Busha, C. E. Cunha, A. Finoguenov, A. Evrard, J. Hao, B. P. Koester, A. Leauthaud, B. Nord, M. Pierre, R. Reddick, T. Sadibekova, E. S. Sheldon, R. H. Wechsler

arXiv:1303.3562v2astro-ph.CO

TL;DR

Photometric cluster surveys need algorithms that handle red-sequence evolution, uncertain centers and redshifts, complex masks, and large datasets. redMaPPer addresses these needs with iterative red-sequence calibration, probabilistic cluster finding, and optimized multicolor richness estimation. The resulting method is reported to improve substantially on previous cluster finders, with near-complete catalogs and approximately 95% purity at the rich end.

  • Problem

    Upcoming photometric surveys require cluster finders that can exploit photometric data despite red-sequence evolution, incomplete spectroscopic training, complex masks, and uncertain redshifts and centers.

  • Method

    redMaPPer uses iterative red-sequence calibration, candidate-center ranking, probabilistic membership and centering, and a multicolor richness estimator.

  • Results

    Completeness is near 100%, while purity is approximately 95% at the rich end and increases at lower richness.

  • Takeaways & Limitations

    redMaPPer provides a survey-oriented optical cluster finder with features intended for DES-, LSST-, and related large photometric-survey analyses.

  • Takeaways & Limitations

    The χ2 filter compresses the full color vector because full multivariate background estimation was too noisy in DR8, sacrificing some color information.

Abstract

from arXiv · show

We describe redMaPPer, a new red-sequence cluster finder specifically designed to make optimal use of ongoing and near-future large photometric surveys. The algorithm has multiple attractive features: (1) It can iteratively self-train the red-sequence model based on minimal spectroscopic training sample, an important feature for high redshift surveys; (2) It can handle complex masks with varying depth; (3) It produces cluster-appropriate random points to enable large-scale structure studies; (4) All clusters are assigned a full redshift probability distribution P(z); (5) Similarly, clusters can have multiple candidate central galaxies, each with corresponding centering probabilities; (6) The algorithm is parallel and numerically efficient: it can run a Dark Energy Survey-like catalog in ~500 CPU hours; (7) The algorithm exhibits excellent photometric redshift performance, the richness estimates are tightly correlated with external mass proxies, and the completeness and purity of the corresponding catalogs is superb. We apply the redMaPPer algorithm to ~10,000 deg^2 of SDSS DR8 data, and present the resulting catalog of ~25,000 clusters over the redshift range 0.08<z<0.55. The redMaPPer photometric redshifts are nearly Gaussian, with a scatter σ_z ~ 0.006 at z~0.1, increasing to σ_z~0.02 at z~0.5 due to increased photometric noise near the survey limit. The median value for |Δz|/(1+z) for the full sample is 0.006. The incidence of projection effects is low (<=5%). Detailed performance comparisons of the redMaPPer DR8 cluster catalog to X-ray and SZ catalogs are presented in a companion paper (Rozo & Rykoff 2014).

1. INTRODUCTION

redMaPPer is introduced as a red-sequence cluster finder designed to meet the requirements of large photometric surveys, including self-training, mask handling, probabilistic redshift and centering information, efficient computation, and optimized richness estimation.

  • Motivation: redMaPPer is proposed because upcoming photometric surveys require cluster finders with capabilities beyond existing approaches.The paper frames the need around surveys such as DES and LSST.
  • Requirements: The algorithm should self-train on available data to reduce biases from incorrect a priori red-sequence or photometric-redshift parameterizations.Biases can worsen for faint galaxies where spectroscopic training and validation samples are incomplete.
  • Requirements: The required survey infrastructure includes efficient computation, complex-mask treatment, and cluster-specific random points for large-scale structure studies.Cluster angular and redshift selection cannot be represented by the galaxy mask alone because clusters are extended objects.
  • Requirements: redMaPPer assigns full P(z) distributions and centering probabilities to represent uncertainty in cluster redshifts and central-galaxy identification.These distributions are intended to support statistical recovery of cluster redshift and angular distributions.
  • Algorithm design: The algorithm uses membership probabilities and optimized detection radii and luminosity cuts to improve richness estimation and its relation to cluster mass.The design also evaluates galaxy counts versus total luminosity and the inclusion of blue galaxies.
  • Application: redMaPPer is designed for catalogs with at least three bands spanning the 4000 Å break and is demonstrated on 10,400 deg2 of SDSS DR8 data.The stated application covers SDSS low and moderate redshift clusters and anticipates DES use at z < 1.

2. DATA

The study constructs a clean SDSS DR8 photometric input catalog over a contiguous, high-quality footprint, while retaining objects that aggressive flag cuts could incorrectly remove from cluster cores.

  • Survey footprint: SDSS DR8 provides more than 14,000 deg2 of imaging, reduced to approximately 10,500 deg2 using a high-quality contiguous survey footprint.The footprint follows the BOSS target-selection survey edge and includes additional masks.
  • Catalog selection: The input catalog applies galaxy classification, an i-band magnitude limit of i < 21.0, and additional quality filters.The limit is approximately the survey’s 10σ detection threshold, though it is not precisely constant across the survey.
  • Catalog selection: Including objects affected by selected SDSS flags increases the input catalog by less than 2% and avoids masking some cluster centers.The final input catalog contains 56.5 million galaxies.
  • Photometry: The photometry uses CMODEL MAG for total i-band magnitudes and MODEL MAG for colors, with DR8 magnitude uniformity of approximately 1% in griz and 2% in u.The faintest objects have characteristic magnitude errors of approximately 0.1.
  • Spectroscopic calibration: Spectroscopic data from SDSS DR9 calibrate the red sequence and validate photometric redshifts, using approximately 20% of the available spectra for training.The catalog contains over 1.3 million galaxy spectra, including over 500,000 CMASS LRGs near z ∼ 0.5.

3. OUTLINE OF THE CLUSTER FINDER

redMaPPer separates iterative red-sequence calibration from cluster finding, then estimates candidate-cluster redshifts and richness using the calibrated model.

  • Algorithm stages: The algorithm has two stages: empirical red-sequence calibration as a function of redshift, followed by cluster identification and richness measurement.The cluster-finding stage uses the calibrated red-sequence model.
  • Calibration: Approximately 40 training clusters per redshift bin of width ±0.025 are sufficient for unbiased richness and photometric-redshift estimation.The calibration begins with training clusters whose red central galaxies have spectroscopic redshifts.
  • Calibration: Training clusters are seeded by red spectroscopic galaxies and used to fit a linear red-sequence model including zero-point, tilt, and scatter.The fit uses high-probability members with luminosity L ≥ 0.2L∗.
  • Cluster finding: Cluster finding tests photometric galaxies as candidate centers, estimates zred, ranks candidates by red-sequence fit and richness likelihood, and refines redshift using member galaxies.The refinement is triggered when at least three red-sequence galaxies are detected within a 500 h−1 kpc aperture.

4. RICHNESS ESTIMATOR λ

The richness estimator models clusters as a mixture of member and background populations, combining positional, luminosity, and multicolor red-sequence information to estimate λ.

  • Estimator framework: The estimator models the observed distribution as S(x) = λu(x|λ) + b(x), separating cluster members from background galaxies.Here λ is the number of cluster galaxies, u is a normalized cluster profile, and b is the background density.
  • Estimator framework: Membership probabilities are modified during percolation for nearby clusters at comparable redshifts, while λ remains constrained by the estimator equation.The modification affects clusters close together along the line of sight.
  • Filter choices: The richness calculation uses a luminosity cut of Lcut = 0.2L∗ and a richness-dependent radial cutoff with R0 = 1.0 h−1 Mpc and β = 0.2.These choices are adopted because they are expected to minimize scatter in the mass–richness relation.
  • Filter construction: The filter combines projected radius, i-band magnitude, and a color variable represented by the red-sequence template’s χ2 goodness of fit.A full color vector is compressed to one χ2 value for practical background estimation.
  • Filter construction: The χ2 filter sacrifices some color information relative to a full multivariate Gaussian filter because estimating a five-dimensional background was too noisy in DR8.The compressed background depends only on mi and χ2 at a given redshift.
  • Background estimation: The background density is modeled as a function of magnitude and χ2 at each redshift, but its variability can produce catastrophic projections.The mean-background bias is described as small, while variability is identified as more problematic and deferred to future work.

5. HANDLING MASKED REGIONS AND LIMITED DEPTH

redMaPPer corrects richness estimates for limited depth and geometrical masking rather than assuming uniform, complete coverage. It estimates the missing contribution and its uncertainty using the cluster model, selection function, and Monte Carlo integration.

  • Mask correction: The method separates observed galaxies inside the mask from an unobserved correction term C for masked regions.The observed contribution is the usual richness estimate, while C accounts for galaxies outside the detectable region.
  • Mask correction: The expected correction depends on richness through both the cutoff radius and the richness-dependent radial filter.The corrected richness is therefore obtained by solving the richness equation with the expected missing contribution included.
  • Uncertainty propagation: The uncertainty in C propagates into richness uncertainty because C is unknown and contributes additional measurement error.The variance uses Poisson galaxy-count statistics and covariance terms, with the richness derivative evaluated numerically.
  • Monte Carlo evaluation: The selection function S(x) identifies detectable and masked regions, allowing integrals over the full cluster region to be evaluated.S(x)=1 denotes detected galaxies and S(x)=0 denotes masked positions; the integration covers the cluster radius, luminosity range, and colors.
  • Monte Carlo evaluation: Monte Carlo draws from the cluster filter estimate the mean masked fraction and the variance of C.The mean correction is the fraction of random model galaxies falling in masked regions, while analogous integrals provide Var(C).
  • Implementation: 5000 template galaxies generally provide accurate richnesses and richness errors, except for clusters that are largely masked out.The same random realizations can be reused across clusters after scaling radius and magnitude appropriately.

6. CALIBRATION OF THE RED SEQUENCE

redMaPPer calibrates its red-sequence model by iteratively expanding a limited spectroscopic seed sample with probabilistically selected cluster members. The resulting model captures color evolution, slope, and scatter, and the third iteration is adopted because richness biases are then small.

  • Training-sample construction: Spectroscopic cluster redshifts are propagated to likely member galaxies, whose membership probabilities allow contamination by nonmembers to be modeled.This leverages one spectroscopic redshift into many training galaxies while accounting for uncertain membership.
  • Calibration procedure: The final seed selection uses galaxies within 2eσint(z) of the model color, using g −r below z=0.35 and r −i above z=0.35.The calibration proceeds through approximate red-galaxy selection, mean-color fitting, scatter estimation, and final seed selection.
  • Training-sample construction: 42,000 seed galaxies associated with λ > 5 clusters expand to over 600,000 final red-sequence training galaxies reaching fainter magnitudes.The enlarged sample enables calibration of red-sequence amplitude, tilt, and scatter across redshift, with modest selection-cut changes leaving the final calibration unchanged.
  • Red-sequence model: A linear color–magnitude relation with roughly constant intrinsic scatter represents the red sequence in both g −r and r −i near z=0.25.The full model uses probabilistically selected galaxies and parameterizes smoothly evolving functions with cubic splines.
  • Red-sequence model: The inferred intrinsic width can exceed the width suggested by the pmem > 0.9 illustration because high-probability members lie near the average color.At high redshift, larger g −r outliers also reflect photometric errors exceeding the intrinsic red-sequence width.
  • Iterative refinement: After the third iteration, richness biases are below 1% at low redshift, below 5% at high redshift, and below 0.1σ in the error-normalized offset.The third iteration is therefore used for the final cluster catalog.

7. PHOTOMETRIC REDSHIFT ESTIMATION

redMaPPer initializes cluster redshifts with a red-sequence galaxy estimator, then iteratively refines cluster richness and redshift. The estimator performs well overall, while filter transitions, projection effects, and miscentering explain the main residual failures.

  • Red-sequence calibration: After the third calibration iteration, richness biases are always < 1% at low redshift and < 5% at high redshift, with normalized deviations < 0.1σ.The final catalog therefore uses the third iteration.
  • Redshift estimation procedure: The cluster redshift procedure iteratively uses an initial guess to estimate richness and likely members, refit the red-sequence model, and converge on an improved redshift.Each iteration maps zin to zout, seeking the stable point where zout = zin.
  • Redshift initialization: The zred estimator maximizes a red-sequence likelihood using galaxy colors, magnitudes, intrinsic scatter, photometric errors, and a volume prior.The search uses a redshift grid with δz = 0.005 and parabolic interpolation, restricted to galaxies brighter than m∗(z) + 2.5.
  • Redshift initialization: zred performs well with low bias and scatter and few gross outliers, but a flare-up near zCG ∼0.35 occurs when the 4000 Å break moves from g −r to r −i.The filter transition also coincides with the limit of the DR8 photometry, aggravating the failures.
  • Redshift initialization: The afterburner improves high-redshift biases, although zred remains biased by ∼0.3σ near z ∼0.4 because scatter at the filter transition is asymmetric and non-Gaussian.The corrected estimate is sufficiently accurate to initialize the photometric cluster redshift estimator.
  • Failure modes: Approximately 1%−2% of clusters exhibit non-convergent redshift curves associated with projection between nearby structures, and the dominant structure depends on the initial photometric redshift.redMaPPer often fragments these systems along the line of sight.
  • Cluster redshift validation: Large ∆z ∼0.1 offsets between zλ and the assigned central-galaxy redshift primarily indicate miscentering rather than incorrect cluster redshifts.In a cleaned sample requiring two spectroscopic members near the central galaxy, the few remaining outliers are ≲0.2% and likely projection systems.

8. CLUSTER CENTERING

redMaPPer treats cluster centering probabilistically because center choice affects cosmological measurements, while its red-galaxy assumption fails for strongly star-forming centrals. The resulting model provides candidate-center probabilities and iterative calibration, but centering remains a major source of systematic failure.

  • Miscentered clusters bias stacked weak-lensing masses, velocity dispersions, and richness estimates, making a characterized centering model essential.
  • ≈85% centering success is achieved, but strong star formation can move a true central off the red sequence and cause a bright red satellite to be selected instead.The cluster remains in the catalog but is miscentered; removing the red-sequence requirement fixes some cases while miscentering ∼10% of clusters on foreground galaxies.
  • redMaPPer assigns each candidate central galaxy a probability rather than forcing a unique center, allowing statistical treatment of uncertain cluster positions.Some clusters have one dominant candidate, whereas others have multiple candidates with comparable probabilities.
  • The centering model uses galaxy magnitude, red-sequence photometric redshift, and local galaxy-density weight to distinguish centrals, satellites, and foreground galaxies.The local-density weight is designed as a pseudo-gravitational potential connecting a candidate to other cluster members.
  • Each cluster is tagged with Pcen and Psat, enabling repeated calibration of the centering model across iterations.The probabilities are used for individual galaxies during subsequent cluster-finder iterations.
  • 4.6% of clusters meet the study’s spectroscopic criterion for possible catastrophic centers, while 0.7% contain such galaxies absent from the member list.Visual inspection indicates most candidates in the broader 4.6% category lie in cluster outskirts rather than being true centrals.

9. THE CLUSTER FINDER

redMaPPer builds a cluster catalog through candidate-center selection, likelihood ranking, and probabilistic percolation. The procedure updates richness, redshift, membership, and centering while accounting for overlapping systems and survey-scale masking.

  • Pipeline: The cluster finder uses three stages: identify overdensities around galaxies, rank candidate clusters by likelihood, and percolate the catalog.The stages combine candidate selection, likelihood evaluation, and iterative galaxy assignment.
  • Candidate centers: Candidate centers are filtered by red-sequence fit, luminosity, richness significance, and the presence of at least three red galaxies above the magnitude limit.The initial cuts reduce 56 million DR8 galaxies to 23 million possible centers, and a later cut rejects roughly 60% of candidates.
  • Likelihood sorting: The total candidate likelihood combines richness and centering likelihoods, with richness typically dominating the initial ranking.Similar richness likelihoods are resolved using the centering likelihood, while the final center is refined during percolation.
  • Probabilistic percolation: The percolation loop recomputes richness and redshift, determines the center, masks galaxies by membership probability, and removes lower-ranked centers with p_mem > 0.5.The procedure repeats these updates for the next ranked cluster to avoid duplicate detections.
  • Probabilistic percolation: Probabilistic percolation updates galaxy availability using p_free = 1 − p_taken, allowing galaxies to contribute probabilistically to lower-ranked clusters.Reported membership is the product p_free p_mem, representing the probability that a galaxy belongs to the cluster under consideration.
  • Masking radius: The default percolation radius is 1.5R_c(λ), and changing it by ±50% affects only ∼5% of clusters, primarily satellites of λ > 100 systems.R_c(λ) optimizes richness signal-to-noise rather than representing a standard halo extent.
  • Cluster example: For Abell 2219 at z = 0.23, the radial distribution of cluster galaxies is broadly consistent with an NFW model.The example displays member locations, radial distribution, and membership probabilities, which are strongly concentrated near p_mem > 0.9.

10. THE REDMAPPER SDSS DR8 CLUSTER

The SDSS DR8 redMaPPer catalog applies conservative richness, redshift, and masking cuts. It contains 25,236 clusters and exhibits redshift-dependent selection effects near the survey depth and filter transition.

  • Catalog selection: The catalog requires λ ≥ 20S(z_λ), z_λ ∈ [0.08,0.55], and f_mask < 0.2.The richness cut corresponds roughly to at least 20 qualifying galaxy counts and an estimated effective mass cut of M_200 ≳ 10^14 M_⊙.
  • Catalog: 25,236 systems comprise the resulting SDSS DR8 cluster catalog.The catalog footprint is shown using a Mangle simple pixelization scheme of depth 7.
  • Selection effects: The comoving density is roughly constant at z_λ ≲ 0.35, where the catalog is volume limited.Near z_λ ∼ 0.35, filter-transition and magnitude-limit effects boost richness and redshift scatter, causing low-richness clusters to scatter into the sample.
  • Catalog examples: A representative unmatched rich cluster has λ = 236 ± 12 and z_λ = 0.396 ± 0.013.The cluster is the richest redMaPPer system not found in the MCXC catalog and is associated with a ROSAT Bright Source Catalog source.

11. PURITY AND COMPLETENESS

The paper evaluates purity and completeness through richness-based comparisons and injection tests that model the survey’s spatial and depth systematics. The DR8 catalog is highly complete and pure, while projection effects mainly reflect richness overestimation from multiple real structures.

  • Injection tests: The injection procedure samples random survey locations, true richness-redshift pairs, and model cluster galaxies before measuring λ_obs repeatedly.This preserves the catalog’s richness-redshift distribution and incorporates survey-mask systematics.
  • Injection tests: The detectability map accounts for large-scale structure already present in the galaxy catalog but not additional effects from correlated structure around clusters.The authors expect those additional effects to be subdominant because projection and cluster-correlation length scales differ.
  • Robustness: Model-cluster and real-redMaPPer-cluster injection results are nearly identical, supporting robustness to changes in the cluster model.The authors also report that cluster ellipticity is largely irrelevant for the richness measurements.
  • Definitions: Completeness and purity are defined using the relationship between observed richness λ_obs and true richness λ_true, with robust statistics used to characterize scatter and outliers.Completeness is evaluated in λ_true bins, while purity is evaluated in λ_obs bins after a specified outlier restriction.
  • Performance: At z < 0.3, completeness is essentially ≳99% for λ > 30.At higher redshift, the richness threshold increases as the DR8 catalog approaches its magnitude limit.
  • Performance: Purity exceeds 95% across all richness and redshift bins, with the richest systems being less pure.The stated impurities correspond to real clusters whose observed richness is overestimated by projection, rather than observational-noise false detections.
  • Performance: Purity can increase toward lower richness and higher redshift because larger richness errors make 4σ projection outliers more extreme and rarer.This trend reflects the purity definition rather than a lower incidence of projections.

12. CLUSTER MASKS

redMaPPer constructs cluster-specific random points by placing model clusters across the real survey mask. This captures detectability changes caused by extended cluster geometry, masking, local depth, and survey structure.

  • Cluster masks: Cluster-appropriate random points are essential because the galaxy mask does not characterize the angular and redshift selection of extended clusters.The method directly constructs random points applicable to the cluster mask for cross-correlation studies.
  • Mask geometry: Around Arcturus, the cluster detectability boundary lies farther from the star than the galaxy-mask edge at low redshift because f_mask < 0.2 limits masked cluster area.The comparison uses model clusters with λ_true = 40 and an appropriate redshift distribution.
  • Detectability: For λ_true = 40 clusters at 0.4 < z < 0.5, detectability is ∼60% ± 30% and varies strongly with local depth and structure.At lower redshift, detectability is essentially 100% outside the Arcturus mask except near some bright stars.

13. SUMMARY

redMaPPer is a red-sequence cluster finder designed for large photometric surveys, combining self-calibration, mask handling, probabilistic redshifts and centering, and efficient computation. Applied to SDSS DR8, it produces a large, accurate cluster catalog with high completeness and purity.

  • Algorithm: redMaPPer extends an optimized richness estimator into a multi-color red-sequence cluster finder.The richness estimator has been shown to be a good mass proxy.
  • Algorithm: Self-training with modest bright-galaxy spectroscopy makes redMaPPer suited to high-redshift surveys and smooths color information across filter transitions.The multi-color model uses all available color data without sharp features as the 4000 Å break moves between filters.
  • Algorithm: redMaPPer handles complex masks, computes mask-corrected richness, and generates cluster-appropriate random catalogs for large-scale-structure studies.
  • Uncertainty modeling: Every cluster receives a redshift probability distribution P(z) and probabilistic centering information, allowing uncertainty in redshift and position to be modeled.P(z) improves reconstruction of the cluster redshift distribution relative to point-redshift estimates.
  • SDSS DR8 catalog: σ_z ranges from 0.006 at z ∼0.1 to 0.020 at z ∼0.5 for DR8 photometric redshifts.The increase is attributed to greater photometric noise.
  • SDSS DR8 catalog: 25,236 clusters satisfying λ/S(z) > 20 are selected over z ∈[0.08,0.55].The threshold is chosen to provide a robust catalog and is volume-limited at z ≲0.35 for M ≳10^14 M⊙.
  • Performance: Completeness is near 100%, purity is ∼95% at the rich end, and projection effects occur in ∼5% of clusters.At lower richness, purity increases because observational uncertainties make projection outliers less prominent.
  • Performance: redMaPPer richness is a low-scatter mass proxy with high completeness and low impurity compared with X-ray and SZ truth tables.

B. HOW MANY TRAINING CLUSTERS?

The calibration study tests how many spectroscopic training clusters redMaPPer needs. Approximately 400 spectra reproduce essentially the full-sample richness calibration, while photometric-redshift calibration requires about 200.

  • Training requirements: ∼40 training clusters per ±0.025 redshift bin are required for unbiased richness estimates below 0.3σ.Five spectra per bin yield a reasonable but ∼1σ richness bias near z ∼0.35.
  • Training requirements: ∼20 training clusters per redshift bin, totaling 200 spectra, suffice for accurate photometric-redshift estimation.
  • Training requirements: ∼400 spectra achieve essentially the same calibration fidelity as millions of SDSS spectra.The minimal sample uses roughly 40 clusters per redshift bin.
  • Training strategy: More than 85% of the DR8 training spectra are brighter than mi < 18.5.Bright central galaxies can therefore be targeted efficiently for spectroscopic follow-up.
  • Training requirements: With at least 40 training clusters per ±0.025 redshift bin, richness biases remain below 0.3σ; with 10, they remain below 0.5σ.

C. COMPARISON OF zred TO SDSS DR8 PHOTO-ZS

The comparison evaluates redMaPPer’s zred against existing DR8 photometric-redshift estimates using high-probability cluster members across bright and faint magnitudes. zred remains competitive to the faint limit while requiring far fewer spectroscopic training galaxies.

  • Training strategy: Assigning central-galaxy spectroscopic redshifts to members extends spectroscopic-quality redshifts to much fainter magnitudes.
  • Low-redshift comparison: zred and zphoto perform well down to the 0.2L∗ limit of redMaPPer’s richness estimation.
  • Higher-redshift comparison: At 0.395 < zCG < 0.405, DR8 zphoto shows slight faint-end biases and p(z) shows a bifurcated distribution.
  • Comparison outcome: zred is at least as good as current photometric-redshift performance while using much smaller spectroscopic training samples.

D. COMPUTING PERFORMANCE BENCHMARKS

The performance benchmarks separate redMaPPer’s calibration and cluster-finding stages. Both are designed for efficient execution, with calibration requiring tens of CPU hours and full DR8 cluster finding about 500 CPU hours.

  • Overview: redMaPPer is designed to be fast, efficient, flexible, and trivially parallelized.The benchmarks separately measure calibration and cluster-finding runtimes.
  • Calibration: ∼30 CPU hr are required for full calibration of the DR8 training sample on 2000 deg2.A minimal training sample of 40 clusters per redshift bin reduces calibration time to ∼13 CPU hr.
  • Cluster finding: ∼500 CPU hr are required to run the cluster finder on the full DR8 catalog.The cluster-finding stage can be divided into arbitrary sky chunks.
  • Parallelization: A 1.5° border region guarantees unique cluster percolation between DR8 sky-processing cells.The overlap must exceed twice the largest cluster size.

E. VALIDATING THE CORRECTION C

The correction for survey holes and bright magnitude limits is validated by comparing restricted and standard richness estimates. The corrected richness remains consistent with the full richness across the tested redshift range.

  • mi < 19.6 corresponds to 0.2L∗ at z = 0.2, so clusters above z = 0.2 require richness correction in this test.The subsample spans 0.15 < z < 0.3 and probes the correction range used in the catalog.
  • The correction richness scales with uncorrected richness, with scatter as expected.The comparison uses λ19.6 for the restricted magnitude limit and λ0.2 for the standard 0.2L∗ cut, with the photometric redshift re-fit for fairness.
  • The corrected richness is consistent with the full λ richness for clusters with 0.15 < z < 0.3.Figure 27 compares richness measured with the mi < 19.6 cut against the standard richness.

F. COMPARING λ TO λcol

The paper compares the multicolor richness λ with single-color estimators using the color appropriate to each redshift range. The appropriate single-color estimates broadly agree with λ, but perform worst near the filter transition around z ∼0.35.

  • F. COMPARING λ TO λcol: The multicolor χ2 filter replaces the Gaussian color filter, while differing background models also affect λ versus λcol.The comparison uses λg−r at low redshift and λr−i at high redshift.
  • F. COMPARING λ TO λcol: At all redshifts, the bias between the appropriate λcol and λ is ≲10%.The appropriate color is g −r for z < 0.35 and r −i for z > 0.35.
  • F. COMPARING λ TO λcol: At low redshift, the median normalized deviation is ∼1σ, while it is smaller at high redshift because richness errors are larger near the magnitude limit.The low-redshift bias is attributed to the different background model, which down-weights redder-than-sequence galaxies in λ.
  • F. COMPARING λ TO λcol: The results show the importance of using the same color model and consistent survey data for best richness estimation.The catalogs and member information are available in FITS and machine-readable formats.
  • F. COMPARING λ TO λcol: The scatter is ≲15% for the appropriate color except at z ∼0.35.At the transition redshift, the 4000˚A break moves between filters, making single-color richness estimation especially poor.
Loading 1303.3562v2…