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MaxBCG: A Red Sequence Galaxy Cluster Finder

Benjamin P. Koester, Timothy A. McKay, James Annis, Risa H. Wechsler, August E. Evrard, Eduardo Rozo, Lindsey Bleem, Erin S. Sheldon, David Johnston

arXiv:astro-ph/0701268v1astro-ph

TL;DR

Cluster abundance cosmology needs cluster samples with understood selection and calibrated links between observables and halo mass. The paper introduces maxBCG, which combines BCG, spatial, and red-sequence information in wide-field imaging. In tests over 0.1 < z < 0.3, it achieves above-90% purity and completeness at specified richness and mass thresholds while recovering halo properties with limited fragmentation and overmerging.

  • Problem

    Reliable cosmological constraints from cluster abundances require understanding observable–mass calibration, scatter, selection effects, and coverage beyond the most massive halos.

  • Method

    MaxBCG combines brightest cluster galaxy, spatial-clustering, and red-sequence information in a redshift-dependent likelihood for wide-field multi-band imaging.

  • Results

    Above 90% purity for clusters with 10 or more red galaxies and above 90% completeness for halos with M200 > 2×10^14 across 0.1 < z < 0.3.

  • Takeaways & Limitations

    MaxBCG extends cluster detection toward group-sized halos and produces richnesses strongly coupled to underlying halo properties.

  • Takeaways & Limitations

    MaxBCG can merge halos within a few Mpc along the line of sight and may fail to center on significantly bluer BCGs in cooling-flow clusters.

Abstract

from arXiv · show

Measurements of galaxy cluster abundances, clustering properties, and mass to- light ratios in current and future surveys can provide important cosmological constraints. Digital wide-field imaging surveys, the recently-demonstrated fidelity of red-sequence cluster detection techniques, and a new generation of realistic mock galaxy surveys provide the means for construction of large, cosmologicallyinteresting cluster samples, whose selection and properties can be understood in unprecedented depth. We present the details of the "maxBCG" algorithm, a cluster-detection technique tailored to multi-band CCD-imaging data. MaxBCG primarily relies on an observational cornerstone of massive galaxy clusters: they are marked by an overdensity of bright, uniformly red galaxies. This detection scheme also exploits classical brightest cluster galaxies (BCGs), which are often found at the center of these same massive clusters. (ABRIDGED)

1. Introduction

Cluster surveys can constrain cosmology, but reliable inference requires broad, well-calibrated samples and control of observable–mass scatter and projection. MaxBCG addresses these needs with an imaging-based method designed to identify rich clusters and extend detection toward group-sized halos.

  • Cluster masses, abundances, and distributions are sensitive probes of cosmology because clusters trace dark matter halos.
  • Reliable cosmological constraints require calibration of the observable–mass relation and its scatter, including Eddington bias from threshold selection.The steep high-mass abundance function makes upward scatter across an observable threshold exceed downward scatter.
  • Broad mass coverage improves cosmological constraints and mitigates some scatter effects where the mass function is shallower.
  • Optical surveys provide wide-field, high signal-to-noise galaxy data that can support mass calibration, weak lensing, spectroscopy, luminosity measurements, and mass-to-light ratios.
  • Red fractions increase with halo mass and luminosity and decrease with halo-centric radius, motivating red-sequence selection beyond the richest systems.
  • MaxBCG combines bright, similarly colored, spatially clustered galaxies and BCGs to build quantitatively characterized catalogs from wide-field imaging.The red sequence supports selection at lower richness, while the method is intended to limit projection effects.

2.1. Outline

MaxBCG evaluates potential BCGs using spatial, color, and brightness properties, then ranks candidates and removes lower-likelihood neighbors to produce cluster centers, redshifts, and richness estimates.

  • MaxBCG exploits spatial clustering, the E/S0 ridgeline, and a distinctive brightest cluster galaxy to identify cluster centers.BCGs are typically luminous, red-sequence members near the galaxy-distribution and halo centers.
  • The redshift-dependent likelihood evaluates each catalog object as a possible BCG in an overdense environment with small color dispersion.
  • Each object receives the redshift maximizing its likelihood, and candidates are ranked by those maximum likelihoods.
  • The highest-likelihood candidate becomes a center, while lower-likelihood objects within z = ±0.02 and a scaled r200 are removed.
  • The final unflagged objects are retained as BCGs identifying clusters.
  • Each cluster receives a photometric redshift and richness estimates based on red galaxies within a fixed aperture or an r200-scaled aperture.Ngals uses a fixed 1 h^-1 Mpc aperture, while Nr200_gals uses r200 = 0.156N_gals^0.6 h^-1 Mpc.

2.2. Likelihood Framework

The likelihood framework combines evidence that a galaxy resembles a BCG with evidence that its surrounding environment matches the E/S0 ridgeline, evaluating this composite across redshift.

  • MaxBCG maximizes a two-part likelihood over redshift to assess whether a galaxy lies at a cluster center.
  • The BCG likelihood measures similarity to a brightest cluster galaxy, while the ridgeline likelihood measures environmental agreement with the E/S0 ridgeline.
  • The full likelihood is evaluated across the galaxy density field, and peaks are labeled as candidate cluster centers.

2.3. Ridgeline Likelihood

The ridgeline likelihood combines spatial and color filtering to identify environments containing galaxies consistent with the E/S0 red sequence. Color–redshift modeling also provides photometric redshifts, while photometric errors and fixed ridgeline-width assumptions shape the filter response.

  • The ridgeline likelihood is a matched filter combining spatial and color components, with parameters set observationally or through an NFW density model.
  • The spatial model weights galaxies by projected distance, strongly emphasizing central objects and down-weighting distant ones; this implementation fixes rs = 150 kpc.
  • The E/S0 ridgeline is a narrow, widespread population dominating the bright end of cluster luminosity functions and extending into group environments.
  • The color filter uses SDSS g − r and r − i relations tied to the 4000 Å break, with intrinsic ridgeline widths of 0.05 and 0.06, respectively.
  • Photometric errors are incorporated into the filter, allowing poorly measured colors to contribute over broader redshift ranges at suppressed weight.
  • For clusters with Nr200_gals > 10 at 0.1 < z < 0.3, maxBCG photometric redshift errors are σz < 0.015.
  • The ridgeline likelihood sums each neighboring galaxy’s spatial and color contribution, producing a measure of the tested galaxy’s environment.

2.4. BCG Likelihood

The BCG likelihood adds information about a galaxy’s brightness, colors, and central-cluster role to improve cluster selection and centering. Tests show BCGs are usually red-sequence galaxies near cluster centers, although poorer systems and complex environments produce exceptions.

  • LBCG models the likelihood that a galaxy is a brightest cluster galaxy, complementing the ridgeline likelihood’s environmental measurement.BCGs are typically luminous early-type galaxies, sometimes giant cD galaxies, and are rare because clusters themselves are rare.
  • 79/99 (≃80%) of visually inspected NORAS-REFLEX clusters exhibit a single distinct BCG.Among the remainder, 15 clusters show two BCG-like galaxies and 5 have no clear BCG and are optically poor.
  • 54h−1 kpc is the median BCG–X-ray-center separation for the 79 clusters with distinct BCGs.The median rises to 77h−1 kpc when clusters with two BCGs are included, and to 220h−1 kpc for all 99 clusters.
  • More than 90% of classical BCGs fall within the E/S0 ridgeline, and BCGs lie quite near cluster centers.These properties support using BCG information for cluster centering, while the authors caution that cluster environments create exceptions.
  • The BCG model is trained observationally from bright LRGs, using their color and magnitude distributions to represent BCG properties.A quadratic magnitude–redshift relation is fitted, while BCG colors evolve similarly to the ridgeline.
  • A smooth i-band magnitude cutoff favors galaxies luminous enough to be BCGs at each redshift and improves centering on manually identified BCGs.The cutoff width is 0.3 magnitudes, and fitting to rich clusters increases sensitivity to the richest systems relative to poorer ones.

2.5. Input Galaxy Catalog

The input catalog is organized into redshift slices with color and luminosity cuts designed to measure comparable cluster populations across redshift. The algorithm uses i-band magnitude limits modeled from observed and synthetic galaxy populations, while comparing candidate BCGs against fainter neighbors.

  • Redshift-dependent slices include galaxies within 3σ of predicted g−r and r−i colors and impose a lower luminosity limit.The cuts are intended to measure richness and find clusters consistently across redshift despite survey flux limits.
  • The cutoff a(z) + M∗ incorporates distance, k-correction, and evolution of M∗ to select objects above Mmin across redshifts.The cuts are applied in the i-band, where galaxies at the catalog’s redshift limit retain small photometric errors and remain within typical cluster ridgelines.
  • 0.4L∗ is adopted as Lmin, corresponding to a catalog redshift limit of approximately 0.40.The observed rich-cluster luminosity function gives M_i∗ = −21.24 and L∗ = 2.3 × 10^10h−1L⊙.
  • A Pegase-2 model predicts the redshift-dependent i-band magnitude limit for Lmin galaxies after matching the observed LRG/BCG color distribution.The model gives M_i∗ = −21.22 and Lmin = 0.9 × 10^10h−1L⊙, with i-band absolute magnitude −20.25.
  • When testing a galaxy as a BCG, the ridgeline likelihood includes only galaxies dimmer than that candidate BCG.This works with LBCG to distinguish similarly likely bright galaxies, favoring the dimmer object’s lower ridgeline likelihood when appropriate.

3. Evaluating Likelihoods

MaxBCG evaluates survey galaxies as possible cluster centers over redshift, using color, luminosity, separation, and likelihood criteria to generate and consolidate candidate clusters. The procedure favors BCG-like centers and absorbs many lower-likelihood group-sized peaks into richer systems, while truncating the catalog below a specified richness.

  • Evaluating Likelihoods: Every survey galaxy is tested as a potential center using a redshift-dependent likelihood for BCG properties and an overdense, low-color-dispersion environment.The output includes each candidate’s maximum-likelihood redshift zmax and maximum cluster likelihood Lmax.
  • Evaluating Likelihoods: Candidate centers must match predicted g−r and r−i colors within ±3σ and be brighter than Lmin.Neighbors must lie within projected 3 h−1 Mpc, satisfy the corresponding color and luminosity cuts, and be dimmer than the candidate BCG.
  • Evaluating Likelihoods: The color, magnitude, and separation cuts remove objects clearly inconsistent with red-sequence membership at the test redshift.Each candidate BCG is evaluated across approximately z ± 0.05 to map the likelihood maximum.
  • Evaluating Likelihoods: The highest-likelihood object becomes the first cluster center, while nearby lower-likelihood candidates are removed within z = ±0.02 and a scaled r200.This consolidation produces a final list from the initially evaluated candidate centers.
  • Evaluating Likelihoods: The studies truncate the process at scaled richness Nr200_gals = 10 and below because selection effects and galaxy properties are less certain for poorer systems.Lower-richness objects trace lower-mass systems, but some groups may have less well-defined red sequences and central galaxies.
  • Evaluating Likelihoods: The reported cluster center lands on a cluster galaxy that typically has characteristic BCG properties.This operationally ties the cataloged center to an observed galaxy rather than an abstract position.
  • Evaluating Likelihoods: Group-sized peaks with 10 < Lmax_tot < 100 are often absorbed as members of higher-likelihood cluster-sized objects.This behavior appears in the one-degree field centered on Abell 1689, where many such peaks are present.

4. maxBCG Selection Function

The maxBCG selection function is highly complete and pure for rich, massive systems, while richness-dependent overmerging and limited fragmentation shape the mapping between detected clusters and halos.

  • Completeness and purity: 90% completeness is reached at N_r200,gals = 20 across all tested redshift ranges, with lower completeness partly reflecting mocks designed for rich Abell clusters.The decline below this richness does not fully represent performance on poorer group-sized systems.
  • Richness scaling: The algorithm recovers the underlying halo abundance approximately and assigns richnesses strongly coupled to halo properties, supporting rich-cluster and rich-halo matching.The authors note that this coupling is encouraging for mass calibration, while detailed mass–richness calibration is pursued elsewhere.
  • Richness scaling: Reverse richness transformation nearly aligns maxBCG and intrinsic halo abundances, indicating that abundance offsets largely arise from differing richness measurements.The forward transformation agrees only at low richness, whereas the reverse relation is more straightforward despite duplication in the comparison.

5. Discussion

The discussion finds that maxBCG provides a broadly characterized optical cluster selection, while identifying projection, model assumptions, photometric errors, and unusual BCG colors as important boundaries.

  • Features of MaxBCG: The algorithm uses local likelihood ranking rather than a global threshold, allowing detection farther down the abundance function at reduced completeness and purity.This design aims to retain lower-richness objects whose selection effects can be modeled for cosmological use.
  • Features of MaxBCG: Galaxy-position likelihoods and percolation help center clusters on visually identified BCGs while avoiding multiple identifications of the same substructure.The approach costs computational time but reduces over-identification and fragmentation in the catalog.
  • Model biases: Richness and likelihood estimates can acquire redshift-dependent biases from L_min modeling, k-corrections, luminosity evolution, and increasing photometric errors.The catalog is approximately volume-limited to z = 0.3, but the treatment of photometric errors remains an active issue.
  • Model biases: The BCG-color model fails for some cooling-flow clusters whose brightest galaxies are significantly bluer than the surrounding red sequence.The cited BCS sample contains emission lines in 27% of BCGs, although the broader prevalence and impact remain open questions.

6. Summary

The paper presents maxBCG as a wide-field optical cluster finder that combines red-sequence, BCG, color, and positional information. Tests indicate high purity and completeness for rich or massive systems, a richness–mass connection, limited fragmentation, and projection-related overmerging, while highlighting challenges for future surveys.

  • Summary: MaxBCG extends optical cluster finding toward group-sized halos, where the broader abundance range contains additional cosmological information.Systems below the stated mass range remain accessible depending on the richness threshold.
  • Summary: MaxBCG has above-90% purity and completeness for N_r200^gals > 10 and M_200 > 2 × 10^14, respectively, across 0.1 < z < 0.3.The thresholds refer separately to richness and halo mass.
  • Summary: The algorithm combines red-sequence galaxy clustering with brightest cluster galaxy and multiple-color information to refine cluster detection.Likelihoods are evaluated at individual galaxy positions rather than on a pixelized sky.
  • Summary: Mock catalogs provide a detailed selection-function analysis and an initial richness–mass mapping in which richer systems preferentially correspond to more massive halos.The mock data enable performance assessment on individual halos using observationally motivated properties coupled to underlying mass.
  • Summary: On approximately 10 h^-1 Mpc scales, the method tends to overmerge line-of-sight systems, whereas fragmentation is not significant.Projection is identified as an effect requiring modeling.
  • Summary: Future multi-band, wide-angle, high-redshift surveys will face challenges involving richness uniformity, multiple colors, photometric errors, and selection-function quantification.The discussion frames these as methodological issues for extending the approach.
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