Robust Optical Richness Estimation With Reduced Scatter

Abstract

Reducing the scatter between cluster mass and optical richness is a key goal for cluster cosmology from photometric catalogs. We consider various modifications to the red-sequence matched filter richness estimator of Rozo et al. (2009b), and evaluate their impact on the scatter in X-ray luminosity at fixed richness. Most significantly, we find that deeper luminosity cuts can reduce the recovered scatter, finding that σ ln LX |λ = 0.63 ± 0.02 for clusters with M 500c 1.6 × 10 14 h −1 70 M ⊙ . The corresponding scatter in mass at fixed richness is σ ln M|λ ≈ 0.2 − 0.3 depending on the richness, comparable to that for total X-ray luminosity. We find that including blue galaxies in the richness estimate increases the scatter, as does weighting galaxies by their optical luminosity. We further demonstrate that our richness estimator is very robust. Specifically, the filter employed when estimating richness can be calibrated directly from the data, without requiring a-priori calibrations of the red-sequence. We also demonstrate that the recovered richness is robust to up to 50% uncertainties in the galaxy background, as well as to the choice of photometric filter employed, so long as the filters span the 4000Å break of red-sequence galaxies. Consequently, our richness estimator can be used to compare richness estimates of different clusters, even if they do not share the same photometric data. Appendix A includes "easy-bake" instructions for implementing our optimal richness estimator, and we are releasing an implementation of the code that works with SDSS data, as well as an augmented maxBCG catalog with the λ richness measured for each cluster. 15 As discussed in Rozo et al. (2009b), throughout this work the word "richness" is meant to be understood as "optical mass tracer," and is not necessarily the actual number of cluster galaxies within the virialized region of a cluster or the total optical luminosity of the cluster.

Observational probes of cosmic acceleration

“…We find that the uncertainty due to errors in the scatter becomes comparable to that from errors in the mass when ∆σ ≈ 1.7∆ ln M . For Stage III experiments with weak lensing calibration, yielding ∆ ln M ≈ 2%, the scatter needs to be known at the ∆σ ≈ 0.04 level, a value in agreement with the more rigorous estimate by Rozo et al (2011b) and likely to be achievable in the near future (see, e.g., Rykoff et al, 2012). If Stage IV experiments reach 0.5% precision, the corresponding uncertainty in the scatter must be below 0.01 (absolute, not fractional), which is difficult to achieve from an ab initio calculation but may be possible with statistical calibration methods.…”

“…For example, a Poisson distribution for a cluster with N = 10 galaxies corresponds to a skewness γ ≈ 0.3. This discussion demonstrates the value of finding improved optical richness estimators that have lower scatter relative to mass (Rozo et al, 2009;Rykoff et al, 2012). Figure 29 shows the impact that various elements of P (X|M, z) can have on the recovered cluster counts.…”

“…For the optical detection threshold we have assumed that projection effects limit useful cluster catalogs to a minimum richness λ = 15 in the algorithm of Rykoff et al (2012), which counts galaxies of luminosity L ≥ 0.2L * . To account for mass-richness scatter, we choose an effective mass threshold that yields approximately the same space density as this richness threshold.…”

Luminous red galaxies in clusters: central occupation, spatial distributions and miscentring

“…In order to convert λ to halo mass, we adopt the richnesshalo mass relation from Rykoff et al (2012). In Section 4, we perform tests to evaluate the impact of the uncertainty in this relation on our results.…”

“…This cosmology is chosen for consistency with the redMaPPer richness-halo mass calibration in Rykoff et al (2012). All distances are expressed in physical Mpc units.…”