Robust optical richness estimation with reduced scatter

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. implemented on the maxBCG cluster catalog and evaluate the impact of these changes on the scatter in X-ray luminosity (L_X) at fixed richness, using L_X from the ROSAT All-Sky Catalog as the best mass proxy available for the large area required. Most significantly, we find that deeper luminosity cuts can reduce the recovered scatter, finding that for clusters with M _500c ≳ 1.6 × 10¹⁴ 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 4000 Å 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 Sloan Digital Sky Survey data, as well as an augmented maxBCG catalog with the λ richness measured for each cluster.