SBCS+3.1+Richness+Measures

3.1.1. Matched Filter Richness (from Rozo et al. 2009 and Rykoff et al. 2011)
__**Notation**:__ math \lambda math

math \lambda math is an estimator of the number of red-sequence galaxies brighter than math 0.2L^* math in a galaxy cluster. The radial aperture within which math \lambda math is computed is itself dependent on the richness math \lambda math because richer objects are also larger. Both the luminosity cut and the relation between math \lambda math and cluster radius are optimized at low redshift to minimize the scatter in X-ray luminosity at fixed richness, with the intention of minimizing the scatter in cluster mass at fixed richness.
 * __Introduction:__** Richness measure [[image:cue_mathinline.gif]]

math \lambda math is conceptually simple, but a detailed description sufficient for writing a code to calculate it would be quite extensive. We therefore summarize the basic idea here and refer readers to Appendix A of [|Rykoff et al. 2011] for the details.
 * __Algorithmic Description:__** The algorithm defining [[image:cue_mathinline.gif]]

Suppose a cluster has math \lambda math red-sequence galaxies above the luminosity limit and within the bounding radius math R_c(\lambda) \;. math In current implementations, this bounding radius is assumed to have a power-law dependence on richness, and the parameters of that relationship are adjusted to minimize the scatter in X-ray luminosity at fixed math \lambda. math Let math \Sigma(R,\lambda) math be the assumed radial distribution of cluster galaxies on the sky, normalized so that its two-dimensional integral within math R_c(\lambda) math is unity. Let math \phi(m) math be the assumed cluster luminosity function, and let math G(c) math be the assumed color distribution. The number of red-sequence cluster galaxies within an annulus of radius math R math and surface area math 2 \pi R \, \Delta R math with magnitude math m math and color math c math is then math n_{\rm cl}(\lambda) = 2 \pi R \, \Delta R \, \lambda \, \Sigma(R,\lambda) \, \phi(m) \, G(c) \;. math

Now let math \bar \Sigma_g(m,c) math be the mean number density of galaxies on the sky as a function of color and magnitude, in galaxies per square degree per magnitude per color interval (in magnitudes). The number of background galaxies with magnitude math m math and color math c math in that same annulus is then math b(m,c) = 2 \pi R \, \Delta R \, \bar \Sigma_g(m,c) \;. math

With these definitions, the probability that a galaxy of magnitude math m math and color math c math at that radius is a cluster member is simply math p = \frac{ n_{\rm cl}(\lambda) }{ n_{\rm cl}(\lambda) + b } \;. math Note that this probability depends only on math \lambda math once a set of functional forms for the distribution functions in space, magnitude, and color have been adopted. The probabilities must also satisfy the constraint equation math \lambda = \sum p_i math where the sum is over all galaxies within the radial aperture. Since math p math is a function only of math \lambda math once choices have been made for the relevant distribution functions, the only unknown in the constraint equation is math \lambda. math The value of math \lambda math that solves this equation is the estimator of cluster richness.

math \lambda math from SDSS data is available [|here].
 * __Links to Software:__** Publicly available software for computing [[image:cue_mathinline.gif]]

__**Strengths:**__
 * Robust to details of the assumed filters.
 * Robust to the choice of bands used to compute galaxy color.
 * Fully optimized to minimize scatter (~20%-30% scatter in mass at fixed richness).
 * Fully independent of cluster finding algorithms.


 * __Weaknesses:__**
 * Only optimized at low redshift. Improvements may be possible at higher redshifts.
 * If one has more than two filters to compute galaxy colors, it disregards that additional information.
 * The optimal radial aperture is not currently set to match a specified overdensity criteria, so that self-similar model evolution does not apply.

math \lambda math is that it is very robust, both to the specific filters used and to the choice of parameters for the filters (within reason). Consequently, one can fairly compare richness measures of galaxy clusters estimated from different data sets, which is an important feature for a standard cluster observable. Of course, sensitivity to ``catalog noise'' (i.e. non-galaxies in the galaxy catalog) is inevitable, so a quality galaxy catalog is still necessary. The definition of math \lambda math has also been fully optimized to minimize the scatter in math L_X math (and therefore presumably mass) at fixed richness. This makes it ideal for estimating cluster masses from cheap photometric data and very useful for cosmological analysis. Because it is still very new, however, a detailed empirical calibration of the mass--observable relation has not yet been carried out. (Eduardo Rozo)
 * __Discussion:__** The most important thing to know about [[image:cue_mathinline.gif]]