# SBCS 3.1 Richness Measures

## 3.1 Richness Measures

### 3.1.1. Matched Filter Richness (from Rozo et al. 2009 and Rykoff et al. 2011)

Notation:
$\lambda$

Introduction: Richness measure
$\lambda$
is an estimator of the number of red-sequence galaxies brighter than
$0.2L^*$
in a galaxy cluster. The radial aperture within which
$\lambda$
is computed is itself dependent on the richness
$\lambda$
because richer objects are also larger. Both the luminosity cut and the relation between
$\lambda$
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.

Algorithmic Description: The algorithm defining
$\lambda$
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.

Suppose a cluster has
$\lambda$
red-sequence galaxies above the luminosity limit and within the bounding radius
$R_c(\lambda) \; .$
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
$\lambda .$
Let
$\Sigma(R,\lambda)$
be the assumed radial distribution of cluster galaxies on the sky, normalized so that its two-dimensional integral within
$R_c(\lambda)$
is unity. Let
$\phi(m)$
be the assumed cluster luminosity function, and let
$G(c)$
be the assumed color distribution. The number of red-sequence cluster galaxies within an annulus of radius
$R$
and surface area
$2 \pi R \, \Delta R$
with magnitude
$m$
and color
$c$
is then
$n_{\rm cl}(\lambda) = 2 \pi R \, \Delta R \, \lambda \, \Sigma(R,\lambda) \, \phi(m) \, G(c) \; .$

Now let
$\bar \Sigma_g(m,c)$
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
$m$
and color
$c$
in that same annulus is then
$b(m,c) = 2 \pi R \, \Delta R \, \bar \Sigma_g(m,c) \; .$

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

Links to Software: Publicly available software for computing
$\lambda$
from SDSS data is available here.

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.

Discussion: The most important thing to know about
$\lambda$
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
$\lambda$
has also been fully optimized to minimize the scatter in
$L_X$
(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)