Galaxies in X-ray Selected Clusters and Groups in Dark Energy Survey
  Data II: Hierarchical Bayesian Modeling of the Red-Sequence Galaxy Luminosity
  Function
  Using $\sim 100$ X-ray selected clusters in the Dark Energy Survey Science
Verification data, we constrain the luminosity function (LF) of cluster red
sequence galaxies as a function of redshift. This is the first homogeneous
optical/X-ray sample large enough to constrain the evolution of the luminosity
function simultaneously in redshift ($0.1<z<1.05$) and cluster mass ($13.5 \le
\rm{log_{10}}(M_{200crit}) \sim< 15.0$). We pay particular attention to
completeness issues and the detection limit of the galaxy sample. We then apply
a hierarchical Bayesian model to fit the cluster galaxy LFs via a Schecter
function, including its characteristic break ($m^*$) to a faint end power-law
slope ($\alpha$). Our method enables us to avoid known issues in similar
analyses based on stacking or binning the clusters. We find weak and
statistically insignificant ($\sim 1.9 \sigma$) evolution in the faint end
slope $\alpha$ versus redshift. We also find no dependence in $\alpha$ or $m^*$
with the X-ray inferred cluster masses. However, the amplitude of the LF as a
function of cluster mass is constrained to $\sim 20\%$ precision. As a
by-product of our algorithm, we utilize the correlation between the LF and
cluster mass to provide an improved estimate of the individual cluster masses
as well as the scatter in true mass given the X-ray inferred masses. This
technique can be applied to a larger sample of X-ray or optically selected
clusters from the Dark Energy Survey, significantly improving the sensitivity
of the analysis.
