Cluster merger variance and the luminosity gap statistic

The presence of multiple luminous galaxies in clusters can be explained by the finite time over which a galaxy sinks to the center of the cluster and merges with the central galaxy. The simplest measurable statistic to quantify the dynamical age of a system of galaxies is the luminosity (magnitude) gap, which is the difference in photometric magnitude between the two most luminous galaxies. We present a simple analytical estimate of the luminosity gap distribution in groups and clusters as a function of dark matter halo mass. The luminosity gap is used to define fossil groups; we expect the fraction of fossil systems to exhibit a strong and model-independent trend with mass: similar to 1%-3% of massive clusters and similar to 5%-40% of groups should be fossil systems. We compare our predictions to the luminosity gap distribution in a sample of 730 clusters in the Sloan Digital Sky Survey C4 Catalog and find good agreement. This suggests that theoretical excursion-set merger probabilities and the standard theory of dynamical segregation are valid on cluster scales.