Cluster Merger Variance and the Luminosity Gap Statistic

The presence of multiple luminous galaxies in clusters can be explained bythe finite time over which a galaxy sinks to the center of the cluster andmerges with the the central galaxy. The simplest measurable statistic toquantify the dynamical age of a system of galaxies is the luminosity(magnitude) gap, which is the difference in photometric magnitude between thetwo most luminous galaxies. We present a simple analytical estimate of theluminosity gap distribution in groups and clusters as a function of dark matterhalo mass. The luminosity gap is used to define "fossil" groups; we expect thefraction of fossil systems to exhibit a strong and model-independent trend withmass: ~1-3% of massive clusters and ~5-40% of groups should be fossil systems.We also show that, on cluster scales, the observed intrinsic scatter in thecentral galaxy luminosity-halo mass relation can be ascribed to dispersion inthe merger histories of satellites within the cluster. We compare ourpredictions to the luminosity gap distribution in a sample of 730 clusters inthe Sloan Digital Sky Survey C4 Catalog and find good agreement. This suggeststhat theoretical excursion set merger probabilities and the standard theory ofdynamical segregation are valid on cluster scales.Comment: 5 pages, 3 figure