Magnitude Gap Statistics and the Conditional Luminosity Function

TL;DR

This paper demonstrates that a conditional luminosity function model explains the observed variation in magnitude gaps among galaxy clusters, aligning with recent findings and supporting the use of the magnitude gap as a calibration tool.

Contribution

It shows that a mass-dependent conditional luminosity function naturally accounts for the magnitude gap distribution, reconciling previous conflicting results and supporting recent observational claims.

Findings

The CLF model predicts a mass-dependent magnitude gap distribution.

Differences in luminosity distributions are subtle and hard to detect with small samples.

The CLF is consistent with multiple independent galaxy observations.

Abstract

In a recent preprint, Hearin et al. (2012,H12) suggest that the halo mass-richness calibration of clusters can be improved by using the difference in the magnitude of the brightest and the second brightest galaxy (magnitude gap) as an additional observable. They claim that their results are at odds with the results from Paranjape & Sheth (2012, PS12) who show that the magnitude distribution of the brightest and second brightest galaxies can be explained based on order statistics of luminosities randomly sampled from the total galaxy luminosity function. We find that a conditional luminosity function (CLF) for galaxies which varies with halo mass, in a manner which is consistent with existing observations, naturally leads to a magnitude gap distribution which changes as a function of halo mass at fixed richness, in qualitative agreement with H12. We show that, in general, the luminosity distribution of the brightest and the second brightest galaxy depends upon whether the luminosities of galaxies are drawn from the CLF or the global luminosity function. However, we also show that the difference between the two cases is small enough to evade detection in the small sample investigated by PS12. This shows that the luminosity distribution is not the appropriate statistic to distinguish between the two cases, given the small sample size. We argue in favor of the CLF (and therefore H12) based upon its consistency with other independent observations, such as the kinematics of satellite galaxies, the abundance and clustering of galaxies, and the galaxy-galaxy lensing signal from the Sloan Digital Sky Survey.