Quantifying the Dominance and Emergence of Brightest Cluster Galaxies in SPT and Planck Clusters: A Novel Application of the Gini Index

Abstract We introduce the Gini index ( G ) as a metric for studying the emergence of brightest cluster galaxies (BCGs) by quantifying the disparity in the light distribution of galaxy cluster cores. The Gini index collapses the skewness of the bright-end luminosity function to a single statistic, and provides a measure for the light dominance of the BCG over other cluster members that is more robust than the magnitude gap. We argue that the Gini index is far less sensitive to short-term changes in the cluster dynamical state than the magnitude gap and solves its known degeneracies by taking the entire cluster membership into account, making G a more sensitive metric to long-term BCG assembly. We apply the magnitude gap and Gini index to a progenitor-selected sample of 92 South Pole Telescope- and Planck-selected cluster cores at 0 < z < 1 with corresponding Hubble Space Telescope F110W and F200LP and Sloan Digital Sky Survey g and r photometry. Based on both parametric and empirical tests, we find no statistical evidence for evolution in G or M 1,2 over 0 < z < 1, and find that BCGs have not grown significantly (2×) faster than the cluster population for z < 1 at 3 σ confidence. This implies that BCGs achieve dominance and establish defining properties early ( z > 1) in massive clusters. In general, we find that the Gini index is a valuable and relatively straightforward measurement, with well-controlled systematics, that could be applied to much larger samples to study BCG dominance and emergence.