Order statistics of the early-type galaxy luminosity function

We apply order statistics (OS) to the bright end (M-r < -22) of the luminosity distribution of early-type galaxies spectroscopically identified in the SDSS DR7 catalogue. We show that an overall normalized luminosity function can be derived from the data that describes the distribution of red elliptical galaxies sufficiently for the purposes of OS in a broad redshift range of 0.1 <= z <= 0.5. We calculate the typical OS quantities of this distribution numerically, measuring the expectation value and variance of the kth most luminous galaxy in a sample with cardinality N over a large ensemble of such samples. From these statistical quantities, we explain why and in what limit the kth most luminous galaxies can be used as standard candles for cosmological studies. As a sample application of OS, we show that galaxy counts in different redshift ranges can be easily estimated if the absolute magnitude of the few most luminous galaxies and the overall shape of the luminosity function is known (and does not change significantly with z). First, we demonstrate that the absolutemagnitude of the kth most luminous early-type galaxies can be estimated from galaxy number counts in the investigated redshift range. By reversing the method, galaxy counts can also be very easily calculated, for example, in redshift ranges where spectroscopic data are available only for the brightest sources. Since our sample contains all bright galaxies including the brightest cluster galaxies (BCGs), based on OS we argue that BCGs can be considered as statistical extremes of a well-established Schechter luminosity distribution when galaxies are binned by redshift and not cluster-by-cluster. We presume that the reason behind this might be that luminous red ellipticals in galaxy clusters are not random samples of an overall luminosity distribution but biased by the fact that they are in a cluster containing the BCG. We show that a simple statistical toy model can reproduce the well-known magnitude gap between the BCG and the second brightest galaxy of the clusters.