The phase-space parameters of the brightest halo galaxies

The brightest galaxy in a dark matter halo is expected to reside at rest at the center of the halo. In this paper we test this `Central Galaxy Paradigm' using group catalogues extracted from the 2dFGRS and the SDSS. For each group we compute a parameter, R, defined as the difference between the velocity of the brightest group galaxy and the average velocity of the other group members (hereafter satellites), normalized by the unbiased estimator of the velocity dispersion of the satellite galaxies. Using detailed mock galaxy redshift surveys, we show that the central galaxy paradigm is inconsistent with the data at high confidence, and that instead the brightest halo galaxies have a specific kinetic energy that is about 50 percent of that of the satellites. This indicates that either central galaxies reside at the minimum of the dark matter potential, but that the halo itself is not yet fully relaxed, or, that the halo is relaxed, but that the central galaxy oscillates in its potential well. The former is consistent with the fact that we find a weak hint that the velocity bias of brightest halo galaxies is larger in more massive haloes, while the latter may be indicative of cored, rather than cusped, dark matter haloes. We discuss several implications of these findings, including mass estimates based on satellite kinematics, strong gravitational lensing, halo occupation models, and the frequency and longevity of lopsidedness in disk galaxies.