Cosmic Homogeneity Demonstrated with Luminous Red Galaxies

We test the homogeneity of the Universe at $zsim 0.3$ with the Luminous Red Galaxy (LRG) spectroscopic sample of the Sloan Digital Sky Survey. First, the mean number $N(R)$ of LRGs within completely surveyed LRG-centered spheres of comoving radius $R$ is shown to be proportional to $R^3$ at radii greater than $Rsim 70 h^{-1} mathrm{Mpc}$. The test has the virtue that it does not rely on the assumption that the LRG sample has a finite mean density; its results show, however, that there emph{is} such a mean density. Secondly, the survey sky area is divided into 10 disjoint solid angular regions and the fractional rms density variations of the LRG sample in the redshift range $0.2