Cosmic homogeneity demonstrated with luminous red galaxies
  We test the homogeneity of the Universe at $z\sim 0.3$ 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 $R\sim 70 h^{-1} \mathrm{Mpc}$. 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 $0.2<z<0.35$ among these ($\sim 2\times10^7 h^{-3} \mathrm{Mpc^3}$) regions is found to be 7 percent of the mean density. This variance is consistent with typical biased \lcdm models and puts very strong constraints on the quality of SDSS photometric calibration. 
David W. Hogg (NYU), Daniel J. Eisenstein (Arizona), Michael R.   Blanton, Neta A. Bahcall, J. Brinkmann, James E. Gunn, and...