Cosmic homogeneity demonstrated with luminous red galaxies

We test the homogeneity of the universe at z similar to 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 similar to 70 h(-1) 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 is such a mean density. Second, 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 ( similar to 2 x 10(7) h(-3) Mpc(3)) regions is found to be 7% of the mean density. This variance is consistent with typical biased Lambda CDM models and puts very strong constraints on the quality of SDSS photometric calibration.