Cosmic Homogeneity Demonstrated with Luminous Red Galaxies

We test the homogeneity of the Universe at z\~ 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 R³ at radii greater than R\~ 70 h⁻¹. 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 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 (\~ 2\×10⁷ h⁻³ ³) regions is found to be 7 percent of the mean density. This variance is consistent with typical biased \ models and puts very strong constraints on the quality of SDSS photometric calibration.