The scale of homogeneity of the galaxy distribution in SDSS DR6

The assumption that the Universe, on sufficiently large scales, is homogeneous and isotropic is crucial to our current understanding of cosmology. In this Letter, we test if the observed galaxy distribution is actually homogeneous on large scales. We have carried out a multifractal analysis of the galaxy distribution in a volume-limited subsample from the Sloan Digital Sky Survey (SDSS) Data Release 6. This considers the scaling properties of different moments of galaxy number counts in spheres of varying radius, r, centred on galaxies. This analysis gives the spectrum of generalized dimension D-q(r), where q > 0 quantifies the scaling properties in overdense regions and q < 0 in underdense regions. We expect D-q(r) = 3 for a homogeneous, random point distribution. In our analysis, we have determined D-q(r) in the range -4 <= q <= 4 and 7 <= r <= 98 h(-1) Mpc. In addition to the SDSS data, we have analysed several random samples which are homogeneous by construction. Simulated galaxy samples generated from dark matter N-body simulations and the Millennium Run were also analysed. The SDSS data is considered to be homogeneous if the measured D-q is consistent with that of the random samples. We find that the galaxy distribution becomes homogeneous at a length-scale between 60 and 70 h(-1) Mpc. The galaxy distribution, we find, is homogeneous at length-scales greater than 70 h(-1) Mpc. This is consistent with earlier works which find the transition to homogeneity at around 70 h(-1) Mpc.