Testing homogeneity on large scales in the Sloan Digital Sky Survey Data Release One

The assumption that the universe is homogeneous and isotropic on large scales is one of the fundamental postulates of cosmology. We have tested the large scale homogeneity of the galaxy distribution in the Sloan Digital Sky Survey Data Release One (SDSS-DR1) using volume limited subsamples extracted from the two equatorial strips which are nearly two dimensional (2D). The galaxy distribution was projected on the equatorial plane and we carried out a 2D multi-fractal analysis by counting the number of galaxies inside circles of different radii r in the range 5 hMpc to 150 hMpc centred on galaxies. Different moments of the count-in-cells were analysed to identify a range of length-scales (60 − 70 hMpc to 150hMpc ) where the moments show a power law scaling behaviour and to determine the scaling exponent which gives the spectrum of generalised dimension Dq. If the galaxy distribution is homogeneous, Dq does not vary with q and is equal to the Euclidean dimension which in our case is 2. We find that Dq varies in the range 1.7 to 2.2. We also constructed mock data from random, homogeneous point distributions and from ΛCDM N-body simulations with bias b = 1, 1.6 and 2, and analysed these in exactly the same way. The values of Dq in the random distribution and the unbiased simulations show much smaller variations and these are not consistent with the actual data. The biased simulations, however, show larger variations in Dq and these are consistent with both the random and the actual data. Interpreting the actual data as a realisation of a biased ΛCDM universe, we conclude that the galaxy distribution is homogeneous on scales larger than 60− 70 hMpc.