Testing homogeneity in the Sloan Digital Sky Survey Data Release Twelve with Shannon entropy

Abstract

We analyze a set of volume limited samples from SDSS DR12 to quantify the degree of inhomogeneity at different length scales using Shannon entropy. We find that the galaxy distributions exhibit a higher degree of inhomogeneity as compared to a Poisson point process at all length scales. Our analysis indicates that signatures of inhomogeneities in the galaxy distributions persist at least upto a length scale of $120 \, h^{-1}\, {\rm Mpc}$. The galaxy distributions appear to be homogeneous on a scale of $140 \, h^{-1}\, {\rm Mpc}$ and beyond. Analyzing a set of mock galaxy samples from a semi analytic galaxy catalogue from the Millennium simulation we find a scale of transition to homogeneity at $\sim 100 \, h^{-1}\, {\rm Mpc}$.