Testing homogeneity of the galaxy distribution in the SDSS using Renyi entropy

TL;DR

This study uses Renyi entropy and divergence to analyze galaxy distribution in SDSS, finding homogeneity beyond 140 Mpc/h and highlighting the importance of analysis method and sample size.

Contribution

It introduces the use of Renyi entropy and divergence to quantify cosmic homogeneity, addressing biases from overlapping spheres and emphasizing larger samples for accurate results.

Findings

Galaxy distribution becomes homogeneous beyond 140 h^{-1} Mpc.

Overlapping spheres underestimate the scale of homogeneity.

Larger samples and independent voxel analysis improve accuracy.

Abstract

We analyze a set of volume limited sample of galaxies from the SDSS to study the issue of cosmic homogeneity. We use the Renyi entropy of different order to probe the inhomogeneties present in the galaxy distributions. We also calculate the Renyi diveregence to quantify the deviations of the galaxy distribution from a homogeneous Poisson distribution on different length scales. We separately carry out the analysis using the overlapping spheres and the independent voxels. Our analysis suggests that the scale of homogeneity is underestimated in the smaller galaxy samples due to the suppression of inhomogeneities by the overlapping of the measuring speheres. We find that an analysis with the independent voxels and/or use of a significantly larger galaxy sample can help to circumvent or mitigate this problem. Combining the results from these analyses, we find that the galaxy distribution in the SDSS becomes homogeneous on a length scale beyond $140 \, h^{-1}\, {\rm Mpc}$.