Abstract:
We quantify the degree of inhomogeneity in the Luminous Red Galaxy (LRG) distribution
from the SDSS DR7 as a function of length scales by measuring the Shannon entropy in
independent and regular cubic voxels of increasing grid sizes. We also analyse the data by
carrying out measurements in overlapping spheres and find that it suppresses inhomogeneities
by a factor of 5–10 on different length scales. Despite the differences observed in the degree
of inhomogeneity both the methods show a decrease in inhomogeneity with increasing length
scales which eventually settle down to a plateau at ∼150 h−1 Mpc. Considering the minus cule values of inhomogeneity at the plateaus and their expected variations we conclude that
the LRG distribution becomes homogeneous at 150 h−1 Mpc and beyond. We also use the
Kullback–Leibler divergence as an alternative measure of inhomogeneity which reaffirms our
findings. We show that the method presented here can effectively capture the inhomogeneity
in a truly inhomogeneous distribution at all length scales. We analyse a set of Monte Carlo
simulations with certain periodicity in their spatial distributions and find periodic variations in
their inhomogeneity which helps us to identify the underlying regularities present in such dis tributions and quantify the scale of their periodicity. We do not find any underlying regularities
in the LRG distribution within the length scales probed
