Probing large scale homogeneity and periodicity in the LRG distribution using Shannon entropy

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 analyze the data by carrying out measurements in overlapping spheres and find that it suppresses inhomogeneities by a factor of 5 to 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 minuscule 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 analyze 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 distributions and quantify the scale of their periodicity. We do not find any underlying regularities in the LRG distribution within the length scales probed.

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

“…If the Poisson noise affect the inhomogeneities in the actual data in the same way as the mock random distributions, then it implies that the SDSS galaxy distribution becomes homogeneous on a length scale ∼ 120 h −1 Mpc. The results of this analysis agree well with our previous analysis of the SDSS main galaxy sample and the LRG sample using Shannon entropy [45,46]. The filaments are known to be the largest known coherrent features present in the galaxy distribution.…”

“…The existence of the fractal nature of the galaxy distribution out to the scale of the survey is clearly in disagreement with the assumption of homogeneity. But many other studies, though confirm the fractal nature of the galaxy distribution on small scales, reported that the galaxy distribution is homogeneous on scales 70 − 150 h −1 Mpc [34][35][36][37][38][39][40][41][42][43][44][45][46].…”

“…But an accurate measurement of generalized dimension or Renyi dimension are hard to achieve due to the finite and discrete nature of the galaxy distributions and the r → 0 limit in these definitions [55]. Pandey [56] propose an alternative statistical measure based on the count-in-spheres statistics and Shannon entropy [57] and use it to measure the scale of homogeneity in the Main Galaxy sample [45], LRG sample [46] and quasar sample [58] from the SDSS. Renyi [59] provided a more generalized concept of entropy which can be used to quantify the uncertainty or randomness of a system.…”

Unravelling the Cosmic Web: An analysis of the SDSS DR14 with the Local Dimension

“…The observed galaxy distribution shows a tendency towards transition to a homogeneous network on larger length scales. This is consistent with the findings that the Universe is homogeneous around a length scales of ∼ 100 h −1 Mpc (Yadav et al 2005;Hogg et al 2005;Scrimgeour et al 2012;Nadathur 2013;Pandey & Sarkar 2015;Pandey & Sarkar 2016;Avila et al 2018).…”

Cosmological backreaction in higher-derivative gravity expansions

“…Assumed by this is the Cosmological Principle, that the universe is homogeneous and isotropic at sufficiently large distance scales and can thus the metric at large scales is accurately approximated by a Friedmann-Lemaître-Robertson-Walker (FLRW) metric. This is supported by astronomical observations, which indicate that the universe transitions to homogeneity at length scales of order 100 h −1 Mpc, see for example the analyses in [1][2][3][4][5][6]. Homogeneity has also been verified at a much larger volume scale of 14 h −3 Gpc in [7].…”