Kullback-Leibler entropy and Penrose conjecture in the Lemaitre-Tolman-Bondi model

TL;DR

This paper investigates measures of cosmic inhomogeneity using Kullback-Leibler entropy and Weyl curvature in the Lemaitre-Tolman-Bondi model, finding proportionality up to second order in both exact and perturbative analyses.

Contribution

It introduces and compares two measures of inhomogeneity in the LTB model, demonstrating their proportionality up to second order.

Findings

Kullback-Leibler entropy and Weyl curvature are proportional up to second order.

Exact and perturbative calculations yield consistent results.

The measures effectively characterize deviations from homogeneity.

Abstract

Our universe hosts various large-scale structures from voids to galaxy clusters, so it would be interesting to find some simple and reasonable measure to describe the inhomogeneities in the universe. We explore two different methods for this purpose: the Kullback-Leibler entropy and the Weyl curvature tensor. These two quantities characterize the deviation of the actual distribution of matter from the unperturbed background. We calculate these two measures in the spherically symmetric Lemaitre-Tolman-Bondi model in the dust universe. Both exact and perturbative calculations are presented, and we observe that these two measures are in proportion up to second order.