Modelling Inhomogeneity in Szekeres Spacetime

TL;DR

This paper analyzes the evolution of density inhomogeneities in Szekeres spacetime, deriving conditions to avoid singularities and examining how initial inhomogeneities evolve over time.

Contribution

It provides analytical expressions for density contrast evolution and establishes conditions for avoiding shell crossing singularities in Szekeres models.

Findings

Density contrast can be analytically modeled as a function of time and radius.

Conditions for avoiding shell crossing singularities are derived.

Small initial inhomogeneities decrease in magnitude over time without changing contrast.

Abstract

We study the behaviour of the density contrast in quasi-spherical Szekeres spacetime and derive its analytical behaviour as a function of $t$ and $r$. We set up the inhomogeneity using initial data in the form of one extreme value of the density and the radial profile. We derive conditions for density extremes that are necessary for avoiding the shell crossing singularity and show that in the special case of a trivial curvature function, the conditions are preserved by evolution. We also show that in this special case if the initial inhomogeneity is small, the time evolution does not influence the density contrast, however its magnitude homogeneously decreases.