Evolution of radial profiles in regular Lemaitre-Tolman-Bondi dust models

TL;DR

This paper provides a rigorous analysis of how radial profiles of density, curvature, and expansion evolve in Lemaitre-Tolman-Bondi dust models, revealing conditions for profile inversions and correcting previous misconceptions.

Contribution

It generalizes prior results on profile inversions, proves that only certain inversions occur under regularity conditions, and offers analytic tools for constructing models with desired profile evolutions.

Findings

Only 'clump to void' density inversions occur in hyperbolic models without shell crossings.

Profiles of spatial curvature exhibit similar inversion patterns as density.

Expansion scalar profiles can invert in elliptic models, unlike density and curvature.

Abstract

We undertake a comprehensive and rigorous analytic study of the evolution of radial profiles of covariant scalars in regular Lemaitre-Tolman-Bondi dust models. We consider specifically the phenomenon of "profile inversions" in which an initial clump profile of density, spatial curvature or the expansion scalar, might evolve into a void profile (and vice versa). Previous work in the literature on models with density void profiles and/or allowing for density profile inversions is given full generalization, with some erroneous results corrected. We prove rigorously that if an evolution without shell crossings is assumed, then only the 'clump to void' inversion can occur in density profiles, and only in hyperbolic models or regions with negative spatial curvature. The profiles of spatial curvature follow similar patterns as those of the density, with 'clump to void' inversions only possible for hyperbolic models or regions. However, profiles of the expansion scalar are less restrictive, with profile inversions necessarily taking place in elliptic models. We also examine radial profiles in special LTB configurations: closed elliptic models, models with a simultaneous big bang singularity, as well as a locally collapsing elliptic region surrounded by an expanding hyperbolic background. The general analytic statements that we obtain allow for setting up the right initial conditions to construct fully regular LTB models with any specific qualitative requirements for the profiles of all scalars and their time evolution. The results presented can be very useful in guiding future numerical work on these models and in revising previous analytic work on all their applications.