Structure formation in the Lemaitre-Tolman model

TL;DR

This paper investigates structure formation in the Lemaitre-Tolman model, demonstrating how initial density perturbations evolve and accrete mass, with exact formulas and numerical examples illustrating the process.

Contribution

It generalizes Bonnor's approach by showing how any two spherically symmetric density profiles can be connected through Lemaitre-Tolman evolution, providing explicit formulas.

Findings

Any two spherically symmetric density profiles can be joined by L-T evolution.

Exact implicit formulas for the functions defining the L-T model are derived.

Numerical examples illustrate the structure formation process.

Abstract

Structure formation within the Lemaitre-Tolman model is investigated in a general manner. We seek models such that the initial density perturbation within a homogeneous background has a smaller mass than the structure into which it will develop, and the perturbation then accretes more mass during evolution. This is a generalisation of the approach taken by Bonnor in 1956. It is proved that any two spherically symmetric density profiles specified on any two constant time slices can be joined by a Lemaitre-Tolman evolution, and exact implicit formulae for the arbitrary functions that determine the resulting L-T model are obtained. Examples of the process are investigated numerically.