Numerical Relativity and Inhomogeneous Cosmologies

TL;DR

This paper develops advanced numerical methods, including adaptive mesh refinement and hyperbolic formulations, to simulate inhomogeneous cosmological models in general relativity, analyzing gravitational waves and singularity formation.

Contribution

It introduces a new two-dimensional code implementing high-resolution schemes and AMR for Einstein's equations, comparing hyperbolic formulations and investigating coordinate singularities.

Findings

Accurate simulation of inhomogeneous cosmologies with steep gradients.

Insights into gravitational wave behavior in vacuum spacetimes.

Analysis of coordinate singularity formation in numerical relativity.

Abstract

In this work numerical methods for solving Einstein's equations are developed and applied to the study of inhomogeneous cosmological models. A two-dimensional computer code is described which implements two advanced numerical methods: LeVeque's multi-dimensional high-resolution integration scheme which allows accurate evolution of solutions containing discontinuities or steep gradients, and an adaptive mesh refinement (AMR) algorithm which enables the local resolution of a simulation to vary dynamically in response to the behaviour of the evolved solution. A family of hyperbolic formulations of the Einstein equations is derived by generalization of an evolution system proposed by Frittelli and Reula, and numerical solutions produced using these formulations are compared to solutions produced using alternative reductions of the evolutions equations. Properties of the harmonic time slicing condition are also investigated, and analytic and numerical results concerning the formation of coordinate singularities are presented. Numerical simulations are performed of planar cosmologies, described using Gowdy's reduction of the Einstein equations, and U(1)-symmetric cosmologies, described using Moncrief's reduction of the Einstein equations, with the spacetimes in both cases being vacuum. Numerical studies follow up on the work of Berger, Moncrief and co-workers, with attention being focused on the small-scale features that develop in the models and the behaviour of linear and nonlinear gravitational waves.