Numerical Investigation of Cosmological Singularities

TL;DR

This paper introduces a numerical method based on operator splitting for studying the approach to singularities in cosmological models, demonstrating its effectiveness on Gowdy T^3 cosmology and supporting the AVTD conjecture.

Contribution

It develops a symplectic numerical scheme that naturally fits Einstein's equations, enabling detailed study of singularity behavior in various cosmological spacetimes.

Findings

Support for the AVTD behavior in generic initial data

Development of small scale spatial structures near singularities

Method generalizes to arbitrary U(1) symmetric spacetimes

Abstract

Although cosmological solutions to Einstein's equations are known to be generically singular, little is known about the nature of singularities in typical spacetimes. It is shown here how the operator splitting used in a particular symplectic numerical integration scheme fits naturally into the Einstein equations for a large class of cosmological models and thus allows study of their approach to the singularity. The numerical method also naturally singles out the asymptotically velocity term dominated (AVTD) behavior known to be characteristic of some of these models, conjectured to describe others, and probably characteristic of a subclass of the rest. The method is first applied to the unpolarized Gowdy T$^3$ cosmology. Exact pseudo-unpolarized solutions are used as a code test and demonstrate that a 4th order accurate implementation of the numerical method yields acceptable agreement. For generic initial data, support for the conjecture that the singularity is AVTD with geodesic velocity (in the harmonic map target space) < 1 is found. A new phenomenon of the development of small scale spatial structure is also observed. Finally, it is shown that the numerical method straightforwardly generalizes to an arbitrary cosmological spacetime on $T^3 \times R$ with one spacelike U(1) symmetry.