Numerical investigation of cosmological singularities

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 (whose dynamical variables are harmonic maps) and thus allows the 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 generic (unpolarized) Gowdy ${\mathit{T}}^{3}$ cosmology. Exact pseudounpolarized solutions are used as a code test and demonstrate that a fourth-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 ${\mathit{T}}^{3}$\ifmmode\times\else\texttimes\fi{}R with one spacelike U(1) symmetry.