Numerical investigation of cosmological singularities.

A primary unresolved issue for cosmological singularities is whether or not their behavior is locally of the Mixmaster type (as conjectured by Belinskii, Khalatnikov, and Lifshitz (BKL)). The Mixmaster dynamics first appears in spatially homogeneous cosmologies of Bianchi Types VIII and IX. A multiple of the spatial scalar curvature acts as a closed potential leading, in the evolution toward the singularity (say τ → ∞), to an (almost certainly) infinite sequence of bounces whose parameters exhibit the sensitivity to initial conditions usually associated with chaos. Other homogeneous cosmologies are characterized by open (or no) potentials leading to a last bounce as τ → ∞ . Such models are called asymptotically velocity term dominated (AVTD). Here we shall describe a numerical approach to address the BKL conjecture. Starting with a symplectic numerical method ideally suited to this problem, we shall consider application of the method to three models of increasing complexity. The first application is to spatially homogeneous (vacuum) Mixmaster cosmologies where we compare the symplectic ODE solver to a Runge-Kutta one. The second application is to the (plane symmetric, vacuum) Gowdy universe on T 3 ×R. The dynamical degrees of freedom satisfy nonlinearly coupled PDE’s in one spatial dimension and time. We demonstrate support for conjectured AVTD behavior for this model and explain its observed nonlinear small scale spatial structure. Finally, we study U(1) symmetric, vacuum cosmologies on T 3 × R. These are the simplest spatially inhomogeneous universes in which local Mixmaster dynamics is allowed. The Gowdy code is easily generalized to this model, although, the spatial differencing needed in the symplectic method is not trivial.For AVTD models,