New Algorithm for Mixmaster Dynamics

TL;DR

This paper introduces a new symplectic numerical algorithm for simulating Mixmaster spacetimes, enabling higher accuracy and longer evolution times, which allows for detailed comparison with BKL dynamics and refinement of the theoretical models.

Contribution

A novel symplectic integration algorithm that improves the accuracy and efficiency of simulating Mixmaster spacetimes, facilitating detailed analysis and comparison with BKL predictions.

Findings

Errors in BKL predictions can be reduced by relaxing assumptions.

The improved map predicts different dynamics for Bianchi Type IX and VI$_0$ models.

Longer simulations match theoretical models more closely.

Abstract

We present a new numerical algorithm for evolving the Mixmaster spacetimes. By using symplectic integration techniques to take advantage of the exact Taub solution for the scattering between asymptotic Kasner regimes, we evolve these spacetimes with higher accuracy using much larger time steps than previously possible. The longer Mixmaster evolution thus allowed enables detailed comparison with the Belinskii, Khalatnikov, Lifshitz (BKL) approximate Mixmaster dynamics. In particular, we show that errors between the BKL prediction and the measured parameters early in the simulation can be eliminated by relaxing the BKL assumptions to yield an improved map. The improved map has different predictions for vacuum Bianchi Type IX and magnetic Bianchi Type VI$_0$ Mixmaster models which are clearly matched in the simulation.