Numerical solutions of Einstein's equations for cosmological spacetimes with spatial topology S3 and symmetry group U(1)

TL;DR

This paper develops and applies a spectral numerical method to solve Einstein's equations for cosmological spacetimes with S3 topology and U(1) symmetry, demonstrating its effectiveness through reproducing known solutions.

Contribution

It extends a spectral method based on spin-weighted spherical harmonics to Einstein's equations for specific cosmological models with S3 topology and symmetry.

Findings

Successfully reproduces exact inhomogeneous solutions

Addresses analytical and numerical implementation issues

Validates the method with known solutions

Abstract

In this paper we consider the single patch pseudo-spectral scheme for tensorial and spinorial evolution problems on the 2-sphere presented in [3,4] which is based on the spin-weighted spherical harmonics transform. We apply and extend this method to Einstein's equations and certain classes of spherical cosmological spacetimes. More specifically, we use the hyperbolic reductions of Einstein's equations obtained in the generalized wave map gauge formalism combined with Geroch's symmetry reduction, and focus on cosmological spacetimes with spatial S3-topologies and symmetry groups U(1) or U(1) x U(1). We discuss analytical and numerical issues related to our implementation. We test our code by reproducing the exact inhomogeneous cosmological solutions of the vacuum Einstein field equations obtained in [7].