Lie sphere-geometry in lattice cosmology

TL;DR

This paper introduces Lie sphere-geometry as a novel mathematical framework to construct and analyze initial data for black-hole configurations in lattice cosmology, revealing similarities to swiss-cheese models.

Contribution

It applies Lie sphere-geometry to systematically generate initial data for complex black-hole lattice cosmologies, a novel approach in the field.

Findings

Solutions resemble swiss-cheese models at maximal expansion

Applicable to configurations with any geometric irregularity

Provides a new geometric perspective in cosmology

Abstract

In this paper we propose to use Lie sphere-geometry as a new tool to systematically construct time-symmetric initial data for a wide variety of generalised black-hole configurations in lattice cosmology. These configurations are iteratively constructed analytically and may have any degree of geometric irregularity. We show that for negligible amounts of dust these solutions are similar (in a sense made precise) to the swiss-cheese models at the moment of maximal expansion. As Lie sphere-geometry has so far not received much attention in cosmology, we will devote a large part of this paper to explain its geometric background in a language familiar to general relativists.