Oscillating 4-Polytopal Universe in Regge Calculus

TL;DR

This paper models a discretized closed FLRW universe with positive cosmological constant using Regge calculus, revealing oscillating behaviors and introducing pseudo-regular 4-polytopes to improve approximation accuracy.

Contribution

It introduces pseudo-regular 4-polytopes with simple Regge equations that better approximate the continuum FLRW universe at high frequencies.

Findings

Numerical solutions match continuum during small edge lengths

The universe oscillates with expansions and contractions in 4D

Pseudo-regular 4-polytopes approach continuum behavior at high frequency

Abstract

The discretized closed Friedmann-Lema\^itre-Robertson-Walker (FLRW) universe with positive cosmological constant is investigated by Regge calculus. According to the Collins-Williams formalism, a hyperspherical Cauchy surface is replaced with regular 4-polytopes. Numerical solutions to the Regge equations approximate well to the continuum solution during the era of small edge length. Unlike the expanding polyhedral universe in three dimensions, the 4-polytopal universes repeat expansions and contractions. To go beyond the approximation using regular 4-polytopes we introduce pseudo-regular 4-polytopes by averaging the dihedral angles of the tessellated regular 600-cell. The degree of precision of the tessellation is called the frequency. Regge equations for the pseudo-regular 4-polytope have simple and unique expressions for any frequency. In the infinite frequency limit, the pseudo-regular 4-polytope model approaches the continuum FLRW universe.