Expanding Polyhedral Universe in Regge Calculus

TL;DR

This paper explores a polyhedral universe model in Regge calculus, demonstrating how regular and pseudo-regular polyhedrons can approximate the continuum FLRW universe, with improvements in the pseudo-regular approach.

Contribution

It introduces pseudo-regular polyhedrons to better approximate the continuum universe in Regge calculus, addressing limitations of regular polyhedral models.

Findings

Pseudo-regular polyhedrons closely match geodesic dome results.

The pseudo-regular model converges to the continuum solution as frequency increases.

Regular polyhedral models expand to infinity in finite time, unlike the continuum universe.

Abstract

The closed Friedmann--Lema\^itre--Robertson--Walker (FLRW) universe of Einstein gravity with positive cosmological constant in three dimensions is investigated by using the Collins--Williams formalism in Regge calculus. A spherical Cauchy surface is replaced with regular polyhedrons. The Regge equations are reduced to differential equations in the continuum time limit. Numerical solutions to the Regge equations approximate well the continuum FLRW universe during the era of small edge length. The deviation from the continuum solution becomes larger and larger with time. Unlike the continuum universe, the polyhedral universe expands to infinite within finite time. To remedy the shortcoming of the model universe we introduce geodesic domes and pseudo-regular polyhedrons. It is shown that the pseudo-regular polyhedron model can approximate well the results of the Regge calculus for the geodesic domes. The pseudo-regular polyhedron model approaches the continuum solution in the infinite frequency limit.