Higher Dimensional Polytopal Universe in Regge Calculus

TL;DR

This paper explores higher-dimensional FLRW universes using Regge calculus, introducing regular and pseudo-regular polytopes to model discretized spacetime, and derives their dynamics and continuum limits.

Contribution

It extends Regge calculus to higher dimensions with polytopal discretizations and introduces pseudo-regular polytopes with fractional Schl"afli symbols as an effective approximation.

Findings

Derivation of Regge action in continuum time limit.

Hamiltonian constraint and evolution equations for higher dimensions.

Pseudo-regular polytope model approaches continuum FLRW universe in infinite frequency limit.

Abstract

Higher dimensional closed Friedmann-Lema\^itre-Robertson-Walker (FLRW) universe with positive cosmological constant is investigated by Regge calculus. A Cauchy surface of discretized FLRW universe is replaced by a regular polytope in accordance with the Collins-Williams (CW) formalism. Polytopes in an arbitrary dimensions can be systematically dealt with by a set of five integers integrating the Schl\"afli symbol of the polytope. Regge action in continuum time limit is given. It possesses reparameterization invariance of the time variable. Variational principle for edge lengths and struts yields Hamiltonian constraint and evolution equation. They describe oscillating universe in dimensions larger than three. To go beyond the approximation by regular polytopes, we propose pseudo-regular polytopes with fractional Schl\"afli symbols as a substitute for geodesic domes in higher dimensions. We examine the pseudo-regular polytope model as an effective theory of Regge calculus for the geodesic domes. In the infinite frequency limit, the pseudo-regular polytope model reduces to the continuum FLRW universe.