Oscillating 4-Polytopal Universe in Regge Calculus

The discretized closed Friedmann-Lemaitre-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.

Automated summary

The discretized closed FLRW universe with positive cosmological constant is studied using Regge calculus. Numerical solutions to the Regge equations approximate well to the continuum solution during the era of small edge length. Pseudo-regular 4-polytopes are introduced as a way to go beyond the approximation using regular 4-polytopes, with simple and unique expressions for the Regge equations for any frequency. In the infinite frequency limit, the pseudo-regular 4-polytope model approaches the continuum FLRW universe.