Higher-dimensional polytopal universe in Regge calculus

Abstract

Abstract The higher-dimensional closed Friedmann–Lemaître–Robertson–Walker (FLRW) universe with a positive cosmological constant is investigated by Regge calculus. A Cauchy surface of the discretized FLRW universe is replaced by a regular polytope in accordance with the Collins–Williams formalism. Polytopes in arbitrary dimensions can be systematically dealt with by a set of five integers integrating the Schläfli symbol of the polytope. The Regge action in the continuum time limit is given. It possesses reparameterization invariance of the time variable. The variational principle for edge lengths and struts yields a Hamiltonian constraint and an evolution equation. They describe an oscillating universe in dimensions larger than three. To go beyond the approximation by regular polytopes, we propose pseudo-regular polytopes with fractional Schläfli 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.