Regge calculus models of the closed vacuumΛ-FLRW universe

Abstract

The Collins-Williams Regge calculus models of Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW) space-times and Brewin's subdivided models are applied to closed vacuum $\mathrm{\ensuremath{\Lambda}}$-FLRW universes. In each case, we embed the Regge Cauchy surfaces into 3-spheres in ${\mathbf{E}}^{4}$ and consider possible measures of Cauchy surface radius that can be derived from the embedding. Regge equations are obtained from both global variation, where entire sets of identical edges get varied simultaneously, and local variation, where each edge gets varied individually. We explore the relationship between the two sets of solutions, the conditions under which the Regge Hamiltonian constraint would be a first integral of the evolution equation, the initial value equation for each model at its moment of time symmetry, and the performance of the various models. It is revealed that local variation does not generally lead to a viable Regge model. It is also demonstrated that the various models do satisfy their respective initial value equations. Finally, it is shown that the models reproduce the correct qualitative dynamics of the space-time. Furthermore, the approximation's accuracy is highest when the universe is small but improves overall as we increase the number of tetrahedra used to construct the Regge Cauchy surface. Eventually though, all models gradually fail to keep up with the continuum FLRW model's expansion, with the models with lower numbers of tetrahedra falling away more quickly. We believe this failure to keep up is due to the finite resolution of the Regge Cauchy surfaces trying to approximate an ever expanding continuum Cauchy surface; each Regge surface has a fixed number of tetrahedra and as the surface being approximated gets larger, the resolution would degrade. Finally, we note that all Regge models end abruptly at a point when the timelike struts of the skeleton become null, though this end point appears to get delayed as the number of tetrahedra is increased.