Lorentzian quantum cosmology goes simplicial

TL;DR

This paper explores Lorentzian quantum cosmology using discrete Regge calculus, comparing different triangulations and their effectiveness in modeling the early universe and the no-boundary proposal, with implications for spin foam approaches.

Contribution

It introduces a discrete Lorentzian quantum cosmology framework employing various triangulations, analyzing their agreement with continuum models and their potential for defining the no-boundary proposal.

Findings

Shell triangulations match continuum results for small scale factors

Simple and subdivided 4-polytopes have limited continuum correspondence

Shell models with dust particles approximate continuum cases well

Abstract

We employ the methods of discrete (Lorentzian) Regge calculus for analysing Lorentzian quantum cosmology models with a special focus on discrete analogues of the no-boundary proposal for the early universe. We use a simple 4-polytope, a subdivided 4-polytope and shells of discrete 3-spheres as triangulations to model a closed universe with cosmological constant, and examine the semiclassical path integral for these different choices. We find that the shells give good agreement with continuum results for small values of the scale factor and in particular for finer discretisations of the boundary 3-sphere, while the simple and subdivided 4-polytopes can only be compared with the continuum in certain regimes, and in particular are not able to capture a transition from Euclidean geometry with small scale factor to a large Lorentzian one. Finally, we consider a closed universe filled with dust particles and discretised by shells of 3-spheres. This model can approximate the continuum case quite well. Our results embed the no-boundary proposal in a discrete setting where it is possibly more naturally defined, and prepare for its discussion within the realm of spin foams.