Cobordism Effects in the Regge Calculus Approach to Quantum Cosmology

TL;DR

This paper investigates the quantum wave function of a universe with lens space topology using Regge calculus, analyzing complex edge lengths, classical extrema, and topology-changing amplitudes, revealing Lorentzian oscillations in the late universe.

Contribution

It introduces a detailed analysis of the ground state wave function in Regge calculus for lens space topologies, including topology change and subdivision effects.

Findings

Wave functions exhibit Lorentzian oscillatory behavior in the late universe.

Classical extrema and steepest descent contours are identified.

Topology-changing amplitudes show a factorization property.

Abstract

We study the ground state wave function for a universe which is topologically a lens space within the Regge calculus approach. By restricting the four dimensional simplicial complex to be a cone over the boundary lens space, described by a single internal edge length, and a single boundary edge length, one can analyze in detail the analytic properties of the action in the space of complex edge lengths. The classical extrema and convergent steepest descent contours of integration yielding the wave function are found. Both the Hartle-Hawking and Linde-Vilenkin type proposals are examined and, in all cases, we find wave functions which predict Lorentzian oscillatory behaviour in the late universe. A factorization property of topology changing amplitudes within a restricted edge length class of cone type complexes is established. The behaviour of the results under subdivision of the boundary universe is also presented.