Numerical Analysis of the Big Bounce in Loop Quantum Cosmology

TL;DR

This paper uses numerical analysis to explore the nature of the big quantum bounce in loop quantum cosmology, revealing how different numerical methods influence the occurrence of bounces.

Contribution

It introduces a numerical perspective on the quantum bounce, showing how explicit and implicit time-updates affect the bounce phenomenon in loop quantum cosmology.

Findings

Explicit time-updates generally produce bounces.

Implicit time-updates can avoid bounces.

Numerical reflections relate to the quantum bounce behavior.

Abstract

Loop quantum cosmology homogeneous models with a massless scalar field show that the big-bang singularity can be replaced by a big quantum bounce. To gain further insight on the nature of this bounce, we study the semi-discrete loop quantum gravity Hamiltonian constraint equation from the point of view of numerical analysis. For illustration purposes, we establish a numerical analogy between the quantum bounces and reflections in finite difference discretizations of wave equations triggered by the use of nonuniform grids or, equivalently, reflections found when solving numerically wave equations with varying coefficients. We show that the bounce is closely related to the method for the temporal update of the system and demonstrate that explicit time-updates in general yield bounces. Finally, we present an example of an implicit time-update devoid of bounces and show back-in-time, deterministic evolutions that reach and partially jump over the big-bang singularity.