Numerical techniques for solving the quantum constraint equation of generic lattice-refined models in loop quantum cosmology

TL;DR

This paper introduces a numerical Taylor expansion method to solve complex difference equations in lattice-refined loop quantum cosmology models, ensuring stability and analyzing wave-function dynamics.

Contribution

It presents a versatile numerical technique applicable to any lattice-refined model, with stability verification and wave-function motion analysis.

Findings

The method accurately computes wave-functions at arbitrary lattice points.

Numerical confirmation of the stability criterion from von Neumann analysis.

Analysis of wave-function dynamics in both constant and refined lattice models.

Abstract

To avoid instabilities in the continuum semi-classical limit of loop quantum cosmology models, refinement of the underlying lattice is necessary. The lattice refinement leads to new dynamical difference equations which, in general, do not have a uniform step-size, implying complications in their analysis and solutions. We propose a numerical method based on Taylor expansions, which can give us the necessary information to calculate the wave-function at any given lattice point. The method we propose can be applied in any lattice-refined model, while in addition the accuracy of the method can be estimated. Moreover, we confirm numerically the stability criterion which was earlier found following a von Neumann analysis. Finally, the `motion' of the wave-function due to the underlying discreteness of the space-time is investigated, for both a constant lattice, as well as lattice refinement models.