Semi-classical and quantum analysis of the isotropic Universe in the polymer paradigm

TL;DR

This paper explores the semi-classical and quantum behavior of the isotropic Universe using Polymer Quantum mechanics, demonstrating a bounce that resolves the initial singularity and linking it to Loop Quantum Cosmology.

Contribution

It identifies the spatial volume as the key variable, establishing a critical energy density and connecting Polymer quantization with Loop Quantum Cosmology.

Findings

The Universe exhibits a bounce replacing the singularity.

Semi-classical states confirm the effective dynamics.

Polymer cut-off scale corresponds to LQG discrete geometry.

Abstract

We analyse the semi-classical and quantum dynamics of the isotropic Universe in the framework of the Polymer Quantum mechanics, in order to implement a cut-off physics on the initial singularity. We first identify in the Universe cubed scale factor (i.e. the spatial volume) the suitable configuration variable, providing a constant critical energy density, such that the Bounce arises as intrinsic geometric feature. We then investigate the obtained semi-classical Bounce dynamics for the primordial Universe, and we outline its impact on the resolution of cosmological paradoxes, as soon as the semi-classical evolution is extended (in the spirit of the Ehrenfest theorem) to the collapsing pre-Bounce Universe. Finally, we validate the use of the semi-classical effective dynamics by investigating the behaviour of the expectation values of a proper semiclassical states. The present analysis has the merit to enforce the equivalence between the Polymer quantization paradigm in the Minisuperspace and the Loop Quantum Cosmology approach. In fact, our study allows to define a precise correspondence between the Polymer cut-off scale and the discrete geometric structure of LQG.