The role of spatial curvature in constraining the Universe anisotropies across a Big Bounce

TL;DR

This paper applies Polymer Quantum Mechanics to cosmological models with spatial curvature, showing that curvature influences anisotropy behavior during a Big Bounce, potentially extending semiclassical bounce scenarios.

Contribution

It generalizes Vilenkin's formalism within the polymer paradigm and analyzes anisotropy dynamics in Bianchi I and IX models near a Big Bounce.

Findings

Anisotropies are regular at the bounce in Bianchi I but grow indefinitely without curvature.

In Bianchi IX with harmonic potential, anisotropies oscillate and remain bounded.

Spatial curvature can stabilize anisotropies during a cosmological bounce.

Abstract

We study the implementation of Polymer Quantum Mechanics (PQM) to a system decomposed into a quasi-classical background and a small quantum subsystem, according to the original Vilenkin proposal. We develop the whole formalism in the momentum representation that is the only viable in the continuum limit of the polymer paradigm and we generalize the fundamental equations of the original Vilenkin analysis in the considered context. Then, we provide a Minisuperspace application of the theory, first considering a Bianchi I cosmology and then extending the analysis to a Bianchi IX model in the limit of small anisotropies. In both these cases, the quasi-classical background is identified with an isotropic bouncing Universe whereas the small quantum subsystem contains the anisotropic degrees of freedom. When the Big Bounce scenario is considered, we obtain that in the Bianchi I model the anisotropies standard deviation is regular at $t=0$ but still increases indefinitely, whereas in the presence of the harmonic Bianchi IX potential such same quantity is bounded and oscillate around a constant value. As a consequence, we demonstrate that the picture of a semiclassical isotropic Bounce can be extended to more general cosmological settings if the spatial curvature becomes relevant when the anisotropic degrees of freedom are still small quantum variables.