Anisotropic models in LQC with GBP polymerisation

TL;DR

This work investigates anisotropic cosmologies in Loop Quantum Cosmology using Gambini-Benítez-Pullin (GBP) polymerisation. By applying the GBP prescription to a Kantowski-Sachs interior and to Bianchi III, the authors derive effective non-singular dynamics in which a Planck-scale bounce replaces the classical singularity, with a minimal area condition tying the polymer scale to the LQG area gap. A positive cosmological constant drives KS models into eternal expansion toward a de Sitter phase, while the hyperbolic Bianchi III case yields a bounce with asymptotically flat geometry and, unlike KS, does not require polymerisation to avoid singularities. The minimal area constraint crucially governs the KS bounce and, in the Bianchi III case, constrains the geometry via a genus-2 surface without fixing the polymer parameters. Overall, the paper clarifies how GBP polymerisation shapes singularity resolution and late-time behavior in anisotropic quantum-corrected spacetimes, linking microphysical area quanta to macroscopic cosmological evolution.

Abstract

Polymer models are effective in describing quantum gravity effects around the initial singularity, leading to its replacement by bouncing surfaces on which the curvature and densities are finite. Their properties depend on the space-time symmetry and on the particular polymerisation scheme adopted. In this article we investigate anisotropic models under the Gambini-Benítez-Pullin polymerisation, recently used to quantise spherically symmetric black-holes, whose interiors are isometric to Kantowski-Sachs (KS) space-times. Demanding that the minimum area defined by the bouncing surface matches the Loop Quantum Gravity area gap, we can find its radius alongside the curvature and effective density and pressures at the bounce. The density is always positive, while the pressures are negative enough to avoid the singularity. Due to the positive spatial curvature, the solution is oscillatory, reaching a maximum radius where a re-collapse occurs. Therefore, a positive cosmological constant is included in order to have an eternal expansion to a late de Sitter phase. We have also considered a Bianchi III metric, showing that the bounce is still present, but the space-time is asymptotically flat in this case, with no re-collapse. In this hyperbolic space, the minimal area constraint can also be imposed on compact $2$-surfaces. Nevertheless, in contrast to the KS case, it is enough for avoiding the singularity, independently of polymerisation procedures.

Figures (2)

• Figure 1: The squared scale factors $p_c$ (dashed red line) and $g_{xx}$ (solid blue line) of metric (\ref{['line']}) as functions of $b(t)$, for $M = 100$, $\gamma = \sqrt{3}/6$ and $\delta_b$ given by (\ref{['area_minima']}). The constant ($cp_c$) in (\ref{['centralterm']}) was arbitrarily fixed.
• Figure 2: Time evolution of $r = \sqrt{p_c}$ for the Bianchi III solution.