Anisotropic models in LQC with GBP polymerisation

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.

💡 Research Summary

The paper investigates anisotropic cosmological models within Loop Quantum Cosmology (LQC) using the Gambini‑Benítez‑Pullin (GBP) polymerisation scheme, which was recently employed to quantise spherically symmetric black holes. The authors focus on two homogeneous but anisotropic space‑times: the Kantowski‑Sachs (KS) metric, which describes the interior of a Schwarzschild black hole (or a closed FRW universe with two scale factors), and a Bianchi III (hyperbolic) metric, the negatively curved counterpart of KS.

Starting from the classical Hamiltonian formulation with canonical pairs ((b,p_b)) and ((c,p_c)) satisfying ({b,p_b}=G\gamma) and ({c,p_c}=2G\gamma), the GBP prescription replaces (b) by (\sin(\delta_b b)/\delta_b) and rescales (p_b) by (\cos(\delta_b b)) while leaving the ((c,p_c)) sector untouched. This non‑bijective canonical transformation preserves the algebra and ensures that the horizon area of the black‑hole interior remains the same as in the classical solution. The resulting effective Hamiltonian (H_{\rm eff}) contains trigonometric functions of (\delta_b b) and a regularisation factor (b_0\equiv\sqrt{1+\gamma^2\delta_b^2}).

From the Hamilton equations the authors obtain a first integral
\