Classical Polymerization of the Bianchi I Model with Deformed Poisson Structure

TL;DR

The paper investigates Bianchi type I cosmology under the joint influence of classical polymerization and a volume-dependent exponential deformation of the Poisson structure. Starting from a polymer-deformed effective Hamiltonian, it derives analytic solutions for the volume variable $\\alpha$ and anisotropies $\\beta_\\pm$ in a convenient gauge, revealing that positive deformation parameters $s_i$ slow the volume evolution and bound anisotropies, though the initial singularity is not removed. The anisotropy suppression arises from the interplay between the polymer scales $\\mu_i$ and the deformation parameters, with a clear threshold $s_\\pm = s_\\alpha$ separating bounded and divergent behavior. The work highlights a mechanism by which quantum-gravity-inspired modifications can temper anisotropic shear in the early universe, offering a path to extended phenomenology in anisotropic cosmologies and motivating future extensions to other Bianchi models and matter couplings.

Abstract

We study the dynamics of the Bianchi~I cosmological model in the presence of both polymer quantization effects and an exponential deformation of the Poisson algebra. Starting from the Hamiltonian formulation, we derive the polymer-deformed equations of motion and analyze their solutions for the contracting branch of the model. In contrast with the undeformed classical dynamics, the exponential deformation with suitable values of deformation parameters, produces a noticeably slower evolution of the volume variable and leads to a stabilization of the anisotropy parameters, which remain bounded throughout the evolution. No removal of the initial singularity is observed; however, the deformation significantly modifies the asymptotic behavior, offering a mechanism to suppress anisotropic shear near the singularity. Our results are illustrated through analytic solutions, highlighting the qualitative differences between the standard and the polymer--deformed Bianchi~I cosmology.

Figures (3)

• Figure 1: Threshold curve in the $(s_\alpha, s_+)$ parameter space. The shaded region indicates the set of initial conditions leading to bounded anisotropy $\beta_+(t)$ throughout the evolution, while the dashed line marks the threshold $s_+ = s_\alpha$.
• Figure 2: Time evolution of $\alpha(t)$ in the polymer-deformed and undeformed Bianchi I models. Parameters: $(\mu_\alpha,\mu_+,\mu_-)=(0.1,0.1,0.1)$, $(P_+,P_-)=(0.2,0.1)$, $(s_\alpha,s_+,s_-)=(0.1,0,0)$; gauge $N=e^{3\alpha}$. Initial data: $\alpha(0)=0$, $\beta_{\pm}(0)=0$, $t_0=0$. Branch choice: principal $\sigma=+1$ for $p_\alpha$ (arcsin branch $k=0$). The polymer-deformed curve is generated from Eq. (47) with $C_0=-1/s_\alpha$ and $K=-\sin(2\mu_\alpha p_\alpha)/\mu_\alpha$, on the time domain where $-s_\alpha(Kt+C_0)>0$. The undeformed curve corresponds to $s_\alpha=0$ with the same remaining parameters. The polymer deformation induces a slower expansion rate and mild oscillations compared to the standard case.
• Figure 3: Time evolution of $\beta_{\pm}(t)$ in the polymer-deformed and undeformed models. Same parameters, gauge and initial data as in Fig. \ref{['fig:alpha_compare']}. Polymer-deformed curve computed from Eq. (53) (since $s_+/s_\alpha\neq 1$) with $D_{\pm}=\sin(2\mu_{\pm}P_{\pm})/\mu_{\pm}$, $K$ as in Fig. \ref{['fig:alpha_compare']}, and $C_0=-1/s_\alpha$. The undeformed curve uses the $s_\alpha=0$ limit (Eq. (57) with $s_{\pm}=0$), keeping the other parameters fixed. The polymer deformation leads to bounded oscillations, stabilizes the anisotropy dynamics at late times, preventing the monotonic divergence observed in the undeformed case.