Bouncing Bianchi Models with Deformed Commutation Relations

TL;DR

The paper tackles the problem of cosmological singularities in general relativity by embedding quantum-gravity-inspired corrections into anisotropic Bianchi I and II spacetimes via deformed commutation relations (DCRs). It uses a semiclassical, Poisson-bracket framework with a cut-off algebra for the volume that reproduces loop quantum cosmology–like bounces, and a Jacobi-consistent deformation for the Misner anisotropy variables, leading to modified Friedmann dynamics. The authors derive effective equations showing a Big Bounce in Bianchi I and a bounce-plus-reflection (Kasner transition) structure in Bianchi II, including multiple Kasner sequences depending on initial conditions; a free function f(π_tot) controls the anisotropy deformation, acting as a global rescaling that persists at late times. The framework provides a simple, adaptable pathway to model quantum-gravity-inspired bounce scenarios beyond isotropy, with connections to LQC and potential extensions to more complex models like Bianchi IX and to observational consequences through the modified relational dynamics. Overall, DCRs offer a unifying, tractable approach to incorporate quantum gravity effects into anisotropic cosmologies and study their phenomenological implications.

Abstract

We study the anisotropic Bianchi I and Bianchi II models in vacuum in the framework of deformed commutation relations (DCRs). Working in a parametrisation of the spatial metric by a volume and two anisotropy variables, we propose modified Poisson brackets that for the volume alone reproduce the bounce dynamics of effective loop quantum cosmology (LQC), with additional modifications for anisotropy degrees of freedom. We derive effective Friedmann equations and observe cosmological bounces both in Bianchi I and Bianchi II. For Bianchi II, we find that the cosmological bounce now interacts with the usual reflection seen in the Kasner indices in various interesting ways, in close similarity again with what had been seen in LQC. This suggests that the DCR framework could model more general quantum-gravity inspired bounce scenarios in a relatively straightforward way.

Figures (5)

• Figure 1: The continuous red lines show the evolution of $v$ (top) and $\beta_+$ (bottom) as functions of $\tau$ in the expanding Bianchi II case, compared with two Bianchi I solutions (shown as dashed blue and black lines).
• Figure 2: Trajectory of the classical Bianchi II model in the $(\beta_+,\beta_-)$ plane. The vertical grey lines are lines of constant potential, growing towards the left for smaller values of $\beta_+$.
• Figure 3: The volume $v$ as function of harmonic time $\tau$ for the bouncing Bianchi I model (red continuous line), compared with the classical Bianchi II (blue dashed line) and Bianchi I (black dotted line) models. Here we have set $\overline{\,\pi_-}=1$, $\overline{\,\pi_+}=2/\sqrt{3\,}$ and $\mu_v=3\sqrt{6/7\,}$; the large value of the deformation parameter was chosen to have the three expanding branches for $\tau>0$ match, highlighting the different smooth transitions.
• Figure 4: Evolution of the volume $v$ as function of harmonic time $\tau$ in the deformed Bianchi II model (red lines) for the case of one reflection (top) or two (bottom), compared with the simple bouncing Bianchi I (black dashed lines).
• Figure 5: The deformed Kasner index $k_3$ (red continuous line) as function of the ratio $r$ in the two cases of $\mu_\beta=1/20$ (top) and $\mu_\beta=1/10$ (bottom), compared with the corresponding classical function (black dashed lines). The thin grey horizontal lines correspond to the values $1$, $1/3$, $-2/7$, and $-1/3$. Notice in particular that the values $k_3<-2/7$ needed to only have one reflection before the bounce are only obtained for a small range of $r$ values (top) or not reached at all (bottom).