Polymer representation of the Bianchi IX cosmology in the Misner variables

Abstract

We analyze the Bianchi IX Universe in the Polymer Quantum Mechanics framework, by facing both semiclassical and purely quantum effects near the cosmological singularity. The peculiar feature of the presented study is that we adopt Misner variables to describe the model dynamics, applying the polymer paradigm simultaneously to the isotropic one (linked to the Universe volume) and to the two anisotropy ones (characterizing the physical gravitational degrees of freedom). Setting two different cut-off scales for the two different variable sets, i.e. the geometrical volume and the gravity tensor modes, we demonstrate how the semiclassical properties of the Bianchi IX dynamics are sensitive to the ratio of the cut-off parameters. In particular, the semiclassical evolution turns out to be chaotic only if the parameter associated to the volume discretization is greater or equal to that one of the anisotropies. Then, we perform, for the chaotic case, a purely quantum polymer analysis, demonstrating that the original Misner result concerning the existence of quasi-classical states near the singularity (in the sense of high occupation numbers) is still valid in the revised approach and able to account for cut-off physics effects. The possibility for a comparison with the original study by Misner is possible because, for all the parameter space, the singularity is still present in the semiclassical evolution of the cosmological model. We interpret this surprising feature as the consequence of a geometrical volume discretization which does not prevent the volume to vanish, i.e. restoring in the Minisuperspace analysis the zero eigenvalue for the volume spectrum.

An Overview on the Nature of the Bounce in LQC and PQM

“…Therefore, its employment in the cosmological sector has great relevance in trying to overcome the singularity issue of GR and making also a comparison with the LQC main results regarding the presence of an initial Big Bounce and its properties. In this sense, the principal feature of the polymer approach is making clear that the properties of the cosmological dynamics are strongly dependent on the set of variables on which the polymer quantization is implemented [27,34,38,42,44,45].…”

“…In this section, we focus on the main applications of the polymer formulation to the FLRW [44], Bianchi I [45] and Bianchi IX models [38,42] that represent the main cosmological scenarios on which the polymer-modified dynamics of the primordial Universe are tested. We will start with a discussion on the main results obtained by treating the polymer quantization of these models in the Ashtekar variables, that constitute the setting more connected to the original LQC formulation from which the polymer formulation is derived.…”

“…In this section, we treat the semiclassical polymer dynamics of the most general anisotropic Universe, i.e., the Bianchi IX model. First, we implement the polymer paradigm in the Misner variables [42], and then we generalize the analysis by considering a inhomogeneous extension of the model in Misner-like variables, i.e., the anisotropies together with the Universe volume v ∝ e 3α [38].…”

“…The aim of this section is to discuss the main features of the Mixmaster semiclassical dynamics in the polymer representation [42]. We choose to define the Misner variables (α, β ± ) as discrete, so we perform the following formal substitutions:…”

“…We will also provide an interesting comparison of the cosmological implementation of the PQM in the metric representation. In particular, we will compare the analyses in [34,38,42], where the homogeneous and inhomogeneous Mixmaster dynamics is studied through the polymerization of the standard Misner isotropic variable α and of the Universe volume, respectively: the difference is only in using or not using a logarithm in the definition of the isotropic variable, but the implication is very deep since only the volume representation ensures a bouncing cosmology. Furthermore, questions concerning the chaotic or non-chaotic nature of the semiclassical Bianchi IX dynamics are addressed in some detail.…”