Semiclassical and quantum features of the Bianchi I cosmology in the polymer representation

We analyze the Bianchi I cosmology in the presence of a massless scalar field and describe its dynamics via a semiclassical and quantum polymer approach. We study the morphology of the Big Bounce by adopting three different sets of configurational variables: the Ashtekar connections, a set of anisotropic volume-like coordinates and the Universe volume plus two anisotropy coordinates (the latter two sets of variables would coincide in the case of an isotropic Universe). In the semiclassical analysis we demonstrate that the value of the critical matter energy density depends on the Cauchy problem for the dynamics when adopting the Ashtekar connections or the anisotropic volume-like coordinates. On the contrary, when the Universe volume is considered as a configurational coordinate, we are able to derive a polymer-modified Friedmann equation for the Bianchi I model, from which the expression of the critical energy density can be derived. This analysis shows that the Big Bounce has universal features only when the Universe volume is defined on the polymer lattice. Then, a cosmological constant is included in the Ashtekar connections' formulation and some interesting results are mentioned making a comparison between the synchronous dynamics and that one when the scalar field is taken as a relational time. From a pure quantum point of view, we investigate the Bianchi I dynamics in terms of the Ashtekar connections. We apply the ADM reduction of the variational principle and then we quantize the system. We study the resulting Schr"{o}dinger dynamics, stressing that the behavior of the wave packet peak over time singles out common features with the semiclassical trajectories, confirming the non universal character of the emerging Big Bounce also on a quantum level.