Dynamical systems in quantum cosmology

The work presented in this Ph.D. thesis has the purpose to provide new quantization procedures for the minisuperspace cosmological models in order to better understand the nature of the initial singularity. The possibility to perform a quantum analysis of the primordial Universe able to provide information about its first instants of life is of absolute interest for the comprehension of the properties of the initial singularity, and, consequently, of the mechanisms at the ground of the birth of our Universe. The original contribution of the thesis starts considering a quadratic correction in the Einstein-Hilbert Action, which in the context of the equivalent scalar-tensor picture behaves as a self-interacting scalar field coupled with the ordinary General Relativity. Once it is considered the Mixmaster Model for this particular f(R) theory, it emerges the existence of a free Kasner regime (formally a Bianchi I model coupled with a scalar field) towards the singularity, i.e. a regime in which the chaos is absent. Always in the context of the extended theories of gravity the presence of the Gauss-Bonnet invariant in the modified action that describe the gravitational field is considered. We demonstrate how, in the Noether Symmetry Approach and following the prescriptions of the Hartle criterion, the framework of Gauss-Bonnet cosmology when a simple flat FRW model is taken into account brings to the selection of possible solutions of the wave function of the Universe from which it is possible to extract the emerging classical cosmological trajectories. Then, the classical and quantum dynamics of a Bianchi I model in the presence of a small negative cosmological constant when a Gaussian Reference Dust Fluid is taken into account. In the framework of the canonical metric approach it is showed that the initial cosmological singularity is removed and it is replaced by a bounce in correspondence to a positive defined value of the dust energy density. A physical interpretation of the Bounce will be provide in term of a correlation between the Cosmological Constant and a characteristic polymer scale related to polymer discretization of the Universe volume. Finally, the final part of this work is focused on the analysis of a homogeneous Bianchi I model in presence of a stiff matter contribution, applying the Vilenkin interpretation of the Wave Function of the Universe when a Polymer quantization procedure is performed to the isotropic component of the spatial metric. The goal of this work is to understand if and how the singularity is avoided, how it changes the behaviour of the anisotropies and if it is possible to extend the results of the above mentioned application to the Bianchi IX model.