Cosmologia não comutativa: dinâmica clássica dos modelos de Bianchi I e III e nucleação quântica no modelo FLRW

Abstract

Understanding the primordial Universe requires extending the framework of General Relativity to regimes in which the e!ects of quantum gravity and possible modifications of the structure of spacetime become relevant. The main objective of this thesis is to investigate the role of noncommutativity in primordial cosmology through two complementary approaches. In the classical regime, we analyze the dynamics of noncommutative locally rotationally symmetric (LRS) Bianchi I and III models during the radiation era, evaluating the impact of the noncommutative (NC) parameter ω on the expansion, isotropization, and its temporal evolution under the assumption of a homogeneous and isotropic Universe. We employ a Hamiltonian formulation within the ADM formalism, with a radiation fluid described through the Schutz formalism, introducing noncommutativity through a deformation of the Poisson algebra between the canonical variables, which allows geometric corrections to be incorporated into the dynamical equations. The results indicate that ω significantly modifies the dynamics, with a stronger influence at early stages, favoring expansion and isotropization for negative values and producing similar behaviors in both Bianchi I and III models. In the quantum regime, we study the quantum nucleation of the Universe in an FLRW model within a noncommutative minisuperspace, analyzing how the NC parameter ε modifies the Wheeler–DeWitt (WDW) equation and the tunneling probability. We formulate a closed FLRW model with a positive cosmological constant ! > 0, in which noncommutativity in the canonical momentum sector leads, after a unitary phase transformation, to a Schrödinger-type WDW equation. By analyzing the tunneling process through the WKB approximation and the Crank–Nicolson numerical method, we observe that values of ε < 0 reduce both the height and width of the potential barrier, increasing the nucleation probability, an e!ect that becomes stronger as the radiation energy E and ! increase. In conclusion, noncommutative deformations of minisuperspace constitute a consistent extension for the study of primordial cosmology, influencing both the mechanisms of classical isotropization and the most probable initial conditions for the “birth” of the Universe.