Modelos cosmológicos não-comutativos nas teorias de Hořava-Lifshitz e relatividade geral

Abstract

In the present thesis we study classical noncommutative cosmological models from the point of view of Hˇorava-Lifshitz gravitational theory and also of General Relativity. Both models were studied with the Friedmann-Robertson-Walker metric, whose spatial sections can have positive (k = 1), negative (k = −1), or zero (k = 0) constant curvature. The material content is described by different perfect fluids. The dynamics of noncommutative models is described in Hamiltonian formalism, with the aid of the ADM formulation and Schutz’s variational formalism. The phase space of the models is given by the variables a(t), T(t), Pa(t) and PT (t), where a(t) is the scale factor of the universe, T(t) is the coordinate associated with the fluid, and Pa(t), PT (t) their respective canonically conjugated moments. The introduction of noncommutativity is done through the FaddeevJackiw formalism. This formalism naturally introduced coordinate transformations that led us to commutative variables, plus a non-commutative parameter α. Combining the Hamilton equations, obtained from the Hamiltonian written in terms of commutative variables, plus the α parameter, we arrive at a first-order differential equation and a second-order differential equation for the scale factor a(t). Such equations describe the dynamics of noncommutative models and depend on several parameters. We obtained numerical solutions for these equations. With the solutions found, we study the new properties introduced by noncommutativity, in order to obtain results that help explain the current accelerated expansion of the universe. In addition to the common parameters, non-commutative solutions have an additional α parameter compared to the corresponding commutative solutions. Such parameter significantly influences the type of behavior of each solution. For certain parameter values involved some solutions can be considered as possible candidates for the explanation of the current expansion of the universe. Finally, we estimate the value of the non-commutative parameter α in the different cases considered.