Não-comutatividade em um modelo cosmológico com fluido de poeira

Abstract

In this dissertation we study a classical noncommutative cosmological model with a Friedmann-Robertson-Walker metric. The spatial sections may have positive (k = 1), negative (k = —1) or zero (k = 0) constant curvature. The matter content is described by a dust perfect fluid. The dynamics of the noncommutative model is described using the Hamilton's formalism, with the aid of the ADM and Schutz's formalisms. The phase space of the model 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 to the fluid and Pa(t), PT(t) are their canonically conjugated momenta. We introduce the noncommutativity through Poisson brackets. In order to study the model, we introduce coordinate transformations from the noncommutative coordinates to the commutative ones plus a noncommutative parameter 'y. Combining the Hamilton's equations, obtained from the Hamiltonian written in terms of the commutative variables plus the 7 parameter, we arrive at a second order differential equation for the scale factor a(t). This equation describes the dynamics of the non-commutative model and depends on several parameters, such as: 7, k, C and B. We obtained analytical solutions for this equation. With the obtained solutions, we study the new properties introduced by noncommutativity, in order to get results that help explaining the accelerated expansion of the Universe. The noncommutative solutions have two additional parameters -y and B, compared to the corresponding commutative solutions, beyond the common parameters k and C, the last one associated to the fluid energy. These parameters significantly influence the behavior of each solution. For certain parameters values some solutions are considered as possible candidates to explain the current expansion of the Universe. Among these cases, for k = 0, the non-commutative solutions showed an exponential increase to infinity, while the corresponding commutative ones showed polynomial growth. For k = —1 both solutions had the same qualitative behavior of expansion to infinity described by hyperbolic functions. For k = 1, expansive solutions, which do not describe the current expansion of the universe, were found. They are important because they are not present in the corresponding commutative model. Such solutions expands linearly in time oscillating between maximum and minimum values. We seek in the literature another non-commutative model in order to verify if different ways of introducing the noncommutativity lead to the same results. Such comparisons result in quite different qualitative behavior of such noncommutative solutions, since the differential equations for the scale factor obtained for each model are different.