Modelos cosmológicos não comutativos em relatividade geral

Abstract

In this work we studied noncommutative cosmological models that may describe the accelerated expansion of the Universe. Through the Hamiltonian formulation of general relativity, we rewrite the geometric part of this theory. To deal with the matter of the Universe, we use Schutz’s formalism to describe this matter as a function of the potential velocities of a relativistic perfect fluid. With these tools we obtained a Hamiltonian corresponding to the isotropic and homogeneous Universe of Friedmann-Lemâitre-Robertson-Walker, coupled to a radiation-like perfect fluid. In this Hamiltonian we apply Faddeev-Jackiw’s (FJ) symplectic formalism, so that noncommutativities were systematically and naturally introduced. Such formalism would induce transformations of variables that would carry noncommutativities. However, the FJ formalism was unable to produce the most general case of noncommutative models, resulting in a case already studied in the literature. We then used a different starting point for our model, proposing that the noncommutative Hamiltonian has the same functional form as the commutative Hamiltonian but its dynamical variables should satisfy deformed Poisson brackets. After that, we obtained a coordinate transformation, relating the noncommutative dynamical variables with commutative variables and some noncommutative parameters α, γ, σ and χ. We proceed to rewrite the noncommutative Hamiltonian as a function of commutative variables and noncommutative parameters and derive dynamical equations for the scale factor in the presence of these noncommutative parameters. We then solved the differential equations for different values of α, γ, σ and χ, for the curvature constant k, the initial fluid energy C and the initial conditions a0, T0 and v0, concluding that noncommutative parameters can, in some cases, describe the accelerated expansion of the Universe. The final result of our work was to estimate the values of α and γ using observational data for different eras in a dust-matter dominated Universe.