A noncommutative Bianchi I model with radiation

Abstract

In this work, we study the dynamical evolution of a homogeneous and anisotropic, noncommutative (NC) Bianchi I (BI) model coupled to a radiation perfect fluid. Our first motivation is determining if the present model tends to a homogeneous and isotropic NC Friedmann–Robertson–Walker (FRW) model, during its evolution. In order to simplify our task, we use the Misner parametrization of the BI metric. In terms of that parametrization the BI metric has three metric functions: the scale factor [Formula: see text] and the two parameters [Formula: see text], which measure the spatial anisotropy of the model. Our second motivation is trying to describe the present accelerated expansion of the universe using noncommutativity (NCTY). The NCTY is introduced by two nontrivial Poisson brackets between some geometrical as well as matter variables of the model. We recover the description in terms of commutative variables by introducing some variables transformations that depend on the NC parameter. Using those variables’ transformations, we rewrite the total NC Hamiltonian of the model in terms of commutative variables. From the resulting Hamiltonian, we obtain the dynamical equations for a generic perfect fluid. In order to solve these equations, we restrict our attention to a model in which the perfect fluid is radiation. We solve, numerically, these equations and compare the NC solutions to the corresponding commutative ones. The comparison shows that the NC model may be considered as a possible candidate for describing the accelerated expansion of the universe. Finally, we obtain estimates for the NC parameter and compare the main results of the NC BI model coupled to radiation with the same NC BI model coupled to other perfect fluids. As our main result, we show that the solutions, after some time, produce an isotropic universe. Based on that result, we can speculate that the solutions may represent an initial anisotropic stage of our Universe, that later, due to the expansion, became isotropic.