Noncommutative cosmological model in the presence of a phantom fluid

Abstract

We study noncommutative classical Friedmann-Robertson-Walker cosmological models. The constant curvature of the spatial sections can be positive ($k=1$), negative ($k=-1$) or zero ($k=0$). The matter is represented by a perfect fluid with negative pressure, phantom fluid, which satisfies the equation of state $p =\alpha \rho$ , with $\alpha < - 1$, where $p$ is the pressure and $\rho$ is the energy density. We use Schutz's formalism in order to write the perfect fluid Hamiltonian. The noncommutativity is introduced by nontrivial Poisson brackets between few variables of the models. In order to recover a description in terms of commutative variables, we introduce variables transformations that depend on a noncommutative parameter ($\gamma$). The main motivation for the introduction of the noncommutativity is trying to explain the present accelerated expansion of the universe. We obtain the dynamical equations for these models and solve them. The solutions have four constants: $\gamma$, a parameter associated with the fluid energy $C$, $k$, $\alpha$ and the initial conditions of the models variables. For each value of $\alpha$, we obtain different equations of motion. Then, we compare the evolution of the universe between the present noncommutative models and the corresponding commutative ones ($\gamma \to 0$). The results show that $\gamma$ is very useful for describing an accelerating universe. We estimate the value of $\gamma$, for the present conditions of the Universe. Then, using that value of $\gamma$, in one of the noncommutative cosmological models, we compute the amount of time this universe would take to reach the {\it big rip}.