  In order to try explaining the present accelerated expansion of the universe,
we consider the most complete noncommutativity, of a certain type, in a
Friedmann-Robertson-Walker cosmological model, coupled to a perfect fluid. We
use the ADM formalism in order to write the gravitational Hamiltonian of the
model and the Schutz's formalism in order to write the perfect fluid
Hamiltonian. The noncommutativity is introduced by four nontrivial Poisson
brackets between all geometrical as well as matter variables of the model. Each
nontrivial Poisson bracket is associated to a noncommutative parameter. We
recover the description in terms of commutative variables by introducing four
variables transformations that depend on the noncommutative parameters. Using
those variables transformations, we rewrite the total noncommutative
Hamiltonian of the model in terms of commutative variables. From the resulting
Hamiltonian, we obtain the scale factor dynamical equations for a generic
perfect fluid. In order to solve these equations, we restrict our attention to
a model where the perfect fluid is radiation. The solutions depend on six
parameters: the four noncommutative parameters, a parameter associated with the
fluid energy $C$, and the curvature parameter $k$. They also depend on the
initial conditions of the model variables. We compare the noncommutative
solutions to the corresponding commutative ones and determine how the former
ones differ from the latter ones. The comparison shows that the noncommutative
model is very useful for describing the accelerated expansion of the universe.
We also obtain estimates for one of the noncommutative parameters.
