  In the present work, we study the noncommutative version of a quantum
cosmology model. The model has a Friedmann-Robertson-Walker geometry, the
matter content is a radiative perfect fluid and the spatial sections have zero
constant curvature. In this model the scale factor takes values in a bounded
domain. Therefore, its quantum mechanical version has a discrete energy
spectrum. We compute the discrete energy spectrum and the corresponding
eigenfunctions. The energies depend on a noncommutative parameter $\beta$. We
compute the scale factor expected value ($\left<a\right>$) for several values
of $\beta$. For all of them, $\left<a\right>$ oscillates between maxima and
minima values and never vanishes. It gives an initial indication that those
models are free from singularities, at the quantum level. We improve this
result by showing that if we subtract a quantity proportional to the standard
deviation of $a$ from $\left<a\right>$, this quantity is still positive. The
$\left<a\right>$ behavior, for the present model, is a drastic modification of
the $\left<a\right>$ behavior in the corresponding commutative version of the
present model. There, $\left<a\right>$ grows without limits with the time
variable. Therefore, if the present model may represent the early stages of the
Universe, the results of the present paper give an indication that
$\left<a\right>$ may have been, initially, bounded due to noncommutativity. We
also compute the Bohmian trajectories for $a$, which are in accordance with
$\left<a\right>$, and the quantum potential $Q$. From $Q$, we may understand
why that model is free from singularities, at the quantum level.
