Noncommutativity in the early Universe

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.