  We study the implications of a noncommutative geometry of the minisuperspace
variables for the FRW universe with a conformally coupled scalar field. The
investigation is carried out by means of a comparative study of the universe
evolution in four different scenarios: classical commutative, classical
noncommutative, quantum commutative, and quantum noncommutative, the last two
employing the Bohmian formalism of quantum trajectories. The role of
noncommutativity is discussed by drawing a parallel between its realizations in
two possible frameworks for physical interpretation: the NC-frame, where it is
manifest in the universe degrees of freedom, and in the C-frame, where it is
manifest through theta-dependent terms in the Hamiltonian. As a result of our
comparative analysis, we find that noncommutative geometry can remove
singularities in the classical context for sufficiently large values of theta.
Moreover, under special conditions, the classical noncommutative model can
admit bouncing solutions characteristic of the commutative quantum FRW
universe. In the quantum context, we find non-singular universe solutions
containing bounces or being periodic in the quantum commutative model. When
noncommutativity effects are turned on in the quantum scenario, they can
introduce significant modifications that change the singular behavior of the
universe solutions or that render them dynamical whenever they are static in
the commutative case. The effects of noncommutativity are completely specified
only when one of the frames for its realization is adopted as the physical one.
Non-singular solutions in the NC-frame can be mapped into singular ones in the
C-frame.
