A free scalar field minimally coupled to gravity model is quantized and the Wheeler-DeWitt equation in minisuperspace is solved analytically, exhibiting positive and negative frequency modes. The analysis is performed for positive, negative and zero values of the curvature of the spatial section. Gaussian superpositions of the modes are constructed, and the quantum Bohmian trajectories are determined in the framework of the Bohm–de Broglie interpretation of quantum cosmology. Oscillating universes appear in all cases, but with a characteristic scale of the order of the Planck scale. Bouncing regular solutions emerge for the flat curvature case. They contract classically from infinity until a minimum size, where quantum effects become important, acting as repulsive forces avoiding the singularity and creating an inflationary phase, expanding afterwards to an infinite size, approaching the classical expansion as long as the scale factor increases. These are nonsingular solutions which are viable models to describe the early Universe.

• Received 3 May 2000

DOI:https://doi.org/10.1103/PhysRevD.62.083507