Just some simple (but nontrivial) analytical solutions for de Broglie-Bohm quantum cosmology

Abstract

In this work, we provide some simple analytical solutions to the Wheeler-DeWitt equation for the minisuperspace applied to de Broglie-Bohmian quantum cosmology for particular potentials of a scalar matter field φ. One solution describes a constantly rolling scalar field with a quadratic potential plus a negative cosmological constant. The scale factor is given by a Gaussian function. The other solution describes a flat Minkowski universe with the potential of a hilltop form.

Uniform rate inflation

“…This can be integrated into α = − λ 2 4 t 2 + λc 2 t + d with an integration constant d. We can freely set d = 0 by imposing α(0) = 0 since only the ratios of the scale factors have physical meaning. The solution was originally found by the author when trying to solve the Wheeler-DeWitt equation through a study of quantum cosmology in the framework of de Broglie-Bohm theory in [8]. Usually, the evolution of the scale factor during slow-roll inflation is described as a "quasi-exponential" expansion.…”