Inflationary models with logarithmic potentials

We examine inflationary universe models driven by scalar fields with logarithmic potentials of the form V(φ)=${\mathit{V}}_0$ ${\mathrm{\phi }}^{\mathit{p}}$(lnφ${)}^{\mathit{q}}$. Combining the slow-roll approximation with asymptotic techniques we identify regions of the potential where inflation may occur and obtain analytic expressions for the evolution of the field and the metric in these cases. We construct a family of exact solutions to the equations of motion with potentials resembling the above form and demonstrate their inflationary nature; deflationary and conflationary cosmological behaviors are also defined and identified. Finally, a computation of scalar density and tensor gravitational perturbations produced by the model is presented.