Evolution of the universe during the inflationary epoch

We often find in the literature solutions to the Friedmann and fluid equations for simple cosmological models during the matter, radiation or cosmological constant dominated epochs. However no solutions appear for the inflationary era dominated by the potential energy of a scalar field due, perhaps, to the fact that we do not have as yet a strongly favored model of inflation; there are, of course, very well motivated models which fit the data. The purpose of this article is to study with some detail the evolution of the Universe during inflation in the slow-roll approximation. Taking the Starobinsky model as an example, we display exact solutions for the time evolution of the scalar field $\phi(t)$, scale factor $a(t)$, Hubble function $H(t)$, equation of state parameter $\omega(t)$ and acceleration of the scale factor $\ddot{a}(t)$ among other quantities of interest.