Mathematical aspects of an exactly solvable inflationary model

The inflationary scenario is nowadays the most natural and simple way to solve the problems of the standard Friedmann-Lemaitre-Robertson-Walker cosmology. Generally, inflation is driven by models of gravity minimally coupled to a scalar field φ rolling on a suitable potential V (φ). We discuss a particular model presenting two scenarios: The vacuum universe and the matter-filled universe, in order to find exact general isotropic and homogeneous cosmological solutions displaying an inflationary behavior at early times and a power-law expansion at late times