A simple nonsingular inflationary quintessential model

In the framework of a flat Friedmann-Lema{\^\i}tre-Robertson-Walker (FLRW) geometry, we present a nonsingular model (no big bang singularity at finite time) of our universe describing its evolution starting from its early inflationary era up to the present accelerating phase. We found that a hydrodynamical fluid with nonlinear equation of state could result in such a non singular scenario, which after the end of this inflationary stage, suffers a sudden phase transition and enters into the stiff matter dominated era, and the universe becomes reheated due to a huge amount of particle production. Finally, it asymptotically enters into the de Sitter phase concluding the present accelerated expansion. We show that the background providing an inflationary potential leads to a power spectrum of the cosmological perturbations which fit well with the latest Planck estimations. At the end, we compared our viable potential with some known inflationary quintessential potentials, which shows that, our model is an improved version of them because it contains an analytic solution that allows us to perform analytic calculations.