In the framework of a flat Friedmann-Lemaître-Robertson-Walker (FLRW) geometry, we present a non-geodesically past complete model of our universe without the big bang singularity at finite cosmic time, 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 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. Using the reconstruction technique, we also show that, this background provides an extremely simple inflationary quintessential potential whose inflationary part is given by the well-known 1-dimensional Higgs potential, i.e., a Double Well Inflationary potential, and the quintessential one by an exponential potential that leads to a deflationary regime after this inflation, and it can depict the current cosmic acceleration at late times. Moreover the Higgs potential leads to a power spectrum of the cosmological perturbations which fit well with the latest Planck estimations. Further, we compared our viable potential with some known inflationary quintessential potential, which shows that our quintessential model, that is, the Higgs potential combined with the exponential one, is an improved version of them because it contains an analytic solution that allows us to perform all analytic calculations. Finally, we have shown that the introduction of a non zero cosmological constant simplifies the potential considerably with an analytic behavior of the background which again permits us to evaluate all the quantities analytically.
INTRODUCTIONThe complete evolution of our universe is still a mystery, and probably, one of the most interesting topics in the history of cosmology. Until now, we have some theories describing different phases of our universe, in agreement with the latest observations, which tell us that our universe underwent a rapid accelerating phase during its very early evolution, namely, the inflation [1, 2], and presently it is going through a phase of accelerated expansion [3,4]. The gap between these two successive accelerating expansions is described by three sequential decelerated phases, the first one is the stiff matter dominated era, then there was a radiation dominated phase, and finally, before its current accelerating phase, the universe was matter dominated. However, since the beginning of modern cosmology, the big bang still remains as one of the controversial issues for cosmologists. Hence, it has been questioned several times, and alternatively, an existence of some kind of "nonsingular" universe (a model of our universe without finite cosmic time singularity) [5] has been proposed just to replace this big bang singularity, but the evolution of the universe will remain same. As a ...