Quintessential inflation studied through semiclassical methods

In this work, we solved the scalar and tensor perturbation equations numerically and using the improved uniform approximation method together with the third-order phase-integral method, for the [Formula: see text]–attractor inflationary model. This inflationary model has become very important because it allows us to describe the initial accelerated expansion of the universe in the inflationary epoch, and the current accelerated expansion with the same potential that depends on one scalar field [Formula: see text]. Once the equations for the scalar and tensor power spectra are found, we calculate the observables: the scalar-to-tensor ratio r, and the scalar spectral index [Formula: see text], concluding that semiclassical methods give excellent results compared to numerical integration. We also compare both observables in the [Formula: see text]–attractor and the Starobinsky inflationary model.