We review the well-known Peebles–Vilenkin (PV) quintessential inflation model and discuss its possible improvements in agreement with the recent observations. The improved PV model depends only on two parameters: the inflaton mass m, and another smaller mass M; where the latter has to be chosen in order to undertake that, at present time, the dark energy density of the universe is approximately about 70% of the total energy budget of the universe. The value of the inflaton mass m is calculated using the observational value of the power spectrum of the scalar perturbations, and the value of mass M, which depends on the reheating temperature, is calculated by solving the corresponding dynamical system whose initial conditions are taken at the matter-radiation equality and are obtained from three observational data: the red shift at the matter-radiation equality, the ratio of the matter energy density to the critical one at the present time and the current value of the Hubble parameter

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