  Post-inflationary evolution and (re)heating of the viable inflationary model,
the $R^2$ one, is made more realistic by including the leptogenesis scenario
into it. For this purpose, right-handed Majorana neutrinos with a large mass
are added to the matter sector of the Standard Model to explain the neutrino
oscillation experiments and the baryon asymmetry of the Universe. We have found
parameters that characterize this model: non-minimal coupling of the Higgs
field $\xi$, the mass of the right-handed Majorana neutrino $M_{N_\alpha}$ and
the Yukawa coupling matrix components for the right-handed Majorana neutrino.
We have analyzed the effect of these parameters on the reheating process and
leptogenesis in this model and how they affect the resultant physical
quantities: spectral parameters of primordial perturbations and baryon
asymmetry.
