Inflation and dark energy from $f(R)$ gravity

Abstract: Inflationary paradigm has several issues, such as the pre-inflationary horizon problem or the eternal inflation. To avoid that the standard Starobinsky inflation has been extended to $R + \alpha R^n$ and $R + \alpha R^n -\delta R^{2-n}$ models as well as Brans-Dicke generalisation of those $f(R)$ models. The region of the parameter space, which provides consistency with PLANCK data and lack of eternal inflation has been founded. The Einstein frame potential has a stable minimum with non-zero vacuum energy, which may be a source of dark energy.