1D Supergravity FLRW Model of Starobinsky

We study two homogeneous supersymmetric extensions for the $f(R)$ modified gravity model of Starobinsky with the FLRW metric. The actions are defined in terms of a superfield $\mathcal{R}$ that contains the FLRW scalar curvature. One model has N=1 local supersymmetry, and its bosonic sector is the Starobinsky action; the other action has N=2, its bosonic sector contains, in additional to Starobinsky, a massive scalar field without self-interaction. As expected, the bosonic sectors of these models are consistent with cosmic inflation, as we show by solving numerically the classical dynamics. Inflation is driven by the $R^2$ term during the large curvature regime. In the N=2 case, the additional scalar field remains in a low energy state during inflation. Further, by means of an additional superfield, we write equivalent tensor-scalar-like actions from which we can give the Hamiltonian formulation.