Inflation and Dark Energy from a Scalar Field in Supergravity

In this paper we present a model for accelerated expansion of the universe, both during inflation and the present stage of the expansion, from four dimensional $\mathcal{N}=1$ supergravity. We evaluate the tensor-to-scalar ratio ($r\approx 0.00034$), the scalar spectral index ($n_s\approx 0.970$) and the running spetral index ($d{n}_s\mathrm{/}dk\approx -6\times {10}^{-5}$), and we notice that these parameters are in agreement with Planck+WP+lensing data and with BICEP2/Keck and Planck joint analysis, at $95\mathrm{\%}$ CL. The number of e-folds is $50$ or higher. The reheating period has an associated temperature $T_R\sim {10}^{12}$ GeV, which agrees with the one required by thermal leptogenesis. Regarding the scalar field as dark energy, the autonomous system for it in the presence of a barotropic fluid provides a stable fixed point that leads to a late-time accelerated expansion of the universe, with an equation of state that mimics the cosmological constant ($w_{\mathrm{\Phi }}\approx -0.997$).