Second-order corrections to Starobinsky inflation

Higher-order theories of gravity are extensions to general relativity (GR) motivated mainly by high-energy physics searching for GR ultraviolet completeness. They are characterized by the inclusion of correction terms in the Einstein-Hilbert action that leads to higher-order field equations. In this paper, we propose investigating inflation due to the GR extension built with all correction terms up to the second-order involving only the scalar curvature $R$, namely, $R^2$, $R^3$, $R\mathrm{\square }R$. We investigate inflation within the Friedmann cosmological background, where we study the phase space of the model, as well as explore inflation in slow-roll leading-order. Furthermore, we describe the evolution of scalar perturbations and properly establish the curvature perturbation. Finally, we confront the proposed model with recent observations from Planck, BICEP3/Keck, and BAO data.