Generalized slow-roll approach for Horndeski theory

The standard realization of inflation is that of a canonical single field slowly rolling down a sufficiently flat potential. In this convention, one usually computes the power spectra in terms of a standard hierarchy of the slow-roll parameters, related to the inflaton potential, which are required to be small and constant. However, there exist well-motivated models, whose inflationary potential exhibits features, where this hierarchy is broken. The Generalized Slow-Roll (GSR) approach provides an efficient tool to overcome the problems of the standard slow-roll procedure for sharp features in the potential. Furthermore, models with noncanonical terms in the Lagrangian arise naturally in other contexts, as in particle physics, and satisfy observations as well. Horndeski theory (and its extensions) is the most general scalar-tensor theory with second-order equations of motion. It covers canonical inflation as well as several noncanonical models proposed in the literature. In this talk I will present the recent and ongoing developments made on GSR and, in particular, on its application to Horndeski inflation.