  We study cosmological $\alpha$-attractors in superconformal/supergravity models, where $\alpha$ is related to the geometry of the moduli space. For $\alpha=1$ attractors \cite{Kallosh:2013hoa} we present a generalization of the previously known manifestly superconformal higher curvature supergravity model \cite{Cecotti:1987sa}. The relevant standard 2-derivative supergravity with a minimum of two chiral multiplets is shown to be dual to a 4-derivative higher curvature supergravity, where in general one of the chiral superfields is traded for a curvature superfield. There is a degenerate case when both matter superfields become non-dynamical and there is only a chiral curvature superfield, pure higher derivative supergravity. Generic $\alpha$-models \cite{Kallosh:2013yoa} interpolate between the attractor point at $\alpha=0$ and generic chaotic inflation models at large $\alpha$, in the limit when the inflaton moduli space becomes flat. They have higher derivative duals with the same number of matter fields as the original theory or less, but at least one matter multiplet remains. In the context of these models, the detection of primordial gravity waves will provide information on the curvature of the inflaton submanifold of the Kahler manifold, and we will learn if the inflaton is a fundamental matter multiplet, or can be replaced by a higher derivative curvature excitation. 
Cosmological Attractor Models and Higher Curvature Supergravity