Multi-field effects in a simple extension of $R^2$ inflation

We consider inflation in the system containing a Ricci scalar squared term and a canonical scalar field with quadratic mass term. In the Einstein frame this model takes the form of a two-field inflation model with a curved field space, and under the slow-roll approximation contains four free parameters corresponding to the masses of the two fields and their initial positions. We investigate how the inflationary dynamics and predictions for the primordial curvature perturbation depend on these four parameters. Our analysis is based on the $\delta N$ formalism, which allows us to determine predictions for the non-Gaussianity of the curvature perturbation as well as for quantities relating to its power spectrum. Depending on the choice of parameters, we find predictions that range from those of $R^2$ inflation to those of quadratic chaotic inflation, with the non-Gaussianity of the curvature perturbation always remaining small. Using our results we are able to put constraints on the masses of the two fields.