Towards distinguishing variants of non-minimal inflation

We study models of inflation where the scalar field $\phi$ that drives inflation is coupled non-minimally to gravity via $\xi \phi^2 R$, or where the gravity sector is enlarged by an $R^2$ term. We consider the original Higgs inflation, Starobinsky inflation, and two different versions of a scenario where the inflaton is a scalar field other than the Higgs, and discuss if they can be distinguished from each other by measuring the tensor-to-scalar ratio and runnings of the spectral index of primordial curvature perturbations, on top of the amplitude and spectral index of the perturbations. We consider both metric and Palatini theories of gravity, showing how detailed studies of non-minimally coupled models can help to identify the inflaton field and how they may provide for a way to also distinguish between different theories of gravity in the present context.