  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.
