General slow-roll inflation in $f(R)$ gravity under the Palatini
  approach
  Slow-roll inflation is analyzed in the context of modified gravity within the
Palatini formalism. As shown in the literature, inflation in this framework
requires the presence of non-traceless matter, otherwise it does not occur just
as a consequence of the non-linear gravitational terms of the action.
Nevertheless, by including a single scalar field that plays the role of the
inflaton, slow-roll inflation can be performed in these theories, where the
equations lead to an effective potential that modifies the dynamics. We obtain
the general slow-roll parameters and analyze a simple model to illustrate the
differences introduced by the gravitational terms under the Palatini approach,
and the modifications on the spectral index and the tensor to scalar ratio
predicted by the model.
