Post-Newtonian constraints on $f(R)$ cosmologies in metric and Palatini formalism

We compute the complete post-Newtonian limit of both the metric and Palatini formulations of $f(R)$ gravities using a scalar-tensor representation. By comparing the predictions of these theories with laboratory and solar system experiments, we find a set of inequalities that any lagrangian $f(R)$ must satisfy. The constraints imposed by those inequalities allow us to find explicit bounds to the possible nonlinear terms of the lagrangian. We conclude that in both formalisms the lagrangian $f(R)$ must be almost linear in $R$ and that corrections that grow at low curvatures are incompatible with observations. This result shows that modifications of gravity at very low cosmic densities cannot be responsible for the observed cosmic speed-up.