Consistency of nonminimally coupled $f(R)$ gravity

Theories with a nonminimal coupling between the space-time curvature and matter fields introduce an extra force due to the nonconservation of the matter energy momentum. In the present work the theoretical consistency of such couplings is studied using a scalar field Lagrangian to model the matter content. The conditions that the coupling does not introduce ghosts, classical instabilities or superluminal propagation of perturbations are derived. These consistency conditions are then employed to rule out or severely restrict the forms of the nonminimal coupling functions considered in the previous literature. For example, a power-law coupling is viable only for sublinear positive power of the curvature scalar.