Scalar-tensor cosmologies with a potential in the general relativity limit

We consider Friedmann-Lemaitre-Robertson-Walker flat cosmological models in the framework of general Jordan frame scalar-tensor theories of gravity with arbitrary coupling functions, in the era when the energy density of the scalar potential dominates over the energy density of ordinary matter. To study the regime suggested by the local weak field tests (i.e. close to the so-called limit of general relativity) we propose a nonlinear approximation scheme, solve for the phase trajectories, and provide a complete classification of possible solutions. We argue that the topology of phase trajectories in the nonlinear approximation is representative of those of the full system, and thus can tell for which scalar-tensor models general relativity functions as an attractor. To the classes of models which asymptotically approach general relativity we give the solutions also in cosmological time and conclude with some observational implications.