General models of Einstein gravity with a non-Newtonian weak-field limit

We investigate Einstein theories of gravity, coupled to a scalar field and point-like matter, which are characterized by a scalar field-dependent matter coupling function . We show that under mild constraints on the form of the potential for the scalar field, there are a broad class of Einstein-like gravity models--characterized by the asymptotic behavior of H--which allow for a non-Newtonian weak-field limit with the gravitational potential behaving for large distances as ln r. The Newtonian term GM/r appears only as sub-leading. We point out that this behavior is also shared by gravity models described by f (R) Lagrangians. The relevance of our results for the building of infrared modified theories of gravity and for modified Newtonian dynamics is also discussed.