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

TL;DR

This paper explores a broad class of Einstein-like gravity models with scalar fields that exhibit a non-Newtonian weak-field limit, where the gravitational potential behaves logarithmically at large distances, relevant for modified gravity theories.

Contribution

It identifies conditions under which Einstein-like gravity models with scalar fields have a non-Newtonian, logarithmic weak-field limit, extending understanding of infrared modifications of gravity.

Findings

Gravitational potential behaves as ln r at large distances.

Newtonian GM/r term appears as a sub-leading correction.

Models include those described by f(R) Lagrangians.

Abstract

We investigate Einstein theories of gravity, coupled to a scalar field \vphi and point-like matter, which are characterized by a scalar field-dependent matter coupling function e^{H(\vphi)}. 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.