# Newtonian approximation and possible time-varying $G$ in nonlocal gravities

## TL;DR

This paper investigates the Newtonian approximation in nonlocal gravity theories, revealing that the effective gravitational constant varies with time in most models, except in the RT model which remains consistent across scales.

## Contribution

It demonstrates that in scalar-tensor and Gauss-Bonnet nonlocal gravities, G is time-varying, and identifies the RT model as uniquely consistent with observations from solar system to cosmological scales.

## Key findings

- G varies over time in most nonlocal gravity models.
- The nonlocal Gauss-Bonnet gravity predicts equal potentials and constant G in de Sitter phase.
- The RT model uniquely matches gravitational phenomena across scales.

## Abstract

The Newtonian approximation with a nonvanishing nonlocal background field is analyzed for the scalar-tensor nonlocal gravity and nonlocal Gauss-Bonnet gravity. For these two theories, our calculations show that the Newtonian gravitational constant $G$ is time-varying and $|\dot{G}/G|=\mathcal{O}(H_0)$ for the general case of cosmological background evolution, which is similar to the results of the Deser-Woodard and Maggiore-Mancarella theories. Therefore, observations about the orbit period of binary star (or star-planet) systems could rule out these theories. One thing worth mentioning is that the nonlocal Gauss-Bonnet gravity gives $\Psi=\Phi$ and a constant $G$ in the de Sitter phase. Our results also highlight the uniqueness of the RT model [M. Maggiore, \href{http://dx.doi.org/10.1103/PhysRevD.89.043008}{Phys. Rev. D {\bf 89}, 043008 (2014)}], which is the only nonlocal gravity theory that can successfully describe the gravitational phenomena from solar system to cosmological scales for now.