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

Understand

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.