Newtonian approximation and possible time-varying G in nonlocal gravities

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 | ˙ G / G | = 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 Ψ = Φ and a constant G in the de Sitter phase. Our results also highlight the uniqueness of the RT model [M. Maggiore, Phys. Rev. D 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.