Cosmology and Newtonian limit in a model of gravity with nonlocally interacting metrics

We investigate the features of the cosmological expansion history described by a recent model of gravity characterised by two nonlocally interacting metrics. We perform a detailed analysis of the dynamical system formed by the field equations and we find no stable critical points at finite and infinite distance. Nonetheless, we show that even if the universe does not evolve towards a de Sitter attractor, the effective equation of state parameter ωeff always tends to −1, independently from the value of the free parameter m2 , which characterises the nonlocality of the theory. We also address the behaviour of gravity on Solar System scales and the growth of small cosmological fluctuations on small scales, in the quasi-static approximation. We find a post-Newtonian γ parameter, a slip parameter and an effective, normalised gravitational coupling different from unity. These differences all depend on m2 and are negligible if one consider the cosmological solution by which m2 ∼ H 2 0 . Finally, we investigate the time evolution of the gravitational coupling and its compatibility with the Lunar Laser Ranging constraints. We find that these are passed for values m2 /H 2 0 ≲ 10−3 .