Solar system, astrophysics, and cosmology from the derivative expansion

In this paper we show how the solar system, the galactic, and the cosmological scales, are accommodated in a single framework, namely, in the derivative expansion framework. We construct a locally inertial static metric, based on the Einstein equations and on the derivative expansion method, which describes a Schwarzschild black hole immersed in dark matter and dark energy. The leading order metric in the expansion corresponds to the solar system, the first order metric to the galaxy, and the second order metric to cosmology. It is shown how this metric captures the main observations at each scale: in the solar system it trivially gives the Keplerian physics, in the galaxy it gives the flat part of the rotation curve and the Baryonic Tully-Fisher relation, and in the cosmological scale it gives the cosmological redshift, the accelerating expansion, and it coincides with the Robertson-Walker spacetime in the appropriate limit and approximation.