The Schwarzschild-de Sitter Metric of Nonlocal $\sqrt{dS}$ Gravity

It is already known that a simple nonlocal de Sitter gravity model, which we denote as $\sqrt{dS}$ gravity, contains an exact vacuum cosmological solution which mimics dark energy and dark matter and is in very good agreement with the standard model of cosmology. This success of $\sqrt{dS}$ gravity motivated us to investigate how it works at lower than cosmic scale -- galactic and the solar system. This paper contains our investigation of the corresponding Schwarzschild-de Sitter metric of the $\sqrt{dS}$ gravity model. To get exact solution, it is necessary to solve the corresponding nonlinear differential equation, what is a very complicated and difficult problem. What we obtained is a solution of linearized equation, which is related to space metric far from the massive body, where gravitational field is weak. The obtained approximate solution is of particular interest for examining the possible role of non-local de Sitter gravity $\sqrt{dS}$ in describing the effects in galactic dynamics that are usually attributed to dark matter. The solution has been tested on the Milky Way and the spiral galaxy M33 and is in good agreement with observational measurements.