The behavior of $f(R)$ gravity in the solar system, galaxies and clusters

Cosmologically interesting $f(R)$ gravity models are in general strongly environment dependent. For these models, we derive the complete sets of the linearized field equations in the Newtonian gauge, under environments of the solar system, galaxies and clusters respectively. Approximating the solar system as the Sun embedded in a uniform background with density $\bar{\rho}$, we find that {\it the constant curvature solution with the PPN parameter $\gamma=1$ is the only solution}. When $\bar{\rho}\to 0$, this solution approaches to the Schwarzschild-de Sitter vacuum solution found in the literature. In the solar system, the matter density is much higher than the cosmological critical density. This results in significant suppression on corrections to the general relativity (GR) induced by $f(R)$ gravity. We show that the behavior of $f(R)$ gravity in the solar system is virtually identical to that of GR. Although the environments in galaxies and clusters differ from that in the solar system, we find that gravitational lensing of galaxies and clusters are virtually identical to that in GR. Fortunately, galaxy rotation curve and intra-cluster gas pressure profile may contain valuable information to distinguish between $f(R)$ gravity and GR