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

For cosmologically interesting $f(R)$ gravity models, we derive the complete set of the linearized field equations in the Newtonian gauge, under environments of the solar system, galaxies, and clusters, respectively. Based on these equations, we confirm the previous $\gamma =1/2$ solution in the solar system. However, $f(R)$ gravity models can be strongly environment dependent and the high density (comparing to the cosmological mean) solar system environment can excite a viable $\gamma =1$ solution for some $f(R)$ gravity models. Although for $f(R)\propto -1/R$, it is not the case; for $f(R)\propto -\exp (-R/{\lambda }_2{H}_0^2)$, such a $\gamma =1$ solution does exist. This solution is virtually indistinguishable from that in general relativity (GR) and the value of the associated curvature approaches the GR limit, which is much higher than the value in the $\gamma =1/2$ solution. We show that for some forms of $f(R)$ gravity, this solution is physically stable in the solar system and can smoothly connect to the surface of the Sun. The derived field equations can be applied directly to gravitational lensing of galaxies and clusters. We find that, despite a significant difference in the environments of galaxies and clusters comparing to that of the solar system, gravitational lensing of galaxies and clusters can be virtually identical to that in GR, for some forms of $f(R)$ gravity. Fortunately, galaxy rotation curve and intracluster gas pressure profile may contain valuable information to distinguish these $f(R)$ gravity models from GR.