Constraining f(R) gravity in solar system, cosmology and binary pulsar systems

The $f(R)$ gravity can be cast into the form of a scalar-tensor theory, and scalar degree of freedom can be suppressed in high-density regions by the chameleon mechanism. In this article, for the general $f(R)$ gravity, using a scalar-tensor representation with the chameleon mechanism, we calculate the parameterized post-Newtonian parameters $\gamma$ and $\beta$, the effective gravitational constant $G_{\mathrm{e}\mathrm{f}\mathrm{f}}$, and the effective cosmological constant ${\mathrm{\Lambda }}_{\mathrm{e}\mathrm{f}\mathrm{f}}$. In addition, for the general $f(R)$ gravity, we also calculate the rate of orbital period decay of the binary system due to gravitational radiation. Then we apply these results to specific $f(R)$ models (Hu-Sawicki model, Tsujikawa model and Starobinsky model) and derive the constraints on the model parameters by combining the observations in solar system, cosmological scales and the binary systems.