Solar system and equivalence principle constraints on $f(R)$ gravity by the chameleon approach

We study constraints on $f(R)$ dark energy models from solar system experiments combined with experiments on the violation of the equivalence principle. When the mass of an equivalent scalar field degree of freedom is heavy in a region with high density, a spherically symmetric body has a thin shell so that an effective coupling of the fifth force is suppressed through a chameleon mechanism. We place experimental bounds on the cosmologically viable models recently proposed in the literature that have an asymptotic form $f(R)=R-\lambda R_c[1-(R_c/R{)}^{2n}]$ in the regime $R\gg R_c$. From the solar system constraints on the post-Newtonian parameter $\gamma$, we derive the bound $n>0.5$, whereas the constraints from the violations of the weak and strong equivalence principles give the bound $n>0.9$. This allows a possibility to find the deviation from the $\Lambda$-cold dark matter ($\Lambda \mathrm{CDM}$) cosmological model. For the model $f(R)=R-\lambda R_c(R/R_c{)}^p$ with $0<p<1$ the severest constraint is found to be $p<{10}^{-10}$, which shows that this model is hardly distinguishable from the $\Lambda \mathrm{CDM}$ cosmology.