We analyze $f(R)$ modifications of Einstein’s gravity as dark energy models in the light of their connection with chameleon theories. Formulated as scalar-tensor theories, the $f(R)$ theories imply the existence of a strong coupling of the scalar field to matter. This would violate all experimental gravitational tests on deviations from Newton’s law. Fortunately, the existence of a matter dependent mass and a thin-shell effect allows one to alleviate these constraints. The thin-shell condition also implies strong restrictions on the cosmological dynamics of the $f(R)$ theories. As a consequence, we find that the equation of state of dark energy is constrained to be extremely close to $-1$ in the recent past. We also examine the potential effects of $f(R)$ theories in the context of the Eöt-wash experiments. We show that the requirement of a thin shell for the test bodies is not enough to guarantee a null result on deviations from Newton’s law. As long as dark energy accounts for a sizeable fraction of the total energy density of the Universe, the constraints that we deduce also forbid any measurable deviation of the dark energy equation of state from $-1$. All in all, we find that both cosmological and laboratory tests imply that $f(R)$ models are almost coincident with a $\Lambda \mathrm{CDM}$ model at the background level.

• Received 1 July 2008

DOI:https://doi.org/10.1103/PhysRevD.78.104021