Solar System constraints to general $f(R)$ gravity

It has been proposed that cosmic acceleration or inflation can be driven by replacing the Einstein-Hilbert action of general relativity with a function $f(R)$ of the Ricci scalar $R$. Such $f(R)$ gravity theories have been shown to be equivalent to scalar-tensor theories of gravity that are incompatible with Solar System tests of general relativity, as long as the scalar field propagates over Solar System scales. Specifically, the parameterized post-Newtonian (PPN) parameter in the equivalent scalar-tensor theory is $\gamma =1/2$, which is far outside the range allowed by observations. In response to a flurry of papers that questioned the equivalence of $f(R)$ theory to scalar-tensor theories, it was recently shown explicitly, without resorting to the scalar-tensor equivalence, that the vacuum field equations for $1/R$ gravity around a spherically symmetric mass also yield $\gamma =1/2$. Here we generalize this analysis to $f(R)$ gravity and enumerate the conditions that, when satisfied by the function $f(R)$, lead to the prediction that $\gamma =1/2$.