Post-Newtonian $\gamma$-like parameters and the gravitational slip in scalar-tensor and $f(R)$ theories

We review the fundamentals and highlight the differences between some commonly used definitions for the PPN gamma parameter ($\gamma$) and the gravitational slip ($\eta$). Here we stress the usefulness of a gammalike parameter used by Berry and Gair (${\gamma }_{\mathrm{\Sigma }}$) that parametrizes the bending of light and the Shapiro time delay in situations in which the standard $\gamma$ cannot be promptly used. First we apply our considerations to two well-known cases, but for which some conflicting results can be found: massive Brans-Dicke gravity and $f(R)$ gravity (both the metric and the Palatini versions). Although the slip parameter is always well defined, it has in general no direct relation to either light deflection or the Shapiro time delay, hence care should be taken on imposing the PPN $\gamma$ bounds on the slip. We stress that, for any system with a well-posed Newtonian limit, Palatini $f(R)$ theories always have $\gamma =1$; while metric $f(R)$ theories can only have two values: either 1 or $1/2$. The extension toward Horndeski gravity shows no qualitative surprises, and ${\gamma }_{\mathrm{\Sigma }}$ is a constant in this context (only assuming that the Horndeski potentials can be approximated by analytical functions). This implies that a precise study on the bending of light for different impact parameters can in principle be used to rule out the complete Horndeski action as an action for gravity. Also, we comment on the consequences for $\gamma$ inferences at external galaxies.