What can Cosmology tell us about Gravity? Constraining Horndeski with Sigma and Mu

Phenomenological functions $\Sigma$ and $\mu$ (also known as $G_{\rm light}/G$ and $G_{\rm matter}/G$) are commonly used to parameterize possible modifications of the Poisson equation relating the matter density contrast to the lensing and the Newtonian potentials, respectively. They will be well constrained by future surveys of large scale structure. But what would the implications of measuring particular values of these functions be for modified gravity theories? We ask this question in the context of general Horndeski class of single field scalar-tensor theories with second order equations of motion. We find several consistency conditions that make it possible to rule out broad classes of theories based on measurements of $\Sigma$ and $\mu$ that are independent of their parametric forms. For instance, a measurement of $\Sigma \ne 1$ would rule out all models with a canonical form of kinetic energy, while finding $\Sigma-1$ and $\mu-1$ to be of opposite sign would strongly disfavour the entire class of Horndeski models. We separately examine the large and the small scale limits, the possibility of scale-dependence, and the consistency with bounds on the speed of gravitational waves. We identify sub-classes of Horndeski theories that can be ruled out based on the measured difference between $\Sigma$ and $\mu$.