What can cosmology tell us about gravity? Constraining Horndeski gravity with Σ and μ

Phenomenological functions Sigma and mu (also known as G(light)/G and G(matter)/G) are commonly used to parametrize 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 the 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 not equal 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 disfavor 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 subclasses of Horndeski theories that can be ruled out based on the measured difference between Sigma and mu.