Large-scale structure phenomenology of viable Horndeski theories

Abstract: Phenomenological functions $\Sigma$ and $\mu$, also known as $G_{\rm light}/G$ and $G_{\rm matter}/G$, are commonly used to parameterize modifications of the growth of large-scale structure in alternative theories of gravity. We study the values these functions can take in Horndeski theories, i.e. the class of scalar-tensor theories with second order equations of motion. We restrict our attention to models that are in a broad agreement with tests of gravity and the observed cosmic expansion history. In particular, we require the speed of gravity to be equal to the speed of light today, as required by the recent detection of gravitational waves and electromagnetic emission from a binary neutron star merger. We examine the correlations between the values of $\Sigma$ and $\mu$ analytically within the quasi-static approximation, and numerically, by sampling the space of allowed solutions. We confirm that the conjecture made in [Pogosian:2016pwr], that $(\Sigma-1)(\mu -1) \ge 0$ in viable Horndeski theories, holds very well. Along with that, we check the validity of the quasi-static approximation within different corners of Horndeski theory. Our results show that, even with the tight bound on the present day speed of gravitational waves, there is room within Horndeski theories for non-trivial signatures of modified gravity at the level of linear perturbations.