  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.
