Phenomenological functions Σ and μ, also known as Glight/G and Gmatter/G, are commonly used to parametrize 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 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 Σ and μ analytically within the quasistatic approximation and numerically by sampling the space of allowed solutions. We confirm that the conjecture made in [L. Pogosian and A. Silvestri, Phys. Rev. D 94, 104014 (2016)PRVDAQ2470-001010.1103/PhysRevD.94.104014], that (Σ-1)(μ-1)≥0 in viable Horndeski theories, holds very well. Along with that, we check the validity of the quasistatic 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 nontrivial signatures of modified gravity at the level of linear perturbations.
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