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In the Horndeski's most general scalar-tensor theories the equations ofscalar density perturbations are derived in the presence of non-relativisticmatter minimally coupled to gravity. Under a quasi-static approximation onsub-horizon scales we obtain the effective gravitational coupling $G_{eff}$associated with the growth rate of matter perturbations as well as theeffective gravitational potential $\Phi_{eff}$ relevant to the deviation oflight rays. We then apply our formulas to a number of modified gravitationalmodels of dark energy--such as those based on f(R) theories, Brans-Dicketheories, kinetic gravity braidings, covariant Galileons, and field derivativecouplings with the Einstein tensor. Our results are useful to test thelarge-distance modification of gravity from the future high-precisionobservations of large-scale structure, weak lensing, and cosmic microwavebackground.

Effective gravitational couplings for cosmological perturbations in the most general scalar-tensor theories with second-order field equations