Effective gravitational couplings for cosmological perturbations in the
  most general scalar-tensor theories with second-order field equations
  In the Horndeski's most general scalar-tensor theories the equations of
scalar density perturbations are derived in the presence of non-relativistic
matter minimally coupled to gravity. Under a quasi-static approximation on
sub-horizon scales we obtain the effective gravitational coupling $G_{eff}$
associated with the growth rate of matter perturbations as well as the
effective gravitational potential $\Phi_{eff}$ relevant to the deviation of
light rays. We then apply our formulas to a number of modified gravitational
models of dark energy--such as those based on f(R) theories, Brans-Dicke
theories, kinetic gravity braidings, covariant Galileons, and field derivative
couplings with the Einstein tensor. Our results are useful to test the
large-distance modification of gravity from the future high-precision
observations of large-scale structure, weak lensing, and cosmic microwave
background.
