Matter density perturbations and effective gravitational constant in modified gravity models of dark energy

We derive the equation of matter density perturbations on subhorizon scales for a general Lagrangian density f(R,phi,X) that is a function of a Ricci scalar R, a scalar field phi, and a kinetic term X=-(del phi)(2)/2. This is useful to constrain modified gravity dark energy models from observations of large-scale structure and weak lensing. We obtain the solutions for the matter perturbation delta(m) as well as the gravitational potential Phi for some analytically solvable models. In an f(R) dark energy model with the Lagrangian density f(R)=alpha R1+m-Lambda, the growth rates of perturbations exhibit notable differences from those in the standard Einstein gravity unless m is very close to 0. In scalar-tensor models with the Lagrangian density f=F(phi)R+2p(phi,X), we relate the models with coupled dark energy scenarios in the Einstein frame and reproduce the equations of perturbations known in the current literature by making a conformal transformation. We also estimate the evolution of perturbations in both Jordan and Einstein frames when the energy fraction of dark energy is constant during the matter-dominated epoch.