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,φ,X) that is a function of a Ricci scalar R, a scalar field φ, and a kinetic term X=-(∇φ)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 δₘ as well as the gravitational potential Φ for some analytically solvable models. In an $f(R)$ dark energy model with the Lagrangian density f(R)=αR¹⁺ᵐ-Λ, 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(φ)R+2p(φ,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.