Cosmological perturbations in general modified gravity theories

We derive the equations of linear cosmological perturbations for the general Lagrangian density f(R, φ,X)/2+L_c, where R is a Ricci scalar, φ is a scalar field, and X = - δ^μφδ_μφ/2 is a field kinetic energy. We take into account a nonlinear self-interaction term L_c = ξ(φ)φ(δ^μφδ_μφ), which keeps the field equations at second order. Taking into account a scalar-field mass explicitly, the equations of matter density perturbations and gravitational potentials are obtained under a quasi-static approximation on sub-horizon scales. We also derive conditions for the avoidance of ghosts and Laplacian instabilities associated with propagation speeds.