Cosmological Perturbations in General Modified Gravity Theories

We derive the equations of linear cosmological perturbations for the general Lagrangian density f</it> (R</it>,ϕ, X</it>)/2 + L</it><inf>c</inf>, where R</it> is a Ricci scalar, ϕ is a scalar field, and X</it> = -∂µ ϕ∂<inf>µ</inf> ϕ/2 is a field kinetic energy. We take into account a nonlinear self-interaction term L</it><inf>c</inf> = ξ(ϕ) □ϕ(∂µ ϕ∂<inf>µ</inf> ϕ), 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.