Density perturbations in general modified gravitational theories

We derive the equations of linear cosmological perturbations for the general Lagrangian density f(R, phi, X)/2 + L-c, where R is a Ricci scalar, phi is a scalar field, and X = -partial derivative(mu) phi partial derivative(mu) phi/2 is a field kinetic energy. We take into account a nonlinear self-interaction term L-c = xi(phi)square phi(partial derivative(mu) phi partial derivative(mu) phi) recently studied in the context of "Galileon'' cosmology, 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 quasistatic approximation on subhorizon scales. We also derive conditions for the avoidance of ghosts and Laplacian instabilities associated with propagation speeds. Our analysis includes most of modified gravity models of dark energy proposed in literature; and thus it is convenient to test the viability of such models from both theoretical and observational points of view.