Evolution of density perturbations in f(R) theories of gravity

In the context of f(R) theories of gravity, we study the evolution of scalar cosmological perturbations in the metric formalism. Using a completely general procedure, we find the exact fourth-order differential equation for the matter density perturbations in the longitudinal gauge. In the case of sub-Hubble modes, the expression reduces to a second-order equation which is compared with the standard (quasistatic) equation used in the literature. We show that for general f(R) functions the quasistatic approximation is not justified. However, for those functions adequately describing the present phase of accelerated expansion and satisfying local gravity tests, it provides a correct description for the evolution of perturbations.