Cosmological density perturbations in modified gravity theories

In the context of f(R) theories of gravity, we study the cosmological evolution of scalar perturbations by using a completely general procedure. We find that the exact fourth‐order differential equation for the matter density perturbations in the longitudinal gauge, reduces to a second‐order equation for sub‐Hubble modes. This simplification is compared with the standard (quasi‐static) equation used in the literature. We show that for general f(R) functions the quasi‐static approximation is not justified. However for those f(R) adequately describing the present phase of accelerated expansion and satisfying local gravity tests, it does give a correct description for the evolution of perturbations.