Scalar perturbations in cosmological $f(R)$ models: the cosmic screening approach

We investigate cosmological perturbations for nonlinear $f(R)$ models within the cosmic screening approach. Matter is considered both in the form of a set of discrete point-like massive bodies and in the form of a continuous pressureless perfect fluid. We perform full relativistic analysis of the first-order theory of scalar perturbations for arbitrary nonlinear $f(R)$ models and demonstrate that scalar potentials $\Phi(t,\mathbf{r})$ and $\Psi(t,\mathbf{r})$ are determined by a system of only two master equations. Our equations are applicable at all spatial scales as long as the approximation $\delta R/\bar R \ll 1$ (which is usually assumed in studies devoted to cosmological perturbations in $f(R)$ models) works.