Cosmological perturbations and quasistatic assumption in f (R) theories

f(R) gravity is one of the simplest theories of modified gravity to explain the accelerated cosmic expansion. Although it is usually assumed that the quasi-Newtonian approach (a combination of the quasistatic approximation and sub-Hubble limit) for cosmic perturbations is good enough to describe the evolution of large scale structure in f(R) models, some studies have suggested that this method is not valid for all f(R) models. Here, we show that in the matter-dominated era, the pressure and shear equations alone, which can be recast into four first-order equations to solve for cosmological perturbations exactly, are sufficient to solve for the Newtonian potential, Ψ, and the curvature potential, Φ. Based on these two equations, we are able to clarify how the exact linear perturbations fit into different limits. We find that the Compton length controls the quasistatic behaviors in f(R) gravity. In addition, regardless the validity of quasistatic approximation, a strong version of the sub-Hubble limit alone is sufficient to reduce the exact linear perturbations in any viable f(R) gravity to second order. Our findings disagree with some previous studies where we find little difference between our exact and quasi-Newtonian solutions even up to k=10c-1H0.