On the evolution of density perturbations in f(R) theories of gravity

TL;DR

This paper derives the exact equations governing the evolution of density perturbations in f(R) gravity theories and evaluates the validity of the quasi-static approximation in different regimes.

Contribution

It provides a general derivation of the fourth-order differential equation for scalar perturbations in f(R) gravity and assesses the applicability of the quasi-static approximation.

Findings

Exact fourth-order differential equation for matter density perturbations.

Quasi-static approximation is valid for certain f(R) models describing accelerated expansion.

The approximation is not justified for all f(R) functions, especially in general cases.

Abstract

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 (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 functions adequately describing the present phase of accelerated expansion and satisfying local gravity tests, it provides a correct description for the evolution of perturbations.