Cosmological Perturbations and Quasi-Static Assumption in $f(R)$ Theories

TL;DR

This paper investigates the validity of the quasi-static approximation in $f(R)$ gravity models, showing that it generally holds under certain conditions and clarifying the role of the Compton length in the behavior of cosmological perturbations.

Contribution

It provides an exact formulation of linear perturbations in $f(R)$ gravity using pressure and shear equations, and clarifies the conditions under which the quasi-static approximation is valid.

Findings

The pressure and shear equations suffice for exact perturbation solutions.

The Compton length controls quasi-static behavior in $f(R)$ gravity.

A strong sub-Hubble limit reduces exact solutions to second order regardless of approximation validity.

Abstract

$f(R)$ 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 quasi-static 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, $\Psi$, and the curvature potential, $\Phi$. 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 quasi-static behaviours in $f(R)$ gravity. In addition, regardless the validity of quasi-static 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 $k=10 c^{-1} \mathcal{H}_0$.