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

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 $k=10c^{-1}{\mathcal{H}}_0$.

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Cited by in corpus (8)

• Horndeski theory and beyond: a review
• Constraining $f(R)$ Gravity Theory Using Weak Lensing Peak Statistics from the Canada-France-Hawaii-Telescope Lensing Survey
• Matter bispectrum beyond Horndeski
• Spatial Curvature in $f(R)$ Gravity
• Equivalence between Scalar-Tensor theories and $f(R)$-gravity: From the action to Cosmological Perturbations
• Reconstruction of f(R) gravity models from observations
• Endowing $\mathbf{\Lambda }$ with a dynamic nature: constraints in a spatially curved Universe
• Scalar perturbation and density contrast evolution in $f(Q,C)$ gravity