Cosmological perturbations and quasistatic assumption inf(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, 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 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 H0.

Horndeski theory and beyond: a review

“…See Refs. [151,152,153] for the limits of the quasi-static assumption and Refs. [150,154,155] for the complete expression of the equations (without using the quasi-static approximation).…”

Spatial Curvature in $f(R)$ Gravity

“…Under the sub-Hubble and 4 Sub-Hubble approximation: we assume that the relevant modes are well within the Hubble radius (the 'horizon') during the time of interest i.e. they have k † H. Quasi-static approximation: can be stated as the condition that |Y (τ )| H|Y |, where Y = Φ, Ψ, H, Φ (τ ), Ψ (τ ) or H (τ ) [61,62]. In other words, the temporal evolution of Y may essentially be attributed to the expansion of the Universe [63], and is thus negligible in comparison to any spatial changes (in Y ).…”

Turnaround radius in modified gravity

“…The parametrized post-Friedmannian approach [27] describes perturbations of a FLRW universe in theories of gravity alternative to General Relativity. The line element (1) below is a rather general parametrization of the metric describing perturbed FLRW universes in modified gravity [28,29,30] (it holds, for example, in f (R) gravity [31,32]). The spacetime metric in the conformal Newtonian gauge is [33,34,35,36,37,38]…”