f(R) gravity on non-linear scales: The post-Friedmann expansion and the vector potential

TL;DR

This paper develops a post-Friedmann framework for non-linear scales in f(R) gravity, deriving equations for scalar and vector potentials, and measures the vector potential from simulations, showing it remains small and Newtonian approximation is valid.

Contribution

It introduces a coherent post-Friedmann approach to f(R) gravity on non-linear scales, including the vector potential, and provides simulation-based measurements of this potential.

Findings

Vector potential can be up to 50% larger than in GR at small scales and redshift zero.

The Newtonian approximation remains highly accurate for f(R) gravity.

Non-linear behavior is primarily described by scalar potentials and the f(R) field.

Abstract

Many modified gravity theories are under consideration in cosmology as the source of the accelerated expansion of the universe and linear perturbation theory, valid on the largest scales, has been examined in many of these models. However, smaller non-linear scales offer a richer phenomenology with which to constrain modified gravity theories. Here, we consider the Hu-Sawicki form of $f(R)$ gravity and apply the post-Friedmann approach to derive the leading order equations for non-linear scales, i.e. the equations valid in the Newtonian-like regime. We reproduce the standard equations for the scalar field, gravitational slip and the modified Poisson equation in a coherent framework. In addition, we derive the equation for the leading order correction to the Newtonian regime, the vector potential. We measure this vector potential from $f(R)$ N-body simulations at redshift zero and one, for two values of the $f_{R_0}$ parameter. We find that the vector potential at redshift zero in $f(R)$ gravity can be close to 50\% larger than in GR on small scales for $|f_{R_0}|=1.289\times10^{-5}$, although this is less for larger scales, earlier times and smaller values of the $f_{R_0}$ parameter. Similarly to in GR, the small amplitude of this vector potential suggests that the Newtonian approximation is highly accurate for $f(R)$ gravity, and also that the non-linear cosmological behaviour of $f(R)$ gravity can be completely described by just the scalar potentials and the $f(R)$ field.