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

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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.
Original language English 051 Journal of Cosmology and Astroparticle Physics 2015 07 https://doi.org/10.1088/1475-7516/2015/07/051 Published - 30 Jul 2015

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Rights statement: This is an author-created, un-copyedited version of an article accepted for publication/published in Journal of Cosmology and Astroparticle Physics. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at http://dx.doi.org/10.1088/1475-7516/2015/07/051.

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