Testing the quasi-static approximation in $f(R)$ gravity simulations

Numerical simulations in modified gravity have commonly been performed under the quasi-static approximation -- that is, by neglecting the effect of time derivatives in the equation of motion of the scalar field that governs the fifth force in a given modified gravity theory. To test the validity of this approximation, we analyse the case of $f(R)$ gravity beyond this quasi-static limit, by considering effects, if any, these terms have in the matter and velocity divergence cosmic fields. To this end, we use the adaptive mesh refinement code ECOSMOG to study three variants ($|f_{R}|= 10^{-4}[$F4$], 10^{-5}[$F5$]$ and $10^{-6}[$F6$]$) of the Hu-Sawicki $f(R)$ gravity model, each of which refers to a different magnitude for the scalar field that generates the fifth force. We find that for F4 and F5, which show stronger deviations from standard gravity, a low-resolution simulation is enough to conclude that time derivatives make a negligible contribution to the matter distribution. The F6 model shows a larger deviation from the quasi-static approximation, but one that diminishes when re-simulated at higher resolution. We therefore come to the conclusion that the quasi-static approximation is valid for the most practical applications in $f(R)$ cosmologies.