Matter Power Spectra in Modified Gravity: A Comparative Study of Approximations and $N$-Body Simulations

Testing gravity and the concordance model of cosmology, $\Lambda$CDM, at large scales is a key goal of this decade's largest galaxy surveys. Here we present a comparative study of dark matter power spectrum predictions from different numerical codes in the context of three popular theories of gravity that induce scale-independent modifications to the linear growth of structure: nDGP, Cubic Galileon and K-mouflage. In particular, we compare the predictions from full $N$-body simulations, two $N$-body codes with approximate time integration schemes, a parametrised modified $N$-body implementation and the analytic halo model reaction approach. We find the modification to the $\Lambda$CDM spectrum is in $2\%$ agreement for $z\leq1$ and $k\leq 1~h/{\rm Mpc}$ over all gravitational models and codes, in accordance with many previous studies, indicating these modelling approaches are robust enough to be used in forthcoming survey analyses under appropriate scale cuts. We further make public the new code implementations presented, specifically the halo model reaction K-mouflage implementation and the relativistic Cubic Galileon implementation.