Relativistic Interpretation of Newtonian Simulations for Cosmic Structure Formation

The standard numerical tools for studying non-linear collapse of matter are Newtonian $N$-body simulations. Previous work has shown that these simulations are in accordance with General Relativity (GR) up to first order in perturbation theory, provided that the effects from radiation can be neglected. In this paper we show that the present day matter density receives more than 1$\%$ corrections from radiation on large scales if Newtonian simulations are initialised before $z=50$. We provide a relativistic framework in which \emph{unmodified} Newtonian simulations are compatible with linear GR even in the presence of radiation. Our idea is to use GR perturbation theory to keep track of the evolution of relativistic species and the relativistic space-time consistent with the Newtonian trajectories computed in $N$-body simulations. If metric potentials are sufficiently small, they can be computed using a first-order Einstein--Boltzmann code such as CLASS. We make this idea rigorous by defining a class of GR gauges, the \emph{Newtonian motion} gauges, which are defined such that matter particles follow Newtonian trajectories. We construct a simple example of a relativistic space-time within which unmodified Newtonian simulations can be interpreted