Perturbed redshifts from $N$-body simulations

In order to keep pace with the increasing data quality of astronomical surveys the observed source redshift has to be modeled beyond the well-known Doppler contribution. In this article I want to examine the gauge issue that is often glossed over when one assigns a perturbed redshift to simulated data generated with a Newtonian $N$-body code. A careful analysis reveals the presence of a correction term that has so far been neglected. It is roughly proportional to the observed length scale divided by the Hubble scale and therefore suppressed inside the horizon. However, on gigaparsec scales it can be comparable to the gravitational redshift and hence amounts to an important relativistic effect.