On the accuracy of N-body simulations at very large scales
  We examine the deviation of Cold Dark Matter particle trajectories from the
Newtonian result as the size of the region under study becomes comparable to or
exceeds the particle horizon. To first order in the gravitational potential,
the general relativistic result coincides with the Zel'dovich approximation and
hence the Newtonian prediction on all scales. At second order, General
Relativity predicts corrections which overtake the corresponding second order
Newtonian terms above a certain scale of the order of the Hubble radius.
However, since second order corrections are very much suppressed on such
scales, we conclude that simulations which exceed the particle horizon but use
Newtonian equations to evolve the particles, reproduce the correct trajectories
very well. The dominant relativistic corrections to the power spectrum on
scales close to the horizon are at most of the order of $\sim 10^{-5}$ at
$z=49$ and $\sim 10^{-3}$ at $z=0$. The differences in the positions of real
space features are affected at a level below $10^{-6}$ at both redshifts. Our
analysis also clarifies the relation of N-body results to relativistic
considerations.
