  Initial conditions for (Newtonian) cosmological N-body simulations are
usually set by re-scaling the present-day power spectrum obtained from linear
(relativistic) Boltzmann codes to the desired initial redshift of the
simulation. This back-scaling method can account for the effect of
inhomogeneous residual thermal radiation at early times, which is absent in the
Newtonian simulations. We analyse this procedure from a fully relativistic
perspective, employing the recently-proposed Newtonian motion gauge framework.
We find that N-body simulations for LambdaCDM cosmology starting from
back-scaled initial conditions can be self-consistently embedded in a
relativistic space-time with first-order metric potentials calculated using a
linear Boltzmann code. This space-time coincides with a simple "N-body gauge"
for z<50 for all observable modes. Care must be taken, however, when simulating
non-standard cosmologies. As an example, we analyse the back-scaling method in
a cosmology with decaying dark matter, and show that metric perturbations
become large at early times in the back-scaling approach, indicating a
breakdown of the perturbative description. We suggest a suitable "forwards
approach" for such cases.
