  Cosmological N-body simulations are now being performed using Newtonian
gravity on scales larger than the Hubble radius. It is well known that a
uniformly expanding, homogeneous ball of dust in Newtonian gravity satisfies
the same equations as arise in relativistic FLRW cosmology, and it also is
known that a correspondence between Newtonian and relativistic dust cosmologies
continues to hold in linearized perturbation theory in the marginally
bound/spatially flat case. Nevertheless, it is far from obvious that Newtonian
gravity can provide a good global description of an inhomogeneous cosmology
when there is significant nonlinear dynamical behavior at small scales. We
investigate this issue in the light of a perturbative framework that we have
recently developed, which allows for such nonlinearity at small scales. We
propose a relatively straightforward "dictionary"---which is exact at the
linearized level---that maps Newtonian dust cosmologies into general
relativistic dust cosmologies, and we use our "ordering scheme" to determine
the degree to which the resulting metric and matter distribution solve
Einstein's equation. We find that Einstein's equation fails to hold at "order
1" at small scales and at "order $\epsilon$" at large scales. We then find the
additional corrections to the metric and matter distribution needed to satisfy
Einstein's equation to these orders. While these corrections are of some
interest in their own right, our main purpose in calculating them is that their
smallness should provide a criterion for the validity of the original
dictionary (as well as simplified versions of this dictionary). We expect that,
in realistic Newtonian cosmologies, these additional corrections will be very
small; if so, this should provide strong justification for the use of Newtonian
simulations to describe relativistic cosmologies, even on scales larger than
the Hubble radius.
