  Both for the background world model and its linear perturbations Newtonian
cosmology coincides with the zero-pressure limits of relativistic cosmology.
However, such successes in Newtonian cosmology are not purely based on Newton's
gravity, but are rather guided ones by previously known results in Einstein's
theory. The action-at-a-distance nature of Newton's gravity requires further
verification from Einstein's theory for its use in the large-scale nonlinear
regimes. We study the domain of validity of the Newtonian cosmology by
investigating weakly nonlinear regimes in relativistic cosmology assuming a
zero-pressure and irrotational fluid. We show that, first, if we ignore the
coupling with gravitational waves the Newtonian cosmology is exactly valid even
to the second order in perturbation. Second, the pure relativistic correction
terms start appearing from the third order. Third, the correction terms are
independent of the horizon scale and are quite small in the large-scale near
the horizon. These conclusions are based on our special (and proper) choice of
variables and gauge conditions. In a complementary situation where the system
is weakly relativistic but fully nonlinear (thus, far inside the horizon) we
can employ the post-Newtonian approximation. We also show that in the
large-scale structures the post-Newtonian effects are quite small. As a
consequence, now we can rely on the Newtonian gravity in analyzing the
evolution of nonlinear large-scale structures even near the horizon volume.
