  It is well known that couplings occur among the scalar-, vector-, and
tensor-type perturbations of Friedmann world model in the second perturbational
order. Here, we prove that, except for the gravitational wave contribution, the
relativistic zero-pressure irrotational fluid perturbed to second order in a
flat Friedmann background coincides exactly with the Newtonian result. Since we
include the cosmological constant, our results are relevant to currently
favoured cosmology. As we prove that the Newtonian hydrodynamic equations are
valid in all cosmological scales to the second order, our result has an
important practical implication that one can now use the large-scale Newtonian
numerical simulation more reliably even as the simulation scale approaches and
even goes beyond the horizon. That is, our discovery shows that, in the
zero-pressure case, except for the gravitational wave contribution, there are
no relativistic correction terms even near and beyond the horizon to the
second-order perturbation.
