Cosmological post-Newtonian approximation compared with perturbation
  theory
  We compare the cosmological first-order post-Newtonian (1PN) approximation
with the relativistic cosmological linear perturbation theory in a
zero-pressure medium with the cosmological constant. We compare equations and
solutions in several different gauge conditions available in both methods. In
the PN method we have perturbation equations for density, velocity and
gravitational potential independently of the gauge condition to 1PN order.
However, correspondences with these 1PN equations are available only in certain
gauge conditions in the perturbation theory. Equations of perturbed velocity
and the perturbed gravitational potential in the zero-shear gauge exactly
coincide with the Newtonian equations which remain valid even to 1PN order (the
same is true for perturbed velocity in the comoving gauge), and equations of
perturbed density in the zero-shear gauge and the uniform-expansion gauge
coincide to 1PN order. We identify other correspondences available in different
gauge conditions of the perturbation theory.
