Second order solutions of cosmological perturbation in the matter
  dominated era
  We present the growing mode solutions of cosmological perturbations to the
second order in the matter dominated era. We also present several
gauge-invariant combinations of perturbation variables to the second order in
most general fluid context. Based on the solutions we study the Newtonian
correspondence of relativistic perturbations to the second order. In addition
to the previously known exact relativistic/Newtonian correspondence of density
and velocity perturbations to the second order in the comoving gauge, here we
show that in the sub-horizon limit we have the correspondences for density,
velocity and potential perturbations in the zero-shear gauge and in the
uniform-expansion gauge to the second order. Density perturbation in the
uniform curvature gauge also shows the correspondence to the second order in
the sub-horizon scale. We also identify the relativistic gravitational
potential which shows exact correspondence to the Newtonian one to the second
order.
