Consistency relations between the source terms in the second-order
  Einstein equations for cosmological perturbations
  In addition to the second-order Einstein equations on four-dimensional
homogeneous isotropic background universe filled with the single perfect fluid,
we also derived the second-order perturbations of the continuity equation and
the Euler equation for a perfect fluid in gauge-invariant manner without
ignoring any mode of perturbations. The consistency of all equations of the
second-order Einstein equation and the equations of motion for matter fields is
confirmed. Due to this consistency check, we may say that the set of all
equations of the second-order are self-consistent and they are correct in this
sense.
