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.