Application of gradient expansion to non-linear gravitational wave in plane-symmetric universe

As an application of a gradient expansion (long-wavelength) approximation, we study the inhomogeneous universe including the gravitational wave (GW). For a plane-symmetric cosmological model, we implement the 2nd order expansion and analyze the purely non-linear perturbation of GW. The result indicates that the non-linear effect plays an important role in the anisotropic stage of the universe. We also confirm that the perturbation from the linear source term in the gradient expansion is consistent with the results of linear perturbation.