Gradient expansion approach to nonlinear superhorizon perturbations

Using the gradient expansion approach, we formulate a nonlinear cosmological perturbation theory on super-horizon scales valid to $O(\epsilon^2)$, where $\epsilon$ is the expansion parameter associated with a spatial derivative. For simplicity, we focus on the case of a single perfect fluid, but we take into account not only scalar but also vector and tensor modes. We derive the general solution under the uniform-Hubble time-slicing. In doing so, we identify the scalar, vector and tensor degrees of freedom contained in the solution. We then consider the coordinate transformation to the synchronous gauge in order to compare our result with the previous result given in the literature. In particular, we find that the tensor mode is invariant to $O(\epsilon^2)$ under the coordinate transformation.