Revisiting the spectrum of a scalar field in an anisotropic universe

We revisit the issue on signatures of pre-inflationary background anisotropy by considering the quantization of a massless and minimally coupled scalar field in an axially symmetric Kasner background, mimicking cosmological perturbations. We show that the power spectrum of the scalar field fluctuation has a negligible difference from the standard inflation in the non-planar directions, but it has a sharp peak around the symmetry plane. For the non-planar high-momentum modes, we use the WKB approximation for the first period and the asymptotic approximation based on the de Sitter solution for the next period. At the boundary, two mode functions have the same accuracy with error of $O({H_{i}}/k)$. We calculate the approximation up to the order of $({H_{i}}/k)^6$ and show that the power spectrum of the scalar field fails to get corrections until we execute the approximation up to $6^{\rm th}$ order.