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.
