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Scalar field in the anisotropic universe

We discuss the primordial spectrum of a massless and minimally coupled scalar field, produced during the initial anisotropic epoch before the onset of inflation. We consider two models of the anisotropic cosmology, the (planar) Kasner de Sitter solution (Bianchi I) and the Taub-NUT de Sitter solution (Bianchi IX), where the 3-space geometry is initially anisotropic, followed by the de Sitter phase due to the presence of a positive cosmological constant. We discuss the behavior of a quantized, massless and minimally coupled scalar field in the anisotropic stage. We focus on the quantization procedure and evolution in the pre-inflationary anisotropic background. The initial condition is set by the requirement that the scalar field is initially in an adiabatic state. In our case, such a vacuum state exists only for limited solutions of the anisotropic Universe, whose spacetime structure is regular in the initial times. In this paper, we call our adiabatic vacuum state the anisotropic vacuum. In the Kasner de Sitter model, for one branch of planar solutions there is an anisotropic vacuum unless $k_3\neq 0$, where $k_3$ is the comoving momentum along the third direction, while in the other branch there is no anisotropic vacuum state. In the first branch, for the moderate modes, $k_3\sim k$, where $k$ is the total comoving momentum, the scalar power spectrum has an oscillatory behavior and its direction dependence is suppressed. For the planar modes, $k_3\ll k$, in contrast, the direction dependence becomes more important, because of the amplification of the scalar amplitude during this interval of the violation of WKB approximation in the initial anisotropic stage. The qualitative behaviors in the Taub-NUT de Sitter models are very similar to the case of the first branch of the planar Kasner de Sitter model.