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. This scalar field is not the inflaton and hence does not contribute to the background dynamics. We focus on the quantization procedure and evolution in the preinflationary anisotropic background. Also, in this paper for simplicity the metric perturbations are not taken into account. The initial condition is set by the requirement that the scalar field is initially in an adiabatic state. Usually, in a quantum harmonic oscillator system, an adiabatic process implies the one where the potential changes slowly enough compared to its size, and the time evolution can be obtained from the zeroth order WKB approximation. 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 $k_3\ne 0$, 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.

Figure 1

The behavior of $ϵ(t)$ with respect to $x$. Here, $r_{\perp }^2=0.999$, 0.99, 0.9, 0.5, and 0.1, respectively, from the top and we choose $H/k=0.05$. The WKB approximation is valid if $ϵ(t)<1$. The matching time $t_{*}$ is almost independent of $k$.

Figure 2

The behavior of $ϵ(\mathcal{T})$. The horizontal axis represents $\mathcal{T}$. Here, we take $K=2$ and $J=30$. The parameter ${\tilde{t}}_0$ is chosen to be ${\tilde{t}}_0=-0.5$, $-0.3$, $-0.1$, $-0.05$, $-0.03$, $-0.01$ respectively from the top. As seen in this figure, the large $J$ approximation gives the best result at $\mathcal{T}\sim \sqrt{J}$ and gives a good approximation even for time $\mathcal{T}\sim J/2$.