Quantum dynamics of the corner of the Bianchi IX model in the WKB approximation

In this paper we analyze the Bianchi IX Universe dynamics within the corner region associated to the potential term which the spatial curvature induces in the minisuperspace. We investigate the dynamics in terms of the WKB scenario: the isotropic Misner variable (α) and one of the two anisotropic variables (β+) are treated as semiclassical, while the remaining anisotropy (β-) is described on a pure quantum level. The quantum dynamics always reduces to the one of a time-dependent Schrödinger equation for a harmonic potential with a time dependent frequency. The study is done in the vacuum and in the presence of a massless scalar field φ and a cosmological constant term Λ. The vacuum case is treated in the limits of a collapsing and an expanding Universe, while the dynamics in presence of φ and Λ is studied only for t→∞. In both analyses the quantum dynamics of the anisotropy variable β- suggests a suppression of the quantum anisotropy associated. In the vacuum case the corner configuration becomes an attractor for the dynamics and the evolution resembles that of a Taub cosmology in the limit of a nonsingular initial Universe. This suggests that if the Bianchi dynamics enters deeply enough in the potential corner the initial singularity is removed and a Taub picture emerges. The case when φ is present well mimics the de Sitter phase of an inflationary Universe. Here we show that both the classical and quantum anisotropies are exponentially suppressed and therefore the resulting dynamics corresponds to an isotropic closed Robertson-Walker geometry.