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Dynamics of quantum anisotropies in a Taub Universe in the WKB approximation

We analyze the dynamics of a Taub cosmological model in the presence of a massless minimally coupled scalar field and a cosmological constant, in the limit when both the Universe volume and the scalar field live in a quasi-classical approximation. By other words, we study the dynamics of a quantum small anisotropy evolving on a de Sitter background and in the presence of a kinetic term of the inflaton field. We demonstrate that the quantum anisotropy exponentially decays during the Universe expansion, approaching a finite and small value. This result suggests that the quantum isotropization of the Universe during a de Sitter phase is much weaker than the corresponding classical evolution, favouring the survival of certain degree of anisotropy to the de Sitter phase. Finally we analyze the case when also the scalar field is considered as quantum variable, by showing how its variance naturally spreads because of no potential term significantly affects its dynamics. This behaviour results to be different from the anisotropy which is subjected to the potential coming out from the spatial curvature.