Biaxial Bianchi type IX quantum cosmology

Abstract

We investigate the quantum cosmology of spatially homogeneous models with compact spatial sections admitting a u(2) isometry algebra. The metric ansatz in these models is that of Bianchi type IX with two scale factors set to be equal. We apply the Hartle-Hawking no-boundary path integral prescription and find the semi-classical contributions to the wave function. Exact formulae are obtainable for certain contributions and otherwise of large and small anisotropy (for the pure vacuum case) and large spatial volume or small anisotropy (for the case with a positive cosmological constant) are considered. For the pure vacuum case we find no rapidly oscillating semiclassical components in the wave function, and hence do recover lorentzian space-time as a prediction of the no-boundary proposal. For the case with a cosmological constant the wave function does contain rapidly oscillating components and thus predicts approximately lorentzian space-times; however, one of the parameters in these lorentzian solutions is formally constrained to be purely imaginary. The possible implications of this imaginary parameter for the breakdown of the lorentzian semiclassical approximation and for the predictions of the no-boundary proposal are discussed.