Exact quantum states for the diagonal Bianchi type IX model with negative cosmological constant

TL;DR

This paper derives quantum states for the diagonal Bianchi type IX cosmological model with negative cosmological constant, revealing a caustic structure and analyzing the properties of specific solutions in relation to classical and quantum boundary conditions.

Contribution

It extends previous methods to the negative cosmological constant case, providing explicit solutions and analyzing their physical and boundary condition properties.

Findings

Five linearly independent solutions obtained.

A caustic separates Euclidean and Lorentzian regimes.

One solution is normalizable but does not satisfy no-boundary conditions.

Abstract

Quantum states of the diagonal Bianchi type IX model with negative cosmological constant $\Lambda$ are obtained by transforming the Chern-Simons solution in Ashtekar's variables to the metric representation. We apply our method developed earlier for $\Lambda>0$ and obtain five linearly independent solutions by using the complete set of topologically inequivalent integration contours in the required generalized Fourier-transformation. A caustic in minisuperspace separates two Euclidean regimes at small and large values of the scale parameter from a single classically interpretable Lorentzian regime in between, corresponding to the fact that classically these model-Universes recollapse. Just one particular solution out of the five we find gives a normalizable probability distribution on both branches of the caustic. However, in contrast to the case of positive cosmological constant, this particular solution neither satisfies the semi-classical no-boundary condition, nor does the special initial condition it picks out for $\hbar \to 0$ evolve into a classically interpretable Universe.