# The no-boundary proposal in biaxial Bianchi IX minisuperspace

## TL;DR

This paper implements and extends the no-boundary proposal in a Bianchi IX minisuperspace model, demonstrating its consistency, normalizability, and relation to tunneling wave functions, with implications for quantum cosmology.

## Contribution

It provides an exact solvable model of the no-boundary wave function including topology contributions and clarifies the role of contour choices and saddle points in its definition.

## Key findings

- Wave function is normalizable and predicts low amplitude for large anisotropies.
- In the isotropic limit, it recovers the Hartle-Hawking wave function.
- The tunneling wave function is essentially equivalent to the no-boundary state in this model.

## Abstract

We implement the no-boundary proposal for the wave function of the universe in an exactly solvable Bianchi IX minisuperspace model with two scale factors. We extend our earlier work (Phys. Rev. Lett. 121, 081302, 2018 / arXiv:1804.01102) to include the contribution from the $\mathbb{C}\text{P}^2 \setminus B^4$ topology. The resulting wave function yields normalizable probabilities and thus fits into a predictive framework for semiclassical quantum cosmology. We find that the amplitude is low for large anisotropies. In the isotropic limit the usual Hartle-Hawking wave function for the de Sitter minisuperspace model is recovered. Inhomogeneous perturbations in an extended minisuperspace are shown to be initially in their ground state. We also demonstrate that the precise mathematical implementation of the no-boundary proposal as a functional integral in minisuperspace depends on detailed aspects of the model, including the choice of gauge-fixing. This shows in particular that the choice of contour cannot be fundamental, adding weight to the recent proposal that the semiclassical no-boundary wave function should be defined solely in terms of a collection of saddle points. We adopt this approach in most of this paper. Finally we show that the semiclassical tunneling wave function of the universe is essentially equal to the no-boundary state in this particular minisuperspace model, at least in the subset of the classical domain where the former is known.