Novel phenomena of the Hartle-Hawking wave function

TL;DR

This paper numerically investigates the Hartle-Hawking wave function, revealing novel phenomena and providing an alternative perspective on the no-boundary proposal, with results consistent with Euclidean and analytical methods.

Contribution

It introduces a numerical solution to the Wheeler-DeWitt equation under $O(4)$ symmetry, uncovering new features of the Hartle-Hawking wave function and discussing its interpretations.

Findings

Numerical solutions align with Euclidean computations.

The wave function's shape matches analytical approximations.

Novel differences in probability computations are identified.

Abstract

We find a novel phenomenon in the solution to the Wheeler-DeWitt equation by solving numerically the equation assuming $O(4)$-symmetry and imposing the Hartle-Hawking wave function as a boundary condition. In the slow-roll limit, as expected, the numerical solution gives the most dominant steepest-descent that describes the probability distribution for the initial condition of a universe. The probability is consistent with the Euclidean computations, and the overall shape of the wave function is compatible with analytical approximations, although there exist novel differences in the detailed probability computation. Our approach gives an alternative point of view of the no-boundary wave function from the wave function point of view. Possible interpretations and conceptual issues of this wave function are discussed.