A Non-Singular One-Loop Wave Function of the Universe From a New Eigenvalue Asymptotics in Quantum Gravity

TL;DR

This paper derives approximate eigenvalue formulas for scalar perturbations in Euclidean quantum gravity on a four-ball, leading to a regularized zeta-function and a vanishing one-loop wave function at small three-geometry, indicating quantum singularity avoidance.

Contribution

It introduces new analytic approximations for eigenvalues of scalar perturbations in quantum gravity with boundary conditions, enabling the first derivation of a vanishing one-loop wave function.

Findings

Eigenvalue approximations for Bessel function roots across all orders.

Construction of a regularized zeta-function confirming regularity at the origin.

Discovery of a zero one-loop wave function in the small-geometry limit.

Abstract

Recent work on Euclidean quantum gravity on the four-ball has proved regularity at the origin of the generalized zeta-function built from eigenvalues for metric and ghost modes, when diffeomorphism-invariant boundary conditions are imposed in the de Donder gauge. The hardest part of the analysis involves one of the four sectors for scalar-type perturbations, the eigenvalues of which are obtained by squaring up roots of a linear combination of Bessel functions of integer adjacent orders, with a coefficient of linear combination depending on the unknown roots. This paper obtains, first, approximate analytic formulae for such roots for all values of the order of Bessel functions. For this purpose, both the descending series for Bessel functions and their uniform asymptotic expansion at large order are used. The resulting generalized zeta-function is also built, and another check of regularity at the origin is obtained. For the first time in the literature on quantum gravity on manifolds with boundary, a vanishing one-loop wave function of the Universe is found in the limit of small three-geometry, which suggests a quantum avoidance of the cosmological singularity driven by full diffeomorphism invariance of the boundary-value problem for one-loop quantum theory.