On the functional determinant of a special operator with a zero mode in cosmology

TL;DR

This paper computes the functional determinant of a specific quantum operator with a zero mode in cosmology, using gauge fixing, relevant for one-loop quantum cosmology calculations involving oscillating instantons.

Contribution

It introduces a method to evaluate the functional determinant of a special operator with a zero mode in a cosmological context, accounting for its unique boundary conditions and zero mode structure.

Findings

Successfully calculated the determinant for an operator with a zero mode with two roots.

Applied the method to cosmological instantons with oscillating scale factors.

Provides a tool for one-loop quantum cosmology computations.

Abstract

The functional determinant of a special second order quantum-mechanical operator is calculated with its zero mode gauged out by the method of the Faddeev-Popov gauge fixing procedure. This operator subject to periodic boundary conditions arises in applications of the early Universe theory and, in particular, determines the one-loop statistical sum in quantum cosmology generated by a conformal field theory (CFT). The calculation is done for a special case of a periodic zero mode of this operator having two roots (nodes) within the period range, which corresponds to the class of cosmological instantons in the CFT driven cosmology with one oscillation of the cosmological scale factor of its Euclidean Friedmann-Robertson-Walker metric.