Cosmological particle production and the precision of the WKB approximation

TL;DR

This paper analyzes the limits of the WKB approximation in quantum field theory in curved spacetime, showing that optimal truncation yields exponentially small errors but cannot fully capture vacuum particle production, which is also exponentially small.

Contribution

It derives a novel formula for the optimal number of terms in the WKB series and demonstrates how to improve the approximation to match particle production.

Findings

Optimal WKB series truncation error is exponentially small.

Imprecision in particle number definition matches expected particle production.

Improved WKB methods can enhance approximation accuracy.

Abstract

Particle production by slow-changing gravitational fields is usually described using quantum field theory in curved spacetime. Calculations require a definition of the vacuum state, which can be given using the adiabatic (WKB) approximation. I investigate the best attainable precision of the resulting approximate definition of the particle number. The standard WKB ansatz yields a divergent asymptotic series in the adiabatic parameter. I derive a novel formula for the optimal number of terms in that series and demonstrate that the error of the optimally truncated WKB series is exponentially small. This precision is still insufficient to describe particle production from vacuum, which is typically also exponentially small. An adequately precise approximation can be found by improving the WKB ansatz through perturbation theory. I show quantitatively that the fundamentally unavoidable imprecision in the definition of particle number in a time-dependent background is equal to the particle production expected to occur during that epoch. The results are illustrated by analytic and numerical examples.