Tunnelling in Quantum Cosmology: WKB vs SWKB

TL;DR

This paper compares the WKB and SWKB methods for tunnelling in quantum cosmology, showing SWKB often provides more accurate tunnelling probabilities especially when WKB assumptions are invalid.

Contribution

It introduces analytic SWKB tunnelling expressions in quantum cosmology and demonstrates their advantages over WKB in certain parameter regimes.

Findings

SWKB and WKB agree under valid WKB conditions.

SWKB yields larger tunnelling probabilities when WKB assumptions break down.

SWKB is a useful complementary method for barrier transmission studies.

Abstract

The WKB approximation is a standard tool for studying tunnelling problems in quantum cosmology. We compare this method to the Supersymmetric WKB (SWKB) applied to a closed FRW minisuperspace model. We consider the transition from a dust towards a dark energy-dominated epoch can be explained by a generalized Chaplygin gas. Using analytic approximations for the superpotential (power-series and a Picard approximation), we derive closed-form SWKB tunnelling expressions and compute transmission probabilities as functions of the Chaplygin parameters $A$, $B$ and $\alpha$. Numerical root-finding locates classical turning points and numerical integration allows comparison with standard WKB results. We find that SWKB and WKB agree when the WKB validity condition holds, while the SWKB yields systematically larger (and plausibly more accurate) tunnelling probabilities for parameter values where the WKB assumptions break down. The results support the SWKB as a useful complementary approach for barrier-transmission studies in quantum cosmology.