Tunnelling in Quantum Cosmology: WKB vs SWKB

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.