Tunnelling in Quantum Cosmology: WKB vs SWKB
This study analyses WKB and Supersymmetric WKB (SWKB) methods in a closed Friedmann-Robertson-Walker (FRW) minisuperspace model to determine whether they yield different results in modelling tunnelling phenomena in quantum cosmology. The transition from a dust-dominated to a dark-energy-dominated epoch driven by a generalised Chaplygin gas is examined. Incorporating supersymmetric quantum mechanics requires modifying the initial potential, influencing late-time evolution: after the dust-dominated phase, the universe enters a dark-energy regime that decays back into a dust-dominated period. Analytic approximations for the superpotential (power series and Picard approximation) yield closed-form SWKB tunnelling expressions, facilitating the calculation of transmission probabilities as functions of the Chaplygin parameters $A$, $B$, and $α$. Both SWKB and WKB methods apply for a concrete range of the scale factor $a$ and Chaplygin parameters, but quantitatively yield different tunnelling probabilities. They diverge for suppressed barriers: standard WKB underestimates tunnelling, while SWKB avoids turning-point singularities but overcompensates due to broken shape-invariance. A backward numerical integration method confirms that true transmission probabilities lie strictly between both semiclassical approximations ($T_{\mathrm{WKB}} < T_{\mathrm{num}} < T_{\mathrm{SWKB}}$). Crucially, by confirming that actual tunnelling probabilities exceed standard WKB estimates, these results demonstrate that quantum transitions into cosmological models with transient late-time acceleration are probabilistically more viable. Therefore, SWKB provides a valuable complementary approach for barrier transmission in quantum cosmology. Deriving a functional form for the decaying dark energy density will enable testing the SQM modified potential against current observational data.