Tunnelling in Quantum Cosmology: WKB vs SWKB

TL;DR

The paper addresses barrier tunnelling in quantum cosmology using a closed FRW minisuperspace with a generalized Chaplygin gas, contrasting the SWKB approach with standard WKB. It develops analytic superpotential approximations via Power-Series and Picard methods to obtain SWKB tunnelling expressions and computes transmission probabilities $T(A,B,\alpha)$, showing SWKB matches WKB where WKB is valid but generally yields larger probabilities when WKB fails. The work demonstrates that SWKB can be a valuable complementary tool for barrier transmission in quantum cosmology and provides insights into how SUSY-inspired potentials influence tunnelling and possibly late-time cosmology. It also discusses limitations of the approximations and outlines directions for refinement and physical interpretation, including potential implications for avoiding or moderating late-time acceleration scenarios.

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 $α$. 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.

Figures (4)

• Figure 1: Generalized Chaplygin gas potential, in orange, and in green, the approximation, for $A=0.01$, $B=1$ and $\alpha=1$. We see that the main features of the original potential, namely the inicial well, the barrier and the sharp decay, are preserved in this approximation.
• Figure 2: Plot of the power-series superpotential $W_p(a)$ (blue), the approximated potential $V_p(a)$ (orange) and the approximated Chaplygin potential $V(a)$ (green), with $B=0.5, \,\,\,\, \alpha=1, \,\,\,\, A=0.01$. We observe a smaller barrier for this approximated potential, and, instead of the sharp, infinite decay, for larger values of the scale factor $a$ the potential grows, which effectively gives us a well.
• Figure 3: Plot of the superpotential $W(a)$ (blue), the approximated potential $V_1(a)$ (orange) and the approximated Chaplygin potential $V(a)$ (green), with $B=1, \,\,\,\, \alpha=1, \,\,\,\, A=0.02$. Comparing with the previous approximation, this one is better fitting near the barrier, for these values of parameters. It maintains a similar qualitative behaviour, the well right after the barrier, which comes from both superpotentials used being polynomials.
• Figure 4: tunnelling probabilities $T(A,B,\alpha)$ versus (a) $A$, with $B=0.8$ and $\alpha=1$ (b) $B$, with $A=0.01$ and $\alpha=1$, and (c) $\alpha$, with $A=0.05$ and $B=0.8$. Blue: SWKB (Picard superpotential); orange: WKB for $V_1(a)$(SUSY-generated potential); green: WKB for the original Chaplygin potential $V(a)$. Turning points were found numerically. We observe in all three plots, the tunnelling probabilities obtained by the different approaches have similar values for small values of the parameters, and start to diverge for larger values. We observe the SWKB tunnelling probabilities to be higher than the WKB ones, something observed as well in the literature SIL1994209.