False Vacuum Decay With Gravity in Non-Thin-Wall Limit

TL;DR

The paper tackles false vacuum decay in curved spacetime by developing a covariant WKB framework in minisuperspace that does not rely on the thin-wall approximation. It constructs a one-parameter tunneling path by smoothly patching the false-vacuum (FV) instanton and the Coleman–De Luccia (CD) bounce, via a turning point at $a=0$, to produce a tunneling wave function describing post-nucleation expansion. The main result is that the tunneling amplitude satisfies $\Gamma\sim\exp[-(S_B-S_F)/\hbar]$, agreeing with naive Euclidean-action extrapolations and extending validity to thick-wall cases. This work provides a quantum-cosmological picture of false vacuum decay with gravity and clarifies how boundary conditions (Hartle–Hawking vs. Vilenkin) relate to the resulting wave function and the evolution after bubble nucleation.

Abstract

We consider a wave-function approach to the false vacuum decay with gravity and present a new method to calculate the tunneling amplitude under the WKB approximation. The result agrees with the one obtained by the Euclidean path-integral method, but gives a much clearer interpretation of an instanton (Euclidean bounce solution) that dominates the path integral. In particular, our method is fully capable of dealing with the case of a thick wall with the radius of the bubble comparable to the radius of the instanton, thus surpassing the path-integral method whose use can be justified only in the thin-wall and small bubble radius limit. The calculation is done by matching two WKB wave functions, one with the final state and another with the initial state, with the wave function in the region where the scale factor of the metric is sufficiently small compared with the inverse of the typical energy scale of the field potential at the tunneling. The relation of the boundary condition on our wave function for the false vacuum decay with Hartle-Hawking's no-boundary boundary condition and Vilenkin's tunneling boundary condition on the wave function of the universe is also discussed.

Figures (5)

• Figure 1: A schematic picture of the potential $V(\phi)$ of a scalar field discussed in the text.
• Figure 2: Schematic pictures of the false vacuum (FV) instanton and the Coleman-DeLuccia (CD) instanton with 2-dimensions suppressed. The arrow indicates the direction of the increase in the parameter $\xi$ defined in the text.
• Figure 3: A picture of the superspace $(\tilde{a},\phi)$ near the matching region. The thick lines are the one-parameter family of configurations $(\bar{h}_{ij}(\xi),\bar{\phi}(\xi))$ in the range $\xi\in[0,\pi/2)$ and $\xi\in(\pi/2,\xi_f]$. The arrow indicates the direction of increase in $\xi$ along the family. The WKB approximation is valid in the hatched region and the wave function (\ref{['eq:WDwfapp']}) is valid in the grayed region. The overlapping region is where the matching is performed.
• Figure 4: A schematic behavior of the tunneling wave function. The point $\xi=\pi/2$ is where the scale factor $a$ vanishes, and towards both right and left directions from there the 3-volume increases. The dotted curve indicates the effective potential barrier.
• Figure 5: Schematic picture of the causal structure of superspace and the one-parameter family of configurations used to construct the wave function. The Euclidean region is shaded. The lines indicated by FV and CD correspond to the configurations of the false vacuum and Coleman-DeLuccia solutions, respectively. The up or down arrows along the lines indicate the expanding or contracting components contained in the wave function.