Thin-Wall Bubble Nucleation
The transition of a metastable false vacuum to a true vacuum is elegantly described by the semiclassical tunneling framework developed by Sidney Coleman and Frank De Luccia. In field theories with multiple local minima, quantum fluctuations can nucleate bubbles of the lower-energy phase. If the energy gained by converting the interior to the true vacuum exceeds the surface energy cost of the wall separating phases, the bubble expands and the universe transitions locally to the new vacuum. In the thin-wall regime—where the energy difference between vacua is much smaller than the barrier height—the bubble wall is narrow compared with its radius, allowing a simple analytic expression for the Euclidean action of the bounce solution.
Neglecting gravitational effects for the moment, the critical bubble that dominates tunneling satisfies a balance between surface and volume energies. The surface contributes while the volume gains . Minimizing the action with respect to the bubble radius yields . Substituting this radius into the action gives the celebrated thin-wall result . Because is typically enormous, the decay rate per unit volume is exponentially suppressed: .
From Inputs to Results
This calculator implements the thin-wall formulas using natural units where ℏ = c = 1 and energies are expressed in giga-electron-volts (GeV). Users specify the surface tension σ in GeV³, the vacuum energy difference ΔV in GeV⁴, and a prefactor A with dimensions of GeV⁴ encapsulating quantum fluctuations. The script computes the bubble radius in both GeV⁻¹ and meters via the conversion factor . It then evaluates the bounce action and the decay rate per unit volume . Because the exponential can underflow to zero for large actions, results are reported using exponential notation.
Interpreting the Bounce Action
The bounce action measures the suppression of tunneling. A value implies an extremely long-lived false vacuum. For example, with σ = 10⁶ GeV³ and ΔV = 10⁸ GeV⁴, one finds , leading to a decay rate utterly negligible on cosmological timescales even with a large prefactor. By contrast, if σ and ΔV are comparable, the action can drop substantially, accelerating the transition. The bubble radius also conveys physical insight: larger surface tension or smaller energy difference produce larger critical bubbles.
Sample Calculations
| σ (GeV³) | ΔV (GeV⁴) | SB | Γ/V (GeV⁴) |
|---|---|---|---|
| 10⁶ | 10⁸ | ≈1.3×10³ | ≈0 |
| 10⁴ | 10⁶ | ≈1.3×10¹ | ≈1×10⁻¹² |
| 10³ | 10⁵ | ≈3.4 | ≈4.0×10³ |
Including Gravity
Coleman and De Luccia extended the thin-wall analysis to incorporate general relativity. Gravity modifies both the bubble radius and the action via the ratio of the vacuum energies to the Planck scale. For ΔV comparable to , the decay can be either enhanced or suppressed depending on the sign of the cosmological constant in each vacuum. Our calculator omits these corrections to remain simple, but the thin-wall expressions provide reasonable estimates whenever ΔV ≪ . Users interested in gravitational effects should consult the original literature for the more elaborate formulae.
Cosmological Implications
False vacuum decay underlies many cosmological phase transitions. During inflation, a high-energy vacuum drives exponential expansion until tunneling or classical rolling triggers reheating. In theories with multiple metastable vacua, such as the string landscape, bubble nucleation can seed pocket universes with varying physical constants. The thin-wall formula helps estimate whether such transitions occurred in the early universe or might threaten our present vacuum. Current measurements of the Higgs mass suggest our vacuum is metastable but extremely long-lived, consistent with a large bounce action.
Limitations and Extensions
The thin-wall approximation assumes a sharp separation between phases, valid when the barrier height greatly exceeds the energy difference. For thick-wall cases, one must solve the bounce equations numerically. Moreover, the prefactor A is often uncertain; dimensional analysis typically replaces it with a power of ΔV or of the characteristic mass scale. Thermal fluctuations at nonzero temperature modify the decay by introducing periodic boundary conditions in imaginary time, effectively changing the problem to a three-dimensional one with its own action.
Despite these caveats, the thin-wall Coleman–De Luccia formula remains a cornerstone of vacuum decay theory, offering intuitive estimates and guiding detailed numerical studies. By exploring how the bounce action scales with surface tension and energy difference, researchers gain insight into the stability of vacua in particle physics models, the likelihood of cosmological phase transitions, and the feasibility of experimental analogues using condensed-matter systems.
The calculator thus serves as both an educational tool and a springboard for deeper inquiry. By experimenting with various parameter choices, users can appreciate the sensitivity of decay rates to the underlying field theory landscape. The exponential nature of the tunneling probability underscores why our universe can remain perched in a metastable state for times vastly exceeding its current age, despite the ever-present possibility of transition.