Coleman–De Luccia Vacuum Decay Calculator
Introduction: Overview
This calculator implements the standard thin‑wall, flat‑space (no gravitational backreaction) formulas for false‑vacuum decay via bubble nucleation as developed in the semiclassical tunneling framework associated with Coleman and (with gravity) Coleman–De Luccia. In a theory with at least two local minima, a metastable (false) vacuum can decay to a lower‑energy (true) vacuum through nucleation of a critical bubble. The critical bubble is the configuration that extremizes the Euclidean action (“bounce”) and dominates the tunneling exponent.
You provide the surface tension of the bubble wall σ (energy per unit area), the vacuum energy density difference ΔV between false and true vacua, and a prefactor A that sets the overall scale of the decay rate. The calculator returns the critical bubble radius R, the bounce action SB, and the decay rate per unit volume Γ/V in natural units.
Thin‑wall, flat‑space formulas
In the thin‑wall regime, the wall thickness is small compared to the bubble radius. The Euclidean action can be approximated by a competition between a surface term and a volume term. In flat spacetime, the critical radius and bounce action are:
Critical radius
The critical radius is R = 3σ/ΔV. Larger surface tension increases the critical size; larger vacuum energy difference decreases it.
Bounce action
Substituting the critical radius into the thin‑wall action yields SB = (27π2 σ4)/(2 ΔV3). This dimensionless quantity controls the exponential suppression of the tunneling rate.
Decay rate per unit volume
The semiclassical estimate for the decay rate per unit volume is Γ/V ≈ A e−SB, where A has units of (energy)4 in ℏ=c=1 units.
Plain-text formula: R = 3*sigma/deltaV; S_B = 27*pi^2*sigma^4 / (2*deltaV^3); Gamma/V = A*exp(-S_B); R_meters = R * 1.97327e-16.
MathML reference (same formulas)
The same key expressions in MathML:
Units and conversions used
- σ is entered in GeV3.
- ΔV is entered in GeV4 and should be positive for decay from false to true vacuum in this sign convention.
- A is entered in GeV4.
- R is produced in GeV−1 and also converted to meters using 1 GeV−1 = 1.97327×10−16 m.
Interpreting the results
Bubble radius R
The critical radius separates subcritical bubbles (which tend to collapse) from supercritical ones (which tend to expand). In the thin‑wall picture, nucleation is dominated by bubbles near the critical size.
Bounce action SB
The bounce action sets the exponential suppression. Small changes in σ or ΔV can change SB dramatically because of the scaling SB ∝ σ4/ΔV3. Values SB ≫ 1 usually imply an extremely long‑lived false vacuum (for reasonable prefactors).
Decay rate Γ/V
The reported quantity Γ/V is a decay rate density in natural units. Interpreting it as a lifetime for a specific physical region requires additional modeling (e.g., integrating over spacetime volume and choosing a cosmological background). Numerically, for very large SB the exponential may underflow in floating‑point arithmetic; log‑space reporting (e.g., log(Γ/V)) is often more stable, but this calculator reports the direct value.
Worked example
Take σ = 106 GeV3, ΔV = 108 GeV4, and A = 108 GeV4.
- Radius: R = 3σ/ΔV = 3×106 / 108 = 3×10−2 GeV−1. In meters, R ≈ 3×10−2 × 1.97327×10−16 m ≈ 5.92×10−18 m.
- Bounce action: SB = (27π2/2) σ4/ΔV3. Here σ4/ΔV3 = 1024/1024 = 1, so SB ≈ 27π2/2 ≈ 133.
- Rate density: Γ/V ≈ A e−SB ≈ 108 e−133 GeV4, which is extremely small.
Quick comparison table (scaling intuition)
| Change | Effect on R = 3σ/ΔV | Effect on SB ∝ σ4/ΔV3 | Qualitative impact on Γ/V |
|---|---|---|---|
| Increase σ | Increases linearly | Increases strongly (fourth power) | Much smaller (more suppressed) |
| Increase ΔV | Decreases linearly | Decreases strongly (third power) | Much larger (less suppressed) |
| Increase A | No change | No change | Scales Γ/V up proportionally |
Including gravity (Coleman–De Luccia) — not implemented here
Coleman–De Luccia (CDL) gravitational corrections modify both the critical radius and the action when the vacuum energies and wall tension are large enough that spacetime curvature is important. This calculator currently uses the flat‑space thin‑wall result only, so treat outputs as an approximation when gravitational backreaction is negligible.
Assumptions & limitations
- Thin‑wall approximation: valid when the energy difference ΔV is small compared to the barrier height and the wall thickness is much smaller than R. Outside this regime, the true bounce must be computed numerically.
- Flat spacetime (no gravity): gravitational backreaction is ignored. If vacuum energies are large (near Planckian scales or in curved backgrounds), CDL corrections can be important.
- Single‑field effective description: σ and ΔV are treated as effective parameters. In multifield settings, the tunneling path and effective tension can differ from naive estimates.
- Sign conventions: this page assumes ΔV > 0 means the false vacuum energy density exceeds the true vacuum energy density by ΔV.
- Prefactor uncertainty: A can vary by many orders of magnitude and depends on fluctuation determinants; the exponential term typically dominates, but A still matters when SB is not huge.
- Numerical underflow: for large SB, e−SB may underflow to 0 in double precision. Consider interpreting results via log(Γ/V) in external analysis.
- From Γ/V to a lifetime: converting Γ/V into a decay probability for “our universe” requires integrating over an appropriate spacetime volume and cosmological history; this calculator does not perform that step.
References
- S. Coleman, “The Fate of the False Vacuum. 1. Semiclassical Theory,” Phys. Rev. D 15 (1977) 2929.
- S. Coleman and F. De Luccia, “Gravitational Effects on and of Vacuum Decay,” Phys. Rev. D 21 (1980) 3305.
Why the exponential term usually dominates
The decay rate density has two ingredients: a prefactor A that sets the overall scale and the exponential e−SB that provides the suppression. In almost every physically interesting case the exponential wins by an overwhelming margin. Consider the worked example, where SB ≈ 133: the factor e−133 is roughly 10−58, so even a prefactor spanning many orders of magnitude barely moves the answer. This is the semiclassical picture in a nutshell — the tunneling probability is set almost entirely by how much Euclidean action the critical bubble costs, and the prefactor becomes a genuine tie-breaker only when SB is not large. It is exactly this exponential structure that lets a false vacuum with a modestly larger action be many orders of magnitude longer-lived than one with a slightly smaller action.
Practically, that means the two levers you care about most are σ and ΔV, not A. Raising the wall tension σ lengthens the critical bubble and, because the action scales with the fourth power of σ, sharply increases the suppression. Raising the vacuum energy difference ΔV shrinks the critical bubble and, through the inverse-cube dependence, sharply decreases the suppression. The comparison table above captures those directions, but the underlying reason they matter so much is that both feed into an exponent.
How to use this vacuum decay calculator
- Enter the wall surface tension σ in GeV3 — the energy per unit area stored in the bubble wall separating the two vacua.
- Enter the vacuum energy difference ΔV in GeV4, taken as positive when the false vacuum sits above the true vacuum.
- Enter the prefactor A in GeV4 for the overall rate scale; when in doubt, a dimensional estimate at the relevant energy scale is common.
- Press Compute Decay to read the critical radius (in GeV−1 and meters), the bounce action, and the decay rate per unit four-volume. Vary σ and ΔV to watch how strongly the exponential responds.
Vacuum decay: frequently asked questions
What is the thin-wall approximation?
The thin-wall approximation applies when the energy difference between the false and true vacuum is small compared to the height of the barrier separating them, so the bubble wall is thin relative to the bubble radius. In that regime the Euclidean action splits cleanly into a surface term and a volume term, giving the closed-form results R = 3 sigma / deltaV and S_B = 27 pi^2 sigma^4 / (2 deltaV^3). Outside this regime the bounce must be found numerically.
Why does a small change in sigma or deltaV change the decay rate so much?
Because the bounce action scales as sigma^4 / deltaV^3 and the rate is proportional to e^(-S_B). A modest change in the inputs moves S_B by a large amount, and that shift is then exponentiated, so decay rates commonly swing by tens or hundreds of orders of magnitude. This extreme sensitivity is why false vacua can be either almost instantly unstable or astronomically long-lived.
Does this calculator include gravity?
No. It uses the flat-space thin-wall result with no gravitational backreaction. The full Coleman-De Luccia treatment adds gravity, which modifies both the critical radius and the action when vacuum energies or wall tension approach the scale where spacetime curvature matters. Treat these outputs as valid only when gravitational effects are negligible.
What does the decay rate per unit volume actually tell me?
Gamma over V is a rate density in natural units: nucleations per unit four-volume. Turning it into a lifetime for a specific region requires integrating over an appropriate spacetime volume and cosmological history, which this tool does not do. For very large bounce actions the exponential can underflow double precision to zero, so serious analysis usually works with log(Gamma/V) instead.
Arcade Mini-Game: Coleman–De Luccia Vacuum Decay Calculator Calibration Run
Use this quick arcade run to practice separating useful scenario inputs from common planning mistakes before you rely on the calculator output.
Start the game, then use your pointer or arrow keys to catch useful inputs and avoid bad assumptions.