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 S_B, 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 S_B = (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^−S_B, where A has units of (energy)⁴ in ℏ=c=1 units.

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 S_B

The bounce action sets the exponential suppression. Small changes in σ or ΔV can change S_B dramatically because of the scaling S_B ∝ σ⁴/ΔV³. Values S_B ≫ 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 S_B 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.

1. 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.
2. Bounce action: S_B = (27π²/2) σ⁴/ΔV³. Here σ⁴/ΔV³ = 10²⁴/10²⁴ = 1, so S_B ≈ 27π²/2 ≈ 133.
3. Rate density: Γ/V ≈ A e^−S_B ≈ 10⁸ e⁻¹³³ GeV⁴, which is extremely small.

Quick comparison table (scaling intuition)

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 S_B is not huge.
• Numerical underflow: for large S_B, e^−S_B 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.

How to use this calculator

1. Enter Surface tension σ (GeV³) using the unit or time period shown by the field.
2. Enter Energy difference ΔV (GeV⁴) using the unit or time period shown by the field.
3. Enter Prefactor A (GeV⁴) using the unit or time period shown by the field.
4. Run the calculation and compare the output with a second scenario before acting on it.