Coleman–De Luccia Vacuum Decay Calculator

False-vacuum bubble nucleation overview

This calculator evaluates the standard thin‑wall, flat‑space (no gravitational backreaction) approximation to false‑vacuum decay by bubble nucleation. In a theory with at least two local minima, a metastable false vacuum can tunnel to a lower-energy true vacuum by forming a critical bubble. The critical bubble extremizes the Euclidean action (the “bounce”) and supplies the dominant tunneling exponent.

For the flat-space false-vacuum calculation, provide the bubble-wall surface tension σ (energy per unit area), the vacuum energy density difference ΔV between false and true vacua, and a prefactor A that fixes the overall decay-rate scale. The calculator returns the critical bubble radius R, the bounce action S_B, and the natural-unit nucleation rate density Γ/V.

Thin‑wall flat‑space vacuum-decay formulas

For a thin-wall false-vacuum bubble, the wall thickness is small compared with the bubble radius, so the Euclidean action is approximated as competing surface and volume contributions. In flat spacetime, this yields the following critical radius and bounce action.

Critical radius of the nucleating bubble

The false-vacuum bubble critical radius is R = 3σ/ΔV. Larger surface tension increases the critical size; a larger vacuum energy difference decreases it.

False-vacuum bounce action

Substituting the critical radius into the thin-wall action gives S_B = (27π² σ⁴)/(2 ΔV³). This dimensionless quantity controls the exponential suppression of tunneling.

Vacuum-decay rate density

The semiclassical false-vacuum decay estimate is Γ/V ≈ A e^−S_B, 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 for vacuum-decay formulas

These MathML expressions reproduce the bubble-radius, bounce-action, and rate-density formulas used above:

Vacuum-decay units and radius conversion

• 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 false-vacuum decay results

Critical bubble radius R

In this false-vacuum decay calculation, the critical radius separates subcritical bubbles, which tend to collapse, from supercritical bubbles, which tend to expand. Thin-wall nucleation is dominated by bubbles close to this critical size.

Vacuum-decay bounce action S_B

For false-vacuum tunneling, the bounce action determines 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 strong suppression for a chosen prefactor.

Computed decay rate Γ/V

The reported false-vacuum quantity Γ/V is a natural-unit rate density. Interpreting it as a lifetime for a particular physical region requires additional modeling, including an appropriate spacetime volume and cosmological background. For very large S_B, the exponential can underflow in floating-point arithmetic; this calculator reports the direct value rather than a logarithm.

Worked example: thin-wall false-vacuum bubble

This thin-wall false-vacuum example uses 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.

Thin-wall vacuum-decay scaling comparison

Coleman–De Luccia gravity corrections not implemented here

The gravitational Coleman–De Luccia treatment of vacuum decay modifies both the critical radius and the action when vacuum energies and wall tension make spacetime curvature important. This calculator uses the flat‑space thin‑wall result only, so its outputs are an approximation when gravitational backreaction is negligible.

Thin-wall vacuum-decay assumptions and 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.

False-vacuum decay 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 vacuum-decay exponential usually dominates

For the false-vacuum rate density calculated here, the prefactor A fixes an overall scale while e^−S_B supplies the tunneling suppression. When the bounce action is large, the exponential determines the result far more strongly than a multiplicative change in A. In the worked thin-wall example, S_B ≈ 133, so e⁻¹³³ is roughly 10⁻⁵⁸. The critical bubble's Euclidean action therefore controls the suppression, while the prefactor becomes comparatively important only when S_B is not large.

For this calculator, σ and ΔV are consequently the main inputs to scrutinize. Raising wall tension σ increases the critical radius and, through the fourth-power dependence of the action, sharply strengthens suppression. Raising the vacuum energy difference ΔV decreases the critical radius and, through the inverse-cube dependence, sharply weakens suppression. The thin-wall scaling table summarizes these directions; their importance comes from their appearance inside the exponential.

How to use this false-vacuum decay calculator

1. Enter the wall surface tension σ in GeV³ — the energy per unit area stored in the bubble wall separating the two vacua.
2. Enter the vacuum energy difference ΔV in GeV⁴, taken as positive when the false vacuum sits above the true vacuum.
3. Enter the prefactor A in GeV⁴ for the overall rate scale; when in doubt, a dimensional estimate at the relevant energy scale is common.
4. 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 see how strongly the tunneling exponent responds.

False-vacuum decay questions