Validity Limits and Thermal and Gravitational Handoffs

A flat-space, zero-temperature bounce supports a defensible decay rate only when the false-vacuum observable, saddle existence, boundary data, instability index, renormalized prefactor, semiclassical expansion, and environmental scale separation all pass quantitative checks. Finite temperature changes the Euclidean domain and the meaning of the prefactor; dynamical gravity changes the field equations, boundary conditions, action subtraction, and mode problem. These are new calculations, not corrections that can always be appended to the flat result.

Required background. Bounce existence, symmetry, and multifield geometry supplies existence and full-field-space stationarity tests. Thin-wall control and corrections supplies approximation and residual diagnostics. Gauge and renormalization-scale dependence supplies fixed-order invariance tests. Decay rates, the negative mode, and prefactors supplies the spectrum, volume factors, and rate normalization.

Helpful background. Renormalized saddles and validity tests supplies the general observable-first semiclassical control logic.

A defensible flat-space decision

Let

$$\gamma_0 := \frac{\Gamma}{V_{d-1}} = A\,e^{-B}$$

denote the zero-temperature decay probability per unit time and spatial volume in $d$ spacetime dimensions. Thus $[\gamma_0]=\text{mass}^d$ in units with $\hbar=c=1$. Before assigning a reliability decision, fix a target accuracy for $\ln\gamma_0$, not merely for $B$ as a percentage.

Use three outcomes:

• Pass: every necessary condition below is demonstrated at the target accuracy.
• Qualified pass: the calculation controls a narrower claim—such as the leading exponent or parametric scaling—but not the requested full rate or precision. State the missing term and do not relabel the narrower result as a rate.
• Fail: a necessary condition is absent or fails at the retained order. Stop the flat zero-temperature interpretation and follow the indicated route.

Check	Condition for a flat zero-temperature rate	If the condition is missed
Observable and state	A normalized false-vacuum persistence probability is defined, with the infinite-volume and long-time limits declared.	Reconstruct the observable; an energy splitting or barrier height is not a decay rate.
Saddle	A finite-action stationary bounce exists in the full admissible field space and returns to the false vacuum.	Solve the full boundary-value problem or report that no canonical bounce has been established.
Symmetry	Any $O(d)$ or field-space reduction satisfies its hypotheses and the discarded equations.	Remove the ansatz or quantify the residual normal to it.
Spectrum and contour	The canonical least-action bounce has exactly one relevant negative mode, $d$ translation zero modes in noncompact flat spacetime, and no unaccounted zero modes.	Reconsider dominance and contour data; do not use the standard one-bounce rate formula.
Semiclassical order	$B/\hbar\gg1$, loop corrections and collective-coordinate interactions are parametrically smaller, and logarithms are controlled.	Reorganize or abandon the saddle expansion.
Prefactor and units	The determinant, negative-mode prescription, zero-mode Jacobians, counterterms, and state normalization give $[A]=\text{mass}^d$.	A result with the wrong dimension or missing spacetime-volume division is not a rate density.
Gauge and scale	Gauge-parameter and RG identities hold through the retained order; residual dependence begins at the first omitted order.	Restore missing derivative, fluctuation, counterterm, or running terms.
EFT domain	Bounce field values, gradients, and fluctuation eigenvalues remain inside the declared EFT domain; higher operators give bounded corrections.	Match to a different EFT or state ultraviolet sensitivity.
Numerics	Boundary, differential-equation, virial, action, determinant, volume, and discretization checks converge more tightly than the error target.	Refine the computation before interpreting it physically.
Thin wall, if used	Wall thickness, potential deformation, radius, and full-bounce comparison support the claimed logarithmic accuracy.	Use the direct bounce; retain thin wall only as qualitative scaling if justified.
Diluteness and stationarity	The expected number of bounces in a bounce-sized $d$-volume, of order $A R^d e^{-B}$, is small, and the background changes slowly over a nucleation event.	Include interactions, depletion, or time dependence; the dilute stationary sum is invalid.
Temperature	The Euclidean time circle is effectively noncompact on all important bounce and fluctuation scales, and competing periodic or static thermal saddles are subleading.	Use the finite-temperature nucleation calculation.
Gravity	Curvature and gravitational backreaction are perturbatively smaller than the target error, with no competing gravitational saddle.	Solve the coupled gravitational bounce and constraint problem.

The “exactly one negative mode” row is tied to the canonical flat-space least-action bounce under the scalar variational hypotheses. Other saddles can have more negative directions and may contribute to a different contour decomposition, but they do not inherit the standard leading decay formula merely by taking an absolute determinant.

Finite temperature changes the problem

At temperature $T$, Euclidean time is periodic with

$$\beta=\frac1T.$$

The zero-temperature $O(4)$ bounce in four spacetime dimensions is reliable only when the periodic images are well separated and thermally assisted saddles are subleading. A large ratio $\beta/R$ is a useful necessary scale-separation diagnostic, but it is not a universal transition criterion: one must compare the actions of the relevant periodic solutions.

In the high-temperature static regime, the Euclidean action factorizes:

$$S_E[\phi_{\mathrm{static}}] = \beta S_3[\phi], \qquad e^{-S_E}=e^{-S_3/T}.$$

The spatial bounce is typically $O(3)$ symmetric, not $O(4)$ symmetric. Its three translation zero modes, thermal determinant, statistical normalization, and real-time growth or damping coefficient form a thermal nucleation prefactor. The Euclidean determinant alone does not supply the complete real-time nucleation rate. The static-bounce limit and its domain were established in Linde 1983, §§2–4, pp. 425–438.

Transfer the calculation to finite-temperature bounces and nucleation rates when any of the following is material:

• $\beta$ is comparable to the bounce radius or wall thickness;
• a periodic or static thermal saddle has lower action than the zero-temperature bounce;
• thermal masses or the thermal effective action change the barrier at the retained order;
• plasma transport, damping, or a real-time growth prefactor enters the observable;
• cosmological cooling, reheating, or percolation is part of the claim.

A factor $e^{-S_3/T}$ by itself is therefore a thermal exponent, not a complete cosmological transition rate.

Gravity changes boundary data and subtraction

Gravity responds to absolute vacuum energy, so the freedom to shift $U$ by a constant—harmless in flat-space scalar dynamics—is lost. In four Euclidean dimensions, an $O(4)$-symmetric scalar–gravity ansatz may be written

$$\mathrm ds_E^2 = \mathrm d\rho^2+a(\rho)^2\mathrm d\Omega_3^2.$$

For a minimally coupled scalar and reduced Planck mass $\bar M_{\mathrm{Pl}}$, the coupled equations include

$$\phi'' +3\frac{a'}{a}\phi' = U'(\phi),$$

and the Euclidean constraint

$$(a')^2 = 1+\frac{a^2}{3\bar M_{\mathrm{Pl}}^2} \left(\frac12(\phi')^2-U(\phi)\right).$$

The false-vacuum geometry supplies the reference action and global boundary conditions. A de Sitter solution is compact and closes at a second pole; Minkowski and anti-de Sitter cases have different asymptotics. The fluctuation problem also contains gravitational constraints, so flat-space mode counting cannot simply be copied. These changes are central to the Coleman–De Luccia construction Coleman and De Luccia 1980, pp. 3306–3312.

Useful perturbative diagnostics include

$$\eta_{\mathrm{curv}}=\frac{R}{L_{\mathrm{curv}}}, \qquad \eta_U= \frac{R^2\sup_{\mathrm{bounce}}\lvert U\rvert} {\bar M_{\mathrm{Pl}}^2}, \qquad \eta_\sigma= \frac{\sigma R}{\bar M_{\mathrm{Pl}}^2},$$

where $L_{\mathrm{curv}}$ is the background curvature radius and $\sigma$ is relevant only when a wall description is controlled. These parameters must be small relative to the requested accuracy, not merely smaller than one. Even then, verify that no compact or Hawking–Moss-type saddle competes and that the perturbative correction to both the action and spectrum is stable.

Use the curved-vacuum-decay regime contract to choose the gravitational regime, and transfer a dynamical calculation to Coleman–De Luccia bounces when backreaction is retained. Do so whenever $R/L_{\mathrm{curv}}$ is not negligible, the action changes at the target precision, the Euclidean manifold is compact, gravitational constraints alter the spectrum, or the claim involves cosmological evolution.

First application: direct versus thin wall

For the four-dimensional tilted quartic with $\lambda=v=1$ and $\epsilon=0.10$, the converged comparison gives

$$\frac{\ell}{R_{\mathrm{tw}}}=0.05, \qquad B_{\mathrm{tw}}=105275.780, \qquad B_{\mathrm{num}}=104766.214.$$

The thin-wall exponent differs by only $0.486\%$, but

$$B_{\mathrm{tw}}-B_{\mathrm{num}}=509.566.$$

For a target uncertainty $\lvert\delta\ln\gamma_0\rvert<1$, the direct numerical exponent passes the stated boundary, virial, and discretization checks, while the leading thin-wall exponent fails. The thin-wall result remains a qualified statement about scaling and a useful initial approximation. Neither exponent alone is yet a full-rate pass: the negative-mode count, determinant, counterterms, and collective-coordinate normalization still have to be supplied.

Now place the same scalar calculation in a declared environment:

• If $\beta/R_{\mathrm{num}}=20$, all other thermal masses are negligible at the target order, and every competing periodic saddle has larger action, the zero-temperature treatment can pass the thermal check.
• If $\beta/R_{\mathrm{num}}=0.7$, the Euclidean circle resolves the bounce and the result transfers to the finite-temperature calculation; substituting $T$ into the zero-temperature prefactor fails.
• If $R_{\mathrm{num}}/L_{\mathrm{curv}}=0.02$ and $\eta_U,\eta_\sigma$ are below the target error with no competing gravitational saddle, gravity can be a bounded correction.
• If $R_{\mathrm{num}}/L_{\mathrm{curv}}=0.8$, the flat boundary conditions and action subtraction fail, and the coupled gravitational problem is required.

The dimensionless ratios classify the environment only after their ingredients and the competing saddles have been computed in the same model.

Shared calculation. The bounce control map summarizes the stop and transfer conditions from boundary data through spectrum, prefactor, thin-wall, thermal, and gravitational checks.

Shared comparison. The instanton–bounce boundary and mode comparison prevents a tunneling level-splitting calculation from being accepted as false-vacuum decay evidence.

Common pitfalls

Using $T\sim R^{-1}$ as an exact crossover formula. It is a scale estimate. The dominant saddle is selected by comparing the allowed periodic configurations and their complete fixed-order contributions.

Declaring gravity negligible because particle energies are below the Planck mass. Vacuum energy, wall tension, bubble size, and background curvature combine into the backreaction parameters. A large bubble can be gravity-sensitive even when local particle masses are sub-Planckian.

Reporting a percentage error in $B$ as the rate uncertainty. The logarithmic rate changes by the absolute action error. A small relative error in a large exponent can imply an exponentially large multiplicative error.

Exercises

1. Show that a static thermal bounce has exponent $S_3/T$, and determine the mass dimension of the thermal prefactor in four spacetime dimensions.

Solution

For a time-independent field,

$$S_E = \int_0^\beta\mathrm d\tau \int\mathrm d^3x\, \left[ \frac12(\boldsymbol\nabla\phi)^2+U_T(\phi) \right] = \beta S_3.$$

Since $\beta=1/T$, the exponent is $S_3/T$. A rate per unit real time and spatial volume has dimension $\text{mass}^4$, so its complete thermal prefactor must also have dimension $\text{mass}^4$.

2. Recover the flat radial equation from the gravitational equations when backreaction is negligible.

Solution

When $U/\bar M_{\mathrm{Pl}}^2$ and $(\phi')^2/\bar M_{\mathrm{Pl}}^2$ are negligible over the bounce, the constraint gives $(a')^2\simeq1$. Regularity at the origin selects $a(\rho)\simeq\rho$. The scalar equation becomes

$$\phi''+\frac{3}{\rho}\phi'=U'(\phi),$$

which is the flat $O(4)$ bounce equation.

3. A calculation has $B=400$, an estimated omitted loop correction $\delta B=0.4$, $AR^4e^{-B}\ll1$, $\beta/R=0.6$, and $R/L_{\mathrm{curv}}=10^{-3}$. Which condition determines the next calculation?

Solution

The semiclassical, dilute, and gravitational diagnostics are compatible with a controlled flat treatment at order-one logarithmic accuracy, but $\beta/R=0.6$ does not support a noncompact Euclidean-time approximation. One must compare periodic and static thermal saddles and transfer to the finite-temperature nucleation calculation. The ratio alone does not select $O(3)$ dominance, but it rules out accepting the zero-temperature result without that comparison.

References

• Coleman, S., and De Luccia, F. (1980). “Gravitational Effects on and of Vacuum Decay.” Physical Review D 21, 3305–3315. doi:10.1103/PhysRevD.21.3305.
• Linde, A. D. (1983). “Decay of the False Vacuum at Finite Temperature.” Nuclear Physics B 216, 421–445; Erratum Nuclear Physics B 223, 544. doi:10.1016/0550-3213(83)90293-6. Erratum.