Gravitational Vacuum Decay and Cosmological Transitions

Gravity changes both vacuum-decay saddles and the spacetime history in which a transition occurs. Connecting a saddle to cosmological completion requires a decay prescription, bubble dynamics, and an evolving expansion rate.

Enter this chapter

Begin with False-Vacuum Decay with Gravity: Regimes and Observables to distinguish a decay exponent, a survival observable, and a cosmological history.

Helpful background. Bounce Solutions and False-Vacuum Boundary Conditions and Decay Rates, the Negative Mode, and Prefactors develop the definitions used by the chapter’s opening pages.

Topics

1. False-Vacuum Decay with Gravity: Regimes and Observables — Which state, geometry, saddle, contour, observable, and scale hierarchies must be fixed before a false-vacuum decay rate with gravity has a definite meaning?
2. Euclidean Gravitational Saddles and Boundary Terms — How is the Euclidean gravitational saddle problem posed with the correct action, constraints, boundary terms, contour data, and subtraction against the false vacuum?
3. Coleman–De Luccia Bounces — When does a Coleman–De Luccia bounce exist and dominate, and how do gravitational constraints, thin-wall control, and Lorentzian continuation shape its interpretation?
4. Hawking–Moss Transitions and the Stochastic Crossover — When is a Hawking–Moss saddle a meaningful transition description, and how is its homogeneous activation picture related to stochastic barrier crossing rather than conflated with it?
5. Negative Modes, Determinants, and Decay Prefactors — Which physical and constrained fluctuation modes determine whether a gravitational bounce contributes to decay, and how far can its determinant prefactor be trusted?
6. Real-Time Vacuum Decay and State Dependence — How can false-vacuum decay be formulated in real time for a specified initial state, and which quantities can legitimately be compared with Euclidean saddle estimates?
7. Thermal Nucleation in an Expanding Universe — How are microscopic finite-temperature nucleation inputs embedded in an expanding universe without rederiving the plasma rate or assuming instantaneous equilibrium?
8. Expansion History, Percolation, and Completion — Given microscopic nucleation and wall inputs, when does a transition percolate and complete in an evolving, reheating, or vacuum-dominated cosmological background?
9. Baryogenesis Inputs and the Cosmological Yield — How are microscopic CP-violating and anomalous-charge rates converted into a cosmological baryon yield through expansion, washout, reheating, and entropy dilution?
10. Bubble Sources and Cosmological Gravitational-Wave Propagation — How are supplied bubble and plasma source correlators converted into a primordial gravitational-wave spectrum and propagated through the later cosmological expansion?
11. Metastability, Eternal Inflation, and Claim Limits — Which local metastability results survive when discussing self-reproduction or eternal inflation, and where do global measures and quantum-gravity assumptions enter irreducibly?

How the ideas fit together

A Coleman–De Luccia bounce solves a coupled Euclidean field–geometry boundary-value problem. Its action difference supplies an exponent only after the relevant saddle and boundary conditions have been identified Coleman and De Luccia 1980, §§ II–IV. Negative modes and prefactors are separate parts of interpreting it as a decay process.

In cosmology, nucleation and growth must then be integrated through expansion and reheating. Baryon yields and gravitational waves require further transport or source information. Neither a small bounce action nor geometric percolation alone proves that the physical false-vacuum volume disappears.

The comparison below separates the assumptions and tests needed for these different conclusions.

Domain and failure conditions

Layer	Required input	Output or observable	Controlled statement	Decisive check	Failure or downgrade
Regime selection	Metastable state, background, region, clock, hierarchy	Proper-volume rate, survival probability, or history statistic	The chosen object has a definite operational meaning	Convert between rate and survival with the actual geometry	Treating foliation-dependent survival as an invariant local rate
Euclidean saddle	Euclidean action, boundary ensemble, contour candidate, matched reference	Action difference $B$	Stationary saddle evidence with declared boundary data	Vanishing boundary variation and invariant matched subtraction	Missing GHY term, unmatched reference, or unspecified conformal contour
Coleman–De Luccia	Scalar potential, gravitational coupling, regular pole data	Inhomogeneous bounce and Lorentzian bubble initial data	Controlled branch where a regular saddle exists	Numerical residual, constraint, thin-wall or weak-gravity limit	Continuing an exponent beyond branch disappearance
Hawking–Moss or stochastic	Barrier top, Hubble scale, coarse-graining, state	Homogeneous action difference or first-passage statistic	Overlap only under slow, light, Markovian, matched-observable conditions	Barrier-curvature scan and matched exponent limit	Calling two unlike observables identical or imposing a sharp universal crossover
Fluctuation rate	Gauge-reduced Hessian, measure, zero modes, contour	Prefactor multiplying $e^{-B}$	Decay interpretation when physical negative directions and determinant are controlled	Gauge and field-redefinition invariant spectrum	Conformal-factor, gauge, or extra-negative-mode ambiguity
Real-time decay	Initial density matrix, false sector, Hamiltonian, clock	$P_F(t)$ and time-dependent hazard	Initial-value result for the prepared state	State normalization, unitarity, and comparison in an exponential window	Universal rate inferred from the potential alone
Thermal nucleation	Versioned $\Gamma(T)$, covariance, equation of state	Nucleation-time or temperature distribution	Cosmological embedding of supplied microphysics	Reproduce adiabatic radiation limit and propagate covariance	Instantaneous equilibrium or rate precision unsupported by inputs
Percolation and completion	$\Gamma(t)$, $a(t)$, wall speed, reheating	False fraction, connected conversion, physical false volume	Separate nucleation, percolation, and completion times	Solve $P_F=e^{-I}$ and test $d(a^3P_F)/dt$	Declaring completion from one bubble per Hubble volume
Baryon yield	CP source, transport output, washout, entropy history	Final $Y_B=n_B/s$	Cosmological propagation of supplied charge microphysics	Time-variable and entropy-normalization invariance	Omitting washout, reheating, or entropy dilution
Gravitational waves	Unequal-time anisotropic-stress correlator, source lifetime, $a(t)$	Primordial and present-day spectra	Source calculation followed by cosmological transfer	Energy conservation, redshift, relativistic-degree count	Forecast precision below source-model or transfer uncertainty
Global self-reproduction	Local rate, patch dynamics, time variable, cutoff measure	Volume-weighted or first-passage statistic	Measure-conditional global statement	Repeat with declared time and cutoff prescriptions	Promoting a local rate to a measure-independent multiverse probability

Where to go next

Continue to Inflationary Correlators and Cosmological Perturbation Theory to identify the clock background and the conditions for decoupling gravitational constraints.

References

• Coleman, S., and F. De Luccia. “Gravitational Effects on and of Vacuum Decay.” Physical Review D 21 (1980): 3305–3315. DOI.