Late-Time Quantum Vacuum Decay

• Late-time quantum vacuum decay is characterized by a transition from exponential to inverse-power law behavior, driven by spectral thresholds, continuum contributions, and finite-time effects.
• The effective Hamiltonian and spectral analysis frameworks reveal that energy relaxation and channel-dependent decay rates critically modify the survival probability of metastable and false vacua.
• Cosmological applications of late-time decay, such as in Λ(t)CDM models, connect quantum energy relaxation to dark sector interactions and the evolution of vacuum energy in the universe.

Late-time quantum vacuum decay denotes the asymptotic regime in which a metastable vacuum, false vacuum, or vacuum-like state ceases to obey the canonical exponential decay law and instead exhibits non-exponential survival, energy relaxation, or effective time dependence of vacuum energy. In the literature, the term encompasses several related but non-identical problems: false-vacuum survival in quantum mechanics and quantum field theory, pair creation from external backgrounds, de Sitter-state instability, and phenomenological cosmologies in which vacuum energy exchanges energy with matter or radiation (Urbanowski et al., 2013, Anderson et al., 2017, Yan et al., 2018). The unifying theme is that the late-time behavior is controlled not by a single resonance pole alone, but by spectral thresholds, continuum contributions, finite-time effects, or environmental couplings that become dominant after the intermediate exponential era.

1. Quantum-mechanical asymptotics of metastable vacua

For an unstable initial state $|M\rangle$, the survival amplitude is

$$A(t)=\langle M|M;t\rangle,\qquad |M;t\rangle=e^{-itH}|M\rangle,$$

and the survival probability is

$P(t)=|A(t)|^2.$

When the energy distribution $\omega(E)$ vanishes below a lower spectral bound $E_{\min}$, the Paley–Wiener theorem implies that purely exponential decay cannot persist for arbitrarily large $t$. The decay law must eventually become slower than any exponential, and for generic threshold behavior it becomes inverse-power-like (Urbanowski et al., 2013, 0711.1821).

This late-time transition can be stated in several equivalent ways. If

$A(t)=A_{\exp}(t)+A_{\text{non}}(t),$

then the canonical regime is controlled by

$A_{\exp}(t)\sim e^{-it(E_M-i\Gamma_M/2)},$

whereas the asymptotic regime is governed by the non-exponential contribution. For threshold behavior of the form

$$\omega(E)=(E-E_{\min})^\lambda \eta(E),\qquad 0<\lambda<1,$$

the amplitude has an inverse-power asymptotic form and the survival probability behaves as

$$P(t)\sim \frac{\text{const}}{t^{2(\lambda+1)}},\qquad t\to\infty.$$

In the simple phase-space estimate emphasized in false-vacuum cosmology, $$A(t)=\langle M|M;t\rangle,\qquad |M;t\rangle=e^{-itH}|M\rangle,$$0, giving

$$A(t)=\langle M|M;t\rangle,\qquad |M;t\rangle=e^{-itH}|M\rangle,$$1

at late times (0711.1821).

The crossover time $$A(t)=\langle M|M;t\rangle,\qquad |M;t\rangle=e^{-itH}|M\rangle,$$2 is defined by

$$A(t)=\langle M|M;t\rangle,\qquad |M;t\rangle=e^{-itH}|M\rangle,$$3

For $$A(t)=\langle M|M;t\rangle,\qquad |M;t\rangle=e^{-itH}|M\rangle,$$4, exponential decay dominates; around $$A(t)=\langle M|M;t\rangle,\qquad |M;t\rangle=e^{-itH}|M\rangle,$$5, exponential and non-exponential parts are comparable; and for $$A(t)=\langle M|M;t\rangle,\qquad |M;t\rangle=e^{-itH}|M\rangle,$$6, the inverse-power tail dominates. This is the regime in which the standard metastable-state approximation fails as a complete description (Urbanowski et al., 2013).

The effective-Hamiltonian formulation makes the energy relaxation explicit. Defining

$$A(t)=\langle M|M;t\rangle,\qquad |M;t\rangle=e^{-itH}|M\rangle,$$7

one obtains the instantaneous energy and decay rate,

$$A(t)=\langle M|M;t\rangle,\qquad |M;t\rangle=e^{-itH}|M\rangle,$$8

At late times,

$$A(t)=\langle M|M;t\rangle,\qquad |M;t\rangle=e^{-itH}|M\rangle,$$9

so that

$P(t)=|A(t)|^2.$0

For false vacuum decay, $P(t)=|A(t)|^2.$1 is identified with the true-vacuum energy, implying

$P(t)=|A(t)|^2.$2

This is the direct quantum-mechanical origin of effective late-time cosmological relations such as

$P(t)=|A(t)|^2.$3

used in decaying-vacuum cosmology (Urbanowski et al., 2013, Szydlowski et al., 2015).

A curved-spacetime variant appears in the finite-time vacuum survival amplitude $P(t)=|A(t)|^2.$4 in de Sitter space. There the large-$P(t)=|A(t)|^2.$5 asymptotics, with $P(t)=|A(t)|^2.$6, is likewise power-law rather than exponential and includes oscillatory factors; at first order in the coupling the asymptotic scaling is proportional to $P(t)=|A(t)|^2.$7 (Álvarez et al., 2011).

2. Spectral structure, finite-time formalisms, and channel dependence

The late-time tail can be reformulated as a statement about spectral analyticity. In a real-time treatment of one-dimensional resonance models, the kernel admits a universal pole-plus-branch decomposition,

$P(t)=|A(t)|^2.$8

with asymptotic form

$P(t)=|A(t)|^2.$9

The pole term produces the intermediate exponential regime, while the branch cut produces the late-time power-law tail. This framework motivates two characteristic times: a “dawn time,” when a single resonance starts dominating, and a “twilight time,” when the branch-cut tail overtakes the exponential. The twilight time can be written in closed form with the Lambert $\omega(E)$0 function (Feng et al., 16 Dec 2025).

This spectral perspective clarifies that exponential decay is only valid in a bounded interval,

$\omega(E)$1

rather than asymptotically. The same work argues that suitably defined dawn and twilight times should exist more generally whenever decay is controlled by isolated complex poles embedded in a continuum, including false-vacuum decay in quantum field theory and cosmology (Feng et al., 16 Dec 2025).

A related technical development replaces the standard $\omega(E)$2 Euclidean projection, which suppresses all excited states, by a bounded-time instanton calculus. In that formulation one computes an endpoint-weighted finite-time amplitude,

$\omega(E)$3

which projects onto a chosen resonant excited state at finite $\omega(E)$4. Because the endpoint wave-function weights are incorporated into a single composite functional integral, the saddle-point expansion yields Robin-type transversality conditions rather than fixed Dirichlet conditions. The resulting excited-state decay widths, including leading quantum corrections, agree with traditional WKB results for arbitrary smooth one-dimensional metastable potentials (Garbrecht et al., 2024).

Late-time asymptotics also depend on decay channel. In multichannel quantum-mechanical and quantum-field-theoretic decay, the exponential lifetime is universal, but the late-time deviations can be channel- or detector-band-dependent. For a near-threshold spectral density with exponent $\omega(E)$5,

$\omega(E)$6

In the multichannel case, the smallest threshold exponent controls the global asymptotic amplitude, but individual channels scale differently. This structure is supported experimentally by erythrosine B fluorescence, where a power-law tail appears after roughly ten lifetimes and two detector bands exhibit the same exponential lifetime, $\omega(E)$7, but different power-law exponents, $\omega(E)$8 and $\omega(E)$9 (Giacosa et al., 21 Sep 2025). Although this is not vacuum decay, it is direct evidence for the generic late-time non-exponential mechanism on which false-vacuum analyses rely.

3. Background-field realizations: electric fields and de Sitter space

In quantum electrodynamics with a static space-dependent electric field,

$E_{\min}$0

the Dirac equation reduces to a Schrödinger-like scattering equation,

$E_{\min}$1

Vacuum pair creation is then mapped to transmission through an effective barrier $E_{\min}$2. In the under-the-barrier regime, real turning points make the problem a tunneling problem; above the barrier, the turning points move into the complex plane and the calculation becomes a coupled-amplitude evolution problem (Dumlu, 2013).

In this scattering approach, the Bogoliubov coefficients encode vacuum persistence and pair creation, and the positron spectrum is read directly from the transmission spectrum. Subcycle structure in the external field produces multiple wells in the effective potential, generating transmission resonances associated with quasi-bound states. When neighboring resonances overlap, their widths broaden, the metastable quasi-bound width $E_{\min}$3 increases, and the tunneling time

$E_{\min}$4

decreases. The collective estimate

$E_{\min}$5

implies that multiple resonant channels accelerate the onset of observable decay. The practical consequence is that vacuum decay may become visible on shorter timescales, potentially before the external field is adiabatically turned off (Dumlu, 2013).

A de Sitter analogue arises in the real-time analysis of particle creation from the Bunch–Davies state. For a massive scalar in spatially flat de Sitter space, the exact Bogoliubov coefficient is

$E_{\min}$6

This implies that the Bunch–Davies state is not the out-vacuum at late times. The associated decay rate of the de Sitter vacuum is

$E_{\min}$7

which for large mass becomes

$E_{\min}$8

In this formulation the created particles are real, persist when the background is switched off adiabatically, contribute positive energy density and pressure, and generate nonzero backreaction already at one loop (Anderson et al., 2017).

Not every late-time vacuum phenomenon in de Sitter should be identified with global vacuum decay. A uniformly accelerated Unruh–DeWitt detector in the Bunch–Davies vacuum emits nonvanishing late-time quantum radiation in the de Sitter radiation zone, but the mechanism is the nonlocal correlation structure of the Bunch–Davies vacuum and its entanglement between different de Sitter regions. The analysis explicitly states that this is not decay of the vacuum in the sense of instability of the de Sitter vacuum itself (Yamaguchi et al., 2018).

4. Cosmological decaying-vacuum frameworks

One cosmological implementation translates the late-time quantum law directly into a time-dependent cosmological term,

$E_{\min}$9

In a flat FLRW universe with matter plus vacuum energy, matter is not separately conserved: $t$0 This construction is presented as equivalent to a class of $t$1CDM models, reducing to standard $t$2CDM when $t$3 or $t$4. Its stated purpose is to connect the quantum-mechanical late-time $t$5 energy relaxation of a false vacuum with the small present value of the cosmological constant and the coincidence problem (Szydlowski et al., 2015).

A second class of models treats the vacuum as an interacting dark-sector component. In the decaying vacuum model,

$t$6

Here $t$7 reproduces standard $t$8CDM scaling, $t$9 means vacuum decays into CDM, and $A(t)=A_{\exp}(t)+A_{\text{non}}(t),$0 means CDM decays into vacuum energy. This directly modifies the late-time matter dilution law and the matter–vacuum balance in the Hubble expansion history (Yan et al., 2018).

A broader running-vacuum family uses the Hubble-dependent ansatz

$A(t)=A_{\exp}(t)+A_{\text{non}}(t),$1

or equivalently

$A(t)=A_{\exp}(t)+A_{\text{non}}(t),$2

In this framework the $A(t)=A_{\exp}(t)+A_{\text{non}}(t),$3 term drives an early quasi-de Sitter inflationary phase, the $A(t)=A_{\exp}(t)+A_{\text{non}}(t),$4 term controls the late-time de Sitter attractor, and the $A(t)=A_{\exp}(t)+A_{\text{non}}(t),$5 term encodes slow running of the vacuum through the radiation and matter eras (Lima et al., 2014, N et al., 2021). One nonsingular realization describes a cosmic history evolving between two de Sitter eras, beginning with an unstable primeval de Sitter phase, passing continuously into radiation and matter domination, and asymptoting to a late-time de Sitter phase. It also yields a continuous “heating up” process rather than sudden reheating, radiation entropy growing from zero to

$A(t)=A_{\exp}(t)+A_{\text{non}}(t),$6

and the vacuum-energy hierarchy

$A(t)=A_{\exp}(t)+A_{\text{non}}(t),$7

from the Gibbons–Hawking temperature normalization (Lima et al., 2014).

Not all decaying-vacuum cosmologies have a de Sitter asymptote. In a horizon-thermodynamic model where the vacuum is coupled to a black-body radiation bath at the apparent-horizon temperature,

$A(t)=A_{\exp}(t)+A_{\text{non}}(t),$8

the late-time evolution approaches

$A(t)=A_{\exp}(t)+A_{\text{non}}(t),$9

that is, an intermediate-inflation-type asymptotic state rather than exact de Sitter. The same model finds an early curvature singularity and, with an additional conserved fluid, the possibility that radiation again dominates over dust at late times (Clifton et al., 2014). This suggests that “late-time vacuum decay” does not select a unique asymptotic cosmology; the late-time fate depends on the interaction mechanism imposed at the effective level.

5. Open-system dynamics, fluctuation-induced decay, and geometric extensions

A different line of work formulates vacuum decay as an open-quantum-system problem. In a closed FRLW universe with scalar-field vacua as the system and spacetime as the environment, the reduced density matrix of the vacuum sector obeys a Redfield equation rather than a strictly Lindblad equation. The resulting master equation describes both the comoving volume fractions of the vacua and the off-diagonal coherence terms. In the Markovian limit, coherence monotonically decreases with time, the reduced state decoheres into a classical mixture, and the diagonal population dynamics reduce to the classical master equation. In the steady state, the dominant vacuum is the one with the smallest cosmological constant (Wang et al., 2023).

This open-system picture is complementary to fluctuation-induced decay mechanisms. For a self-coupled scalar field in a false vacuum, spacetime-averaged fluctuations of the linear operator $A_{\exp}(t)\sim e^{-it(E_M-i\Gamma_M/2)},$0 can act like an effective initial velocity and trigger a classical fly-over of the barrier. The corresponding probability distribution is Gaussian. By contrast, spacetime-averaged quadratic operators such as $A_{\exp}(t)\sim e^{-it(E_M-i\Gamma_M/2)},$1 have a tail that falls more slowly than an exponential when the averaging functions are compactly supported. The analysis concludes that the resulting fluctuation-induced contribution to the decay rate can exceed both ordinary instanton tunneling and the contribution from linear field fluctuations, particularly in the thin-wall regime (Huang et al., 2020).

Vacuum decay can also generate nontrivial background geometry and perturbation spectra. In the pseudo-conformal scenario realized as decay of a conformally invariant metastable vacuum, spontaneous bubble nucleation and subsequent growth reproduce the late-time symmetry-breaking pattern $A_{\exp}(t)\sim e^{-it(E_M-i\Gamma_M/2)},$2. Perturbations of the radial mode yield a red spectrum,

$A_{\exp}(t)\sim e^{-it(E_M-i\Gamma_M/2)},$3

while a spectator dimension-zero field acquires a flat spectrum. The paper’s central claim is that these late-time properties coincide with those of the original homogeneous rolling constructions (Libanov et al., 2015).

A more geometric extension arises in dimension-changing vacuum decay, where a parent vacuum with fewer large dimensions tunnels to a daughter vacuum with an additional decompactified direction. The resulting universe is homogeneous but anisotropic. In this setting, the late-time CMB signal receives both late-time geometric effects and early-time primordial-anisotropy effects. Scalar-sector early-time effects are subdominant, but tensor-sector early-time effects can grow with multipole and dominate over late-time effects, making $A_{\exp}(t)\sim e^{-it(E_M-i\Gamma_M/2)},$4, $A_{\exp}(t)\sim e^{-it(E_M-i\Gamma_M/2)},$5, and $A_{\exp}(t)\sim e^{-it(E_M-i\Gamma_M/2)},$6 correlations particularly sensitive probes of such decay histories (Scargill, 2015).

6. Observational status and present constraints

Phenomenological decaying-vacuum models are already constrained by precision cosmology. In the interacting dark-sector decaying vacuum model, parameter estimation with Planck 2015 temperature and polarization data, BAO, the JLA supernova sample of 740 supernovae, and 30 $A_{\exp}(t)\sim e^{-it(E_M-i\Gamma_M/2)},$7 cosmic chronometer points gives

$A_{\exp}(t)\sim e^{-it(E_M-i\Gamma_M/2)},$8

with

$A_{\exp}(t)\sim e^{-it(E_M-i\Gamma_M/2)},$9

for the full dataset combination. The negative best-fit $$\omega(E)=(E-E_{\min})^\lambda \eta(E),\qquad 0<\lambda<1,$$0 corresponds to dark matter transferring energy into the vacuum sector. The inferred $$\omega(E)=(E-E_{\min})^\lambda \eta(E),\qquad 0<\lambda<1,$$1 reduces the stated tension from about $$\omega(E)=(E-E_{\min})^\lambda \eta(E),\qquad 0<\lambda<1,$$2 to $$\omega(E)=(E-E_{\min})^\lambda \eta(E),\qquad 0<\lambda<1,$$3. In the effective equation-of-state description, the model is generally quintessence-like, with $$\omega(E)=(E-E_{\min})^\lambda \eta(E),\qquad 0<\lambda<1,$$4, although a phantom-like region remains allowed at $$\omega(E)=(E-E_{\min})^\lambda \eta(E),\qquad 0<\lambda<1,$$5 (Yan et al., 2018).

In the quantum-mechanics-motivated $$\omega(E)=(E-E_{\min})^\lambda \eta(E),\qquad 0<\lambda<1,$$6CDM parametrization, fits to Union 2.1 supernovae, $$\omega(E)=(E-E_{\min})^\lambda \eta(E),\qquad 0<\lambda<1,$$7, BAO, CMB, and Alcock–Paczynski data yield $$\omega(E)=(E-E_{\min})^\lambda \eta(E),\qquad 0<\lambda<1,$$8, close to but not exactly equal to unity. The analysis interprets this as a possible signature of decaying vacuum energy, but it simultaneously emphasizes that the effect is too small to be clearly detected with current data and that the present universe remains very close to flat $$\omega(E)=(E-E_{\min})^\lambda \eta(E),\qquad 0<\lambda<1,$$9CDM (Szydlowski et al., 2015).

A more literal late-time tunneling phenomenology has now been confronted with post-recombination distance data and compressed CMB information. In minimal quantum-tunneling models, current BAO and supernova measurements still allow

$$P(t)\sim \frac{\text{const}}{t^{2(\lambda+1)}},\qquad t\to\infty.$$0

for $$P(t)\sim \frac{\text{const}}{t^{2(\lambda+1)}},\qquad t\to\infty.$$1, corresponding to up to a $$P(t)\sim \frac{\text{const}}{t^{2(\lambda+1)}},\qquad t\to\infty.$$2 decrease in total vacuum energy. Once CMB anisotropy constraints from bubble nucleation and domain walls are included, the minimal model allows only an $$P(t)\sim \frac{\text{const}}{t^{2(\lambda+1)}},\qquad t\to\infty.$$3 decrease in total vacuum energy for a transition redshift $$P(t)\sim \frac{\text{const}}{t^{2(\lambda+1)}},\qquad t\to\infty.$$4 of order unity. Nonminimal models with dark-matter conversion and domain-wall production fit the data more successfully, with a preferred transition around

$$P(t)\sim \frac{\text{const}}{t^{2(\lambda+1)}},\qquad t\to\infty.$$5

and about $$P(t)\sim \frac{\text{const}}{t^{2(\lambda+1)}},\qquad t\to\infty.$$6 of dark matter participating in the transition (Bai et al., 28 May 2026).

Several controversies remain active. One concerns extrapolation: the late-time power-law tail is rigorous in the quantum-mechanical setting of unstable states, but its full implementation in field theory and in gravity remains model-dependent (0711.1821). Another concerns terminology: false-vacuum survival, external-field pair creation, effective $$P(t)\sim \frac{\text{const}}{t^{2(\lambda+1)}},\qquad t\to\infty.$$7 cosmologies, and detector-induced late-time radiation are mathematically connected by non-exponential vacuum persistence or vacuum response, but they are not interchangeable observables (Anderson et al., 2017, Yamaguchi et al., 2018). A plausible implication is that progress in late-time quantum vacuum decay will continue to depend on keeping these distinct notions separate while identifying the spectral and dynamical structures they share.