High-Temperature Vacuum Decay

• High-temperature vacuum decay is the transition from a metastable (false) state to a stable state, driven by quantum tunneling, thermal activation, and spacetime effects.
• It uses methodologies like the path-integral formalism, sphaleron transitions, and lattice simulations to quantify decay rates and bubble nucleation dynamics.
• The insights gained imply observable consequences, including gravitational wave signatures and modified cosmological expansion in early-universe scenarios.

High-temperature vacuum decay is the phenomenon in which a metastable (false) vacuum state transitions to a more stable (true) vacuum due to quantum and thermal fluctuations, with the rate and dynamics of this process substantially modified by high-temperature environments, spacetime curvature, and field interactions. This topic intersects quantum field theory, statistical mechanics, cosmology, and gravitational physics, with direct relevance for early-universe phase transitions, vacuum metastability, and gravitational wave signatures from first-order transitions.

1. Quantum Survival Amplitudes and Non-Exponential Decay

The survival amplitude ${\mathcal A}(z, z')$ is central to the quantitative analysis of vacuum energy decay. It is defined as the overlap of the vacuum state at two time-slices (or space-like hypersurfaces):

$$A(t_1, t_2) = \langle \text{vac}, t_1 | \text{vac}, t_2 \rangle$$

Within a path-integral formalism:

$$A(t_1, t_2) = \int [D\phi]\, \Psi_{t_1}^*[\phi]\, \Psi_{t_2}[\phi]\, e^{i S[\phi]}$$

This amplitude, for free Minkowski field theory, scales exponentially with time:

$A(t_f, t_i) \propto \exp[ -i V_{n-1} E_0 T ]$

For interacting or curved backgrounds (notably de Sitter space), conformal transformations $\phi \to a^{(n-2)/2}\phi$ are used to absorb the geometry's time dependence into an effective potential $V(z)$, introducing explicit time-dependence in both the mass and couplings.

At late times, the finite-time transition amplitude exhibits power-law suppression:

$\Gamma(Z) \sim A I_{00}(Z) + B I_{01}(Z)$

$$I_{00}(Z) \sim i Z^{-2}, \quad I_{01}(Z) \sim C_n(Z) Z^{-2}$$

This $Z^{-2}$ decay signifies a non-exponential (inverse power-law) tail dominating at late times. Quantum mechanical analysis in generic potentials, due to the boundedness of the spectrum, implies that the survival probability always departs from a strict exponential at very late times, instead falling as $P(t) \sim 1/t^{2(\lambda+1)}$ (Urbanowski et al., 2013).

2. Thermal and Cosmological Effects on Vacuum Decay

Coupling to a thermal bath (or curved backgrounds with horizon temperature, such as de Sitter) induces two key effects:

• Thermal Fluctuations: At high temperature, the decay mechanism progresses from quantum tunneling (instantons) to thermal activation over the barrier (sphaleron transitions) (Masoumi, 2015, Mohamadnejad, 24 Sep 2025). The path integral must be formulated with Euclidean time compactified to period $\beta = 1/T$, causing the dominant bounce to become time-independent at high $T$:

  $$\Gamma_{\text{decay}} \sim T^4 \left( \frac{S_3}{2\pi T} \right)^{3/2} \exp\left( -\frac{S_3}{T} \right)$$

with $S_3$ the three-dimensional action of a static (sphaleron) solution.

• Energy Flow and Cosmological Evolution: When the vacuum is in thermal equilibrium with radiation, there is a continual energy transfer:

  $$\dot{\rho}_r + 4 H \rho_r = Q, \quad \dot{\rho}_v = -Q$$

leading to non-standard expansion. In such scenarios, asymptotic solutions show the Hubble parameter scaling as $H \propto t^{-1/3}$ ("intermediate inflation"), and the vacuum energy (possibly negative at early times) can become positive through continued decay (Clifton et al., 2014). False vacuum remnants give rise to time-dependent contributions to the dark sector energy density:

$\rho_{de}(t) = \Lambda + \frac{\alpha^2}{t^2}$

with the $1/t^2$ term reflecting the long-time tail of vacuum survival (Urbanowski et al., 2013).

3. Decoherence, Quantum Zeno Effect, and Gravitational Suppression

In the presence of a thermal environment and gravity, environmental decoherence can strongly alter the dynamics of bubble nucleation. Gravitationally coupled massless thermal modes (such as photons in de Sitter) effectively "measure" the existence of a critical bubble, suppressing off-diagonal elements of the bubble density matrix. The result is an efficient quantum Zeno effect: the act of frequent decohering "measurements" hinders tunneling transitions.

For the canonical Coleman–de Luccia (CDL) process, the decay probability per unit time receives an extra exponential suppression; the net decay rate obeys:

$\Gamma_{\text{dec}} \sim \Gamma_{CDL}^2$

This dramatic doubling of the exponent ($S_E \to 2S_E$) arises because decoherence events reset the tunneling process (Bachlechner, 2012). The effect is polynomially enhanced by temperature even for $T$ as low as the de Sitter value ($T = H/2\pi$), with the decoherence rate scaling as $\Gamma_{\text{dec}}\propto T^9$ for photon-induced effects.

In contrast, Hawking–Moss decay—where the entire Hubble patch transitions homogeneously—is not efficiently suppressed, since external modes cannot distinguish between false- and true-vacuum patches.

4. Bubble Nucleation: Field Theory, Symmetry, and Exotic Scenarios

• Instanton and Sphaleron Formalism: The nucleation rate in field theory is given semiclassically by the exponential of the bounce (instanton) action $S_E$ derived from the Euclideanized equations of motion, or, at high $T$, by the sphaleron energy divided by temperature.

  $\Gamma \sim A\, e^{-B/\hbar}$

  $$B = \frac{27\pi^2 \sigma^4}{2\epsilon^3} \quad \text{(thin-wall limit)}$$

  with $\sigma$ the bubble wall tension and $\epsilon$ the vacuum energy difference.

• Role of Symmetry and Additional Fields: In multi-field and vector field models, domain wall tension and bubble configurations can be orientation-dependent and may favor non-spherical or "kinky" domain walls, especially if anisotropic "sound speeds" are significant (Masoumi, 2015). This increases the complexity of the nucleation process and can lower the action for decay.
• Compactification and Landscape Decay: In higher-dimensional and flux compactification models, decay corresponds to transitions between moduli vacua or decompactification. The effective four-dimensional potentials exhibit rich structure, and thermal and gravitational corrections strongly influence the decay rates.
• Non-exponential Decay in Low-Dimensional Models: Studies in scaling Ising and tricritical Ising field theories confirm that, while the latent heat determines the exponential sensitivity of the decay rate (as in higher dimensions), prefactors are model-dependent and can vary from standard predictions; exotic vacuum degeneracies (multiple or asymmetric vacua) lead to novel tunneling processes (Lencsés et al., 2022).

5. Black Hole Catalysis and High-Temperature Transitions

• Catalysis by Black Holes: Primordial or astrophysical black holes can locally lower the barrier for vacuum decay, especially if the field couples nonminimally to curvature ($\xi \mathcal{R} h^2$). For negative $\xi$, the energy barrier for bubble nucleation can be nearly eliminated in certain parameter regimes, greatly enhancing decay probability near black holes (Canko et al., 2017). However, in the presence of black holes at high temperature, only $\tau$-independent (static, sphaleron-like) bounces typically contribute; periodic bounces with nontrivial Euclidean time-structure are not dominant (Briaud et al., 2022). This finding is robust beyond the thin-wall limit and extends to multiplet scalar field models.
• Quantum State Dependence: The quantum vacuum around a black hole (Boulware, Hartle–Hawking, Unruh) influences the tunneling suppression exponent. For instance, the Hartle–Hawking vacuum allows for reduced suppression (by up to half in the high-temperature limit) compared to flat space, while the Unruh vacuum can be even less suppressed near the horizon (Shkerin et al., 2021).

6. Real-Time Dynamics, Lattice Simulations, and Experimental Analogs

• Nonequilibrium Real-Time Frameworks: Beyond Euclidean methods, real-time formalisms (two-particle irreducible—2PI—effective actions or parametrized path approaches) treat false vacuum decay as an initial-value problem. The evolution is governed by the field and its fluctuations:

  $$S[\phi] = \int d^4x\, \left[\frac{1}{2}\dot{\phi}^2 - \frac{1}{2} (\nabla \phi)^2 - V(\phi)\right]$$

  $\frac{d p(t)}{dt} = -\Gamma(t) p(t)$

This approach reveals time-dependent decay rates, effective potential evolution, and complex bubble dynamics (Batini et al., 2023, Braden et al., 2018, Michel, 2019). At high temperatures, classical-statistical lattice simulations suffice, but quantum corrections can qualitatively change the decay process—permitting transitions forbidden in the classical approximation.

• Cold-Atom Analog Systems: Ultracold atomic gases can simulate vacuum decay, including both quantum and thermal nucleation, using coupled Bose–Einstein condensates under modulated RF fields. The effective quantum fluctuation spectrum matches relativistic Klein–Gordon theory in the IR, and bubble nucleation rates scale exponentially with the fluctuation amplitude or temperature. By tuning system parameters, both quantum (tunneling) and thermal (high-$T$) regimes can be probed in the laboratory (Jenkins et al., 2023).

7. Gravitational Wave Signatures and Observational Consequences

A first-order (strong) high-temperature vacuum decay proceeds via bubble nucleation, with expanding and colliding bubbles generating a stochastic gravitational wave (GW) background. The key parameters are:

• Transition Strength:

  $$\alpha = \frac{\rho_{\text{vac}}}{\rho_{\text{rad}}}$$

• Inverse Duration:

  $\beta = T \frac{d}{dT} \left(\frac{S_3}{T}\right)$

• Source mechanisms: Collisions, sound waves, and turbulence, with peak frequencies and amplitudes highly sensitive to the parameters above.

Distinct regimes emerge:

• Nanohertz GW signals, detectable by Pulsar Timing Arrays, with large amplitudes but constrained by cosmological observations.
• Millihertz GW signals, within the range of LISA and other space-based interferometers, with amplitudes up to $h^2\Omega_{GW} \sim 10^{-10}$ (Mohamadnejad, 24 Sep 2025).

Thermal (sphaleron-dominated) transitions at high temperature, rather than quantum tunneling, set the dominant vacuum decay rate in these scenarios.

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In summary, high-temperature vacuum decay is characterized by a rich interplay of quantum field theory, statistical and thermal fluctuations, spacetime background effects (including gravity and black holes), non-equilibrium real-time evolution, and phenomenological signatures ranging from late-time energy density relaxation to gravitational wave production. The field continues to integrate developments in analytic, numerical, and experimental approaches, with ongoing connections to cosmic phase transitions, the stability of our vacuum, and the search for new physics via multimessenger astrophysical observables.