Supercooled First-Order Phase Transitions

• Supercooled first-order phase transitions are defined by a system remaining in a metastable high-temperature phase until delayed nucleation occurs far below the equilibrium temperature.
• They critically influence cosmological dynamics by triggering vacuum domination, altering gravitational wave signals, and potentially leading to primordial black hole formation.
• These transitions are studied in both cosmological and condensed matter contexts using thermal and quantum nucleation theories alongside advanced computational models.

Supercooled first-order phase transitions (FOPTs) are transitions in which a system is metastably trapped in a high-temperature (false-vacuum) phase even after the free energy of the low-temperature (true-vacuum) phase has dropped below that of the false vacuum. In both condensed matter and cosmology, this scenario results when the nucleation rate for the true-vacuum bubbles is so small that the system cools significantly (well below the equilibrium, or “critical,” temperature) before the transition actually occurs. Such supercooling leads to substantial consequences for the phase-transition dynamics, gravitational wave (GW) signatures, early-universe expansion, and the possible formation of cosmological relics.

1. Key Concepts and Phenomenology of Supercooled First-Order Transitions

A first-order phase transition proceeds via thermal (or quantum) nucleation of true-vacuum bubbles in a metastable false vacuum. The scalar potential $V(\phi,T)$ has two minima with a barrier in between; at $T=T_c$ the minima are degenerate, but transitions occur only when a critical bubble nucleates and expands. If the nucleation rate $\Gamma_V(T)$ is suppressed, the system supercools to a temperature $T_n \ll T_c$ (nucleation temperature), often remaining trapped until the vacuum energy $\Delta V = V_\text{false} - V_\text{true}$ dominates the energy budget.

In cosmology, this results in an epoch of vacuum domination (“thermal inflation”), which is terminated only when bubble percolation occurs and the released latent heat (from $\Delta V$) reheats the universe. The strength of the transition is characterized by

$$\alpha = \frac{\Delta V}{\rho_R(T_p)}, \quad \rho_R(T) = \frac{\pi^2}{30} g_*(T) T^4$$

with $\alpha \gg 1$ in strong supercooling ($T_p$ is the percolation temperature). The rapidity of the transition is captured by

$$\beta = \left. \frac{d}{dt} \ln \Gamma_V \right|_{t_*}, \qquad \beta/H_* \gg 1$$

where $H_*$ is the Hubble scale at $T_p$.

Typical scenarios in the early universe involve classically scale-invariant or radiatively-broken models, which naturally support large supercooling due to a flat potential and slow logarithmic evolution of the bounce action with $T$ (Levi et al., 2022, Christiansen et al., 4 Nov 2025, Kierkla et al., 2023, Gonçalves et al., 3 Dec 2024).

2. Thermodynamic Evolution and Reheating Dynamics

2.1. Expansion Regimes and Energy Partition

During supercooling, the system typically evolves through three regimes:

• Thermal inflation: The universe becomes vacuum-energy dominated, with negligible radiation. Scale factor grows quasi-exponentially, $a(t) \propto e^{H_\text{vac} t}$, $H_\text{vac}^2 = \Delta V/(3M_P^2)$.
• Matter-dominated reheating: If the field responsible for the transition (e.g., a scalar $\phi$) decays slowly, its oscillations redshift as matter ($\rho_\phi \propto a^{-3}$), and pre-existing radiation is diluted ($\rho_R \propto a^{-4}$). Reheating occurs when $H \sim \Gamma_\phi$, at a temperature $T_\text{reh}$, with $T_\text{reh} \ll T_c$. The maximal temperature ($T_\text{max}$) is typically higher and occurs shortly after percolation.
• Radiation domination: Once $\phi$ decays, the universe is reheated, and customary radiation-dominated expansion resumes.

The coupled Boltzmann-Friedmann equations describing this system are:

$$\frac{d\rho_R}{dt} + 4H\rho_R = \Gamma_\phi \rho_\phi, \quad \frac{d\rho_\phi}{dt} + 3H\rho_\phi = -\Gamma_\phi \rho_\phi, \quad H^2 = \frac{\rho_R + \rho_\phi}{3M_P^2}$$

where initial conditions are $\rho_\phi \approx \Delta V$ at percolation, and $\rho_R \ll \Delta V$ (Gonstal et al., 25 Feb 2025, Li et al., 24 Jan 2025).

2.2. Duration and Completion of Supercooled Transitions

With strong supercooling, percolation can be hindered by sustained vacuum domination: the expansion of the universe can dilute the nucleation probability such that transition completion (i.e., filling most of space with true vacuum) fails, reminiscent of the “graceful exit problem” in old inflation (Athron et al., 2023). Successful transitions require careful tracking of both the fraction of unbroken phase and the decrease in its physical volume.

2.3. Impact on Cosmological Parameters

The duration parameter $\beta/H_*$ directly affects bubble size ($R_* \sim (8\pi)^{1/3}/\beta$ (Christiansen et al., 4 Nov 2025)), and thereby both GW spectra and the possible seeding of inhomogeneities and primordial black holes (PBHs).

3. Gravitational Wave Signatures from Supercooled FOPTs

The supercooled regime provides characteristic GW signals, generated as the bubbles expand and collide:

3.1. GW Sources and Spectral Templates

The main GW emission sources are:

• Bubble collisions (envelope approximation): Dominates when walls “run away.” The peak amplitude scales as $$\sim (H_*/\beta)^2 \left[\kappa \alpha/(1+\alpha)\right]^2$$.
• Sound waves in plasma: Usually dominant unless most energy is in the walls; amplitude $\sim (H_*/\beta) [\kappa_v \alpha/(1+\alpha)]^2$.
• MHD turbulence: Subdominant; amplitude scales as $$(1-H_*\tau_\text{sw}) (\kappa_\text{sw}\alpha/(1+\alpha))^{3/2}$$.

The spectral shape exhibits a broken power law,

$$S(f) \sim \frac{(f/f_\text{peak})^a}{[1+(f/f_\text{peak})^b]^c}$$

with low-frequency rising as $f^3$, and high-frequency falling as $f^{-1}$ for bubble collisions (Yamada, 16 Sep 2025, Kierkla et al., 2023).

The peak frequency, redshifted to today, is given by

$$f_0^\text{peak} \simeq 1.65 \times 10^{-5} \, \text{Hz} \; \left(\frac{\beta}{H_*}\right) \left(\frac{T_\text{reh}}{100\,\text{GeV}}\right) \left(\frac{g_*}{100}\right)^{1/6}$$

while the peak amplitude is

$$\Omega^\text{peak}_\text{GW} h^2 \lesssim 10^{-7}~\kappa^2 \alpha^2$$

for strong supercooling, with $\kappa$ efficiency (Yamada, 16 Sep 2025, Ellis et al., 2019).

3.2. Expansion History Imprint

An early period of matter domination (from slow reheating) imprints a spectral “knee”:

• For $f < f_*$, the slope changes from $f^3$ (radiation) to $f^1$ (matter) and then back to $f^3$ when radiation-dominated era resumes.
• The low-frequency plateau encodes the duration and decay rate $\Gamma_\phi$, allowing GW observatories to probe post-transition expansion (Gonstal et al., 25 Feb 2025).

3.3. GW Parameter Reconstruction

The spectrum is parametrized by either geometric (e.g., $\{\Omega_p, f_p, d\}$) or thermodynamic ($\{\beta/H_*, T_\text{max}, \Gamma_\phi/H_*\}$) variables. Fisher-matrix analysis determines that measurements of the peak and the spectral “knee” enable extraction of these parameters. For a 10% determination of $\Gamma_\phi/H_*$, $SNR \gtrsim 50$ at the peak is required.

4. Supercooled FOPTs in Cosmological and Laboratory Contexts

4.1. Dark-Sector and Conformal Models

Supercooled FOPTs arise naturally in theories featuring radiative symmetry breaking (e.g., classically scale-invariant Abelian Higgs or $U(1)_{B-L}$ models) or hidden $U(1)$, $SU(2)$ sectors (Gonçalves et al., 3 Dec 2024, Costa et al., 26 Jan 2025, Christiansen et al., 4 Nov 2025). This often produces phase transitions at the MeV–TeV scale, compatible with Pulsar Timing Array (PTA) stochastic GW background measurements (NANOGrav, PPTA, EPTA), and leads to distinctive mass hierarchies in the dark sector (e.g., $m_{Z'} \gg m_\phi$) (Costa et al., 26 Jan 2025, Gonçalves et al., 20 Jan 2025).

Laboratory and future collider searches (e.g., NA64, LHCb, intensity-frontier experiments) can probe regions of parameter space where dark-sector couplings are sufficiently large, but many relevant decay widths ($\Gamma_\phi$) are inaccessible except via cosmological probes such as GW observations (Gonstal et al., 25 Feb 2025).

4.2. Model-Building Constraints and Supercooling Window

Realizing a strongly supercooled transition is nontrivial. The “supercooling window” in nearly scale-invariant theories is bounded from below (by the requirement that tunneling completes) and above (by completion during radiation domination) (Levi et al., 2022). Analytical control using 3D dimensionally-reduced EFT (with 2-loop thermal masses, resummed thermal corrections, and RG-improved couplings) is essential for quantitative reliability in the strong-supercooling regime, where high-$T$ expansions break down (Christiansen et al., 4 Nov 2025, Kierkla et al., 2023).

4.3. Phase Transition Completion and Observational Challenges

Incomplete transitions can occur if supercooling is too extreme—exponential expansion may prevent percolation and fail to “gracefully exit” the false vacuum (Athron et al., 2023). Even when percolation occurs, reheating can erase the memory of the low percolation temperature, resetting the universe to the higher energy scale of the underlying physics. This complicates matching the GW signal frequency to the desired nHz range relevant for PTA observations (Athron et al., 2023).

5. Primordial Black Hole Formation from Supercooled FOPTs

Statistical fluctuations in the nucleation time across Hubble patches, enhanced during supercooling, can lead to significant inhomogeneities after reheating. Patches which remain in the false vacuum longest become overdense and can collapse into primordial black holes (PBHs) as the vacuum energy is converted to radiation (Gouttenoire et al., 2023, Lewicki et al., 2023, Flores et al., 20 Feb 2024).

The abundance of PBHs is largely determined by the inverse transition duration $\beta/H$, with a threshold at $\beta/H \lesssim 3.8$ for significant PBH production. The mass scale is controlled by the vacuum energy scale; e.g., $\Delta V^{1/4} \sim 10^5 - 10^8$ GeV yields asteroid-mass PBHs, potentially sufficient for dark matter if $\beta/H \simeq 3.8$ (Lewicki et al., 2023).

6. Laboratory Systems and Condensed-Matter Analogues

Supercooled first-order transitions are not limited to cosmology. In condensed matter:

• Superfluid $^3$He: The A–B superfluid transition provides a clean environment for first-order kinetics, exhibiting large, tunable supercooling. The degree of metastability and supercooling is highly path-dependent and controlled by both bulk and interface nucleation dynamics, sensitive to system thermodynamic history and defect landscapes (Tian et al., 2022).
• Liquid–liquid and glass transitions: Liquid–liquid transitions (e.g., in water, ionic solutions, glasses) can be first-order, as revealed by heat-capacity spikes and latent heat, with supercooling leading to novel glassy phases. Extensions of classical nucleation theory (e.g., including enthalpy-saving terms) quantitatively reproduce observations in diverse glass-formers (Woutersen et al., 2016, Tournier, 2018, Tournier, 2019).

7. Outlook and Observational Prospects

Supercooled FOPTs can have profound consequences for early-universe cosmology, the spectrum of relic gravitational waves, and the possible existence and properties of primordial black holes. Current and future GW detectors (LISA, Einstein Telescope, SKA, etc.) have the potential to extract not only the thermal-history parameters (like $\beta/H_*$ and $T_\text{reh}$) but also microphysical quantities such as the scalar decay width $\Gamma_\phi$, which are typically inaccessible to accelerator experiments (Gonstal et al., 25 Feb 2025, Yamada, 16 Sep 2025). Quantitative agreement between different computational methods (4D high-$T$ expansion, 3D EFT, RG-improved potentials) is essential for precise inference (Christiansen et al., 4 Nov 2025, Kierkla et al., 2023).

The phenomena of supercooled first-order phase transitions thus provide a unique probe of both the microphysics of new sectors beyond the Standard Model and the cosmological evolution of the early universe, tightly connecting theoretical modeling with GW astrophysics and experimental particle physics.