Subcritical bubble prehistory in weak first-order phase transition

Abstract: Standard calculations of cosmological first-order phase transitions usually assume critical bubbles to nucleate on a homogeneous symmetric vacuum background. However, this assumption can fail in weak transitions, where thermal fluctuations trigger subcritical bubbles before the standard nucleation temperature $T_n$. Motivated by this possibility, we systematically examine whether the homogeneous nucleation background approximation is self-consistent. By evolving the Gelmini-Gleiser subcritical bubble kinetics and comparing it with the standard critical bubble nucleation picture, we identify the parameter regions in which the background becomes apparently mixed. A detailed scan of these regions shows that sizable subcritical volume fractions arise when the two phases are nearly degenerate at $T_n$, the potential barrier is low, the difference of free energy between the symmetric and broken phases is moderate and the transition strength is weak. Our analysis further yields a simple criterion, $\log_{10}\hat f_ξ(T_n)\simeq -1.95$, for a percent level subcritical bubble volume fraction. Parameter points above this boundary should be treated as mixed background candidates rather than as ordinary homogeneous bounce points.

• The paper demonstrates how thermal fluctuations form subcritical bubbles that challenge the homogeneous nucleation assumption in weak first-order phase transitions.
• The paper employs the Gelmini-Gleiser kinetic framework and a detailed parameter scan to quantify subcritical bubble volume fractions, classifying regimes from safe to dilute breakdown.
• The paper identifies significant cosmological implications, influencing gravitational wave spectra, baryogenesis, dark matter, and primordial black hole formation.

Subcritical Bubble Prehistory in Weak First-Order Phase Transitions

Overview

"Subcritical bubble prehistory in weak first-order phase transition" (2605.24891) systematically addresses the validity of homogeneous nucleation backgrounds in cosmological phase transition calculations, specifically in the presence of subcritical bubbles generated by thermal fluctuations before the critical nucleation temperature $T_n$. The work quantitatively evaluates the extent to which the standard assumption—that critical bubbles nucleate within a homogeneous symmetric phase—is self-consistent, and establishes numerical criteria identifying when a mixed background must be considered. The analysis leverages the Gelmini-Gleiser kinetic framework for subcritical bubble formation/erasure, implements a detailed parameter scan, and highlights phenomenological implications across cosmological observables.

Motivation and Conceptual Framework

Standard treatments of cosmological first-order phase transitions (e.g., electroweak) assume that critical bubble nucleation occurs in a spatially homogeneous symmetric (false vacuum) background. This is justified when phase transitions are strong; however, in weak transitions, thermal fluctuations induce subcritical bubbles—broken phase regions smaller than the critical size—which may persist and overlap, yielding a mixed background. Such prehistory challenges the standard calculation's initial condition, possibly impacting gravitational waves, baryogenesis, dark matter, and primordial black hole (PBH) formation.

Figure 1: Conceptual comparison of homogeneous nucleation versus nucleation with subcritical bubble prehistory; critical bubbles can nucleate on a mixed background of subcritical bubbles.

Theoretical Formalism

Finite-Temperature Scalar Potential

The analysis employs a minimal single scalar field model:

$$V(\phi, T) = D(T^2 - T_0^2)\phi^2 - E T \phi^3 + \frac{\lambda}{4}\phi^4$$

Here, the structure of the finite-temperature potential determines the emergence of a broken-phase minimum, the barrier, and eventual bubble nucleation.

Critical Bubble Nucleation

The critical bubble profile ($O(3)$ bounce) and its Euclidean action $S_3(T)$ produce a nucleation rate $\Gamma_c(T) \sim T^4 \exp[-S_3(T)/T]$, leading to standard determination of $T_n$ (where one critical bubble forms per Hubble volume). The fraction of converted volume via critical bubbles and the transition strength $\alpha_n$ (vacuum energy released) are tracked for diagnostic purposes.

Subcritical Bubble Kinetics

The homogeneous nucleation assumption is challenged by subcritical bubbles: thermal fluctuations producing localized broken phase regions below the critical size. The Gelmini-Gleiser kinetic equation is solved for the number density $n(R, t)$ of subcritical bubbles of radius $R$, incorporating expansion, formation, shrinkage, reverse nucleation, and erasure. The broken-phase fraction due to subcritical bubbles is given by:

$$p_b^{\rm sub}(T) = 1 - e^{-f_b(T)}, \quad f_b(T) = \int dR \left(\frac{4\pi R^3}{3} n(R, t)\right)$$

The criterion for the validity of the homogeneous assumption is then linked to $$V(\phi, T) = D(T^2 - T_0^2)\phi^2 - E T \phi^3 + \frac{\lambda}{4}\phi^4$$0.

Parameter Scan Methodology

A comprehensive parameter scan explores weak first-order transitions, varying $$V(\phi, T) = D(T^2 - T_0^2)\phi^2 - E T \phi^3 + \frac{\lambda}{4}\phi^4$$1 across theoretical and phenomenological plausible ranges. Each point is retained only if a first-order transition occurs, and multiple scan diagnostics are employed to assess subcritical bubble volume fractions and the extent of mixed backgrounds.

Figure 2: Survey of scan points showing the relationship between potential properties and subcritical bubble volume fraction at nucleation.

Numerical Results and Classification

A fourfold classification emerges from scan results:

• Safe Critical ($$V(\phi, T) = D(T^2 - T_0^2)\phi^2 - E T \phi^3 + \frac{\lambda}{4}\phi^4$$2): Homogeneous background remains valid
• Subcritical Correction ($$V(\phi, T) = D(T^2 - T_0^2)\phi^2 - E T \phi^3 + \frac{\lambda}{4}\phi^4$$3): Minor corrections apply
• Prehistory Relevant ($$V(\phi, T) = D(T^2 - T_0^2)\phi^2 - E T \phi^3 + \frac{\lambda}{4}\phi^4$$4): Mixed background required
• Dilute Breakdown ($$V(\phi, T) = D(T^2 - T_0^2)\phi^2 - E T \phi^3 + \frac{\lambda}{4}\phi^4$$5): Dilute assumption fails, nonlinear effects dominate

Scan results identify the regions in parameter space where subcritical bubbles constitute a non-negligible portion of the volume at $$V(\phi, T) = D(T^2 - T_0^2)\phi^2 - E T \phi^3 + \frac{\lambda}{4}\phi^4$$6. Large subcritical fractions coincide with nearly degenerate vacua at nucleation, low barrier heights, modest free energy splitting, and weak transition strength ($$V(\phi, T) = D(T^2 - T_0^2)\phi^2 - E T \phi^3 + \frac{\lambda}{4}\phi^4$$7).

Figure 3: Correlation between fast single-bin criterion and full kinetic evaluation; threshold for percent-level subcritical bubbles established.

Analytical Criterion for Subcritical Prehistory

A fast, physically motivated single-bin criterion successfully predicts the subcritical fraction:

$$V(\phi, T) = D(T^2 - T_0^2)\phi^2 - E T \phi^3 + \frac{\lambda}{4}\phi^4$$8

This threshold demarcates the transition to mixed backgrounds (percent level volume fraction). The criterion encodes the combined effects of bubble volume, formation rate, and erasure rates, and correlates strongly with the full kinetic numerical results.

Implications for Cosmology

Gravitational Waves

Mixed backgrounds from subcritical bubble prehistory introduce additional small-scale structure, which can impact the high-frequency tail of the gravitational wave spectrum. In weak transitions, where $$V(\phi, T) = D(T^2 - T_0^2)\phi^2 - E T \phi^3 + \frac{\lambda}{4}\phi^4$$9 is small and signals are sound-wave dominated, the subcritical contribution could produce discernible deviations from standard predictions [Bian:2026xdm].

Baryogenesis

In the present parameter scan, sizable subcritical fractions primarily arise in weak transitions ($O(3)$0), so they are unlikely to assist electroweak baryogenesis, which demands stronger transitions to avoid sphaleron washout.

Dark Matter and PBH Formation

Subcritical bubbles modify particle filtering histories during phase transitions, potentially altering relic abundances and exposure timing in filtered dark matter models [Baker:2019ndr]. For PBH formation, stochastic subcritical bubbles can disturb regions that would otherwise persist longer in the symmetric phase, possibly altering PBH abundance and clustering.

Numerical Robustness and Model Dependence

Extensive numerical validation and robustness checks show the classification boundaries are mildly sensitive to kinetic details (e.g., erasure rates, source normalization, shrink velocity), but the qualitative behavior is stable. The scan utilizes a single-field benchmark, but the methodology applies readily to more realistic multi-field models.

Figure 4: Numerical validation of kinetic assumptions; classification robustness against variations in kinetic parameters.

Conclusion

The homogeneous nucleation background assumption underlying standard calculations for cosmological first-order phase transitions fails for sufficiently weak transitions. In these cases, subcritical thermal fluctuations populate the broken phase before the standard nucleation temperature—and the critical bubble can nucleate in a mixed background. The percent-level volume fraction threshold, $O(3)$1, provides a practically useful criterion for distinguishing between regimes of validity. The implications cascade to gravitational wave spectra, filtered dark matter, and PBH formation, motivating the inclusion of subcritical prehistory in cosmological modeling. Future work should extend this formalism to realistic multi-field potentials and implement full real-time simulations in regimes of dilute breakdown.