Bubble Wall Terminal Lorentz Factor

• Bubble wall terminal Lorentz factor defines the steady-state relativistic boost of an expanding phase transition bubble wall, balancing the vacuum-energy driving pressure and frictional forces.
• The determination of γ_w employs quantum kinetic theory, hydrodynamic equations, and non-local collision integrals to quantify friction from multiple sources.
• Calculated terminal boosts directly affect predictions for gravitational wave spectra, baryogenesis efficiency, and the generation of heavy relic particles in the early universe.

A bubble wall terminal Lorentz factor is a key dynamical parameter for cosmological phase transitions, denoting the steady-state relativistic boost γ_w ≡ (1–v_w²)^–½ of an expanding bubble wall after it equilibrates under the competing influences of vacuum-energy driving pressure and frictional backreaction from surrounding plasma or background fields. Its quantitative determination requires a precise accounting of all friction sources (hydrodynamic, kinetic, and quantum) and highly model-dependent microphysics. Terminal wall boost directly affects the efficiency of gravitational wave production, baryogenesis, and out-of-equilibrium particle generation. Recent research has advanced the theoretical computation of γ_w using first-principle quantum kinetic theory, non-local collision effects, and hydrodynamic obstruction constraints.

1. Theoretical Framework: Quantum Kinetic and Fluid Dynamics

Modern calculations of the bubble wall terminal Lorentz factor employ a combined kinetic and fluid treatment (Ramsey-Musolf et al., 18 Apr 2025). The dynamics is encoded in coupled Schwinger–Dyson (Kadanoff–Baym) equations for the two-point correlators of wall and plasma fields—cast in Wigner space to yield transport equations for quasiparticle distributions f(k,X). Under gradient expansion and appropriate approximations, these equations reduce to a local Boltzmann equation: $$\frac{p^{\mu}}{E_{p}}\partial_{\mu}f + \partial_{p^{i}}\left(F^{i}f\right) = \mathcal{C}_{\rm fluid}[f]+\mathcal{C}_{\rm micro}[f]$$ where the ‘fluid’ collision integral $\mathcal{C}_{\rm fluid}$ preserves four-momentum and encapsulates local equilibrium restoring processes, while ‘microscopic’ collisions $\mathcal{C}_{\rm micro}$ (such as non-conserving momentum transfer due to mode-function effects) generate genuinely nonlocal friction. The force term arises from particle-mass variation across the wall $F^{z}=-\frac{\partial_{z}m^{2}(z)}{2E_{p}}$.

The equation-of-motion for the wall background field, typically obtained from the two-particle-irreducible effective action or direct QFT derivations, includes both elastic (mass-gain), inelastic (production/mixing), and transition-radiation friction terms (Ai et al., 18 Apr 2025).

2. Sources and Scaling of Bubble Wall Friction

All frictional pressures contributing to the γ_w terminal value can be organized as follows:

Friction Mechanism	Scaling with γ_w	Dominance Regime
Hydrodynamic (fluid)	saturates/γ_w^0	subsonic, LTE
1→1 Transmission (mass-gain)	γ_w^0	ballistic regime
1→2 Splitting (transition-radiation)	log(γ_w)	intermediate/large γ_w
2→2 Scattering (nonlocal)	linear: γ_w	ultra-relativistic
Condensate-particle vertex production	log(γ_w)	ultrarelativistic, thin wall
Gauge boson emission (LL resummed)	γ_w^2	ultra-relativistic, thin wall

In the ballistic (ultra-relativistic) regime ($\gamma_w \gg 1$), the 2→2 scattering (from non-conserving Δp_z processes) dominates friction and grows linearly: $P_{2\to2}(\gamma_w) \simeq C_{2\to2}\gamma_w T^4$ where $C_{2\to2}$ is process-dependent and typically $\sim10^{-3}$ for weak-scale scalar portals (Ramsey-Musolf et al., 18 Apr 2025).

In models with soft gauge boson emissions, the friction can scale quadratically: $P(\gamma) \sim C\,g^2\,T^4\,\gamma^2$ with C ≈ 5×10^–3 for SM electroweak content after leading-log resummation (Hoeche et al., 2020).

Splitting and gauge radiation processes lead to subdominant $\ln\gamma_w$ corrections.

3. Equilibrium Force Balance and Terminal γ_w Formulas

The steady-state wall velocity is found by solving the pressure balance: $$P_{\rm drive} = P_{\rm friction}(\gamma_{w, \rm term})$$ For generic first-order phase transitions, the driving pressure is

$$P_{\rm drive} = \Delta V = V(\phi_{\rm sym}) - V(\phi_{\rm brk})$$

and friction, as constructed above, includes all relevant contributions. In the large-γ_w limit,

$$\gamma_{w,\rm term} \simeq \frac{\Delta V/T^4 - c_{\rm fluid}}{C_{2\to2}}$$

or, when 2→2 linear friction dominates over the logarithmic and non-relativistic terms,

$$\gamma_{w,\rm term} \approx \frac{\Delta V}{C_{2\to2}\,T^4}$$

assuming negligible fluid friction or incorporating it into ΔV (Ramsey-Musolf et al., 18 Apr 2025).

For scenarios dominated by transition radiation (soft gauge boson emissions), the leading-log resummed formula is

$$\gamma_{\rm term} = \sqrt{\frac{\Delta V}{C\,g^2\,T^4}}$$

with C and g model-dependent (Hoeche et al., 2020).

4. Phenomenological Model Dependence and Scaling Estimates

Numerically, weak-scale scalar portal models with couplings λ∼1 and mass differences Δm_Φ∼m_Φ∼10T yield C_2→2∼10^–3, driving terminal boosts

$$\gamma_{w,\rm term} \sim 10^3 \times (\Delta V/T^4)$$

For ΔV/T^4 ≈ 0.1–1, this gives $\gamma_w \sim 10^2$ – $10^3$. Symmetry-restoring transitions can introduce negative friction at intermediate boosts—prolonging acceleration and pushing γ_term up by O(10)–20× compared to symmetry-breaking cases (Long et al., 13 Nov 2025).

In scenarios where 1→2 splitting friction is the dominant term, the terminal boost is found from transcendental/logarithmic equations, e.g.,

$$\gamma_w = \frac{L_w\,m_\phi^2}{0.26\,T}\, \Biggl[ \exp\!\left(\frac{\Delta V}{1.4\times10^{-5}\,\lambda^2\,v_b^2\,T^2}\right) -1 \Biggr]$$

Though in realistic models, 2→2 or gauge emission friction is usually larger, setting the main velocity scale (Ai, 2023).

Hydrodynamic constraints, including the nucleation-to-critical temperature ratio, can cap the maximal possible steady-state v_w and hence γ_w beyond which no stationary wall solution exists—in particular for deflagration/hybrid modes (Krajewski et al., 2023).

5. Runaway Criterion and Its Exclusion by Genuine Nonlocal Friction

A central question for cosmological transitions is whether the bubble wall can ‘run away’ (γ_w → ∞) or is always stabilized. The criterion is: $\Delta V > P_{\rm friction}^{(\infty)}$ where $P_{\rm friction}^{(\infty)}$ is the friction saturated value as γ_w → ∞. When genuine nonlocal friction (e.g., from 2→2 scattering or gauge transition radiation) grows with γ_w, it always balances any fixed driving pressure, excluding true runaway solutions (Ramsey-Musolf et al., 18 Apr 2025, Hoeche et al., 2020). For splitting-only friction ($\ln\gamma_w$), the wall can in principle run away for sufficiently large ΔV.

6. Cosmological and Observational Implications

The value of γ_w directly influences bubble collision energy budgets, the resulting gravitational wave spectrum, baryogenesis scenarios (e.g., suppression of charge transport for large γ_w), and production thresholds for superheavy particles in the early universe (Long et al., 13 Nov 2025). In scenarios with symmetry-restoring transitions, enhanced terminal boosts can elevate high-frequency gravitational wave peaks and allow access to heavier relic states.

Accurate first-principles computation of γ_w—especially incorporating all microphysical friction sources—remains vital for precise predictions in cosmological phenomenology. The terminal Lorentz factor is thus an indispensable input for theoretical and numerical forecasts of gravitational wave signals and baryogenesis yields in models of strong first-order transitions.