First-Order Phase-Transition Backgrounds

A cosmological first-order phase transition does not have one universal gravitational-wave spectrum. Bubble nucleation supplies a length scale, but the amplitude and shape depend on where the released energy goes: coherent scalar-field gradients, acoustic motion, or vortical and magnetohydrodynamic turbulence. The central task is therefore to translate a transition model into a source stress tensor and to test whether the assumed source lifetime and hydrodynamic regime are self-consistent.

Helpful background. The Einstein Equation supplies Einstein Equations for sourced tensor modes; Relativistic Matter and Compact Objects supplies stress-energy and fluid language; Gravitational Radiation and Waveforms supplies stochastic-spectrum conventions; Relativistic Cosmology supplies the expansion and thermal history; and Partial Differential Equations and Green Functions supplies the retarded-source construction.

Nucleation fixes a scale, not a complete spectrum

For a thermal transition, the bubble nucleation rate per physical volume is

$$\Gamma(t)=A(t)e^{-S(t)}, \qquad \beta\equiv\frac{d\ln\Gamma}{dt} =\frac{d\ln A}{dt}-\frac{dS}{dt} \simeq-\frac{dS}{dt}.$$

The last approximation requires the prefactor to vary slowly compared with the exponential, $\lvert d\ln A/dt\rvert\ll\lvert dS/dt\rvert$. The usual thermal estimate applies when $S\gg1$ and $\beta\gg H_*$. In that regime, an exponentially rising rate gives the effective source-scale estimate

$$R_*\simeq(8\pi)^{1/3}\frac{\max(v_w,c_s)}{\beta}$$

used by the acoustic model at percolation. Here $H_*$ is the Hubble rate, $v_w$ the wall speed, and $c_s$ the plasma sound speed at the transition. This prescription replaces $v_w$ by $c_s$ for subsonic deflagrations to represent the fluid disturbance ahead of the wall. It must not be mistaken for an exact count of bubble centers; a separation computed from the actual bubble number density is a distinct input. The formula is not a definition of $R_*$. It fails for sufficiently slow or strongly supercooled transitions with $\beta/H_*\sim1$; then the nucleation and false-vacuum volume histories must be integrated directly. A candidate model must also demonstrate that bubbles percolate before vacuum-driven expansion prevents completion. These distinctions are developed in Caprini et al. 2020, §2.1, printed pp. 5–7, eqs. (1)–(6) (Open PDF).

The second input is an energy budget. In the bag equation of state one often writes

$$\alpha=\frac{\epsilon}{a_+T_*^4}, \qquad K=\frac{\kappa\alpha}{1+\alpha},$$

where $\epsilon$ is the bag-model vacuum-energy difference, $a_+$ is the high-temperature phase’s radiation constant (not the cosmological scale factor), and $K$ is the fraction of total energy placed in bulk-fluid kinetic energy. Outside the bag model, a useful strength variable is $\alpha_\theta=4\Delta\theta/(3w_+)$, where $\theta=(e-3p)/4$ and $w_+=e_++p_+$ is the enthalpy density ahead of the wall, not the cosmological equation-of-state ratio $p/e$. An efficiency fit $\kappa(\alpha,v_w)$ must be paired with the same definition of $\alpha$ used to construct it. Mixing a latent-heat definition, a trace-anomaly definition, and a bag-model efficiency can shift an amplitude without changing any displayed formula. Caprini et al. 2020, §2.2, printed pp. 7–9, eqs. (7)–(13) (Open PDF) gives this dictionary and its equation-of-state boundary.

The acoustic template has a lifetime condition

For a non-runaway transition in a coupled relativistic plasma, simulations often support an acoustic source. A commonly used present-day template is

$$\Omega_{{\rm GW},0}(f) =0.687\,F_{{\rm GW},0}K^2 \left(\frac{H_*R_*}{c_s}\right) \widetilde\Omega_{\rm GW} \,C\!\left(\frac{f}{f_{p,0}}\right),$$

with

$$C(s)=s^3\left(\frac{7}{4+3s^2}\right)^{7/2}, \qquad f_{p,0}\simeq26\,\mu{\rm Hz}\, \frac{1}{H_*R_*} \left(\frac{z_p}{10}\right) \left(\frac{T_*}{100\,{\rm GeV}}\right) \left(\frac{g_*}{100}\right)^{1/6}.$$

$F_{{\rm GW},0}$ is the radiation-and-entropy redshift factor, $\widetilde\Omega_{\rm GW}$ is the simulation-calibrated source efficiency, and $z_p$ locates the dimensionless peak. The $f^3$ infrared slope is the causal acoustic limit of this fit; $C(s)\propto f^{-4}$ at high frequency. The cited source writes this per-log-frequency spectrum as $d\Omega_{{\rm GW},0}/d\ln f$; here it is translated to the site’s convention

$$\Omega_{{\rm GW},0}(f) \equiv\rho_{c,0}^{-1}\frac{d\rho_{{\rm GW},0}}{d\ln f},$$

so the template display is not a second logarithmic derivative. Neither exponent is a model-independent fit to every first-order transition. In particular, thin sound shells near the Chapman–Jouguet velocity introduce another scale and invalidate the single-shape ansatz. The fit and its simulation domain are stated in Caprini et al. 2020, §3.3, printed pp. 16–17, eqs. (29)–(32) (Open PDF).

The formula above assumes acoustic stresses persist for roughly a Hubble time. Shocks instead form after approximately

$$\tau_{\rm sh}\sim\frac{R_*}{\overline U_{\parallel}}, \qquad \overline U_{\parallel}\sim K^{1/2}.$$

If $H_*\tau_{\rm sh}\lt1$, multiplying the long-lived estimate by the lifetime factor gives the parametric replacement

$$K^2\frac{H_*R_*}{c_s} \longrightarrow K^{3/2}\frac{(H_*R_*)^2}{c_s}.$$

This is not a small cosmetic correction: it changes the dependence on both transition strength and duration. It also marks the onset of nonlinear flow, where some acoustic energy may feed turbulence. Caprini et al. 2020, §§3.1–3.2, printed pp. 11–16, eqs. (20)–(28) and Table 1 (Open PDF) derives the relevant coherence, shock, and eddy-turnover timescales.

Competing source pictures

Source description	Control parameters	Principal failure mode
Hubble-lived sound waves	$K$, $H_*R_*$, $c_s$, sound-shell shape	Overpredicts when shocks form at $H_*\tau_{\rm sh}\lt1$
Short-lived acoustic source	The same inputs plus $\tau_{\rm sh}$	Nonlinear conversion to vortical flow is not captured by a mere lifetime factor
MHD turbulence	Vortical fraction, eddy scale, unequal-time correlator, magnetic helicity	Decay and decorrelation prescriptions give materially different amplitudes and slopes
Scalar-field collisions	Wall-gradient energy and collision geometry	Envelope-like models are unreliable when plasma friction gives a terminal wall speed
Model-independent broken power law	Peak, amplitude, slopes, and widths	Fit parameters need not map uniquely to $\alpha$, $v_w$, $R_*$, or a particle model

A runaway wall can retain enough scalar-gradient energy for collision emission to dominate. A non-runaway wall instead transfers much of the released energy to the plasma. The wall speed is an out-of-equilibrium friction problem, not a quantity fixed by equilibrium thermodynamics. Turbulence is physically plausible after shocks, but its unequal-time stress correlator and magnetic evolution remain less securely calibrated than the acoustic template. The mechanisms and their older analytic/simulation boundaries are reviewed in Caprini and Figueroa 2018, §§8.1–8.7, printed pp. 134–151 (Open PDF).

Results and limitations through 2026-08-19

Formulation or evidence	Result	Scope or limitation
Analytic causality and energy budget	Relates the infrared rise and overall scale to a finite causal source with $K\leq1$	Does not determine the wall friction, lifetime, or unequal-time correlator
Relativistic fluid simulations summarized by Caprini et al. 2020	Support a sound-shell peak and calibrate $\widetilde\Omega_{\rm GW}$ in simulated regimes	Finite boxes, simultaneous nucleation approximations, limited run times, and thin-shell regimes restrict extrapolation
Shock-lifetime analysis in Caprini et al. 2020	Replaces the Hubble-lived scaling when $H_*R_*/\sqrt K\lt1$	A lifetime factor alone does not predict the subsequent turbulent spectrum
Scalar-field collision calculations reviewed by Caprini–Figueroa 2018	Supply a distinct template when wall gradients remain an important stress	Applicability depends on runaway behavior and plasma coupling
Turbulence models and early simulations	Show that vortical and magnetic stresses can contribute	Amplitude, decorrelation law, helicity dependence, and high-frequency slope remain unresolved
LVK O1–O4a first-order-transition search	Finds no evidence for either tested sound-wave-plus-CBC or bubble-collision-plus-CBC component	The result is conditional on the detector band, fixed CBC slope, transition templates, and Table I priors; it is not a source-independent exclusion

The LVK analysis fixes the compact-binary power-law slope, varies its amplitude with the transition parameters, and reports no evidence for an added sound-wave or bubble-collision component under those priors (LIGO–Virgo–KAGRA 2025, §II.C, printed pp. 4–5, and Table I (Open PDF)). The detector likelihood and mutable numerical contours remain within Gravitational-Wave Astronomy.

The comparison is deliberately conditional. A spectrum computed from one parameter regime is not evidence that the same template applies to a strongly supercooled transition, a hidden sector with different hydrodynamics, or a runaway wall.

What could change the conclusion. The interpretation could change after long-duration relativistic hydrodynamic or magnetohydrodynamic simulations, controlled wall-friction calculations, or new detector likelihoods jointly recalibrate the acoustic lifetime, source partition, peak amplitude, or spectral slopes beyond the boundaries stated above.

An energy-budget and redshift check

Before trusting a forecast, verify four linked statements: percolation completes; the chosen $\alpha$ and $\kappa$ define the displayed $K$; the source lifetime satisfies the template used; and $T_*$ is the temperature after energy release that controls redshifting. For strong supercooling, $T_*$ need not equal the nucleation temperature. The dimensionless spectrum must also respect that no source contribution can use more kinetic or gradient energy than the transition supplies.

These checks constrain a theory prediction without importing an instrument likelihood. Current search sensitivities and source-confusion analyses belong to Gravitational-Wave Astronomy.

Exercises

Take $H_*R_*=0.01$, $K=0.04$, $c_s=1/\sqrt3$, $T_*=100\,{\rm GeV}$, $g_*=100$, and $z_p=10$. Decide whether the Hubble-lived acoustic formula is self-consistent. Find the peak frequency and the ratio of the short-lived to long-lived amplitude scalings.

Solution

The shock estimate gives

$$H_*\tau_{\rm sh}\sim\frac{H_*R_*}{\sqrt K} =\frac{0.01}{0.2}=0.05.$$

The acoustic source therefore shocks well within one Hubble time. The long-lived template is not self-consistent for these inputs. The peak is

$$f_{p,0}\simeq26\,\mu{\rm Hz}\,(0.01)^{-1} =2.6\,{\rm mHz}.$$

At fixed shape and redshift factor, the ratio is

$$\frac{K^{3/2}(H_*R_*)^2/c_s} {K^2(H_*R_*)/c_s} =\frac{H_*R_*}{\sqrt K}=0.05.$$

Thus the naive Hubble-lived amplitude is twenty times larger than the short-lifetime scaling. That factor is a diagnostic of the template assumption, not a complete turbulence calculation.

References

• Caprini, Chiara, and Daniel G. Figueroa. 2018. “Cosmological Backgrounds of Gravitational Waves.” Classical and Quantum Gravity 35, 163001. DOI.
• Caprini, Chiara, et al. 2020. “Detecting Gravitational Waves from Cosmological Phase Transitions with LISA: An Update.” Journal of Cosmology and Astroparticle Physics 03, 024. DOI.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration. 2025. “Cosmological and High Energy Physics Implications from Gravitational-Wave Background Searches in LIGO–Virgo–KAGRA’s O1–O4a Runs.” arXiv:2510.26848v2. Open preprint.