Gravitational wave energy budget in strongly supercooled phase transitions

TL;DR

This work develops a quantitative framework to dissect gravitational-wave production during strong first-order phase transitions by separating bubble-wall (collisions) and plasma (sound waves and turbulence) sources. It combines lattice simulations with analytic energy-budget arguments and applies the formalism to SM+|H|^6 and a classically conformal U(1)_B-L model, finding wall-driven GW signals are typically subdominant unless strong supercooling is present. The study emphasizes that plasma dynamics largely govern the GW spectrum, though turbulence can enhance off-peak regions and, in strongly supercooled conformal scenarios, bubble collisions can become relevant. The results refine the cosmological evolution during transitions and map detector prospects across current and future gravitational-wave observatories.

Abstract

We derive efficiency factors for the production of gravitational waves through bubble collisions and plasma-related sources in strong phase transitions, and find the conditions under which the bubble collisions can contribute significantly to the signal. We use lattice simulations to clarify the dependence of the colliding bubbles on their initial state. We illustrate our findings in two examples, the Standard Model with an extra $|H|^6$ interaction and a classically scale-invariant $U(1)_{\rm B-L}$ extension of the Standard Model. The contribution to the GW spectrum from bubble collisions is found to be negligible in the $|H|^6$ model, whereas it can play an important role in parts of the parameter space in the scale-invariant $U(1)_{\rm B-L}$ model. In both cases the sound-wave period is much shorter than a Hubble time, suggesting a significant amplification of the turbulence-sourced signal. We find, however, that the peak of the plasma-sourced spectrum is still produced by sound waves with the slower-falling turbulence contribution becoming important off-peak.

Figures (9)

• Figure 1: The Lorentz $\gamma$ factor of the wall of an expanding bubble as a function of time. Different lines correspond to different values of $(\Delta V - \Delta P_{\rm LO})/\Delta P_{\rm NLO}$.
• Figure 2: The solid lines show the energy stored in the bubble wall $E_{\rm wall}$ as a function of time calculated through a lattice evolution for three different initial energies. The dashed lines show the corresponding results obtained by the approximation \ref{['eq:Ewall2']} with $R(t)$ solved from \ref{['eq:gammasol']}, and the dot-dashed one shows the simplest possible approximation \ref{['eq:Ewall2']} with $R(t)=t$ and $\Delta V(t) = \Delta V$.
• Figure 3: The strength of the transition $\alpha$ in the SM+$H^6$ model as a function of the percolation temperature $T_*$ together with $\alpha_\infty$ and $\alpha_{\rm eq}$ from Eqs. \ref{['eq:gammaeq2']} and \ref{['eq:gamma_star']} (upper left panel) . The final size of bubbles at collision $H_* R_*$ and RMS velocity of the plasma $U_f$ used to compute fraction of Hubble time $H_*R_*/U_f$ at which shocks develop in the plasma ending the sound-wave period and starting the turbulence period (upper right panel). The gamma factor a runaway wall would have,$\gamma_*$, together with the terminal gamma factor the bubbles reach in their expansion $\gamma_{eq}$ (lower left panel). The resulting efficiency factors for the sound wave GW signal $\kappa_{sw}$ and the bubble collision signal $\kappa_{col}$ (lower right panel). The gray areas in each panel indicates the area of the parameter space excluded because the bubbles do not percolate, while the light gray area separated by a dashed line indicates where percolation is questionable.
• Figure 4: The GW spectra for representative points in the parameter space of the $SM+H^6$ model. The left-hand panel shows the strongest signal not excluded by lack of percolation, while the strongest signal for which percolation is assured is shown on the right-hand panel. The solid line shows the total signal. The dot-dashed and dotted lines show separately the contributions from sound waves and turbulence. The colored lines and regions show the power-law integrated sensitivities of various current and future detectors.
• Figure 5: The signal-to-noise ratio (SNR) of the GW signals in the $SM+H^6$ model as could be observed in the planned LISA and AION/MAGIS experiments.