Gravitational waves from first order cosmological phase transitions in the Sound Shell Model

TL;DR

The paper develops the Sound Shell Model to predict gravitational wave spectra from first-order cosmological phase transitions, focusing on acoustic gravitational wave production from sound waves generated by expanding bubbles.It identifies two fundamental length scales, the mean bubble separation $R_*$ and the fluid-shell thickness $\Delta R_*$, and shows how the nucleation-rate history parameterized by $\beta$ shapes the spectrum, with the overall frequency scale set by the nucleation temperature $T_n$.An improved double-broken power-law fit is derived to describe the GW spectrum across the relevant wavenumber range, and predictions are compared to 3D hydrodynamic simulations, showing strong agreement for detonations but systematic over-prediction for deflagrations due to kinetic-energy suppression and collision-phase effects.The work highlights the importance of nucleation history (exponential vs simultaneous) for peak amplitude and position, discusses limitations from non-linear evolution, and outlines directions for future simulations and model refinements to enable robust LISA-era parameter estimation.

Abstract

We calculate gravitational wave power spectra from first order early Universe phase transitions using the Sound Shell Model. The model predicts that the power spectrum depends on the mean bubble separation, the phase transition strength, the phase boundary speed, with the overall frequency scale set by the nucleation temperature. There is also a dependence on the time evolution of the bubble nucleation rate. The gravitational wave peak power and frequency are in good agreement with published numerical simulations, where bubbles are nucleated simultaneously. Agreement is particularly good for detonations, but the total power for deflagrations is predicted higher than numerical simulations show, indicating refinement of the model of the transfer of energy to the fluid is needed for accurate computations. We show how the time-dependence of the bubble nucleation rate affects the shape of the power spectrum: an exponentially rising nucleation rate produces higher amplitude gravitational waves at a longer wavelength than simultaneous nucleation. We present an improved fit for the predicted gravitational wave power spectrum in the form of a double broken power law, where the two breaks in the slope happen at wavenumber corresponding to the mean bubble separation and the thickness of the fluid shell surrounding the expanding bubbles, which in turn is related to the difference of the phase boundary speed from the speed of sound.

Figures (10)

• Figure 1: Self-similar radial velocity $v$ and enthalpy density $w$ profiles as functions of the scaled radius $\xi = R/T$, where $R$ is the distance from the bubble centre and $T$ is the time since nucleation, for wall speeds $v_\text{w} = [0.44,0.56,0.68,0.80,0.92]$ and phase transition strengths $\alpha_n = 0.0046$ (left) and $\alpha_n = 0.05$ (right, where the third wall speed is adjusted to $v_\text{w} = 0.731$ to better match the simulations in Hindmarsh:2017gnf). The black dashed line is the curve in $(\xi,v)$ and $(\xi,w)$ planes where shocks must occur. The dash-dot line indicates the maximum possible fluid velocity behind a wall in all cases, and the maximum possible enthalpy behind a detonation. See Appendix \ref{['s:HydExpBub']} for more details.
• Figure 2: Single-bubble plane wave power spectra. Left are weak strength phase transitions, with (top to bottom) $v_\text{w} = 0.92$, $0.56$ and $0.44$. Right are intermediate phase transitions, with $v_\text{w} = 0.92$, $0.56$ and $v_\text{w} = 0.44$. See Eqs. (\ref{['e:fDef']}), (\ref{['e:LamDef']}) and (\ref{['e:ADef']}) for definitions of the quantities plotted.
• Figure 3: Bubble collision. A bubble nucleated at time $t'$ at distance $L$ from the advancing phase boundary $S(t')$ (thin dashed line). The bubble wall and the phase boundary both move at speed $v_\text{w}$. At time $t' + L/2v_\text{w}$ the bubble makes contact with the boundary (thick dashed lines). At time $t' + L/v_\text{w}$ the boundary would have reached the nucleation site, had the nucleation not taken place, and approximately half the bubble has been destroyed (solid lines). We take this time to mark the end of the bubble, so that its lifetime is $T = L/v_\text{w}$, when its radius is $R = L$.
• Figure 4: Velocity power spectra for detonations. Predictions of the sound shell model with simultaneous nucleation are shown in blue, with exponential nucleation in red. Left are weak strength phase transitions, with $v_\text{w} = 0.92$, $0.80$ and $0.68$. Right are intermediate phase transitions, with $v_\text{w} = 0.92$, $0.80$ and $v_\text{w} = 0.731$.
• Figure 5: Velocity power spectra for deflagrations. Predictions of the sound shell model with simultaneous nucleation are shown in blue, with exponential nucleation in red. Left are weak phase transitions, with $v_\text{w} = 0.56$ and $0.44$. Right are intermediate transitions with the same wall speeds.