Gravitational waves from strong first order phase transitions

TL;DR

The paper investigates gravitational wave production from strong first-order phase transitions by performing large-scale 3D simulations of a scalar order parameter coupled to a relativistic fluid, examining two representative transitions: a detonation with $v_w=0.92$ and $\alpha_n=0.67$, and a deflagration with $v_w=0.44$ and $\alpha_n=0.5$. By analyzing velocity and shear-stress power spectra, unequal-time correlators, and the evolution of enthalpy-weighted velocities, the authors develop a Gaussian-velocity-based framework to predict the GW power spectrum from velocity UETCs, finding a robust asymptotic GW efficiency of about $\tilde{\Omega}_{gw}^\infty \simeq 0.017$ for both cases. They show compressional modes dominate GW production while vortical modes are subdominant, and that non-linear effects such as shocks and reheating significantly shape the kinetic energy decay and the resulting GW signal, with implications for the sound-shell model and potential electroweak baryogenesis scenarios. The results yield present-day GW density predictions scaling with $H_n R_*$ and the peak frequency near $f_p \approx 26\,(H_n R_*)^{-1}$ μHz, highlighting the importance of flow lifetime and scale evolution in modeling GWs from strong phase transitions.

Abstract

We study gravitational wave production at strong first order phase transitions, with large-scale, long-running simulations of a system with a scalar order parameter and a relativistic fluid. One transition proceeds by detonations with asymptotic wall speed $v_\text{w}=0.92$ and transition strength $α_n=0.67$, and the other by deflagrations, with a nominal asymptotic wall speed $v_\text{w}=0.44$ and transition strength $α_n=0.5$. We investigate in detail the power spectra of velocity and shear stress and - for the first time in a phase transition simulation - their time decorrelation, which is essential for the understanding of gravitational wave production. In the detonation, the decorrelation speed is larger than the sound speed over a wide range of wavenumbers in the inertial range, supporting a visual impression of a flow dominated by supersonic shocks. Vortical modes do not contribute greatly to the produced gravitational wave power spectra even in the deflagration, where they dominate over a range of wavenumbers. In both cases, we observe dissipation of kinetic energy by acoustic turbulence, and in the case of the detonation an accompanying growth in the integral scale of the flow. The gravitational wave power approaches a constant with a power law in time, from which can be derived a gravitational wave production efficiency. For both cases this is approximately $\tildeΩ^\infty_\text{gw} \simeq 0.017$, even though they have quite different kinetic energy densities. The corresponding fractional density in gravitational radiation today, normalised by the square of the mean bubble spacing in Hubble units, for flows which decay in much less than a Hubble time, is $Ω_{\text{gw},0}/(H_\text{n} R_*)^2=(4.8\pm1.1)\times 10^{-8}$ for the detonation, and $Ω_{\text{gw},0}/(H_\text{n} R_*)^2=(1.3\pm0.2)\times 10^{-8}$ for the deflagration.

Figures (10)

• Figure 1: Slices through simulations. Left: detonation at time $t=5000/T_c$. Right: deflagration at time $t=7700/T_c$. Top panels show the magnitude of the 3-velocity $|{\mathbf{v}}|$, middle panels the magnitude of the divergence of spatial components of the 3-velocity $|\nabla \cdot {\mathbf{v}}|$, and the bottom panels showcase the magnitude of the curl $|\nabla \times {\mathbf{v}}|$.
• Figure 2: Selected volume-averaged quantities for the detonation ($\alpha_\text{n}=0.67$, $v_\text{w}=0.92$, left-hand-side) and deflagration ($\alpha_\text{n}=0.50$, $v_\text{w}=0.44$, right-hand-side). Top: wall speed estimates $v_\text{w}$. The velocity estimate $v_{w,\phi}$ is computed from the ratio of kinetic and gradient energy of the scalar field (purple dotted line), and $v_{w,N}$ through the rate of change of number of sites in broken phase and number of links (purple solid line). The fraction of sites in the broken phase $f_\text{b}$ is plotted in olive green, and a dot-dashed vertical green line indicates when 95% of the sites are in the broken phase (this also marks $t_{\text{ref}}$, the reference time for unequal time correlators). Middle: root mean square enthalpy-weighted 4-velocities for compressional $\bar{U}_\parallel$ (in red) and vortical $\bar{U}_\perp$ (in blue) components, along with their non-relativistic equivalents $\bar{v}_{\parallel,\perp}$. The total RMS weighted 4-velocity $\bar{U}$ is also shown. Bottom: integral scales, for the compressional and vortical modes, defined in Eq. \ref{['e:IntScaDef']}. We also display fitted decay and growth indexes ($\zeta, \lambda$, respectively) for compressional kinetic energy and integral scale.
• Figure 3: Power spectra for the detonation ($\alpha_\text{n}=0.67$, $v_\text{w}=0.92$, left column) at times $tT_\text{c} = 880, 1720, 2560, 3400,4240$, and the deflagration ($\alpha_\text{n}=0.50$, $v_\text{w}=0.44$, right column), at times $t T_\text{c} = 1440, 2800, 4160, 5520, 6880$. The times in units of the shock appearance time are given in the legend. Top: power spectra of compressional modes $\mathcal{P}_{v_\parallel}$ (pastel blue to cyan) and vortical modes $\mathcal{P}_{v_\perp}$ (pastel pink to pink). Middle: power spectra of the shear stress $\mathcal{P}_\Pi$ (light teal to teal) in units of $\bar{\epsilon}^2$. Bottom: the gravitational wave power spectra $\mathcal{P}_\text{gw}$ (pastel orange to orange) in units of $(H_*R_*)^2$.
• Figure 4: Decorrelation functions (defined in Eq. \ref{['e:DecFunDef']}) of $v_\parallel$ (top panels), $v_\perp$ (middle row) and shear stress $\Pi$ (bottom panels) for the detonation ($\alpha_\text{n}=0.67$, $v_\text{w}=0.92$, left-hand-side) and deflagration ($\alpha_\text{n}=0.50$, $v_\text{w}=0.44$, right-hand-side). $\overline{v}_\text{ref}$ is the volume averaged 3-velocity measured at the UETC reference time.
• Figure 5: Top row: comparison of the measured shear stress power spectrum normalised by energy density squared ${\mathcal{P}}_\Pi/\bar{\epsilon}^2$ with that obtained by convolving the measured weighted 4-velocity power spectra (see Eq. \ref{['e:shstCon']}, \ref{['e:ConDef']}; in dashed lines) and the 3-velocity power spectra (in dash-dotted lines). Left panel: detonation ($\alpha_\text{n}=0.67$, $v_\text{w}=0.92$) at time $3600/T_c$; right panel: deflagration ($\alpha_\text{n}=0.5$, $v_\text{w}=0.44$) at time $4400/T_c$. Bottom row: comparison of the instantaneous rate of change of the gravitational wave power spectra $\dot{\mathcal{P}}_\text{gw}$ computed via the convolution model prediction \ref{['e:dGWdt']}. Left panel: detonation ($\alpha_\text{n}=0.67$, $v_\text{w}=0.92$) with growth between times $2400/T_c$ and $3600/T_c$ ; right panel: deflagration ($\alpha_\text{n}=0.5$, $v_\text{w}=0.44$) with growth between times $3600/T_c$ and $4400/T_c$. Convolution predictions are shown with dashed lines in four colours, representing the three terms (purely compressional, mixed compressional-vortical, and purely vortical) and their sum. In the bottom row we show with a dotted line the sound shell model prediction, including the phenomenological suppression factor Cutting:2019zws, whose value is given in the legend.