Numerical simulations of acoustically generated gravitational waves at a first order phase transition

TL;DR

This work uses large-scale 3D simulations of a scalar field coupled to a relativistic fluid to study gravitational wave production during a first-order cosmological phase transition. It shows that acoustic sound waves are the principal GW source, producing a GW spectrum with a steep high-k tail that differs from the envelope approximation. The GW energy density scales with the fourth power of the fluid velocity, the source lifetime (set by the Hubble time), and the source length scale (roughly the bubble separation), and is governed by a quasi-universal parameter $8\pi\tilde{\Omega}_{\rm GW} \approx 0.8 \pm 0.1$; extrapolations suggest electroweak-scale transitions could yield much larger signals than previously estimated. A scaling approach links limited simulations to physically large bubble separations, supporting the acoustic model as a robust framework for predicting gravitational wave signatures of first-order phase transitions.

Abstract

We present details of numerical simulations of the gravitational radiation produced by a first order thermal phase transition in the early universe. We confirm that the dominant source of gravitational waves is sound waves generated by the expanding bubbles of the low-temperature phase. We demonstrate that the sound waves have a power spectrum with a power-law form between the scales set by the average bubble separation (which sets the length scale of the fluid flow $L_\text{f}$) and the bubble wall width. The sound waves generate gravitational waves whose power spectrum also has a power-law form, at a rate proportional to $L_\text{f}$ and the square of the fluid kinetic energy density. We identify a dimensionless parameter $\tildeΩ_\text{GW}$ characterising the efficiency of this "acoustic" gravitational wave production whose value is $8π\tildeΩ_\text{GW} \simeq 0.8 \pm 0.1$ across all our simulations. We compare the acoustic gravitational waves with the standard prediction from the envelope approximation. Not only is the power spectrum steeper (apart from an initial transient) but the gravitational wave energy density is generically larger by the ratio of the Hubble time to the phase transition duration, which can be 2 orders of magnitude or more in a typical first order electroweak phase transition.

Figures (13)

• Figure 1: Layout of quantities simulated. The positions of quantities related to simulating an ideal relativistic fluid are standard WilsonMatthews. Because the field and fluid are coupled together, it is important that the scalar field $\phi$ and its conjugate momentum $\pi$ are correctly centred. We take $\phi$ to reside in zones (like pressure, temperature, etc.), so that no centring is required to compute, for example, the equation of state.
• Figure 2: Single bubble test simulation, with the correlation length $\ell$ shown as an indication of the wall width. Only the fluid source is shown here; discretisation errors for the field source are the same or smaller. There is good agreement between $1/L$ and $1/\ell$, as desired.
• Figure 3: Comparison of radial fluid velocity profiles for simulations at approximate collision times for $N_\text{b}=1000$ ($t=500/T_\text{c}$; gray), $N_\text{b}=37$ ($t=1000/T_\text{c}$; black) and at late times (dashed red).
• Figure 4: Slices of fluid kinetic energy density $E/T_\text{c}^4$ at $t=500\, T_\mathrm{c}^{-1}$, $t=1000 \,T_\mathrm{c}^{-1}$ and $t=1500\, T_\mathrm{c}^{-1}$ respectively, for the $\eta/T_c=0.15$, $N_\text{b}=988$ simulation.
• Figure 5: Root mean square fluid velocity $\overline{U}_\text{f}$ and root mean square scalar gradients $\overline{U}_\phi$ for $N_\text{b} = 988$ (top row) and $N_\text{b} = 37$ (bottom row).