Shape of the acoustic gravitational wave power spectrum from a first order phase transition

TL;DR

The paper demonstrates, through large-scale simulations, that acoustic waves from a first-order cosmological phase transition generate a characteristic GW spectrum shaped by two key length scales: the mean bubble separation and the sound-shell thickness. The results largely support the sound shell model for detonations, reveal nuances for deflagrations, and provide a practical broken-power-law fit to forecast LISA detectability. It shows how spectral features encode the wall speed and transition strength, offering a path to infer microphysical phase-transition parameters from a stochastic GW background. The study also outlines the regimes where the acoustic approximation holds and where shocks/turbulence will modify the spectrum, guiding future simulations and data interpretation.

Abstract

We present results from large-scale numerical simulations of a first order thermal phase transition in the early universe, in order to explore the shape of the acoustic gravitational wave and the velocity power spectra. We compare the results with the predictions of the recently proposed sound shell model. For the gravitational wave power spectrum, we find that the predicted $k^{-3}$ behaviour, where $k$ is the wavenumber, emerges clearly for detonations. The power spectra from deflagrations show similar features, but exhibit a steeper high-$k$ decay and an extra feature not accounted for in the model. There are two independent length scales: the mean bubble separation and the thickness of the sound shell around the expanding bubble of the low temperature phase. It is the sound shell thickness which sets the position of the peak of the power spectrum. The low wavenumber behaviour of the velocity power spectrum is consistent with a causal $k^{3}$, except for the thinnest sound shell, where it is steeper. We present parameters for a simple broken power law fit to the gravitational wave power spectrum for wall speeds well away from the speed of sound where this form can be usefully applied. We examine the prospects for the detection, showing that a LISA-like mission has the sensitivity to detect a gravitational wave signal from sound waves with an RMS fluid velocity of about $0.05c$, produced from bubbles with a mean separation of about $10^{-2}$ of the Hubble radius. The shape of the gravitational wave power spectrum depends on the bubble wall speed, and it may be possible to estimate the wall speed, and constrain other phase transition parameters, with an accurate measurement of a stochastic gravitational wave background.

Figures (10)

• Figure 1: RMS velocity and scalar gradient energy $\overline{U}_\text{f}$ and $\overline{U}_\phi$. Solid lines denote the RMS fluid velocity, dashed lines the RMS scalar field gradients. The simulations are separated into three plots (a)-(c) according to phase transition strength and bubble radius.
• Figure 1: Corrected version of Fig. \ref{['f:contour']}, corresponding to Eq. \ref{['eq:corrected']} above. The missing factor of 3 in Eq. \ref{['eq:final']} was already included, but the 0.687 numerical prefactor was missing. Furthermore, an additional factor of 2 was mistakenly present. The overall effect was to reduce the SNR for a given parameter choice by a factor of approximately 0.34 compared to the original figure. Note that the mission profile for LISA used here is no longer current, so this plot is for qualitative comparison with Fig. \ref{['f:contour']} only.
• Figure 2: Fluid radial velocity profiles as a function of scaled radius $\xi=r/t$. In red are curves taken at the peak of $\overline{U}_\phi$ (see Fig. \ref{['fig:ubars']}), at times $t_\text{pc}$ given in Table \ref{['t:SimVelStats']}. In black are fluid velocities at late times, $t \gtrsim 10000/T_\text{c}$. Note that the wall speeds can be read off from the positions of the phase transition fronts. Although the $v_\text{w}$ quoted in the main text comes from simulations with a smaller lattice spacing ($dx T_\text{c} = 0.2$), the discrepancy is small -- at most 3% for the fastest detonations.
• Figure 3: Velocity power spectra for detonations. Left are weak strength phase transitions, with $v_\text{w} = 0.92$, $0.80$ and $0.68$. Right are intermediate phase transitions, with $v_\text{w} = 0.92$ and $v_\text{w} = 0.72$. All have $N_b = 84$ bubbles (average separation $R_\text{c} = 1918/T_\text{c}$) and a lattice spacing $dx = 2/T_\text{c}$, with the exception of the $v_\text{w}=0.72$ intermediate transition which has $N_b = 11$ ($R_* = 1889/T_\text{c}$) and $dx = 1/T_\text{c}$. Note there is no intermediate strength transition with $v_\text{w}=0.80$.
• Figure 4: Velocity power spectra for deflagrations with $v_\text{w}=0.44$. Left is a weak phase transition, right is an intermediate transition. Both have $N_b = 84$ bubbles (mean separation $R_\text{c} = 1918/T_\text{c}$).