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

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 $k^{-3}$ behavior, where $k$ is the wave number, 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 wave number behavior 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.

Figure 1

RMS velocity and scalar gradient energy ${\overline{U}}_{\mathrm{f}}$ and ${\overline{U}}_{\varphi }$. 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 2

Fluid radial velocity profiles as a function of scaled radius $\xi =r/t$. In red are curves taken at the peak of ${\overline{U}}_{\varphi }$ (see Fig. 1), at times $t_{\mathrm{pc}}$ given in Table 3. In black are fluid velocities at late times, $t\gtrsim 10000/T_{\mathrm{c}}$. Note that the wall speeds can be read off from the positions of the phase transition fronts. Although the $v_{\mathrm{w}}$ quoted in the main text comes from simulations with a smaller lattice spacing ($dx{T}_{\mathrm{c}}=0.2$), 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 $v_{\mathrm{w}}=0.92$, 0.80, and 0.68. Right are intermediate phase transitions, with $v_{\mathrm{w}}=0.92$ and $v_{\mathrm{w}}=0.72$. All have $N_b=84$ bubbles (average separation $R_{\mathrm{c}}=1918/T_{\mathrm{c}}$) and a lattice spacing $dx=2/T_{\mathrm{c}}$, with the exception of the $v_{\mathrm{w}}=0.72$ intermediate transition which has $N_b=11$ ($R_{*}=1889/T_{\mathrm{c}}$) and $dx=1/T_{\mathrm{c}}$. Note there is no intermediate strength transition with $v_{\mathrm{w}}=0.80$.

Figure 4

Velocity power spectra for deflagrations with $v_{\mathrm{w}}=0.44$. Left is a weak phase transition, right is an intermediate transition. Both have $N_b=84$ bubbles (mean separation $R_{\mathrm{c}}=1918/T_{\mathrm{c}}$).

Figure 5

Velocity power spectra for near-Jouguet deflagrations. All are weak transitions with $v_{\mathrm{w}}=0.56$, and $N_b=11$, 84, and 5376 ($R_{\mathrm{c}}=1889/T_{\mathrm{c}}$, $1918/T_{\mathrm{c}}$, and $480/T_{\mathrm{c}}$).

Figure 6

Power spectra of fractional energy density in gravitational waves for detonations, divided by the ratio of the mean bubble separation $R_{*}$ to the Hubble length at the transition $H_{\mathrm{n}}$, and the ratio of the time to the Hubble time. The wave number is scaled by the mean bubble separation. Left are weak phase transitions, showing detonations with $v_{\mathrm{w}}=0.92$, 0.80, and 0.68 (top to bottom). Right are intermediate phase transitions, with wall speeds $v_{\mathrm{w}}=0.92$ and $v_{\mathrm{w}}=0.72$. All have $N_b=84$ bubbles, giving a mean bubble separation of $R_{*}=1918/T_{\mathrm{c}}$, and a lattice spacing $dx=2/T_{\mathrm{c}}$, with the exception of the $v_{\mathrm{w}}=0.72$ intermediate transition which has $N_b=11$ ($R_{\mathrm{c}}=1889/T_{\mathrm{c}}$) and $dx=1/T_{\mathrm{c}}$.

Figure 7

Power spectra of fractional energy density in gravitational waves for deflagrations with $v_{\mathrm{w}}=0.44$, divided by the ratio of the mean bubble separation $R_{*}$ to the Hubble length at the transition $H_{\mathrm{n}}$, and the ratio of the time to the Hubble time. The wave number is scaled by the mean bubble separation. Left is a weak phase transition, right is an intermediate transition. Both have $v_{\mathrm{w}}=0.44$ and $N_b=84$ ($R_{*}=1918/T_{\mathrm{c}}$).

Figure 8

Power spectra of fractional energy density in gravitational waves for near-Jouguet deflagrations, $v_{\mathrm{w}}=0.56$, divided by the ratio of the mean bubble separation $R_{*}$ to the Hubble length at the transition $H_{\mathrm{n}}$, and the ratio of the time to the Hubble time. The wave number is scaled by the mean bubble separation. All are weak transitions with $N_b=11$, 84, and 5376 ($R_{*}=1889/T_{\mathrm{c}}$, $1918/T_{\mathrm{c}}$, and $480/T_{\mathrm{c}}$).

Figure 9

Signal-to-noise ratio (solid contours) for the model power spectrum (35), using our acoustic peak gravitational wave power estimate (39), with mission configuration described in the text, and taking the nucleation temperature to be $T_{\mathrm{n}}=100\mathrm{GeV}$. Dashed lines show the ratio of the order of magnitude of the shock appearance timescale $R_{*}{H}_{\mathrm{n}}/{\overline{U}}_{\mathrm{f}}$. For ratios of order 1 the fluid flow can be assumed to be purely acoustic, while for values much less than 1, the fluid flow may become turbulent within a Hubble time, and further investigation is required.