Gravitational Waves from the Sound of a First Order Phase Transition

We report on the first three-dimensional numerical simulations of first-order phase transitions in the early Universe to include the cosmic fluid as well as the scalar field order parameter. We calculate the gravitational wave (GW) spectrum resulting from the nucleation, expansion, and collision of bubbles of the low-temperature phase, for phase transition strengths and bubble wall velocities covering many cases of interest. We find that the compression waves in the fluid continue to be a source of GWs long after the bubbles have merged, a new effect not taken properly into account in previous modeling of the GW source. For a wide range of models, the main source of the GWs produced by a phase transition is, therefore, the sound the bubbles make.

Figure 1

Slices of fluid energy density $E/T_c^4$ at $t=400T_c^{-1}$, $800T_c^{-1}$, and $1200T_c^{-1}$, respectively, for the $\eta =0.2$ simulation. The slices correspond roughly to the end of the nucleation phase, the end of the initial coalescence phase, and the end of the simulation.

Figure 2

Top: time series of ${\overline{U}}_{\varphi }$ and ${\overline{U}}_f$ (9), showing the progress of the phase transition; the curves for ${\overline{U}}_{\varphi }$ and ${\overline{U}}_f$ are individually identified for the “intermediate” case. Bottom: time series of ${\rho }_{\mathrm{GW}}{R}_{*}^{-1}[{(\overline{\epsilon }+\overline{p})}^{-2}{\overline{U}}_f^{-4}{]}_{t_{\text{end}}}$, showing the evolution of the gravitational wave energy density relative to an estimate of the square of the final fluid shear stresses.

Figure 3

Gravitational wave power spectra during the phase transition, for the intermediate strength transition, from fluid only (black lines) and both fluid and field (gray lines). From bottom to top, the times are $t=600$, 800, 1000, 1200, and $1400T_c^{-1}$. The red dashed line indicates the expected $k^{-1}$ behavior.

Figure 4

Fluid velocity power spectra for the intermediate strength transition, separated into longitudinal (compressional) and transverse (rotational) components; shown in gray and black lines, respectively. Times shown are the same as in Fig. 3.