## Author(s)

Dahl, Jani, Hindmarsh, Mark, Rummukainen, Kari, Weir, David## Abstract

Gravitational waves (GWs) generated by a first-order phase transition at the electroweak scale are detectable by future space-based detectors like LISA. The lifetime of the resulting shock waves plays an important role in determining the intensity of the generated GWs. We have simulated decaying primordial acoustic turbulence in three dimensions and make a prediction for the universal shape of the energy spectrum by using its self-similar decay properties and the shape of individual shock waves. The shape for the spectrum is used to determine the time dependence of the fluid kinetic energy and the energy containing length scale at late times. The inertial range power law is found to be close to the classically predicted $k^{-2}$ and approaches it with increasing Reynolds number. The resulting model for the velocity spectrum and its decay in time is combined with the sound shell model assumptions about the correlations of the velocity field to compute the GW power spectrum for flows that decay in less than the Hubble time. The decay is found to bring about a convergence in the spectral amplitude and the peak power law that leads to a power law shallower than the $k^9$ of the stationary case.

## Figures

:

: A zoom in of the non-uniform energy density $\delta \rho$ slice in Run V at $t \approx 2.5 t_s$. Both plots use the same normalization for the color scheme, red regions corresponding to positive values, blue regions to negative ones, and regions close to zero are white. In Figure (a) the additional terms in equations (\ref{continuity}) and (\ref{NS}) are disabled, corresponding to the fluid equations in Ref.~\cite{Dahl_2022}. Figure (b) contains the energy density with all terms included, displaying the smoothing effect around the shock waves resulting from these terms. : Caption not extracted

The maximum velocity in the flow for Run V as a function of the number of shock formation times. The red curve shows the value in the case where the additional terms are disabled, and the blue curve in the case where all terms are included. The smoothing effect of the additional terms is seen here at initial times when the shocks are at their strongest, as the maximum velocity in this range is substantially smaller when the terms are enabled.

The mean energy density as a function of the number of shock formation times in Run V. The red curve shows the value in the case where the additional terms are disabled, and the blue curve in the case where all terms are included. While the energy density is no longer conserved as well as before as a result of the new term that emerges in the continuity equation, the deviation from the mean value still remains small and approaches the initial mean value over time after the initial stage.

: The total kinetic energy as a function of the number of shock formation times in Run V. With the additional terms included, the kinetic energy decays more slowly after the shocks form, but both curves still follow the same decay power law after $t > 10 t_s$.

: The integral length scale as a function of the number of shock formation times in Run V. After the initial phase when the shocks form, its value ends up being initially slightly larger with the additional terms included, but both curves still follow the power law after $t > 10 t_s$.

The energy spectrum $E(k)$ in Run V at $t \approx 15 t_s$. The additional terms do not have an effect on the power law values at low wavenumbers or at the inertial range. Instead, they only affect the spectral amplitude due to the difference between the kinetic energies in Figure~\ref{fig:kin_en_dec}, and the shape of the spectrum at the very high-$k$ end.

A right moving shock in a shock tube run at early times. The blue curve is the simulation data, and the red curve is the analytical prediction for the shock shape using equations (\ref{fac_a})-(\ref{shock_width_k}). The values used for the analytical prediction are $\eta=0.4$, $V_-=0.149$, $V_+=0.0$, and $x_0=51.9$, resulting in a shock velocity ${u_s \approx 0.625 \approx 1.083 c_s}$.

Fit (dashed red line) on the function $\Psi (\kappa)$ obtained from the longitudinal energy spectrum data of Run I at ${t=20 t_s}$ using Eq.~(\ref{specshape}) as the fitting function. The obtained fit parameters are $\widetilde{\Psi}_0=0.61$, $\beta=3.31$, $\alpha=7.72$, $\kappa_p=1.03$, and $\kappa_s=2.65$.

The fitting results for the parameter $\kappa_s$ of Eq.~\ref{specshape} listed in Table \ref{tab:table1} plotted against the Reynolds number at the fitting time. The red line is a linear fit $\kappa_s = d \text{Re}$ with ${d = 0.630 \pm 0.014}$.

The scaled inertial range power law values $\left\langle \beta - \alpha \right\rangle_t$ of Table~\Ref{tab:table2} plotted against the Reynolds number at the middle of the averaging interval at $t = 10 t_s$. The standard deviations resulting from the time fluctuations are shown as error bars for each case.

The dissipation rate of the kinetic energy fraction $\mathcal{E}_r (t) = \mathcal{E}(t)/\mathcal{E}_0$ resulting from the viscous dissipation, the pressure gradient, and the non-linear contributions of Equation~(\ref{kin_en_dis}) measured in Run VII. At early times before the shocks form, the steepening of the initial conditions into shocks causes strong oscillation in the pressure gradient contribution. After shock formation, the pressure gradient term is positive and generates kinetic energy, while the other two contributions dissipate it. At late times the viscous dissipation becomes the dominant term and sets the mean decay rate while the pressure gradient contribution induces oscillations into it.

The kinetic energy fraction of Run VII obtained from simulation data (blue curve) plotted with the fraction (red curve) computed from the measured values of the terms of the energy dissipation equation (\ref{kin_en_dis}), also visualized in Figure~\ref{fig:disterms}.

: $\left| \mathbf{v} \right|$ : $L \left| \mathbf{\omega} \right|$

: The magnitude of the velocity (a) and the magnitude of the vorticity (b) in a slice of run XII during the period of strong shocks at $t=4.15 t_s$. The vorticity has been plotted with a logarithmic color bar to make the different scales in the background vorticity more visible. The generated vorticity remains small, being largest at shock crests and in regions with overlapping shock waves. The maximum values for the quantities in the plots are 0.416 for the velocity, and 1.17 for $L \left| \mathbf{\omega} \right|$. : Caption not extracted

: Gravitational wave power spectrum. : Contribution from the $w_{--}$ terms.

: Contribution from the $w_{+-}$ terms. : Contribution from the $w_{++}$ terms.

: The total gravitational wave power spectrum for decaying acoustic turbulence at various times obtained numerically from Equation~(\Ref{Gwspec}), and the contributions from each of the four terms in the sum of Eq.~(\Ref{delta_kernel}). The times in question are approximately $t_{H_\star}/t_s \in [0.5, 1, 2, 5, 10, 21, 52, 104, 260, 520, 1039, 1559, 2078]$, with dark lines corresponding to late times. The chosen values for the free parameters are $\beta=4$, $\bar{v}_0 = 0.2$, and $C=0.2$. The spectrum converges to a constant value at late times. : Caption not extracted

: Caption not extracted

The converged spectra of Figure~\Ref{fig:GW_specs_dec} colored in blue plotted together with the stationary spectra ($C=0$), colored in black, at times $t_{H_\star}/t_s \in [0.5, 2, 10, 20, 100, 520]$. The stationary spectrum at $t_{H_\star}/t_s=2$ (green curve) coincides with the converged spectra of the decaying case at large wavenumbers.

The magnitude of the decaying GW power spectrum at different wavenumbers as a function of the amount of shock formation times. The wavenumbers have been indicated in Figure~\ref{fig:GW_specs_nondec} with colored ticks in the horizontal axis. The vertical axis has been normalized by the value of the spectrum at the first data point ($t_{H_\star}/t_s=1/2$). The spectrum has converged at $t_{H_\star}/t_s \geq 20$ to a reasonable degree in the vicinity of the peak.

The maximum velocity measured in the fluid of run XII as a function of shock formation times (red line) and the same run with the velocity limiter active with a threshold velocity of $v_t=0.4$ (blue line). The limiter activates for the first time at $t \approx 1.71 t_s$, and the last time at $t \approx 15.75 t_s$. It can be seen that after the limiter is no longer active, the maximum velocity coincides with the original run, indicating that the limiter does not leave a lasting imprint on the shock waves.

The energy spectrum for Runs XI (red curve) and XI NUV (blue curve), the latter of which is the same run but with the non-uniform viscosity of Eqs.~(\ref{NUV_RHS}) and (\ref{NUV}) enabled. The reduction in the viscosity at the shock waves leads to a slightly increased bottleneck effect at the high wavenumber end of the spectrum.

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