Vorticity, kinetic energy, and suppressed gravitational wave production in strong first order phase transitions

TL;DR

The paper investigates strong first-order phase transitions in the early Universe using first-ever 3D simulations of a coupled scalar field and relativistic fluid. It quantifies how fluid vorticity and kinetic energy transfer evolve with transition strength, showing that deflagrations develop substantial rotational motion and that reheating droplets slow bubble walls, reducing gravitational-wave production. The results indicate that previous models overestimate the GW signal for strong transitions, by up to three orders of magnitude for deflagrations, with detonations less affected. The work provides implications for LISA observability and motivates larger simulations to precisely map suppression effects across parameter space.

Abstract

We have performed the first 3-dimensional simulations of strong first-order thermal phase transitions in the early Universe. For deflagrations, we find that the rotational component of the fluid velocity increases as the transition strength is increased. For detonations, however, the rotational velocity component remains constant and small. We also find that the efficiency with which kinetic energy is transferred to the fluid falls below theoretical expectations as we increase the transition strength. The probable origin of the kinetic energy deficit is the formation of reheated droplets of the metastable phase during the collision, slowing the bubble walls. The rate of increase in the gravitational wave energy density for deflagrations in strong transitions is suppressed compared to that predicted in earlier work. This is largely accounted for by the reduction in kinetic energy. Current modelling therefore substantially overestimates the gravitational wave signal for strong transitions with deflagrations, in the most extreme case by a factor of $10^{3}$. Detonations are less affected.

Figures (12)

• Figure 1: Proportion of mean square fluid velocity in the rotational modes. We plot the ratio of $\overline{v}_{\perp\text{,max}}$ to $\overline{v}_\text{max}$ against $\alpha$. Dashed lines give a linear fit for the last four simulation points. The fits are extrapolated to $\alpha_{\mathrm{max}}$ for deflagrations, or to the largest $\alpha$ for which a wall speed corresponds to a detonation (hollow circles).
• Figure 2: The evolution of $\overline{U}_\phi$ (dashed lines) and $\overline{U}_\text{f}$ (solid lines) for simulations with increasing $\alpha$ (darker shades). In blue we show deflagrations with $v_{\text{w}}=0.44$ whereas red lines show detonations with $v_{\text{w}}=0.92$.
• Figure 3: Comparison between the maximum value of $\overline{U}_\text{f}$ in each simulation and that predicted by Espinosa:2010hh for the given $v_{\text{w}}$ and $\alpha$. Dashed lines give a linear fit for the last four simulation points. Hollow circles show the extrapolation to $\alpha_{\mathrm{max}}$ for deflagrations, or up to to the largest $\alpha$ for which the wall speed corresponds to a detonation.
• Figure 4: Comparison of the gravitational waves produced in our simulations against that predicted by Eq. (\ref{['eqn:OmGWExp']}) using $\overline{U}_{\text{f,exp}}$ found from $v_{\text{w}}$ and $\alpha$.
• Figure 5: Variation of gravitational wave energy density with $\delta x$ for $v_\text{w}=0.44$ and $v_\text{w}=0.92$ and transition strength of $\alpha=0.5$. We normalise the $y$-axis by dividing by the result from the simulation presented in the paper ($\delta x=1.0$). Note that $\overline{\left(\Omega_\text{gw}/ H_\text{n} t \right)}$ signifies that we average the quantity inside the brackets over the final $\Delta t= 2 R_*$ of the simulation.