Gravitational waves from vacuum first-order phase transitions: from the envelope to the lattice

TL;DR

The paper investigates gravitational-wave production from vacuum first-order phase transitions using large-scale 3D lattice simulations to test the envelope approximation.It demonstrates that as bubble walls reach ultra-relativistic speeds, the high-k slope of the GW spectrum steepens to about $k^{-1.5}$, while the peak is similar in location and amplitude to envelope-based predictions, though shifted to slightly lower $k$ in the full scalar-field treatment.A distinct UV feature arises from post-collision scalar field oscillations, creating a bump near the bubble-wall thickness scale $l_0$ that grows linearly but contributes negligibly to $\Omega_{gw}$ for sub-Planckian scalar masses.The authors provide a robust fit for the collision-driven GW spectrum and show that the envelope approximation remains only approximately accurate for the peak, with notable deviations in the UV, highlighting the importance of non-linear oscillations in shaping the full spectrum.

Abstract

We conduct large scale numerical simulations of gravitational wave production at a first order vacuum phase transition. We find a power law for the gravitational wave power spectrum at high wavenumber which falls off as $k^{-1.5}$ rather than the $k^{-1}$ produced by the envelope approximation. The peak of the power spectrum is shifted to slightly lower wave numbers from that of the envelope approximation. The envelope approximation reproduces our results for the peak power less well, agreeing only to within an order of magnitude. After the bubbles finish colliding the scalar field oscillates around the true vacuum. An additional feature is produced in the UV of the gravitational wave power spectrum, and this continues to grow linearly until the end of our simulation. The additional feature peaks at a length scale close to the bubble wall thickness and is shown to have a negligible contribution to the energy in gravitational waves, providing the scalar field mass is much smaller than the Planck mass.

Figures (15)

• Figure 1: Values of the scalar field along the collision axis during a two bubble collision where $R_\text{c} M=7.15$. Here the $x$ axis is the collision axis which connects the two bubble centres. The $y$ axis is time since the nucleation of the bubbles. The bubbles are separated by a distance $D$. This figure can be compared with Fig. 1 of PhysRevD.26.2681 and Fig. 7 of Braden:2014cra.
• Figure 2: Slices through a simultaneous nucleation simulation with parameters $R_\text{c} M=7.15$, $N_\text{b}=64$ and $R_* M=56.32$ showing the expansion (a), collision (b), and oscillatory (c and d) phase of the scalar field. The scalar field value is shown in blue, and the gravitational wave energy density is shown in red. Note that the range of the colourbar for the gravitational wave energy density changes for each plot. During the oscillatory phase the gravitational wave energy density becomes very uniform and the "hotspots" are deviations on the sub percent level. The full set of parameters for this run is shown in Table \ref{['table:sim']}. A movie based on this simulation is included in the supplemental material.
• Figure 3: The Lorentz factor $\gamma$ of the bubble wall for different values lattice spacings plotted against the radius of the bubble in units of the critical radius. This is for a bubble with $R_\text{c} M=7.15$. The dashed black line shows $\gamma=R/R_\text{c}$.
• Figure 4: Energy densities in the scalar field over time for a simultaneous nucleation run with $R_\text{c} M=7.15$, $R_* M=56.3$ and $N_\text{b}=4096$. The full set of parameters for this run is shown in Table \ref{['table:sim']}.
• Figure 5: Energy conservation for several simulations of the same physical volume. Runs with exponential nucleation are plotted with dashed lines, and simultaneous nucleation runs are shown with solid lines. See Tables \ref{['table:sim']} and \ref{['table:exp']} for the full set of parameters of each run.