Gravitational waves from vacuum first order phase transitions II: from thin to thick walls

TL;DR

The paper investigates gravitational waves from vacuum first-order phase transitions using a single-parameter toy potential, $\overline{\lambda}$, controlling bubble-wall thickness. Through large-scale 3D lattice simulations with simultaneous nucleation, it shows that thinner walls ( $\overline{\lambda}\to 1$ ) and thicker walls ( $\overline{\lambda}\to 0$ ) yield distinct scalar-field dynamics and TT-stress distributions, with the UV GW spectrum steepening as walls become thicker. The GW spectrum during bubble collisions deviates from both envelope and bulk-flow predictions, displaying a UV slope that depends on $\overline{\lambda}$ and an IR slope approaching bulk-flow values at late times; oscillation-phase contributions near $k\sim M_b$ provide an additional, longer-lasting feature. Overall, the results indicate that the underlying potential shape can imprint on the GW signal, offering a potential diagnostic for early-universe phase transitions, while highlighting the need for higher-$\gamma$ studies and damping mechanisms for realistic extrapolation.

Abstract

In a vacuum first-order phase transition, gravitational waves are generated from collision of bubbles of the true vacuum. The spectrum from such collisions takes the form of a broken power law. We consider a toy model for such a phase transition, where the dynamics of the scalar field depends on a single parameter $\overlineλ$, which controls how thin the bubble wall is at nucleation and how close to degenerate the vacua are relative to the barrier. We extend on our previous work by performing a series of simulations with a range of $\overlineλ$. The peak of the gravitational-wave power spectrum varies by up to a factor of $1.3$, which is probably an unobservable effect. We find that the ultraviolet (UV) power law in the gravitational-wave spectrum becomes steeper as $\overlineλ \rightarrow 0$, varying between $k^{-1.4}$ and $k^{-2.2}$ for the $\overlineλ$ considered. This provides some evidence that the form of the underlying effective potential of a vacuum first-order phase transition could be determined from the gravitational-wave spectrum it produces.

Figures (20)

• Figure 1: The effect on the potential due to the variation of $\overline{\lambda}$.
• Figure 2: The critical profile for a series of potentials with different values of $\overline{\lambda}$.
• Figure 3: Field profiles of bubbles when the bubble walls have accelerated up to various $\gamma$ factors. Note that $\gamma=1$ corresponds to the critical bubble profile.
• Figure 4: Evolution of $\gamma$ as defined in Eq. \ref{['eq:gammath']} for a series of values of $\overline{\lambda}$.
• Figure 5: The collision of two bubbles of the true vacuum plotted for a thin wall (a) and thick wall (b) potential. The $x$ axis corresponds to the line joining the two bubble centres, with $D$ being the separation between bubbles. On the $y$ axis we plot the time $t$ since the bubbles were nucleated. For both these simulations, the bubbles collide when the Lorentz factors of the bubble walls are $\gamma = 4.0$.