Gravitational Radiation from First-Order Phase Transitions

TL;DR

This work models a scalar-field driven first-order cosmological phase transition using a Coleman-Weinberg–type potential in a radiation-dominated expanding universe and extracts the gravitational-wave spectrum from the TT metric perturbations. Using high-resolution 3D lattice simulations (512^3), the authors show that bubble collisions establish the initial GW peak at a scale set by the mean bubble separation $L_*\\approx\\beta^{-1}$, while a subsequent coalescence phase amplifies the signal by roughly an order of magnitude and shifts the peak to higher frequencies, even in the absence of a turbulent fluid. The results reproduce expected scalings with the energy scale $\\mu$ and bubble count $n$, and reveal a robust, scale-invariant spectrum shape with respect to $\\mu$ across several orders of magnitude. The finding that coalescence enhances the GW signal has important implications for detectability of electroweak-scale first-order transitions and informs future detector design; future work will incorporate dynamical fluids to assess turbulence effects and parameter dependencies, with EW-scale predictions near $f\\sim\\text{few}\\times 10^{-5}$ Hz for $\\beta/H_*\\approx 5$.

Abstract

It is believed that first-order phase transitions at or around the GUT scale will produce high-frequency gravitational radiation. This radiation is a consequence of the collisions and coalescence of multiple bubbles during the transition. We employ high-resolution lattice simulations to numerically evolve a system of bubbles using only scalar fields, track the anisotropic stress during the process and evolve the metric perturbations associated with gravitational radiation. Although the radiation produced during the bubble collisions has previously been estimated, we find that the coalescence phase enhances this radiation even in the absence of a coupled fluid or turbulence. We comment on how these simulations scale and propose that the same enhancement should be found at the Electroweak scale; this modification should make direct detection of a first-order electroweak phase transition easier.

Figures (7)

• Figure 1: Four time-slices of a first order phase transition when the energy density of the Universe was $\rho\approx (10^{-4}\,m_{\rm pl})^4$. The first slice shows the nucleation of five bubbles at $\tau=0$, followed by a slice taken as the bubbles initially collide ($\tau \approx 0.5 \beta^{-1}$), a slice at the end of the phase transition ($\tau \approx \beta^{-1}$) and finally a slice when $\tau \approx 2\beta^{-1}$. Contours are drawn at $\phi=-0.83\phi_0$ (gold) and $\phi=0.83\phi_0$ (red) to guide the eye.
• Figure 2: Gravitational wave spectra from a first-order phase transition when the energy density of the Universe was $\rho\approx (10^{-4}\,m_{\rm pl})^4$: the spectrum immediately after the nucleation of five bubbles (red, solid), at $\tau \approx 0.5 \beta^{-1}$ (blue, dotted), when $\tau \approx \beta^{-1}$(green, dashed), and at $\tau \approx 2\beta^{-1}$ (black, dot-dashed). The bump at high frequencies is a numerical artifact.
• Figure 3: The maximum intensity of the gravitational wave spectrum for a $\mu = 10^{-4}\,m_{\rm pl}$ simulation initialized with 40 (red, solid), 32 (blue, dotted), 24 (green, dashed), or 16 (black, dot-dashed) bubbles per Hubble volume, $\tau<\beta^{-1}$.
• Figure 4: The present-day gravitational wave spectrum produced by time $\tau=\beta^{-1}$. This is for a simulation with $\mu=10^{-4}\,m_{\rm pl}$ and $n=16$ bubbles per Hubble volume (red, solid), $n=24$ (blue, dotted), $n=32$ (green, dashed), and $n=40$ (black, dot-dashed). The bump at high frequencies is a numerical artifact.
• Figure 5: The maximum intensity of the gravitational wave spectrum for a $\mu = 10^{-4}\,m_{\rm pl}$ simulation initialized with 40 (red, solid), 32 (blue, dotted), 24 (green, dashed), or 16 (black, dot-dashed) bubbles per Hubble volume, $\beta^{-1}<\tau<2.5\beta^{-1}$.