Gravitational waves from high-temperature vacuum decay in scale-invariant models: nanohertz vs. millihertz regimes

TL;DR

This work analyzes high-temperature vacuum decay in a general class of scale-invariant models by constructing a comprehensive finite-temperature effective potential that includes tree-level, Coleman–Weinberg, thermal, and Daisy-resummed contributions. It demonstrates that vacuum decay is dominantly sphaleron-driven rather than instanton tunneling and computes the full stochastic gravitational-wave spectrum from bubble collisions, sound waves, and turbulence. A model-independent scan reveals two distinct phenomenological regimes set by the parameter $b$: a low-$b$ regime producing nanohertz GWs in tension with cosmological constraints, and a high-$b$ regime producing millihertz GWs within the sensitivity of LISA. The results map fundamental scale-invariant parameters to observable GW signatures, offering robust guidance for building cosmologically viable models and for interpreting future GW observations in PTA and space-based detectors.

Abstract

We present a comprehensive analysis of high-temperature vacuum decay and the resulting stochastic gravitational wave (GW) background within the framework of general scale-invariant models. The effective potential is constructed to include tree-level contributions, the Coleman-Weinberg correction, finite-temperature effects, and the Daisy resummation technique, culminating in a high-temperature form. We investigate the dynamics of the first-order phase transition, calculating the critical and nucleation temperatures, the supercooling parameter, and the key transition parameters $α$ (transition strength) and $β$ (inverse duration). The vacuum decay is found to be dominated by sphaleron transitions rather than quantum tunneling. We compute the full GW spectrum arising from bubble collisions, sound waves, and turbulence. An extensive numerical scan reveals two distinct phenomenological regimes: one produces nanohertz-frequency GW signals potentially detectable by Pulsar Timing Arrays (PTA), while the other yields millihertz-frequency signals that are prime targets for future space-based interferometers like the Laser Interferometer Space Antenna (LISA).

Figures (8)

• Figure 1: The ratio $(S_3/T)/S_4$ plotted in the $\delta$-$d$ plane, indicating that quantum tunneling is less significant than sphaleron processes, as reflected by $(S_3/T) < S_4$ in all regions. This supports the use of equation (\ref{['3-19']}) for estimating the vacuum decay rate.
• Figure 2: Regions where the decay rate via sphalerons exceeds 100 times that of instantons for three different $\lambda_T$. In nearly all areas where $0 < \delta < 2$, sphalerons dominate over instantons.
• Figure 3: Representation of the relationship between the parameters $\delta_1$ and $\delta_2$ under varying values of $\gamma$. In both cases ${\lim_{\gamma \to \infty}} \delta_{1,2} = 2$.
• Figure 4: Left: visualization of the allowable parameter space for $c$ and $d$ constrained by the condition $\gamma > 2.07345$. The shaded regions represent the combinations of $c$ and $d$ that satisfy the given criteria, allowing for a better understanding of how different parameter values influence the system's behavior in conjunction with the inequalities associated with $\delta_1$ and $\delta_2$. Right: $\lambda_{T_n}$ as a function of the parameter space $c$ and $d$.
• Figure 5: Illustration of the relationship between the critical temperature $T_c$ and the nucleation temperature $T_n$. This figure presents how variations in the parameters $b$, $c$, and $d$ influence the supercooling parameter $\delta_{\text{sc}}$.