Maximal GW amplitude from bubble collisions in supercooled phase transitions

TL;DR

This work extends analytic formulas for gravitational-wave spectra from bubble collisions in first-order phase transitions to an expanding FLRW universe using thin-wall and envelope approximations. By introducing conformal-time nucleation parameters defined at a characteristic collision time $ au_*$, the authors show that cosmic expansion redshifts tensor modes and impose an upper bound on the GW amplitude, even for strongly supercooled transitions. Numerical results across delta-function, power-law, and power-exponential nucleation histories reveal a nearly universal spectral shape when expressed in terms of the dimensionless ratio $k/ ilde{eta}$, with the spectrum largely insensitive to nucleation details after appropriate rescaling; the amplitude scales as $( ilde{eta}/ ilde{H}_*)^{-2}( ext{κ}_* ext{α}_*)^2$ and peaks at $k oughly H_*$. The findings yield a practical criterion for estimating signals from supercooled phase transitions and provide a robust upper bound on present-day gravitational-wave energy density, relevant for future GW experiments.

Abstract

We extend analytic formulas for the gravitational-wave (GW) spectrum from first-order phase transitions to include cosmic expansion under the thin-wall and envelope approximations. We demonstrate that even for strongly supercooled transitions the GW amplitude is bounded from above. This conclusion is explicitly verified for several representative nucleation histories, including delta-function, power-law, and power-exponential types. Moreover, the spectral shape, amplitude, and peak frequency remain largely unaffected by the details of the nucleation rate once expressed in terms of the conformal variables evaluated at an appropriately defined characteristic collision time.

Figures (7)

• Figure 1: GW spectra $\Delta^{(s)}$ (solid curves) and $\Delta^{(d)}$ (dashed curves) as functions of $k/\tilde{\beta}$ for $\tilde{\beta} \tau_{\rm nuc} = 0.1$ (blue), $0.5$ (orange), $1$ (green), $2$ (brown), and $10$ (red), in the case of delta-function nucleation rate.
• Figure 2: Peak amplitude $\Delta^{(s)} (k_{\rm peak},\tilde{\beta})$ (top panel) and wavenumber $k_{\rm peak}/\tilde{\beta}$ (bottom panel) as a function of $\tilde{\beta} \tau_{\rm nuc}$ for the case of a delta-function nucleation rate.
• Figure 3: Same as Fig. \ref{['fig:delta1']} but for a power-law nucleation rate. We take $n =0$ (blue), $2$ (orange), and $4$ (green).
• Figure 4: Same as Fig. \ref{['fig:delta1']} but for a power-exponential nucleation rate. We take $n=0$ (top), $2$ (middle), and $4$ (bottom) with $\tilde{\Gamma}_0/\tilde{\beta}'^4 = 10^{-15}$ (blue), $10^{-11}$ (orange), $10^{-7}$ (green), and $10^1$ (brown). The case with $\tilde{\Gamma}_0/\tilde{\beta}'^4 =10^1$ is omitted in the lower two panels.
• Figure 5: Same as the top panel of Fig. \ref{['fig:delta2']} but for a power-exponential nucleation rate. We take $n=0$ (top), $2$ (middle), and $4$ (bottom).