Gravitational waves from first-order phase transitions: Towards model separation by bubble nucleation rate

TL;DR

GW from cosmic first-order phase transitions is studied with a focus on whether a Gaussian correction to the bubble nucleation rate leaves a detectable imprint on the spectrum. The authors relate the spectrum to the unequal-time energy-momentum tensor correlator and derive analytic expressions for single- and double-bubble contributions under thin-wall and envelope approximations, incorporating a Gaussian term in the nucleation rate. They find approximate 10% deviations in spectral shape for typical values of the Gaussian parameter, and discuss detector sensitivities needed to distinguish these shapes. This work demonstrates a concrete pathway for model separation via gravitational-wave spectral features and motivates further studies beyond the idealized setup to extract more detailed physics from future GW observations.

Abstract

We study gravitational-wave production from bubble collisions in a cosmic first-order phase transition, focusing on the possibility of model separation by the bubble nucleation rate dependence of the resulting gravitational-wave spectrum. By using the method of relating the spectrum with the two-point correlator of the energy-momentum tensor $\left< T(x)T(y) \right>$, we first write down analytic expressions for the spectrum with a Gaussian correction to the commonly used nucleation rate, $Γ\propto e^{βt}\rightarrow e^{βt-γ^2t^2}$, under the thin-wall and envelope approximations. Then we quantitatively investigate how the spectrum changes with the size of the Gaussian correction. It is found that the spectral shape shows ${\mathcal O}(10)\%$ deviation from $Γ\propto e^{βt}$ case for some physically motivated scenarios. We also briefly discuss detector sensitivities required to distinguish different spectral shapes.

Figures (13)

• Figure 1: Relation (\ref{['eq:rel']}) between dimensionless combinations $\gamma/\beta$ and $\gamma/\beta'$. In this plot $\beta/H_*$ is taken to be $1, 10, 100$ from bottom to top.
• Figure 2: (Left) Spectral shape for $v = 1$. (Right) Spectral shape for $v = 0.3$.
• Figure 3: (Left) Ratio $R$ between the spectra with $\gamma = 0$ and $\gamma \neq 0$ defined in Eq. (\ref{['eq:R']}). This figure shows $v = 1$ case for various values of $\gamma/\beta'$. (Right) Log plot of the left panel.
• Figure 4: The same as Fig. \ref{['fig:SpectralShape_v=1']} except that $v = 0.3$.
• Figure 5: Setup in Eqs. (\ref{['eq:Detector']})--(\ref{['eq:Distinguish']}). The blue line denotes the detector sensitivity curve (\ref{['eq:Detector']}), while the red lines correspond to the signal $\Omega_{{\rm GW}, \gamma = 0}$ (solid) and $\Omega_{\rm GW}$ with $\gamma / \beta' = 5$ (dashed). The bubble wall velocity is taken to be $v = 1$, and we have extrapolated the normalized spectrum $R(f/f_{\rm peak})$ below $f/f_{\rm peak} < 0.1$ and $f/f_{\rm peak} > 4$ by assuming that it is constant for these frequencies. The extrapolation is expected to give conservative estimate for the deviation.