Analytic derivation of GW spectrum from bubble collisions in FLRW Universe

TL;DR

This work extends the analytic framework for gravitational waves from bubble collisions during cosmological first-order phase transitions from Minkowski space to a general expanding FLRW background, incorporating the effects of conformal time, scale-factor evolution, and a time-dependent nucleation rate. The authors derive complete analytic expressions for the GW spectrum, decomposed into single-bubble and double-bubble contributions, and provide non-oscillatory forms suitable for efficient numerics. They systematically develop asymptotic formulas in large- and small-wavenumber limits and perform a Minkowski-limit expansion in powers of $ ilde{eta}/ ilde{calH}_*$, highlighting when expansion effects become non-negligible (e.g., breakdown around $eta/H_* oughly 10$ and $O(10 ext{–}20)$ for next-to-leading corrections). Numerical results for delta-function and exponential nucleation rates corroborate the analytic limits, show the dominance of the single-bubble part, and quantify how expansion and nucleation history shape the present-day GW signals, with direct implications for upcoming detectors.

Abstract

We generalize the analytic formula for the gravitational-wave spectrum from bubble collisions during a cosmological first-order phase transition, under the thin-wall and envelope approximations, by incorporating the effect of cosmic expansion in the FLRW metric. Along with presenting the complete analytic expression and corresponding numerical results, we also derive simplified formulas valid in the large- and small-$k$ limits, as well as in the Minkovski limit. The latter expansion reveals that the Minkovski approximation breaks down for $β/ H_* \lesssim 10$, where $β$ denotes the inverse duration of the phase transition and $H_*$ the Hubble parameter at its completion. Furthermore, the next-to-leading-order term contributes about a $10\%$ correction for $β/ H_* \sim 140$, a typical value for the electroweak phase transition.

Figures (9)

• Figure 1: Schematic illustration of the hierarchy among energy densities. The SM energy density $\rho_{\mathrm{SM}}$ ($\simeq \rho_{\rm tot}$) dominates the total energy of the Universe and determines the Hubble expansion rate. The vacuum energy in the bubble-nucleation sector $\rho_0$ is much smaller than $\rho_{\mathrm{SM}}$, ensuring $\alpha_* \ll 1$. However, within the bubble-nucleation sector, $\rho_0$ dominates over its plasma energy $\rho_{\mathrm{b, rad}}$, allowing the phase transition to be strongly supercooled.
• Figure 2: Schematic illustration of bubbles under the envelope approximation at different times. Points $x$ and $y$ lie within the world volume of the same bubble, while point $y'$ lies within that of a different bubble. The correlation function between $x$ and $y$ corresponds to the single-bubble contribution, whereas that between $x$ and $y'$ corresponds to the double-bubble contribution.
• Figure 3: Schematic illustration of variables. Left: causality of points $x$ and $y$ on a one-dimensional slice. Right: unit vectors on a two-dimensional slice of a constant-time hypersurface $\Sigma_\tau$.
• Figure 4: $\tilde{\beta} \tau_*$ (blue solid curve) as a function of $\tilde{\beta} \tau_{\rm nuc}$ for the case of a delta-function nucleation rate. The dashed and dot-dashed lines represent the asymptotic forms for large and small values of $\tilde{\beta} \tau_{\rm nuc}$, respectively. Note that $\tau_* = \mathcal{H}_*$.
• Figure 5: 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. The dot-dashed lines represent the asymptotic behavior corresponding to the leading-order result in the Minkovski limit, as derived in Eqs. (\ref{['eq:delta-flat-largek']}) and (\ref{['eq:delta-flat-smallk']}).