Analytic derivation of GW spectrum from bubble collisions in FLRW Universe

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.

💡 Research Summary

This paper presents a significant theoretical advancement in the modeling of the gravitational-wave (GW) spectrum generated during cosmological first-order phase transitions. Traditionally, the calculation of GW spectra from bubble collisions has relied heavily on the Minkowski approximation, which assumes a static spacetime. While this approximation simplifies the complex dynamics of bubble nucleation and collision, it neglects the fundamental role of cosmic expansion, which is intrinsic to the Friedmann-Lemaître-Robertson-Walker (FLRW) metric of our universe.

The authors address this gap by generalizing the analytic formula for the GW spectrum to incorporate the effects of the expanding FLRW universe. Utilizing the thin-wall and envelope approximations—standard frameworks for modeling the energy-momentum tensor of colliding bubbles—the researchers have derived a complete analytic expression that accounts for the expansion-induced evolution of the GW signal. The study provides not only the comprehensive formula but also simplified asymptotic expressions for the large-$k$ and small-$k$ limits, as well as the limit where the expansion becomes negligible (the Minkowski limit).

A critical contribution of this work is the quantitative identification of the breakdown of the Minkowski approximation. The researchers demonstrate that the approximation fails when the ratio $\beta/H_$ is approximately 10 or less, where $\beta$ represents the inverse duration of the phase transition and $H_$ is the Hubble parameter at the completion of the transition. This indicates that in regimes where the phase transition duration is comparable to the Hubble time, the expansion of the universe significantly alters the spectral shape, making static models unreliable.

Furthermore, the paper quantifies the impact of expansion on typical cosmological scenarios. For the Electroweak Phase Transition (EWPT), where $\beta/H_*$ is typically around 140, the authors show that the next-to-leading-order (NLO) term, which accounts for expansion effects, contributes a roughly 10% correction to the spectrum. This finding is particularly vital for the era of precision gravitational-wave astronomy. As next-generation detectors like LISA (Laser Interferometer Space Antenna) prepare to probe the early universe, having highly accurate theoretical templates that include expansion-induced corrections is essential for distinguishing primordial signals from astrophysical backgrounds. By providing a more rigorous mathematical framework, this paper establishes a new standard for interpreting the gravitational-wave signatures of the early universe.