⚛️ general relativity

Fluid perturbations from expanding bubbles in first-order phase transitions

This paper presents a refined template for the velocity power spectrum generated by expanding bubbles during first-order phase transitions, demonstrating that spectral breaks are determined by velocity profile discontinuities rather than geometric scales and providing these results via the CosmoGW Python package for gravitational wave studies.

Original authors: Chiara Caprini, Antonino S. Midiri, Simona Procacci, Alberto Roper Pol

Published 2026-10-05

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CC BY 4.0

Original authors: Chiara Caprini, Antonino S. Midiri, Simona Procacci, Alberto Roper Pol

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the earliest moments of the universe, fractions of a second after the Big Bang, the cosmos was not the smooth, cooling expanse we see today. Instead, it was a seething, superheated soup of particles and energy. As this primordial fire cooled, it underwent a dramatic transformation known as a phase transition, similar to how water freezes into ice. In a first-order phase transition, this change does not happen everywhere at once. Instead, new regions of the universe, like bubbles of ice forming in a freezing pond, begin to nucleate and expand rapidly into the old, unstable state. As these bubbles grow, they push against the surrounding fluid, creating ripples and shockwaves that travel outward.

These violent collisions and the resulting ripples are more than just a historical curiosity; they are a potential source of gravitational waves, the faint ripples in the fabric of spacetime itself. Scientists hope to detect these ancient whispers using future observatories like the Laser Interferometer Space Antenna. To find them, researchers must first understand exactly how the fluid moved during those first moments. The shape of the signal they are looking for depends entirely on the speed and pattern of the expanding bubbles and the fluid shells they drag along with them. If the theoretical models used to predict these signals are slightly off, the search for these cosmic echoes could miss the mark entirely.

A team of researchers has now refined the map of these fluid movements, correcting long-held assumptions about how the energy from these ancient bubbles spreads. They focused on the precise moment when the expanding bubbles begin to collide, a chaotic phase that sets the stage for the sound waves that eventually generate gravitational waves. By analyzing the fluid dynamics of these expanding bubbles, the team discovered that the patterns of motion are governed by something much more specific than previously thought. For decades, models assumed that the key features of the resulting signal were determined by the overall size of the bubbles and the thickness of the fluid shells surrounding them. However, the new analysis shows that this is not the whole story.

The researchers found that the critical features of the fluid's motion are actually dictated by the sharp edges where the fluid properties change abruptly. In the case of certain types of bubble expansion, specifically those that move faster than the speed of sound, the fluid does not just form a smooth shell. Instead, it develops distinct discontinuities, or sharp jumps, in velocity at specific locations. One such jump occurs at the bubble wall itself, while another forms where a shockwave develops ahead of the wall. The team demonstrated that the distance between these two sharp jumps, rather than the total thickness of the fluid shell, is what determines the specific frequencies of the gravitational waves produced. This distinction is particularly important for a type of expansion called a "hybrid" deflagration, where the bubble wall moves at a speed that creates a significant gap between the wall and the shock front. In these scenarios, the traditional models would miss the true scale of the signal, potentially leading observers to look for the waves in the wrong part of the spectrum.

To verify these findings, the team developed a new set of mathematical templates that describe the velocity of the fluid with high precision. They tested these templates against detailed computer simulations of the fluid dynamics. The results showed that the new models accurately capture the complex behavior of the fluid, including the subtle slopes and breaks in the energy spectrum that previous approximations smoothed over. They also showed that the specific timing of when bubbles appear—whether they all form at once or spread out over time—shifts the location of these signal features. By accounting for these details, the researchers have provided a more reliable tool for predicting what gravitational wave detectors should see.

This work is part of a larger effort to decode the universe's history. The researchers have made their new models and calculations available in a public software package, allowing other scientists to use these refined templates to search for the signatures of first-order phase transitions. By pinpointing the exact scales and shapes of the fluid perturbations, this study helps narrow the search for the gravitational waves that could confirm theories about physics beyond our current understanding. It turns the abstract mathematics of the early universe into a concrete guide for the next generation of cosmic exploration, ensuring that when the detectors finally listen, they know exactly what to expect.

Technical Summary: Fluid Perturbations from Expanding Bubbles in First-Order Phase Transitions

Problem Statement
First-order phase transitions in the early Universe generate gravitational waves (GWs) primarily through the anisotropic stresses induced by fluid perturbations surrounding expanding bubbles of the broken phase. While the "sound shell model" has been widely used to estimate the resulting GW spectrum, the precise origin of the characteristic scales and spectral breaks in the velocity power spectrum has been a subject of debate. Previous literature often attributed these breaks to the mean bubble size at collision ( $R_*$ ) and the fluid shell thickness ( $R_* \Delta \xi$ ). However, the relationship between the microscopic hydrodynamics of single bubbles and the macroscopic spectral shape, particularly regarding the role of discontinuities in velocity profiles and the distribution of nucleation times, requires a more rigorous analytical treatment to improve the accuracy of GW predictions.

Methodology
The authors investigate the velocity field perturbations during the phase of bubble expansion, prior to collisions, and at the onset of the sound-wave regime. The study employs the following methodological framework:

Hydrodynamic Modeling: The analysis assumes a radiation-dominated era with a bag equation of state. The fluid perturbations are modeled as a superposition of spherically symmetric, self-similar single-bubble profiles ( $v_{ip}(\xi)$ ), where $\xi = r/t$ . The study covers subsonic deflagrations, supersonic deflagrations (hybrids), and detonations.
Fourier Analysis of Single Bubbles: The authors analyze the function $f'(z)$ , which represents the Fourier transform of the single-bubble velocity profile. They derive the asymptotic limits of $f'(z)$ and its square $f'^2(z)$ (the spectral density envelope) for large scales ( $z \to 0$ ) and small scales ( $z \to \infty$ ).
Statistical Averaging: The study computes the two-point correlation function of the velocity field by averaging over random nucleation locations and times. Two nucleation time distributions are considered: exponential (characterized by a rate $\beta$ ) and simultaneous (delta-function limit).
Energy Conservation and Effective Time: By enforcing the conservation of kinetic energy density across the collision phase, the authors define an effective deterministic start time for the sound-wave regime ( $t_{sw}$ ). This allows them to compute the velocity spectrum at the beginning of the sound-wave phase without relying on the specific bubble lifetime distribution function used in previous sound shell models.
Toy Models: To validate analytical results, simplified toy models (quadratic profiles for deflagrations/detonations and linear-constant profiles for hybrids) are constructed to reproduce the asymptotic limits and spectral breaks of the full numerical profiles.

Key Contributions and Results

Reinterpretation of Spectral Breaks: The paper establishes that the breaks in the velocity power spectrum are not determined by the bubble size and shell thickness as previously proposed. Instead, they are determined by the positions of the discontinuities in the velocity profiles.
- For deflagrations (subsonic and supersonic), the two spectral breaks correspond to the sum ( $\xi_+ = \xi_{sh} + \xi_w$ ) and the difference ( $\tilde{\Delta}\xi = \xi_{sh} - \xi_w$ ) of the discontinuity locations (the bubble wall $\xi_w$ and the shock front $\xi_{sh}$ ).
- For detonations, with a single discontinuity at the wall, the breaks are determined by the sum and difference of the front and back of the fluid shell ( $\xi_w$ and $c_s$ ).
- Crucially, for hybrids (supersonic deflagrations), the second break is determined by the distance between discontinuities ( $\tilde{\Delta}\xi$ ), which differs significantly from the fluid shell thickness ( $\Delta \xi$ ). This distinction becomes particularly pronounced as the wall velocity approaches the Chapman-Jouget speed.
Analytical Templates: The authors provide refined templates for the velocity power spectrum at the beginning of the sound-wave phase. These templates take the form of single or double broken power laws (SBPL/DBPL), characterized by:
- Asymptotic Amplitudes: Determined by the integral of the velocity profile (large scales) and the magnitude of velocity jumps at discontinuities (small scales).
- Spectral Breaks ( $k_1, k_2$ ): Analytically derived from the discontinuity positions and scaled by factors dependent on the nucleation time distribution (exponential vs. simultaneous).
- Intermediate Slope: A derived slope $\sigma$ that connects the two asymptotic branches, which depends on the separation of the spectral breaks.
Nucleation Time Dependence: The study quantifies how the nucleation history affects the spectrum. For exponential nucleation, the resulting spectrum closely matches the sound shell model results (within ~8% relative difference). For simultaneous nucleation, the deterministic approach yields a spectrum with persistent oscillations (inherited from $f'^2(z)$ ) that are smoothed out in the standard sound shell model due to the averaging over bubble lifetimes. However, the integrated kinetic energy remains consistent across approaches.
Toy Model Validation: The paper demonstrates that simple quadratic and linear-constant toy models can accurately reproduce the asymptotic limits, spectral breaks, and intermediate slopes of the full numerical velocity profiles, providing a computationally efficient tool for spectral estimation.

Significance and Claims
The authors claim that this work provides a more fundamental and accurate analytical description of the velocity spectrum generated by first-order phase transitions. By identifying the spectral breaks with the specific discontinuities in the fluid profiles rather than geometric shell properties, the paper refines the input parameters for GW spectrum calculations.

The refined templates and the definition of an effective start time $t_{sw}$ allow for the reconstruction of the velocity spectrum with minimal reliance on complex numerical simulations or specific assumptions about bubble lifetime distributions. The authors state that these results serve as the necessary initial conditions for the sound-wave regime and will be used in a separate publication to determine the precise impact on the resulting gravitational wave spectrum, particularly regarding the location of spectral peaks and the shape of the signal for experiments like LISA and PTAs. The models and computations are made available via the public Python package CosmoGW.

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