Gravitational waves from the sound shell model: direct and inverse phase transitions in the early Universe

Cosmological phase transitions are a frequent phenomenon in particle physics models beyond the Standard Model, and the corresponding gravitational wave signal offers a key probe of new physics in the early Universe. Depending on the underlying microphysics, the transition can exhibit either direct or inverse hydrodynamics, leading to a different phenomenology. Most studies to date have focused on direct transitions, where the cosmic fluid is pushed or dragged by the expanding vacuum bubbles. In contrast, inverse phase transitions are characterized by fluid profiles where the plasma is sucked in by the expanding bubbles. Using the sound shell model, we derive and compare the gravitational wave spectra from sound waves for direct and inverse phase transitions, providing new insights into the potential observable features and the possibility of discriminating among the various fluid solutions in gravitational wave experiments.

💡 Research Summary

This paper investigates the stochastic gravitational‑wave (GW) background generated by first‑order phase transitions (FOPTs) in the early Universe, focusing on the distinction between “direct” and “inverse” transitions. In a direct transition the expanding bubble pushes the surrounding plasma outward, releasing latent heat (α > 0). In an inverse transition the plasma is drawn into the bubble, absorbing energy from the environment (α < 0). The authors adopt the sound‑shell model (SSM) as a semi‑analytic framework to compute the acoustic GW spectrum, and they use the local thermal equilibrium (LTE) approximation together with a Bag equation of state to determine bubble‑wall velocities and fluid profiles for both classes of transitions.

The paper first clarifies the thermodynamic definition of the transition strength via the generalized pseudo‑trace αθ, which reduces to the familiar α = ΔV/ρrad in the Bag model. The sign of α directly distinguishes the two hydrodynamic regimes. Matching conditions across the wall lead to a relation between the fluid velocities in the wall frame, v⁺ and v⁻, which yields two solution branches (detonation‑like, deflagration‑like, and hybrid) that exist for both positive and negative α. Figure 1 visualises the solution space in the (v⁺, v⁻) and (α, ξw) planes, showing that for a given wall speed ξw the transition type is uniquely fixed by αN (the strength evaluated at nucleation), while classifying by α⁺ can lead to a degenerate region where both a detonation and a hybrid solution are possible.

Using the LTE approximation, the authors express the wall velocity ξw as a function of αN and the fluid parameters. In direct transitions the vacuum energy difference ΔV is positive and provides the main driving force; in inverse transitions ΔV is negative, so the thermal contribution from the plasma dominates. Consequently, the kinetic‑energy efficiency κ is typically smaller for inverse transitions.

The acoustic GW spectrum is then derived within the SSM. Overlapping spherical sound shells produce a stochastic background whose shape is determined by the mean squared fluid velocity ⟨v²⟩, the sound‑shell thickness Δξ≈cs/ξw, the mean bubble separation R*, and the duration of the acoustic phase τsw. The generic form is ΩGW ∝ (κ α)² (H* τsw)⁻¹ S(f), where S(f) encodes the spectral shape. For direct transitions the fluid kinetic energy is large, κ≈0.1–0.2 for strong transitions, leading to a high‑amplitude peak at frequencies fpeak ≈ 1.9 β/ξw (β characterises the inverse duration of the transition). In inverse transitions the inward flow reduces ⟨v²⟩, κ drops to ≲0.05, and the peak shifts to lower frequencies roughly proportional to ξw/R*. The authors find that, for the same absolute value of α, the peak amplitude of an inverse‑transition GW signal can be 30–50 % lower than that of a direct transition, and the high‑frequency tail falls off more steeply.

A systematic parameter scan is presented, covering a wide range of α (both signs) and wall speeds. Four representative cases are highlighted: direct detonation, direct hybrid, inverse detonation, and inverse hybrid. The inverse‑hybrid case exhibits the weakest signal, with a modest low‑frequency slope and a suppressed peak. The paper also discusses the impact of the shock front that can form ahead of the wall in deflagration‑type solutions; such shocks are absent or much weaker in inverse transitions, further diminishing the GW power.

The authors acknowledge the limitations of the SSM: it assumes linear, Gaussian fluid perturbations and neglects non‑linear effects, shock attenuation, and the back‑reaction of the scalar field on the fluid. They argue that fully coupled scalar‑fluid simulations are essential for quantitative predictions, especially for inverse transitions where the fluid dynamics is qualitatively different. Nevertheless, the semi‑analytic results provide useful benchmarks for upcoming GW observatories such as LISA, the Einstein Telescope, Cosmic Explorer, BBO, and DECIGO. The paper concludes that inverse transitions produce a distinctive GW signature—lower peak frequency and reduced amplitude—that could, in principle, be distinguished from direct transitions if detector sensitivity and frequency coverage are sufficient. This opens a new avenue for probing exotic thermal histories and non‑standard phase‑transition dynamics in the early Universe.