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

TL;DR

This work analyzes early-Universe first-order phase transitions, contrasting direct (outward) and inverse (inward) hydrodynamics. It combines local thermal equilibrium (LTE) estimates for bubble-wall velocity with the sound-shell model (SSM) to predict the gravitational-wave spectra generated by acoustic modes, accounting for cosmic expansion and the expansion history. A key finding is that, while inverse transitions produce qualitatively different fluid profiles, their GW spectra share strong shape similarities with direct transitions, making discrimination based on spectral shapes challenging unless amplitude information is exploited. The study provides a framework for interpreting stochastic GW backgrounds from inverse PTs and highlights the need for field-fluid simulations to go beyond LTE and improve discriminative power for future GW experiments. Overall, the paper lays out the parameter space, LTE limitations, and GW signatures relevant for distinguishing inverse from direct phase-transition dynamics in the early Universe.

Abstract

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.

Figures (15)

• Figure 1: Hydrodynamic solution space for direct and inverse first-order phase transitions. Left (branch structure in the $(v_+,v_-)$ plane): the two hydrodynamic branches from \ref{['eq:vplus_vminus_relation']} are shown in the wall frame with $v_-$ on the horizontal axis and $v_+$ on the vertical axis. The color coding indicates the corresponding value of the transition strength parameter $\bar{\alpha}_+$ normalized to the range $[-1,1]$, $\bar{\alpha}_+=\alpha_+/(1+|\alpha_+|)$, showing inverse (negative $\bar{\alpha}_+$) and direct (positive $\bar{\alpha}_+$) transitions in red and blue, respectively. The locus annotated as "shock front" marks the region where a shock forms ahead of the wall in deflagrations, consistent with the standard classification into deflagration, hybrid, and detonation (and their inverse counterparts). Right (classification in the $(\alpha,\xi_w)$ plane): the horizontal axis is the wall speed $\xi_w$ in the plasma rest frame; the vertical axis is the strength $\alpha$. Solid curves refer to contours/boundaries for $\alpha_N$ (evaluated at $T_N$), while dashed curves refer to $\alpha_+$ (evaluated just ahead of the wall). For a given point $(\alpha_N,\xi_w)$ the solution is unique, whereas for $(\alpha_+,\xi_w)$ there exists a region (hatched) in which two distinct hydrodynamic solutions (detonation or hybrid) share the same pair $(\alpha_+,\xi_w)$. Colors follow the legend in the panel; the shaded gray area denotes kinematically forbidden configurations.
• Figure 2: Hydrodynamic profile of the inverse phase transition. Left Panel: inverse detonations. Middle Panel: inverse hybrid. Right Panel: inverse deflagration.
• Figure 3: Profile pressure for different values of $\Psi, \alpha_N$. Left is for direct PT with $\Psi=0.85$, while right is for inverse PT with $\Psi=1.8$. Dot-dashed lines correspond to runaway solutions, where the friction never vanishes and remains negative throughout the entire wall evolution.
• Figure 4: Phase diagram of the LTE wall velocity in the $(\alpha_N,\Psi)$ plane. Coloured squares show the numerical wall speed. The solid black boundary is the full $\mathcal{P}_{\mathrm{LTE}}=0$ contour: its straight left segment is the static-wall limit $\mathcal{P}_{\mathrm{LTE}}(\xi_w=0)=0$ (a newly nucleated bubble does not expand). In the direct region, the curved branch marks the onset of runaway, equivalently $\mathcal{P}_{\mathrm{LTE}}(\xi_w=\xi_J)<0$; in the inverse region, it is set by the slowest hybrid just past the inverse Jouguet speed, $\mathcal{P}_{\mathrm{LTE}}(\xi_w=\xi_{J,\mathrm{inv}}^+)<0$. The green band highlights parameters satisfying the runaway criterion. The dotted vertical line at $\alpha_N\simeq-0.082$ indicates the point at which the kinematically forbidden region opens up, and the slowest inverse hybrid solution is always a stable solution.
• Figure 5: Kinetic spectrum. Time-independent component $E_{\rm kin}(k)/R_*$ of the fluid velocity UETC for wall speeds of $\xi_w=0.4$ (left) and $\xi_w=0.7$ (right), comparing direct (blue) and inverse (red) branches. Solid lines correspond to (direct or inverse) deflagrations and detonations, whereas dashed lines are hybrids. The corresponding values of $\alpha_N$ are indicated by the respective colors, where shades of blue (red) are direct (inverse) transitions.