⚛️ phenomenology

Tachyonic particle production in strongly supercooled phase transitions

This paper investigates how particle production driven by temporarily tachyonic masses during strongly supercooled cosmological phase transitions diverts energy from gradient motion, thereby altering gravitational-wave spectra, modifying bubble dynamics, and potentially facilitating primordial black hole formation.

Original authors: Mateusz Kulejewski, Bogumila Swiezewska, Jorinde van de Vis

Published 2026-09-23

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CC BY 4.0

Original authors: Mateusz Kulejewski, Bogumila Swiezewska, Jorinde van de Vis

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

The universe is not a static stage; it is a dynamic history of cooling and changing states, much like water turning to ice. In the earliest moments after the Big Bang, the cosmos was a seething soup of energy and particles. As it expanded and cooled, it underwent dramatic shifts known as phase transitions. Most of these changes happened smoothly, but some were violent and sudden, creating bubbles of a new, lower-energy state that expanded through the old one. When these bubbles collide and crash into each other, they create ripples in the fabric of space-time called gravitational waves. For decades, scientists have looked for these ripples, hoping to hear the echo of the universe's birth. Recently, the focus has shifted to a specific, extreme scenario: a transition that was "supercooled." This means the universe stayed in its high-energy, unstable state for much longer than expected, cooling far below the point where the change should have naturally occurred. When this supercooled state finally gave way, it released a massive amount of energy, and theorists believed this energy would mostly go into the motion of the bubble walls, creating a distinct, loud signal for gravitational wave detectors to find.

A team of researchers has now challenged this long-held picture by looking more closely at what happens inside those expanding bubbles. They studied a specific model of particle physics where a new, hidden force interacts with the known forces of nature. In this model, when the universe undergoes a supercooled phase transition, the field driving the change does not simply settle into its new home. Instead, it overshoots the mark, rolling past the stable point and then bouncing back and forth. During this chaotic rolling, the mathematical description of the particles associated with this field temporarily becomes unstable, a condition physicists call "tachyonic." This instability acts like a trigger, causing the field to spontaneously generate a vast number of new particles right at the moment the bubble forms. The researchers found that this process is incredibly efficient, converting a significant portion of the energy that was supposed to power the bubble's expansion into a swarm of these newly created particles.

The authors of this study used computer simulations to track the evolution of these bubbles from the moment they appear. They discovered that the energy budget of the transition is fundamentally different from what was previously assumed. Instead of all the released energy going into the kinetic motion of the bubble walls, a large fraction is siphoned off to create these fluctuations. This has a direct impact on how the bubbles expand. Because the energy is being drained to create particles, the bubbles grow more slowly than expected. Furthermore, the newly created particles exert a kind of friction against the moving walls, further slowing their progress. This is a crucial detail because the speed of the bubble walls determines the shape and strength of the gravitational wave signal we hope to detect. If the walls are moving slower and the energy is distributed differently, the signal predicted by older models will be incorrect.

The study also revealed a new source of gravitational waves that had not been considered before. While the collisions of bubble walls create one type of signal, the rapid, resonant production of particles creates a second, distinct signal. This new signal is expected to be much higher in frequency than the one from the collisions. The researchers noted that the energy of these particle fluctuations grows rapidly over time, and the most energetic waves are produced at the very end of the process. This suggests that the final gravitational wave spectrum will be dominated by these high-energy modes, creating a signature that is quite different from the standard prediction. The team calculated that for the specific models they examined, the temperature at which these transitions occur can be as high as 10,000, and the energy released is substantial enough to make these effects observable.

Beyond the gravitational waves, these findings have implications for the formation of primordial black holes, which are tiny black holes that could have formed in the early universe. The presence of these abundant particles changes the equation of state, a measure of how the pressure and density of the universe relate to each other. The researchers found that the mixture of the background field and the new particles creates a state that is less "stiff" than previously thought. This change lowers the threshold required for a region of space to collapse under its own gravity, potentially making it much easier for primordial black holes to form during these supercooled transitions. This could explain the existence of certain black holes that are difficult to account for with current theories.

The researchers emphasize that their work is a first step. They used a simplified version of the physics, focusing on the initial stages of the bubble's life and ignoring some complex interactions that would require even more powerful computer simulations to track. They acknowledge that a full, non-linear simulation of the entire process is needed to confirm the exact shape of the gravitational wave spectrum. However, their results are robust enough to show that the old assumption—that all energy goes into the bubble walls—is wrong. The universe, in these extreme moments, is far more efficient at creating matter than previously realized. This discovery forces a re-evaluation of what we expect to see when we listen to the gravitational waves of the early universe. It suggests that the signals we are hunting for may be quieter, slower, and higher in pitch than we thought, and that the early universe may have been a factory for both gravitational waves and primordial black holes in ways we are only just beginning to understand.

Technical Summary: Tachyonic Particle Production in Strongly Supercooled Phase Transitions

Problem Statement
In cosmological models featuring strongly supercooled first-order phase transitions (PTs), the scalar field typically transitions to a value significantly below the minimum of its potential. While these transitions are known to generate strong gravitational-wave (GW) backgrounds, standard models often assume that the released vacuum energy is entirely converted into the gradient energy of the bubble walls or fluid shells. This paper investigates the evolution of the scalar field immediately following bubble nucleation, specifically focusing on the period where the field rolls through a region of negative effective mass (tachyonic mass). The authors posit that this evolution leads to efficient particle production via tachyonic resonance, a mechanism previously overlooked in the context of early bubble expansion. This production could convert a significant fraction of the released energy into particle fluctuations rather than gradient energy, thereby altering the dynamics of the transition, the equation of state (EoS), and the resulting GW signatures.

Methodology
The study utilizes the $SU(2)_c$ SM model, a conformal Standard Model extended with a dark $SU(2)$ sector and a scalar doublet, as a concrete working example of radiative symmetry breaking. The analysis proceeds through the following steps:

Vacuum Transition Approximation: To test the hypothesis of early-stage particle production, the authors consider an $O(4)$ -symmetric vacuum transition rather than the more realistic $O(3)$ -symmetric thermal transition. This choice yields a thinner bubble and a conservative estimate of particle production, as the dynamics depend on a single parameter and mode equations decouple.
Field Evolution and Decomposition: The bounce solution is analytically continued to real time, resulting in an $SO(1,3)$ -symmetric solution. The quantum field is decomposed into a background field $\phi(s)$ and quantum fluctuations $\delta\hat{\phi}$ using Milne coordinates $(s, \chi, \Omega)$ .
Coupled Equations of Motion: The authors solve a closed set of coupled equations:
- The background equation of motion (EoM) includes a backreaction term $\langle \delta\hat{\phi}^2 \rangle$ to ensure energy conservation.
- The fluctuation modes are expanded on a three-dimensional hyperboloid, leading to a set of linearized mode equations for each momentum mode $p$ .
Initial Conditions: Fluctuations are seeded by solving the mode equations in the Euclidean region (metastable vacuum) and analytically continuing the solution to real time at $s \to 0$ .
Numerical Implementation: The coupled Eqs. (3) and (6) are solved numerically in Mathematica for discretized momentum lattices across several benchmark points (BM) with varying gauge couplings ( $g_X$ ) and temperatures ( $T$ ). The study includes thermal corrections via the Arnold-Espinosa daisy-resummation and uses RG-improved potentials.

Key Contributions

Application of Backreaction Formalism: The paper applies a formalism for particle production sourced by background evolution (previously developed in [54, 55]) to the specific case of tachyonic production for the first time. Crucially, the authors modify this formalism to include backreaction in the background EoM, ensuring energy conservation between the background field and the produced fluctuations.
Identification of Tachyonic Resonance: The study demonstrates that during the rolling phase of the scalar field, modes with temporarily tachyonic masses are efficiently produced.
Energy Budget Redistribution: The authors quantify the conversion of vacuum energy into particle fluctuations, showing that this process competes with the generation of gradient energy.
Equation of State Modification: The paper calculates the resulting EoS parameter ( $\omega = P/\rho$ ) for the system, distinguishing between the background and the produced fluctuations.

Results

Efficient Particle Production: Numerical results indicate that a significant fraction (of order $O(1)$ ) of the released vacuum energy is converted into fluctuations immediately after nucleation. This production is most intense during the rolling phase and the first oscillation of the background field.
Damping of Background Oscillations: The backreaction of the produced fluctuations visibly dampens the amplitude of the background field $\phi(s)$ .
Energy Transfer: A substantial portion of the energy budget is transferred to fluctuations. This transfer increases as the temperature decreases, as lower temperatures result in thicker bounce solutions where the field remains in the slow-rolling phase longer.
Equation of State: The produced fluctuations exhibit an EoS parameter of $\omega \simeq 0.2$ , which is lower than the relativistic value ( $\omega = 1/3$ ) of the background field. This lowers the overall EoS parameter of the bubble.
Spectral Evolution: The energy spectrum of the fluctuations shows that as the bubble expands, the peak momentum shifts to higher values. While the peak momentum increases by a factor of roughly 3 during the simulation time, the energy density increases by 2–3 orders of magnitude.
Bubble Dynamics: The authors note that the production of particles exerts friction on the bubble wall and consumes background energy, suggesting that bubble expansion velocities may be slower than in models without particle production.

Significance and Claims
The paper claims that this previously overlooked phenomenon of tachyonic particle production has several phenomenological consequences:

Gravitational Waves: The redistribution of energy alters the amplitude and potentially the shape of the stochastic GW background expected from bubble collisions. Furthermore, the resonant particle production sources a new, high-frequency GW signal, analogous to signals from resonant preheating after inflation, though with a spectrum dominated by the most energetic modes produced at the end of the evolution.
Bubble Wall Velocity: The friction from produced particles and the consumption of kinetic energy may slow down bubble expansion, affecting the GW spectrum's shape and mechanisms relying on supercooled PTs for heavy dark matter production or baryogenesis.
Primordial Black Holes (PBHs): The altered EoS and the accumulation of energy contrast between false-vacuum patches and transitioned regions may facilitate the formation of PBHs. Specifically, the lower density contrast threshold required for collapse in this scenario could significantly ease PBH formation during supercooled PTs.

The authors conclude that while their current semi-analytical study does not include mode-mode scattering (which would require lattice simulations), it provides an essential benchmark for future non-linear studies and suggests that standard predictions for GW signals from supercooled phase transitions require revision.

Technical Summary: Tachyonic Particle Production in Strongly Supercooled Phase Transitions

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