⚛️ phenomenology

Can the universe be matter-dominated after a supercooled first-order phase transition?

This paper demonstrates that a supercooled first-order phase transition cannot immediately result in a matter-dominated universe because bubble collisions generate persistent relativistic gradients that create an intermediate equation of state between matter and radiation, with matter domination only emerging later once these high-momentum modes redshift or the field thermalizes.

Original authors: Henda Mansour, Yann Gouttenoire, Felix Kahlhoefer

Published 2026-07-23

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Henda Mansour, Yann Gouttenoire, Felix Kahlhoefer

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

Imagine the universe as a giant, expanding balloon. Inside this balloon, there is a cosmic fluid that dictates how fast the balloon grows and what kind of "stuff" fills it. For most of its history, scientists have assumed that after the very first moments of the universe, this fluid behaved like a chaotic soup of light-speed particles (radiation) or, later, like slow-moving clumps of matter (dust). But there's a tricky scenario where the universe might have gotten stuck in a weird middle ground. This happens during a "phase transition," a cosmic event similar to water freezing into ice, but for the fundamental fields that make up reality. If the universe gets too cold too fast—a state called "supercooling"—bubbles of a new, lower-energy state of reality can form and crash into each other. The big question physicists have been asking is: right after these bubbles smash together, does the universe instantly settle down into a calm, matter-dominated state (like a quiet room), or does it stay wild and energetic (like a mosh pit)? This matters because the answer changes how we calculate the history of the universe, how black holes might have formed, and how we listen for the faint echoes of the Big Bang in the form of gravitational waves.

In this paper, the authors tackle this question by running massive computer simulations of these bubble collisions. They set up a digital universe where bubbles of "true vacuum" (the new, stable state) nucleate and expand at incredible speeds until they collide. The key discovery is that the universe does not immediately become matter-dominated after these collisions. Instead, the aftermath is a highly chaotic, inhomogeneous mess. When the ultra-fast bubble walls smash together, they don't just stop; they shatter into a jumble of ripples and waves that carry a lot of momentum. Because of this, the energy in the universe behaves more like radiation (light) than matter (dust) for a surprisingly long time.

The authors found that the speed of the bubbles is the deciding factor. If the bubbles are moving at relativistic speeds (close to the speed of light), the collision leaves behind a fluid that acts like radiation. It takes a long time—specifically, the universe has to expand by a factor equal to the bubble's speed factor (Lorentz factor, $\gamma_\star$ )—before these fast-moving ripples slow down enough to act like matter. If the bubbles are moving even faster, this "radiation-like" phase lasts much longer. The paper explicitly rules out the idea that the universe instantly becomes a calm, matter-dominated era right after the bubbles merge. Instead, the transition is delayed. The only way to skip this long, wild phase is if the particles created in the crash can quickly interact with each other to share energy and settle down (thermalize) or if they can change their numbers through specific particle reactions (a process called "cannibalism"). Without these specific interactions, the universe stays in a "warm," radiation-like state for a long time, which changes the timeline for when matter finally takes over.

To visualize this, think of the bubble walls as two giant, super-fast trapeze artists swinging toward each other. When they collide, they don't just gently land; they crash, sending a shockwave of debris flying in every direction. If they are moving slowly, they might just tumble to the ground and sit there (acting like matter). But if they are moving at near-light speed, the crash sends a massive spray of sparks and shrapnel flying outward. For a long time, the energy of the system is dominated by these flying sparks (radiation), not the artists sitting on the ground (matter). The universe has to expand enough for those sparks to slow down and lose their speed before the system can finally settle into a "matter-dominated" state. The authors used a code called "CoolBubble" to simulate this crash in 1, 2, and 3 dimensions, confirming that the faster the crash, the longer the universe stays in this energetic, spark-filled phase.

This finding has a ripple effect on our understanding of the early universe. If the universe stays radiation-like for longer than expected, it changes how we calculate the formation of primordial black holes and the strength of gravitational waves we might detect today. It also means that if dark matter was produced during this time, its abundance might be different than previously thought because the "dilution" effect of the expanding universe works differently in a radiation-like era compared to a matter-dominated one. The paper doesn't claim to have solved the mystery of the universe's entire history, but it provides a very specific, simulation-backed correction to a common assumption: the universe doesn't just snap into a calm state after a supercooled phase transition; it has to cool down from a very high-speed crash first.

Technical Summary: Can the Universe be Matter-Dominated after a Supercooled First-Order Phase Transition?

Problem Statement
The paper addresses a critical uncertainty in the cosmology of supercooled first-order phase transitions (FOPTs). While it is often assumed that the universe enters an early matter-dominated (eMD) era immediately following the completion of such a transition (percolation), this assumption relies on the scalar field behaving as a coherent, homogeneous oscillator. However, in supercooled transitions, the scalar field configuration post-collision is highly inhomogeneous, dominated by the gradients of ultra-relativistic bubble walls. The central question is whether this inhomogeneous state can be treated as matter (equation of state $w \approx 0$ ) immediately, or if it retains radiation-like characteristics ( $w \approx 1/3$ ) for a significant period, thereby altering the thermal history and cosmological predictions.

Methodology
The authors employ a combination of analytical formalism and numerical lattice simulations to investigate the equation of state (EoS) of a classical scalar field following a FOPT.

Analytical Framework: The authors derive the EoS ( $w$ ) for a scalar field in $d$ spatial dimensions using the virial theorem and Fourier mode analysis. They establish that $w$ depends on the ratio of gradient, kinetic, and potential energy densities. Crucially, they analyze the redshifting behavior of Fourier modes in an expanding universe, distinguishing between relativistic ( $k/a \gg m_\phi$ ) and non-relativistic ( $k/a \ll m_\phi$ ) regimes.
Numerical Simulations: Using a custom lattice code, CoolBubble, the authors simulate the nucleation, expansion, and collision of vacuum bubbles in 1+1, 2+1, and 3+1 dimensions.
- Setup: They utilize a specific scalar potential with two minima (metastable and true vacuum) separated by a barrier. Bubbles are nucleated with a critical radius slightly larger than the bounce solution to ensure expansion.
- Parameters: The simulations vary the bubble separation (and thus the collision Lorentz factor $\gamma_\star$ ) and the spatial dimension $d$ . The simulations are performed in both static backgrounds (to determine the immediate post-collision EoS) and expanding backgrounds (to study late-time evolution).
- Diagnostics: The code computes the time-averaged EoS parameter $\langle w \rangle$ and the scalar power spectrum $\Delta_\phi(k)$ to track energy distribution among modes.

Key Contributions and Results

Dependence on Wall Lorentz Factor ( $\gamma_\star$ ):
The primary finding is that the post-collision EoS is not universally matter-like. Instead, it is strongly controlled by the Lorentz factor of the bubble walls at the moment of collision, $\gamma_\star$ .
- High $\gamma_\star$ : For ultra-relativistic walls (typical of strongly supercooled transitions), the collision populates high-momentum modes up to $k \sim \gamma_\star m_\phi$ . These relativistic modes dominate the energy density, resulting in an EoS that is radiation-like ( $w \approx 1/d$ ).
- Low $\gamma_\star$ : For slower walls, the EoS is closer to matter-like, but even then, gradient energy prevents an immediate transition to $w=0$ .
- Quantitative Fit: The authors derive an analytical fitting function for the EoS in a static universe:
  $\omega(\gamma_\star) = \frac{1}{d} \frac{b + \frac{\gamma_\star}{4} \text{erfc}(\sqrt{\pi}/\gamma_\star)}{c + \frac{\gamma_\star}{4} \text{erfc}(\sqrt{\pi}/\gamma_\star)}$
  where $b$ and $c$ are dimensionless parameters fitted to simulation data. This function confirms that for $\gamma_\star \gtrsim 100$ , the fluid is effectively radiation-like.
Delayed Matter Domination:
In an expanding universe, the transition to matter domination is significantly delayed if the scalar field does not thermalize immediately.
- Free-Streaming Case: If the field modes redshift independently (free-streaming), the relativistic modes populated at collision ( $k \sim \gamma_\star m_\phi$ ) remain relativistic until the physical momentum redshifts below the mass scale ( $k/a \sim m_\phi$ ). This requires an expansion factor of $a/a_\star \simeq \gamma_\star$ . Thus, for supercooled transitions with $\gamma_\star \sim 10^8 - 10^{10}$ , the universe remains radiation-like for an enormous duration before becoming matter-dominated.
- Thermalization Scenarios: The delay can be shortened if self-interactions efficiently thermalize the field.
  - Prompt Thermalization (No Cannibalism): If thermalization is instantaneous and number-changing processes are inefficient, matter domination begins after a factor of a few ( $a/a_\star \sim \mathcal{O}(1-10)$ ).
  - Cannibalism: If number-changing processes ( $3 \to 2$ ) are active, the gas enters a "cannibal" phase where rest mass is converted to kinetic energy, slowing cooling. This can delay matter domination by factors of $10^3 - 10^4$ , though this is still vastly shorter than the free-streaming delay for large $\gamma_\star$ .
Power Spectrum Evolution:
The authors characterize the post-collision power spectrum as a broken power law with a turnover at $k_p \sim 0.5-0.7 m_\phi$ and a UV cutoff at $k_\star \sim \gamma_\star m_\phi$ . Unlike thermal equilibrium spectra, the post-collision spectrum retains significant power in high-momentum modes, preventing immediate thermalization and sustaining the radiation-like EoS.

Significance and Claims
The paper concludes that the answer to whether the universe is matter-dominated immediately after a supercooled FOPT is generally no.

Correction to Standard Assumptions: The common assumption of an immediate early matter-dominated (eMD) era is physically unjustified for strongly supercooled transitions where $\gamma_\star \gg 1$ . The universe typically undergoes a prolonged radiation-like phase driven by the persistent relativistic gradients of the scalar field.
Cosmological Implications: This delay has direct consequences for:
- Gravitational Waves (GWs): The spectrum of GWs generated during the transition is sensitive to the expansion history. A prolonged radiation-like era alters the redshifting of modes entering the horizon, changing the spectral slope in the infrared (scaling as $f^1$ instead of $f^3$ ) and diluting the overall amplitude.
- Primordial Black Holes (PBHs): The formation and abundance of PBHs are highly sensitive to the EoS. A delayed matter era suppresses PBH formation compared to scenarios assuming immediate eMD.
- Dark Matter: The production and dilution of dark matter relics depend on the entropy evolution during this epoch. A prolonged radiation-like phase modifies the entropy dilution factors and the thermal production history of dark matter candidates.

The authors emphasize that the specific outcome depends on the microphysics of the scalar sector (specifically the efficiency of thermalization and number-changing processes), but in the absence of rapid thermalization, the kinematic delay imposed by large $\gamma_\star$ is a robust feature of supercooled FOPTs.

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