⚛️ phenomenology

Black holes and gravitational waves from phase transitions in realistic models

This paper demonstrates that including second-order corrections in the bubble nucleation rate expansion is essential for accurately predicting primordial black hole abundances and gravitational wave spectra in realistic models, as these corrections alter the fluctuation distribution toward Gaussianity and decouple PBH yields from GW signals.

Original authors: Marek Lewicki, Piotr Toczek, Ville Vaskonen

Published 2026-02-27

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Marek Lewicki, Piotr Toczek, Ville Vaskonen

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 Big Picture: A Cosmic "Pop" and a "Crunch"

Imagine the very early universe as a giant pot of water on a stove. Usually, water boils smoothly. But in this specific scenario, the universe was supercooled—like water that has dropped below freezing but hasn't turned into ice yet. It was stuck in a "false vacuum" state, full of tension.

Suddenly, the universe decided to snap out of this state. It didn't happen all at once; instead, tiny bubbles of the "true" (stable) universe started popping into existence, expanding, and crashing into each other until the whole universe was filled with them. This event is called a First-Order Phase Transition.

This paper is about two things that happened during this cosmic "pop":

Primordial Black Holes (PBHs): Tiny, ancient black holes that might make up the invisible "Dark Matter" holding galaxies together.
Gravitational Waves (GWs): Ripples in the fabric of space-time, like sound waves from a drum, that we can try to detect today.

The Old Way vs. The New Way

The Old Way (The "Exponential" Guess):
Previous scientists tried to predict how many black holes formed and what the "sound" (gravitational waves) would look like. They used a simple math shortcut. They assumed the bubbles popped at a rate that just kept getting faster and faster, like a snowball rolling down a hill. They used a simple formula: Rate = $e^{time}$ .

The New Way (The "Curved" Reality):
The authors of this paper say, "Wait a minute. In realistic models, things aren't that simple." They found that for the universe to form enough black holes to be interesting, the transition has to be slow and supercooled.

In these slow scenarios, the simple "snowball" formula breaks down. The rate at which bubbles pop changes in a more complex way. It's like driving a car:

Old Model: You press the gas pedal, and you accelerate forever in a straight line.
New Model: You press the gas, but then you hit a slight hill or a curve. You have to account for that second-order change (the curve) to know exactly where you'll end up.

The authors show that if you ignore this "curve" (the second-order correction), your predictions are wrong.

The Two Main Discoveries

1. The "Shape" of the Chaos (Gaussian vs. Skewed)

When the bubbles pop, they create density fluctuations (clumps of energy).

The Analogy: Imagine throwing darts at a board.
- If the transition is fast, the darts are scattered in a perfect bell curve (Gaussian). Most are near the middle, fewer at the edges.
- If the transition is slow (the new model), the darts get "skewed." There are way more darts in one direction than the other.

Why does this matter?

Black Holes: To make a black hole, you need a huge clump of energy (a dart way out in the tail of the distribution). If the distribution is skewed (non-Gaussian), the odds of getting that massive clump change drastically. The authors found that the "skew" makes it harder to form black holes than the old simple models predicted.
Gravitational Waves: The "sound" of the waves depends mostly on the average size of the clumps, not the extreme outliers. So, the waves look mostly the same whether the distribution is skewed or not.

The Big Twist: You can have two different universes that produce the exact same number of black holes, but they produce completely different gravitational wave sounds. This is a game-changer for astronomers. If we detect black holes but hear a different "song" than expected, we know our models need to be updated.

2. The "Pop" and the "Crash" (Two Peaks in the Sound)

The gravitational waves from this event have a unique signature: Two Peaks.

Peak 1 (High Pitch): This comes from the bubbles smashing into each other. Think of it like the sound of popcorn popping.
Peak 2 (Low Pitch): This comes from the giant clumps of energy (the ones that might become black holes) collapsing. Think of this as the deep rumble of a landslide.

The authors show that the "curved" math (the second-order term) changes the volume of the Low Pitch peak relative to the High Pitch peak. It's like turning the bass knob on a stereo.

The Real-World Test: A Particle Physics Example

To prove this isn't just math theory, the authors plugged these ideas into a real-world particle physics model (involving a field called a scalar field). They checked if the "curved" math held up.

Result: Yes. Even in complex, realistic models, this second-order correction is necessary to get the numbers right.
The "Percolation" Rule: They also simplified a complex rule about when the bubbles finish taking over the universe. They found it's as simple as saying: "The bubbles must be smaller than the horizon (the visible universe) when they collide." If they are bigger, the transition fails.

Why Should We Care?

Dark Matter Mystery: If Primordial Black Holes are the Dark Matter, we need to know exactly how many formed. This paper tells us we've been overestimating or underestimating them because we used the wrong math.
Listening to the Universe: We have new telescopes coming online (like LISA, Einstein Telescope, and AEDGE) that will listen for these gravitational waves. This paper gives them a better "map" of what to listen for.
The "Smoking Gun": Because the paper shows that Black Hole counts and Gravitational Wave sounds can be decoupled (they don't always match up in the way we thought), scientists can use the combination of these two signals to figure out exactly what happened in the first split-second of the universe.

Summary

The universe had a slow, dramatic phase transition where bubbles of "new reality" popped into existence. The authors realized that previous math was too simple for these slow transitions. By adding a more complex "curve" to the math, they found that:

The distribution of energy clumps becomes "skewed."
This skew changes how many black holes form but leaves the "sound" of the waves mostly intact.
Therefore, same black hole count $\neq$ same gravitational wave sound.

This helps us build a more accurate picture of the universe's infancy and tells us exactly what to listen for with our future cosmic microphones.

1. Problem Statement

Primordial Black Holes (PBHs) are a compelling candidate for Dark Matter (DM), particularly in the asteroidal mass window. While PBHs can form from density fluctuations generated during inflation, they can also arise from first-order phase transitions in the early universe.

Previous studies (e.g., Refs. [19–24]) modeled the bubble nucleation rate during these transitions using a first-order exponential approximation: $\Gamma(t) \propto e^{\beta t}$ . However, this approximation assumes the transition is relatively fast. The authors identify a critical gap: in realistic models featuring slow, strongly supercooled phase transitions, the nucleation rate deviates significantly from a simple exponential. The standard first-order expansion fails to capture the dynamics necessary for accurate predictions of PBH abundance and the resulting Gravitational Wave (GW) spectrum. Specifically, the non-Gaussian nature of density fluctuations, which is crucial for PBH formation, was not accurately captured in previous simplified models.

2. Methodology

The authors employ a comprehensive approach combining analytical expansions, numerical simulations, and realistic particle physics models:

Second-Order Expansion of Nucleation Rate:
Instead of $\Gamma(t) \propto e^{\beta t}$ , they expand the nucleation rate to the second order:
$\Gamma(t) = H_0^4 \exp\left[ \beta t - \frac{1}{2}\gamma^2 t^2 \right]$
where $H_0$ is the Hubble rate at nucleation, $\beta$ is the inverse duration, and $\gamma$ represents the second-order correction. This term accounts for the rapid damping of the nucleation rate in slow transitions.
Numerical Simulation (deltaPT):
They utilize their publicly available C++ code, deltaPT, to simulate the evolution of energy densities across the phase transition. This code tracks:
- The fraction of the universe in the false vacuum ( $\bar{F}(t)$ ).
- The expansion rate of the universe (solving Friedmann equations with vacuum energy $\rho_v$ and radiation $\rho_r$ ).
- The generation of density fluctuations ( $\delta$ ) in different Hubble patches.
Statistical Analysis of Fluctuations:
They generate $10^6$ realizations of the false vacuum fraction to compute the probability distribution function (PDF), $P_k(\delta)$ , of the density contrast. They specifically analyze how the parameter $\gamma$ affects the non-Gaussianity of this distribution.
PBH and GW Calculation:
- PBHs: They calculate the PBH mass function ( $\psi(M)$ ) by integrating the PDF over the critical threshold $\delta_c$ , applying critical scaling laws.
- GWs: They compute the total GW spectrum as the sum of:
  1. Primary GWs ( $\Omega_{PGW}$ ): From bubble wall collisions and fluid shells.
  2. Secondary GWs ( $\Omega_{SGW}$ ): Induced by large curvature perturbations (scalar-induced).
Realistic Model Application:
To validate their findings, they apply this framework to a classically scale-invariant scalar electrodynamics model. They calculate the decay rate $S_3/T$ and derive the coefficients $\beta$ and $\gamma$ directly from the potential, rather than treating them as free parameters.

3. Key Contributions

Necessity of Second-Order Correction: The paper demonstrates that for slow, supercooled transitions, the second-order term ( $\gamma$ ) in the nucleation rate expansion is necessary and sufficient for accurate predictions. The deviation between the full decay rate and the second-order expansion is negligible (a few percent), but the first-order approximation is insufficient.
Non-Gaussianity and PBH Abundance: The authors show that the distribution of density fluctuations becomes more Gaussian as the second-order term ( $\gamma$ $γ$ ) increases.
- Small $\gamma$ (fast transition) $\rightarrow$ Strong negative non-Gaussianity $\rightarrow$ Suppressed PBH formation.
- Large $\gamma$ (slow transition) $\rightarrow$ Near-Gaussian distribution $\rightarrow$ Enhanced PBH formation probability for the same fluctuation amplitude.
Decoupling of PBH and GW Signals: A major finding is that models predicting the same PBH abundance can produce vastly different GW spectra.
- The PBH abundance is highly sensitive to the shape of the fluctuation distribution (non-Gaussianity).
- The GW amplitude is primarily determined by the typical size of fluctuations (variance).
- Consequently, two models with identical PBH yields can have different ratios of primary to secondary GW peaks.
Refined Percolation Criterion: By consistently treating the impact of bubble nucleation on the expansion rate, they show that the standard percolation criterion coincides with a simple physical requirement: the mean bubble separation at the time of collision must be smaller than the Hubble horizon ( $R_p H_p < 1$ ).

4. Key Results

Fluctuation Distribution (Fig. 1): As $\gamma/\beta$ increases, the distribution $P_k(\delta)$ shifts from having a strong negative tail (non-Gaussian) to being almost Gaussian.
PBH vs. GW Contours (Fig. 2): The contours for constant PBH abundance ( $f_{PBH}$ $f_{P B H}$ ) and constant GW peak amplitude do not overlap.
- For small $\gamma$ , non-Gaussianity suppresses PBHs, but the secondary GW signal remains largely unaffected.
- For large $\gamma$ , PBH production is efficient, and the secondary GW signal can dominate or be comparable to the primary signal depending on parameters.
GW Spectrum Structure (Fig. 3): The resulting GW spectrum exhibits a characteristic double-peak structure:
- High-frequency peak: Originates from primary GWs (bubble collisions), related to the mean bubble separation $R_p$ .
- Low-frequency peak: Originates from secondary GWs (curvature perturbations), related to the horizon size at the end of thermal inflation ( $k_{max}$ ).
Realistic Model Constraints (Fig. 5 & 6):
- In the classically scale-invariant model, the parameter space allowing for $f_{PBH} \approx 1$ (all DM) is viable.
- The predicted GW signals fall within the sensitivity ranges of future experiments like LISA, AEDGE, AION, and the Einstein Telescope (ET).
- The secondary GWs dominate the signal-to-noise ratio (SNR) in specific regions of the parameter space where PBHs are abundant.

5. Significance

Observational Discrimination: The paper provides a crucial tool for distinguishing between different early-universe scenarios. Future GW detectors will not only detect the presence of a phase transition but, by analyzing the ratio of the two GW peaks and comparing it with PBH abundance constraints, could potentially infer the nucleation history (specifically the value of $\gamma$ ).
Dark Matter Viability: It reinforces the viability of PBHs as Dark Matter in the asteroidal mass window, provided the phase transition is sufficiently slow and supercooled to generate the necessary fluctuations without violating GW constraints.
Methodological Improvement: It establishes a new standard for calculating PBH and GW yields from phase transitions, moving beyond the simplistic first-order approximation to a more rigorous second-order treatment that is computationally efficient yet highly accurate.
Note on Gauge Fixing: The authors acknowledge a "Note added" regarding the potential impact of gauge fixing (flat gauge vs. comoving gauge) on the PBH collapse threshold, indicating that while their current results are robust, future work will address this specific systematic uncertainty.

In summary, this work bridges the gap between simplified theoretical models and realistic particle physics scenarios, revealing that the non-Gaussian nature of fluctuations is the key differentiator between PBH formation and GW generation in first-order phase transitions.