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Primordial black holes spin from cosmological first-order phase transitions

This paper investigates the spin angular momentum of primordial black holes formed from nonspherical collapse during cosmological first-order phase transitions, deriving a quantitative relationship showing that the Kerr parameter increases with latent heat strength and decreases with the phase transition rate, potentially reaching values as high as $10^{-3}$ .

Original authors: Yu-Shi Hao

Published 2026-06-02

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CC BY 4.0

Original authors: Yu-Shi Hao

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" That Spins

Imagine the very early universe as a giant pot of water that is about to boil. In physics, this "boiling" is called a First-Order Phase Transition. Instead of water turning smoothly into steam, the universe is like a super-cooled liquid that suddenly starts forming bubbles of a new, stable state (like steam) inside an old, unstable state (like super-cooled water).

This paper asks a specific question: When these bubbles form and collapse into Black Holes, do those black holes spin?

The authors say: Yes, and they spin faster than we previously thought.

Here is how they figured it out, broken down into three simple steps.

1. The "Late Bloomer" Problem (Why things get lumpy)

In a perfect world, every part of the universe would start "boiling" (forming bubbles) at the exact same moment. But in reality, it's random.

The Analogy: Imagine a stadium full of people. If a whistle blows to start a game, everyone starts at once. But if the whistle is broken and people start clapping randomly, some sections clap early, and some sections are late.
The Physics: In the early universe, some regions started their phase transition (forming bubbles) later than their neighbors. Because they waited, they held onto more "energy" (like a battery that wasn't used yet).
The Result: These "late" regions became denser and heavier than the areas around them. When these heavy, lumpy regions eventually collapsed under their own gravity to form a Black Hole, they didn't collapse perfectly evenly.

2. The "Squashed Ball" (Why it spins)

If a cloud of gas collapses perfectly into a sphere, it doesn't spin. But if it's a bit squashed or stretched, it starts to rotate (just like a figure skater pulling in their arms, or a pizza dough being tossed).

The Analogy: Imagine trying to squeeze a perfectly round water balloon. If you squeeze it evenly, it stays round. But if you squeeze it harder on the left side than the right, it turns into an oval (an ellipsoid). As it collapses, that uneven shape causes it to twist.
The Paper's Claim: The authors calculated that because the "late" regions were lumpy and uneven (ellipsoidal) rather than perfect spheres, the Black Holes formed from them would naturally acquire spin (angular momentum).

3. The New Math vs. The Old Math

For a long time, scientists used a method called "Peak Theory" to guess how much these Black Holes spin.

The Old Way (Peak Theory): This method assumed the universe's density was perfectly smooth and random (like static on a TV screen), following a "Gaussian" distribution. It predicted these Black Holes would spin very slowly (a tiny amount).
The New Way (Nucleation History Integration): The authors realized that the "boiling" process in the early universe isn't smooth; it's jagged and "non-Gaussian." They invented a new way to add up the history of when bubbles popped in different places.
The Result: Their new math shows that the spin is much stronger than the old math predicted.

The Key Findings (The "Spin" Numbers)

The authors found that the speed of the spin (called the Kerr parameter, $a_*$ ) depends on two things about the "boiling" process:

How much energy is released (Latent Heat, $\alpha$ ): More energy = Faster spin.
How fast the transition happens (Rate, $\beta$ ): A slower transition = Faster spin.

The Numbers:

Old Prediction: Spin was tiny ( $\sim 10^{-5}$ ).
New Prediction: Spin is about 100 times stronger ( $\sim 10^{-3}$ ).
Comparison:
- It is still slower than Black Holes formed in a "Matter-Dominated" era (which can spin very fast, up to $0.1$).
- But it is significantly faster than the "Radiation-Dominated" era predictions we used to have.

Why Does This Matter?

The paper concludes that these Primordial Black Holes (formed right after the Big Bang) are likely spinning much faster than we thought. This is important because:

It changes how they might evaporate (Hawking radiation).
It could trigger instabilities that we might be able to detect with future telescopes.
It suggests that the "lumpy" nature of the early universe's phase transitions is a key ingredient in creating spinning Black Holes.

In short: The universe didn't just "pop" into existence; it popped unevenly. Those uneven pops created Black Holes that are spinning faster than our old models predicted, and the authors have provided a new mathematical recipe to calculate exactly how fast.

Technical Summary: Primordial Black Hole Spin from Cosmological First-Order Phase Transitions

Problem Statement
Primordial Black Holes (PBHs) formed during cosmological first-order phase transitions (FOPTs) are a compelling candidate for dark matter and a probe of new physics. While the mass and abundance of such PBHs have been extensively studied, their spin (angular momentum) remains less explored. Existing literature typically calculates PBH spin within the "peak theory" framework, assuming density perturbations follow a Gaussian distribution. However, density perturbations generated during FOPTs are inherently non-Gaussian due to the stochastic nature of vacuum bubble nucleation. This paper addresses the gap in understanding the spin angular momentum of PBHs formed via the "accumulation mechanism" in FOPTs, specifically investigating how the non-Gaussian, asynchronous nature of the phase transition influences the Kerr parameter ( $a_*$ ).

Methodology
The authors employ a model-independent approach based on the accumulation mechanism, where PBHs form in regions where vacuum bubble nucleation is delayed relative to the background, leading to local energy density overdensities. The methodology diverges from standard peak theory by avoiding Gaussian assumptions and instead utilizing a nucleation history integration method.

Key methodological steps include:

Geometric Origin of Spin: The study models the collapse of an overdense region as a non-spherical process. By expanding density perturbations to the second order, the collapse region is treated as an ellipsoid. The deviation of the ellipsoid's semi-axes ( $a_i$ ) from the background horizon scale ($R = 1/aH$) creates the necessary asymmetry for tidal torque generation.
False-Vacuum Island Model: The authors quantify density perturbations arising from stochastic bubble nucleation. Regions with delayed nucleation retain higher vacuum energy, creating overdensities. The evolution of these regions is described using coupled integro-differential equations for the false-vacuum fraction and energy densities, solved by transforming the system into a set of first-order ordinary differential equations using auxiliary functions.
Nucleation History Integration: Instead of analyzing peaks at a single snapshot in time, the paper integrates over the nucleation history. It calculates the expectation values and variances of the semi-axis lengths of the overdense ellipsoidal regions by integrating over the probability distribution of nucleation times ( $t_n$ ).
Angular Momentum Calculation: The total angular momentum ( $S$ ) is derived by combining the statistical properties of the semi-axes lengths with the velocity shear tensor. The velocity shear is decomposed into a time-evolving factor and a time-independent variance component.
Kerr Parameter Derivation: The dimensionless Kerr parameter ( $a_* = S/M_{PBH}^2$ ) is calculated by relating the angular momentum expectation values to the PBH mass, which is determined by the horizon mass at the time of collapse.

Key Contributions

Non-Gaussian Framework: The paper introduces a formalism to calculate PBH spin without assuming a Gaussian distribution of density perturbations, which is critical for FOPT scenarios where perturbations are non-Gaussian.
Quantitative Relationship: It derives a quantitative relationship between the Kerr parameter and the fundamental phase transition parameters: the latent heat strength ( $\alpha$ ) and the phase transition rate ( $\beta$ ).
Comparison with Peak Theory: The study explicitly contrasts its results with those derived from peak theory under Gaussian assumptions, highlighting the limitations of the latter in the context of FOPTs.

Results

Dependence on Phase Transition Parameters: The study finds that the Kerr parameter $a_*$ increases with the latent heat strength $\alpha$ and decreases with the phase transition rate $\beta$ .
Magnitude of Spin: For typical parameters (e.g., $\alpha = 0.3, \beta = 2$ ), the estimated magnitude of the Kerr parameter is $a_* \sim 10^{-3}$ .
Comparative Analysis:
- This value ( $10^{-3}$ ) is significantly higher than the spin predicted for PBHs formed in the radiation-dominated era under standard peak theory ( $a_* \sim 10^{-5}$ ).
- However, it remains lower than the spin expected for PBHs formed during a matter-dominated era ( $a_* \sim 10^{-1}$ ).
Conservative Estimates: The authors note that their calculated spin represents a conservative lower bound. They acknowledge that hydrodynamic effects during FOPTs (bubble collisions, sound waves, turbulence) could inject additional angular momentum, and subsequent astrophysical processes (hierarchical mergers, gas accretion) could further amplify the spin toward the Thorne limit ( $a_* \approx 0.998$ ).

Significance and Claims
The paper claims that the accumulation mechanism in FOPTs provides a robust pathway for generating spinning PBHs with angular momentum significantly higher than previously estimated for radiation-dominated scenarios. By utilizing the nucleation history integration method, the study provides a more accurate description of PBH spin in the context of non-Gaussian perturbations.

The authors emphasize that while their calculated $a_* \sim 10^{-3}$ is a baseline, the actual observable spin could be substantially larger due to the dynamic environment of the phase transition and subsequent cosmological evolution. This has implications for the Hawking evaporation spectrum and the potential for superradiant instabilities. The study also imposes necessary constraints on the parameter space, excluding regions where the PBH abundance fraction $f_{PBH} > 1$ or where $a_* > 1$ (which would imply naked singularities), and requires consistency with observational constraints such as CMB spectral distortions.

In summary, the paper establishes that the stochastic nature of FOPTs naturally leads to non-spherical collapse and significant primordial spin, offering a distinct signature for PBHs formed in the early universe compared to those formed via other mechanisms.