Reviving primordial black hole formation in slow first-order phase transitions

This paper demonstrates that primordial black holes can still form during slow first-order phase transitions via an early matter-dominated era induced by slow reheating, a mechanism previously thought to be ruled out, resulting in the production of black holes with large spins.

The Gist

The Big Picture: Fixing a "Broken" Theory

Imagine the early universe as a giant pot of soup. Sometimes, this soup undergoes a sudden change, like water turning into ice. In physics, this is called a First-Order Phase Transition.

For a long time, scientists thought these transitions could create Primordial Black Holes (PBHs)—tiny black holes born at the very beginning of time. The idea was that bubbles of the "new" phase (ice) would form inside the "old" phase (water). Because these bubbles formed at different times in different places, some spots would get crowded with energy, creating heavy clumps that collapsed into black holes.

However, a recent study threw cold water on this idea. They argued that when you calculate the math correctly, the "clumps" aren't heavy enough to collapse. It was like realizing the ice bubbles were too light to crush the water around them. The mechanism seemed dead.

This paper says: "Not so fast." The authors argue the mechanism is still alive, but it needs a specific set of conditions to work. They found a way to revive the idea of these black holes forming, and they discovered something surprising about the black holes they create: they spin incredibly fast.

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The Problem: A Matter of Perspective (Gauge Dependence)

To understand why the idea was thought to be dead, imagine looking at a landscape through two different pairs of glasses:

1. Flat Glasses: You see the hills as very tall and steep.
2. Moving Glasses: You see the same hills as much smaller and flatter.

In the previous failed attempt, scientists used the "Flat Glasses" (a specific mathematical viewpoint called the flat gauge) to measure how heavy the energy clumps were. They found the clumps were heavy enough to form black holes.

But then, other scientists pointed out that the "Moving Glasses" (the comoving gauge, which is the standard way to measure the universe's expansion) showed the clumps were actually much lighter—too light to collapse. It was like realizing the hills were just optical illusions. If you measure the weight correctly, the black holes shouldn't form.

The Solution: The "Slow Reheating" Rescue

The authors of this paper didn't just re-measure the hills; they changed the story of what happens after the transition.

The Old Story (Fast Reheating):
Usually, scientists assume that after the phase transition, the universe instantly heats up and fills with radiation (like light and heat). Radiation acts like a stiff, pressurized gas. If you try to squeeze a gas cloud, it pushes back hard. In this scenario, the small, light clumps (the ones we saw through the "Moving Glasses") get pushed apart before they can collapse.

The New Story (Slow Reheating):
The authors propose a scenario where the universe doesn't heat up immediately. Instead, the energy gets stuck in a "holding pattern" for a while.

• The Analogy: Imagine a crowd of people (energy) that suddenly stops dancing. Instead of immediately running around (radiation), they stand still and sway gently (matter).
• The Result: When the universe is filled with this "swaying" matter instead of "running" radiation, there is no pressure pushing back. Gravity becomes the only boss. Even the small, light clumps that were previously too weak to collapse can now slowly grow, pull themselves together, and eventually collapse into black holes.

This period is called an Early Matter-Dominated Era. It gives the small clumps the extra time they need to grow up and become black holes.

The Twist: Spinning Black Holes

Here is the most interesting part of their discovery.

When black holes form in a normal, fast-expanding universe, they are usually born spinning slowly. But in this "slow reheating" scenario, the collapse happens differently.

• The Analogy: Think of a figure skater. If they pull their arms in while spinning, they spin faster. In this early universe scenario, the collapsing clumps of matter are not perfectly round; they are lumpy and uneven. As they collapse, this unevenness, combined with the lack of pressure, causes them to spin up to extreme speeds.
• The Claim: The paper suggests these black holes are born with "near-extremal spin." They are spinning as fast as physics allows. This is a unique fingerprint that could help us identify them later.

How They Proved It

The authors didn't just guess; they ran complex computer simulations.

1. Bubble Simulation: They simulated how bubbles of the new phase form and grow in the early universe.
2. Math Check: They used a new, more accurate way of doing the math (gauge-invariant equations) to confirm that the clumps are indeed small when measured correctly.
3. The "Slow" Test: They showed that if the universe stays in the "swaying matter" phase long enough (specifically, if the reheating is slow), those small clumps grow big enough to become black holes.

They found that for this to work, the universe needs to stay in this "matter mode" for a specific amount of time. If it switches to radiation too quickly, the black holes don't form. If it stays long enough, they form in abundance.

A Real-World Example

To prove this isn't just math, they built a simple model using a hypothetical "Dark Sector" (a hidden part of the universe we don't see yet).

• They imagined a new type of particle that interacts very weakly with our normal world.
• In their model, this particle causes the phase transition.
• Because it interacts so weakly, it decays very slowly, naturally creating the "slow reheating" condition needed to make the black holes.
• This proves that such a scenario is possible within the rules of physics we already know.

Summary

• The Problem: Recent math suggested primordial black holes from phase transitions are impossible because the energy clumps are too small.
• The Fix: If the universe cools down slowly after the transition, the lack of pressure allows even small clumps to collapse.
• The Result: This creates a population of primordial black holes that are spinning incredibly fast.
• The Significance: This revives a promising theory for the origin of dark matter and provides a unique way to test it: if we ever detect a black hole spinning at the absolute maximum speed, it might be a relic from this specific type of early universe event.

Technical Summary

Technical Summary: Reviving Primordial Black Hole Formation in Slow First-Order Phase Transitions

Problem Statement
Primordial black holes (PBHs) are a candidate for dark matter, and their formation during supercooled first-order phase transitions (FOPTs) has been proposed as a promising mechanism. In supercooled FOPTs, stochastic bubble nucleation creates regions where the transition is delayed, leading to local overdensities relative to the surrounding plasma. However, recent analyses (Refs. [36, 37]) suggested this mechanism is non-viable. These studies highlighted a critical gauge-dependence issue: the density contrast ($\delta$) extracted from bubble simulations corresponds to the spatially flat gauge ($\delta^{(F)}$), whereas the physical threshold for PBH formation is defined in the comoving gauge ($\delta^{(C)}$). The relationship $\delta^{(F)} \simeq 10 \delta^{(C)}$ implies that the actual comoving density contrast is significantly suppressed, falling well below the critical threshold ($\delta_c \approx 0.45$) required for collapse in a radiation-dominated era, effectively ruling out PBH production in standard scenarios.

Methodology
The authors revisit this problem using a new, gauge-invariant computational framework that does not rely on the superhorizon approximation used in previous critiques.

1. Gauge-Invariant Formalism: The authors employ relativistic perturbation theory to define gauge-invariant variables, specifically the comoving density contrast $\Delta$ (equivalent to $\delta^{(C)}$) and the Bardeen potential $\Phi$. They derive a closed system of equations (continuity and Euler equations) for these variables, treating the universe during the FOPT as a two-fluid system (vacuum energy and a second component $f$).
2. Bubble Simulations: They modify the deltaPT code to simulate bubble nucleation histories. In a spherical comoving volume, the first $j_c$ bubbles are generated individually to capture nucleation time and position fluctuations, while the remainder are treated as an averaged background. This allows them to extract the source term $\delta\rho_V$ and evolve the comoving density contrast $\Delta_k$ up to horizon re-entry.
3. Reheating Scenarios: The study distinguishes between fast reheating ($\Gamma_\phi > H_*$) and slow reheating ($\Gamma_\phi < H_*$). In the slow reheating scenario, the scalar field oscillates as non-relativistic matter before decaying, inducing a transient early matter-dominated (EMD) era.
4. PBH Formation Probability: For the EMD era, the authors extend the formalism of Refs. [43, 44] to account for the non-Gaussian nature of the density contrast distribution $P(\delta_k)$ derived from simulations. They assume a factorized joint distribution for the deformation tensor, combining the simulated $P(\delta_k)$ with a standard Gaussian distribution for the traceless part. They calculate the PBH formation probability $\beta_k$ by integrating over the density contrast and deformation parameters, subject to angular momentum constraints (Kerr parameter $\chi \leq 1$).

Key Results

1. Confirmation of Gauge Suppression: The authors confirm that in the standard fast-reheating scenario, the comoving density contrast $\delta^{(C)}$ is indeed significantly smaller than the flat-gauge value, suppressing PBH formation to negligible levels. This validates the gauge-dependence issue identified in recent literature.
2. Viability via Slow Reheating: The central finding is that PBH formation remains viable if the post-FOPT reheating is slow ($\Gamma_\phi < H_*$). This inefficiency leads to an EMD era where the absence of radiation pressure allows even small overdensities to grow and collapse.
3. Formation Conditions: The authors derive that abundant PBH production occurs when the transition rate parameter satisfies $\beta/H_n \lesssim 10$ and the EMD era lasts for a duration $\gtrsim 10^2 H_*^{-1}$. These conditions translate to a bound on the reheating temperature: $T_{\text{reh}} \leq T_{\text{reh}}^{\text{max}}$.
4. PBH Profile:
   • Mass: The resulting PBHs fall into the "asteroid-mass" window ($10^{20} - 10^{22}$ g), a range capable of explaining all dark matter while satisfying current observational constraints.
   • Spin: A distinctive feature of this scenario is that PBHs are born with near-extremal spins ($\chi \sim 1$). This arises because the collapse in an EMD era is sensitive to the specific shape of the perturbation and angular momentum accumulation, leading to a "pancake collapse" mechanism.
5. Model Realization: The authors demonstrate that these conditions are naturally realized in a classically conformal dark $U(1)_X$ gauge theory. In this model, radiative corrections induce a supercooled FOPT, and feeble kinetic mixing with the Standard Model ensures a sufficiently small decay width ($\Gamma_\phi$) to sustain the necessary EMD era.

Significance and Claims
The paper claims to revive the mechanism of PBH formation via supercooled FOPTs, which was previously thought to be ruled out by gauge artifacts. The authors emphasize that their conclusion relies on a consistent gauge-invariant treatment and the physical possibility of a slow reheating phase.

• Mechanism Viability: They assert that the suppression of $\delta^{(C)}$ does not preclude PBH formation; rather, it necessitates an EMD era to amplify the perturbations.
• Distinctive Signatures: The scenario predicts a unique combination of observables: a population of asteroid-mass PBHs with near-extremal spins, a strong stochastic gravitational wave (GW) background from the FOPT, and potential spectral imprints on the GW background from the subsequent EMD era.
• Theoretical Contribution: The work provides the first calculable estimate of PBH formation probability using simulated non-Gaussian perturbations combined with standard Gaussian statistics for the deformation tensor's traceless part. The authors note that while the factorization ansatz is a strong assumption, it opens a new avenue for studying PBH formation under non-Gaussian perturbations and motivates future refined simulations of the full deformation tensor.

The paper concludes that this mechanism is general and can be realized in a broad class of new physics models, particularly those involving feeble couplings between a dark sector and the Standard Model.