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

arXiv:2605.11332 · hep-ph, astro-ph.CO, gr-qc · Submitted 2026-05-11 · Read on arXiv

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Introduction to the show: ident: Astrophysics Radio. Generated commentary on the latest astrophysics papers.

Vera: Today's paper: "Reviving primordial black hole formation in slow first-order phase transitions".

Jocelyn: The gist:

Vera: First, who's behind it and why it matters.

Title and authors: Vera: So, we’re looking at how this paper revives primordial black hole formation even when the phase transition isn't super fast anymore.

Jocelyn: Right, so instead of needing that rapid bubble nucleation to make black holes, this study shows that if reheating is slow afterward, those small density bumps can still grow into PBHs.

Subrahmanyan: Basically, they’re arguing that the universe gets a chance to be matter-dominated after the phase transition finishes because the decay of the scalar field takes time.

Vera: It hinges on this concept of an early matter-dominated era where those small initial fluctuations get pushed to collapse into black holes.

Jocelyn: And they tackle that tricky problem with gauge dependence in the math, showing that what you measure in a simulation actually corresponds to a specific physical setup.

Subrahmanyan: They do this by looking at how the vacuum energy perturbation sources the density contrast, which is key because it links the phase transition dynamics directly to those black hole seeds.

Vera: The numbers they use show that if the decay width of that scalar field is smaller than some critical Hubble rate, reheating is slow enough for this process to happen.

Jocelyn: So, what does this mean for us observing the universe? It suggests that PBHs from these slower transitions might be a real possibility and not just something ruled out by faster models.

Subrahmanyan: It opens up a whole class of new physics scenarios because it shows this mechanism is general and can happen in lots of different models, especially those with weak dark sector couplings.

Vera: And the authors have proposed some really solid ways to test this, like looking for strong gravitational waves from both the transition itself and that subsequent matter-dominated era.

Jocelyn: They even give us some constraints on the mass of these black holes and how much dark matter they could account for, which is pretty neat.

Subrahmanyan: They’re using scaling relations to map out where these black holes might fit in our dark matter budget, linking the initial formation to current observations.

Vera: It seems like this work is moving us closer to a more concrete picture of PBH production from cosmic phase transitions, and it’s not just a theoretical exercise anymore.

Jocelyn: Exactly, because they’re giving us tools to predict what kind of gravitational waves we might actually see if this scenario is happening.

Subrahmanyan: And that leads perfectly into how we can use these constraints to look for those specific GW signatures in future experiments.

The paper's summary: Vera: So, we’re looking at how this paper suggests ways to make the research on primordial black holes even more solid and testable.

Jocelyn: I mean, they want to get rid of that assumption about the width parameter sigma three in Eq:twenty-five because it’s not really robust.

Subrahmanyan: That assumption is shaky because it doesn't account for the full complexity of the deformation tensor distribution, and they want to extract that directly from the simulation data instead.

Vera: Right, so they want a more direct extraction of the full deformation tensor distribution rather than relying on a simple correlation between variables.

Jocelyn: And then there's this idea about using those scaling relations in Eq:twelve and Eq:sixteen to set bounds on the reheating temperature, T reh.

Subrahmanyan: That’s important because it lets us test if this slow transition actually produces that strong gravitational wave background they mentioned earlier.

Vera: So we can connect the physics of the phase transition directly to something we might be able to measure with future detectors.

Jocelyn: They also have a tool for calculating PBH spin using a statistical anticorrelation, chi = (delta th/delta k) cubed, which helps them predict if these black holes will have near-extremal spin or lower spins.

Subrahmanyan: That's interesting because the spin of the black hole depends on how those initial density perturbations behave during the collapse in that matter-dominated era.

Vera: It’s showing us that even in this slow reheating scenario, we can predict things like black hole spin based on the underlying perturbation statistics.

Jocelyn: So, it sounds like they’re building a framework for actually predicting observable signatures, not just theoretical possibilities.

Subrahmanyan: They are trying to connect the microphysics of the phase transition directly to macro-level observables like gravitational waves and black hole spins.

The paper's improvements: Vera: So, we’re wrapping up this look at "Reviving primordial black hole formation in slow first-order phase transitions" by Vera, Jocelyn, and Subrahmanyan.

Jocelyn: Basically, they’ve shown that PBH formation from slow first-order phase transitions is still possible if reheating is gentle enough to allow for an early matter-dominated era.

Subrahmanyan: This work shows the mechanism is general, meaning it can happen in many different new physics models with weak couplings between dark matter and our Standard Model particles.

Vera: And they’ve given us concrete tools to test this through gravitational waves and accretion mass bounds, which changes how we think about these early black holes.

Jocelyn: It really means we have a better way to predict observable signatures from the very beginning of the universe, which is crucial for our pulsar and sky surveys.

Subrahmanyan: The implications are that we need to look for dual gravitational wave signals, one from the transition and one from that subsequent matter-dominated phase.

Vera: Exactly, so we’re not just looking at black holes forming in isolation anymore; we're looking at a whole dynamic process.

Jocelyn: This paper is a really solid piece for anyone trying to connect high-energy theory with the actual cosmological relics we observe.

Subrahmanyan: It provides the necessary theoretical bridge between the early universe dynamics and what we might see in gravitational wave detectors.

Conclusion: Vera: So we’re wrapping up our look at "Reviving primordial black hole formation in slow first-order phase transitions." This paper shows that PBH formation from slow first-order phase transitions is still possible if reheating is gentle enough to allow for an early matter-dominated era.

Jocelyn: It’s about using that slow reheating period to let small density bumps grow into black holes, which addresses a problem we thought was pretty closed off.

Subrahmanyan: This work shows the mechanism is general, meaning it can happen in many different new physics models with weak couplings between dark matter and our Standard Model.

Vera: And they’ve given us concrete tools to test this through gravitational waves and accretion mass bounds, which changes how we think about these early black holes.

Jocelyn: It really means we have a better way to predict observable signatures from the very beginning of the universe, which is crucial for our pulsar and sky surveys.

Subrahmanyan: The implications are that we need to look for dual gravitational wave signals, one from the transition and one from that subsequent matter-dominated phase.

Vera: Exactly, so we’re not just looking at black holes forming in isolation anymore; we're looking at a whole dynamic process.

Jocelyn: This paper is a really solid piece for anyone trying to connect high-energy theory with the actual cosmological relics we observe.

Subrahmanyan: It provides the necessary theoretical bridge between the early universe dynamics and what we might see in gravitational wave detectors.

Vera: That’s the gist of it. We have a clearer picture of how these black holes might have been made in the early universe, even when things didn't go super fast during the phase transition.

Jocelyn: Thanks for joining us on this look at "Reviving primordial black hole formation in slow first-order phase transitions." We’ll be back next week with another paper from arXiv.

State Key Laboratory of Dark Matter Physics, Tsung-Dao Lee Institute and School of Physics and Astronomy, Shanghai Jiao Tong University · Key Laboratory for Particle Astrophysics and Cosmology (MOE), Shanghai Jiao Tong University · School of Physics, Beihang University

hep-ph, astro-ph.CO, gr-qc

Submitted: 2026-05-11

Updated: 2026-10-08

Comments: 6 pages + 2 figures + references + appendix; v3: to match the published version

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 79/100

The gist: The gist: This work shows that primordial black hole formation from slow first-order phase transitions remains viable because reheating can be sufficiently slow to allow an early matter-dominated era

Key concepts

Gauge Dependence
In physics calculations involving tiny fluctuations, the result can change depending on how you choose to describe the coordinate system (the 'gauge'). The paper notes that different ways of calculating density contrasts lead to different answers, confirming this problem exists in standard theories.
Early Matter-Dominated (EMD) Era
If reheating is slow after the phase transition, the universe enters an EMD era. During this time, because there is no pressure from radiation or vacuum energy dominating, small initial density variations can grow very quickly and collapse to form black holes.
Vacuum Energy Perturbation
The process of a supercooled first-order phase transition creates ripples in the vacuum energy density. These ripples ($\delta\rho_V$) act as seeds, causing specific regions to have slightly higher or lower energy densities, which then source the density contrasts needed for PBH formation.

Terminology

Summary

The gist: This work shows that primordial black hole formation from slow first-order phase transitions remains viable because reheating can be sufficiently slow to allow an early matter-dominated era where small overdensities grow and collapse into PBHs.

Gauge Dependence and Viability

The paper addresses the issue that the density contrast is gauge-dependent in relativistic perturbation theory, noting that the quantity extracted from bubble simulations corresponds to the spatially flat gauge, δ(F), while the conventionally cited value of PBH formation threshold is defined in the comoving gauge, δ(C). They reproduce the results of Refs. [37, 38], confirming the gauge-dependence issue. Crucially, they then show that PBH formation remains viable because a post-FOPT reheating stage may be inefficient, leading to a transient early matter-dominated (EMD) era during which small density perturbations grow rapidly and collapse into PBHs.

Vacuum Energy Perturbation Dynamics

The bubble nucleation rate is parameterized as Γ(t) = H 4/n e β(t−tn), where Hn is the Hubble rate at nucleation time tn, and β−1 sets the FOPT duration. In a supercooled FOPT, the comoving Hubble radius (aH)−1 first shrinks during vacuum domination and later expands after the vacuum energy is released, with its minimum defining percolation moment t∗. The average false vacuum volume fraction is F¯(t) = exp "− 4π/3 Z t−∞ dt′ Γ(t′) Z t t′ dt'' a(t′) a(t′'') cubed, where the bubble wall velocity vw ≈ 1, as appropriate for a supercooled transition. This induces a vacuum energy perturbation δρV = (Fk − F¯)∆V, which sources the density contrast δ.

Reheating Scenarios and EMD Era

The dynamics factorize into two successive energy transfers: from vacuum to a matter-like scalar field, and subsequently from scalar to radiation. If the decay width satisfies Γϕ ≳ H∗, reheating is fast, and vacuum energy converts directly into radiation. However, if Γϕ < H∗, reheating is slow and the oscillating ϕ behaves as nonrelativistic matter, causing an EMD era. Between tk and treh, the Universe is in an EMD era during which the absence of pressure allows a small δk to grow efficiently and form PBHs.

PBH Formation Probability

The PBH formation probability βk is obtained by combining the simulated P(δk) with standard Gaussian statistics for the traceless modes, assuming a factorization of the deformation tensor PD(Dij) = P(δk)PD˜ (D˜ij). The resulting probability is given by βk = fq(qc) Z ∞ −∞ dδk Z ∞ dy Z ∞ 2y dz θ (1 − h(δk, y, z)) × P(δk). The hoop-conjecture condition is enforced by the step function enforcing h ⩽ 1, where h(δk, y, z) = 36z E(ξ)/[π(δk + 2y + 3z) 2].

PBH Profile and Constraints

The final PBH mass satisfies Mmin ≤ 4π/3/45 T0 k cubed gs(T0) gs(Treh) / (gρ(Treh)). The upper limit on the reheating temperature is Treh ⩽ Tmax reh, where Tmax reh = 0.4 Iσ˜k gs(Treh) gs(T0)(1/3) s 45 / (π 2gρ(Treh) k mPl T0). The resulting PBH fraction of dark matter is fpbh = omegapbh/omegadm = 1/omegadmh squared Mpbh Ypbhs0 / (3mPl h H0 2).

Illustrative Model

An illustrative model involves a classically conformal dark U(1)X gauge theory where the FOPT occurs at a low temperature T∗ ≈ 387 GeV with β/Hn ≈ 11.7, yielding Γϕ/H∗ ≈ 0.7 × 10−3. This scenario naturally contains all necessary ingredients for abundant PBH production. The analysis shows that the upper panel of Figure 2 fixes Treh = Tmax reh while varying β/Hn from 8 to 18, and Mpbh lies within an “asteroid-mass” window that can explain all of the dark matter while satisfying current constraints.

Conclusion

The mechanism is general and can be naturally realized in a wide class of new physics models, particularly those with feeble couplings between the dark sector and the SM. The slow and supercooled FOPT that produces the PBHs also generates a strong stochastic GW background, and the subsequent EMD may leave a further spectral imprint on it. In addition, the non-spherical collapse in the EMD era itself could be an independent GW source.

Acknowledgements

The authors thank Yann Gouttenoire, Zhaofeng Kang, Kazunori Kohri, Jing Liu, Chi Tian, and Chen Yuan for helpful discussions and comments. KPX thanks the hospitality of the Tsung-Dao Lee Institute, where part of this work was carried out. WA is supported by startup funds from the Tsung-Dao Lee Institute and Shanghai Jiao Tong University. KPX is supported by the National Natural Science Foundation of China under Grant No. 12305108.

References

[1] A. Mazumdar and G. White, “Review of cosmic phase transitions: their significance and experimental signatures,” Rept. Prog. Phys. 82 (2019) no. 7, 076901, arXiv:1811.01948 [hep-ph].

[2] C. Caprini et al., “Detecting gravitational waves from cosmological phase transitions with LISA: an update,” JCAP 03 (2020) 024, arXiv:1910.13125 [astro-ph.CO].

[3] P. Athron, C. Balázs, A. Fowlie, L. Morris, and L. Wu, “Cosmological phase transitions: From perturbative particle physics to gravitational waves,” Prog. Part. Nucl. Phys. 135 (2024) 104094, arXiv:2305.02357 [hep-ph].

[4] J. Liu, L. Bian, R.-G. Cai, Z.-K. Guo, and S.-J. Wang, “Primordial black hole production during first-order phase transitions,” Phys. Rev. D 105 (2022) no. 2, L021303, arXiv:2106.05637 [astro-ph.CO].

[5] I. Musco, J. C. Miller, and L. Rezzolla, “Computations of primordial black hole formation,” Class. Quant. Grav.

Improvements for AI systems

  1. To develop a more robust PBH formation prediction system, integrate Eq. (25) directly into simulation pipelines to extract non-Gaussian PDF statistics from bubble nucleation histories, moving beyond the assumption that the width parameter σ3 entering this expression is independent of P(δk). This allows for a direct extraction of the full deformation tensor distribution rather than relying on the correlation assumption.

  2. Implement a parameterized model-independent analysis to constrain new physics parameters, specifically using Eq. (12) and (16) to derive bounds on reheating temperature, Treh ⩽ Tmax reh, which directly tests whether slow and supercooled FOPT that produces the PBHs also generates a strong stochastic GW background. This allows AI systems to predict observable gravitational wave signatures associated with PBH formation.

  3. Enhance the dark matter fraction prediction by incorporating accretion effects into the mass bounds, utilizing "Mmin ≡ 4πγ m2 Pl/Hk < Mpbh < Mmax ≡ 4πγ m2 Pl/Hreh and their corresponding fractions, fmin = βk2/2omegadmh2 s0Treh m2/Pl," to provide a more nuanced dark matter budget prediction based on the EMD era duration.

  4. Integrate the analytic estimates for PBH mass and fraction into a parameter estimation framework, using the scaling relations derived from Eq. (17) and (18), which define Mmin = 4π/3 45 T0 k3 gs(T0) gs(Treh) gρ(Treh), Mmax = 4m3 Pl T2 reh s45 gρ(Treh), to rapidly map the parameter space of new physics models.

  5. Develop a tool for calculating PBH spin, incorporating the statistical anticorrelation χ = (δth/δk)3 and the resulting condition χ ≤ 1 imposes δk ≥ δth, to predict whether PBHs are born with near-extremal spin χ ∼ 1 or lower spins based on the underlying perturbation statistics.

Abstract

Large curvature perturbations generated during slow first-order phase transitions are a promising source of primordial black holes. However, recent analyses suggested that the mechanism is ruled out once the density contrast and the formation threshold are evaluated in the same gauge. In this work, we show that this mechanism remains viable: after a supercooled transition, reheating can be sufficiently slow that the Universe enters an early matter-dominated era, during which even small overdensities grow and collapse into primordial black holes.

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