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Primordial Black Holes (as Dark Matter) from the Supercooled Phase Transitions with Radiative Symmetry Breaking

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Pith's one-line read This paper claims that radiative symmetry breaking with strong supercooling generically produces primordial black holes, with model-independent predictions for their abundance, mass, and initial spin.

desk verdict Useful model-independent PBH formulas from RSB phase transitions, but the proof of exponential Γ(t) is backed by an order-of-magnitude argument with an unquantified O(1) correction in the paper's own benchmark. read the letter →

arxiv 2412.06889 v2 pith:S4AUUGVQ submitted 2024-12-09 hep-ph astro-ph.COgr-qchep-th

Indra Kumar Banerjee , Francesco Rescigno , Alberto Salvio This is my paper · ORCID
classification hep-phastro-ph.COgr-qchep-th
keywords primordialblackholesradiativesymmetrybreakingsupercooledphasetransitionlate-bloomingmechanismfalsevacuumdecaydarkmatterabundancemicrolensinganomaliesB-Lextension
topics Dark Matter
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper works to establish that primordial black hole (PBH) production is a generic, model-independent consequence of radiative symmetry breaking (RSB) whenever the associated first-order phase transition supercools strongly. It shows that in such theories the false-vacuum decay rate grows exponentially with time, $\Gamma(t)\approx H_n^4 e^{\beta(t-t_n)}$, to high accuracy, and that this exponential law makes the late-blooming mechanism the dominant channel for PBH formation. It then gives ready-to-use formulas for the PBH dark matter fraction $f_{\rm PBH}$, the typical mass $M_{\rm PBH}$, and the initial spin in terms of the few supercool-expansion parameters, and it maps the region of parameter space where PBHs are produced with observable abundance. A concrete Standard Model extension with gauged $B-L$ and right-handed neutrinos is shown to reproduce reported microlensing anomalies, so the mechanism is testable. A sympathetic reader would care because the mechanism demands no fine-tuning and because $f_{\rm PBH}\le1$ becomes a model-independent bound on the parameter space of RSB theories.

What carries the argument

The load-bearing object is the late-blooming mechanism combined with the supercool expansion of RSB theories. The late-blooming mechanism treats a region that nucleates late as a separate homogeneous patch with its own scale factor $a_l(t)$ and Hubble rate $H_l(t)$, while the background evolves with $a_b(t)$ and $H_b(t)$; the false-vacuum fraction $F(t,t_{ni})=e^{-I(t,t_{ni})}$, with $I$ given by Eq. (2.10), controls how vacuum energy converts into radiation. The supercool expansion rewrites the effective potential using the parameters $\chi_0$, $\bar\beta$, $g$, and $\tilde g$, so that the nucleation temperature and $\beta/H_n$ follow from closed formulas such as Eqs. (3.13)-(3.15). The paper's new step is to prove, from this expansion, that $\Gamma(t)\approx H_n^4 e^{\beta(t-t_n)}$ is accurate on the time intervals that matter; that exponential law is what reduces $P_{\rm coll}$, $f_{\rm PBH}$, $M_{\rm PBH}$, and $\sqrt{\langle a_*^2\rangle}$ to simple functions of $\beta/H_n$, $\delta_c$, and $T_{\rm eq}$, and it also explains why smaller $\beta/H_n$ makes PBH production more efficient.

What would settle it

A full numerical-relativity simulation of the collapse of a late-blooming Hubble patch in a supercooled RSB transition, computing the compaction-function threshold directly, would settle the central claim: if that threshold departs significantly from $\delta_c=0.45$, the predicted $f_{\rm PBH}$ moves by orders of magnitude and the claimed generic-production region in the parameter plane would not hold.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central claim is that PBH production is generic across the model-independent parameter space of perturbative RSB theories with large supercooling, operating through the late-blooming mechanism: some Hubble patches remain in the false vacuum longer than the average background, their radiation density is diluted less, and the resulting mass excess makes them collapse into black holes. The demonstration has two linked parts. First, in the supercool and improved supercool expansions the decay rate $\Gamma$ is dominated by the time-independent bounce, and the nearly de Sitter expansion before nucleation gives $\Gamma(t)\approx H_n^4 e^{\beta(t-t_n)}$ with corrections suppressed by the supercooling parameter $X=\log(\chi_0/T_n)$. Second, solving the two-patch Friedmann equations with this $\Gamma$ yields the mass excess $\delta(t,t_{ni})$; when its maximum reaches the threshold $\delta_c=0.45$ the patch collapses, and the resulting collapse probability gives $f_{\rm PBH}$, $M_{\rm PBH}$, and the RMS initial spin $\sqrt{\langle a_*^2\rangle}$ as explicit functions of the model-independent parameters. The paper finds that the PBH abundance scans a broad range up to $f_{\rm PBH}>1$ over the parameter plane, that the requirement $f_{\rm PBH}\le1$ therefore constrains RSB theories whenever the PBHs survive until today, and that a $U(1)_{B-L}$ extension with right-handed neutrinos can fit the HSC microlensing anomaly at $\chi_0\sim2\times10^4$ GeV.

Load-bearing premise

The calculation treats the late-blooming patch as a homogeneous region and assumes it collapses into a black hole when its space-averaged mass excess $\delta$ reaches $\delta_c=0.45$, a threshold taken from spherical-collapse studies; the true collapse threshold and the inhomogeneities that develop after $t_{\max}$ are not computed, so the predicted $f_{\rm PBH}$ shifts if that threshold is inaccurate.

Editorial extensions

If this is right

  • Any RSB model with large supercooling can be tested for PBH dark matter without a dedicated simulation: one plugs the model's $(\chi_0,\bar\beta,g)$ into the provided formulas for $f_{\rm PBH}$, $M_{\rm PBH}$, and spin.
  • Requiring $f_{\rm PBH}\le1$ yields a model-independent upper bound on the RSB parameter space for $\chi_0\lesssim10^9$ GeV, where the produced PBHs survive Hawking evaporation until today.
  • The predicted mass is concentrated around a value set by $\chi_0$ through $M_{\rm PBH}\approx M_J (2\pi^2/3\bar\beta)^{1/2}(280\,{\rm MeV}/\chi_0)^2$, so PBH searches translate directly into constraints on the symmetry-breaking scale.
  • The initial PBH spin is predicted to be very small, and the paper notes that accretion, mergers, close hyperbolic encounters, and scalar-driven Hawking evaporation can later spin PBHs up, so a large observed spin would not contradict the mechanism's initial conditions.
  • A concrete Standard Model extension with gauged $B-L$ and right-handed neutrinos is shown to fit the HSC microlensing anomalies for $\chi_0\sim2\times10^4$ GeV and $g\simeq0.95$, making the mechanism testable with optical surveys.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Not stated in the paper: if the exponential decay law is accurate, the same parameters that set $f_{\rm PBH}$ also fix the stochastic gravitational-wave background from the transition, so a future detection of a strongly supercooled RSB transition carries a definite PBH-abundance prediction to cross-check.
  • Not stated in the paper: the paper's spin result is only the RMS value; applying peak statistics to the shear field would yield the full spin distribution, which binary-merger data could test.
  • Not stated in the paper: the threshold $\delta_c=0.45$ is imported from spherical-collapse studies; non-linear collapse simulations of late-blooming patches would sharpen it and could shift the boundaries of the claimed production region.
  • Not stated in the paper: the same formulas can be applied to other supercooling sectors (axion, dark scalar, or otherwise) to decide whether they produce PBHs without repeating the two-patch integration.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies primordial black hole (PBH) formation via the late-blooming mechanism in supercooled first-order phase transitions with radiative symmetry breaking (RSB). It reformulates the mechanism in Sec. 2, assumes an exponentially growing false-vacuum decay rate Gamma(t), and derives fitting formulas for the collapse probability, PBH abundance, mass, and initial spin. In Sec. 3 it attempts to justify the exponential time dependence of Gamma using the large-supercooling expansion of RSB, and it scans the model-independent parameters chi0, beta-bar, g, and g-tilde to map f_PBH, M_PBH, and the initial spin, including observational constraints. As an application, it identifies a region of a gauged B-L extension of the Standard Model that can fit recently reported HSC microlensing anomalies.

Significance. If the central claims hold, the paper provides a useful model-independent package: for any RSB model with large supercooling one could read off PBH abundance, mass, and spin, and compare with constraints, with a concrete B-L model as a test case. The paper is transparent about some limitations, notably the homogeneous-patch approximation in Sec. 2, and it cites a broad set of PBH constraints. Its main novelty is the attempted justification of Eq. (2.4) and the systematic parameter scan, which extends earlier work in Refs. [20,22,33]. However, the exponential-decay proof is an order-of-magnitude estimate whose controlling second-order term is not small for the parameter range used, and the collapse probability inherits unquantified uncertainties from a fitted function and the collapse threshold. The qualitative conclusion that PBHs are generically produced may survive, but the quantitative predictions do not yet match the strength of the claims made in the abstract and conclusions.

major comments (3)
  1. [Sec. 3.1, Eq. (3.22)] The proof that the non-linear terms in Eq. (3.22) are negligible is quantitatively incomplete. The second-order term in the exponent of Gamma is -(a'/X)(H_I(t-t_n)/X)^2. For the paper's own example with beta/H_n = 9 and tau_max - tau_n = 2.08, one has H_I t_eq = gamma/sqrt(3) = 0.4406, so H_I(t_max-t_n) = 0.916. For representative RSB parameters consistent with Eq. (3.15) and the nucleation condition a'/X = 4 ln(T_n/H_n), one finds X = 8.6 and a'/X = 112, giving a correction at t_max of approximately -1.3 in the exponent. This suppresses Gamma by e^{-1.3} = 0.27 relative to Eq. (2.4), which is about 15% of the linear exponent beta(t_max-t_n). Because this correction enters P_coll through Eqs. (2.18)-(2.21), f_PBH can shift by an order of magnitude or more. The claim that the corrections are 'always very small' is therefore not established by the displayed order-of-magnitude estimate, and the numerical check in the improved supercool expansion is not shown. Please provide residuals or a controlled upper bound on the second- and higher-order terms over the parameter range used in the figures.
  2. [Sec. 2, Eqs. (2.13)-(2.14)] The collapse criterion is set by requiring the space-averaged homogeneous mass excess delta to reach delta_c = 0.45. The paper candidly states after Eq. (2.13) that 'after t_max one must start including inhomogeneities' and that curvature perturbations cannot be captured. That limitation is load-bearing: Eq. (2.14) fixes the late-blooming time t_PBH_ni through this homogeneous-space criterion, and Eq. (2.18) then converts t_PBH_ni into a collapse probability. Neither the threshold delta_c nor the averaging scale is derived within the mechanism. Consequently, the absolute normalization of f_PBH and the positions of the f_PBH = 1 and constraint contours in Figs. 4-10 carry a systematic uncertainty that is not displayed. A sensitivity analysis over delta_c and over the patch-size choice (for example, Hubble radius versus sound horizon) should be presented, or the central claims should be restricted to relative comparisons between parameter regions.
  3. [Sec. 2, Eq. (2.21)] Equation (2.21) is a fitting function whose coefficients a_P, b_P, c_P are quoted to four significant figures, but no residuals, fit range, or uncertainties are reported. Since P_coll enters f_PBH exponentially through Eq. (2.22), even a few percent error in b_P or c_P can change f_PBH by orders of magnitude in parts of the parameter space shown in Figs. 5-10. The text says the fit is obtained 'varying delta_c around 0.45,' but it is not clear whether the quoted coefficients are refitted for each delta_c or whether the delta_c dependence is captured by the final factor. Please show the fit residuals, state the validity range of Eq. (2.21), and provide a sensitivity estimate. Without this, the advertised 'ready-to-use' formulas do not have a stated quantitative reliability.
minor comments (5)
  1. [Appendix A, after Eq. (A.5)] The line defining tau_ni contains a typo: 'tau_ni = t_ni/t_ni' should read 'tau_ni = t_ni/t_eq', and similarly for tau_PBH_ni.
  2. [Captions of Figs. 5, 6, 8, 9] The captions use the notation 'q <a_*^(1/2)>'; according to Eq. (2.23) the plotted quantity is sqrt(<a_*^2>), the RMS dimensionless Kerr parameter. Please make the notation consistent.
  3. [Sec. 2, after Eq. (2.13)] The phrase 'estimate of thedelta, which includes non-linear effects' contains a typo ('thedelta') and should read 'estimate of the delta'.
  4. [Abstract and Sec. 4] The abstract and conclusions say the paper 'demonstrates' and provides a 'full justification' of the exponential time dependence of Gamma, while Sec. 3.1 itself describes the estimate as an order-of-magnitude argument. The wording should be softened to match the strength of the evidence actually presented.
  5. [Eq. (2.2)] The coefficients beta_2 and beta_3 in Eq. (2.2) are not explicitly defined with the factorial factors of a Taylor expansion. A brief defining sentence would avoid ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Sec. 3.1 derives the exponential decay law from the RSB action and the definition of beta; the PBH abundances are obtained by solving the stated IDEs with literature thresholds, not by fitting.

full rationale

The paper's central derivation is self-contained. In Sec. 3.1, the exponential time dependence (2.4) is not assumed; it is obtained by inserting the RSB form S3/T = a'/(X + HI(t-tn)) and T(t) approximately Tn exp(-HI(t-tn)) into Gamma approximately T^4 exp(-S3/T) and expanding 1/(1 + HI(t-tn)/X) to first order. The resulting coefficient a'HI/X^2 - 4HI is identified with beta via the independent definition (2.3) and the prior RSB result (3.15); neither of those inputs contains the exponential law as an assumption, and the identification is a direct Taylor expansion rather than a circular consistency check. The PBH abundance, mass, and spin predictions in Sec. 3.2 follow from solving the two IDEs (2.7)-(2.8) with the survival probability (2.18) and the literature value delta_c = 0.45; the fitting function (2.21) is an empirical fit to the authors' numerical IDE solutions and is not fitted to the PBH observables it is used to predict. The B-L application in Sec. 3.2 uses the same model-independent formulas. The remaining self-citations ([22], [33], [57]) are to prior derivations of the RSB parameterization and spin formulas, which are stated with explicit assumptions and do not include the target exponential-growth claim; they are therefore real evidence, not load-bearing circularity. The skeptical concern about the size of the omitted O((t-tn)^2) term in (3.22) is a quantitative accuracy question about the approximation, not a reduction of the prediction to its own input, and so does not constitute circularity under the rules of this review.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The central calculation rests on the model-independent supercool expansion taken from the authors' prior work (Refs. [22,33]), on standard thermal tunneling formulas, and on a set of externally chosen or numerically fitted constants (delta_c, fit coefficients aP,bP,cP, spin formula coefficients). No new particles or entities are introduced.

free parameters (7)
  • chi0 = varies; HSC fit ~2e4 GeV
    Radiatively generated scale; the paper scans over it and selects a value to match microlensing anomalies.
  • beta_bar = varies
    Beta function coefficient; part of the model-independent parameter set (Eq. (3.4)).
  • g = HSC fit g=0.95
    Coupling in the thermal mass (Eq. (3.6)); scanned and chosen in the HSC application.
  • g_tilde = varies, g_tilde <= g
    Coupling in the cubic term (Eq. (3.6)); scanned in the improved supercool expansion.
  • collapse threshold delta_c = 0.45
    Collapse threshold from literature; though externally set, it is varied in the fit (2.21) and directly controls fPBH.
  • fit coefficients aP, bP, cP = 0.5646, 1.266, 0.6639
    Fit coefficients in Eq. (2.21), fitted to the paper's numerical integration results.
  • spin formula coefficients = 2.1e-3, 23.484, 1.25, 0.625
    Constants in Eq. (2.23) imported from Ref. [58]; they encode the assumed power spectrum and affect the spin prediction.
assumptions (6)
  • domain assumption Effective potential admits the supercool expansion (3.5)-(3.10) with m^2, k, lambda as in (3.6)
    Taken from Refs. [22,33] (Salvio); validity requires large supercooling (small epsilon).
  • standard math False-vacuum decay rate is Gamma ~ T^4 exp(-S3/T) with the time-independent bounce dominating (Sec. 3.1, Eq. (3.16))
    Standard thermal tunneling formula (Coleman, Linde); the paper assumes it for RSB.
  • domain assumption The universe is radiation-dominated before the transition and vacuum-dominated at nucleation, so H ~ H_I in the relevant era (Sec. 3.1, Eq. (3.19))
    Requires large supercooling; this underpins the derivation of the exponential time dependence.
  • domain assumption Bubble walls propagate at the speed of light in supercooled PTs (Sec. 2, Eq. (2.10) footnote 6)
    Used in the false-vacuum fraction F(t,tni); if walls are slower, the PBH abundance changes.
  • domain assumption The PBH mass equals the mass inside the sound horizon H^{-1}/sqrt(3) at collapse (Sec. 2, before Eq. (2.22))
    Taken from Ref. [20]; determines MPBH and the spin normalization.
  • domain assumption Gaussian statistics for the velocity shear and inertia tensor in spin calculations (Sec. 2, text after Eq. (2.23))
    Needed for the spin formula from Ref. [58].

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Cite this review

Pith. "Pith review of Primordial Black Holes (as Dark Matter) from the Supercooled Phase Transitions with Radiative Symmetry Breaking." pith.science (2026). https://pith.science/paper/S4AUUGVQ

@misc{pith2026241206889,
  author       = {Pith},
  title        = {Pith review of: Primordial Black Holes (as Dark Matter) from the Supercooled Phase Transitions with Radiative Symmetry Breaking},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S4AUUGVQ}},
  note         = {Machine review of arXiv:2412.06889}
}
abstract

We study in detail the production of primordial black holes (PBHs), as well as their mass and initial spin, due to the phase transitions corresponding to radiative symmetry breaking (RSB) and featuring a large supercooling. The latter property allows us to use a model-independent approach. In this context, we demonstrate that the decay rate of the false vacuum grows exponentially with time to a high degree of accuracy, justifying a time dependence commonly assumed in the literature. Our study provides ready-to-use results for determining the abundance, mass and initial spin of PBHs generated in a generic RSB model with large supercooling. We find that PBHs are generically produced in a broad region of the model-independent parameter space. As an application, we identify the subregion that may explain recently reported microlensing anomalies. Additionally, we show that a simple Standard-Model extension, with right-handed neutrinos and gauged $B-L$ featuring RSB, may explain an anomaly of this sort in a region of its parameter space.

Figures

Figures reproduced from arXiv: 2412.06889 by the authors.

Figure 1
Figure 1. Evolution of the scale factors (left) and Hubble rates (right) for the background and the LP. to know the probability Psurv(tni , tmax) that in a LP at time tmax (the maximal time before the collapse for tni = t PBH ni ) no bubble formed before tni . This probability was originally determined for a Hubble patch of radius H−1 (tmax) (the Hubble radius at tmax) in [7] and re-derived in [20]: Psurv(tni , tmax) = exp  … view at source ↗
Figure 2
Figure 2. Evolution of the Hubble radius for the background and the LP. We find that our numerical results (see Appendix A for our method of integration) are well described for α ≳ 102 by the following fitting function: Pcoll ≈ exp " −aP  β Hn bP (1 + δc) cP β Hn # , (2.21) with aP ≈ 0.5646, bP ≈ 1.266, cP ≈ 0.6639 in agreement with the result in [20], varying δc around 0.45. Note that the large-α limit (α ≳ 102 ) is all we… view at source ↗
Figure 3
Figure 3. Left: Evolution of the vacuum and radiation energy densities (ρˆV ≡ ρV /∆V, ρˆR ≡ ρR/∆V ) for the background and the LP. Right: Evolution of the mass excess in the LP, δ. units) aK = J/MPBH, which has the dimension of the inverse of a mass. The dimensionless Kerr parameter a∗ can be expressed as8 a∗ = aK/(GNMPBH), where GN is Newton’s constant. We will refer to a∗ as the spin of the black hole. The expression in (2.… view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: β/Hn calculated using the supercool expansion at LO for fixed values of χ0. It is then natural to ask whether there is a specific RSB model able to account for at least some of these anomalous events. Ref. [22] constructed14 a phenomenological completion of the SM with…
Figure 5
Figure 5. Figure 5: fPBH, MPBH and q ⟨a 1/2 ∗ ⟩ calculated with the supercool expansion at LO for fixed values of g. content is the SM one plus the B − L gauge field and a complex scalar. In the plot on the bottom of [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: Like in [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]
Figure 7
Figure 7. Figure 7: β/Hn calculated using the improved supercool expansion where the effective potential is approximated as in (3.10) with g = ˜g and for fixed values of χ0. more efficient. Note that the plasma-driven superradiant instability may decrease the spin of the PBH as well, howe…
Figure 8
Figure 8. Figure 8: fPBH, MPBH and q ⟨a 1/2 ∗ ⟩ calculated using the improved supercool expansion where the effective potential is approximated as in (3.10) for fixed values of g and g˜ = g. However, if there is an accretion disk around the PBH, then the non-spherical accretion can become…
Figure 9
Figure 9. Figure 9: Like in [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]
Figure 10
Figure 10. Figure 10: Like in [PITH_FULL_IMAGE:figures/full_fig_p022_10.png]

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