Primordial Black Hole Formation via Inverted Bubble Collapse

Fuminobu Takahashi; Kai Murai; Kodai Sakurai

REVIEW 4 major objections 4 minor 4 cited by

Primordial Black Hole Formation via Inverted Bubble Collapse

T0 review · 4 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read An incomplete first-order phase transition followed by a bulk transition can turn sparse bubbles into false-vacuum pockets that collapse into primordial black holes, preserving spherical symmetry and producing a monochromatic mass spectrum.

desk verdict Genuinely new PBH formation mechanism with a coherent derivation; the quantitative claims rest on an uncomputed runaway-collapse efficiency the authors themselves flag. read the letter →

arxiv 2502.02291 v3 pith:NV6FF55T submitted 2025-02-04 astro-ph.CO hep-ph

Kai Murai , Kodai Sakurai , Fuminobu Takahashi This is my paper · ORCID

classification astro-ph.COhep-ph

keywords primordialblackholesinvertedbubblecollapsefirst-orderphasetransitionnucleationfalsevacuumsingletextensionoftheStandardModeldarkmattermicrolensing

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 proposes that primordial black holes can form when an incomplete first-order phase transition leaves behind sparse, isolated bubbles, and a later bulk phase transition turns the bubble interiors into false-vacuum regions of higher energy. Those interiors then sit at higher energy than their surroundings, so the bubbles contract and, if the collapsing walls concentrate enough energy, become black holes. The authors argue that this inverted bubble collapse preserves spherical symmetry automatically, avoiding a long-standing uncertainty in bubble- and domain-wall-based PBH formation. Applied to a singlet extension of the Standard Model, they find highly monochromatic PBHs with masses around $10^{-7}$--$10^{-5}\,M_\odot$, a range that could explain observed microlensing events and is being probed by ongoing surveys.

What carries the argument

The load-bearing object is the two-stage inverted bubble collapse geometry. First, a low nucleation rate $\Gamma(T)\ll H^4$ keeps bubbles isolated so they never collide; second, a bulk phase transition at $T_{\rm PT}$ stops nucleation and makes the bubble interiors false vacuum. The size distribution of bubbles at the collapse temperature $T_c$, derived from $dn_b/dt=-3Hn_b+\Gamma(T)$, together with the energy estimate $M_b\simeq(4\pi/3)R^3\Delta V$ and the collapse condition $R_s=2G\epsilon M_b\gtrsim\delta$, determines the PBH mass function. The near-monochromaticity comes from $t_c$ being delayed well past $t_{\rm PT}$: existing bubbles all approach the sound-horizon size, so their collapse masses cluster near $M_{\max}\sim(32\pi/3)\epsilon v^3t_c^3\Delta V$. The efficiency parameter $\epsilon$ carries the crucial assumption that the wall's kinetic energy stays concentrated.

What would settle it

A numerical relativity simulation of a collapsing false-vacuum bubble that includes plasma friction would settle the formation condition: if the wall fails to reach the runaway regime and the shrinking radius never satisfies $R_s=2G\epsilon M_b\gtrsim\delta$, the central PBH-formation criterion is false. Observationally, a high-cadence microlensing survey that resolves the $10^{-7}$--$10^{-5}\,M_\odot$ window and finds a broad, smooth mass distribution instead of a sharp monochromatic peak would test the predicted spectrum.

Watch

Extended reading notes

Core claim

The paper's central claim is that PBHs can be produced by inverted bubble collapse: a first-order phase transition with an extremely low nucleation rate creates isolated true-vacuum bubbles that never percolate; before that transition completes, a second, bulk phase transition makes the outside of each bubble the true vacuum and leaves the bubble interior as a false-vacuum region of higher energy. The bubbles then shrink, and if the energy released into the wall is efficiently concentrated, each bubble collapses once its Schwarzschild radius $R_s=2G\epsilon M_b$ exceeds the wall width $\delta$. Because each nucleated bubble is spherical and never collides, spherical symmetry is preserved through collapse, which the authors contrast with domain-wall or bubble-coalescence mechanisms where the overdense region is aspherical. In the singlet-extended Standard Model example, the predicted mass function is sharply peaked and lies near $\mathcal{O}(10^{-7}\text{--}10^{-5})\,M_\odot$, with an abundance that can account for the observed microlensing events; different parameters can make much lighter PBHs that could constitute all dark matter.

Load-bearing premise

The mechanism's load-bearing premise is that a shrinking bubble's wall accelerates almost without friction and packs essentially all of the bubble's vacuum energy into a region smaller than its Schwarzschild radius before the energy dissipates or the wall bounces; the paper itself notes that this runaway criterion is still under debate.

Editorial extensions

If this is right

PBH production would no longer require specially flat inflaton potentials or large curvature perturbations; ordinary phase-transition physics would suffice.
The singlet-extension example predicts an essentially monochromatic PBH population around $10^{-7}$--$10^{-5}\,M_\odot$, so a future microlensing survey that finds a narrow cluster of events in that mass range would support the scenario.
The mass function is controlled directly by phase-transition parameters ($\Delta V$, $v$, $\epsilon$, $T_c$, and the nucleation rate), giving a direct map from particle physics to PBH observables.
Other parameter choices produce much lighter PBHs that could account for all of the dark matter, and if either transition is strongly first-order the setup would also produce gravitational waves detectable by future space-based interferometers.
The paper's alternative route--a short period of late-time inflation stretching the false-vacuum regions before they reenter the horizon--keeps the mechanism viable even if the runaway collapse criterion fails.

Reading between the lines

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

One extension the authors leave implicit: the same two-transition sequence could operate in hidden sectors at different energy scales, scanning PBH masses from asteroid-scale up to stellar-scale.
The sharp monochromatic peak is a distinguishing signature, because inflation-generated PBH mass functions are generally broad; a narrow spike would point specifically to this mechanism.
A first-principles numerical study of collapsing bubbles in the singlet model would be decisive, since the $\epsilon\simeq1$ assumption is the part most likely to break; the paper itself flags the runaway criterion as unsettled.
The anthropic argument for the parameter choice implies a testable population-level consequence: universes without the required parameter values would have no PBH dark matter from this channel, which bears on the broader question of why dark matter exists.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague—the one thing to know: this paper describes a PBH formation channel I haven't seen before, and the derivation is clean; but the abundance numbers that touch OGLE/HSC depend on an efficiency parameter epsilon that is assumed near unity, not computed. The authors say so in their own Discussion. So read it as a promising mechanism paper, not as a settled prediction.

What's actually new: the sequence where an incomplete FOPT leaves isolated true-vacuum bubbles, a later bulk transition flips their status to false-vacuum regions, and they then contract quasispherically and collapse to PBHs directly in a radiation-dominated universe. That is not in the literature they cite; the closest, Ref. [72], has shrinking bubbles seeding accretion, a different story. The spherical-symmetry point is a real advance over domain-wall and bubble-coalescence mechanisms, where ellipticity is a known obstacle.

The paper does well where it counts for a mechanism paper. Equations (1)-(15) are internally consistent; I checked the bubble-size distribution and the mass-function mapping and did not find an algebraic slip. The singlet-extension example is concrete: full potential, one-loop effective potential, thermal masses, a parameter point checked against unitarity, stability, and LHC constraints, and tunneling evaluated with CosmoTransitions. Reproducible enough to build on.

Now the soft spots, in proportion. The load-bearing step is Eq. (9): PBH formation is identified with the Schwarzschild radius of the bubble's energy exceeding the wall width, with epsilon the fraction of vacuum energy converted to wall kinetic energy. Nothing computes epsilon, the wall Lorentz factor, or the turnaround radius for the singlet model; v=1 and epsilon=1 are inputs. The Discussion concedes the runaway criterion is debated. If epsilon is small or the wall bounces before concentrating the energy, the mechanism gives no PBHs—not fewer, none. The paper offers a post-inflation fallback in that case, but it is unquantified. The constant-DeltaV treatment in Sec. II C is also a simplification, acknowledged, since DeltaV runs with T. These are physical-input concerns, not errors in the math. The parameter choices near the microlensing hints are choices, not circularity.

Who's this for: anyone working on PBH formation from phase transitions, and model builders looking for sharp monochromatic mass predictions. The mechanism deserves a serious referee—the criticism to referee is the collapse criterion, and the authors have left that door open themselves. My own verdict is conditional: the mechanism is plausible and new, the abundance is not yet robust.

Recommendation: send it to review, with a referee who knows bubble wall dynamics.

Referee Report

4 major / 4 minor

Summary. The manuscript proposes a new primordial black hole (PBH) formation mechanism called inverted bubble collapse (IBC). In this scenario, an incomplete first-order phase transition nucleates isolated, spherically symmetric true-vacuum bubbles; before percolation, a second bulk phase transition makes the bubble interiors false-vacuum regions, so the bubbles stop expanding and collapse. The paper derives the bubble size distribution and maps it to a PBH mass function, Eqs. (1)-(15), then applies the mechanism to a singlet extension of the Standard Model. It claims highly monochromatic PBHs with masses around 10^-7 to 10^-5 solar masses that could explain OGLE and Subaru HSC microlensing events, with the possibility of much lighter PBHs constituting all dark matter for other parameters. The central quantitative result depends on the assumption that collapsing bubble walls run away with high efficiency and that an O(1) fraction of the vacuum energy ends up inside the Schwarzschild radius.

Significance. If the collapse dynamics is established, the IBC mechanism is an attractive alternative to PBH formation from domain walls or bubble collisions because it preserves spherical symmetry by construction and maps phase-transition parameters directly to a nearly monochromatic PBH mass function. The formal derivation in Eqs. (1)-(15) is internally consistent and transparent, and the singlet-model example is worked out in considerable detail, including effective-potential construction, thermal corrections, and collider constraints. The connection to OGLE/HSC microlensing hints is timely. However, the quantitative predictions rest on the uncomputed and admittedly debated runaway-collapse assumption, so the significance of the numerical claims is conditional on that physical input being supplied.

major comments (4)

[Sec. II C, Eq. (9)] The PBH formation condition R_s = 2G epsilon M_b ≳ delta is asserted rather than derived. All abundance and mass predictions depend on an O(1) efficiency epsilon with which false-vacuum energy is converted into wall kinetic energy and remains concentrated inside the Schwarzschild radius before the wall reaches width delta, but epsilon, the wall Lorentz factor, and the turnaround radius/time are not computed for the singlet model. The Discussion itself concedes that 'there is currently ongoing debate about establishing a definite criterion for the runaway bubble dynamics' and only states that singlet-sector bubbles are 'likely' to run away. Because Eq. (15) is proportional to v(epsilon Delta V)^{1/3} and because the mechanism produces no PBHs at all if the wall bounces or dissipates before R_s ~ delta, this point must be supported by a dynamical calculation or a controlled estimate before the quantitative claims can be accepted.
[Sec. II C, Eq. (8), and Sec. III C] Delta V is treated as constant in the collapse analysis, while Sec. III C states that in the singlet model Delta V depends on temperature and gives T_c = T_t/sqrt(2) with T_t ≈ 60-70 GeV. Since Eq. (15) depends on Delta V both through the mass-radius relation and through the cutoff M_max, the temperature variation of Delta V over the collapse interval should be quantified or shown to be negligible; otherwise the predicted peak height and peak position are not controlled at the claimed level.
[Sec. III C, paragraph 2] The requirement that the S1 FOPT remain incomplete (Gamma << H^4, isolated bubbles) is confirmed only by the statement that CosmoTransitions gives an increasing nucleation rate. No numerical value of Gamma/H^4 at T_PT or of the bubble volume fraction is reported. Since percolation would break spherical symmetry and invalidate the size-distribution derivation, the paper should present the computed nucleation rate or the nucleation probability per Hubble volume for the parameters in Eq. (26).
[Sec. III C, text after Fig. 4] The delay between the S2-vacuum becoming the true vacuum and the onset of bubble collapse is set by the ad hoc choice T_c = T_t/sqrt(2), i.e., one Hubble time. No calculation of the actual wall dynamics or friction determines this timescale, although t_c directly sets M_max and the peak amplitude in Eq. (15). This assumption should be replaced or bounded by a dynamical estimate of the turnaround time.

minor comments (4)

[Sec. II B, Eq. (2)] Equation (2) is difficult to parse as printed: the solution to Eq. (1) should be the integral n_b a^3 = ∫^t dt' a^3 Γ, which is the form actually used in Eq. (4). Please correct the typesetting if an extraneous exponential is present.
[Figs. 2 and 5] The green dots in Fig. 2 represent f_PBH, while the curves are df_PBH/d ln M, and Fig. 5 repeats this dual usage; the captions and text explain the factor of about two orders of magnitude, but a reader could easily misread the plots. Consider using separate panels or a clear annotation on the vertical axis.
[Sec. III C] There is a typo in the phrase 'satisfies the releavnt theoretical conditions'; it should read 'relevant.'
[Eq. (15) and text after it] The statement that v, epsilon, and Delta V appear only in the combination v(epsilon Delta V)^{1/3} should be made explicit as a degeneracy among model inputs, since the illustrative curves set v=1 and epsilon=1 without a dedicated justification for those values.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the IBC mass function is a genuine mapping from nucleation rate and vacuum-energy inputs, and the unproven runaway-collapse assumption is a physics risk rather than a logical circularity.

full rationale

The derivation chain is self-contained. The bubble size distribution is computed from the nucleation rate via Eqs. (3)-(6), the PBH mass is defined as M = epsilon M_b with M_b from the false-vacuum energy (Eq. (8)), and the formation condition (Eq. (9)) is a physical threshold criterion rather than a restatement of the desired abundance. Equation (15) maps the input parameters (Gamma, DeltaV, v, epsilon, c, alpha) to df_PBH/dlnM; no equation in the chain has the target PBH abundance as an input. In the singlet-extension example, Gamma and DeltaV are computed from the scalar potential using CosmoTransitions, not fitted to the OGLE/HSC microlensing data. The illustrative curves in Fig. 2 use hand-picked c and alpha, but that is parameter choice, not circular reasoning. The Discussion explicitly flags the runaway-bubble criterion as an open question ('There is currently ongoing debate about establishing a definite criterion for the runaway bubble dynamics'); that is a correctness/robustness concern, not a logical circularity. Self-citations appear only as background PBH literature and do not carry the load-bearing argument. No circular step can be exhibited.

Assumptions & free parameters 7 free parameters · 9 assumptions · 0 invented entities

The central claim rests on a set of model assumptions: radiation domination, isolated spherical bubbles, constant wall velocity, inversion of interiors by the bulk transition, a simple Schwarzschild-radius collapse threshold, and runaway wall motion. The first three are standard approximations. The last two, especially the collapse threshold and the runaway efficiency, carry most of the uncertainty and are explicitly flagged by the authors as debated. No new particles or exotic entities are introduced; the two-singlet extension is a well-studied framework.

free parameters (7)

bubble wall velocity v = v = 1 (used in model figures); v = 0.5 in one scan
Assumed constant in Eq. (3); no microphysical calculation for the singlet model. It enters the mass function and peak PBH mass directly.
energy conversion efficiency epsilon = epsilon = 1 (assumed)
Introduced in Sec. II C as the fraction of vacuum energy converted into wall kinetic energy; set to unity in the examples, matching the runaway assumption.
nucleation-rate normalization c = c = 2.0e-8 (set a), 1.7e-11 (set b)
Free parameter in the parametrization of Gamma(T) in Eq. (13) for the illustrative mass functions. In the model it should be fixed by the potential, but the resulting value is not reported.
nucleation-rate exponent alpha = alpha = 0 (baseline), alpha = 10 (variant)
Free parameter in Eq. (13) controlling how quickly nucleation shuts off below TPT.
Delta V (vacuum energy difference during collapse) = DeltaV/rho_tot(Tc) = 0.1 (toy sets)
Treated as constant in Eqs. (8)-(15); in the singlet model it is temperature dependent, and the paper does not specify the effective constant value used.
delay between true-vacuum onset and collapse start = one Hubble time; Tc = Tt/sqrt(2) approximately 46 GeV
The paper assumes the bubbles start shrinking one Hubble time after the S2-vacuum becomes the true vacuum; this choice sets the bubble size and resulting PBH mass.
singlet-model input parameters = mu1 in [-190.49, -190.48] GeV, lambda1 = 0.335, lambda12 = 0.5, lambdaPhi1 = -0.14
Chosen by hand in Eq. (26) to realize the required phase transition sequence and to place the PBH abundance near observational hints. These are Lagrangian parameters, but the selection is tuned.

assumptions (9)

standard math Radiation domination with scale factor a proportional to t^(1/2) during PBH formation.
Used to derive bubble size distribution in Eqs. (3)-(6) and the Hubble relation t = 1/(2H).
domain assumption Bubble nucleation rate is so low (Gamma much less than H^4) that bubbles remain isolated and do not collide or percolate.
Required for spherical symmetry and for Eq. (1) to describe an isolated population.
domain assumption Each bubble expands at constant velocity v after nucleation until collapse begins.
Used in Eq. (3) to map nucleation time to bubble radius at tc.
domain assumption The bulk S2 transition occurs outside bubbles but not inside them, so bubble interiors become false vacuum.
Central to the inversion idea; supported by Fig. 4 but not proven dynamically.
ad hoc to paper The vacuum energy difference DeltaV is constant during the collapse.
Explicit simplification in Sec. II C; inconsistent with later temperature-dependent DeltaV use.
ad hoc to paper A PBH forms when the Schwarzschild radius 2G epsilon M_b exceeds the bubble wall width delta.
Order-of-magnitude collapse criterion in Eq. (9), not derived from a relativistic simulation.
ad hoc to paper Runaway bubble wall acceleration with efficiency epsilon approximately 1.
Assumed to justify energy concentration; the authors note the runaway criterion is debated.
domain assumption Bubble wall width delta is small enough that M_min is negligible for the mass range of interest.
Stated as typically satisfied in Sec. II C; used to drop the lower cutoff.
domain assumption CosmoTransitions reliably computes the bounce action and nucleation rate for the singlet model.
Used to obtain TPT, Tc, and DeltaV; no cross-check with another code is reported.

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Cite this review

Pith. "Pith review of Primordial Black Hole Formation via Inverted Bubble Collapse." pith.science (2026). https://pith.science/paper/NV6FF55T

@misc{pith2026250202291,
  author       = {Pith},
  title        = {Pith review of: Primordial Black Hole Formation via Inverted Bubble Collapse},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NV6FF55T}},
  note         = {Machine review of arXiv:2502.02291}
}

abstract

We propose a novel mechanism of primordial black hole (PBH) formation through inverted bubble collapse. In this scenario, bubbles nucleate sparsely in an incomplete first-order phase transition, such that they remain isolated and do not percolate or collide with each other due to the extremely low nucleation rate. This is followed by a bulk phase transition in the rest of the universe that inverts these pre-existing bubbles into false vacuum regions. These spherically symmetric false-vacuum bubbles subsequently collapse to form PBHs. Unlike conventional PBH formation mechanisms associated with domain wall collapse or bubble coalescence, our inverted bubble collapse mechanism naturally ensures spherical collapse. We demonstrate that, when applied to the singlet extension of the Standard Model, this mechanism can produce highly monochromatic PBHs with masses up to ${\cal O}(10^{-7}\,\text{-}\,10^{-5}) M_\odot$, which potentially explain the microlensing events observed in the OGLE and Subaru HSC data.

Figures

Figures reproduced from arXiv: 2502.02291 by the authors.

**Figure 1.** FIG. 1. Schematic illustration of the IBC mechanism. Left panel: At high temperatures, bubbles nucleate at a low rate, and [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** FIG. 2. PBH fraction in dark matter, [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3. Temperature dependence of three different minima. The blue (green) line shows the local minima appearing in the [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. Contours of the effective potential in the [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: FIG. 5. The highly monochromatic PBH mass spectrum in the singlet-extension model. The spectra from top to bottom are [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

Discussion (0). Continue with ORCID to comment.

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