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Primordial black hole formation from a type II perturbation in the absence and presence of pressure

T0 review · 2 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read For pressureless spherical collapse, type II initial perturbations and type B spacetimes with a bifurcating trapping horizon are exactly the same phenomenon, and pressure breaks this equivalence.

desk verdict A clean proof of the dust equivalence in the long-wavelength limit, with a new necessity argument; the abstract's blanket 'dust fluid system' overstates the domain. read the letter →

arxiv 2505.00366 v1 pith:6ZWFOEVD submitted 2025-05-01 gr-qc

Koichiro Uehara , Albert Escrivà , Tomohiro Harada , Daiki Saito , Chul-Moon Yoo This is my paper · ORCID
classification gr-qc
keywords primordialblackholeformationtypeIIperturbationbifurcatingtrappinghorizonLemaitre-Tolman-Bondisolutionlong-wavelengthapproximationdustcollapsepressureeffectsclassification
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

This paper asks when a very large primordial curvature fluctuation—one whose areal radius has a stationary "neck" or "belly" on the initial slice (type II)—necessarily produces a black hole whose past and future trapping horizons meet in a bifurcating trapping horizon (type B). The authors prove that for a pressureless, spherically symmetric dust fluid the two classifications are exactly equivalent: type II initial data always yield a type B spacetime, and a type B spacetime can only come from type II initial data, for any fluctuation profile. The proof lives inside the Lemaitre-Tolman-Bondi exact solution, where both conditions reduce to the single algebraic statement $E = -1/2$ on the energy function. This matters because it identifies a sharp, profile-independent boundary in dust-dominated PBH formation, and the paper shows numerically that pressure destroys the equivalence: type II fluctuations can then produce type A spacetimes without bifurcating horizons.

What carries the argument

The load-bearing object is the Lemaitre-Tolman-Bondi solution, the general spherically symmetric dust spacetime, with free functions $E(r)$ and $M(r)$. In the long-wavelength limit the curvature perturbation fixes $E(r) = \frac{1}{2}[-1 + (1 + r\,\partial_r\zeta)^2]$, and trapping horizons are located by $R = 2M$, giving explicit time functions $t_{TH\pm}(r)$. The bifurcation condition $t_{TH+}(r_b) = t_{TH-}(r_b)$ reduces to $y(x) = 0$ with $x = 1 + 4E(r_b)$ and $y(x) = \pi - \cos^{-1}x + \cos^{-1}(-x) + 2\sqrt{1-x^2}$; monotonicity of $y$ forces $x = -1$, i.e., $E(r_b) = -1/2$. The same value of $E$ is equivalent, through the long-wavelength map, to $\partial_r R = 0$, closing the if-and-only-if chain. The numerical part then varies the equation-of-state parameter $w = p/\rho$ to show the chain breaks once pressure is present.

What would settle it

Compute, for an arbitrary dust LTB solution, the initial areal radius $R(t_i,r)$ on a synchronous slice close to the big bang for a configuration that already has a bifurcating trapping horizon ($E(r_b) = -1/2$). The equivalence predicts $\partial_r R(t_i,r_b) = 0$ must appear; finding such a solution without the stationary point would falsify the necessity direction. A companion numerical test is to initialize dust collapse with a Gaussian curvature profile of finite wavelength, so that $\epsilon = k/(aH) \sim 1$, and check whether the bifurcating horizon still tracks the stationary point; losing that tracking would show where the long-wavelength assumption carries the proof.

Watch

Extended reading notes

Core claim

The central claim is an equivalence theorem for spherically symmetric dust collapse. On the initial-data side, a fluctuation is type II if the areal radius $R(r) = a e^{\zeta(r)} r$ has a stationary point $\partial_r R = 0$; on the spacetime side, a formed black hole is type B if its future and past trapping horizons meet at a bifurcating trapping horizon. Within the Lemaitre-Tolman-Bondi solution, the paper proves that a stationary point exists if and only if the free function $E(r)$ takes the value $E = -1/2$, and that $E = -1/2$ is exactly the unique solution of the bifurcation condition $t_{TH+}(r_b) - t_{TH-}(r_b) = 0$. Therefore, for dust, type I/II and type A/B coincide for every fluctuation profile. The paper also shows numerically that the equivalence fails in the presence of pressure: stiffer equations of state make stationary points disappear and reappear, and sufficiently stiff fluids produce type II-A spacetimes with no bifurcating horizon.

Load-bearing premise

The proof assumes the long-wavelength map $E = \frac{1}{2}[-1 + (1 + r\,\partial_r\zeta)^2]$ between the curvature perturbation and the LTB energy function holds exactly, and that the stationary points are regular rather than shell-crossing singular; if either assumption fails, the equivalence is not established.

Editorial extensions

If this is right

  • In a dust-dominated epoch, every PBH formed from a type II fluctuation has a bifurcating trapping horizon; type II-A dust collapse does not exist.
  • The amplitude threshold for the initial profile to develop a stationary point, $\mu_{II}$, also equals the threshold $\mu_B$ for a bifurcating horizon, so one threshold calculation serves both classifications.
  • The equivalence is independent of the fluctuation profile, so it covers any spherically symmetric long-wavelength curvature perturbation, not just the Gaussian example plotted in the paper.
  • With pressure, the link splits: the same type II initial data can end as a type A spacetime for $w > 0$, and stiffer equations of state raise the threshold for type B formation.
  • For dust, a bifurcating trapping horizon in the final spacetime forces the initial areal radius to have a stationary point, giving a constraint on which initial data can produce such spacetimes.

Reading between the lines

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

  • Because the whole equivalence collapses into the single condition $E = -1/2$, the same algebraic criterion could be tested in any matter model with a known long-wavelength map between $\zeta$ and the analogue of $E$, giving a unified test for when type II implies type B.
  • The dust result implies that PBH abundance estimates in dust-dominated or matter-dominated phases can treat type II fluctuations as one class, whereas in radiation-dominated phases the type II-A cases require tracking the two classifications separately.
  • A concrete next calculation is to initialize dust collapse with the same Gaussian profile at finite $k/(aH)$ and evolve it with full numerical relativity; if a bifurcating horizon appears without a stationary initial point, the long-wavelength map is the limiting step.
  • The monotonic function $y(x)$ used in the proof transfers naturally to other horizon-bifurcation problems, such as cosmological wormholes, wherever two trapping-horizon branches are defined by matching times.
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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

2 major / 5 minor

Summary. This paper studies the relation between two classifications used in primordial black hole formation: type I/II initial curvature perturbations, defined by the presence or absence of a stationary point of the areal radius at an initial long-wavelength slice, and type A/B horizon configurations, defined by the absence or presence of a bifurcating trapping horizon. Working with LTB dust in the long-wavelength limit, the authors prove both directions of equivalence: a stationary point implies E(rp) = -1/2 and hence tTH+(rp) = tTH-(rp), and a bifurcating horizon implies, through a monotonicity argument for y(x), that E(rb) = -1/2 and hence a stationary point. The proof is illustrated with a Gaussian profile and with numerical evolutions for equations of state w = 0, 0.15, 1/3, and 1. The paper concludes that for dust the type II threshold equals the type B threshold, while for fluids with pressure the equivalence fails and type B is suppressed.

Significance. The result is a clean analytic contribution: it explains and generalizes the earlier KHW observation for dust and sharpens the recent numerical finding that type II-A configurations exist for radiation. The proof is self-contained, uses no fitted parameters, and gives a transparent monotonicity argument in Eq. (3.4). The explicit example and the numerical comparison across equations of state strengthen the paper's message. The main caveat is that the theorem's proven domain is the long-wavelength initial-data class, not all LTB dust solutions; the abstract's phrasing is broader than the proof. Even with that qualification, the paper is useful for PBH classification and threshold estimates.

major comments (2)
  1. [§3 (first paragraph), §3.1, §3.2, Eq. (2.10)] The equivalence proof relies on Eq. (2.10), which the paper states is valid only in the long-wavelength limit ϵ ≪ 1. The claim in Section 3 that the equivalence holds 'regardless of the functional forms of M(r) and E(r)' is therefore not established for general LTB dust solutions. In a general LTB solution with tB = 0, the areal radius near the big bang behaves as R ≈ (9M/2)^{1/3} t^{2/3}, so the initial stationary-point condition is M'(r0) = 0, which is independent of E(r0). For example, take E(r) ≡ -0.4 and M(r) = M0 + A(r - r0)^2; then ∂rR = 0 at r0 for all sufficiently small t, but x = 1 + 4E(r0) = -0.6 gives y(x) > 0 in Eq. (3.3), so tTH+(r0) - tTH-(r0) > 0 and the configuration is type A. Thus the equivalence is a property of the long-wavelength parametrization of initial data by a single profile ζ(r), not of dust collapse in general. The abstract and conclusion should be qualified to this domain.
  2. [§3.2, Eq. (3.3)] In the necessity proof, Eq. (3.3) is divided by M(rb)/(-2E(rb))^{3/2} to conclude y(x) = 0. The argument implicitly assumes M(rb) > 0 (and, from the restriction E < 0, (-2E)^{3/2} > 0). If M(rb) = 0, the equation holds for any x and the unique conclusion E(rb) = -1/2 does not follow. Since the paper does not state this assumption, the theorem as written is formally incomplete; it should explicitly restrict to black-hole-forming configurations with M(rb) > 0.
minor comments (5)
  1. [Footnote 2 and §3] Footnote 2 excludes shell-crossing singularities, but Section 3 does not repeat this restriction; the theorem statement should mention that the stationary points and the bifurcation point are assumed regular.
  2. [Figure 5] The horizontal axis is z/L, but z is defined only by reference to Eq. (2.3) of Ref. [25]; a brief definition of z in this paper would improve readability.
  3. [Title block] The arXiv ePrint line reads '2505.XXXXX'; this placeholder should be replaced with the actual arXiv identifier.
  4. [§4.2] The text says 'a stiffer w increases the threshold µB for type B PBH formation, similar to the effect on the threshold µA for black hole formation,' but µA is not defined in this paper; it should be defined or replaced by a reference to the black-hole-formation threshold.
  5. [General] The notation 'type I/II' and 'type A/B' is used throughout; a short table summarizing the four combinations and which ones are realized for w = 0 and w > 0 would help readers.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the type I/II and type A/B equivalence is derived from the LTB and trapping-horizon equations with no fitted inputs; self-citations are contextual only.

full rationale

The central claim is proved in Section 3 without fitting or renaming. In Section 3.1, a stationary point ∂_r R(r_p)=0 gives E(r_p)=-1/2 through Eq. (2.10), and direct substitution into the trapping-horizon times (2.18)-(2.19) gives t_TH+(r_p)=t_TH-(r_p)=πM(r_p), so type II implies type B. In Section 3.2, the bifurcation condition t_TH+(r_b)-t_TH-(r_b)=0 is reduced to M(r_b)(-2E(r_b))^{-3/2}y(x)=0 with x=1+4E(r_b); the paper proves y(x) is continuous and monotone on [-1,1], so the only root is x=-1, i.e. E(r_b)=-1/2, and Eq. (2.10) then yields ∂_r R(r_b)=0. Both directions are algebraic consequences of the stated LTB equations; no parameter is fitted to the conclusion and no external result is assumed. The citation of Refs. [15,17] in Section 3.1 is not load-bearing because the two-line calculation is shown. Self-citations such as Ref. [25] are used mainly for motivation, context, and the numerical demonstration of pressure effects in Section 4.2, not to establish the dust equivalence. The paper itself notes that the key relation (2.10) 'is valid only in the limit' (Section 2.2), so the equivalence is proven for the long-wavelength adiabatic growing-mode subclass of LTB models; this limits the abstract's unconditioned phrasing but is a domain restriction, not circularity. The explicit redefinition of type II in Section 2.1 as the existence of a stationary point is transparent and fixes the theorem's hypotheses.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters or invented entities are introduced. The proof is a derivation from the LTB metric and standard horizon identities, with domain assumptions listed.

assumptions (5)
  • domain assumption Long-wavelength approximation ϵ = k/(aH) << 1 holds for initial data.
    Invoked in Section 2.1; Eq. (2.10) relating E(r) to ζ(r) is derived in this limit and is used throughout Section 3.
  • domain assumption Dust fluid (p=0) and growing mode only, with t_B(r)=0.
    Section 2.2 sets t_B(r)=0 to discard decaying modes; the equivalence theorem is for this class of LTB solutions.
  • domain assumption Stationary points of R are regular; shell-crossing singularities are excluded.
    Footnote 2 explicitly excludes singular extrema.
  • ad hoc to paper M(rb) > 0 at the bifurcating trapping horizon.
    Eq. (3.3) divides by M(rb) to conclude x=-1; the proof does not state this bound, though it holds for PBH formation away from the center.
  • standard math Standard identities for spherical trapping horizons: R=2M and monotonic properties of inverse cosine.
    Used in Eqs. (2.18)-(2.19) and in the monotonicity argument for y(x).

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Pith. "Pith review of Primordial black hole formation from a type II perturbation in the absence and presence of pressure." pith.science (2026). https://pith.science/paper/6ZWFOEVD

@misc{pith2026250500366,
  author       = {Pith},
  title        = {Pith review of: Primordial black hole formation from a type II perturbation in the absence and presence of pressure},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6ZWFOEVD}},
  note         = {Machine review of arXiv:2505.00366}
}
read the original abstract

We investigate primordial black holes (PBHs) formed from extremely large amplitudes of primordial curvature fluctuations, classified as type II. Type II fluctuations differ from type I by the presence of a stationary point on the initial time slice, when we see the areal radius as a function of the radial coordinate. Starting from these type II perturbations to form black holes, the nonlinear evolution governed by the Einstein equations generally results in two distinct types, A and B, of horizon configurations, respectively characterized by the absence and presence of a bifurcating trapping horizon where past and future trapping horizons meet. In this paper, we use the Lemaitre-Tolman-Bondi solution to show that type I/II and type A/B classifications are equivalent for a spherically symmetric dust fluid system, regardless of the fluctuation profile. However, this equivalence does not generally hold in the presence of pressure.

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