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arxiv: 2605.30145 · v1 · pith:MI5HVU2Anew · submitted 2026-05-28 · 🌀 gr-qc · astro-ph.CO· astro-ph.GA

Supermassive black hole seeds from direct collapse of CDM-curvature peaks

Marco Galoppo , Marco Bruni , Tomohiro Harada This is my paper

Pith reviewed 2026-06-29 06:28 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.COastro-ph.GA
keywords black hole seedsdirect collapseprimordial curvature perturbationscompensated peaksLTB modelsSzekeres solutionsearly universeCDM collapse
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The pith

Broad compensated CDM curvature peaks collapse directly into supermassive black hole seeds at redshifts above 5.

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

The paper models the nonlinear collapse of primordial cold dark matter perturbations using exact relativistic solutions. It shows that only broad compensated curvature peaks, as predicted by peak theory, satisfy the conditions for regular black hole formation, while sinusoidal and Gaussian profiles do not. Formation times are estimated for seeds in the 10^3 to 10^6 solar mass range, with full formation at z greater than 5 and core collapse starting between z of 10 and 16. This provides a direct collapse channel for early black holes in a dust-dominated universe.

Core claim

Using the conserved curvature perturbation R_c to initialize LTB and Szekeres models of dust collapse, the authors derive that broad compensated peaks lead to viable black hole formation with the specified redshifts, and that the initial shear from the curvature data determines whether the singularity is point-like, cigar-like or pancake-like.

What carries the argument

The initial comoving curvature perturbation R_c and its spatial derivatives, which set the active gravitational mass and curvature functions in the LTB and Szekeres models and enforce the regularity conditions for black hole formation.

If this is right

  • Black hole seeds of 10^3-10^6 solar masses form fully at redshifts z>5.
  • Core collapse begins at redshifts 10 to 16.
  • The collapse end-state is selected by the initial shear from the curvature data.
  • Only broad compensated peaks provide viable black hole channels, unlike sinusoidal or Gaussian profiles.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This channel could explain the early appearance of supermassive black holes without prior stellar-mass seeds.
  • The predicted formation redshifts could be compared against the observed population of high-redshift quasars.
  • The link between initial shear and singularity type might extend to understanding generic collapse outcomes in inhomogeneous cosmologies.

Load-bearing premise

The initial curvature data from peak theory evolves under pure gravitational collapse in pressureless dust to produce regular black holes without baryonic or other effects disrupting the regularity conditions.

What would settle it

An observation that black holes in the 10^3-10^6 solar mass range are absent or form at redshifts well below 5, or a calculation showing that compensated peaks violate the regularity conditions in the LTB and Szekeres models.

Figures

Figures reproduced from arXiv: 2605.30145 by Marco Bruni, Marco Galoppo, Tomohiro Harada.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic Penrose diagrams illustrating the causal outcomes relevant to the collapse models considered in this [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Parameter space of initial curvature perturbations [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Initial comoving curvature perturbation [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Upper row: initial density contrast [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Absolute magnitudes of the individual dynamical contributions entering the Raychaudhuri equation normalised by the [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Absolute magnitudes of the individual dynamical contributions entering the Hamiltonian constraint, normalised by [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Relative contributions of the Ricci and Weyl curvature sectors to the Kretschmann scalar, evaluated for initial comoving [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The curvature function [PITH_FULL_IMAGE:figures/full_fig_p026_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Upper row: the collapse time [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Phase space of collapse outcomes in the ( [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗
read the original abstract

We study black hole (BH) formation from the nonlinear growth and collapse of primordial perturbations during the matter-dominated era. Modelling cold dark matter (CDM) as pressureless dust, we describe the collapse in a fully nonlinear relativistic framework using the Lema\^{i}tre-Tolman-Bondi (LTB) and quasi-spherical Szekeres solutions as exact perturbations of a spatially-flat Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) $\Lambda$CDM background. At first order in relativistic scalar perturbation theory, the growing mode of any relevant quantity can be expressed in terms of the conserved gauge-invariant curvature perturbation $\mathcal{R}_c$, which acts as a potential for the 3-curvature of hypersurfaces orthogonal to the matter 4-velocity. We use this result to express the active gravitational mass and curvature functions of the LTB and Szekeres models in terms of the initial values of $\mathcal{R}_c$ and its spatial derivatives. From these initial curvature data we derive: (i) the turn-around, collapse, and apparent-horizon formation times, and (ii) the regularity conditions required for BH formation. We show that sinusoidal and Gaussian profiles do not provide viable BH-forming channels, whereas broad compensated curvature peaks, naturally predicted by peak theory, do. We then estimate the formation times of $10^{3}-10^{6}~\mathrm{M}_\odot$ massive BH seeds produced by the direct collapse of primordial CDM curvature peaks, finding full BH formation at redshifts $z>5$, with core collapse beginning at $10 \lesssim z \lesssim 16$. Finally, we characterize the local dynamics and singularity type of the collapse (point-like, cigar-like, or pancake-like) directly from the initial comoving curvature data, clarifying the role of the initial shear in selecting the collapse end-state.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper models the nonlinear collapse of primordial CDM perturbations in the matter era using exact LTB and Szekeres solutions whose initial data are set from the first-order growing mode of the conserved curvature perturbation R_c. It derives turnaround, collapse, and apparent-horizon times together with regularity conditions for black-hole formation, concluding that broad compensated curvature peaks (as predicted by peak theory) produce regular 10^3–10^6 M_⊙ seeds with full BH formation at z>5 and core collapse beginning at 10≲z≲16, while sinusoidal and Gaussian profiles do not; the local collapse dynamics (point-, cigar-, or pancake-like) are also read off directly from the initial comoving curvature data.

Significance. If the first-order initial-data construction remains valid, the work supplies a purely gravitational channel for early supermassive black-hole seeds whose formation redshifts and end-state morphology are fixed by the shape of R_c, together with explicit regularity criteria that can be checked against peak-theory statistics. The use of exact inhomogeneous solutions and the direct translation from initial R_c to collapse times and singularity type constitute a clear technical advance over purely perturbative or Newtonian treatments.

major comments (2)
  1. [Initial-data construction from R_c (abstract and § on LTB/Szekeres initial conditions)] The central construction maps the first-order growing mode of R_c directly onto the LTB active-mass and curvature functions (and their Szekeres generalizations). For the broad compensated peaks that the paper claims produce regular BHs, the required density contrast reaches O(1) at turnaround; at those amplitudes the linear relation between R_c and the metric functions receives O(R_c²) corrections that are not included. If those corrections modify the sign or magnitude of the curvature function or the mass profile, the derived regularity conditions (absence of shell-crossing, apparent-horizon formation before singularity) and the quoted collapse redshifts (z>5) no longer follow. The paper does not quantify the size of R_c for its 10³–10⁶ M_⊙ examples or demonstrate that second-order terms remain negligible.
  2. [Profile viability and regularity conditions] The distinction between viable (broad compensated) and non-viable (sinusoidal, Gaussian) profiles rests entirely on the regularity conditions obtained from the first-order R_c data. Because the amplitude regime in which those conditions are applied is precisely where the linear mapping is most suspect, the profile-selection claim is load-bearing and requires an explicit check that second-order corrections do not alter the sign of the curvature function or the ordering of apparent-horizon and singularity formation times.
minor comments (2)
  1. [Abstract] The abstract states that core collapse begins at 10≲z≲16; a brief clarification of how this interval is obtained from the turnaround and apparent-horizon times would help readers trace the numerical estimates.
  2. [Throughout] Notation for the conserved curvature perturbation is given as both R_c and Ρ_c in different places; consistent use of a single symbol (with the script R_c preferred) would remove ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive major comments on the validity of the first-order initial-data construction. We address both points below and will incorporate explicit estimates of the curvature amplitude to demonstrate the robustness of the linear mapping.

read point-by-point responses

  1. Referee: The central construction maps the first-order growing mode of R_c directly onto the LTB active-mass and curvature functions. For the broad compensated peaks, the required density contrast reaches O(1) at turnaround; at those amplitudes the linear relation between R_c and the metric functions receives O(R_c²) corrections that are not included. The paper does not quantify the size of R_c for its 10³–10⁶ M_⊙ examples or demonstrate that second-order terms remain negligible.

    Authors: We agree that an explicit quantification strengthens the presentation. In the matter-dominated era the density contrast relates to the conserved curvature perturbation by δ ≈ −(2/3)(k/(aH))² R_c (comoving gauge, leading order). For the comoving wavenumbers corresponding to 10³–10⁶ M_⊙ objects collapsing at z ≳ 5, k/(aH) ≳ 10³, so |R_c| ≲ 10^{-6} when δ reaches O(1). Second-order corrections are therefore O(10^{-12}) and cannot change the sign of the curvature function, the mass profile, or the ordering of apparent-horizon and singularity times. We will add a short subsection (or paragraph in the initial-conditions section) that computes the representative R_c values for the quoted mass range and confirms the linear approximation remains valid throughout the evolution. revision: yes

  2. Referee: The distinction between viable (broad compensated) and non-viable (sinusoidal, Gaussian) profiles rests entirely on the regularity conditions obtained from the first-order R_c data. Because the amplitude regime is where the linear mapping is most suspect, the profile-selection claim requires an explicit check that second-order corrections do not alter the sign of the curvature function or the ordering of apparent-horizon and singularity formation times.

    Authors: The same amplitude estimate applies uniformly to all profiles examined. Because |R_c| remains ≪ 1 for every case, the O(R_c²) corrections are negligible and preserve both the sign of the curvature function and the relative ordering of the characteristic times. The distinction between profiles originates from the spatial structure and compensation of R_c, which are leading-order quantities. We will insert a brief statement in the revised manuscript noting that the profile viability conclusions are insensitive to second-order corrections at the amplitudes relevant to the 10³–10⁶ M_⊙ seeds. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation starts from external standard perturbation result

full rationale

The paper begins from the standard first-order result that the growing mode is expressed in terms of the conserved curvature perturbation ℛ_c (an input from relativistic perturbation theory, not derived or fitted here). It then maps this directly to the LTB/Szekeres mass and curvature functions via the quoted first-order expressions, and derives turnaround/collapse times and regularity conditions as consequences. No steps match the enumerated circularity patterns: there are no self-definitional loops, no fitted inputs renamed as predictions, no load-bearing self-citations, and no ansatz smuggled via prior work. The central claims about which profiles produce BHs and the quoted redshifts follow from the external ℛ_c data without reducing to the inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Abstract-only review; limited details available on parameters or axioms beyond those explicitly stated.

free parameters (1)
  • Mass range for seeds
    Estimates provided specifically for 10^3-10^6 solar masses to match potential early BH seeds.
axioms (2)
  • domain assumption CDM modeled as pressureless dust
    Stated explicitly for the collapse modeling in the abstract.
  • standard math Growing mode expressed via conserved ℛ_c
    From first-order relativistic scalar perturbation theory on FLRW background.

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Reference graph

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