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arxiv: 2606.30641 · v1 · pith:AKGGMINYnew · submitted 2026-06-29 · 🌌 astro-ph.CO · gr-qc

Primordial Black Holes in a Radiation-Dominated Universe

Thomas W. Baumgarte , Katy Clough , Mary Gerhardinger , John T. Giblin Jr. , Amanda Miller This is my paper

Pith reviewed 2026-06-30 04:28 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qc
keywords primordial black holesradiation-dominated epochgeneral relativistic simulationsdensity thresholdcosmological horizon re-entryBSSN formalismoverdense collapseperiodic boundary conditions
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The pith

Numerical simulations establish that over-densities above a threshold between 0.77 and 0.83 collapse to form primordial black holes in a radiation-dominated universe.

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

The paper performs full 3+1 general relativistic simulations of overdense regions re-entering the cosmological horizon during the radiation-dominated epoch. It employs the BSSN formalism together with periodic boundary conditions while letting the matter content drive the spacetime dynamics self-consistently. The central result is an identified collapse threshold in the interval 0.77 to 0.83 above which over-densities form black holes rather than disperse. This supplies an independent numerical check on earlier analytic estimates and on simulations that used different codes or gauges. A reader would care because the precise value directly controls the expected abundance of primordial black holes produced by early-universe fluctuations.

Core claim

We perform simulations of the direct collapse of over-dense regions re-entering the horizon during a radiation-dominated epoch, using full 3+1 general relativistic simulations with the BSSN formalism. Building on previous studies, we impose periodic boundary conditions and allow the matter content of the Universe to self-consistently drive its dynamics. We analyze the evolution of over-densities in both the collapse and dispersal regimes and find a threshold, 0.77<δ_c<0.83, above which over-densities collapse and form primordial black holes. Our findings are consistent with previous analytic predictions as well as numerical studies that use different formalisms and computational approaches.

What carries the argument

BSSN 3+1 evolution of horizon-reentering over-densities under periodic boundaries with self-consistent matter-driven dynamics, used to bracket the collapse threshold δ_c.

If this is right

  • Over-densities with contrast below 0.77 disperse without forming black holes.
  • The same threshold emerges from simulations that employ distinct numerical formalisms.
  • The value can be used to convert an initial fluctuation spectrum into a predicted primordial-black-hole mass function in the radiation era.
  • Analytic estimates of the threshold are confirmed to lie inside the numerically bracketed interval.

Where Pith is reading between the lines

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

  • The reported threshold supplies a fixed number that can be inserted into semi-analytic calculations of the primordial black hole dark-matter fraction without further tuning.
  • Changing the equation of state away from pure radiation would require a new set of runs to determine whether the same numerical interval still holds.
  • The periodic-box setup implicitly assumes that neighboring patches do not exert net tidal influence on the collapsing region.
  • If the threshold remains stable under mesh refinement, it becomes a standard input for forecasts of gravitational-wave signals from primordial black hole binaries.

Load-bearing premise

The numerical evolution with periodic boundary conditions and self-consistent matter-driven dynamics accurately captures the threshold behavior without significant artifacts from the chosen gauge, resolution, or boundary implementation.

What would settle it

A higher-resolution run or an evolution in a different gauge that places the threshold outside the interval 0.77–0.83 would falsify the reported range.

Figures

Figures reproduced from arXiv: 2606.30641 by Amanda Miller, John T. Giblin Jr., Katy Clough, Mary Gerhardinger, Thomas W. Baumgarte.

Figure 1
Figure 1. Figure 1: FIG. 1. The initial conformal factor, [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. A plot of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Two-dimensional slices of the lapse, [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. One-dimensional slices of the energy density contrast (top row) and mean curvature contrast (bottom row) at [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. One-dimensional slices of the energy density contrast (top row) and mean curvature contrast (bottom row) at [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The expansion, Θ( [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
read the original abstract

Primordial fluctuations frozen out during inflation re-enter the cosmological horizon and can collapse, leading to the formation of primordial black holes. We perform simulations of the direct collapse of over-dense regions re-entering the horizon during a radiation-dominated epoch, using full 3+1 general relativistic simulations with the BSSN formalism. Building on previous studies, we impose periodic boundary conditions and allow the matter content of the Universe to self-consistently drive its dynamics. We analyze the evolution of over-densities in both the collapse and dispersal regimes and find a threshold, $0.77<\delta_c<0.83$, above which over-densities collapse and form primordial black holes. Our findings are consistent with previous analytic predictions as well as numerical studies that use different formalisms and computational approaches, and hence provide independent validation of those results.

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 / 0 minor

Summary. The paper performs 3+1 BSSN simulations of horizon re-entry of over-dense regions in a radiation-dominated universe, imposing periodic boundary conditions and allowing the radiation fluid to drive the dynamics self-consistently. From the evolution of collapse versus dispersal cases the authors report a threshold interval 0.77 < δ_c < 0.83 above which primordial black holes form, stating consistency with prior analytic and numerical results obtained with different formalisms.

Significance. If the quoted threshold interval is shown to be robust, the work supplies an independent numerical confirmation of the PBH collapse criterion using the BSSN 3+1 approach with periodic boundaries, strengthening the foundation for calculations of PBH abundance from inflationary fluctuations.

major comments (2)
  1. [Abstract] Abstract (results paragraph): the central claim is the numerically bracketed threshold 0.77 < δ_c < 0.83, yet the manuscript supplies no convergence tests, resolution studies, or error bars on the bounds, nor any description of the diagnostic used to decide collapse versus dispersal; this information is required to establish that the interval is not an artifact of the chosen resolution, gauge, or boundary implementation.
  2. [Methods] Methods description (numerical setup): the weakest assumption—that periodic boundaries and the self-consistent radiation fluid evolution introduce no significant artifacts—is stated without supporting tests (e.g., comparison runs at multiple resolutions or with different gauges), leaving the load-bearing threshold result without demonstrated numerical control.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive report. The major comments correctly note the absence of explicit numerical validation details in the current manuscript. We respond point-by-point below and will revise the manuscript to address these issues.

read point-by-point responses

  1. Referee: [Abstract] Abstract (results paragraph): the central claim is the numerically bracketed threshold 0.77 < δ_c < 0.83, yet the manuscript supplies no convergence tests, resolution studies, or error bars on the bounds, nor any description of the diagnostic used to decide collapse versus dispersal; this information is required to establish that the interval is not an artifact of the chosen resolution, gauge, or boundary implementation.

    Authors: We agree that the manuscript as submitted lacks an explicit description of the collapse diagnostic and supporting convergence information. In the revised version we will add a dedicated subsection detailing the diagnostic (apparent-horizon formation together with the central value of the 3-Ricci scalar exceeding a fixed threshold) and will present a resolution study performed at three different grid spacings. The threshold interval will be reported with an estimated uncertainty derived from the variation across those resolutions. revision: yes

  2. Referee: [Methods] Methods description (numerical setup): the weakest assumption—that periodic boundaries and the self-consistent radiation fluid evolution introduce no significant artifacts—is stated without supporting tests (e.g., comparison runs at multiple resolutions or with different gauges), leaving the load-bearing threshold result without demonstrated numerical control.

    Authors: We acknowledge that the current text does not contain explicit tests of the periodic-boundary implementation or gauge sensitivity. The revised manuscript will include a short numerical-validation subsection that reports (i) the maximum constraint violation as a function of resolution and (ii) a comparison of the threshold obtained with the default 1+log slicing against a limited set of runs using harmonic slicing. These tests will be used to argue that the quoted interval is not an artifact of the chosen setup. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The central result is a numerically bracketed threshold 0.77 < δ_c < 0.83 obtained from direct 3+1 BSSN evolution of the Einstein equations with self-consistent radiation fluid and periodic boundaries. This threshold is not defined in terms of any fitted parameter, self-referential quantity, or ansatz within the paper; it emerges from the simulation outcomes in the collapse versus dispersal regimes. The paper cites prior analytic and numerical work only for consistency checks, not as load-bearing justification for the threshold itself. No self-definitional, fitted-input, or uniqueness-imported steps appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the validity of the BSSN evolution, the assumption that the universe remains radiation-dominated, and the adequacy of periodic boundaries; no free parameters or new entities are introduced.

axioms (2)
  • domain assumption The cosmological background is radiation-dominated during the epoch studied.
    Explicitly stated as the setting for the simulations.
  • standard math The BSSN formulation with the chosen gauge and boundary conditions faithfully evolves the Einstein equations for this problem.
    The method is invoked without further justification in the abstract.

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Reference graph

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