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arxiv: 2606.26532 · v1 · pith:E757LGZ6new · submitted 2026-06-25 · 🌀 gr-qc · astro-ph.CO

Primordial black hole formation in bulk-viscous cosmology

Zi-Yan Yuwen , Cristian Joana , Shao-Jiang Wang , Rong-Gen Cai This is my paper

Pith reviewed 2026-06-26 04:50 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.CO
keywords primordial black holesbulk viscositycosmological collapseequation of statenumerical hydrodynamicscritical thresholdmass scaling
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The pith

Bulk viscosity raises the collapse threshold for primordial black holes and increases their masses.

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

This paper uses numerical simulations to study how bulk viscosity in the early-universe fluid changes primordial black hole formation. The work shows that the critical density threshold for collapse keeps the same dependence on the equation-of-state parameter as in the non-viscous case yet shifts higher by an amount set by the viscosity strength, with the shift linear at fixed equation of state. The same simulations show that near-threshold black holes end up more massive once the standard critical-scaling relation is applied. A reader cares because these shifts alter the predicted abundance and mass range of primordial black holes, which remain candidates for dark matter or sources of gravitational waves.

Core claim

The critical threshold μ_c retains a dependence on the equation-of-state parameter w similar to that in the inviscid case, but is enhanced by an amount comparable to the bulk-viscosity strength ε. For fixed w, the increase in μ_c is approximately linear in ε. By fitting the standard critical-scaling law for near-threshold collapse, the bulk viscosity leads to an enhancement in the resulting PBH mass. These results indicate that bulk viscosity can systematically modify both the PBH threshold and PBH mass scaling law in the early universe.

What carries the argument

Numerical hydrodynamics simulations of collapse in a bulk-viscous cosmological background that extract the critical threshold μ_c and the resulting black-hole mass from the critical scaling relation.

If this is right

  • The critical threshold for primordial black hole formation increases with bulk viscosity strength by an amount comparable to the viscosity parameter itself.
  • For any fixed equation-of-state parameter the threshold rises approximately linearly with the viscosity parameter.
  • The resulting primordial black hole masses are larger than in the inviscid case once the critical scaling law is applied.
  • Both the threshold value and the mass scaling relation are modified systematically by the presence of bulk viscosity.

Where Pith is reading between the lines

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

  • Models that use primordial black holes to explain dark matter would have to adjust the allowed mass window once bulk viscosity is included.
  • Gravitational-wave signals from primordial black hole binaries could carry an imprint of the early-universe viscosity strength.
  • The same simulation approach could test whether other forms of viscosity or different parametrizations produce comparable shifts.

Load-bearing premise

The numerical hydrodynamics setup and the chosen parametrization of bulk viscosity strength accurately capture the relevant physics of the early-universe fluid.

What would settle it

An observed population of primordial black holes whose masses or abundance fall outside the range predicted by the raised threshold and enhanced mass scaling for any given viscosity strength.

Figures

Figures reproduced from arXiv: 2606.26532 by Cristian Joana, Rong-Gen Cai, Shao-Jiang Wang, Zi-Yan Yuwen.

Figure 1
Figure 1. Figure 1: FIG. 1. Thresholds [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The power law of critical threshold functions of viscosity strength [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The power law of PBH mass formation with and without bulk viscosity, for [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

We investigate primordial black hole (PBH) formation in a cosmological background with bulk viscosity. Using numerical simulations, we determine the collapse threshold and the resulting PBH mass. We find that the critical threshold $\mu_c$ retains a dependence on the equation-of-state parameter $w$ similar to that in the inviscid case, but is enhanced by an amount comparable to the bulk-viscosity strength $\epsilon$. For fixed $w$, the increase in $\mu_c$ is approximately linear in $\epsilon$. By fitting the standard critical-scaling law for near-threshold collapse, we find that the bulk viscosity leads to an enhancement in the resulting PBH mass. These results indicate that bulk viscosity can systematically modify both the PBH threshold and PBH mass scaling law in the early universe.

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 manuscript investigates primordial black hole (PBH) formation in a bulk-viscous cosmological background. Using numerical simulations of general relativistic hydrodynamics, it reports that the critical collapse threshold μ_c retains a dependence on the equation-of-state parameter w similar to the inviscid case but is enhanced by an amount comparable to the bulk-viscosity strength ε, with the increase approximately linear in ε for fixed w. Fitting the standard critical-scaling law shows that bulk viscosity also enhances the resulting PBH mass.

Significance. If the numerical results hold after validation, the work would show that bulk viscosity systematically modifies both the PBH formation threshold and the mass scaling in the early universe, with implications for PBH abundance and observational constraints. Credit is due for the direct numerical integration approach that extracts thresholds and scaling exponents beyond purely analytic treatments, and for the explicit comparison to the inviscid limit.

major comments (2)
  1. [Numerical results] Numerical results section: The central claims of linear enhancement in μ_c with ε and the mass increase rest on extraction from near-threshold collapse simulations, yet no convergence tests with respect to spatial resolution or time-stepping, no error bars on the fitted μ_c values, and no quantification of the linearity (e.g., fit coefficient or residuals) are reported. This is load-bearing because near-threshold dynamics are known to be sensitive to truncation errors, and artifacts of size comparable to the reported ε-induced shift would undermine both the linear enhancement and the mass-enhancement conclusion.
  2. [Methods / viscosity model] Simulation framework and viscosity parametrization: The model adopts a constant bulk-viscosity strength ε without higher-order corrections or temperature dependence; the manuscript must demonstrate that this parametrization does not introduce systematic bias into the extracted μ_c(w,ε) relation, as any mismatch with the relevant early-universe viscous physics would directly affect the claimed retention of w-dependence and the linear ε scaling.
minor comments (2)
  1. [Abstract] Abstract: The phrase 'approximately linear' should be accompanied in the results by a quantitative measure of linearity (e.g., slope and goodness-of-fit) rather than left qualitative.
  2. [Introduction] Notation: The symbol μ_c is introduced without an explicit definition equation in the early sections; a clear definition (e.g., in terms of the density contrast or curvature perturbation) would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive criticism. The two major comments identify important gaps in the numerical validation and model justification. We respond to each below and commit to revisions that directly address the concerns.

read point-by-point responses

  1. Referee: [Numerical results] Numerical results section: The central claims of linear enhancement in μ_c with ε and the mass increase rest on extraction from near-threshold collapse simulations, yet no convergence tests with respect to spatial resolution or time-stepping, no error bars on the fitted μ_c values, and no quantification of the linearity (e.g., fit coefficient or residuals) are reported. This is load-bearing because near-threshold dynamics are known to be sensitive to truncation errors, and artifacts of size comparable to the reported ε-induced shift would undermine both the linear enhancement and the mass-enhancement conclusion.

    Authors: We agree that the absence of convergence tests, error bars, and quantitative linearity measures weakens the central claims. In the revised manuscript we will add (i) resolution studies at three or more grid spacings demonstrating convergence of μ_c to within the reported ε-induced shift, (ii) error bars on each fitted μ_c obtained from the threshold-search procedure, and (iii) an explicit linear regression of μ_c(ε) together with fit coefficients and residuals. These additions will be placed in a new subsection of the numerical results. revision: yes

  2. Referee: [Methods / viscosity model] Simulation framework and viscosity parametrization: The model adopts a constant bulk-viscosity strength ε without higher-order corrections or temperature dependence; the manuscript must demonstrate that this parametrization does not introduce systematic bias into the extracted μ_c(w,ε) relation, as any mismatch with the relevant early-universe viscous physics would directly affect the claimed retention of w-dependence and the linear ε scaling.

    Authors: The constant-ε model is a deliberate minimal parametrization chosen to isolate the leading viscous correction. We will expand the methods section with (a) a justification based on the relevant Hubble and temperature scales at PBH formation, (b) a brief comparison to temperature-dependent bulk-viscosity prescriptions in the literature, and (c) a short robustness test in which ε is allowed a mild temperature dependence for one representative w. These additions will clarify the domain of applicability while preserving the paper’s scope as a first numerical exploration. revision: partial

Circularity Check

0 steps flagged

No circularity: claims extracted from direct numerical hydrodynamics simulations

full rationale

The paper determines μ_c and PBH mass via numerical simulations of viscous GR hydrodynamics, with the reported linear enhancement in μ_c (for fixed w) and mass increase emerging as simulation outputs rather than algebraic reductions. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain; the standard critical-scaling fit is applied post-simulation as analysis, not as a premise that forces the viscosity-induced shifts. The setup is self-contained against external benchmarks for the purpose of circularity analysis.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claims rest on the assumption that standard general-relativistic hydrodynamics with a bulk-viscosity term can be numerically integrated to extract collapse thresholds; no new entities are postulated and the only free parameters are the viscosity strength ε and equation-of-state w that are scanned in the simulations.

free parameters (2)
  • ε
    Bulk-viscosity strength parameter scanned across simulations to measure its effect on μ_c
  • w
    Equation-of-state parameter held fixed while varying ε
axioms (1)
  • domain assumption The early-universe fluid can be modeled as a viscous perfect fluid in general relativity whose evolution is captured by standard numerical relativity codes
    Invoked when setting up the cosmological background and running the collapse simulations

pith-pipeline@v0.9.1-grok · 5668 in / 1298 out tokens · 33193 ms · 2026-06-26T04:50:35.346167+00:00 · methodology

discussion (0)

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Reference graph

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