pith. sign in

arxiv: 2606.29590 · v1 · pith:UKCUO5VAnew · submitted 2026-06-28 · 🌀 gr-qc

Saturation Equations of State in Critical Gravitational Collapse: The Primordial Black Hole Threshold

Benaoumeur Bakhti This is my paper

Pith reviewed 2026-06-30 01:54 UTC · model grok-4.3

classification 🌀 gr-qc
keywords primordial black holescritical collapseequation of statesaturation densitylattice gasgravitational collapseprimordial black hole threshold
0
0 comments X

The pith

A lattice-gas saturation equation of state raises the primordial black hole formation threshold by 0.50 percent relative to radiation.

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

The paper studies how pressure stiffening near a maximum density changes the threshold and scaling of critical gravitational collapse. It adopts an exactly solvable lattice-gas equation of state that saturates density while reducing to radiation at low density. Spherically symmetric general-relativistic simulations then measure the effect on the critical overdensity needed for primordial black hole formation. The nonlinear pressure feedback produces a small but measurable upward shift in the threshold while leaving the mass-scaling exponent unchanged. The result supplies a concrete example that dense-matter features absent from the radiation fluid can alter primordial black hole production.

Core claim

Within the causal regime of the model, the lattice equation of state p = -T ln(1-ρ) increases the PBH formation threshold by 0.50 ± 0.02 percent compared with the radiation equation of state; simultaneously the critical exponent remains γ = 0.357 ± 0.001, consistent with the radiation value to numerical precision.

What carries the argument

The closed-form single-occupancy lattice-gas equation of state p = -T ln(1-ρ), which supplies a density-dependent sound speed that stiffens as density approaches saturation.

If this is right

  • Saturation-induced pressure stiffening can stabilize gravitational collapse and raise the minimum density required for black-hole formation.
  • The critical exponent is insensitive to this mild, density-dependent perturbation because the equation of state recovers radiation behavior at low density.
  • A linear-response approach can be used to estimate the effect of any realistic high-density equation of state on the PBH threshold.
  • Primordial black hole abundance calculations that assume a pure radiation fluid may slightly overestimate formation rates.

Where Pith is reading between the lines

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

  • Stronger saturation effects in more realistic equations of state could produce threshold shifts large enough to affect observational bounds on PBH dark matter.
  • The same saturation mechanism may alter the collapse dynamics of other compact objects formed in the early universe.
  • Extending the lattice model to include temperature dependence or multiple species would test how robust the half-percent shift remains.

Load-bearing premise

The lattice equation of state approaches the radiation fluid at low density and stays only a mild perturbation throughout the near-critical regime.

What would settle it

A numerical simulation that finds either a threshold shift larger than a few percent or a statistically significant change in the scaling exponent γ when the same lattice equation of state is used.

read the original abstract

The threshold and scaling laws of gravitational critical collapse depend sensitively on the matter equation of state. We investigate how these quantities are modified by a generic feature of dense matter that is absent from the radiation fluid commonly assumed in primordial black hole (PBH) studies: pressure stiffening as a maximum density is approached. As an analytically tractable proxy, we adopt the closed-form equation of state of a single-occupancy lattice gas, \(p=-T\ln(1-\rho)\), which exhibits a density-dependent sound speed and a saturation density. Using general-relativistic simulations of spherically symmetric collapse, we show that this nonlinear pressure feedback increases the PBH formation threshold by \(0.50\pm0.02\%\) relative to the radiation equation of state within the causal regime of the model. At the same time, the critical mass-scaling exponent remains \(\gamma=0.357\pm0.001\), consistent with the radiation-fluid value to within our numerical precision. This agreement reflects the fact that the lattice equation of state approaches the radiation fluid at low density and remains only a mild perturbation over the near-critical regime, rather than indicating a universal critical exponent. Our results provide a proof of principle that saturation-induced stiffening can stabilize gravitational collapse and shift the PBH threshold, while introducing a linear-response framework for assessing the impact of more realistic equations of state on primordial black hole formation.

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

1 major / 0 minor

Summary. The manuscript examines how a density-dependent equation of state modeling pressure stiffening near saturation modifies the threshold and scaling exponent of critical gravitational collapse for primordial black hole formation. Adopting the closed-form lattice-gas EOS p = −T ln(1 − ρ) as an analytically tractable proxy that reduces to radiation at low density, the authors perform spherically symmetric general-relativistic simulations and report that this EOS raises the PBH formation threshold by 0.50 ± 0.02 % relative to the radiation fluid while leaving the critical exponent unchanged at γ = 0.357 ± 0.001. The small shift is attributed to the mild perturbation the lattice EOS produces in the near-critical regime, and the work is presented as a proof-of-principle for assessing more realistic saturation effects via a linear-response framework.

Significance. If the reported threshold shift is robust, the result supplies a controlled demonstration that nonlinear pressure feedback from saturation can stabilize collapse and alter the PBH mass threshold, even when the effect remains small. The explicit statement that the lattice EOS approaches radiation at low density and induces only a mild perturbation supplies a clear physical explanation for both the 0.50 % shift and the unchanged γ, strengthening the internal consistency of the comparison. The work also introduces a concrete linear-response approach that could be applied to other realistic equations of state, which is a useful methodological contribution for the PBH literature.

major comments (1)
  1. [Numerical methods / simulation setup] The abstract states concrete numerical results with uncertainties (0.50 ± 0.02 % threshold shift and γ = 0.357 ± 0.001), yet the manuscript provides no description of grid resolution, convergence tests, or the precise definition of the causal regime. These details are load-bearing for the central claim because the reported threshold difference is only 0.50 % and must be distinguished from numerical truncation error.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment of its significance. We address the major comment below and will incorporate the requested details into a revised version of the paper.

read point-by-point responses

  1. Referee: [Numerical methods / simulation setup] The abstract states concrete numerical results with uncertainties (0.50 ± 0.02 % threshold shift and γ = 0.357 ± 0.001), yet the manuscript provides no description of grid resolution, convergence tests, or the precise definition of the causal regime. These details are load-bearing for the central claim because the reported threshold difference is only 0.50 % and must be distinguished from numerical truncation error.

    Authors: We agree that the current manuscript does not provide sufficient detail on the numerical methods to support the reported precision of the 0.50% threshold shift. In the revised manuscript we will add a dedicated subsection to the Methods section that specifies the grid resolutions employed (including the number of radial zones and adaptive mesh refinement levels), presents explicit convergence tests showing that the threshold value is stable to better than 0.05% under refinement, and defines the causal regime via the precise criterion on the lapse function and apparent-horizon formation time used to classify supercritical versus subcritical runs. These additions will allow independent verification that the observed shift exceeds numerical truncation error. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper obtains its central results—the 0.50±0.02% threshold shift and γ=0.357±0.001 scaling exponent—directly from numerical integration of the Einstein equations with the chosen lattice-gas EOS p=-T ln(1-ρ). The manuscript states that this EOS reduces to the radiation fluid at low density and remains a mild perturbation in the near-critical regime; this physical property, not any fitted parameter or self-referential definition, explains both the small shift and the unchanged exponent. No step in the reported derivation chain reduces by construction to a self-citation, an ansatz smuggled via prior work, or a fitted input renamed as a prediction. The numerical comparison is externally falsifiable and independent of the target quantities.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on the standard assumptions of general relativity and spherical symmetry plus the modeling choice of the lattice-gas equation of state as an analytically tractable proxy; no free parameters are fitted to data in the reported results and no new physical entities are introduced.

axioms (3)
  • standard math General relativity governs the dynamics of the gravitational collapse
    Basis for all simulations described in the abstract
  • domain assumption The collapse remains spherically symmetric
    Simplification used to reduce the problem to one spatial dimension
  • ad hoc to paper The lattice-gas equation of state serves as a suitable proxy for saturation-induced pressure stiffening
    Chosen explicitly as an analytically tractable model absent from standard radiation-fluid studies

pith-pipeline@v0.9.1-grok · 5782 in / 1582 out tokens · 61581 ms · 2026-06-30T01:54:56.646007+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

42 extracted references · 11 canonical work pages · 9 internal anchors

  1. [1]

    Oertel, M., Hempel, M., Kl¨ ahn, T., Typel, S.: Equations of state for supernovae and compact stars. Rev. Mod. Phys.89, 015007 (2017). arXiv:1610.03361

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  2. [2]

    Zel’dovich, Y.B., Novikov, I.D.: The hypothesis of cores retarded and expansion of hot universe. Sov. Astron.10, 602 (1967)

    1967

  3. [3]

    Hawking, S.: Gravitationally Collapsed Objects of Very Low Mass. Mon. Not. R. Astro. Soc.152(1), 75–78 (1971) 29

    1971

  4. [4]

    Astrophys

    Carr, B.J.: The primordial black hole mass spectrum. Astrophys. J.201, 1–19 (1975)

    1975

  5. [5]

    Polnarev, A.G., Musco, I.: Curvature profiles as initial conditions for primordial black hole formation. Class. Quant. Grav.24(6), 1405–1431 (2007)

    2007

  6. [6]

    Musco, I.: Threshold for primordial black holes: Dependence on the shape of the cosmological perturbations. Phys. Rev. D100, 123524 (2019)

    2019

  7. [7]

    Choptuik, M.W.: Universality and scaling in gravitational collapse of a massless scalar field. Phys. Rev. Lett.70, 9–12 (1993)

    1993

  8. [8]

    Gundlach, C.: Critical phenomena in gravitational collapse. Phys. Rev. D55, 6002 (1997)

    1997

  9. [9]

    Bakhti, B.: D-dimensional self-gravitating lattice gas in general relativity. Int. J. Mod. Phys. A36(29), 2150217 (2021)

    2021

  10. [10]

    Bakhti, B., Boukari, D., Karbach, M., Maass, P., M¨ uller, G.: Density profiles of a self-gravitating lattice gas in one, two, and three dimensions. Phys. Rev. E97, 042131 (2018)

    2018

  11. [11]

    Musco, I., Miller, J.C., Rezzolla, L.: Computations of primordial black-hole formation. Class. Quant. Grav.22(7), 1405–1424 (2005)

    2005

  12. [12]

    Nakama, T., Harada, T., Polnarev, A.G., Yokoyama, J.: Identifying the most cru- cial parameters of the initial curvature profile for primordial black hole formation. J. Cosmol. Astropart. Phys.2014(01), 037–037 (2014)

    2014

  13. [13]

    Harada, T., Yoo, C.-M., Kohri, K.: Threshold of primordial black hole formation. Phys. Rev. D88, 084051 (2013)

    2013

  14. [14]

    Jedamzik, K.: Primordial black hole formation during the qcd epoch. Phys. Rev. D55, 5871–5875 (1997). arXiv:astro-ph/9605152

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  15. [15]

    Byrnes, C.T., Hindmarsh, M., Young, S., Hawkins, M.R.S.: Primordial black holes with an accurate qcd equation of state. J. Cosmol. Astropart. Phys.2018(08), 041 (2018). arXiv:1801.06138

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  16. [16]

    Maison, D.: Non-universality of critical behaviour in spherically symmetric gravitational collapse. Phys. Lett. B366, 82–84 (1996). arXiv:gr-qc/9504008

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  17. [17]

    Koike, T., Hara, T., Adachi, S.: Critical behaviour in gravitational collapse of radiation fluid: A renormalization group (linear perturbation) analysis. Phys. Rev. Lett.74, 5170–5173 (1995). arXiv:gr-qc/9503007

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  18. [18]

    Critical phenomena in gravitational collapse

    Gundlach, C., Mart´ ın-Garc´ ıa, J.M.: Critical phenomena in gravitational collapse. Living Rev. Relativity10, 5 (2007). arXiv:0711.4620 30

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  19. [19]

    Evans, C.R., Coleman, J.S.: Observation of critical phenomena and self-similarity in the gravitational collapse of radiation fluid. Phys. Rev. Lett.72, 1782–1785 (1994). arXiv:gr-qc/9402041

    work page internal anchor Pith review Pith/arXiv arXiv 1994

  20. [20]

    Choptuik, M.W., Hirschmann, E.W., Marsa, R.L.: New critical behavior in einstein–yang–mills collapse. Phys. Rev. D60, 124011 (1999). arXiv:gr- qc/9903081

    work page arXiv 1999

  21. [21]

    Hirschmann, E.W., Eardley, D.M.: Criticality and bifurcation in the gravita- tional collapse of a self-coupled scalar field. Phys. Rev. D56, 4696–4705 (1997). arXiv:gr-qc/9511052

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  22. [22]

    Musco, I., De Luca, V., Franciolini, G., Riotto, A.: Threshold for primordial black holes. ii. a simple analytic prescription. Phys. Rev. D103, 063538 (2021). arXiv:2011.03014

    work page arXiv 2021

  23. [23]

    Bakhti, B., Karbach, M., Maass, P., Mokim, M., M¨ uller, G.: Statistically interacting vacancy particles. Phys. Rev. E89, 012137 (2014)

    2014

  24. [24]

    Bakhti, B., Karbach, M., B., Maass, P., M¨ uller, G.: Monodisperse hard rods in external potentials. Phys. Rev. E92, 042112 (2015)

    2015

  25. [25]

    Carr, B., K¨ uhnel, F., Sandstad, M.: Primordial black holes as dark matter. Phys. Rev. D94, 083504 (2016)

    2016

  26. [26]

    In: Essig, R., Feng, J., Zurek, K

    Carr, B.: Primordial black holes as dark matter and generators of cosmic struc- ture. In: Essig, R., Feng, J., Zurek, K. (eds.) Illuminating Dark Matter, pp. 29–39. Springer, Cham (2019)

    2019

  27. [27]

    Sasaki, M., Suyama, T., Tanaka, T., Yokoyama, S.: Primordial black hole scenario for the gravitational-wave event gw150914. Phys. Rev. Lett.117, 061101 (2016)

    2016

  28. [28]

    Physics of the Dark Universe18, 105–114 (2017)

    Clesse, S., Garc´ ıa-Bellido, J.: Detecting the gravitational wave background from primordial black hole dark matter. Physics of the Dark Universe18, 105–114 (2017)

    2017

  29. [29]

    Bartolo, N., De Luca, V., Franciolini, G., Lewis, A., Peloso, M., Riotto, A.: Pri- mordial black hole dark matter: Lisa serendipity. Phys. Rev. Lett.122, 211301 (2019)

    2019

  30. [30]

    Misner, C.W., Sharp, D.H.: Relativistic equations for adiabatic, spherically symmetric gravitational collapse. Phys. Rev.136, 571–576 (1964)

    1964

  31. [31]

    Astrophys

    Hernandez, W.C., Misner, C.W.: Observer Time as a Coordinate in Relativistic Spherical Hydrodynamics. Astrophys. J.143, 452 (1966)

    1966

  32. [32]

    Formalism for Primordial Black Hole Formation in Spherical Symmetry

    Bloomfield, J., Bulhosa, D., Face, S.: Formalism for primordial black hole formation in spherical symmetry. (arXiv:1504.02071v1 ) (2014) 31

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  33. [33]

    Physics of the Dark Universe27, 100466 (2020)

    Escriv` a, A.: Primordial black holes: A brief review of their formation and properties. Physics of the Dark Universe27, 100466 (2020)

    2020

  34. [34]

    Harada, T., Yoo, C.-M., Nakama, T., Koga, Y.: Cosmological long-wavelength solutions and primordial black hole formation. Phys. Rev. D91, 084057 (2015)

    2015

  35. [35]

    Faraoni, V., Ellis, G.F.R., Firouzjaee, J.T., Helou, A., Musco, I.: Foliation depen- dence of black hole apparent horizons in spherical symmetry. Phys. Rev. D95, 024008 (2017)

    2017

  36. [36]

    Shibata, M., Sasaki, M.: Black hole formation in the friedmann universe: For- mulation and computation in numerical relativity. Phys. Rev. D60, 084002 (1999)

    1999

  37. [37]

    Nadezhin, D.K., Novikov, I.D., Polnarev, A.G. Sov. Astron.22, 129 (1978)

    1978

  38. [38]

    Nadezhin, D.K., Novikov, I.D., Polnarev, A.G. Astron. Zh.55, 216 (1978)

    1978

  39. [39]

    Novikov, I.D., Polnarev, A.G. Sov. Astron.24, 147 (1980)

    1980

  40. [40]

    Harada, T., Jhingan, S.: Spherical and nonspherical models of primordial black hole formation: exact solutions. Prog. Theor. Exp. Phys.2016, 093–04 (2016)

    2016

  41. [41]

    He, M., Suyama, T.: Formation threshold of rotating primordial black holes. Phys. Rev. D100, 063520 (2019)

    2019

  42. [42]

    Garc´ ıa-Bellido, J.: Massive primordial black holes as dark matter and their detection with gravitational waves. J. Phys.: Conference Series840, 012032 (2017) 32

    2017