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Primordial Black Hole Formation in a Scalar Field Dominated Universe: Investigation of the Critical nature of the Collapse

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Pith's one-line read A quartic scalar field and a radiation fluid collapse into primordial black holes with nearly identical critical exponents.

desk verdict Useful first measurement of the quartic-scalar critical exponent, but the abstract overstates the difference from radiation and the gauge-defining fluid is never tested. read the letter →

arxiv 2509.10431 v1 pith:Y3J7Z3DH submitted 2025-09-12 astro-ph.CO

Luis E. Padilla , Ethan Milligan , David J. Mulryne , Juan Carlos Hidalgo This is my paper · ORCID
classification astro-ph.CO
keywords primordialblackholescriticalgravitationalcollapseexponentquarticscalarfieldtypeIIMisner-Sharpformalismradiationfluidnearuniversality
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

The paper asks whether a universe dominated by a self-interacting scalar field with a quartic potential forms primordial black holes in the same critical way as a radiation-filled universe. Using fully nonlinear, spherically symmetric simulations in the Misner-Sharp formalism, it finds that the black hole mass near threshold follows the same type II power law in both cases, with critical exponents agreeing to within about $2\sigma$. If correct, this extends the common shortcut of modelling oscillating scalar fields as perfect fluids into the strongly nonlinear collapse regime and supports the near universality of the critical exponent in primordial black hole formation. The paper reports a small residual difference between the exponents rather than claiming exact equality.

What carries the argument

The central object is the power-law scaling of the black hole mass near threshold, $M_{\rm BH}\propto(p-p_{\rm th})^\gamma$, in type II critical collapse. The machinery that extracts it is the Misner-Sharp formulation of spherical relativistic collapse, evolved with a fourth-order Runge-Kutta method and finite differences, with a scalar field plus an extremely diluted homogeneous perfect fluid that defines the comoving frame. Black hole formation is detected through the apparent-horizon condition $\Theta_+=0$; the final mass is read from the Misner-Sharp mass once the apparent-horizon velocity satisfies $v_{\rm AH}\approx 1$, and $\gamma$ is obtained by fitting the mass against $p-p_{\rm th}$.

What would settle it

Run the same scalar-field collapse with the homogeneous background fluid density reduced by factors of 10 while holding the scalar perturbation fixed; if the fitted exponent $\gamma$ moves, or if the fluid's share of the Misner-Sharp mass near horizon formation grows, the quoted scalar-field exponent is contaminated by the frame-defining fluid.

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Extended reading notes

Core claim

The paper's central claim is that a quartic self-interacting scalar field, $V(\psi)\propto\psi^4$, exhibits type II critical collapse (black hole mass becoming arbitrarily small near threshold and scaling as a power law) with critical exponent $\gamma = 0.3401 \pm 0.0071$, statistically close to the radiation-fluid value $\gamma = 0.3474 \pm 0.004$ measured with the same code. The threshold amplitudes are also close: $p_{\rm th}^{\rm sf}=0.0271205$ versus $p_{\rm th}^{\rm rad}=0.0270895$. The authors present this as the first explicit verification that the scalar-field/perfect-fluid correspondence extends into the critical collapse regime, at least for the quartic potential, while emphasising that the two exponents differ by about $2\sigma$ and that the quadratic case departs from dust-like behaviour.

Load-bearing premise

The scalar-field simulations carry a tiny homogeneous perfect fluid purely to define the comoving frame, and the paper does not show that this fluid stays dynamically negligible at the critical threshold.

Editorial extensions

If this is right

  • For quartic-potential scenarios, primordial black hole abundance calculations that approximate the field as a radiation fluid should capture the near-threshold scaling, with the $\sim2\%$ exponent difference entering as a small systematic uncertainty.
  • The same code reproduces the known radiation critical exponent, so the comparison baseline is not an artefact of the numerical setup.
  • The quadratic scalar-field case departs from dust-like behaviour, so the fluid analogy is potential-dependent and cannot be assumed for all scalar-field models.
  • The $2\sigma$ separation between the two exponents defines a concrete target: higher-resolution runs will either close the gap or establish a genuine matter-model dependence of $\gamma$.

Reading between the lines

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

  • Varying the potential exponent between the quadratic and quartic cases, or changing the quartic coupling, could map how $\gamma$ drifts with the effective equation of state and reveal whether the $2\sigma$ gap is a physical trend rather than a numerical artefact.
  • The frame-defining homogeneous fluid could be eliminated by reformulating the evolution in a different gauge; if the same exponent survives, the near-universality claim would no longer rest on that technical crutch.
  • The compaction function oscillates near threshold in the scalar-field runs, so peak-theory estimates of primordial black hole abundance that rely on a single maximum of the compaction function may need oscillation-aware prescriptions.
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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

3 major / 6 minor

Summary. This paper uses a Misner–Sharp, spherically symmetric, fully nonlinear relativistic code to study critical collapse of a quartic self-interacting scalar field and a radiation fluid in an expanding universe. It reports type II critical behavior in both cases, with critical exponents γ = 0.3474 ± 0.004 (radiation) and γ = 0.3401 ± 0.0071 (scalar field), and interprets the difference as about 2σ, supporting the near universality of the critical exponent in primordial black hole formation. The numerical methodology includes horizon detection, excision, Hamiltonian-constraint monitoring, and resolution tests.

Significance. If the result is robust, this is a useful first direct numerical test of the scalar-field/perfect-fluid correspondence in the critical collapse regime, with implications for PBH mass calculations in scalar-field-dominated early-universe scenarios. The paper's strengths are that it uses a fully nonlinear evolution with constraint monitoring, reports resolution dependence, and compares two matter models with the same code. However, the central quantitative claim depends on an unquantified artificial fluid in the scalar-field runs, and the stated 2σ separation is not supported by the quoted errors.

major comments (3)
  1. [II.C.2 and Eqs. (12e), (12h)] The diluted perfect fluid is not demonstrated to be dynamically negligible. The fluid is not a passive marker: it contributes to the stress-energy tensor and, through Eq. (12h), it directly fixes the lapse gradient. Eq. (12e) sources δρ_pf/ρ_pf through terms independent of the fluid's initial amplitude because the source is proportional to (ρ_pf+P_pf), so fractional perturbations of order unity can develop even for an 'extremely diluted' initial fluid. The manuscript gives no value for ρ_b0pf, does not include fluid perturbations in the initial-data list, and shows no diagnostic for the fluid density in Fig. 4. Please quantify ρ_b0pf and demonstrate by explicit convergence tests, for example by varying ρ_b0pf over several orders of magnitude, that γ_sf is unaffected; otherwise the measured exponent may be contaminated by the gauge fluid.
  2. [Abstract and Sections V.B, VI] The claimed 'about 2σ' difference between the exponents is not consistent with the quoted errors. The difference is 0.3474 − 0.3401 = 0.0073; the quadrature-summed error is sqrt(0.004^2 + 0.0071^2) ≈ 0.0082, giving about 0.9σ. Please correct this statement and avoid interpreting the result as evidence for a statistically significant difference.
  3. [IV.B and V.A] The quoted uncertainties in γ are fit errors only. The conclusion of 'near universality' rests on a small difference between exponents, so systematic uncertainties from the choice of fitting range, resolution, threshold bracket in Eq. (36), horizon-mass extrapolation, and the excision criterion need to be estimated. Without such estimates, the comparison between γ_sf and γ_rad cannot be assigned a reliable significance.
minor comments (6)
  1. [Throughout] There are typos: 'dominateed' in the Introduction, 'collaps' in the Introduction, and 'threashold' in Section V.A.
  2. [II.C.2, Eq. (16)] The initial condition for the scalar field is normalized as ~Π_b0^2/2 = 1 − ρ_b0pf, but ρ_b0pf is never specified; please state its value and justify that it is too small to alter the background dynamics.
  3. [V.A, Eq. (36)] The threshold is defined as the midpoint between p_min_bh and p_max_no-bh, but the bracketing procedure and how close to threshold the simulations are run are not described; please specify the bisection or bracketing algorithm.
  4. [Figs. 2 and 5] The set of resolutions used for the convergence tests is not stated in the captions; please list the dA values and the corresponding line styles.
  5. [III.C and IV.B] The notation for the horizon velocity criterion is inconsistent: Section IV.B uses v_c − 1 < 3 × 10^-3, while Section III.C and Appendix A define v_AH; please unify the notation.
  6. [IV.A, footnote 2] The footnote acknowledges that w = 1/3 is only an effective equation of state for the quartic scalar field; near criticality the field may not oscillate rapidly, so please discuss how this could affect the interpretation of the comparison with radiation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the critical exponents are dynamical outputs of independent numerical evolutions, and self-citations are methodological rather than load-bearing.

full rationale

The central claim—gamma_sf = 0.3401 +/- 0.0071 versus gamma_rad = 0.3474 +/- 0.004—is obtained by evolving the full Misner-Sharp equations (12) for different initial amplitudes p, locating apparent horizons, measuring PBH masses, and fitting the scaling law M proportional to (p - p_th)^gamma. The critical exponent is not an input to the code; it is a fitting output. The initial conditions (16)-(19) are physical setup choices (purely kinetic scalar field, Gaussian curvature profile, background equation of state w = 1/3, quartic coupling lambda4 = 10) and do not encode the target scaling exponent. The paper's reliance on Ref. [21] for the derivation of initial conditions and the numerical code is a self-citation, but it is methodological rather than load-bearing: the scalar-field result is compared with a radiation-fluid run performed with the same code and with the external value of Musco et al. The 'inherently circular' remark about determining lambda by iteration is a numerical fixed-point procedure, not a logical circularity. The homogeneous, extremely diluted fluid used to define the Misner-Sharp comoving frame is an admitted modeling choice; whether it contaminates the dynamics is a robustness/correctness concern, not a case of the prediction being equivalent to its inputs by construction. No fitted parameter is renamed as a prediction, and no uniqueness theorem from the authors' prior work is invoked to forbid alternatives. Therefore no circular step is identified.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The critical exponent measurement does not rely on fitting the target result, but the physical setup contains several hand-fixed parameters (coupling, profile width, diluted fluid density, initial phase) that could in principle affect the exponent. The paper does not vary these to demonstrate universality.

free parameters (4)
  • p (initial amplitude of Gaussian overdensity) = p_rad_th = 0.0270895, p_sf_th = 0.0271205
    Control parameter tuned to bracket the collapse threshold; the critical exponent is fitted as a function of p - p_th.
  • Sigma (Gaussian width) = 5 R_H
    Fixed for all runs; the critical exponent is assumed independent of initial profile shape, but this is not tested.
  • lambda_hat_4 (quartic coupling) = 10
    Sets the strength of the scalar field self-interaction relative to the Hubble scale; only one value is simulated.
  • rho_b0pf (diluted fluid density) = not specified
    A diluted perfect fluid is added in scalar field runs to define the comoving frame; its value is not reported and its effect is not tested.
assumptions (5)
  • domain assumption The Misner-Sharp formalism with spherical symmetry is sufficient to capture critical collapse in an expanding universe.
    The evolution equations are solved in spherical symmetry using a comoving fluid frame; no non-spherical or back-reaction effects are considered.
  • domain assumption Initial conditions are valid to first order in gradient and amplitude perturbations, selecting the growing mode.
    Initial data are taken from gradient expansion results (Refs. [21,34,35]) without testing higher-order corrections for the scalar field case.
  • domain assumption The scalar field starts purely kinetic at the initial time (psi0 = 0, chi0 = 0).
    Fixes the initial phase of the scalar field oscillations; no exploration of phase dependence is performed.
  • domain assumption The background universe obeys an effective barotropic equation of state w = 1/3 for the quartic scalar field.
    Uses the averaged equation of state for a rapidly oscillating quartic scalar field, while the full dynamics include oscillations.
  • ad hoc to paper The diluted perfect fluid in scalar field runs is dynamically negligible.
    The fluid is required for the comoving Misner-Sharp frame but its dynamical effect near criticality is not validated by tests.

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Pith. "Pith review of Primordial Black Hole Formation in a Scalar Field Dominated Universe: Investigation of the Critical nature of the Collapse." pith.science (2026). https://pith.science/paper/Y3J7Z3DH

@misc{pith2026250910431,
  author       = {Pith},
  title        = {Pith review of: Primordial Black Hole Formation in a Scalar Field Dominated Universe: Investigation of the Critical nature of the Collapse},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y3J7Z3DH}},
  note         = {Machine review of arXiv:2509.10431}
}
abstract

In this paper, we investigate the critical collapse leading to primordial black hole (PBH) formation in a universe dominated by a self-interacting scalar field with a quartic potential, comparing it to the well-known radiation-dominated case. Using fully relativistic nonlinear numerical simulations in spherical symmetry, based on the Misner--Sharp formalism, we analyze the dynamics near the collapse threshold and track the scaling of the black hole mass. Our results confirm that both the scalar field and radiation cases exhibit type II critical behavior with similar -- though not identical -- critical exponents, differing by about $2\sigma$. This suggests that, while a quartic scalar field effectively mimics a radiation fluid even in the nonlinear collapse regime, small differences in the critical exponent persist. Our findings provide direct numerical evidence for the near universality of the critical exponent in PBH formation, with only mild dependence on whether the collapse is driven by a scalar field or a perfect fluid.

Figures

Figures reproduced from arXiv: 2509.10431 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. shows the evolution of the Hamiltonian con￾straint and the maximum of the compaction function for different initial conditions. As before, the compaction function clearly distinguishes between collapsing and dis￾persing configurations, while the Hamiltonian constraint remains well controlled, showing significant growth only near criticality. We note, however, that in the scalar field case the maximum of the compacti… view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

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