Cosmological discrete self-similarity in primordial black hole formation

David Mulryne; Ethan Milligan; Luis E. Padilla; Tomohiro Harada

arxiv: 2604.21520 · v1 · submitted 2026-04-23 · 🌌 astro-ph.CO · gr-qc

Cosmological discrete self-similarity in primordial black hole formation

Luis E. Padilla , Tomohiro Harada , Ethan Milligan , David Mulryne This is my paper

Pith reviewed 2026-05-08 14:13 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qc

keywords primordial black holesdiscrete self-similaritycritical collapsemassless scalar fieldFLRW cosmologylog-periodic oscillationsmass scaling relationinduced gravitational waves

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The pith

Discrete self-similarity from flat-spacetime collapse persists in primordial black hole formation inside an expanding universe.

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

The paper demonstrates that discrete self-similarity, first seen in the collapse of a massless scalar field in asymptotically flat spacetime, also occurs when the same field collapses to form primordial black holes inside a Friedmann-Lemaître-Robertson-Walker background. Fully relativistic numerical simulations resolve the critical regime to deviations of order 10 to the minus 8 from threshold and detect clear log-periodic oscillations in the black-hole mass as a function of the initial-data parameter. A reader would care because these oscillations imply regular, repeating modulations in the primordial black hole mass spectrum that would affect both their overall abundance and the gravitational-wave background they source during formation.

Core claim

Discrete self-similarity survives in primordial black hole formation within an expanding cosmological background. Using fully relativistic simulations of massless scalar-field collapse in a Friedmann-Lemaître-Robertson-Walker universe, the authors resolve the critical regime down to |p - p_c| ~ 10^{-8} and find clear log-periodic oscillations in the PBH mass scaling relation. The oscillations display a more pronounced asymmetry between peaks and troughs than in the asymptotically flat case. Critical exponents and DSS periods remain broadly consistent, though slightly different, across Gaussian and piecewise rational families of initial data.

What carries the argument

Discrete self-similarity (DSS) of the near-critical solution, which imprints log-periodic oscillations on the black-hole mass scaling relation near the threshold value p_c.

If this is right

The PBH mass spectrum carries characteristic log-periodic modulations near the critical threshold.
These modulations alter the predicted abundance of primordial black holes formed from near-critical collapse.
The spectrum of gravitational waves induced by PBH formation acquires corresponding periodic features.
The asymmetry between oscillation peaks and troughs is stronger than in asymptotically flat collapse.
Critical exponents and DSS periods stay consistent within uncertainties for Gaussian and piecewise rational initial data.

Where Pith is reading between the lines

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

The persistence of DSS indicates that the echoing structure of critical collapse is robust against the addition of cosmological expansion.
Observable features in the PBH mass function could serve as a diagnostic for whether a given population formed through near-critical collapse.
The modified asymmetry in the oscillations may encode information about how expansion stretches the critical solution.
Similar numerical campaigns in other backgrounds, such as de Sitter, could test how the DSS period responds to different expansion rates.

Load-bearing premise

The numerical simulations maintain sufficient resolution and remain free of artifacts at deviations of order 10^{-8} from threshold, and the two chosen families of initial data capture the general critical behavior in the FLRW background.

What would settle it

A higher-resolution simulation or a third family of initial data that shows the log-periodic oscillations in the mass scaling either disappear or dampen significantly below |p - p_c| ~ 10^{-8} would falsify the survival of discrete self-similarity.

Figures

Figures reproduced from arXiv: 2604.21520 by David Mulryne, Ethan Milligan, Luis E. Padilla, Tomohiro Harada.

**Figure 1.** Figure 1: FIG. 1 view at source ↗

**Figure 2.** Figure 2: FIG. 2 view at source ↗

**Figure 3.** Figure 3: shows representative profiles of ρψ and Π for a near-critical subcritical evolution, evaluated at successive values of the self-similar time coordinate T. As the collapse approaches the threshold, the central region exhibits repeated episodes of contraction in which the profiles approximately reproduce the same shape after a logarithmic shift in time and scale, with spacing ∆/2 in T for the energy dens… view at source ↗

**Figure 4.** Figure 4: FIG. 4 view at source ↗

**Figure 6.** Figure 6: shows the numerical data of the initial Gaussian profile together with the best-fit power-law envelope and the corresponding sinusoidally modulated model. PHENOMENOLOGICAL FIT OF THE OSCILLATORY MODULATION Although the simple sinusoidal template discussed above captures the overall oscillatory behavior, it does not fully account for the detailed shape of the residual signal as a function of x = ln(p − pth… view at source ↗

**Figure 7.** Figure 7: FIG. 7 view at source ↗

**Figure 8.** Figure 8: FIG. 8 view at source ↗

**Figure 9.** Figure 9: FIG. 9 view at source ↗

read the original abstract

We demonstrate that discrete self-similarity (DSS), originally discovered in the collapse of a massless scalar field in an asymptotically flat system, survives in primordial black hole (PBH) formation within an expanding cosmological background. Using fully relativistic numerical simulations of massless scalar-field collapse in an Friedmann-Lema\^{i}tre-Robertson-Walker universe, we resolve the critical regime down to $|p-p_c|\sim 10^{-8}$, where $p$ and $p_c$ respectively are a parameter of the family of initial data and its threshold value, and find clear log-periodic oscillations in the PBH mass scaling relation. The detailed structure of these oscillations differs from that previously reported in the asymptotically flat case, exhibiting a more pronounced asymmetry between peaks and troughs. Analyzing two distinct families of initial data (Gaussian and piecewise rational curvature profiles), we find critical exponents and DSS periods that differ slightly but are broadly consistent within uncertainties. The presence of DSS implies characteristic log-periodic modulations in the PBH mass spectrum, with potential consequences for PBH abundances and the spectrum of induced gravitational waves.

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.

Desk Editor's Note private letter to a colleague

The paper shows DSS and log-periodic mass scaling survive the move to FLRW backgrounds with some asymmetry changes, but the numerics at 10^{-8} closeness lack reported convergence support.

read the letter

The central result is that discrete self-similarity from flat-space critical collapse of a massless scalar field carries over to PBH formation in an expanding FLRW universe. They reach |p - pc| ~ 10^{-8} in fully relativistic simulations and extract clear log-periodic oscillations in the mass scaling, with a more asymmetric peak-trough pattern than in the asymptotically flat case. Two initial-data families (Gaussian and piecewise rational) give broadly consistent exponents and periods, which is a reasonable check against setup dependence. The potential knock-on effects for PBH mass spectra and induced gravitational waves follow directly from the scaling and are worth noting for abundance calculations. What the work does cleanly is demonstrate the phenomenon is not confined to flat space and that the cosmological expansion does not destroy the discrete scaling. The soft spot is the numerical evidence. Reaching that proximity to criticality requires extreme dynamic range, yet the abstract supplies no grid resolutions, AMR settings, convergence tests, or error bars on the extracted period and asymmetry. Without those, truncation or gauge effects could contribute to the oscillations, and the slight family-to-family differences might partly reflect setup choices rather than universal behavior. The stress-test concern on artifact control is on target until the full methods section is examined. This is for people working on critical collapse and PBH formation in cosmology. A reader already following the flat-space literature will see the extension as a concrete step, but only if the numerics hold up. It deserves peer review because the claim is specific and the simulations are non-trivial, though referees will need to press for the missing convergence data and any available code.

Referee Report

2 major / 2 minor

Summary. The paper uses fully relativistic numerical simulations of massless scalar-field collapse in an FLRW cosmological background to show that discrete self-similarity (DSS) persists near the threshold for primordial black hole formation. It reports reaching |p - p_c| ~ 10^{-8} in two families of initial data (Gaussian and piecewise rational), observing clear log-periodic oscillations in the PBH mass scaling relation that differ in asymmetry from the asymptotically flat case, with broadly consistent critical exponents and DSS periods between families. The presence of DSS is argued to imply log-periodic modulations in the PBH mass spectrum with consequences for abundances and induced gravitational waves.

Significance. If the numerical results hold, the demonstration that DSS survives the cosmological expansion would be a significant extension of critical phenomena from asymptotically flat spacetimes to FLRW backgrounds. This could provide a concrete mechanism for log-periodic features in PBH mass spectra, affecting predictions for PBH dark matter fractions and the spectrum of induced gravitational waves, and would strengthen the case for universal critical behavior in gravitational collapse.

major comments (2)

[Numerical methods and results sections] The central claim of surviving DSS with log-periodic mass scaling at |p - p_c| ~ 10^{-8} is load-bearing on the numerical evidence, yet the abstract and reported results provide no quantitative convergence tests, error bars, or details on grid resolution, AMR thresholds, domain size, or gauge choices. Without these, truncation errors or artifacts cannot be ruled out as the source of the observed oscillations, particularly given the extreme dynamic range required.
[Analysis of two initial-data families] The slight differences in critical exponents and DSS periods between the Gaussian and piecewise-rational families are presented as broadly consistent, but the manuscript does not quantify the uncertainties or demonstrate that the differences are smaller than numerical errors; this weakens the claim of universality within the cosmological setting.

minor comments (2)

[Abstract] The abstract contains a grammatical error: 'an Friedmann-Lemaître-Robertson-Walker' should read 'a Friedmann-Lemaître-Robertson-Walker'.
[Results] The manuscript would benefit from explicit tabulation of the extracted periods, exponents, and their uncertainties for both families to allow direct comparison with the asymptotically flat literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting these important points regarding the numerical evidence and analysis. We address each major comment below and will revise the manuscript to strengthen the presentation of our results.

read point-by-point responses

Referee: [Numerical methods and results sections] The central claim of surviving DSS with log-periodic mass scaling at |p - p_c| ~ 10^{-8} is load-bearing on the numerical evidence, yet the abstract and reported results provide no quantitative convergence tests, error bars, or details on grid resolution, AMR thresholds, domain size, or gauge choices. Without these, truncation errors or artifacts cannot be ruled out as the source of the observed oscillations, particularly given the extreme dynamic range required.

Authors: We agree that the original submission lacks sufficient quantitative documentation of the numerical setup and convergence tests. In the revised manuscript we will add a dedicated subsection to the Numerical Methods section that specifies the grid resolutions employed, AMR refinement thresholds and criteria, computational domain sizes, and gauge choices. We will also include explicit convergence tests at multiple resolutions, together with error estimates on the extracted critical exponents and DSS periods, to demonstrate that the log-periodic oscillations persist and are not numerical artifacts. revision: yes
Referee: [Analysis of two initial-data families] The slight differences in critical exponents and DSS periods between the Gaussian and piecewise-rational families are presented as broadly consistent, but the manuscript does not quantify the uncertainties or demonstrate that the differences are smaller than numerical errors; this weakens the claim of universality within the cosmological setting.

Authors: We acknowledge that the uncertainties on the critical exponents and DSS periods were not quantified in the submitted version. In the revision we will report error bars obtained from the convergence studies and explicitly compare the differences between the two families against these uncertainties, confirming that they lie within the estimated numerical errors and thereby supporting the claim of broad consistency. revision: yes

Circularity Check

0 steps flagged

No significant circularity in numerical extraction of DSS for cosmological PBH formation

full rationale

The paper reports results from fully relativistic numerical simulations of massless scalar-field collapse in FLRW spacetime, resolving the critical regime to |p-pc|~10^{-8} and directly measuring log-periodic oscillations in the PBH mass scaling. Critical exponents and DSS periods are outputs extracted from the simulation data for two independent families of initial data (Gaussian and piecewise rational), with no equations or derivations that reduce by construction to fitted inputs or self-definitions. No load-bearing self-citations, uniqueness theorems, or ansatzes are invoked to force the central claim; the presence of DSS is presented as an observed feature of the numerics. The derivation chain is therefore self-contained computational evidence rather than tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard general-relativistic assumptions for FLRW cosmology and massless scalar fields; no new free parameters, axioms, or invented entities are introduced beyond those required for the numerical setup.

axioms (1)

domain assumption Spacetime is described by the Friedmann-Lemaître-Robertson-Walker metric with a massless scalar field source.
Standard cosmological background and matter content stated in the abstract.

pith-pipeline@v0.9.0 · 5500 in / 1349 out tokens · 37127 ms · 2026-05-08T14:13:40.854351+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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2020

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S. Hod and T. Piran, “Fine structure of Choptuik’s mass scaling relation,” Phys. Rev. D, vol. 55, pp. 440–442

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Understanding critical collapse of a scalar field,

C. Gundlach, “Understanding critical collapse of a scalar field,” Phys. Rev. D, vol. 55, pp. 695–713, Jan 1997

1997

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P. R. Brady, M. W. Choptuik, C. Gundlach, and D. W. Neilsen, “Black hole threshold solutions in stiff fluid col- lapse,” Class. Quant. Grav., vol. 19, pp. 6359–6376, 2002

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Y. Gouttenoire, G. Servant, and P. Simakachorn, “Kina- tion cosmology from scalar fields and gravitational-wave signatures,” 11

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E. Milligan, L. E. Padilla, D. J. Mulryne, and J. C. Hidalgo, “Primordial Black Hole Formation in a Scalar Field Dominated Universe,” 4 2025

2025

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L. E. Padilla, E. Milligan, D. J. Mulryne, and J. C. Hidalgo, “Primordial Black Hole Formation in a Scalar Field Dominated Universe: Investigation of the Critical nature of the Collapse,” 9 2025

2025

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M. Shibata and M. Sasaki, “Black hole formation in the friedmann universe: Formulation and computation in nu- merical relativity,” Physical Review D, vol. 60, Sept. 1999

1999

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Cos- mological long-wavelength solutions and primordial black hole formation,

T. Harada, C.-M. Yoo, T. Nakama, and Y. Koga, “Cos- mological long-wavelength solutions and primordial black hole formation,” Phys. Rev. D, vol. 91, no. 8, p. 084057, 2015

2015

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O. Rinne, “Type II critical collapse on a single fixed grid: a gauge-driven ingoing boundary method,” Gen. Rel. Grav., vol. 52, no. 12, p. 117, 2020. 5 Supplemental Material for: Cosmological discrete self-similarity in primordial black hole formation EVOLUTION EQUA TIONS The simulations are performed using the extended Misner–Sharp formulation described ...

2020