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arxiv: 2606.02053 · v1 · pith:G4YHAZVPnew · submitted 2026-06-01 · 🌀 gr-qc · astro-ph.CO· hep-ph· hep-th

COSMOS: A numerical relativity code specialized for PBH formation

Chul-Moon Yoo , Hirotada Okawa , Albert Escriv\`a , Tomohiro Harada , Hayami Iizuka , Taishi Ikeda , Yasutaka Koga , Daiki Saito
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Masaaki Shimada Koichiro Uehara
This is my paper

Pith reviewed 2026-06-28 13:28 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.COhep-phhep-th
keywords primordial black holesnumerical relativityEinstein equationsmesh refinementscalar fieldcosmological simulationsblack hole formation
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The pith

COSMOS is a C++ package that solves the Einstein equations in 3+1 dimensions to simulate primordial black hole formation.

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

The paper introduces COSMOS as a numerical relativity code specialized for modeling the collapse of super-horizon primordial fluctuations into black holes. It solves the Einstein equations using non-Cartesian scale-up coordinates and fixed mesh refinement to capture the dynamics in the collapsing region. The code supports a massless scalar field and a perfect fluid with linear equation of state as matter content, with OpenMP parallelization and no external dependencies for straightforward use. A sympathetic reader would care because analytic methods fail for the required nonlinear gravitational evolution, so a dedicated solver makes concrete calculations of PBH formation possible.

Core claim

COSMOS is a C++ package for solving the Einstein equations in 3+1 dimensions, providing simple tools for the simulation of PBH formation. In order to resolve the collapsing region, non-Cartesian scale-up coordinates and a fixed mesh-refinement procedure are implemented. In COSMOS, a massless scalar field and a perfect fluid with a linear equation of state are implemented as matter fields. To achieve a practically acceptable computational speed, OpenMP is used for the parallelization. COSMOS has no other dependencies, which makes for an easier installation.

What carries the argument

The COSMOS code, which implements 3+1 dimensional numerical relativity with non-Cartesian scale-up coordinates and fixed mesh refinement to handle collapsing regions in PBH formation simulations.

If this is right

  • Researchers can now run three-dimensional simulations of PBH formation with either a massless scalar field or a linear-EOS fluid.
  • Fixed mesh refinement combined with scale-up coordinates allows the code to focus computational effort on the collapsing region without adaptive mesh overhead.
  • OpenMP parallelization and the absence of external library dependencies reduce the barrier to running the simulations on standard hardware.
  • The same framework can be applied to both scalar-field and fluid descriptions of the early-universe matter content.

Where Pith is reading between the lines

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

  • Results from COSMOS could be used to map the threshold amplitude for PBH formation as a function of the equation-of-state parameter.
  • The code's simplicity may allow direct comparison with analytic approximations in the near-threshold regime to test their validity.
  • Extensions to include more realistic initial spectra or additional matter species would follow naturally from the current modular structure.

Load-bearing premise

The implemented matter models and coordinate plus refinement scheme suffice to capture the essential nonlinear dynamics of PBH formation from super-horizon fluctuations.

What would settle it

A side-by-side comparison of a COSMOS run for a standard initial fluctuation against an independent numerical relativity code using the same initial data and matter model would show whether the results agree within numerical error.

Figures

Figures reproduced from arXiv: 2606.02053 by Albert Escriv\`a, Chul-Moon Yoo, Daiki Saito, Hayami Iizuka, Hirotada Okawa, Koichiro Uehara, Masaaki Shimada, Taishi Ikeda, Tomohiro Harada, Yasutaka Koga.

Figure 1
Figure 1. Figure 1: The time evolution of the trace of the extrinsic curvature tr [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The lapse function (“tt-component” of the metric) on the xy-plane at the time when an apparent horizon is found. The blue, green, and purple meshes show the region covered by the lowest, 1st, and 2nd mesh refinement layers, respectively. −0.02 0 0.02 x/L 0.04 0.02 0.04 y/L 0 0.02 0.04 z/L [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The shape of the apparent horizon when it is found. [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
read the original abstract

Primordial black holes (PBHs) are black holes generated in the early universe without having gone through stellar evolution. In the standard formation process, PBHs are formed from super-horizon primordial fluctuations with non-linearly large initial amplitude. In order to simulate the non-linear gravitational dynamics of PBH formation, one has to rely on numerical relativity solvers to approximate the solution of the Einstein equations. COSMOS is a C++ package for solving the Einstein equations in 3+1 dimensions, providing simple tools for the simulation of PBH formation. In order to resolve the collapsing region, non-Cartesian scale-up coordinates and a fixed mesh-refinement procedure are implemented. In COSMOS, a massless scalar field and a perfect fluid with a linear equation of state are implemented as matter fields. To achieve a practically acceptable computational speed, OpenMP is used for the parallelization. COSMOS has no other dependencies, which makes for an easier installation.

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 / 1 minor

Summary. The manuscript introduces COSMOS, a C++ package implementing a 3+1 numerical relativity solver specialized for primordial black hole formation. It uses non-Cartesian scale-up coordinates with fixed mesh refinement, supports a massless scalar field and a perfect fluid with linear equation of state as matter models, employs OpenMP for parallelization, and has no external dependencies.

Significance. A correctly implemented and validated code with these features could provide a practical, lightweight tool for studying the nonlinear collapse of super-horizon fluctuations into PBHs, addressing a niche in early-universe numerical relativity. The choice of coordinates and matter models is well-motivated for the target application, and the lack of dependencies is a practical strength for accessibility.

major comments (2)
  1. [Abstract] Abstract and manuscript body: the central claim that COSMOS solves the Einstein equations for the stated matter models and coordinate system is unsupported, as the text supplies only a feature list with no evolution equations, gauge choices, constraint-damping terms, initial-data construction, or any numerical output (convergence tests, comparison to known solutions such as Choptuik critical collapse or Tolman-Oppenheimer-Volkoff stars, or PBH threshold amplitudes).
  2. The absence of any validation or benchmark results means the assertion that the implementation correctly captures the nonlinear dynamics of PBH formation cannot be evaluated; this is load-bearing for the paper's purpose as a specialized solver.
minor comments (1)
  1. [Abstract] The abstract and introduction could more explicitly list the specific numerical methods (e.g., finite-difference order, time integrator, boundary conditions) rather than only high-level features.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We agree that the current manuscript version is primarily a high-level description and lacks the supporting technical details and validation results needed to substantiate the central claims. We will perform a major revision to address these points by adding the requested equations, implementation specifics, and benchmark results.

read point-by-point responses

  1. Referee: [Abstract] Abstract and manuscript body: the central claim that COSMOS solves the Einstein equations for the stated matter models and coordinate system is unsupported, as the text supplies only a feature list with no evolution equations, gauge choices, constraint-damping terms, initial-data construction, or any numerical output (convergence tests, comparison to known solutions such as Choptuik critical collapse or Tolman-Oppenheimer-Volkoff stars, or PBH threshold amplitudes).

    Authors: We acknowledge that the present manuscript provides only a feature overview without the underlying equations or numerical demonstrations. In the revised version we will add the complete 3+1 evolution system (including the specific BSSN or similar formulation employed), the chosen gauge conditions and constraint-damping terms, the construction of initial data for super-horizon fluctuations, and the full set of numerical results (convergence tests, Choptuik critical collapse, TOV stars, and PBH threshold amplitudes) that were omitted from the initial submission. revision: yes

  2. Referee: [—] The absence of any validation or benchmark results means the assertion that the implementation correctly captures the nonlinear dynamics of PBH formation cannot be evaluated; this is load-bearing for the paper's purpose as a specialized solver.

    Authors: We agree that validation results are essential for a code paper. The revised manuscript will incorporate the missing benchmark suite—convergence studies, comparisons against known analytic and semi-analytic solutions (Choptuik, TOV), and PBH formation threshold measurements—to allow readers to assess the code’s correctness for the target application. revision: yes

Circularity Check

0 steps flagged

No circularity: code description paper with no derivation chain

full rationale

The manuscript is a straightforward description of a C++ numerical relativity code package (COSMOS) implementing 3+1 Einstein equations with specific coordinates, refinement, matter models, and parallelization for PBH simulations. No predictions, fitted parameters, uniqueness theorems, ansatzes, or derivations are claimed or presented; the text consists solely of feature lists and implementation notes. No self-citations or load-bearing reductions to inputs exist. This matches the default expectation of no significant circularity for non-derivational papers.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard numerical relativity infrastructure and common matter models; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)
  • standard math The 3+1 decomposition of the Einstein equations is an appropriate starting point for numerical evolution of PBH formation.
    Implicit in any 3+1 NR code; stated in the abstract's description of the solver.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Primordial Black Holes: A Review of Formation and Evolution

    gr-qc 2026-06 unverdicted novelty 3.0

    Review of PBH formation via compaction function and relativistic thresholds in FLRW backgrounds, arguing that memory burden and curvature corrections halt evaporation to leave Planck-scale relics.

Reference graph

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