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arxiv: 2604.22213 · v1 · submitted 2026-04-24 · 🌌 astro-ph.CO · gr-qc

Caustic Skeleton and the Local Cosmic Web: the Coma Cluster node and the Pisces-Perseus ridge

Amelie Read , Job Feldbrugge , Celine Boehm , Rien van de Weygaert , Benjamin Hertzsch This is my paper

Pith reviewed 2026-05-08 10:19 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qc
keywords caustic skeletoncosmic weblarge-scale structureComa ClusterPisces-PerseusfilamentssingularitiesManticore simulations
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The pith

The caustic skeleton network accurately reproduces the large-scale galaxy organisation in the Coma Cluster and Pisces-Perseus ridge.

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

The paper applies caustic skeleton theory to Bayesian reconstructions of the local universe from the 2M++ galaxy catalogue. It demonstrates that the predicted network of caustic singularities matches the observed distribution of galaxies in redshift space for the Coma Cluster and Pisces-Perseus ridge. The hierarchy of features supports a multi-scale classification of environments and identifies two distinct filament types based on their formation. A sympathetic reader would care because this supplies a topological description of how the nearby cosmic web is organized beyond density or visual inspection.

Core claim

The central claim is that the Caustic Skeleton network of caustic singularities accurately reproduces the observed large-scale organisation of galaxies in redshift space for one of the Manticore realisations. The hierarchy of caustic features allows multi-scale classification of the large-scale environment in which observed galaxies reside. The Pisces-Perseus Supercluster is revealed to be a distinctly D4-dominated structure compared to the extended structure around the Coma Cluster.

What carries the argument

The caustic skeleton network of A4 swallowtail and D4 umbilic caustics, which are singularities formed by different folding histories of the density field and used to trace filaments and nodes in the cosmic web.

If this is right

  • The hierarchy of caustic features enables multi-scale classification of the large-scale environments where galaxies reside.
  • Filaments are divided into two topologically distinct classes (A4 and D4) that form through different histories but can appear similar.
  • The Pisces-Perseus Supercluster is characterized as distinctly D4-dominated, unlike the area around the Coma Cluster.
  • A novel topological characterisation of filamentary complexes in the local universe becomes possible.

Where Pith is reading between the lines

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

  • The classification could be applied to other local superclusters to check if D4 dominance is a general feature of certain ridges.
  • It offers a route to connect observed filament topologies back to the initial density fluctuations in cosmological models.
  • Extending the analysis to larger volumes or different reconstruction methods could test consistency across the wider cosmic web.

Load-bearing premise

The Manticore-Local Bayesian reconstructions from the 2M++ catalogue faithfully capture the true three-dimensional density and velocity fields without significant biases from galaxy selection, redshift-space distortions, or reconstruction priors.

What would settle it

A systematic mismatch between the predicted positions of caustic singularities and the actual galaxy locations from the 2M++ catalogue in the Coma Cluster and Pisces-Perseus ridge would disprove the accurate reproduction.

Figures

Figures reproduced from arXiv: 2604.22213 by Amelie Read, Benjamin Hertzsch, Celine Boehm, Job Feldbrugge, Rien van de Weygaert.

Figure 1
Figure 1. Figure 1: Example illustration of a Lagrangian mapping of a 2D continuous particle distribution. The vertical direction 𝑞1 represents one of the initial (Lagrangian) positions of the distribution. The 𝑥1 − 𝑥2 plane represents the final (Eulerian) distribution. The top panel corresponds to 𝑡 = 0 before the distribution has evolved, such that the mapping is 1-1, so it appears as a flat sheet. Subsequent panels show ho… view at source ↗
Figure 2
Figure 2. Figure 2: Example of caustic skeleton theory applied to an unconstrained 2D 𝑁-body simulation. The Lagrangian-space initial particle displacements are shown in the top left panel, with the first eigenvalue field and caustics shown, also in Lagrangian space, in the panel below. To the right, the final particle displacements are shown in the top panel with the Eulerian-space caustics overlaid in the panel below. On th… view at source ↗
Figure 3
Figure 3. Figure 3: This figure shows the Coma Cluster with associated Stickman structure in a 100 Mpc square region of the galactic (𝑋, 𝑌 ) plane for three different Manticore-Local simulations (left to right: 𝑀0, 𝑀1, 𝑀2). This is obtained by taking a 2D slice through the calculated 3D simulation density field at 𝑍Coma = 49.56 Mpc, the observed galactic Z-coordinate of the Coma Cluster. Top panels: Dark Matter only density f… view at source ↗
Figure 4
Figure 4. Figure 4: This figure shows the caustic skeleton around the Coma Cluster in both Lagrangian space (top panels) and Eulerian space (bottom panels) in a 𝐿 = 100 Mpc box extracted from one example of a Manticore-Local Universe re-simulation (refer to as 𝑀1). We use the smoothing scale of 𝜎 = 3 Mpc. Left: 𝐴3 walls with 𝐴4 and 𝐷4 filaments. Middle: Both 𝐴4 and 𝐷4 filaments. Right: 𝐷4 filaments alone. The 3-dimensional fi… view at source ↗
Figure 5
Figure 5. Figure 5: Filamentary caustics of Stickman for 𝑀1 simulation within 10 Mpc of 𝑍Coma = 49.56 Mpc projected onto X-Y galactic plane in 100 Mpc square region. Z-coordinates of filaments are shown for 𝐴4 filaments and 𝐷4 filaments on the bottom left and right, respectively. 2D real-space density field slice of 𝑀1 simulations underlaid, same as upper middle panel of fig. 3. The density is estimated with the Phase-Space D… view at source ↗
Figure 6
Figure 6. Figure 6: This figure shows the caustic skeleton at a 𝜎 = 3 Mpc smoothing scale of the Stickman in a 100 Mpc square region of the galaxtic (𝑌, 𝑋) plane. in real space for three different Manticore-Local simulations (𝑀0, 𝑀1, 𝑀2). This is obtained by taking an infinitesimally 2D slice through the calculated 3D skeleton and simulation density field in real space at 𝑍Coma = 49.56 Mpc, which is the observed Z-coordinate … view at source ↗
Figure 7
Figure 7. Figure 7: 2D slice through caustic skeleton of Coma Cluster at 𝑍Coma = 49.56 Mpc in 𝑀1, one example of a Manticore Local Universe re-simulation. Red lines show the intersection of 3D walls with the thin 2D slice. Intersection of 𝐴4 and 𝐷4 filaments with slice is shown by green points and purple triangles, respectively. Left to right from larger scales at 𝜎 = 3 Mpc to progressively smaller scales at 𝜎 = 2 Mpc, then 𝜎… view at source ↗
Figure 8
Figure 8. Figure 8: This figure shows the caustic skeleton of the Pisces-Perseus Supercluster in a 60 Mpc box for 𝑀1 Local Universe re-simulation. Coordinates are rotated so that the positive 𝑍𝑅-direction is roughly along the line of sight and RA and DEC correspond to positive 𝑌𝑅 and negative 𝑋𝑅 directions. Left: 𝐴3 walls with 𝐴4 and 𝐷4 filaments, Middle: 𝐴4 and 𝐷4 filaments, Right: Only 𝐷4 filaments. We use the smoothing sca… view at source ↗
Figure 9
Figure 9. Figure 9: Infinitesimally thin slices of caustic skeleton of Pisces-Perseus cluster in 60 Mpc square region for 𝑀1 Local Universe simulation in real space on the left and redshift space on the right. Top panels: 2D (𝑌𝑅, 𝑋𝑅 ) slice of density field and caustic skeleton at 𝑍𝑅 = 73.3 Mpc. Bottom panels: 2D 𝑌𝑅 − 𝑍𝑅 slice at 𝑋𝑅 = 0 Mpc. These slices are taken according to the observed location of the A426 cluster in the … view at source ↗
Figure 10
Figure 10. Figure 10: Filamentary caustics of Pisces-Perseus cluster for 𝑀1 simulation within 10 Mpc of 𝑋𝑅 = 0 Mpc projected onto (𝑌𝑅, 𝑍𝑅 ) rotated coordinate plane in 60 Mpc square region. 𝑋𝑅-coordinate of filaments are shown for 𝐴4 filaments and 𝐷4 filaments on the bottom left and right respectively. 2D real-space density field slice of 𝑀1 simulations underlaid, same as the field underlaid in the bottom left panel of fig. 9.… view at source ↗
Figure 11
Figure 11. Figure 11: Classification scheme outlined in section 5 applied to 2M++ galaxies in 10 Mpc thick slice around 𝑍Coma (this is the same subset of galaxies shown in fig. 7) using caustic skeleton extracted from the 𝑀1 realisation. Since the large-scale environment of a galaxy is scale dependent, the galaxy classification scheme is applied at smoothing scales 𝜎 = 5, 3, 2, 1, shown in the top left, top right, bottom left … view at source ↗
Figure 12
Figure 12. Figure 12: Classification scheme outlined in section 5 applied to 2M++ galaxies in 10 Mpc thick slice around 𝑍𝑅 = 73.3 Mpc in a rotated coordinate system (this is the same subset of galaxies shown in top right panel of fig. 9) using the caustic skeleton extracted from the 𝑀1 realisation. Since the large-scale environment of a galaxy is scale dependent, the galaxy classification scheme is applied at smoothing scales … view at source ↗
Figure 13
Figure 13. Figure 13: Stacked histograms displaying classification of dominant associ￾ated caustic structure for each 2M++ galaxy within a 100 Mpc box around Stickman structure (top panel) and a 60 Mpc box (extracted in a rotated coor￾dinate system) around Pisces-Perseus Supercluster structure (bottom panel) in realisation 𝑀1. The classification scheme, outlined in section 5, is repeat￾edly applied with the caustic skeleton at… view at source ↗
Figure 15
Figure 15. Figure 15: Right: 2D (𝑌, 𝑍) projection of 3D filament from fig. 14 in Eulerian space. Filament’s extent in X direction is limited, so the slice of the simulation density field for comparison is taken around halfway through the 𝑋-extent at 𝑋 = −91 Mpc. Left: Filament traced back to initial conditions (Lagrangian space) projected into the same (𝑌, 𝑍) plane. Following the same logic as in Eulerian space, a slice of the… view at source ↗
Figure 16
Figure 16. Figure 16: Formation time (𝐷 𝑓 𝑜𝑟𝑚) as a function of distance along the length of example filament from fig. 14 in Lagrangian space (left) and Eulerian space (right). Distance along filament is defined from the bottom left to top right of the filament in fig. 15. Formation time is defined as the time at which shell-crossing first occurred and the relevant caustic condition is used to show this is equivalent to the s… view at source ↗
Figure 17
Figure 17. Figure 17: Formation times (𝐷 𝑓 𝑜𝑟𝑚) mapped onto a 10 Mpc thick slice of 𝐴4 filaments around the Coma Cluster in a 100 Mpc square region (this is the same slice of filaments at 𝜎 = 3 Mpc shown in the left panel of fig. 5). The slice is taken in real space on the left and redshift space on the right. Due to the impact of RSDs, this is not exactly the same slice of the filamentary caustics around Coma, as some segment… view at source ↗
Figure 18
Figure 18. Figure 18: Formation times (𝐷 𝑓 𝑜𝑟𝑚) of 𝐴4 filaments in 10 Mpc thick slices around Pisces-Perseus Supercluster at smoothing scale 𝜎 = 3 Mpc in a 60 Mpc square region. Top panels: Slice taken about 𝑍𝑅 = 73.3 Mpc and filaments projected onto (𝑌𝑅, 𝑋𝑅 ) plane. Bottom panels: Slice taken about 𝑋𝑅 = 0 Mpc and projected onto (𝑌𝑅, 𝑍𝑅 ) plane, such that the bottom left panel is the same section of filaments shown in fig. 10.… view at source ↗
Figure 19
Figure 19. Figure 19: Cumulative probability distributions of formation times for Eulerian-space caustic skeleton structures in a 100 Mpc region around Coma Cluster (left) and 60 Mpc region around Pisces-Perseus Supercluster (right) for the 𝑀1 realisation. 𝐴3 walls, 𝐴4 filaments and 𝐷4 filaments are shown separately in the top, middle and bottom panels, respectively. The cumulative distribution for each caustic structure is gi… view at source ↗
read the original abstract

We apply caustic skeleton theory to the Manticore-Local simulations, which are Bayesian constrained reconstructions of the Local Universe from the 2M++ galaxy catalogue, and extract the three-dimensional multi-scale caustic skeleton of two canonical weblike structures in our Local Universe, namely the Coma Cluster and the Pisces-Perseus ridge as they represent the most prominent cluster node and filamentary artery in the nearby Universe. We show that the Caustic Skeleton network of caustic singularities accurately reproduces the observed large-scale organisation of galaxies in redshift space for one of the Manticore realisations. The hierarchy of caustic features allows us to establish a multi-scale classification of the large-scale environment in which observed 2M++ galaxies reside. One of the most interesting aspects of the theory is that it predicts two topologically distinct classes of filaments (A_4 swallowtail and D_4 umbilic caustics) that form through fundamentally different folding histories yet appear morphologically similar enough, on the surface, to be overlooked by conventional structure identifiers. We find that the influence of D_4 filaments only becomes increasingly relevant towards smaller scales, and the Pisces-Perseus Supercluster in particular is revealed to be a distinctly D_4-dominated structure compared to the extended Stickman structure around the Coma Cluster. In other words, caustic skeleton theory enables a novel topological characterisation of one of the most studied filamentary complexes in the nearby Universe. [Shortened]

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 applies caustic skeleton theory to the Manticore-Local Bayesian constrained reconstructions of the local universe from the 2M++ galaxy catalogue. It extracts the three-dimensional multi-scale caustic skeleton for the Coma Cluster node and the Pisces-Perseus ridge, claiming that this network accurately reproduces the observed large-scale galaxy organization in redshift space for one realization. The hierarchy of caustic singularities (A4 swallowtail and D4 umbilic) is used to establish a multi-scale environmental classification of 2M++ galaxies, with the conclusion that the Pisces-Perseus Supercluster is distinctly D4-dominated compared to structures around Coma.

Significance. If the reproduction and classification hold under quantitative scrutiny, the work introduces a topological framework for the cosmic web that distinguishes filament classes by their folding histories rather than morphology alone. This could refine characterizations of local structures like the Pisces-Perseus ridge beyond density-field or conventional web finders, leveraging constrained simulations for direct observational ties. The distinction between A4 and D4 features is a conceptual strength with potential for broader application in structure formation studies.

major comments (2)
  1. [§4] §4 (comparison to observations): The central claim that the caustic skeleton 'accurately reproduces' the observed galaxy organization in redshift space rests on visual inspection for a single Manticore realization. No quantitative agreement metrics (e.g., overlap fractions, Hausdorff distances between caustic surfaces and galaxy positions, or residual statistics) are reported, and the result is not shown to be stable across the ensemble of reconstructions. This is load-bearing for the accuracy assertion, particularly since the input density and velocity fields derive from Bayesian priors, galaxy bias modeling, and RSD corrections that could induce apparent matches.
  2. [§5] §5 (classification results): The conclusion that Pisces-Perseus is a distinctly D4-dominated structure inherits the same limitation, as the dominance ratio is extracted from caustics in the reconstructed fields without tests of sensitivity to reconstruction choices (e.g., prior variations or selection function). The paper does not quantify how changes in these inputs shift the A4/D4 balance, undermining the topological classification claim.
minor comments (2)
  1. [Figures] Figure captions (e.g., those showing the caustic network overlay) could explicitly state the viewing projection, redshift-space mapping, and color scheme for A4 versus D4 features to improve clarity for readers unfamiliar with the formalism.
  2. [Methods] The methods section would benefit from a brief summary of how the multi-scale caustic extraction thresholds are chosen and whether they are held fixed across the two structures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major comment below and indicate the revisions we will implement to strengthen the quantitative support for our claims.

read point-by-point responses

  1. Referee: [§4] §4 (comparison to observations): The central claim that the caustic skeleton 'accurately reproduces' the observed galaxy organization in redshift space rests on visual inspection for a single Manticore realization. No quantitative agreement metrics (e.g., overlap fractions, Hausdorff distances between caustic surfaces and galaxy positions, or residual statistics) are reported, and the result is not shown to be stable across the ensemble of reconstructions. This is load-bearing for the accuracy assertion, particularly since the input density and velocity fields derive from Bayesian priors, galaxy bias modeling, and RSD corrections that could induce apparent matches.

    Authors: We acknowledge that the reproduction is demonstrated through visual comparison for a single realization. The Manticore-Local reconstructions are Bayesian constrained by the 2M++ catalogue, so the underlying density and velocity fields are designed to reproduce the observed galaxy distribution rather than being arbitrary. Nevertheless, we agree that quantitative metrics would make the accuracy claim more robust. In the revised manuscript we will add overlap statistics (fraction of 2M++ galaxies within a fixed distance of the caustic surfaces) and a basic alignment measure between the extracted caustics and the observed galaxy positions. We will also include a comparison with a second independent realization to illustrate stability against reconstruction variations. revision: yes

  2. Referee: [§5] §5 (classification results): The conclusion that Pisces-Perseus is a distinctly D4-dominated structure inherits the same limitation, as the dominance ratio is extracted from caustics in the reconstructed fields without tests of sensitivity to reconstruction choices (e.g., prior variations or selection function). The paper does not quantify how changes in these inputs shift the A4/D4 balance, undermining the topological classification claim.

    Authors: We agree that the D4-dominance result would be strengthened by explicit sensitivity tests. In the revision we will compute the A4/D4 ratio for at least one additional Manticore realization and briefly discuss how modest changes in the Bayesian prior or selection function affect the balance. This will provide a quantitative check on the robustness of the topological classification for the Pisces-Perseus ridge. revision: yes

Circularity Check

0 steps flagged

No circularity in the paper's derivation chain

full rationale

The paper applies an established caustic skeleton theory to the Manticore-Local Bayesian reconstructions (treated as external inputs derived from the 2M++ catalogue) and performs an empirical comparison of the resulting caustic network against observed galaxy positions in redshift space. No load-bearing step reduces by the paper's equations or definitions to a fitted parameter, self-referential input, or self-citation chain; the multi-scale classification (A4/D4) follows from the independent mathematical structure of the theory, and the reported reproduction for one realization is a test rather than a tautology. The derivation remains self-contained against external observational benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the applicability of prior caustic skeleton theory to the density fields of the Manticore simulations; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption Caustic skeleton theory provides an accurate description of the folding singularities in the cosmic density field that correspond to observed large-scale structures.
    The paper invokes this established framework to extract and classify the caustic network from the simulations.

pith-pipeline@v0.9.0 · 5578 in / 1367 out tokens · 38680 ms · 2026-05-08T10:19:06.291707+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

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    Abdul Karim M., et al., 2025a, Phys. Rev. D, 112, 083514 Abdul Karim M., et al., 2025b, Phys. Rev. D, 112, 083515 AbelT.,HahnO.,KaehlerR.,2012,MonthlyNoticesoftheRoyalAstronom- ical Society, 427, 61 Alpaslan M., et al., 2014, MNRAS, 440, L106 Aragón-CalvoM.A.,PlatenE.,vandeWeygaertR.,SzalayA.S.,2010,ApJ, 723, 364 Aragón-Calvo,M.A.Jones,B.J.T.vandeWeygaert...

    work page doi:10.5281/zenodo.16637561 2012

  2. [2]

    Significant merging (𝐴− 4: blue down triangles) and formation (𝐴+ 4: pink triangles) are plotted in all panels

    These slices are taken in real space on the left and redshift space with 2M++ galaxies included on the right. Significant merging (𝐴− 4: blue down triangles) and formation (𝐴+ 4: pink triangles) are plotted in all panels. Same as fig. 17, some simply represent the points at which the filament first enters the slice. The density is estimated with the Phase...

    work page doi:10.5281/zenodo.4158731 2024