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

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]

• The paper demonstrates caustic skeleton theory’s ability to map initial Gaussian fields to observable cosmic structures like walls, filaments, and nodes.
• It employs Bayesian Manticore reconstructions of the 2M++ galaxy catalogue to classify filament types (A3, A4, D4) and assign environmental signatures.
• The study identifies distinct formation chronologies and merging histories for filament types, providing new insights into galaxy–cosmic web interactions.

Caustic Skeleton Theory and the Local Universe’s Cosmic Web: A Technical Analysis

Introduction

"Caustic Skeleton and the Local Cosmic Web: the Coma Cluster node and the Pisces-Perseus ridge" (2604.22213) demonstrates the application of caustic skeleton theory—a phase-space formalism for predicting galaxy-scale structure formation—to observationally constrained reconstructions of the local Universe. Specifically, the work leverages the Manticore-Local Bayesian reconstructions matched to the 2M++ galaxy catalogue to extract and analyze the multi-scale caustic skeleton organizing two canonical structures: the Coma Cluster (and its Stickman configuration) and the Pisces-Perseus Supercluster. The study delivers new insights into cosmic web topology, filament classification, structure formation chronology, and the environmental characterization of observed galaxies.

Theoretical Foundations: Caustic Skeleton Formalism

Caustic skeleton theory analyzes the gravitational collapse of the cosmic matter field through the evolution of the dark matter sheet in phase-space. Catastrophe theory, particularly in the context of Lagrangian fluid dynamics, identifies stable singularities—caustics—corresponding to distinct large-scale structures: walls ($A_3$ cusps), filaments ($A_4$ swallowtails, $D_4$ umbilics), and cluster nodes ($A_5$, $D_5$). Shell-crossing and higher-order derivatives of the deformation tensor's eigenvalue fields yield analytic criteria for these caustics.

Figure 1: Cartoon illustration of sheet folding in phase-space and the resulting formation of $A_2$ fold caustics, showing the origin of multi-stream regions.

While conventional $N$-body or heuristic cosmic web classifiers provide only diagnostic decomposition in Eulerian space, the caustic skeleton formalism anchors predictions directly to primordial conditions within the Zel'dovich approximation. This enables an explicit analytical mapping of the initial Gaussian field to the spatial organization of cosmic walls, filaments (of both $A$ and $D$ types), and nodes, supplying topological and chronological information missing in standard approaches.

Figure 2: Caustic skeleton theory correctly predicts multi-stream structure in a 2D $N$-body simulation, with caustic loci extracted solely from the initial conditions and mapped to Eulerian space.

Data and Simulation Framework

Bayesian field-level analyses of the 2M++ galaxy catalogue via the Manticore project reconstruct the initial density field in the Local Universe and propagate it forward using the SWIFT $A_4$0-body code. MonofonIC is employed to provide small-scale consistency via power-spectrum conformant Gaussian fluctuations. This enables direct computation of the initial Lagrangian deformation tensor, smoothing it at varying scales (typically $A_4$1–$A_4$2 Mpc). The work explicitly examines realizations $A_4$3, $A_4$4, and $A_4$5.

Caustic Skeleton in Key Local Volume Structures

Coma Cluster and the Stickman Configuration

A detailed caustic extraction reveals the Stickman structure—the archetypal cosmic web configuration centered on the Coma Cluster. The identification is robust across all Manticore realizations analyzed.

Figure 3: Simulated density field slices reveal the morphology of the Stickman and Coma Cluster in real and redshift space, with 2M++ galaxies overlaid for validation.

Caustic skeleton mapping demonstrates excellent correspondence between predicted caustic spines and observed galaxy positions, surviving even quasi-linear redshift distortions. $A_4$6 walls map the prominent Stickman limbs, while $A_4$7 filaments populate the high-density ridges. $A_4$8 filaments, although abundant only at smaller smoothing scales or high densities, provide environments with distinct folding histories.

Figure 4: Three-dimensional caustic skeleton around the Coma Cluster, separating $A_4$9 walls, $D_4$0 filaments, and $D_4$1 filaments; density and eigenvalue structure are visualized in Lagrangian and Eulerian space.

Figure 5: 2D slice of caustic skeleton and density for all three Manticore realizations demonstrates overall robustness and convergence in Eulerian space.

Pisces-Perseus Supercluster Analysis

A complementary analysis of the Pisces-Perseus Supercluster leverages coordinate rotation to align with the structure’s observed orientation. The caustic skeleton, resolved at multiple scales, captures both the principal filamentary backbone and the more intricate $D_4$2 structure embedded within high-density knots.

Figure 6: Three-dimensional caustic skeleton and density field in the Pisces-Perseus region, clearly distinguishing $D_4$3- and $D_4$4-type filaments.

Figure 7: Infinitesimal 2D slice of caustic filaments—separating $D_4$5 and $D_4$6—showing relative spatial segregation near dense supercluster cores.

A bold result is that the Pisces-Perseus Supercluster, while morphologically similar to standard cosmic web filaments, is topologically characterized by dominant $D_4$7 umbilic filaments at smaller scales—an environmental distinction undetectable through purely morphological algorithms such as DisPerSE.

Galaxy Environmental Classification via Caustic Topology

The authors introduce a hierarchical classification scheme, asserting that $D_4$8 filaments prevail locally over $D_4$9, which in turn supersede $A_5$0 walls, down to unclassified void environments. For each 2M++ galaxy, proximity to a caustic of each type at multiple smoothing scales determines a nested environmental signature.

Figure 8: Scale-resolved environmental classification (by dominant caustic type) of 2M++ galaxies in the Coma Cluster region, directly linking galaxy position to multi-scale cosmic web topology.

Scale-dependent classification histograms reveal that, as the skeleton is probed at finer ($A_5$1) resolution, void assignments drop and filamentary ($A_5$2, $A_5$3) designations rise. In the Stickman region, $A_5$4 predominates at large scale, becoming comparable to $A_5$5 at $A_5$6 Mpc. For Pisces-Perseus, $A_5$7 classification dominates at all but the coarsest scales.

Figure 9: Stacked histograms reveal the evolution of galaxy environmental association with different caustic types as a function of smoothing scale, highlighting the $A_5$8 dominance in Pisces-Perseus.

Chronology and Merging of Caustic Filaments

The formation times of caustic features are accessible via the Zel'dovich growth factor: $A_5$9. This approach exposes the highly inhomogeneous and merging nature of filament formation: $D_5$0 filaments typically emerge in disjoint segments ($D_5$1) which subsequently merge ($D_5$2) at separate spacetime loci.

Figure 10: Spatial structure of a long $D_5$3 filament in the Coma Cluster, with color-coding by formation time showing complex merging and stretching as one moves from Lagrangian to Eulerian space.

A direct mapping of filamentary skeletons colored by formation epoch reveals significant temporal and topological heterogeneity, with late-forming $D_5$4 filaments constituting a non-negligible fraction in dense supercluster cores. Cumulative formation time distributions further quantify systematically earlier formation for walls and for coarser-scale features, consistent with hierarchical $D_5$5CDM growth.

Figure 11: Cumulative formation time distributions for walls, $D_5$6, and $D_5$7 filaments in the Coma and Pisces-Perseus regions at varying smoothing scales.

Implications and Future Prospects

The work's most significant implication is the ability to identify and characterize two topologically distinct classes of filaments ($D_5$8 and $D_5$9), with differing formation histories and environmental dominance. This not only augments the theoretical understanding of cosmic web connectivity but also opens observational avenues for linking galaxy properties—such as morphology, spin alignment, and star formation—to detailed caustic environment signatures.

The explicit environmental chronology (via shell-crossing) and multi-scale nesting invite further studies integrating galaxy evolution models with caustic skeleton theory. As large redshift surveys (Euclid, DESI, SKA) deliver ever deeper and broader galaxy catalogues, the techniques presented are well-placed for leveraging these datasets.

On the theoretical side, a systematic extension to nodes ($A_2$0, $A_2$1), more robust statistical samples over the multiple Manticore-Local realizations, and incorporation of baryonic effects in structure formation will be critical frontiers. Additionally, there is ample scope to extend these analyses to machine learning-based classification of cosmic environments using caustic-derived topological features.

Conclusion

This study presents a robust implementation of caustic skeleton theory on observationally constrained simulations, demonstrating the technique's predictive power for the multi-scale, multistream connectivity of the local cosmic web. The separation and environmental tracing of $A_2$2 versus $A_2$3 filaments in key cosmic structures yield new information not accessible to heuristic methods, and the chronological mapping of the formation and merger of these features provides a new axis for the study of galaxy–cosmic web interactions. This work lays out a solid theoretical and algorithmic framework, with significant implications for future analyses in both observational cosmology and theoretical galaxy formation.