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Enabling Cosmic Web Analysis at Gigaparsec Scales: A Multi Block Approach for DisPerSE

This paper introduces a multi-block "frozen-core" method that overcomes DisPerSE's memory limitations to enable gigaparsec-scale cosmic web analysis, successfully reconstructing the global filament network of the MDPL2 simulation and confirming the theoretically predicted power-law mass-connectivity relation across three decades of halo mass.

Original authors: Ankit Singh, Frazer Pearce, Meghan Gray, Gustavo Yepes

Published 2026-07-17

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CC BY 4.0

Original authors: Ankit Singh, Frazer Pearce, Meghan Gray, Gustavo Yepes

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Universe's Invisible Skeleton

Imagine the universe not as a random scattering of stars, but as a giant, three-dimensional spiderweb. This is the "cosmic web," a structure so vast it spans billions of light-years. In this cosmic web, matter isn't spread out evenly; it clumps together into massive knots (galaxy clusters), stretches out into long, thin bridges (filaments), flattens into sheets, and leaves behind huge empty bubbles called voids. Think of it like a giant, invisible city where the skyscrapers are galaxy clusters, the roads connecting them are the filaments, and the empty suburbs are the voids.

Scientists have known about this structure for decades, but they've struggled to map it perfectly, especially when looking at the biggest chunks of the universe. The main tool they use to find these invisible roads is a clever mathematical trick called "topology," which studies how things are connected. However, this tool has a major problem: to draw the map, it needs to remember every single piece of data at once. If you try to map the whole universe at once, the computer's memory explodes, like trying to hold a library of a billion books in your head all at once. For a long time, this meant scientists could only map small neighborhoods of the universe, missing the big picture of how the entire cosmic web is stitched together.

The Big Breakthrough: Freezing the Core

In this paper, a team of researchers has solved that memory problem. They developed a new method called the "frozen-core" approach, which allows them to map the cosmic web across a volume of space so huge it would have previously crashed any computer.

Here is how they did it: Imagine you are trying to draw a map of a massive, foggy city, but your sketchbook is too small to hold the whole thing. Instead of giving up, you decide to draw the city in overlapping squares. The trick is that when you draw one square, you also draw a little bit of the surrounding area (the "padding") so you know how the roads connect to the next square. But here's the genius part: once you've drawn the roads in the very center of your square, you "freeze" them. You know for a fact that those central roads are correct because you had enough context from the surrounding fog to draw them right. You then throw away the messy edges of your drawing where the fog was too thick to be sure.

The researchers applied this to the MDPL2 simulation, a massive computer model of the universe containing 92 million galaxy halos (the dark matter "homes" where galaxies live) spread across a cube 1 h⁻¹ Gpc on each side. That's a cube of space a billion light-years wide.

What they found:
By using this "frozen-core" method, they successfully reconstructed the entire filament network of this giant cube. They compared their new, tiled map against a perfect (but impossible to run) single-map version of a smaller section of the universe. The results were incredibly accurate:

They recovered 99.6% of the total length of all the cosmic filaments.
They found 100% of the density peaks (the "knots" where clusters form) and valleys (the "voids").
They matched 94.7% of the individual filaments perfectly. The few they missed were mostly very short, unimportant wiggles.

What they ruled out:
The paper explicitly shows that a "naive" approach—just chopping the universe into small pieces and mapping them separately without the extra padding and filtering—fails miserably. If you do that, you get a broken map where roads end abruptly at the edges of your squares, and you miss the long bridges that connect different parts of the web. The authors proved that without their specific "circumsphere" filtering (checking if a road segment is valid based on the geometry of the surrounding space), the map is inconsistent and wrong.

The Scientific Discovery:
With this new, giant map in hand, the team measured something called "connectivity" (denoted as κ). This is simply the number of filament roads that connect to a specific galaxy cluster. Think of it as counting how many highways lead into a major city.

They measured this for 22,900 different galaxy groups and clusters, ranging from small groups to the most massive clusters in the simulation. They discovered a clear pattern: the more massive a cluster is, the more filaments connect to it.

Specifically, they found a "power-law" relationship. This means that as the mass of a cluster increases, the number of connecting filaments increases in a predictable way. They confirmed this relationship holds true across three different orders of magnitude in mass (from 10¹² to 10¹⁵.⁵ h⁻¹ M⊙). This is the first time this scaling has been confirmed in a simulation using dark matter halos, matching what was previously only seen in gas-based simulations and theoretical predictions.

How sure are they?
The authors are very confident in their method. They didn't just guess; they ran rigorous tests. They showed that their "frozen-core" map is statistically identical to a perfect map for the parts that matter most (the long, important roads and the central knots). They also tested their results at different levels of "strictness" (called persistence thresholds, nσ = 6.0, 6.5, and 7.0) and found that while the number of filaments changes slightly depending on how strict you are, the relationship between mass and connectivity stays the same. The slope of this relationship is roughly 0.27 to 0.32, which matches other studies using different methods.

In short, this paper didn't just find a new way to draw the universe; it proved that the biggest, heaviest clusters in the cosmos are the busiest intersections, connected to the most cosmic highways, and it did so by inventing a clever trick to make the computer math work without running out of memory.

Technical Summary: Enabling Cosmic Web Analysis at Gigaparsec Scales via a Multi-Block Frozen-Core Approach for DisPerSE

Problem Statement
The cosmic web, characterized by filaments connecting massive nodes, is a dominant structural element of the Universe. Accurate reconstruction of this network across gigaparsec volumes is critical for cosmology and galaxy evolution studies. However, the topological filament finder DisPerSE (Discrete Persistent Structures Extractor), widely used for its rigorous Morse-theoretic approach, faces a severe memory bottleneck. DisPerSE requires a global Delaunay tessellation of the entire input point set to construct the Morse-Smale complex. For large cosmological simulations, such as the MDPL2 simulation containing $\sim 9.2 \times 10^7$ haloes, a monolithic Delaunay tessellation requires $\sim 460\text{--}920$ GB of RAM, exceeding the capacity of standard compute nodes.

A naive solution—splitting the volume into independent sub-volumes and running DisPerSE separately—fails because the Delaunay tessellation and the resulting persistence pairs (which define filament topology) depend on the global point distribution. Processing sub-volumes independently yields inconsistent tessellations and filament networks, even in overlapping regions, making simple concatenation invalid.

Methodology: The Frozen-Core Approach
The authors present a "frozen-core" method that enables DisPerSE to analyze gigaparsec-scale halo catalogues while preserving global topological integrity. The method proceeds in three main stages:

Volume Decomposition and Padding: The simulation volume is decomposed into overlapping spatial blocks (tiles). Each tile consists of an inner "core" region (where filaments are extracted) surrounded by a "padding" layer of width $P$ . The padding provides the necessary spatial context for the Delaunay tessellation. The width $P$ is determined empirically from the distribution of tetrahedron circumradii; for the MDPL2 halo catalogue, a padding of $P = 60 \text{ Mpc } h^{-1}$ was chosen to capture 99.33% of tetrahedra, ensuring that the circumsphere of any core tetrahedron fits entirely within the padded region.
Independent Tessellation and Circumsphere Filtering: DisPerSE is run independently on each padded tile. A circumsphere-based filter is then applied: any tetrahedron whose circumsphere extends beyond the padded region is discarded. The remaining tetrahedra constitute the "frozen core," guaranteed to be identical to those in a global monolithic tessellation because all points influencing them are present within the padded tile.
Post-Processing Pipeline: The tiled outputs are merged via a three-stage pipeline:
- Frozen Core Filtering: Retains filament segments where at least one endpoint lies within the core region, discarding segments entirely within the padding.
- Spatial Deduplication: Uses a KDTree to identify and remove duplicate filaments detected in overlapping tile regions.
- Boundary Stitching: Reconnects filament fragments split at tile boundaries using a proximity tolerance ( $\delta = 0.5 \text{ Mpc } h^{-1}$ ).

Key Contributions

Algorithmic Innovation: The development of a frozen-core method that overcomes the memory bottleneck of global Delaunay tessellation without compromising topological accuracy.
Validation: Rigorous validation against a monolithic reference run on a $300 \text{ Mpc } h^{-1}$ $300 Mpc h^{- 1}$ subvolume of MDPL2. The method achieves:
- 99.6% total filament length recovery.
- 100% recovery of density maxima (peaks) and minima (void centers).
- 94.7% individual filament matching.
- The unmatched $\sim 5\%$ of filaments are predominantly short, low-significance structures, indicating a conservative bias rather than the introduction of spurious features.
Scalability: The method is "embarrassingly parallel," allowing the processing of arbitrarily large volumes by increasing the number of tiles while keeping per-node memory usage constant (limited to $\sim 10$ GB for the tile configuration used).

Results
The authors applied the method to the full $(1 \text{ Gpc } h^{-1})^3$ MDPL2 box, containing 92 million haloes, producing the first gigaparsec-scale filament catalogue derived via topologically rigorous methods.

Filament Statistics: At a fiducial persistence threshold ( $n\sigma = 6.5$ ), the catalogue contains $\sim 1.16 \times 10^6$ filaments with a total length of $\sim 7.5 \times 10^6 \text{ Mpc } h^{-1}$ . The analysis confirms that increasing the persistence threshold preferentially removes short filaments and merges shorter arcs into longer structures, consistent with persistence theory.
Mass-Connectivity Relation: As a primary scientific application, the authors measured the connectivity ( $\kappa$ $κ$ , the number of filaments attached to a halo) for $\sim 22,900$ $\sim 22, 900$ haloes spanning masses from $10^{12}$ $1 0^{12}$ to $10^{15.5} h^{-1} M_{\odot}$ $1 0^{15.5} h^{- 1} M_{⊙}$ .
- They found a power-law mass-connectivity relation: $\log_{10} \kappa = A \log_{10} M_{200c} + B$ .
- The slope $A$ was found to be stable across persistence thresholds, with values of $A \approx 0.27\text{--}0.32$ for an aperture of $1.5 R_{200}$ .
- This result confirms the theoretically predicted scaling across three decades in halo mass, extending from galaxy groups to the most massive clusters. The slope is consistent with previous gas-based measurements (e.g., Santoni et al. 2024) and topological analyses (Bermejo et al. 2024).

Significance and Claims
The paper claims to provide the first confirmation in an N-body halo catalogue that the theoretically predicted mass-connectivity scaling holds across a vast range of halo masses. By enabling topologically rigorous filament detection in gigaparsec volumes, the method opens access to the statistical power of large cosmological volumes previously unavailable for such analyses. The authors emphasize that the method preserves the deterministic nature of DisPerSE, avoiding the stochastic variance of other methods like the Bisous model, while making the analysis of next-generation simulations (e.g., Euclid Flagship, FLAMINGO) feasible. The work establishes a robust framework for studying the geometry of the cosmic web and the accretion history of clusters on the largest scales.

The Universe's Invisible Skeleton

The Big Breakthrough: Freezing the Core

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