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Cylindrical cosmological simulations with StePS

This paper introduces a novel cylindrical cosmological simulation framework called StePS, which utilizes an $\mathrm{S}^1\times\mathbb{R}^2$ topology and inverse stereographic projection to enable self-consistent simulations of systems with cylindrical symmetries while avoiding the periodic-image artifacts inherent in standard cubic box models.

Original authors: Gábor Rácz, Viola H. Varga, Balázs Pál, István Szapudi, István Csabai, Till Sawala

Published 2026-06-03

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CC BY 4.0

Original authors: Gábor Rácz, Viola H. Varga, Balázs Pál, István Szapudi, István Csabai, Till Sawala

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

Imagine you are trying to simulate how the universe grows, like watching a time-lapse video of a forest sprouting from a single seed. For decades, scientists have done this by building a giant, invisible cube and filling it with digital particles representing matter. They then wrap the walls of this cube so that if a particle flies out the right side, it instantly reappears on the left. This is called a "periodic box."

While this cube is easy to build on a computer, it has a problem: it forces the universe to look like a box. In reality, the universe isn't a cube; it's round and open. The cube shape creates weird "ghost" effects where gravity behaves strangely because of the sharp corners and edges, potentially messing up the results.

The New Idea: A Cosmic Tunnel
In this paper, the authors introduce a new way to simulate the universe: instead of a cube, they use a giant, infinite tube (a cylinder).

The Tube: Imagine a long, endless highway. If a car drives off the front, it doesn't loop back to the back (like in the cube); instead, the road just keeps going forever. However, if you look to the left or right, the road loops around in a circle.
The Benefit: This shape is perfect for studying things that naturally look like tubes or long strands, such as cosmic filaments (the giant threads of matter that connect galaxies). It removes the "boxy" bias and lets the universe behave more naturally in those directions.

The "Zoom-In" Trick
There is a catch: computers have limited power. You can't simulate the entire infinite tube with high detail everywhere.

The Analogy: Think of a camera lens. When you zoom in on a flower in a garden, the flower is crystal clear, but the trees in the background are blurry.
The Solution: The authors built their simulation so that the "center" of the tube (the axis) is super sharp and detailed. As you move further out toward the edges of the tube, the resolution gets "blurry" (the particles get bigger and fewer). This allows them to study tiny details near the center while still covering a massive volume of space without needing a supercomputer the size of a city.

How They Did It

The Map: They used a mathematical trick (like projecting a globe onto a flat map) to squeeze an infinite space into a finite computer memory.
The Gravity: They had to rewrite the rules of gravity for this tube shape. In a normal box, gravity pulls in all directions equally. In this tube, gravity pulls differently depending on whether you are looking along the tube or across it. They calculated these new rules precisely.
The Test: They ran a simulation of our standard universe model (Lambda Cold Dark Matter) inside this new tube. They checked if the "cosmic web" (the network of galaxies and voids) formed correctly. It did. The structures looked just like they do in real life and in previous cube simulations.

What They Found

It Works: The simulation successfully created a universe with filaments, walls, and empty spaces, just like we see in the real sky.
No Hidden Bias: They checked if the tube shape made galaxies spin in a weird, specific direction (which some recent telescope data suggested might be happening). Their simulation showed that, for a tube of this size, the universe remains perfectly random and isotropic (the same in all directions). The tube shape itself did not create a fake "spin" signal.
The Trade-off: The main downside is that because the resolution changes (sharp in the middle, blurry on the edges), analyzing the data is harder. You have to be careful not to count a "blurry" galaxy as a "sharp" one.

In Summary
The authors have built a new "cosmic laboratory" shaped like a tube instead of a box. It allows them to study long, string-like structures in the universe with high precision in the center, while avoiding the artificial "boxy" effects of traditional simulations. It's a specialized tool for specific jobs, not a replacement for the old cube, but a powerful new option for understanding the shape and spin of our universe.

Technical Summary: Cylindrical Cosmological Simulations with StePS

Problem Statement
Standard cosmological $N$ -body simulations typically evolve matter within a periodic cubic box, imposing a three-torus ( $T^3 = S^1 \times S^1 \times S^1$ ) topology. While numerically convenient and preserving statistical homogeneity, this topology breaks statistical isotropy because the cubic geometry is invariant only under the octahedral group ( $O_h$ ), not the continuous rotation group $SO(3)$. Consequently, angular momentum is generally not conserved in periodic boxes, and the cubic symmetry can introduce subtle anisotropies in the gravitational field, potentially biasing large-scale clustering statistics, particularly when the simulation volume is not significantly larger than the scales of interest. Furthermore, standard periodic boundaries are ill-suited for simulating systems with inherent cylindrical symmetries, such as cosmic filaments or specific anisotropic cosmological models.

Methodology
The authors introduce a compactified simulation framework based on the $S^1 \times \mathbb{R}^2$ (slab) topological manifold. In this geometry, the simulation domain is periodic along a single axis (the $z$ -axis, with period $L_z$ ) but open in the radial direction ( $\mathbb{R}^2$ ).

Geometric Compactification: The authors employ an inverse stereographic projection along the radial direction to compactify the non-compact $\mathbb{R}^2$ plane. This maps the infinite radial coordinate to a finite domain, allowing for a "zoom-in" strategy where spatial and mass resolution are high near the central axis and decrease smoothly toward the outer boundary. This suppresses edge artifacts at the free outer boundary.
Equations of Motion: The Newtonian equations of motion are solved in comoving coordinates. The gravitational force calculation accounts for the $S^1 \times \mathbb{R}^2$ topology by summing over periodic images along the $z$ -axis. To handle the slow convergence of direct summation, the authors implement an Ewald summation method adapted for cylindrical geometry, splitting the force into real-space and reciprocal-space components.
Force Calculation Algorithms: The implementation within the StePS (STEreographically Projected cosmological Simulations) framework supports two force calculation methods:
1. Direct $O(N^2)$ : Maps efficiently to GPUs, suitable for smaller particle counts or smaller cylindrical volumes.
2. Octree $O(N \log N)$ : Implemented for CPU clusters. To mitigate systematic errors arising from the cubic nature of the octree grid in a cylindrical domain, the code applies random shifts and rotations to the domain at every timestep. A critical calibration step corrects for radial force errors to prevent spurious radial collapse or expansion, ensuring background density stability.
Initial Conditions: Initial conditions are generated using a "glass" particle distribution created via repulsive gravity in the $S^1 \times \mathbb{R}^2$ topology. Perturbations are imprinted using the Zel'dovich approximation or second-order Lagrangian perturbation theory (2LPT), adapted to the cylindrical geometry and the non-uniform sampling of the zoom-in setup. The power spectrum is estimated using the Feldman–Kaiser–Peacock (FKP) estimator to account for the non-trivial window function.

Key Contributions

Novel Topology Implementation: The paper presents the first implementation of cosmological $N$ -body simulations in the $S^1 \times \mathbb{R}^2$ manifold, extending the StePS code beyond its previous $T^3$ and spherical $\mathbb{R}^3$ capabilities.
Conservation Laws: The authors demonstrate that in this topology, the Lagrangian is invariant under translation along the cylinder axis and rotation in the transverse plane ($SO(2)$). Consequently, linear momentum along the $z$ -axis and angular momentum along the same axis are conserved, addressing the symmetry-breaking issues of the $T^3$ topology.
Algorithmic Adaptations: The work details the necessary modifications to force calculation (cylindrical Ewald summation), initial condition generation (cylindrical glass), and halo finding (a custom spherical-overdensity finder adapted for $S^1$ periodicity and radial resolution variation).
Validation: A full $\Lambda$ CDM simulation is performed to validate the method, demonstrating the formation of a cosmic web consistent with standard cosmological expectations.

Results

Simulation Execution: A simulation was run with $2.4 \times 10^7$ particles in a cylinder of height $L_z = 1.0$ Gpc and radius $R_{sim} = 500$ Mpc, using Planck 2018 $\Lambda$ CDM parameters.
Structure Formation: The simulation successfully reproduces the non-linear cosmic web (voids, walls, filaments, and halos). The zoom-in strategy allows for high-fidelity resolution of low-mass halos near the central axis, while larger volumes resolve more massive halos at greater radii.
Power Spectra: The estimated matter power spectra agree well with linear CAMB predictions on large scales and the non-linear Mead et al. (2021) "halofit" emulator on small scales, confirming the method's accuracy in capturing structure growth.
Halo Properties: The authors identified $\sim 1.1 \times 10^5$ dark matter halos at $z=0$ and calculated 68 parameters for each, including mass, concentration, and angular momentum.
Anisotropy Test: Motivated by recent claims of galaxy spin anisotropy, the authors tested the distribution of halo spin axes relative to the periodic $z$ -axis. Using Kolmogorov-Smirnov tests, they found that for $L_z = 1.0$ Gpc, the spin distribution is consistent with statistical isotropy. They conclude that this topology, at this scale, does not naturally explain reported asymmetries in galaxy spins.

Significance and Claims
The paper claims that the $S^1 \times \mathbb{R}^2$ topology offers a specialized alternative to standard cubic simulations, particularly for studying systems with cylindrical symmetries (e.g., filaments) or models where the conservation of angular momentum along a specific axis is physically relevant. The authors emphasize that while the method introduces distinct systematics—such as radially varying resolution and selection effects—it provides a self-consistent framework free from the cubic anisotropy of $T^3$ boxes.

The authors are modest in their claims, stating they do not expect this algorithm to replace well-tested cubic $T^3$ simulations for general-purpose cosmology. Instead, it serves as a complementary tool for specific scientific questions where cubic symmetry is a limiting factor. The work highlights that the anisotropic force field of the topology can imprint artifacts if the periodic length is too small ( $L_z \lesssim 200$ Mpc), a topic reserved for future detailed characterization. The code is released as open-source (StePS v2) to facilitate further exploration of these geometries.

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