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STEPSIC: Initial condition generator for stereographic cosmological simulations

This paper introduces STEPSIC, an open-source initial condition generator that extends Lagrangian perturbation theory to non-periodic stereographic and arbitrary aspect ratio domains, enabling accurate, variance-reduced cosmological simulations for the StePS N-body code with sub-percent agreement against standard periodic methods.

Original authors: Balázs Pál, Gábor Rácz, István Csabai, István Szapudi

Published 2026-05-12

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CC BY 4.0

Original authors: Balázs Pál, Gábor Rácz, István Csabai, István Szapudi

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 trying to simulate the entire history of the universe, from the Big Bang to today, on a computer. To do this, scientists need to create a "starting line" for the simulation: a digital snapshot of the early universe filled with tiny particles representing matter.

For decades, the standard way to do this has been like building a giant, invisible cube and filling it with particles. The walls of this cube are magical: if a particle hits the left wall, it instantly reappears on the right. This is called a "periodic box." It's easy to build and works well for many things, but it's not perfect. It forces the universe to repeat itself in a grid, which can create fake patterns, and it doesn't allow for a "center" or a specific viewpoint, which is crucial if you want to simulate what an observer (like us) actually sees.

Enter StePS, a new type of simulation software that tries to fix these problems. Instead of a cube, StePS uses a sphere or a cylinder. It's like looking at the universe through a fisheye lens: things near the center (where the observer is) are shown in high detail, while things far away are squished together. This is much more realistic for studying how the universe looks from Earth, but it breaks the old rules. The standard "cube" tools can't build starting conditions for these round or cylindrical shapes.

Enter stepsic: The New Architect

This paper introduces stepsic, a new open-source tool designed specifically to build these tricky starting conditions for the StePS software. Think of stepsic as a specialized 3D printer that can now print starting points for spheres and cylinders, not just cubes.

Here is how it works, broken down into simple concepts:

1. The "Fisheye" Challenge (The Geometry)

In a normal cube, every part of the space is the same size. In the StePS sphere, the center is high-definition, but the edges are low-resolution (like a map where the city center is huge and the countryside is tiny).

The Problem: If you just throw particles randomly into this sphere, the ones in the center would be too heavy compared to the ones on the edge, or vice versa.
The stepsic Solution: It uses a "concentric shell" method. Imagine an onion. stepsic builds layers of the onion. The inner layers (high detail) get many tiny particles. The outer layers (low detail) get fewer, heavier particles. It carefully calculates the weight of each particle so the total amount of "stuff" in the universe is correct, even though the resolution changes.

2. The "Blueprint" (The Physics)

Before the simulation starts, the universe isn't just a random mess; it has a specific pattern of clumps and voids based on the laws of physics.

The Process: stepsic starts with a "white noise" field (like static on an old TV). It then uses a mathematical recipe called Lagrangian Perturbation Theory (LPT) to turn that static into a realistic pattern of clumps.
The Analogy: Imagine you have a flat sheet of rubber with a random pattern drawn on it. LPT is the set of instructions that tells you exactly how to stretch, pull, and twist that rubber sheet so it forms the specific shapes of galaxies and voids we expect to see.
The Upgrade: The paper shows that stepsic uses a "second-order" version of these instructions. This is like giving the rubber sheet a more precise twist. If you only use the basic instructions (first-order), the simulation starts with a slight "glitch" or extra energy that takes a long time to fade away. Using the advanced instructions fixes this glitch immediately.

3. The "Translation" (Interpolation)

The math happens on a grid (like graph paper), but the particles are individual dots.

The Problem: The grid tells you where the "force" is, but the particles need to know exactly where to move.
The stepsic Solution: It uses a technique called "B-spline interpolation." Imagine the grid is a trampoline. If you push down on the center, the whole trampoline moves, but the center moves the most. stepsic calculates exactly how much each particle should move based on how close it is to the grid points, ensuring a smooth transition.

4. Did it Work? (The Proof)

The authors tested stepsic rigorously:

The Cube Test: They ran it on a standard cube and compared it to other famous tools. The results matched almost perfectly (within 0.5% error), proving the math is solid.
The Shape Test: They tried making "slab" shapes (very flat boxes) and found the tool worked just as well, proving it doesn't get confused by weird shapes.
The Cylinder Test: They ran a full simulation in a cylindrical shape (like a soda can) and compared a "basic" start vs. an "advanced" start. The advanced start (using the second-order math) removed the 2–3% "glitch" that usually plagues these simulations, confirming the physics is correct.

Summary

In short, stepsic is a new tool that allows cosmologists to stop using the "box" model of the universe and start using "sphere" and "cylinder" models that better match how we actually observe the cosmos. It handles the tricky math of changing particle sizes across the simulation and ensures the starting conditions are physically accurate, removing common errors that have plagued simulations for years. It's a bridge that lets scientists simulate the universe the way it really looks, rather than the way it's easiest to calculate.

Technical Summary of stepsic: Initial condition generator for stereographic cosmological simulations

Problem Statement
Conventional cosmological initial condition (IC) generators are designed exclusively for fully periodic cubic domains ( $T^3$ topology). While successful for standard structure formation studies, the $T^3$ topology introduces artificial correlations at scales comparable to the box size and fails to conserve angular momentum due to a lack of invariance under the full rotation group $SO(3)$. These limitations render standard tools unsuitable for non-periodic, observer-centric simulations or phenomena requiring rotational invariance. Specifically, the StePS (STEreographically Projected cosmological Simulations) code, which utilizes inverse stereographic projection to map the $R^3$ manifold into finite spherical ( $R^3$ ) and cylindrical ( $S^1 \times R^2$ ) domains with radially varying particle resolution, lacks a compatible public IC generator. Existing codes (e.g., 2lptic, music, monofonic) cannot produce the necessary non-standard geometries or handle the intrinsic mass resolution gradients of stereographic projections.

Methodology
The paper presents stepsic, an open-source initial condition generator designed to bridge this gap. The code extends Lagrangian Perturbation Theory (LPT) to support spherical, cylindrical, and arbitrary aspect-ratio cuboid domains. The technical workflow involves:

Particle Load Generation: stepsic supports four loading strategies:
- Regular Lattice: Cubic voxels for periodic $T^3$ domains.
- Poisson Distribution: Homogeneous random sampling for testing or glass generation.
- Concentric Shells/Annuli: Specific to StePS geometries, filling the volume with shells (spherical) or annuli (cylindrical) containing fixed particle counts. This creates the radially varying mass resolution intrinsic to stereographic projections. Two binning modes are supported: constant stereographic angle ( $\Delta\omega$ ) and constant compact-space volume. A constant-resolution inner zone is defined to mitigate force accuracy degradation caused by interacting particles of vastly different masses.
- Glass Inputs: Capability to read and rescale externally generated glass configurations.
Field Construction:
- Fourier Grid: Anisotropy-free Fourier grids with cubic voxels are constructed. For non-cubical domains, the grid is conditioned by the box geometry, which modifies the ensemble of accessible modes (suppressing variance along short axes) rather than simply cropping a larger isotropic realization.
- Density Fields: Gaussian random fields are generated using white noise, rescaled to the target linear matter power spectrum $P(k, z)$ at the initial redshift. The code supports standard realizations, paired realizations (phase-shifted), and fixed-amplitude realizations to reduce cosmic variance.
- LPT Implementation: The code implements first-order (Zel'dovich approximation) and second-order (2LPT) Lagrangian perturbation theory. Displacement and velocity fields are computed in Fourier space and transformed to physical space. The implementation follows conventions established by 2lptic and monofonic, ensuring compatibility with standard workflows while extending to non-standard topologies.
Interpolation and Multiresolution:
- Displacement and velocity fields are interpolated onto particle positions using B-spline mass-assignment kernels (NGP, CIC, TSC) with Fourier-space deconvolution to correct for smoothing.
- A multiresolution scheme maps displacement fields across the radially varying mass resolution. Particles are assigned displacement fields from grids of varying resolution corresponding to their mass, ensuring that high-mass particles (representing larger volumes) do not receive unresolved small-scale fluctuations that would act as noise.

Key Results and Validation
The authors validate stepsic through three levels of testing: reproducibility, numerical correctness, and statistical agreement with theory and other codes.

Power Spectrum Recovery: In periodic cubic boxes, the recovered matter power spectrum agrees with the input linear theory prediction to better than 0.5% up to half the Nyquist wavenumber. This accuracy holds independent of the box aspect ratio (tested up to 10:1).
Cross-Validation: When compared against the established monofonic code using identical white noise fields and cosmological parameters, stepsic achieves sub-percent power spectrum agreement. A small residual offset ( $\approx 0.6\%$ ) is attributed to differences in the implementation of linear growth factor calculations (using the colossus library in stepsic vs. an internal solver in monofonic).
Transient Suppression: Full N-body evolution of matched cylindrical StePS runs confirms that second-order LPT correctly suppresses the 2–3% transient power excess present in first-order initial conditions at quasi-linear scales.
Geometric Consistency: Tests on slab geometries demonstrate that the cubic-voxel Fourier grid construction introduces no geometry-dependent systematic bias for aspect ratios up to 10:1, aside from the expected reduction in sample variance for smaller volumes.

Significance and Claims
The paper positions stepsic as the first public initial condition generator capable of supporting the non-standard geometries ( $R^3$ and $S^1 \times R^2$ ) and radially varying mass resolutions required by the StePS code. By successfully extending LPT to these domains and validating the output against theoretical expectations and existing codes, the authors demonstrate that accurate, observer-centric cosmological simulations can be performed without the artificial correlations inherent in periodic $T^3$ topologies. The code is presented as a necessary tool for studying filamentary structures and anisotropic cosmological models within the StePS framework. The authors modestly note that while third-order LPT and primordial non-Gaussianity are planned for future releases, the current implementation focuses on providing a robust, second-order foundation for stereographic simulations.

1. The "Fisheye" Challenge (The Geometry)

2. The "Blueprint" (The Physics)

3. The "Translation" (Interpolation)

4. Did it Work? (The Proof)

Summary

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