The shape of pancakes: Catastrophe theory and Gaussian statistics in 2D

¹ Sorbonne Université, CNRS, UMR7095, Institut d’Astrophysique de Paris, 98bis boulevard Arago, F-75014 Paris, France
² Institute for Advanced Research, Nagoya University, Furo-cho Chikusa-ku, Nagoya 464-8601, Japan
³ Kobayashi-Maskawa Institute for the Origin of Particles and the Universe, Nagoya University, Chikusa-ku, Nagoya 464-8602, Japan
⁴ Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
⁵ Kavli Institute for the Physics and Mathematics of the Universe (WPI), Todai institute for Advanced Study, University of Tokyo, Kashiwa, Chiba 277-8568, Japan
⁶ Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, Seoul 02455, Republic of Korea

^★ Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 30 September 2025
Accepted: 15 April 2026

Abstract

Cold dark matter (CDM) can be thought of as a 2D (or 3D) sheet of particles in 4D (or 6D) phase-space because its velocity dispersion is negligible. The large-scale structure, also called the cosmic web, is thus a result of the topology of the CDM manifold. Initial crossings of particle trajectories occurs at the critical points of this manifold, forming singularities that seed most of the collapsed structures. The cosmic web can thus be characterized using the points of singularities. In this context, we employed catastrophe theory in 2D to study the motion around such singularities and to analytically model the shape of the emerging structures, particularly the pancakes, which later evolve into halos and filaments that are the building blocks of the 2D web. We computed higher-order corrections to the shape of the pancakes, including properties such as the curvature and the scale of transition of their shapes from C to S. Using Gaussian statistics (with the assumption of a Zeldovich flow) for our model parameters, we also computed the distributions of observable features related to the shape of pancakes and their variation across halo and filament populations in 2D cosmologies. We found that a larger fraction of pancakes evolves into filaments, they are more strongly curved when they evolve into halos, are dominantly C-shaped, and the nature of the shell crossing is highly anisotropic. An extension of this work to 3D will allow us to test the predictions against actual observations of the cosmic web and to search for signatures of non-Gaussianity at corresponding scales.