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

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.