On the shape of pancakes: catastrophe theory and Gaussian statistics in 2D

TL;DR

This work develops a 2D catastrophe-theory framework to understand pancake formation at shell-crossing in the cosmic web, using a Taylor expansion around the shell-crossing point and Lagrangian perturbation theory to derive the $A_3^+$ spine and the $A_2$ caustic that seed pancakes. It couples this morphology with Gaussian statistics of the displacement potential under Zel'dovich flow to predict distributions of observable pancake features—shell-crossing times, curvature of the spine, C-to-S transition scale, and axis ratios—across pancake, halo, and filament populations. Key results show pancakes are predominantly C-shaped with flat, anisotropic $A_2$ boundaries, halos exhibiting higher spine curvature and earlier shell-crossing than filaments, and distinct trends with the power-spectrum parameters $n$ and smoothing scale $R_c$. The framework is validated against 1-LPT simulations and offers a path to extending to 3D and testing for non-Gaussian signatures in the observed cosmic web.

Abstract

Cold dark matter (CDM) can be thought of as a 2D (or 3D) sheet of particles in 4D (or 6D) phase-space due to its negligible velocity dispersion. The large-scale structure, also called the cosmic web, is thus a result of the topology of the CDM manifold. Initial crossing 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 employ catastrophe theory in 2D to study the motion around such singularities and analytically model the shape of the emerging structures, particularly the pancakes, which later evolve into halos and filaments-the building blocks of the 2D web. We compute higher-order corrections to the shape of the pancakes, including properties such as the curvature and the scale of transition from their C to S shape. Using Gaussian statistics (with the assumption of Zeldovich flow) for our model parameters, we also compute the distributions of observable features related to the shape of pancakes and their variation across halo and filament populations in 2D cosmologies. We find that a larger fraction of pancakes evolve into filaments, they are more curved if they are to evolve into halos, are dominantly C-shaped, and the nature of shell-crossing is highly anisotropic. Extending this work to 3D will allow testing of predictions against actual observations of the cosmic web and searching for signatures of non-Gaussianity at corresponding scales.

Figures (15)

• Figure 1: Illustration of 2D shell-crossing in Eulerian $x-y$ space (top row) along with an $q_x$-axis slice through the shell-crossing point in $q-x$ space (bottom row) to help visualize the folding of the CDM sheet. The left column depicts a snapshot during the single stream flow prior to shell-crossing. The middle column shows the moment of shell-crossing. It is an event of catastrophe marked by the appearance of an $A_3^+$ singularity (red dot) that seeds the 2D pancake-like structure, shown in the right column. The boundary of the pancake encompassing the multi-streaming region is defined by $A_2$ singularities (blue), which grows along the $A_3^+$ spine (green).
• Figure 7: Marginal probability densities of $\phi^{2,0}$ (left) and $\phi^{0,2}$ (right) scaled by $\sigma_2$.
• Figure 8: Marginal probability densities of $\phi^{1,2}$ (left) and $\phi^{0,3}$ (right) scaled by $\sigma_3$. Eq. \ref{['eq:MP_phi03']} is shown in dashed black curve for comparison.
• Figure 9: Marginal probability densities of $\phi^{4,0}$ (solid) and $\phi^{2,2}$ (dashed) in the top-left panel, $\phi^{3,1}$ in the top-right panel, $\phi^{1,3}$ in the bottom-left panel, and $\phi^{0,4}$ in the bottom-right panel, all scaled by $\sigma_4$.
• Figure 10: The $A_2$ caustic (solid lines) and the $A_3$ spine (dashed lines) of the average pancake at $D_+(t)-D_+(t_c) \in \{ 0.001, 0.05, 0.1, 0.15 \}$ in Lagrangian (left) and Eulerian (right) spaces. The $x$-axis in Eulerian space has been zoomed 200% to make the features distinguishable.