A New Recipe for Caustic Pancakes: On the Reality of Walls in the Cosmic Web

TL;DR

This work develops and applies a rigorous phase-space caustic-skeleton framework to identify and study cosmic walls, the first collapsed structures in the Zel'dovich pancake picture, within fully constrained 3D simulations. It introduces a novel wall-centre constraint anchored to cusp-sheet geometry, translates it into primordial-field derivatives via an eigenframe transformation, and implements a two-step sampling+BHR procedure to generate constrained initial conditions. Through extensive simulations, it characterises wall formation times, Lagrangian-area densities, and the embedded halo population, revealing wall-dominated, three-streaming planar regions interconnected within a scale-space caustic network and showing filamentary halo patterns consistent with observed walls. By contrast, conventional saddle-point constraints on the primordial potential or density perturbation fail to robustly produce realistic walls, underscoring the necessity of phase-space-based constraints. The results advance a physically grounded, predictive description of cosmic walls and their role in the cosmic web, with implications for halo statistics, mass transport, and Local Universe reconstructions.

Abstract

The caustic skeleton model is a mathematically rigorous framework for studying the formation history of the emerging cosmic web from the caustics in the underlying dark matter flow. In a series of two papers, we use constrained N-body simulations to investigate the different cosmic web environments. For the current study, we focus on the cosmic walls. We derive the conditions of the centres of proto-walls and analyse their evolution with N-body simulations. Next, we investigate the statistical properties of Zel'dovich pancakes by studying the number density of the cosmic wall centres in scale space and, for the first time, we calculate the Lagrangian-space volume of cosmic walls. Finally, we infer the mean density and velocity fields and the distribution of haloes around cosmic walls with a suite of physically realistic dark-matter-only simulations. We compare the cosmic walls obtained with the caustic skeleton framework with previously proposed saddle point conditions on the primordial potential and density perturbation.

Figures (30)

• Figure 1: High-resolution density field from a $256^3$-particle constrained simulation of cosmic wall formation from the caustic skeleton model. Shown is a volume of side length $50 \,h^{-1}\textrm{Mpc}$, with the large-scale cusp sheets, swallowtail and umbilic filaments and butterfly clusters in red, blue, green and yellow respectively. In the central white point we imposed the novel constraint presented in this paper, which hence resides at the centre of the simulated cosmic wall. A fly-through and rotating view animation is provided on the additional materials page at .
• Figure 2: Haloes identified in a high-resolution constrained simulation of cosmic wall formation. The left panel shows the density field sliced along the face of the wall, along with the haloes identified in a thin volume (thickness $\epsilon = 1.5 \,h^{-1}\textrm{Mpc}$) around the wall, projected into the same slice. The right panel shows the caustic skeleton evaluated at different length scales $\sigma$ (see \ref{['subsec:theory-scale_space']}), with the $A_3$ cusp walls in red, the $A_4$ swallowtail filaments in blue and the $D_4$ umbilic filaments in green respectively.
• Figure 3: Different views of the cosmic web from slices of width $70\,h^{-1 }\textrm{Mpc}$ and height $20\,h^{-1 }\textrm{Mpc}$ slices through a $256^3$-particle $N$-body simulation in a box of side length $100\,h^{-1 }\textrm{Mpc}$. The upper panel shows the density field, the middle shows a slice of through the folding particle mesh and the lower panel shows corresponding the number of streams coming into the Eulerian positions. The upper and lower panels were evaluated using the PS-DTFE method.
• Figure 4: Comparison of the density estimates for zoomed region of the simulation of \ref{['fig:sim_256_triple_plot']} with the ordinary DTFE method (left panel) and the PS-DTFE method (middle panel). The corresponding number of streams evaluated from the PS-DTFE method is displayed in the right panel.
• Figure 5: Caustics making up the structural elements of the cosmic web, illustrated using a 2D mesh simulation of cosmic structure formation. The left panel shows a multistream region (Zel'dovich pancake) bisected by the cusp line, corresponding to a wall in the 3D cosmic web. The middle panel shows two swallowtail caustics emerging from a cusp sheet folding onto itself. The right panel shows the umbilic caustic, corresponding to a filamentary junction of three incoming walls. For an animation of the Zel'dovich pancake formation in a 2D mesh simulation, see .

Theorems & Definitions (2)

• Theorem 1
• proof