Post-collapse Lagrangian perturbation theory in three dimensions

TL;DR

This work introduces three-dimensional post-collapse perturbation theory (PCPT) for cold dark matter, extending the one-dimensional framework to realistic 3D pancake-like collapse. It combines high-order Lagrangian perturbation theory for the pre-collapse background with a perturbative, analytically tractable backreaction treatment derived from a reduced one-dimensional Poisson problem along the collapse axis, yielding explicit corrections to displacement and velocity shortly after shell-crossing. Validation against high-resolution ColDICE Vlasov-Poisson simulations shows PCPT accurately captures early multistream dynamics and caustic structure, outperforming standard LPT in the post-collapse regime. The approach provides a deterministic bridge between Zel'dovich flow and nonlinear multi-stream evolution, with implications for understanding the initial stages of halo formation, while acknowledging limitations related to symmetry, transverse corrections, and LPT convergence that guide future refinements.

Abstract

The gravitational collapse of collisionless matter leads to shell-crossing singularities that challenge the applicability of standard perturbation theory. Here, we present the first fully perturbative approach in three dimensions by using Lagrangian coordinates that asymptotically captures the highly nonlinear nature of matter evolution after the first shell-crossing. This is made possible essentially thanks to two basic ingredients: (1) We employ high-order standard Lagrangian perturbation theory to evolve the system until shell-crossing, and (2) we exploit the fact that the density caustic structure near the first shell-crossing begins generically with pancake formation. The latter property allows us to exploit largely known one-dimensional results to determine perturbatively the gravitational backreaction after collapse, yielding accurate solutions within our post-collapse perturbation theory (PCPT) formalism. We validate the PCPT predictions against high-resolution Vlasov-Poisson simulations and demonstrate that PCPT provides a robust framework for describing the early stages of post-collapse dynamics.

Figures (6)

• Figure 1: Schematic illustration of particle locations in the $q_{x}$-$x$ plane for $(q_{y},q_{z})$-tuples that are located within the caustic.
• Figure 2: Lagrangian phase-space slices $x(q_x,q_y)$-$v_{x}(q_x,q_y)$ for two-sine waves initial conditions. The top and bottom sets of panels correspond to Q1D-2SIN and ANI-2SIN cases, respectively. For each set of panels, time increases from top to bottom, and the fixed value of $q_y$ increases from left to right, that is, from the center to the outer part of the system. The solid blue lines represent the analytical predictions based on PCPT with a 15LPT background flow (see Appendix \ref{['sec: app LPT dependence']} for the effects of changing the LPT order $n$ on the solution), the black solid lines correspond to the measurements in the Vlasov-Poisson simulations, and the dotted lines give standard 15LPT predictions.
• Figure 3: Same as Fig. \ref{['fig: x-vx 2SIN']} but for Q1D-3SIN (top panels) and ANI-3SIN cases (bottom panels), for $q_{z}=0$.
• Figure 4: Density slices at times indicated in the figures for two- and three-sine waves initial conditions. Top-left, bottom-left, top-right, and bottom-right panels, show density slices for Q1D-2SIN, Q1D-3SIN, ANI-2SIN, and ANI-3SIN cases, respectively. In each panel, from left to right, we present the results from standard 15LPT, PCPT with a 15LPT background flow, and the Vlasov-Poisson simulations, respectively. We note that for three-sine waves initial conditions, we show the two-dimensional slice with $z=0$.
• Figure 5: Lagrangian phase-space slices $x(q_x,q_y)$-$v_{x}(q_x,q_y)$ in the 2D configurations, similarly as in Fig. \ref{['fig: x-vx 2SIN']}. Here, we test how the results depend on the LPT order $n$. From light to dark blue, PCPT is shown with the background flow modelled with 2LPT, 5LPT, 9LPT, 13LPT, and 15LPT, respectively. The standard LPT results are also displayed in grey dashed, while the black curve corresponds to the Vlasov simulations.