# Shell-crossing in quasi-one-dimensional flow

## TL;DR

This paper develops a perturbative approach in Lagrangian coordinates to analyze shell-crossing in quasi-one-dimensional cosmological flows, providing a way to predict the timing and location of singularities more accurately than in purely 1D models.

## Contribution

It introduces an all-order recursion relation for displacement field coefficients, demonstrating the convergence of the series and enabling precise prediction of shell-crossing in Q1D flows.

## Key findings

- All-order recursion relations derived for displacement field coefficients.
- The Taylor series for shell-crossing has an infinite radius of convergence.
- Shell-crossing occurs earlier in Q1D flows than in pure 1D models.

## Abstract

Blow-up of solutions for the cosmological fluid equations, often dubbed shell-crossing or orbit crossing, denotes the breakdown of the single-stream regime of the cold-dark-matter fluid. At this instant, the velocity becomes multi-valued and the density singular. Shell-crossing is well understood in one dimension (1D), but not in higher dimensions. This paper is about quasi-one-dimensional (Q1D) flow that depends on all three coordinates but differs only slightly from a strictly 1D flow, thereby allowing a perturbative treatment of shell-crossing using the Euler--Poisson equations written in Lagrangian coordinates. The signature of shell-crossing is then just the vanishing of the Jacobian of the Lagrangian map, a regular perturbation problem. In essence the problem of the first shell-crossing, which is highly singular in Eulerian coordinates, has been desingularized by switching to Lagrangian coordinates, and can then be handled by perturbation theory. Here, all-order recursion relations are obtained for the time-Taylor coefficients of the displacement field, and it is shown that the Taylor series has an infinite radius of convergence. This allows the determination of the time and location of the first shell-crossing, which is generically shown to be taking place earlier than for the unperturbed 1D flow. The time variable used for these statements is not the cosmic time $t$ but the linear growth time $\tau \sim t^{2/3}$. For simplicity, calculations are restricted to an Einstein--de Sitter universe in the Newtonian approximation, and tailored initial data are used. However it is straightforward to relax these limitations, if needed.