Shell-crossing in quasi-one-dimensional flow
  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.
