Zel'dovich approximation

The Zel'dovich approximation maps a fluid element from its Lagrangian coordinate $\mathbf{q}$ to the comoving position $\mathbf{x}=\mathbf{q}+D(t)\mathbf{s}(\mathbf{q})$, where $D$ is the linear growth factor. It follows the initial gravitational displacement along a straight comoving trajectory until trajectories cross.

For an irrotational displacement $\mathbf{s}=-{\nabla }_q\Phi$, the deformation tensor is the symmetric Hessian $d_{ij}=-{\partial }_i{\partial }_j\Phi$. Its real eigenvalues determine the principal collapse directions.

Shell crossing occurs when the Lagrangian-to-Eulerian map ceases to be one-to-one. Its Jacobian determinant vanishes, the pressureless single-stream density formally diverges, and several streams subsequently occupy the same position.

A cosmological caustic is the image of a singular point of the Lagrangian map. The ideal collisionless density diverges there, although velocity dispersion and other small-scale physics regularize the singularity.