Zeldovich approximation

The Zeldovich 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 $\mathbf{r}=a[\mathbf{q}+b(t)(p(q_1),0,0)]$, put $J=1+b{p}^{\prime }$. Before shell crossing, mass conservation gives $\rho =\bar{\rho }/J$. The dust background equation $3\ddot{a}/a=-4\pi G\bar{\rho }$ and the growing-mode equation $\ddot{b}+2H\dot{b}=4\pi G\bar{\rho }b$ imply ${\nabla }_r\cdot \ddot{\mathbf{r}}=-4\pi G\bar{\rho }+4\pi G\bar{\rho }b{p}^{\prime }/J=-4\pi G\rho$. Thus planar Zeldovich approximation trajectories satisfy the full nonlinear dust force-divergence equation until the map loses invertibility. All three physical coordinates are included in the divergence; the two unperturbed directions still contribute background acceleration.

The cosmological Lagrangian displacement sends an initially labelled matter element from its Lagrangian coordinate $\mathbf{q}$ to $\mathbf{x}(\mathbf{q},t)=\mathbf{q}+\mathbf{\psi }(\mathbf{q},t)$. Its time derivative determines the peculiar velocity through $\mathbf{v}=a\dot{\mathbf{\psi }}$.

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.

For the Zeldovich approximation $\mathbf{x}=\mathbf{q}-b{\nabla }_q\psi$, let ${\lambda }_i$ be the eigenvalues of the potential Hessian, with this sign convention. Before shell crossing, mass conservation gives the displayed density. The first positive eigenvalue reaches $b{\lambda }_i=1$ and collapses one direction, creating a sheetlike cosmological caustic. Extrapolating straight trajectories after crossing lets particles pass through without self-consistent gravitational trapping, so the predicted multistream sheet spreads excessively. Collisionless gravitational evolution keeps matter more tightly bound; the linear displacement extrapolation cannot determine its late thickness reliably.

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.