Zel'dovich pancake

Builds on Zel'dovich approximation Caustic

Definition

Consider the Zel'dovich approximation $x(q, t) = q - D(t) \nabla_q \phi(q)$ , with growth factor $D(t)$ increasing in $t$ . Let $\lambda_1(q)$ be the largest eigenvalue of the deformation tensor $\partial^2 \phi / \partial q_i \partial q_j$ at $q$ , and assume that its maximum over $q$ is positive, attained at $q_0$ . At the time $t_0$ with $D(t_0) \lambda_1(q_0) = 1$ , the factor $1 - D \lambda_1$ of the density formula vanishes at $q_0$ , and the density becomes infinite there. The Zel'dovich pancake is the thin sheet of high density that forms around $x(q_0, t_0)$ at and after $t_0$ . In it, matter has been compressed along the eigendirection of $\lambda_1$ . Along the two other directions it is still expanding or slowly contracting.

Remarks

The surface where the density is formally infinite is a caustic of the map $q \mapsto x(q, t)$ . For a Gaussian random initial field, three equal eigenvalues occur with probability zero, so the first collapse is generically one-dimensional.

Example: in one dimension with $\phi(q) = -A\cos(kq)/k^2$ and $A > 0$ , the deformation is $\lambda_1(q) = A \cos(kq)$ , which is maximal at $q_0 = 0$ . At $D = 1/A$ the density $1/(1 - DA\cos(kq))$ is infinite at $q = 0$ , and in three dimensions this is a plane sheet.

Non-example: a spherically symmetric overdensity has $\lambda_1 = \lambda_2 = \lambda_3$ at its centre. All three factors vanish at once, so it collapses to a point and does not form a pancake.

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Tag	`CME3WQDFKN`
URI	`m2://CME3WQDFKN`
Kind	definition
Writing	W0 stub
MSC	85A40
Defines	Zel'dovich pancake
Uses	Zel'dovich approximation, Caustic
Changed	3 November 2026

Zel'dovich pancake

Definition

Remarks

Reading path

Further reading

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