Zel'dovich approximation

Builds on Lagrangian and Eulerian description Jacobian determinant

Definition

Consider pressureless matter in an expanding universe, described in comoving coordinates. Label each mass element by its initial (Lagrangian) position $q \in \R^3$ . Let $x(q, t)$ be its comoving (Eulerian) position at time $t$ , as in the Lagrangian and Eulerian description. Let $D(t)$ be the growing mode of linear perturbation theory, and let $\phi(q)$ be a potential such that the linear density contrast is $\delta_L(q, t) = D(t) \nabla_q^2 \phi(q)$ . The Zel'dovich approximation is the trajectory $x(q, t) = q - D(t) \, \nabla_q \phi(q).$ The density then follows from conservation of mass: $\frac{\rho(x, t)}{\bar\rho} = \frac{1}{\abs{\det(\partial x / \partial q)}} = \frac{1}{\abs{(1 - D \lambda_1)(1 - D \lambda_2)(1 - D \lambda_3)}},$ where $\det$ is the Jacobian determinant. Here $\lambda_1 \ge \lambda_2 \ge \lambda_3$ are the eigenvalues of the deformation tensor $\partial^2 \phi / \partial q_i \partial q_j$ at $q$ .

Remarks

To first order in $D$ , the density formula gives $\rho/\bar\rho - 1 \approx D(\lambda_1 + \lambda_2 + \lambda_3) = \delta_L$ , so the approximation agrees with linear theory. The formula holds while the map $q \mapsto x(q, t)$ is one to one; the first time $D(t) \lambda_1(q) = 1$ is called shell crossing.

Example: in one dimension take $\phi(q) = -A \cos(kq)/k^2$ with $A > 0$ . Then $x = q - D A \sin(kq)/k$ and $\partial x/\partial q = 1 - D A \cos(kq)$ , so $\rho/\bar\rho = 1/(1 - DA\cos(kq))$ . At $q = 0$ the density becomes infinite when $DA = 1$ . For plane-parallel motion such as this one, the approximation is exact up to shell crossing.

Non-example: after shell crossing the map $q \mapsto x$ folds over and several streams overlap at one point. The formula then makes particles pass freely through each other, while real gravity binds them into a structure, so it no longer describes the dynamics.

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Euclidean space object
Function concept
Lagrangian and Eulerian description concept
Total derivative concept
Function of several variables concept
Partial derivative concept
Jacobian matrix concept
Jacobian determinant concept
Zel'dovich approximation definition

Tag	`Z0PVBZC8KX`
URI	`m2://Z0PVBZC8KX`
Kind	definition
Writing	W0 stub
MSC	85A40, 85A05
Defines	Zel'dovich approximation
Uses	Lagrangian and Eulerian description, Jacobian determinant
Changed	3 November 2026

Zel'dovich approximation

Definition

Remarks

Reading path

Further reading

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