The equations of motion are nonlinear, and we have only solved them in the limit of linear perturbations. We now discuss evolution beyond the linear regime, first for a few special density models, and then considering full numerical solution of the equations of motion.

**3.1. The Zeldovich approximation**

Zeldovich (1970)
invented a *kinematical* approach to the
formation of structure. In this method, we work out the
initial displacement of particles and assume
that they continue to move in this initial direction.
Thus, we write for the proper coordinate of a given particle

(55) |

This looks like Hubble expansion with some
perturbation, which will become negligible as
*t*
0. The coordinates
**q** are therefore equal to the usual
comoving coordinates at *t* = 0, and *b*(*t*) is a function
that scales the time-independent displacement field **f(q)**.
In fluid-mechanical terminology, **x** is said to be the Eulerian
position, and **q** the Lagrangian position.

To get the Eulerian density, we need the Jacobian
of the transformation between **x** and **q**,
in which frame
is constant. This strain tensor is symmetric, provided
we assume that the density perturbation originated from a
growing mode. The displacement field is then irrotational, so that
we can write it in terms of a potential

(56) |

The strain tensor is thus characterized by its three eigenvalues, and the density becomes infinite when the most negative eigenvalue reaches -1.

If we linearize the density relation, then the relation to density perturbations is

(57) |

This is first-order Lagrangian perturbation theory, in contrast to the
earlier approach, which carried out perturbation theory in
Eulerian space (higher-order Lagrangian theory is discussed by
Bouchet et al. 1995).
When the density fluctuations are small, a first-order treatment from
either point of view should give the same result. Since the
linearized density relation is
= - (*b* /
*a*)
^{.} **f** , we can tell immediately that
[*b*(*t*) / *a*(*t*)] = *D*(*t*), where
*D*(*t*) is the linear density growth law.
Without doing any more work, we therefore know that the first-order
form of Lagrangian perturbations must be

(58) |

so that *b*(*t*) = *a*(*t*)*D*(*t*).
The advantage of the Zeldovich approximation is
that it normally breaks down later than
Eulerian linear theory - i.e. first-order
Lagrangian perturbation theory can give results
comparable in accuracy to Eulerian theory with higher-order
terms included. This method is therefore commonly used to set up
quasi-linear initial conditions for *N*-body
simulations, as discussed below.
The same arguments that we used earlier in discussing peculiar
velocities show that the growing-mode
comoving displacement field **f** is parallel to **k**
for a given Fourier mode, so that

(59) |

Given the desired linear density mode amplitudes, the corresponding displacement field can then be constructed.