Here is the cosmic expansion factor and the growing rate of linear fluctuations, as provided by eq.(13). Moreover, the velocity term , which provides the particle displacement with respect to the initial (Laplacian) position, is related to the potential originated by the initially linear fluctuations, according to

In order to better visualize the meaning of eq.(15), let us consider a pressureless and viscosity-free, homogeneous medium without any gravitational interaction. For this system, the Eulerian positions of the particles at time are related to the Lagrangian positions by the linear relation

being the initial velocity. The above expression is essentially analogous to the Zeldovich approximation (15), apart from the presence of the term, which accounts for the background cosmic expansion, and of the term, which accounts for the presence of gravity, giving a deceleration of particles along the trajectories (actually, in a matter dominated Universe).

Since at density inhomogeneities are created, mass conservation
requires that
, so that the
density field as a function of Lagrangian coordinates reads

so that and we recover (the growing mode of) the linear solution.

More in general, since the expression (16) for
makes the deformation tensor a real symmetric matrix, its eigenvectors
define a set of three principal (orthogonal) axes. After
diagonalization, eq.(18) can be written in terms of its
eigenvalues
,
and
, which
give the contraction or expansion along the three principal axes:

The Zeldovich approximation predicts the first non-linear structure to arise in correspondence of the high peaks of the field and represents a significant step forward with respect to linear theory and in fact it has been successfully applied to describe the large scale clustering in the distribution of galaxy clusters.

However, within the Zeldovich prescription, after a pancake forms in correspondence of crossing of particle orbits, such particles continue travelling along straight lines, according to eq.(15). Viceversa, in the framework of a realistic description of gravitational dynamics, we expect that the potential wells, that correspond to non-linear structures, should be able to retain particles and to accrete from surrounding regions.