In Section 21 we were able to get some analytical understanding of linear perturbation growth. Subsequent sections used a combination of techniques, involving graininess, energy principles, numerical simulations and thermodynamics, to determine non-linear properties of clustering. In this section, we extend the general approach of Section 21 into the non-linear regime. Our aim is to see how long it takes a perturbation to begin contracting, to break away from the expansion of the surrounding universe.

If we were to extend the detailed linear Fourier perturbation technique of Section 21 into the non-linear regime, in a brute force fashion, it would rapidly become too complicated to provide much insight. Instead of that Eulerian technique it becomes simpler to take a Lagrangian point of view, as in the discussion of ‘pancakes’ in Section 35.

The Lagrangian technique follows the motion of a particular object, or set of objects, in the perturbation. (By contrast the Eulerian technique describes what happens as a function of spatial position). A simple inhomogeneity, whose growth typifies many more complicated inhomogeneities to order of magnitude, is just a spherical region of constant density in an otherwise uniformly expanding universe. If the inhomogeneity is to contract, and not merely expand more slowly than the rest of the universe, its density must be greater than the critical density ρc, just necessary to close the universe.