Alternative perturbation theory schemes
It should be noted that there is no single way of doing perturbation theory. Indeed, the choice of fields to represent the cosmic quantities is not unique. It is always possible to change to linearly related fields, for instance the potential instead of the density contrast, or the velocity potential instead of the velocity divergence, but this does not change the structure of the perturbation series. A more dramatic change is to introduce nonlinear transforms of the field. A straightforward example is to replace the peculiar velocity field by the momentum field, p(x, n) = p(x, n)u(x, n) (as exploited in a series of papers applied to the redshift space distortions, starting with Seljak and McDonald 2011). This makes the continuity equation very simple, the divergence of p is the time derivative of the density to all orders, but it makes the Euler equation more cumbersome to manipulate. In particular, the momentum field is no longer potential to all orders.
An even more dramatic transformation is a change of the coordinate system itself. This is what the Lagrangian approach does. This is a very popular, and useful, approach that we present in more detail in the following.