Lagrangian theory of gravitational instability of Friedman-Lematre cosmologies and the 'Zel'dovich approximation'

The aim of this paper is to clarify the connection between the so-called 'Zel'dovich approximation' and perturbative solutions of the Euler–Poisson system for the motion of a self-gravitating dust continuum, evaluated in the Lagrangian picture of fluid dynamics. Solutions of the Lagrangian equations, linearized at an isotropically expanding background, are derived. This approximation is investigated; it contains the 'Zel'dovich approximation' as well as a generalized form of it as subclasses, and allowed treatment of the non-linear evolution of vortical perturbations consistently within the framework of self-gravitating motions. In contrast to the prediction of the standard linear approximation, vorticity is coupled to the density enhancement and is amplified in the present approximation. The transport equation for vorticity is examined and applied. Besides the generalization to vortical flows, this work gives a straightforward derivation and consistent definition of Zel'dovich's approach. Relations to the fully non-linear equations are discussed in terms of the eigenvalues of the velocity gradient of the inhomogeneous deformation. In particular, the approximation, although based on linearized equations, provides the exact solution for the evolution of plane-symmetric inhomogeneities as well as a class of three-dimensional solutions, as was shown in an earlier paper.