We investigate three different local approximations for nonlinear gravitational instability in the framework of cosmological Lagrangian fluid dynamics of cold dust. By local we mean that the evolution is described by a set of ordinary differential equations in time for each mass element, with no coupling to other mass elements aside from those implied by the initial conditions. We first show that the Zel'dovich approximation (ZA) can be cast in this form. Next, we consider extensions involving the evolution of the Newtonian tidal tensor. We show that two approximations can be found that are exact for plane-parallel and spherical perturbations. The first one (" nonmagnetic " approximation, or NMA) neglects the Newtonian counterpart of the magnetic part of the Weyl tensor in the fluid frame and was investigated previously by Bertschinger & Jain. A new approximation ("local tidal," or LTA) involves neglecting still more terms in the tidal evolution equation. It is motivated by the analytic demonstration that it is exact for any perturbations whose gravitational and velocity equipotentials have the same constant shape with time. Thus, the LTA is exact for spherical, cylindrical, and plane-parallel perturbations. It corresponds physically to neglecting the curl of the magnetic part of the Weyl tensor in the comoving threading as well as an advection term in the tidal evolution equation. All three approximations can be applied up to the point of orbit crossing. We tested them in the case of the collapse of a homogeneous triaxial ellipsoid, for which an exact solution exists for an ellipsoid embedded in empty space and an excellent approximation is known in the cosmological context. We find that the LTA is significantly more accurate in general than the ZA and the NMA. Like the ZA, but unlike the NMA, the LTA generically leads to pancake collapse. For a randomly chosen mass element in an Einstein-de Sitter universe, assuming a Gaussian random field of initial density fluctuations, the LTA predicts that at least 78% of initially underdense regions collapse owing to nonlinear effects of shear and tides.
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