LOCAL APPROXIMATIONS TO THE GRAVITATIONAL COLLAPSE OF COLD MATTER

We investigate three different local approximations for nonlinear grav itational instability in the framework of cosmological Lagrangian flui d dynamics of cold dust. By local we mean that the evolution is descri bed by a set of ordinary differential equations in time for each mass element, with no coupling to other mass elements aside from those impl ied by the initial conditions. We first show that the Zel'dovich appro ximation (ZA) can be cast in this form. Next, we consider extensions i nvolving the evolution of the Newtonian tidal tenser. We show that two approximations can be found that are exact for plane-parallel and sph erical perturbations. The first one (''nonmagnetic'' approximation, or NMA) neglects the Newtonian counterpart of the magnetic part of the W eyl tensor in the fluid frame and was investigated previously by Berts chinger & Jain. A new approximation (''local tidal,'' or LTA) involves neglecting still more terms in the tidal evolution equation. It is mo tivated by the analytic demonstration that it is exact for any perturb ations whose gravitational and velocity equipotentials have the same c onstant shape with time. Thus, the LTA is exact for spherical, cylindr ical, and plane-parallel perturbations. It corresponds physically to n eglecting the curl of the magnetic part of the Weyl tensor in the como ving threading as well as an advection term in the tidal evolution equ ation. All three approximations can be applied up to the point of orbi t crossing. We tested them in the case of the collapse of a homogeneou s triaxial ellipsoid, for which an exact solution exists for an ellips oid 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 t he NMA, the LTA generically leads to pancake collapse. For a randomly chosen mass element in an Einstein-de Sitter universe, assuming a Gaus sian random field of initial density fluctuations, the LTA predicts th at at least 78% of initially underdense regions collapse owing to nonl inear effects of shear and tides.