Local Approximations to the Gravitational Collapse of Cold Matter

We investigate three different local approximations for nonlineargravitational instability in the framework of cosmological Lagrangian fluiddynamics of cold dust. They include the Zel'dovich approximation (ZA), the``non-magnetic'' approximation of Bertschinger \& Jain (1994, NMA), and a new``local tidal'' approximation (LTA). The LTA is exact for any perturbationswhose gravitational and velocity equipotentials have the same constant shapewith time, including spherical, cylindrical, and plane-parallel perturbations.We tested all three local approximations with the collapse of a homogeneoustriaxial ellipsoid, for which an exact solution exists for an ellipsoidembedded in empty space and an excellent approximation is known in thecosmological context. We find that the LTA is significantly more accurate ingeneral than the ZA and the NMA. Like the ZA, but unlike the NMA, the LTAgenerically leads to pancake collapse. For a randomly chosen mass element in anEinstein-de Sitter universe, assuming a Gaussian random field of initialdensity fluctuations, the LTA predicts that at least 78\% of initiallyunderdense regions collapse owing to nonlinear effects of shear and tides.Comment: 29 pages of latex, uses aaspp4.sty (AASTeX v4.0), submitted to Ap