GRAVITATIONAL-INSTABILITY OF COLD MATTER

We solve the nonlinear evolution of pressureless, irrotational density fluctuations in a perturbed Robertson-Walker spacetime using a new La grangian method based on the velocity gradient and gravity gradient te nsors. Borrowing results from general relativity, we obtain a set of N ewtonian ordinary differential equations for these quantities followin g a given mass element. Using these Lagrangian fluid equations we prov e the following collapse theorem: A mass element-whose density exceeds the cosmic mean at high redshift collapses to infinite density at lea st as fast as a uniform spherical perturbation with the same initial d ensity and velocity divergence. Velocity shear invariably speeds colla pse-the spherical top-hat perturbation, having zero shear, is the slow est configuration to collapse for a given initial density and growth r ate. Two corollaries follow: (1) Initial density maxima are not genera lly the sites where collapse first occurs. The initial velocity shear (or tidal gravity field) also is important in determining the collapse time. (2) Initially underdense regions undergo collapse if the shear is sufficiently large. If the magnetic part of the Weyl tensor vanishe s, the nonlinear evolution is described purely locally by these equati ons. This condition is exact for highly symmetrical perturbations (e.g ., with planar, cylindrical, or spherical symmetry) and may be a good approximation in many other circumstances. Assuming the vanishing of t he magnetic part of the Weyl tensor we compute the exact nonlinear gra vitational evolution of cold matter. We find that 56% of initially und erdense regions collapse in an Einstein-de Sitter universe for a homog eneous and isotropic random field. We also show that, given this assum ption, the final stage of collapse is generically two-dimensional, lea ding to strongly prolate filaments rather than Zel'dovich pancakes. Wh ile this result may explain the prevalence of filamentary collapses in some N-body simulations, it is not true in general suggesting that th e magnetic part of the Weyl tensor does not necessarily vanish in the Newtonian limit.