THE DOMINANT LOCAL INSTABILITIES IN AN EINSTEIN-DESITTER UNIVERSE

We investigate the nonlinear evolution of compensated aspherical perturbations in an Einstein-de Sitter universe with cold, collisionless matter. Newtonian gravity is used throughout. Perturbations with dipolar and quadrupolar (both two polar caps and an equatorial ring) symmetry are considered. N-body simulations show that the perturbations quickly develop into spherical expanding voids surrounded by shells with local mass concentrations that reflect the initial perturbation symmetry. These nonlinear structures evolve in a self-similar fashion with their dimensions expanding as powers of time. We derive quasi-analytic, self-similar solutions that are in good agreement with the N-body results, both in terms of the density profiles and the expansion rates. Our aspherical solutions, which have zero total energy, expand more rapidly than positive energy spherical blast waves. The mass in the nonlinear region, MNL, grows as aγ, where a is the scale factor and γ= 0.6 for an energy-conserving compensated spherical void. As shown by Vishniac, momentum conservation implies that γ = 0.75 for the dipole solution. There is no conservation law which determines the expansion rates of the quadrupole solutions, and we find γ= 0.77 and 0.69 for the quadrupole polar cap and equatorial ring solutions, respectively. Due to limitations in numerical accuracy, it is not clear that the quadrupole polar cap solution is truly faster than the dipole. We suspect that one of these is the dominant local disturbance in an Einstein-de Sitter universe. In the development of structure from Gaussian density fluctuations, the mass scale that goes nonlinear increases as aγ, with γ= 6/(n + 3) for a power spectrum \γk\2 ∝ kn. Thus, even for the steepest natural spectrum (n = 4), the linear growth of fluctuations produces a more rapid increase of MNL than that due to nonlinear mode coupling associated with dipolar voids.