Gravitational Instability in Collisionless Cosmological Pancakes

The gravitational instability of cosmological pancakes composed of collisionless dark matter in an Einstein-de Sitter universe is investigated numerically to demonstrate that pancakes are unstable with respect to fragmentation and the formation of filaments. A ``pancake'' is defined here as the nonlinear outcome of the growth of a 1D, sinusoidal, plane-wave, adiabatic density perturbation. We have used high resolution, 2D, N-body simulations by the Particle-Mesh (PM) method to study the response of pancakes to perturbation by either symmetric (density) or antisymmetric (bending or rippling) modes, with corresponding wavevectors k_s and k_a transverse to the wavevector k_p of the unperturbed pancake plane-wave. We consider dark matter which is initially ``cold'' (i.e. with no random thermal velocity in the initial conditions). We also investigate the effect of a finite, random, isotropic, initial velocity dispersion (i.e. initial thermal velocity) on the fate of pancake collapse and instability. Pancakes are shown to be gravitationally unstable with respect to all perturbations of wavelength l<l_p (where l_p= 2pi/k_p). These results are in contradiction with the expectations of an approximate, thin-sheet energy argument.