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 one-dimensional, sinusoidal, plane-wave, adiabatic density perturbation. We have used high-resolution, two-dimensional, 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 ks and ka transverse to the wavevector kp of the unperturbed pancake plane wave. We consider dark matter that 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. Our results include the following: (1) For "cold" initial conditions, pancakes are gravitationally unstable with respect to all perturbations of wavenumber k 1 (where k = λp/λ, and λp and λ are the wavelengths of the unperturbed pancake and of the perturbation, respectively). This is contrary to the expectations of an approximate, thin-sheet energy argument applied to the results of one-dimensional pancake simulations. The latter predicts that unstable wavenumbers are restricted to the range kmin kmax > 1 are stabilized by the one-dimensional velocity dispersion of the collisionless particles along the direction of pancake collapse, within the region of shell crossing. (2) Shortly after the pancake first reaches a nonlinear state of collapse, the dimensionless growth rate of the perturbation of pancake surface density by unstable transverse modes rises rapidly from the value of 2/3 for a linear density fluctuation in the absence of the primary pancake and levels off. This signals the onset of a new, linear instability of the nonlinear pancake, which grows as a power law in time. (3) The index of this power-law time dependence scales as kn, where n ≈ 0.2-0.25 for both the symmetric and antisymmetric modes. (4) The power spectrum of the perturbation in pancake surface density is strongly peaked during the linear phase of growth of unstable modes, at wavenumber k = ks for symmetric modes and k = 2ka for antisymmetric modes. (5) Eventually, the onset of nonlinearity is signaled by a decline of the growth rate and the production of clumps of large overdensity relative to that of the unperturbed pancake, located in or near the plane of the unperturbed pancake. In two dimensions, these "clumps" correspond to filaments, one per λs (two per λa) for symmetric (antisymmetric) modes. (6) For equal initial amplitudes, the antisymmetric mode reaches nonlinearity later than the symmetric mode. (7) These filaments have azimuthally averaged density profiles ρ ~ r-m, m = 1.1 ± 0.1. (8) Contrary to the expectations of the thin-sheet energy argument, pancakes in a collisionless gas of finite temperature are also gravitationally unstable with respect to all perturbations of wavenumber k 1. Finite temperature does not stabilize the pancake against perturbations of large wavenumber as predicted by the energy argument. Finite temperature can cause perturbations to decay prior to the collapse of the primary pancake, but once pancake caustics form, transverse perturbations grow at the same linear growth rate as for cold initial conditions.