Weakly nonlinear gravitational instability for arbitrary Omega

Abstract

We investigate the weakly nonlinear evolution of particle trajectories in Friedmann-Lemaitre models with zero cosmological constant. The matter is assumed to be a nonrelativistic pressureless fluid. We use a perturbative expansion in Lagrangian coordinates and derive analytic expressions for the second-order solutions with arbitary density parameter {OMEGA}. This perturbative expansion is valid provided the gradients of the displacement field are small, a much weaker condition than the usual Eulerian requirement of smallness of the density perturbations. We then calculate the gravitationally induced skewness in the distribution of an initially gaussian density contrast field δ. The statistic S_3_ = <δ^3^><δ^2^>^-2^ is found to be very weakly sensitive to {OMEGA}. This property improves the prospects of using observational estimates of S_3_ to distinguish models with Gaussian initial conditions from various non-Gaussian alternatives.