THE OMEGA-DEPENDENCE IN EQUATIONS OF MOTION

The equations of motion governing the evolution of a collisionless gra vitating system of particles in an expanding universe can be cast in a form which is almost independent of the cosmological density paramete r, Omega, and the cosmological constant, A. The new equations are expr essed in terms of a time variable tau=ln D, where D is the linear rate of growth of density fluctuations. The dependence on the density para meter is proportional to epsilon=Omega(-0.2) - 1 times the difference between the peculiar velocity (with respect to tau) of particles and t he gravity field (minus the gradient of the potential); or, before she ll-crossing, times the sum of the density contrast and the velocity di vergence. In a one-dimensional collapse or expansion, the equations ar e fully independent of Omega and Lambda before shell crossing. In the general case, the effect of this weak Omega dependence is to enhance t he rate of evolution of density perturbations in dense regions. In a f lat universe with Lambda not equal 0, this enhancement is less pronoun ced than in an open universe with Lambda=0 and the same Omega. Using t he spherical collapse model, we find that the increase of the rms dens ity fluctuations in a low-Omega universe relative to that in a flat un iverse with the same linear normalization is similar to 0.01 epsilon(O mega)[delta(3)](R)(63), where delta is the density field in the flat u niverse. The equations predict that the smooth average velocity field scales like Omega(0.6), while the local velocity dispersion (rms value ) scales, approximately, like Omega(0.5). High-resolution N-body simul ations confirm these results and show that density fields, when smooth ed on scales slightly larger than clusters, are insensitive to the cos mological model. Haloes in an open model simulation are more concentra ted than haloes of the same M/Omega in a flat model simulation.