The $\Omega$ Dependence in the Equations of Motion

We show that the equations of motion governing the evolution of a collisionless gravitating system of particles in an expanding universe can be cast in a form which is almost independent of the cosmological density parameter, $\Omega$, and the cosmological constant, $\Lambda$. The new equations are expressed in terms of a time variable $\tau\equiv \ln D$, where $D$ is the linear rate of growth of density fluctuations. The weak dependence on the density parameter is proportional to $\epsilon=\Omega^{-0.2}-1$ times the difference between the peculiar velocity (with respect to $\tau$) of particles and the gravity field. In the general case, the effect of this weak $\Omega$ dependence is to enhance the rate of evolution of density perturbations in dense regions. In a flat universe with $\Lambda\ne 0$, this enhancement is less pronounced than in an open universe with $\Lambda=0$ and the same $\Omega$. Using the spherical collapse model, we find that the increase of the $rms$ density fluctuations in a low $\Omega$ universe relative to that in a flat universe with the same linear normalization is $\sim 0.01 \epsilon(\Omega) < \delta^3 >$, where $\delta$ is the density field in the flat universe. 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 simulations confirm these results and show that density fields, when smoothed on scales slightly larger than clusters, are insensitive to the cosmological model. Halos in an open model simulation are more concentrated than halos of the same $M/\Omega$ in a flat model simulation.