Pattern in Nonlinear Gravatational Clustering: A Numerical Investigation

The nonlinear clustering of dark matter particles in an expanding universe is usually studied by N-body simulations. One can gain some insight into this complex problem if simple relations between physical quantities in the linear and nonlinear regimes can be extracted from the results of N-body simulations. Hamilton et al. (1991) and Nityananda and Padmanabhan (1994) have made an attempt in this direction by relating the mean relative pair velocities to the mean correlation function in a useful manner. We investigate this relation and other closely related issues in detail for the case of six different power spectra: power laws with spectral indexes $n=-2,-1$, cold dark matter (CDM), and hot dark matter models with density parameter $\Omega=1$; CDM including a cosmological constant ($\Lambda$) with $\Omega_{CDM}=0.4$, $\Omega_{\Lambda}=0.6$; and $n=-1$ model with $\Omega=0.1$. We find that: (i) Power law spectra lead to self-similar evolution in an $\Omega=1$ universe. (ii) Stable clustering does not hold in an $\Omega=1$ universe to the extent our simulations can ascertain. (iii) Stable clustering is a better approximation in the case of $\Omega<1$ universe in which structure formation freezes out at some low redshift. (iv) The relation between dimensionless pair velocity and the mean correlation function, $\bar\xi$, is only approximately independent of the shape of the power spectrum. At the nonlinear end, the asymptotic value of the dimensionless pair velocity decreases with increasing small scale power, because the stable clustering assumption is not universally true. (v) The relation between the evolved $\bar\xi$ and the linear regime $\bar\xi$ is also not universal but shows a weak spectrum dependence. We present simple theoretical arguments for these conclusions.