Quantification of discreteness effects in cosmological N-body simulations: II. Evolution up to shell crossing

We apply a recently developed perturbative formalism which describes the evolution under their self-gravity of particles displaced from a perfect lattice to quantify precisely, up to shell crossing, the effects of discreteness in dissipationless cosmological N-body simulations. We give simple expressions, explicitly dependent on the particle density, for the evolution of power in each mode as a function of red-shift. For typical starting red-shifts the effect of finite particle number is to {\it slow down} slightly the growth of power compared to that in the fluid limit (e.g. by about ten percent at half the Nyquist frequency), and to induce also dispersion in the growth as a function of direction at a comparable level. Further, above the Nyquist frequency, purely discrete power generated in the initial conditions is amplified. We note that, at fixed particle number, the effects of discreteness increase as the initial red-shift $z_{\rm init}$ is increased, with divergence from the fluid limit as $z_{\rm init} \to \infty$. We also study how these effects are modified when there is a small-scale regularization of the gravitational force. We show that such a smoothing may reduce the anisotropy of the discreteness effects, but it then {\it increases} their average effect. This behaviour illustrates the fact that the discreteness effects described here are distinct from those usually considered in this context, due to two-body collisions. Indeed the characteristic time for divergence from the collisionless limit is proportional to $N^{2/3}$, rather than $N/ \log N$ in the latter case.