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We apply a recently developed perturbative formalism which describes theevolution under their self-gravity of particles displaced from a perfectlattice to quantify precisely, up to shell crossing, the effects ofdiscreteness in dissipationless cosmological N-body simulations. We give simpleexpressions, explicitly dependent on the particle density, for the evolution ofpower in each mode as a function of red-shift. For typical starting red-shiftsthe effect of finite particle number is to {\it slow down} slightly the growthof power compared to that in the fluid limit (e.g. by about ten percent at halfthe Nyquist frequency), and to induce also dispersion in the growth as afunction of direction at a comparable level. In the limit that the initialred-shift tends to infinity, at fixed particle density, the evolution in factdiverges from that in the fluid limit (described by the Zeldovichapproximation). Contrary to widely held belief, this means that a simulationstarted at a red-shift much higher than the red-shift of shell crossingactually gives a worse, rather than a better, result. We also study how theseeffects are modified when there is a small-scale regularization of thegravitational force. We show that such a smoothing may reduce the anisotropy ofthe discreteness effects, but it then {\it increases} their average effect.This behaviour illustrates the fact that the discreteness effects describedhere are distinct from those usually considered in this context, due totwo-body collisions. Indeed the characteristic time for divergence from thecollisionless limit is proportional to $N^{2/3}$, rather than $N/ \log N$ inthe latter case.

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