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 redshift. For typical starting redshifts the effect of finite particle number is to slow down slightly the growth of power compared to that in the fluid limit (e.g., by about 10% at half the Nyquist frequency), and to induce also dispersion in the growth as a function of direction at a comparable level. In the limit that the initial redshift tends to infinity, at fixed particle density, the evolution in fact diverges from that in the fluid limit (described by the Zeldovich approximation). Contrary to widely held belief, this means that a simulation started at a redshift much higher than the redshift of shell crossing actually gives a worse, rather than a better, result. 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 increases their average effect. This behavior 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.

• Received 30 April 2007

DOI:https://doi.org/10.1103/PhysRevD.76.103505