Fundamental physical considerations and past tests suggest that there may be a problem with discreteness error in N-body methods widely used in cosmological clustering studies. This could cause problems with accuracy when coupled to hydrodynamics codes. We therefore investigate some of the effects that discreteness and two-body scattering may have on N-body simulations with "realistic" cosmological initial conditions.

We use an identical subset of particles from the initial conditions for a 128³ particle-mesh (PM) calculation as the initial conditions for a variety of particle-particle-particle mesh (P³M) and tree code runs. The force softening length and particle number in the P³M and tree code runs are varied, and results are compared with those of the PM run. In particular, we investigate the effect of mass resolution (or equivalently the mean interparticle separation) since most "high-resolution" codes only have high resolution in gravitational force, not in mass. We show the evolution of a wide variety of statistical measures. The phase-insensitive two-point statistics, P(k) and ξ(R), are affected by the number of particles when the force resolution is held constant and differ in different N-body codes with similar parameters and the same initial conditions. Phase-sensitive statistics show greater differences. Results converge at the mean interparticle separation scale of the lowest mass-resolution code. As more particles are added but the absolute scale of the force resolution is held constant, the P³M and the tree runs agree more and more strongly with each other and with the PM run that had the same initial conditions, suggesting that the time integration is converging. However, they do not particularly converge to a PM run that continued the power-law fluctuations to small scales. This suggests high particle density is necessary for correct time evolution, since many different results cannot all be correct. Our results showing the effect of the presence or absence of small-scale initial power suggest that leaving it out is a considerable source of error on comoving scales of the missing wavelengths, which can be resolved by putting in a high particle density.

Since the codes never agree well on scales below the mean comoving interparticle separation, we find little justification to use results on these scales to make quantitative predictions in cosmology. The range of values found for some quantities spans 50%, but others, such as the amount of mass in high-density regions, can be off by a factor of 3 or more. Our results have strong implications for applications such as the density of galaxy halos, early generation objects such as QSO absorber clouds, etc.