]{}method itself, but makes it more complicated to evaluate the possible errors of [$N$-body ]{}simulations quantitatively, as there are basically no independent results to compare with. For instance [@Ludlow2011] find a non-negligible fraction of halos in CDM simulations that cannot be matched to peaks in the initial density distribution and are possible artefacts of the [$N$-body ]{}method. The different techniques used to calculate the forces are, of course, different and can lead to marginally different results for the same initial sampling of the field when the resolution limit is reached. They do, however, all share the decomposition of $f(x,p)$ in a set of $N$ macroscopic particles and will, hence, share the consequences of this Ansatz.

Spurious effects due to the discretization become more apparent when looking at simulations of warm dark matter (WDM) or hot dark matter (HDM) cosmologies. The initial matter power-spectrum entering such simulations is truncated below a certain free-streaming scale related to the dark matter particle rest mass. Those particles having a small mass, they also have a finite velocity distribution function at every point in space, making the problem effectively 6 dimensional. In practice, these velocities are neglected and the DM fluid is treated in the 