$\mathcal{O}(N\log N)$ or even better [@Dehnen2000]. All these techniques (tree-code, particle-mesh, P$^3$M, AMR, tree-PM,...) do however rely on particles and do, hence, share the same initial assumptions leading to the two following challenges.

Firstly, since the dark matter fluid is supposed to be collisionless, one has to manually suppress artificial two-body collisions arising between the pseudo-particles introduced to sample the phase space distribution. This is usually done by introducing an ad-hoc softening length and suppressing the gravitational force at scales below it [@Dehnen2001]. [$N$-body ]{}simulations are run under the assumption that for a suitable choice of the smoothing, the evolution of the $N$ pseudo-particles under the softened force should be the same as the gravitational evolution of the elementary dark matter particles.

The second challenge is to relate the particle distribution to the theoretical Vlasov-Poisson the particles are supposed to model. Despite its obvious relevance, it seems that the question of the precise quantitative importance of the discretization (\[eq:nbody\]) and its effects is still not settled [@Joyce2008].

As a matter of fact, there are no alternative tools to study the cosmic structure formation with the same resolution as [$N$-body ]{}simulations. This is of course not a limitation of the [$N$-body 