Effects of the size of cosmological N-Body simulations on physical   quantities -- I: Mass Function
  N-Body simulations are a very important tool in the study of formation of large scale structures. Much of the progress in understanding the physics of galaxy formation and comparison with observations would not have been possible without N-Body simulations. Given the importance of this tool, it is essential to understand its limitations as ignoring these can easily lead to interesting but unreliable results. In this paper we study the limitations due to the finite size of the simulation volume. We explicitly construct the correction term arising due to a finite box size and study its generic features for clustering of matter and also on mass functions. We show that the correction to mass function is maximum near the scale of non-linearity, as a corollary we show that the correction to the number density of haloes of a given mass changes sign at this scale; the number of haloes at small masses is over estimated in simulations. This over estimate results from a delay in mergers that lead to formation of more massive haloes. The same technique can be used to study corrections to other physical quantities. The corrections are typically small if the scale of non-linearity is much smaller than the box-size. However, there are some cases of physical interest in which the relative correction term is of order unity even though a simulation box much larger than the scale of non-linearity is used. Within the context of the concordance model, our analysis suggests that it is very difficult for present day simulations to resolve mass scales smaller than 100 solar masses accurately and the level of difficulty increases as we go to even smaller masses, though this constraint does not apply to multi-scale simulations. 
J. S. Bagla and Jayanti Prasad