Comments on the size of the simulation box in cosmological N-body simulations

N-body simulations are a very important tool in the study of the formation of large-scale structures. Much of the progress in understanding the physics of the high-redshift universe 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 the limitations can easily lead to interesting but unreliable results. In this paper we study the limitations arising out of the finite size of the simulation volume. This finite size implies that modes larger than the size of the simulation volume are ignored and a truncated power spectrum is simulated. If the simulation volume is large enough then the mass in collapsed haloes expected from the full power spectrum and from the truncated power spectrum should match. We propose a quantitative measure based on this approach that allows us to compute the minimum box size for an N-body simulation. We find that the required box size for simulations of the Lambda CDM model at high redshifts is much larger than is typically used. We can also use this approach to quantify the effect of perturbations at large scales for power-law models and we find that if we fix the scale of non-linearity, the required box size becomes very large as the index becomes small. The appropriate box size computed using this approach is also an appropriate choice for the transition scale when tools like MAP, which add the contribution of the missing power, are used.