Subhaloes in $\Lambda$CDM cosmological simulations: I. Masses and   abundances
  If the concordance $\Lambda$CDM model is a true description of the universe, it should also properly predict the properties and structure of dark matter haloes, where galaxies are born. Using N-body simulations with a broad scale of mass and spatial resolution, we study the structure of dark matter haloes, the distribution of masses and the spatial distribution of subhaloes within the main haloes.   We carry out three $\Lambda$CDM simulations with different resolutions using the AMIGA code. Dark matter haloes are identified using an algorithm that is based on the adaptive grid structure of the simulation code. The haloes we find encompass the mass scales from $10^8\mathrm{M}_{\sun}$ to $10^{15}\mathrm{M}_{\sun}$.   We find that if we have to study the halo structure (search for subhaloes), the haloes have to contain at least $10^4$ particles. For such haloes, where we can resolve substructure, we determined the subhalo mass function and found that it is close to a power law with the slope -0.9 (at present time), consistent with previous studies. This slope depends slightly on the redshift and it is approximately the same for main haloes.   The subhalo mass fraction ($M_{\s subH}/M_{\s MH}$) is between 0.08 and 0.2, increasing slightly with redshift and with the mass of the main halo. Its distribution is approximated using the Weibull distribution at different epochs. The mean values of subhalo mass are independent of the main halo mass. The spatial density of subhaloes, scaled to the virial radius of the main halo ($r_{\s vir}$), is independent of redshift and follows the $r^{1/3}$ rule. 
P. Nurmi, P. Hein\"am\"aki, E. Saar, M. Einasto, J. Holopainen, V. J.   Mart\'inez and J. Einasto