The statistics of Λ CDM halo concentrations

We use the Millennium Simulation (MS) to study the statistics of Lambda cold dark matter (Lambda CDM) halo concentrations at z = 0. Our results confirm that the average halo concentration declines monotonically with mass; the concentration-mass relation is well fitted by a power law over three decades in mass, up to the most massive objects that form in a Lambda CDM universe (similar to 10(15) h(-1)M(circle dot)) Bullock et al. for these rare objects, and agrees better with the original predictions of Navarro, Frenk & White. The large volume surveyed, together with the unprecedented numerical resolution of the MS, allows us to estimate with confidence the distribution of concentrations and, consequently, the abundance of systems with unusual properties. About one in a hundred cluster haloes (M-200 >= 3 x 10(14) h(-1) M-circle dot) have concentrations exceeding c(200) = 7.5, a result that may be useful in interpreting the likelihood of unusually strong massive gravitational lenses, such as Abell 1689, in the Delta CDM cosmogony. A similar fraction of about 1 per cent of galaxy-sized haloes (M-200 similar to 10(12) h(-1)M(circle dot)) have c(200) < 4.5 and this could be relevant to models that attempt to reconcile the Delta CDM cosmology with rotation curves of low surface brightness galaxies by appealing to haloes of unexpectedly low concentration. We find that halo concentrations are independent of spin once haloes manifestly out of equilibrium have been removed from the sample. Compared to their relaxed brethren, the concentrations of out-of-equilibrium haloes tend to be lower and have more scatter, while their spins tend to be higher. A number of previously noted trends within the halo population are induced primarily by these properties of unrelaxed systems. Finally, we compare the result of predicting halo concentrations using the mass assembly history of the main progenitor with predictions based on simple arguments regarding the assembly time of all progenitors. The latter are typically as good or better than the former, suggesting that halo concentration depends not only on the evolutionary path of a halo's main progenitor, but on how and when all of its constituents collapsed to form non-linear objects.