Dissecting the spin distribution of dark matter haloes

The spin probability distribution of dark matter haloes has often been modelled as being very near to a lognormal. Most of the theoretical attempts to explain its origin and evolution invoke some hypotheses concerning the influence of tidal interactions or merging on haloes. Here we apply a very general statistical theorem introduced by Cramér (1936) to study the origin of the deviations from the reference lognormal shape: we find that these deviations originate from correlations between two quantities entering the definition of spin, namely the ratio J/M^5/2 (which depends only on mass) and the modulus E of the total (gravitational + kinetic) energy.To reach this conclusion, we have made usage of the results deduced from two high spatial- and mass-resolution simulations. Our simulations cover a relatively small volume and produce a sample of more than 16 000 gravitationally bound haloes, each traced by at least 300 particles. We verify that our results are stable to different systematics, by comparing our results with those derived by the gif2 and by a more recent simulation performed by Macciò et al.We find that the spin probability distribution function shows systematic deviations from a lognormal, at all redshifts z ≲ 1. These deviations depend on mass and redshift: at small masses they change little with redshift, and also the best lognormal fits are more stable. The J -M relationship is well described by a power law of exponent α very near to the linear theory prediction (α = 5/3), but systematically lower than this at z ≲ 0.3. We argue that the fact that deviations from a lognormal PDF are present only for high-spin haloes could point to a role of large-scale tidal fields in the evolution of the spin PDF.