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 \la 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 \alpha very near to the linear theory prediction ( \alpha = 5 / 3 ) , but systematically lower than this at z \la 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 .