We investigate the spins and shapes of over a million dark matter haloes identified at z= 0 in the Millennium simulation. Our sample spans halo masses ranging from dwarf galaxies to rich galaxy clusters. The very large dynamic range of this Λ cold dark matter cosmological simulation enables the distribution of spins and shapes and their variation with halo mass and environment to be characterized with unprecedented precision. We compare results for haloes identified using three different algorithms, and investigate (and remove) biases in the estimate of angular momentum introduced both by the algorithm itself and by numerical effects. We introduce a novel halo definition called the TREE halo, based on the branches of the halo merger trees, which is more appropriate for comparison with real astronomical objects than the traditional ‘friends-of-friends’ and ‘spherical overdensity’ (SO) algorithms. We find that for this many objects, the traditional lognormal function is no longer an adequate description of the distribution, P(λ), of the dimensionless spin parameter λ, and we provide a different function that gives a better fit for TREE and SO haloes. The variation in spin with halo mass is weak but detectable, although the trend depends strongly on the halo definition used. For the entire population of haloes, we find median values of λmed= 0.0367–0.0429, depending on the definition of a halo. The haloes exhibit a range of shapes, with a preference for prolateness over oblateness. More-massive haloes tend to be less spherical and more prolate. We find that the more-spherical haloes have less coherent rotation in the median, and those closest to being spherical have a spin independent of mass (λmed≈ 0.033). The most-massive haloes have a spin independent of shape (λmed≈ 0.032). The majority of haloes have their angular momentum vector aligned with their minor axis and perpendicular to their major axis. We find a general trend for higher spin haloes to be more clustered, with a stronger effect for more-massive haloes. For galaxy cluster haloes, this can be larger than a factor of ∼2.

The spin and shape of dark matter haloes in the Millennium simulation of a Λ cold dark matter universe