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

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 simulation enables the distribution of spins and shapes and their variation with halo mass and environment to be characterised with unprecedented precision. We compare results for haloes identified using different algorithms, including a novel one based on the branches of the halo merger trees. We investigate (and remove) biases in the estimate of angular momentum introduced by both the algorithms themselves and by numerical effects. We find that for this many objects, the traditional lognormal function is no longer an adequate description of the distribution of the dimensionless spin parameter lambda, and we provide a different function that gives a better fit. The variation of spin with halo mass is weak but detectable, although the trend depends strongly on the halo definition used. 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 spin in the median, and those closest to spherical have a spin independent of mass. The most massive have a spin independent of shape. The majority of haloes have their angular momentum vector aligned with their minor axis and perpendicular to their major axis. We find that higher tend to be more clustered, with a stronger effect for more massive haloes. (abridged)