The shape of dark matter haloes: dependence on mass, redshift, radius and formation

Using six high resolution dissipationless simulations with a varying box size in a flat LCDM universe, we study the mass and redshift dependence of dark matter halo shapes for Mᵥir = 9. 0e11 - 2. 0e14, over the redshift range z=0-3, and for two values of sigma₈=0. 75 and 0. 9. Remarkably, we find that the redshift, mass, and sigma₈ dependence of the mean smallest-to-largest axis ratio of halos is well described by the simple power-law relation = (0. 54 +- 0. 02) (Mᵥir/M_*) ^ (-0. 050 +- 0. 003), where s is measured at 0. 3 Rᵥir and the z and sigma₈ dependences are governed by the characteristic nonlinear mass, M_*=M_* (z, sigma₈). We find that the scatter about the mean s is well described by a Gaussian with sigma ~ 0. 1, for all masses and redshifts. We compare our results to a variety of previous works on halo shapes and find that reported differences between studies are primarily explained by differences in their methodologies. We address the evolutionary aspects of individual halo shapes by following the shapes of the halos through ~100 snapshots in time. We determine the formation scalefactor ac as defined by Wechsler et al. (2002) and find that it can be related to the halo shape at z = 0 and its evolution over time.