On the spin parameter of dark-matter haloes

The study by White (1984) on the growth of angular momentum in dark haloes is extended towards a more detailed investigation of the spin parameter L p E=GM 2:5. Starting from the Zel'dovich approximation to structure formation, a dark halo is approximated by a homogeneous ellipsoid with the inertial tensor of the (highly irregular) Lagrangian region from which the dark halo forms. Within this approximation, an expression for the spin parameter can be derived, which depends on the geometry of , the cosmological density parameter 0 , the overdensity of the dark halo, and the tidal torque exerted on it. For Gaussian random elds, this expression can be evaluated statistically. As a result, we derive a probability distribution of the spin parameter which gives ' 0:07 +0:04 0:05 , consistent with numerical investigations. This probability distribution steeply rises with increasing spin parameter, reaching its maximum at ' 0:025. The 10 (50,90) percentile values are = 0:02 (0.05,0.11, respectively). There is a weak anticorrelation of the spin parameter with the peak height of the density uctuation eld / 0:29. The dependence on 0 and the variance of the density-contrast eld is very weak; there is only a marginal tendency for the spin parameter to be slightly larger for late-forming objects in an open universe. Due to the weak dependence on , our results should be quite generally applicable and independent on the special kind of the uctuation spectrum.