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/G M^{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 fields, 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 fluctuation field \\ \^-0. 29. The dependence on \₀ and the variance \ of the density-contrast field 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