On the spin parameter of dark-matter haloes - Steinmetz, Matthias; Bartelmann, Matthias (Arxiv, 1994)
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 $λ\equiv L\sqrt{E}/{G M^{2.5}}$. 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 $Υ$ from which the dark halo forms. Within this approximation, an expression for the spin parameter can be derived, which depends on the geometry of $Υ$, the cosmological density parameter $Ω_0$, 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 $λ\simeq0.07^{+0.04}_{-0.05}$, consistent with numerical investigations. This probability distribution steeply rises with increasing spin parameter, reaching its maximum at $λ\simeq0.025$. The 10 (50,90) percentile values are $λ=0.02$ (0.05,0.11, respectively). There is a weak anticorrelation of the spin parameter with the peak height $ν$ of the density fluctuation field $λ\propto ν^{-0.29}$. The dependence on $Ω_0$ and the variance $σ$ 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 $σ$, our results should be quite generally applicable and independent on