Angular momentum - mass relation for dark matter haloes

We study the empirical relation between an astronomical object's angular momentum $J$ and mass $M$, $J=\beta M^\alpha$, the $J-M$ relation, using N-body simulations. In particular, we investigate the time evolution of the $J-M$ relation to study how the initial power spectrum and cosmological model affect this relation, and to test two popular models of its origin - mechanical equilibrium and tidal torque theory. We find that in the $\Lambda$CDM model, $\alpha$ starts with a value of 1.5 at high redshift $z$, increases monotonically, and finally reaches 5/3 near $z=0$, whereas $\beta$ evolves linearly with time in the beginning, reaches a maximum and decreases, and stabilizes finally. A three-regime scheme is proposed to understand this newly observed picture. We show that the tidal torque theory accounts for this time evolution behaviour in the linear regime, whereas $\alpha=5/3$ comes from the virial equilibrium of haloes. The $J-M$ relation in the linear regime contains the information of the power spectrum and cosmological model. The $J-M$ relations for haloes in different environments and with different merging histories are also investigated to study the effects of a halo's non-linear evolution. An updated and more complete understanding of this relation is thus obtained.