An analytical approach to the spin distribution of dark halos

We derive the distribution of the dimensionless specific angular momentum of dark matter halos P(j) in the frameworks of tidal-torque theory as well as the nonspherical collapse model and explain the characteristic shape of P(j) commonly observed in N-body simulations. A scalar quantity ( shear scalar) r is introduced for measuring the effective strength of the tidal-torque force acting on the halo from the surrounding matter. It is found that the ubiquitous and broad shape of P(j) can be mostly attributed to the unique property of the shear scalar r and also that P(j) is in fact insensitive to the underlying collapse dynamics. Our result demonstrates that although the shape of P(j) is ubiquitous and close to lognormal, the distribution is not exactly lognormal, but decreases exponentially at the high angular momentum end and drops as a power law with an index of 2 at the low angular momentum end.