Probability Distribution of Primordial Angular Momentum and Formation of Massive Black Holes

We consider the joint probability distribution function for the mass contrast and angular momentum of over-density regions on the proto- galactic scale and investigate the formation of massive black holes at redshift $z\text{gsim}10$. We estimate the growth rate of the angular momentum by the linear perturbation theory and the decay rate by the Compton drag and apply the Press-Schechter theory to obtain the formation rate of massive black holes, assuming the full reionization of the universe at $z=z_{ion}\gg 10$. We find the correlation between the mass contrast and angular momentum vanishes in the linear theory. However, application of the Press-Schechter theory introduces a correlation between the mass contrast and angular momentum of bound objects. Using thus obtained probability distribution, we calculate the mass fraction of black holes with $M\sim {10}^6-{10}^8{M}_{\odot }$ in the universe. We find that it crucially depends on the reionization epoch $z_{ion}$. Specifically, for the standard CDM power spectrum with the COBE normalization, the condition $z_{ion}\text{gsim}500$ must be satisfied to reproduce the observed number density of QSOs.