The role of stellar relaxation in the formation and evolution of the first massive black holes

We present calculations on the formation of massive black holes of 10⁵ M_⊙ at z > 6, which can be the seeds of supermassive black holes at z ≳ 6. Under the assumption of compact star cluster formation in merging galaxies, star clusters in haloes of ~ 10⁸-10⁹M_⊙ can undergo rapid core collapse, leading to the formation of very massive stars (VMSs) of ~1000M_⊙ that collapse directly into black holes with similar masses. Star clusters in haloes of ≳10⁹M_⊙ experience Type II supernovae before the formation of VMSs, due to long core-collapse timescales. We also model the subsequent growth of black holes via accretion of residual stars in clusters. Two-body relaxation refills the loss cones of stellar orbits efficiently at larger radii and resonant relaxation at small radii is the main driver for accretion of stars on to black holes. As a result, more than 90 percent of stars in the initial cluster are swallowed by the central black holes before z = 6. Using dark matter merger trees, we derive black hole mass functions at z =6-20. Themass function ranges from 10³-10⁵M_⊙ at z ≲15. Majormerging of galaxies of ≳4 × 10⁸M_⊙ at z ~ 20 leads successfully to the formation of ≳ 10⁵M_⊙ black holes by z ≳ 10, which could be the potential seeds of supermassive black holes seen today.