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 with 10^5 Msun at z > 6 that 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 10^8 ~ 10^9 Msun can undergo rapid core-collapse leading to the formation of very massive stars (VMSs) with ~1000 Msun which directly collapse into black holes with similar masses. Star clusters in halos of > 10^9 Msun experience type-II supernovae before the formation of VMSs due to long core-collapse time scales. We also model the subsequent growth of black holes via accretion of residual stars in clusters. 2-body relaxation efficiently re-fills the loss cones of stellar orbits at larger radii and resonant relaxation at small radii is the main driver for accretion of stars onto black holes. As a result, more than ninety 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. The mass function ranges from 10^3 to 10^5 Msun at z <~ 15. Major merging of galaxies of >~ 4*10^8 Msun at z ~ 20 successfully leads to the formation of >~ 10^5 Msun BHs by z >~ 10 which can be the potential seeds of supermassive black holes seen today.