We present a model for the formation of massive black holes ( \sim 1000 M _ { \odot } ) due to stellar-dynamical processes in the first stellar clusters formed at early cosmic times ( z \sim 10 - 20 ) .
These black holes are likely candidates as seeds for the supermassive black holes detected in quasars and nearby quiescent galaxies .
The high redshift black hole seeds form as a result of multiple successive instabilities that occur in low metallicity ( Z \sim 10 ^ { -5 } Z _ { \odot } ) protogalaxies .
We focus on relatively massive halos at high redshift ( T _ { vir } > 10 ^ { 4 } K , z \mathrel { \hbox to 0.0 pt { \lower 3.0 pt \hbox { $ \sim$ } } \raise 2.0 pt \hbox { $ > $ } } 10 ) after the very first stars in the Universe have completed their evolution .
This set of assumptions ensures that ( i ) atomic hydrogen cooling can contribute to the gas cooling process , ( ii ) a UV field has been created by the first stars , and ( iii ) the gas inside the halo has been mildly polluted by the first metals .
The second condition implies that at low density H _ { 2 } is dissociated and does not contribute to cooling .
The third condition sets a minimum threshold density for fragmentation , so that stars form efficiently only in the very inner core of the protogalaxy .
Within this core , very compact stellar clusters form .
The typical star cluster masses are of order 10 ^ { 5 } M _ { \odot } and the typical half mass radii \sim 1 pc .
A large fraction of these very dense clusters undergo core collapse before stars are able to complete stellar evolution .
Runaway star-star collisions eventually lead to the formation of a very massive star , leaving behind a massive black hole remnant .
Clusters unstable to runaway collisions are always the first , less massive ones that form .
As the metallicity of the Universe increases , the critical density for fragmentation decreases and stars start to form in the entire protogalactic disk so that i ) accretion of gas in the centre is no longer efficient and ii ) the core collapse timescale increases .
Typically a fraction \sim 0.05 of protogalaxies at z \sim 10 - 20 form black hole seeds , with masses \sim 1000 - 2000 M _ { \odot } , leading to a mass density in seeds of a few \simeq 10 ^ { 2 } M _ { \odot } / { Mpc } ^ { -3 } .
This density allows enough room for black hole growth by accretion during the quasar epoch .