We study the Initial Mass Function ( IMF ) and hosting halo properties of Intermediate Mass Black Holes ( IMBH , 10 ^ { 4 - 6 } M _ { \odot } ) formed inside metal-free , UV illuminated atomic cooling haloes ( virial temperature T _ { vir } \geq 10 ^ { 4 } K ) either via the direct collapse of the gas or via an intermediate Super Massive Star ( SMS ) stage .
These IMBHs have been recently advocated as the seeds of the supermassive black holes observed at z \approx 6 .
We achieve this goal in three steps : ( a ) we derive the gas accretion rate for a proto-SMS to undergo General Relativity instability and produce a direct collapse black hole ( DCBH ) or to enter the ZAMS and later collapse into a IMBH ; ( b ) we use merger-tree simulations to select atomic cooling halos in which either a DCBH or SMS can form and grow , accounting for metal enrichment and major mergers that halt the growth of the proto-SMS by gas fragmentation .
We derive the properties of the hosting haloes and the mass distribution of black holes at this stage , and dub it the “ Birth Mass Function ” ; ( c ) we follow the further growth of the DCBH by accreting the leftover gas in the parent halo and compute the final IMBH mass .
We consider two extreme cases in which minihalos ( T _ { vir } < 10 ^ { 4 } K ) can ( fertile ) or can not ( sterile ) form stars and pollute their gas leading to a different IMBH IMF .
In the ( fiducial ) fertile case the IMF is bimodal extending over a broad range of masses , M \approx ( 0.5 - 20 ) \times 10 ^ { 5 } M _ { \odot } , and the DCBH accretion phase lasts from 10 to 100 Myr .
If minihalos are sterile , the IMF spans the narrower mass range M \approx ( 1 - 2.8 ) \times 10 ^ { 6 } M _ { \odot } , and the DCBH accretion phase is more extended ( 70 - 120 Myr ) .
We conclude that a good seeding prescription is to populate halos ( a ) of mass 7.5 < \log ( M _ { h } / M _ { \odot } ) < 8 , ( b ) in the redshift range 8 < z < 17 , ( c ) with IMBH in the mass range 4.75 < ( \log M _ { \bullet } / M _ { \odot } ) < 6.25 .