Initial mass function of intermediate mass black hole seeds
  We study the Initial Mass Function (IMF) and host halo properties of
Intermediate Mass Black Holes (IMBH, 10^{4-6} Msun) formed inside metal-free,
UV illuminated atomic cooling haloes (virial temperature T_vir > 10^4 K) either
via the direct collapse of the gas or via an intermediate Super Massive Star
(SMS) stage. 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 host halos 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 due to accretion of
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 cannot
(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= (0.5-20)x10^5 Msun, and the DCBH accretion phase lasts from 10 to
100 Myr. If minihalos are sterile, the IMF spans the narrower mass range M=
(1-2.8)x10^6 Msun, 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/Msun) < 8, (b) in the redshift range 8 < z < 17, (c) with IMBH
in the mass range 4.75 < log (M_BH/Msun) < 6.25.
