Abstract

We study the initial mass function (IMF) and hosting halo properties of intermediate-mass black holes (IMBHs, 10$^{4-6}$ M$_{⊙}$) formed inside metal-free, UV-illuminated atomic-cooling haloes (virial temperature T$_{vir}$ {\ge} 10$^{4}$ K) either via the direct collapse of the gas or via an intermediate supermassive star (SMS) stage. These IMBHs have been recently advocated as the seeds of the supermassive black holes observed at z {\ap} 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 zero-age main sequence and later collapse into an IMBH; (b) we use merger-tree simulations to select atomic-cooling haloes 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 minihaloes (T$_{vir}$ $\lt$ 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 {\ap} (0.5-20) {\times} 10$^{5}$ M$_{⊙}$, and the DCBH accretion phase lasts from 10 to 100 Myr. If minihaloes are sterile, the IMF spans the narrower mass range M {\ap} (1-2.8) {\times} 10$^{6}$ M$_{⊙}$, and the DCBH accretion phase is more extended (70-120 Myr). We conclude that a good seeding prescription is to populate haloes (a) of mass 7.5 $\lt$ log (M$_{h}$/ M$_{⊙}$) $\lt$ 8, (b) in the redshift range 8 $\lt$ z $\lt$ 17, (c) with IMBH in the mass range 4.75 $\lt$ (log M$_{•}$/ M$_{⊙}$) $\lt$ 6.25.