A lower limit on the halo mass to form supermassive black holes

We consider a scenario where supermassive black holes form through direct accumulation of gas at the centre of proto-galaxies. In the first stage, the accumulated gas forms a super-massive star whose core collapses when the nuclear fuel is exhausted, forming a black hole of M_BH≈ 100M_⊙. As the black hole starts accreting, it inflates the surrounding dense gas into an almost hydrostatic self-gravitating envelope, with at least 10-100 times the mass of the hole. We find that these 'quasi-stars' suffer extremely high rates of mass loss through winds from their envelopes, in analogy to very massive stars such as η-Carinae. Only for envelope masses greater than 2.8 × 10⁵(M_BH/100M_⊙)^9/11 is the envelope evaporation time-scale longer than the accretion time-scale of the black hole. This relation thus constitutes a 'threshold-growth line' above which quasi-stars can grow their internal black holes. Accretion rates can be 10 to 100 times the Eddington rate. The quasi-stars born in this 'growth region' with 10⁷-10⁸M_⊙ can grow black holes with masses between 10⁴ and 10⁵M_⊙, before crossing the threshold-growth line and dispersing their envelopes in less than 10⁴yr. This scenario therefore predicts that massive black hole seeds can be found only in dark matter haloes with total masses larger than about 10⁹M_⊙, which can provide sufficiently high accretion rates to form such massive quasi-stars.