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 $M_{\rm BH} \approx 100
M_{\sun}$. 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 "quasistars" suffer
extremely high rates of mass loss through winds from their envelopes, in
analogy to very massive stars such as eta-Carinae. Only for envelope masses
greater than 2.8 \times 10^{5} (M_{\rm BH}/100 M_{\sun})^{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 quasistars
can grow their internal black holes. Accretion rates can be 10 to 100 times the
Eddington rate. The quasistars born in this "growth region" with 10^7-10^8
M_{\sun} can grow black holes with masses between 10^4$ to $10^5 M_{\sun},
before crossing the threshold growth line and dispersing their envelopes in
less than $10^4$ yr. This scenario therefore predicts that massive black hole
seeds can be found only in dark matter halos with total masses larger than
about 10^9 M_{\sun}, which can provide sufficiently high accretion rates to
form such massive quasistars.
