QSO’s have been discovered at high redshift (Wu et al. 2015). The three QSOs having the highest redshift are J1061+3922 at z = 6.61, J1120+0641 at z = 7.08 and J1342+0928 at z = 7.54. The later corresponds to an age of the universe of only 0.69 Gyr. Since most of these high redshift QSOs are associated with SMBHs having masses around $10^8-10^9 ~ M_\odot$, their growth was probably not gradual but rather fast in other they could shine so early in the history of the universe. Thus, a possible issue is to admit the existence of two evolutionary paths leading to the formation of SMBHs: one in which seeds grow intermittently as their host galaxies are assembled and another in which seeds grow very fast in a timescale of less than 1 Gyr. These two possibilities will be discussed in the next sections of this article.

Spherical Accretion and the Eddington limit
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Many authors still consider in their investigations spherical accretion processes in which the mass inflow rate is controlled by the Eddington luminosity. In this case, it seems judicious to recall the physical assumptions that permit the derivation of the Eddington limit. The Euler equation describing a 