We discuss the link between dark matter halos hosting the first PopIII stars and the rare , massive , halos that are generally considered to host bright quasars at high redshift ( z \approx 6 ) .
The main question that we intend to answer is whether the super-massive black holes powering these QSOs grew out from the seeds planted by the first intermediate massive black holes created in the universe .
This question involves a dynamical range of 10 ^ { 13 } in mass and we address it by combining N-body simulations of structure formation to identify the most massive halos at z \approx 6 with a Monte Carlo method based on linear theory to obtain the location and formation times of the first light halos within the whole simulation box .
We show that the descendants of the first \approx 10 ^ { 6 } M _ { \sun } virialized halos do not , on average , end up in the most massive halos at z \approx 6 , but rather live in a large variety of environments .
The oldest PopIII progenitors of the most massive halos at z \approx 6 , form instead from density peaks that are on average one and a half standard deviations more common than the first PopIII star formed in the volume occupied by one bright high-z QSO .
The intermediate mass black hole seeds planted by the very first PopIII stars at z \gtrsim 40 can easily grow to masses m _ { BH } > 10 ^ { 9.5 } M _ { \sun } by z = 6 assuming Eddington accretion with radiative efficiency \epsilon \lesssim 0.1 .
Quenching of the black hole accretion is therefore crucial to avoid an overabundance of supermassive black holes at lower redshift .
This can be obtained if the mass accretion is limited to a fraction \eta \approx 6 \cdot 10 ^ { -3 } of the total baryon mass of the halo hosting the black hole .
The resulting high end slope of the black hole mass function at z = 6 is \alpha \approx - 3.7 , a value within the 1 \sigma error bar for the bright end slope of the observed quasar luminosity function at z = 6 .