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Understanding the formation and growth of supermassive black holes (SMBHs) athigh redshift represents a major challenge for theoretical models. In this workwe investigate the early evolution of the first SMBHs by constraining theirdistribution in mass and luminosity at $z > 4$. In particular, we focus on thepoorly explored low-mass end of the nuclear black hole (BH) distribution downto $z \simeq 4$, and explore its connection with the nature of the first BHseeds and the processes governing their mass growth. To this aim, we havedeveloped CAT (Cosmic Archaeology Tool), a new semi-analytic model thatdescribes the formation of the first stars and black holes in a self-consistentway and follows the co-evolution of nuclear BHs and their host galaxies for arepresentative population at $z > 4$. We find that current observationalconstraints favour models where the growth of BH seeds is Eddington limited andoccurs at the Bondi-Hoyle-Lyttleton rate or where super-Eddington accretionoccurs via a slim disk during gas rich galaxy mergers. The main differencebetween these two model variants lies at the low-end of the predicted mass andluminosity functions at $4 \le z \le 6$, where a clear gap appears in the firstmodel, reflecting the stunted growth of light BH seeds formed as remnants ofthe first stars. Detecting this signature will be extremely challenging evenfor the future generation of space observatories, such as JWST, Athena andLynx.

The low-end of the black hole mass function at cosmic dawn