  In this paper we provide predictions for the BH population that would be
observable with planned JWST surveys at $5 \le z \le 15$. We base our study on
the recently developed Cosmic Archaeology Tool (CAT), which allows us to model
BH seeds formation and growth, while being consistent with the general
population of AGNs and galaxies observed at $4 \le z \le 7$. We find that JWST
planned surveys will provide a complementary view on active BHs at $z > 5$,
with JADES-Medium/-Deep being capable of detecting the numerous BHs that
populate the faint-end of the distribution, COSMOS-Web sampling a large enough
area to detect the rarest brightest systems, and CEERS/PRIMER bridging the gap
between these two regimes. The relatively small field of view of the above
surveys preferentially selects BHs with masses $6 \leq {\rm Log} (M_{\rm
BH}/M_\odot) < 8$ at $7 \le z < 10$, residing in relatively metal poor (${\rm
Log} (Z/Z_\odot) \ge -2$) and massive ($8\leq {\rm Log} (M_*/M_\odot) < 10$)
galaxies. At $z \ge 10$, only JADES-Deep will have the sensitivity to detect
growing BHs with masses $4 \leq {\rm Log} (M_{\rm BH}/M_\odot) < 6$, hosted by
more metal poor ($-3 \leq {\rm Log} (Z/Z_\odot) < -2$) and less massive ($6
\leq {\rm Log} (M_*/M_\odot) < 8$) galaxies. In our model, the latter
population corresponds to heavy BH seeds formed by the direct collapse of
super-massive stars in their earliest phases of mass growth. Detecting these
systems would provide invaluable insights on the nature and early growth of the
first BH seeds.
