  JWST has recently revealed a large population of accreting black holes (BHs)
in the early Universe. Even after accounting for possible systematic biases,
the high-z $M_*-M_{\rm \rm bh}$ relation derived from these objects by Pacucci
et al. (2023 P23 relation) is above the local scaling relation by $>3\sigma$.
To understand the implications of potentially overmassive high-z BH
populations, we study the BH growth at $z\sim4-7$ using the
$[18~\mathrm{Mpc}]^3$ BRAHMA suite of cosmological simulations with systematic
variations of heavy seed models that emulate direct collapse black hole (DCBH)
formation. In our least restrictive seed model, we place $\sim10^5~M_{\odot}$
seeds in halos with sufficient dense and metal-poor gas. To model conditions
for direct collapse, we impose additional criteria based on a minimum Lyman
Werner flux (LW flux $=10~J_{21}$), maximum gas spin, and an environmental
richness criterion. The high-z BH growth in our simulations is merger
dominated, with a relatively small contribution from gas accretion. For the
most restrictive simulation that includes all the above seeding criteria for
DCBH formation, the high-z $M_*-M_{\rm bh}$ relation falls significantly below
the P23 relation (by factor of $\sim10$ at $z\sim4$). Only by excluding the
spin and environment based criteria, and by assuming $\lesssim750~\mathrm{Myr}$
delay times between host galaxy mergers and subsequent BH mergers, are we able
to reproduce the P23 relation. Overall, our results suggest that if high-z BHs
are indeed systematically overmassive, assembling them would require more
efficient heavy seeding channels, higher initial seed masses, additional
contributions from lighter seeds to BH mergers, and / or more efficient modes
for BH accretion.
