Co-evolution of massive black holes and their host galaxies at high
  redshift: discrepancies from six cosmological simulations and the key role of
  JWST
  The James Webb Space Telescope will have the power to characterize
high-redshift quasars at z>6 with an unprecedented depth and spatial
resolution. While the brightest quasars at such redshift (i.e., with bolometric
luminosity L_bol> 10^46 erg/s) provide us with key information on the most
extreme objects in the Universe, measuring the black hole (BH) mass and
Eddington ratios of fainter quasars with L_bol= 10^45-10^46 erg/s opens a path
to understand the build-up of more normal BHs at z>6. In this paper, we show
that the Illustris, TNG100, TNG300, Horizon-AGN, EAGLE, and SIMBA large-scale
cosmological simulations do not agree on whether BHs at z>4 are overmassive or
undermassive at fixed galaxy stellar mass with respect to the M_BH-M_star
scaling relation at z=0 (BH mass offsets). Our conclusions are unchanged when
using the local scaling relation produced by each simulation or empirical
relations. We find that the BH mass offsets of the simulated faint quasar
population at z>4, unlike those of bright quasars, represent the BH mass
offsets of the entire BH population, for all the simulations. Thus, a
population of faint quasars with L_bol= 10^45-10^46 erg/s observed by JWST can
provide key constraints on the assembly of BHs at high redshift. Moreover, this
will help constraining the high-redshift regime of cosmological simulations,
including BH seeding, early growth, and co-evolution with the host galaxies.
Our results also motivate the need for simulations of larger cosmological
volumes down to z=6, with the same diversity of sub-grid physics, in order to
gain statistics on the most extreme objects at high redshift.
