Constraining the AGN luminosity function from JWST with the X-ray Background

We predict the X-ray background (XRB) expected from the population of quasars detected by the JWST spectroscopic surveys over the redshift range $z \sim 4-7$. We find that the measured UV emissivities, in combination with a best-fitting quasar SED template, imply a $\sim 10$ times higher unresolved X-ray background than constrained by current experiments. We illustrate the difficulty of simultaneously matching the faint-end of the quasar luminosity function and the X-ray background constraints. We discuss possible origins and consequences of this discrepancy.