JWST ERS Program Q3D: The pitfalls of virial BH mass constraints shown in a z = 3 quasar with an ultramassive host

We present JWST MIRI/NIRSpec observations of the extremely red quasar SDSS J165202.64+172852.3 at z~3, one of the most luminous quasars known to date, driving powerful outflows and hosting a clumpy starburst, amidst several interacting companions. We estimate the black hole (BH) mass of the system based on the broad H$\alpha$ and H$\beta$ lines, as well as the Pa$\beta$ emission in the IR and MgII in the UV. We recover a very broad range of mass estimates, with constraints ranging between log $M_{\rm BH}$=9 and 10.1, which is exacerbated if imposing a uniform BLR geometry at all wavelengths. Several factors may contribute to the large spread: measurement uncertainties (insufficient sensitivity to detect the broadest component of the faint Pa$\beta$ line, spectral blending, ambiguities in the broad/narrow component distinction), lack of virial equilibrium (in a system characterised by powerful outflows and rapid accretion), and uncertainties on the luminosity-inferred size of the broad line region, a.o. given central dust obscuration. We constrain the stellar mass via SED fitting, suggesting the host to be extremely massive at $10^{12.8\pm 0.5} M_\odot$ - ~2 dex above the characteristic mass of the Schechter fit to the z=3 stellar mass function. Notably, J1652's central BH might be interpreted as being either undermassive, overmassive, or in line with the BH mass-stellar mass relation, depending on the choice of assumptions. The recovered Eddington ratio varies accordingly, but exceeds 10% in any case. We put our results into context by providing an extensive overview and discussion of recent literature results and their associated assumptions. Our findings provide an important demonstration of the uncertainties inherent in virial BH mass estimates, which are of particular relevance in the JWST era given the growing number of studies on rapidly accreting quasars at high redshift.