JWST Spectroscopic View of a Rapidly Growing Dust-obscured Quasar at $z\sim 4$: Effect of Dust Extinction Correction on Black Hole Mass and Eddington Ratio Estimation

In the merger-driven galaxy evolution scenario, the central supermassive black holes (SMBHs) in dust obscured galaxies grow rapidly. Interestingly, a recent work \citep{suh24} on a dust-obscured galaxy, LID-568 at $z=3.965$, has shown that its SMBH is growing extremely fast at about 40 times of the Eddington-limited accretion rate (i.e., super-Eddington accretion). However, the heavy dust extinction of the host galaxy could affect the result if not corrected properly. Here, we analyze James Webb Space Telescope (JWST) NIRSpec/IFU and MIRI spectra of LID-568. By measuring its bolometric luminosity ($L_{\mathrm{b}\mathrm{o}\mathrm{l}}$) and BH mass ($M_{\mathrm{B}\mathrm{H}}$) using an extinction-free estimator based on mid-infrared spectra, we obtain $L_{\mathrm{b}\mathrm{o}\mathrm{l}}={10}^{46.83\pm 0.07}\mathrm{e}\mathrm{r}\mathrm{g}{\mathrm{s}}^{-1}$ and $M_{\mathrm{B}\mathrm{H}}={10}^{8.43\pm 0.15}{M}_{\odot }$. The measured Eddington ratio (${\lambda }_{\mathrm{E}\mathrm{d}\mathrm{d}}$) is 1.97$\pm$0.88, consistent with the accretion rate at the Eddington limit; in other words, not in super-Eddington in a significant manner. This result underscores challenges and the importance of carefully considering dust extinction when analyzing the BH growth in dust-obscured quasars.