Black Hole Mass Estimates and Rapid Growth of Supermassive Black Holes in Luminous $z \sim$ 3.5 Quasars

We present new near-infrared (IR) observations of the H$β \lambda4861$ and MgII $\lambda2798$ lines for 32 luminous quasars with $3.2<z<3.9$ using the Palomar Hale 200 inch telescope and the Large Binocular Telescope. We find that the MgII Full Width at Half Maximum (FWHM) is well correlated with the H$β$ FWHM, confirming itself as a good substitute for the H$β$ FWHM in the black hole mass estimates. The continuum luminosity at 5100 Å well correlates with the continuum luminosity at 3000 Å and the broad emission line luminosities (H$β$ and MgII). With simultaneous near-IR spectroscopy of the H$β$ and MgII lines to exclude the influences of flux variability, we are able to evaluate the reliability of estimating black hole masses based on the MgII line for high redshift quasars. With the reliable H$β$ line based black hole mass and Eddington ratio estimates, we find that the $z\sim3.5$ quasars in our sample have black hole masses $1.90\times10^{9} M_{\odot} \lesssim M_{\rm BH} \lesssim 1.37\times10^{10} M_{\odot}$, with a median of $\sim 5.14\times10^{9} M_{\odot}$ and are accreting at Eddington ratios between 0.30 and 3.05, with a median of $\sim1.12$. Assuming a duty cycle of 1 and a seed black hole mass of $10^{4} M_{\odot}$, we show that the $z\sim3.5$ quasars in this sample can grow to their estimated black hole masses within the age of the Universe at their redshifts.