Black hole mass estimates and rapid growth of supermassive black holes in luminous z ∼ 3.5 quasars

We present new near-infrared (IR) observations of the Hβ λ4861 and Mg II λ2798 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 Mg II FWHM is well correlated with the Hβ FWHM, confirming itself as a good substitute for the Hβ FWHM in the black hole mass estimates. The continuum luminosity at 5100 A˚ well correlates with the continuum luminosity at 3000 A˚ and the broad emission line luminosities (Hβ and Mg II). With simultaneous near-IR spectroscopy of the Hβ and Mg II lines to exclude the influences of flux variability, we are able to evaluate the reliability of estimating black hole masses based on the Mg II line for high redshift quasars. With the reliable Hβ line based black hole mass and Eddington ratio estimates, we find that the z ∼ 3.5 quasars in our sample have black hole masses 1.90 × 10⁹ M_⊙ ≲ M_BH ≲ 1.37 × 10¹⁰ M_⊙, with a median of ∼5.14 × 10⁹ M_⊙ and are accreting at Eddington ratios between 0.30 and 3.05, with a median of ∼1.12. Assuming a duty cycle of 1 and a seed black hole mass of 10⁴ M_⊙,we show that the z ∼ 3.5 quasars in this sample can grow to their estimated black hole masses within the age of the universe at their redshifts.