Recently , it has been suggested that the CO line width ( FWHM ( CO ) ) is a surrogate for the bulge velocity dispersion ( \sigma ) of the host galaxies of high-redshift quasars , and the black hole – bulge ( M _ { BH } - \sigma ) relation obtained with this assumption departs significantly from the M _ { BH } - \sigma relation in the local universe .
In this study , we first present an investigation of the correlation between the CO line width and the bulge velocity dispersion using a sample of 33 nearby Seyfert galaxies .
We find that the formula adopted in previous studies , \sigma = FWHM ( CO ) / 2.35 , is generally not a good approximation .
Using it , one may underestimate the value of bulge velocity dispersion significantly when the CO line is narrower than 400 km~ { } s ^ { -1 } .
By involving the galactic inclination angle i as an additional parameter , we obtain a tight correlation between the inclination-corrected CO line width and the bulge velocity dispersion , namely , FWHM ( CO ) / \sin i = -67.16 \pm 80.18 + ( 3.62 \pm 0.68 ) \sigma .
Using this new relation , we can better estimate the bulge velocity dispersion from the CO line width if the galactic inclination is known .
We apply this new relation to nine high-redshift quasars with CO line detections and find that they are consistent with the local M _ { BH } - \sigma relation if their inclination angles are around 15 ^ { o } .
The possible smaller inclinations of the high-redshift quasars are preferred because of their relatively greater likelihood of detection , and are also consistent with their relatively smaller CO line widths compared to submillimeter galaxies ( SMGs ) at high redshift having a similar total amount of molecular gas .
Future observations are needed to confirm these results .