Measuring the Virial Factor in SDSS DR7 Active Galactic Nuclei with Redshifted Hβ and Hα Broad Emission Lines

Black hole mass, M_•, is an important fundamental parameter of black holes. Reliable measurement of M_• is always a key issue of black hole–related research. For active galactic nuclei (AGNs), the reverberation mapping (RM) method or the relevant secondary methods based on single-epoch spectra were widely used to measure M_• by a virial mass ${M}_{\mathrm{RM}}={{fv}}_{\mathrm{FWHM}}^{2}{r}_{\mathrm{BLR}}/G$ when clouds in the broad-line region (BLR) are in virialized motion, where f is the virial factor, v_FWHM is the full width at half-maximum of the broad emission line, r_BLR is the radius of the BLR, and G is the gravitational constant (e.g., Peterson et al. 2004). However, f is very uncertain due to the unclear kinematics and geometry of the BLR (e.g., Peterson et al. 2004; Woo et al. 2015). f is commonly considered to be the main source of uncertainty in M_RM. The reverberation-based masses are themselves uncertain, typically by a factor of ∼2.9 (Onken et al. 2004), and the absolute uncertainties in M_RM given by the secondary methods are typically around a factor of 4 (Vestergaard & Peterson 2006). If v_FWHM is replaced with the line width σ_line, the second moment of emission line, f becomes f_σ . Based on the photoionization assumption (e.g., Blandford & McKee 1982; Peterson 1993), r_BLR = τ_ob c/(1 + z), where c is the speed of light, z is the cosmological redshift of the source, and τ_ob is the observed time lag of the broad-line variations relative to the continuum ones. For non-RM AGNs studied by the secondary methods, r_BLR can be estimated with the empirical r_BLR–L₅₁₀₀ relation for the Hβ emission line of the RM AGNs, where L₅₁₀₀ is the AGN continuum luminosity at rest-frame wavelength 5100 Å (e.g., Kaspi et al. 2000; Bentz et al. 2013; Du et al. 2018b; Du & Wang 2019; Yu et al. 2020).

RM surveys have been carried out (e.g., Shen et al. 2015a, 2015b, 2016, 2019; King et al. 2015; Grier et al. 2017; Hoormann et al. 2019). Nonsurvey RM observation research has been done for more than 100 AGNs over the last several decades (e.g., Kaspi & Netzer 1999; Kaspi et al. 2000, 2007; Peterson et al. 2005; Bentz et al. 2006,2010, 2016, 2021, 2022, 2023; Denney et al. 2010; Barth et al. 2011, 2015; Haas et al. 2011; Pozo Nuñez et al. 2012; Du et al. 2014, 2015, 2016, 2018a, 2018b; Pei et al. 2014, 2017; Wang et al. 2014; Hu et al. 2015, 2020, 2021; Lu et al. 2016, 2021; Xiao et al. 2018a, 2018b; Zhang et al. 2019; Feng et al. 2021a, 2021b; Li et al. 2021, 2022). The single-epoch spectra had been widely used to estimate M_RM in studies of high-z quasars (e.g., Willott et al. 2010; Wu et al. 2015; Wang et al. 2019; Eilers et al. 2023) and on statistics of AGNs, such as the Sloan Digital Sky Survey (SDSS) quasars (e.g., Hu et al. 2008; Liu et al. 2019). Based on the M_•–σ_* relation for the low-z inactive and quiescent galaxies, where σ_* is the stellar velocity dispersion of the galaxy bulge (e.g., Tremaine et al. 2002; Onken et al. 2004; Piotrovich et al. 2015; Woo et al. 2015), these derived averages of 〈f〉 ≈ 1 and/or 〈f_σ 〉 ≈ 5 were usually used to estimate M_RM by the RM and/or single-epoch spectra of AGNs. Therefore, measuring f and/or f_σ independently by a new method for individual AGNs is necessary and important to understand the physics of the BLR and the issues related to the masses of the supermassive black holes (SMBHs), e.g., the formation and growth of SMBHs at z ≳ 6 (e.g., Wu et al. 2015; Fan et al. 2023), the coevolution (or not) of SMBHs and host galaxies (e.g., Tremaine et al. 2002; Kormendy & Ho 2013; Woo et al. 2013; Caglar et al. 2020), etc.

Some efforts have been made on an object-by-object basis for small samples of AGNs using high-fidelity RM techniques (e.g., Pancoast et al. 2014a, 2014b) or by using spectral fitting methods (Mejía-Restrepo et al. 2018 and references therein). Liu et al. (2017) proposed a new method to measure f based on the widths and shifts of redward-shifted broad emission lines for the RM AGNs. Based on SDSS DR5 quasars with redward-shifted Hβ and Fe ii broad emission lines, Liu et al. (2022) made further efforts toward researching f and f_σ . The Fe iii λ λ2039–2113 UV line blend comes from an inner region of the BLR (Mediavilla et al. 2018; Mediavilla & Jiménez-Vicente 2021), and for 10 lensed quasars of higher Eddington ratio, the redward-shifted Fe iii blend was used to estimate f with 〈f〉 = 14.3 much larger than 〈f〉 ≈ 1 (Mediavilla et al. 2020). However, the origins of broad emission lines and BLRs are yet unclear for AGNs (e.g., Wang et al. 2017). Thus, the origin of redward shifts of broad emission lines is unclear. The redward shifts of broad emission lines are commonly believed to be from inflow (e.g., Hu et al. 2008). Inflow can generate the redward shifts of broad absorption lines, but the broad absorption and emission lines may be from different gas regions due to their distinct velocities (Zhou et al. 2019).

RM observations of Mrk 817 suggest that the redward shifts of broad emission lines do not originate from inflow because of their redward asymmetric velocity-resolved lag maps (Lu et al. 2021), which are not consistent with the blueward asymmetric maps expected from inflow. Redward shifts of broad emission lines in the RM observations of Mrk 110 follow the gravitational redshift prediction (Kollatschny 2003). The gravitational interpretation of the redward shift of the Fe iii blend is preferred over alternative explanations, such as inflow, that will need additional physics to explain the observed correlation between the width and redward shift of the blend (Mediavilla et al. 2018). A sign of the gravitational redshift z_g was found in a statistical sense for broad Hβ in the single-epoch spectra of SDSS DR7 quasars (Tremaine et al. 2014). Based on the widths and asymmetries of Hα and Hβ broad emission line profiles in a sample of type 1 AGNs taken from SDSS DR16, Rakić (2022) showed that the BLR gas seems to be virialized. The velocity-resolved lag maps of the Hβ broad emission lines for Mrk 50 and SBS 1518+593 show characteristics of Keplerian disks or virialized motion (Barth et al. 2011; Du et al. 2018a). Thus, it is likely that the redward shifts of broad emission lines originate from the gravity of the central black hole.

Radiation pressure from the accretion disk has a significant influence on the stability and dynamics of clouds in the BLR (e.g., Marconi et al. 2008; Netzer & Marziani 2010; Krause et al. 2011, 2012; Naddaf et al. 2021). The dynamics of clouds can determine the three-dimensional geometry of the BLR (Naddaf et al. 2021). However, radiation pressure was not considered in estimating M_RM, and the virial factor was believed to be only from the geometric effect of the BLR. Lu et al. (2016) found that the BLR of NGC 5548 could be jointly controlled by the radiation pressure force from the accretion disk and the gravity of the central black hole. Krause et al. (2011) found that stable orbits of clouds in the BLR exist for very sub-Keplerian rotation, for which the radiation pressure force contributes substantially to the force budget. Thus, the radiation pressure force may result in significant influence on the virial factor. Based on redward-shifted Hβ and Fe ii broad emission lines for a sample of 1973 z < 0.8 SDSS DR5 quasars, Liu et al. (2022) found a positive correlation of the virial factor with the dimensionless accretion rate or the Eddington ratio. They suggested that the radiation pressure force is a significant contributor to the virial factor, and that the redward shift of the Hβ broad emission line is mainly from the gravity of the black hole. In this work, more than 8000 SDSS DR7 AGNs with redward-shifted Hβ and Hα broad emission lines from Table 2 in Liu et al. (2019) will be adopted to investigate the virial factor, the relations of the virial factor with other physical quantities, the origin of the redward shifts of the broad Balmer emission lines, and the implications of the f-correction.

The structure is as follows. Section 2 presents the method. Section 3 describes the sample selection. Section 4 presents the analysis and results. Section 5 is the potential influence on quasars at z ≳ 6. Section 6 is the potential influence on the M_•–σ_* map of AGNs. Section 7 presents the discussion, and Section 8 is the conclusion. Throughout this paper, we assume a standard cosmology with H₀ = 70 km s⁻¹ Mpc⁻¹, Ω_M = 0.3, and Ω_Λ = 0.7 (Spergel et al. 2007).

A BLR cloud is subject to the gravity of the black hole, F_g, and radiation pressure force, F_r, due to central continuum radiation. The total mechanical energy and angular momentum are conserved for the BLR clouds because F_g and F_r are central forces. Under various assumptions, F_r can be calculated for more than hundreds of thousands of lines, with detailed photoionization, radiative transfer, and energy balance calculations (e.g., Dannen et al. 2019). In principle, M_• could be estimated by the BLR cloud motions when the numerical calculation methods give F_r. However, the various assumptions may significantly influence the reliability of F_r, especially the many unknown physical parameters that likely vary for different AGNs. Thus, a new method was proposed to measure f and then M_RM, avoiding the use of the averages of the virial factor or the numerical calculation of F_r (Liu et al. 2017, 2022).

The virial factor formula in Liu et al. (2022) was derived from the Schwarzschild metric for clouds in the virialized motion,

$\begin{eqnarray}&&f=\displaystyle \frac{1}{3}\displaystyle \frac{{c}^{2}}{{v}_{\mathrm{FWHM}}^{2}}\left[1-{\left(1+{z}_{{\rm{g}}}\right)}^{-2}\right],\end{eqnarray} \tag{ 1 }$

where the gravitational and transverse Doppler shifts are taken into account. If v_FWHM is replaced with σ_line, f becomes f_σ . As z_g ≪ 1 or r_g/r_BLR ≪ 1 for broad emission lines (the gravitational radius r_g = GM_•/c²), we have

$\begin{eqnarray}&&f=\displaystyle \frac{2}{3}\displaystyle \frac{{c}^{2}}{{v}_{\mathrm{FWHM}}^{2}}{z}_{{\rm{g}}}.\end{eqnarray} \tag{ 2 }$

Mediavilla & Jiménez-Vicente (2021) pinpointed that the observed redward shift of the Fe iii λ λ2039–2113 emission line blend in quasars originates from the gravity of the black hole, while these Fe iii λ λ2039–2113 emission lines are broad emission lines. Furthermore, their redward shifts and line widths follow the gravitational redshift prediction (see Figure 4 in Mediavilla et al. 2018). So, the broad emission line position and width should be determined not only by the kinematics of the BLR but also by the gravity of the black hole. The virialized assumption in measuring M_RM will ensure that the line position should be governed by the gravity of the black hole for the redward-shifted broad emission lines in AGNs (this will be tested in the next section). The same method as in Liu et al. (2017, 2022) was used to estimate the virial factor in Mediavilla et al. (2018; see their Equation 5). Thus, the method in this work, evolved from Liu et al. (2017), is reliable, and the assumption of the gravitational redshift is reasonable.

The Schwarzschild metric is valid at the optical BLR scales, and Equation (1) is valid for a disklike BLR (see Liu et al. 2022). The disklike BLR is preferred by some RM observations of AGNs, e.g., NGC 3516 (e.g., Denney et al. 2010; Feng et al. 2021a) and the VLTI instrument GRAVITY observations of quasar 3C 273 (Sturm et al. 2018). For rapidly rotating BLR clouds, the relativistic beaming effect can give rise to a profile asymmetry with an enhanced blue side in broad emission lines, i.e., blueshifts of broad emission lines (Mediavilla & Insertis 1989). Thus, the relativistic beaming effect should be neglected for the redward-shifted broad emission lines, which should be dominated by the gravitational redshift and transverse Doppler effects. The reliability of the redward shift method was confirmed by the consistent masses estimated from Equations (4) and (7) based on four broad emission lines for Mrk 110 (see Figure 2 of Liu et al. 2017). Hereafter, M_RM denotes M_• measured with the RM method and/or the relevant secondary methods, f_g denotes 〈f〉 = 1 for v_FWHM or 〈f_σ 〉 = 5.5 for σ_line, M_RM ≡ M_RM(f_g = 1), the Eddington luminosity L_Edd ≡ L_Edd(f_g = 1), the Eddington ratio L_bol/L_Edd ≡ L_bol/L_Edd(f_g = 1), and r_g ≡ r_g(f_g = 1).

Liu et al. (2019) reported a comprehensive and uniform sample of 14,584 broad-line AGNs with z < 0.35 from the SDSS DR7. The stellar continuum was properly removed for each spectrum with significant host absorption line features, and careful analyses of emission line spectra, particularly in the Hα and Hβ wave bands, were carried out. The line widths and line centroid wavelengths of the Hα, Hβ, and [O iii] spectra are given in Table 2 of Liu et al. (2019). The redward shifts of the broad emission lines Hβ and Hα are defined as

$\begin{eqnarray}\begin{array}{rl}{z}_{{\rm{g}}} & =\displaystyle \frac{{\lambda }_{{\rm{b}}}({\rm{H}})}{{\lambda }_{{\rm{n}}}({\rm{H}})}-1\\ & =\displaystyle \frac{{\lambda }_{{\rm{b}}}({\rm{H}})}{{\lambda }_{{\rm{n}}}([{\rm{O}}\,{\rm\small{III}}])}\displaystyle \frac{{\lambda }_{0}([{\rm{O}}\,{\rm\small{III}}])}{{\lambda }_{0}({\rm{H}})}-1,\end{array}\end{eqnarray} \tag{ 3 }$

where λ_b is the centroid wavelength of the broad emission line corrected by the cosmological redshift z_SDSS given by the SDSS site (Liu et al. 2019), λ_n is the centroid wavelength of the narrow emission line corrected by z_SDSS, and λ₀ is the vacuum wavelength of the spectrum line (λ₀ = 4862.68 Å for Hβ, λ₀ = 6564.61 Å for Hα, and λ₀ = 5008.24 Å for [O iii] λ5007). ⁵

Because of the absence of the uncertainty of λ_n for Hβ in Table 2 of Liu et al. (2019), and in order to unify the standard of estimating z_g for the broad Hβ and Hα, the [O iii] λ5007 line is used in Equation (3). First, one of the choice criteria is the AGN's flag = 0, which means no emission lines with multiple peaks (Liu et al. 2019), because the multiple peaks of the emisson lines may be from dual AGNs (e.g., Wang et al. 2009). Second, AGNs are selected on the basis of z_g > 0 and z_g − σ(z_g) > 0 for the broad Hβ and Hα, where σ(z_g) is the uncertainty of z_g. Third, AGNs are selected on the basis of v_FWHM > 0 and v_FWHM − σ(v_FWHM) > 0 for the broad Hβ and Hα, where σ(v_FWHM) is the uncertainty of v_FWHM. The selection conditions of z_g > 0 and z_g − σ(z_g) > 0 make sure that the shifts of the broad emission lines are redward within 1σ uncertainties. Because the empirical r_BLR–L₅₁₀₀ relation is established for broad emission line Hβ, the relevant research on the virial factor is made with the broad Hβ and Hα in this work. A total of 9185 AGNs are selected out of the 14,584 AGNs as sample 1 only for the broad Hβ. A total of 9271 AGNs are selected out of the 14,584 AGNs as sample 2 only for the broad Hα. The cross-identified AGNs in samples 1 and 2 are used as sample 3, which contains 8169 AGNs with the z_g of the broad Hβ and Hα.

Some physical quantities are taken or estimated from Table 2 in Liu et al. (2019), including v_FWHM(Hβ), v_FWHM(Hα), z_g(Hβ), z_g(Hα), L₅₁₀₀, M_RM, L_bol/L_Edd, and the dimensionless accretion rate ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$. The bolometric luminosity L_bol was estimated in Liu et al. (2019) using L_bol = 9.8L₅₁₀₀ (McLure & Dunlop 2004). The details of samples are listed in Tables 1–3. The virial factors, f(Hβ) and f(Hα), are estimated by Equation (1) for the broad Hβ and Hα (see Tables 1–3). ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}={L}_{\mathrm{bol}}/{L}_{\mathrm{Edd}}/\eta$, where η is the efficiency of converting rest-mass energy to radiation. Hereafter, in addition to a special statement, we adopt η = 0.038 (Du et al. 2015).

Table 1. The Relevant Parameters for 9185 AGNs in SDSS DR7 for Sample 1

Designation	$\tfrac{{v}_{\mathrm{FWHM}}({\rm{H}}\beta )}{\mathrm{km}\ {{\rm{s}}}^{-1}}$	zg(Hβ)	$\mathrm{log}{L}_{5100}$	$\mathrm{log}\tfrac{{M}_{\mathrm{RM}}}{{M}_{\odot }}$	$\mathrm{log}\tfrac{{L}_{\mathrm{bol}}}{{L}_{\mathrm{Edd}}}$	f(Hβ)	$\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$	$\tfrac{{r}_{\mathrm{BLR}}}{{r}_{{\rm{g}}}}$	RFe II	$\mathrm{log}{\tfrac{{M}_{\mathrm{RM}}}{{M}_{\odot }}}^{\dagger }$	${\tfrac{{r}_{\mathrm{BLR}}}{{r}_{{\rm{g}}}}}^{\dagger }$	$\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}^{\dagger }$
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)
J000048.16−095404.0	2132.9 ± 71.0	0.00089 ± 0.00005	43.39 ± 0.00	7.26	−1.086	11.7 ± 0.8	0.334	15,381.3	−999	−999	−999	−999
J000102.19−102326.9	4695.0 ± 69.2	0.00062 ± 0.00049	44.36 ± 0.00	8.46	−1.356	1.7 ± 0.0	0.064	3191.6	0.188	8.49	3191.7	0.037
J000154.29+000732.5	1813.8 ± 98.9	0.00046 ± 0.00023	43.48 ± 0.00	7.14	−0.972	8.4 ± 0.9	0.448	22,644.5	0.115	7.27	22,645.3	0.322

Note. Column (1): object name. Column (2): v_FWHM of Hβ broad emission line. Column (3): redward shift of Hβ. Column (4): logarithm of L₅₁₀₀ in units of erg s⁻¹. Column (5): logarithm of M_RM in units of M_⊙. Column (6): logarithm of L_bol/L_Edd. Column (7): f estimated from v_FWHM of Hβ. Column (8): logarithm of ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$. Column (9): r_BLR in units of r_g, where ${r}_{\mathrm{BLR}}=33.65{L}_{44}^{0.533}$ lt-day with L₄₄ = L₅₁₀₀/(10⁴⁴ erg s⁻¹). Column (10): R_{Fe II} is the line ratio of Fe ii to Hβ. Columns (2)–(6) and (10) are taken from Table 2 of Liu et al. (2019) or converted from the relevant quantities in Table 2 of Liu et al. (2019). An amount of −999 denotes no data of R(Fe ii), which results in no data for the latter three quantities. † denotes the values estimated from Equations (6) and (7).

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Table 2. The Relevant Parameters for 9271 AGNs in SDSS DR7 for Sample 2

Designation	$\tfrac{{v}_{\mathrm{FWHM}}({\rm{H}}\alpha )}{{\mathrm{kms}}^{-1}}$	zg(Hα)	zg(Hα)(b-n)	$\mathrm{log}{L}_{5100}$	$\mathrm{log}\tfrac{{M}_{\mathrm{RM}}}{{M}_{\odot }}$	$\mathrm{log}\tfrac{{L}_{\mathrm{bol}}}{{L}_{\mathrm{Edd}}}$	f(Hα)	f(Hα)(b-n)	$\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$	$\tfrac{{r}_{\mathrm{BLR}}}{{r}_{{\rm{g}}}}$
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)
J000048.16−095404.0	2132.9 ± 282.9	0.00089 ± 0.00022	0.00063 ± 0.00022	43.39 ± 0.00	7.26	−1.086	11.7 ± 3.1	8.3 ± 2.2	0.334	15,381.3
J000102.19−102326.9	4695.0 ± 172.5	0.00062 ± 0.00050	0.00073 ± 0.00050	44.36 ± 0.00	8.46	−1.356	1.7 ± 0.1	2.0 ± 0.1	0.064	3191.6
J000111.15−100155.5	1937.4 ± 84.2	0.00015 ± 0.00012	0.00013 ± 0.00012	43.15 ± 0.04	6.37	−0.327	2.4 ± 0.2	2.1 ± 0.2	1.093	88,935.0

Note. Column (1): object name. Column (2): v_FWHM of Hα broad emission line. Column (3): redward shift of broad Hα with respect to [O iii] λ5007. Column (4): redward shift of broad Hα with respect to narrow Hα. Column (5): logarithm of L₅₁₀₀ in units of ergs⁻¹. Column (6): logarithm of M_RM in units of M_⊙. Column (7): logarithm of L_bol/L_Edd. Column (8): virial factor estimated from v_FWHM of broad Hα and z_g(Hα). Column (9): virial factor estimated from v_FWHM of broad Hα and z_g(Hα)(b-n). Column (10): logarithm of ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$. Column (11): r_BLR in units of r_g, where ${r}_{\mathrm{BLR}}=33.65{L}_{44}^{0.533}$ lt-day with L₄₄ = L₅₁₀₀/(10⁴⁴ erg s⁻¹). Columns (2)–(7) are taken from Table 2 of Liu et al. (2019) or converted from the relevant quantities in Table 2 of Liu et al. (2019). An amount of -999 indicates a null value in columns (4) and (9).

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Table 3. The Relevant Parameters for 8169 AGNs in SDSS DR7 for Sample 3

Designation	$\tfrac{{v}_{\mathrm{FWHM}}({\rm{H}}\beta )}{\mathrm{km}\ {{\rm{s}}}^{-1}}$	zg(Hβ)	$\tfrac{{v}_{\mathrm{FWHM}}({\rm{H}}\alpha )}{\mathrm{km}\ {{\rm{s}}}^{-1}}$	zg(Hα)	$\mathrm{log}{L}_{5100}$	$\mathrm{log}\tfrac{{M}_{\mathrm{RM}}}{{M}_{\odot }}$	$\mathrm{log}\tfrac{{L}_{\mathrm{bol}}}{{L}_{\mathrm{Edd}}}$	f(Hβ)	f(Hα)	$\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$	$\tfrac{{r}_{\mathrm{BLR}}}{{r}_{{\rm{g}}}}$	RFe II	$\mathrm{log}{\tfrac{{M}_{\mathrm{RM}}}{{M}_{\odot }}}^{\dagger }$	${\tfrac{{r}_{\mathrm{BLR}}}{{r}_{{\rm{g}}}}}^{\dagger }$	$\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}^{\dagger }$
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)	(11)	(12)	(13)	(14)	(15)	(16)
J000048.16–095404.0	2132.9 ± 71.0	0.00089 ± 0.00005	2132.9 ± 282.9	0.00089 ± 0.00022	43.39 ± 0.00	7.26	−1.086	11.7 ± 0.8	11.7 ± 3.1	0.334	15,381.3	−999	–999	−999	–999
J000102.19–102326.9	4695.0 ± 69.2	0.00062 ± 0.00049	4695.0 ± 172.5	0.00062 ± 0.00050	44.36 ± 0.00	8.46	−1.356	1.7 ± 0.0	1.7 ± 0.1	0.064	3191.6	0.188	8.49	6663.8	0.037
J000154.29+000732.5	1813.8 ± 98.9	0.00046 ± 0.00023	1813.8 ± 178.9	0.00046 ± 0.00023	43.48 ± 0.00	7.14	−0.972	8.4 ± 0.9	8.4 ± 1.7	0.448	22,644.5	0.115	7.27	52,991.6	0.322

Note. Column (1): object name. Column (2): v_FWHM of Hβ broad emission line. Column (3): redward shift of Hβ. Column (4): v_FWHM of Hα broad emission line. Column (5): redward shift of Hα. Column (6): logarithm of L₅₁₀₀ in units of erg s⁻¹. Column (7): logarithm of M_RM in units of M_⊙. Column (8): logarithm of L_bol/L_Edd. Column (9): f estimated from v_FWHM of Hβ. Column (10): f estimated from v_FWHM of Hα. Column (11): logarithm of ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$. Column (12): r_BLR in units of r_g, where ${r}_{\mathrm{BLR}}=33.65{L}_{44}^{0.533}$ lt-day with L₄₄ = L₅₁₀₀/(10⁴⁴ erg s⁻¹). Column (13): R_{Fe II} is the line ratio of Fe ii to Hβ. Columns (2)–(8) and (13) are taken from Table 2 of Liu et al. (2019) or converted from the relevant quantities in Table 2 of Liu et al. (2019). The number −999 and † are the same as in Table 1.

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For our selected AGNs, L₅₁₀₀ spans 4 orders of magnitude, M_RM spans more than 4 orders of magnitude, and L_bol/L_Edd spans more than 3 orders of magnitude. These parameters are at least 1 order of magnitude wider than those in Liu et al. (2022). The measured values of f span more than 3 orders of magnitude, which are at least 1 order of magnitude wider than those in Liu et al. (2022). These much wider parameters can ensure that this work is feasible.

In order to study the correlation between f, ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$, L₅₁₀₀, and v_FWHM, as well as z_g and r_BLR/r_g, we will perform the Spearman's rank test and/or the Pearson's correlation analysis. The bisector linear regression (Isobe et al. 1990) is performed to obtain the slope and intercept coefficients of y = a + bx in fitting our samples, if needed for some quantities. The partial correlation analysis is used to further verify the presence of a correlation between f and ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$. All correlation analyses are calculated in log-space. The SPEAR (Press et al. 1992) is used to calculate the Spearman's rank correlation coefficient r_s and the p-value P_s of the hypothesis test. The PEARSN (Press et al. 1992) is used to give the Pearson's correlation coefficient r and the p-value P of the hypothesis test.

The Spearman's rank correlation test is run for samples 1–3, and the analysis results are listed in Table 4. There are positive correlations between the virial factor and ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$ or L_bol/L_Edd for samples 1–3 (see Figure 1 and Table 4). The results from the Pearson's correlation analysis are listed in Table 5. The bisector regression fit can give a and b, as well as their uncertainties Δa and Δb, but it does not take into account the obervational errors of the data (Isobe et al. 1990). So, based on Monte Carlo simulated data sets from the obervational values and errors, we calculate the best parameters using the bisector regression and repeat this procedure 10⁴ times to generate the distributions of a_MC, b_MC, Δa_MC, and Δb_MC. The means of the a_MC and b_MC distributions are taken to be the final best parameters of a and b, respectively. The corresponding uncertainties are given by the combinations of the means of the Δa_MC and Δb_MC distributions with the standard deviations of the a_MC and b_MC distributions, respectively. The bisector regression is run for $\mathrm{log}f=a+b\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$, and the fitting results are

$\begin{eqnarray}&&\mathrm{log}f=0.76(\pm 0.01)+0.88(\pm 0.01)\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1},\end{eqnarray} \tag{ 4a }$

$\begin{eqnarray}&&\mathrm{log}f=0.76(\pm 0.01)+0.77(\pm 0.01)\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1},\end{eqnarray} \tag{ 4b }$

$\begin{eqnarray}&&\mathrm{log}f=0.80(\pm 0.01)+0.78(\pm 0.01)\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1},\end{eqnarray} \tag{ 4c }$

$\begin{eqnarray}&&\mathrm{log}f=0.77(\pm 0.01)+0.78(\pm 0.01)\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1},\end{eqnarray} \tag{ 4d }$

where the p-values of the hypothesis test are <10⁻⁴⁰, and in the fittings, the uncertainty of $\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$ is taken to be 0.4, determined by the uncertainty of 0.4 dex usually used in M_RM. Equations (4a–d) correspond to the best fits to the data sets (f,${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$) for Hβ in sample 1, Hα in sample 2, Hβ in sample 3, and Hα in sample 3, respectively. It is clear that $f\propto {\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}^{0.8-0.9}$, and then $f\propto {({L}_{\mathrm{bol}}/{L}_{\mathrm{Edd}})}^{0.8-0.9}$.

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Table 4. Spearman's Rank Analysis Results

X	Y	Line	rs	Ps	Sample
$\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$	$\mathrm{log}f$	Hβ	0.615	<10−40	1
$\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$	$\mathrm{log}f$	Hα	0.578	<10−40	2
$\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$	$\mathrm{log}f$	Hβ	0.626	<10−40	3
$\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$	$\mathrm{log}f$	Hα	0.590	<10−40	3
$\mathrm{log}{L}_{5100}$	$\mathrm{log}f$	Hβ	0.089	1.0 × 10−17	1
$\mathrm{log}{L}_{5100}$	$\mathrm{log}f$	Hα	0.047	6.0 × 10−6	2
$\mathrm{log}{L}_{5100}$	$\mathrm{log}f$	Hβ	0.113	8.3 × 10−25	3
$\mathrm{log}{L}_{5100}$	$\mathrm{log}f$	Hα	0.044	5.8 × 10−5	3
$\mathrm{log}[{r}_{\mathrm{BLR}}/{r}_{{\rm{g}}}]$	$\mathrm{log}{z}_{{\rm{g}}}$	Hβ	−0.440	<10−40	1
$\mathrm{log}[{r}_{\mathrm{BLR}}/{r}_{{\rm{g}}}]$	$\mathrm{log}{z}_{{\rm{g}}}$	Hα	−0.318	<10−40	2
$\mathrm{log}[{r}_{\mathrm{BLR}}/{r}_{{\rm{g}}}]$	$\mathrm{log}{z}_{{\rm{g}}}$	Hβ	−0.447	<10−40	3
$\mathrm{log}[{r}_{\mathrm{BLR}}/{r}_{{\rm{g}}}]$	$\mathrm{log}{z}_{{\rm{g}}}$	Hα	−0.335	<10−40	3
$\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}(\eta )$	$\mathrm{log}f$	Hβ	0.654	<10−40	1
$\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}(\eta )$	$\mathrm{log}f$	Hα	0.647	<10−40	2
$\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}(\eta )$	$\mathrm{log}f$	Hβ	0.662	<10−40	3
$\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}(\eta )$	$\mathrm{log}f$	Hα	0.658	<10−40	3

Note. X and Y are the relevant quantities presented in samples 1–3. The first part: ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}={L}_{\mathrm{bol}}/{L}_{\mathrm{Edd}}/\eta$ and η = 0.038. The last part: ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}(\eta )={L}_{\mathrm{bol}}/{L}_{\mathrm{Edd}}/\eta$ and $\eta =0.089{M}_{8}^{0.52}$.

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Table 5. Pearson's Analysis Results

X	Y	Line	r	P	Sample
$\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$	$\mathrm{log}f$	Hβ	0.601	<10−40	1
$\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$	$\mathrm{log}f$	Hα	0.568	<10−40	2
$\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$	$\mathrm{log}f$	Hβ	0.618	<10−40	3
$\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$	$\mathrm{log}f$	Hα	0.588	<10−40	3
$\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$	$\mathrm{log}{L}_{5100}$	Hβ	0.236	<10−40	1
$\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$	$\mathrm{log}{L}_{5100}$	Hα	0.265	<10−40	2
$\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$	$\mathrm{log}{L}_{5100}$	Hβ	0.247	<10−40	3
$\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$	$\mathrm{log}{L}_{5100}$	Hα	0.247	<10−40	3
$\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$	$\mathrm{log}{v}_{\mathrm{FWHM}}$	Hβ	−0.747	<10−40	1
$\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$	$\mathrm{log}{v}_{\mathrm{FWHM}}$	Hα	−0.757	<10−40	2
$\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$	$\mathrm{log}{v}_{\mathrm{FWHM}}$	Hβ	−0.753	<10−40	3
$\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$	$\mathrm{log}{v}_{\mathrm{FWHM}}$	Hα	−0.787	<10−40	3
$\mathrm{log}{L}_{5100}$	$\mathrm{log}f$	Hβ	0.072	5.4 × 10−12	1
$\mathrm{log}{L}_{5100}$	$\mathrm{log}f$	Hα	0.032	2.1 × 10−3	2
$\mathrm{log}{L}_{5100}$	$\mathrm{log}f$	Hβ	0.096	4.6 × 10−18	3
$\mathrm{log}{L}_{5100}$	$\mathrm{log}f$	Hα	0.028	1.0 × 10−2	3
$\mathrm{log}{L}_{5100}$	$\mathrm{log}{v}_{\mathrm{FWHM}}$	Hβ	0.133	1.5 × 10−37	1
$\mathrm{log}{L}_{5100}$	$\mathrm{log}{v}_{\mathrm{FWHM}}$	Hα	0.073	2.7 × 10−12	2
$\mathrm{log}{L}_{5100}$	$\mathrm{log}{v}_{\mathrm{FWHM}}$	Hβ	0.117	3.9 × 10−26	3
$\mathrm{log}{L}_{5100}$	$\mathrm{log}{v}_{\mathrm{FWHM}}$	Hα	0.090	3.3 × 10−16	3
$\mathrm{log}{v}_{\mathrm{FWHM}}$	$\mathrm{log}f$	Hβ	−0.647	<10−40	1
$\mathrm{log}{v}_{\mathrm{FWHM}}$	$\mathrm{log}f$	Hα	−0.686	<10−40	2
$\mathrm{log}{v}_{\mathrm{FWHM}}$	$\mathrm{log}f$	Hβ	−0.666	<10−40	3
$\mathrm{log}{v}_{\mathrm{FWHM}}$	$\mathrm{log}f$	Hα	−0.693	<10−40	3

Note. X and Y are the relevant quantities presented in samples 1–3. ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}={L}_{\mathrm{bol}}/{L}_{\mathrm{Edd}}/\eta$ and η = 0.038.

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Since f may be affected by F_r, it is possible that f is correlated with L₅₁₀₀. In fact, there are weak correlations between f and L₅₁₀₀ (see Tables 4 and 5). Also, the dependence of f and ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$ on v_FWHM may result in a false correlation. Thus, the partial correlation analysis is needed to test the $\mathrm{log}f$–$\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$ correlation when excluding the influence of v_FWHM and/or L₅₁₀₀. Based on the Pearson's correlation coefficients in Table 5, the first-order partial correlation analysis gives a confidence level of >99.99% for the $\mathrm{log}f$–$\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$ correlation when excluding the dependence on L₅₁₀₀ or v_FWHM (see Table 6). The second-order partial correlation analysis gives a confidence level of >99.99% for the $\mathrm{log}f$–$\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$ correlation when excluding the dependence on v_FWHM and L₅₁₀₀, except for the broad Hα in sample 3 at a confidence level of 99.84% (see Table 6). Thus, a positive correlation exists between f and ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$. This positive correlation is qualitatively consistent with the logical expectation when the overall effect of F_r on the BLR clouds is taken into account to estimate M_RM. In addition, f > f_g = 1 for Hβ and Hα in most of the AGNs (see Figure 1): >96.5% for Hβ in sample 1, >97.2% for Hα in sample 2, and >97.7% for Hβ and >97.4% for Hα in sample 3.

Table 6. Partial Correlation Analysis Results

Name	Order	Line	rp	Pp	Sample
${r}_{\mathrm{log}f\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1},\mathrm{log}{L}_{5100}}$	1	Hβ	0.603	<10−4	1
${r}_{\mathrm{log}f\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1},\mathrm{log}{v}_{\mathrm{FWHM}}}$	1	Hβ	0.232	<10−4	1
${r}_{\mathrm{log}f\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1},\mathrm{log}{v}_{\mathrm{FWHM}}\mathrm{log}{L}_{5100}}$	2	Hβ	0.151	<10−4	1
${r}_{\mathrm{log}f\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1},\mathrm{log}{L}_{5100}}$	1	Hα	0.581	<10−4	2
${r}_{\mathrm{log}f\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1},\mathrm{log}{v}_{\mathrm{FWHM}}}$	1	Hα	0.102	<10−4	2
${r}_{\mathrm{log}f\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1},\mathrm{log}{v}_{\mathrm{FWHM}}\mathrm{log}{L}_{5100}}$	2	Hα	0.055	<10−4	2
${r}_{\mathrm{log}f\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1},\mathrm{log}{L}_{5100}}$	1	Hβ	0.616	<10−4	3
${r}_{\mathrm{log}f\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1},\mathrm{log}{v}_{\mathrm{FWHM}}}$	1	Hβ	0.237	<10−4	3
${r}_{\mathrm{log}f\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1},\mathrm{log}{v}_{\mathrm{FWHM}}\mathrm{log}{L}_{5100}}$	2	Hβ	0.141	<10−4	3
${r}_{\mathrm{log}f\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1},\mathrm{log}{L}_{5100}}$	1	Hα	0.600	<10−4	3
${r}_{\mathrm{log}f\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1},\mathrm{log}{v}_{\mathrm{FWHM}}}$	1	Hα	0.096	<10−4	3
${r}_{\mathrm{log}f\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1},\mathrm{log}{v}_{\mathrm{FWHM}}\mathrm{log}{L}_{5100}}$	2	Hα	0.035	1.6 × 10−3	3

Note. Based on the Pearson's correlation coefficient r in Table 5, the partial correlation coefficient r_p and the p-value P_p of the hypothesis test are estimated using the website for Statistical Computation (http://vassarstats.net/index.html). Orders 1 and 2 denote the first- and second-order partial correlation coefficients, respectively. ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}={L}_{\mathrm{bol}}/{L}_{\mathrm{Edd}}/\eta$ and η = 0.038.

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In order to test the gravitational origin of the redward shift of the broad emission line, we compare z_g to r_BLR/r_g, the dimensionless radius of the BLR in units of r_g. The Spearman's rank correlation test shows a negative correlation between z_g and r_BLR/r_g (see Table 4). This negative correlation is qualitatively consistent with the expectation that z_g is mainly from the gravity of the central black hole, because M_RM cannot be corrected individually for each AGN due to the absence of the individual virial factor that is independent of z_g. So, we can make the overall correction of M_RM to be M_RM/〈f〉, where 〈f〉 is the average of f in samples 1–3 (see Figure 2). This overall correction is equivalent to the parallel shift of the data point in Figure 2. The negative correlation expectation is basically consistent with the trend between z_g and r_BLR/r_g/〈f〉 in Figure 2. This indicates that z_g is dominated by the gravity of the central black hole. In addition, r_g/r_BLR ≲ 0.01 ≪ 1 and 〈f〉r_g/r_BLR < 0.1 for AGNs in Tables 1–3. 〈f〉r_g/r_BLR ≲ 0.01 for >96% of AGNs in sample 1, >97% of AGNs in sample 2, and >96% of AGNs in sample 3. Thus, the Schwarzschild metric is valid and matches the weak-field limit at the optical BLR scales of AGNs in our samples, which are conditions on the validity of Equation (1) (see Liu et al. 2022).

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Since f > 1 for most of the SDSS AGNs in samples 1–3, the f-correction might result in substantial influence on M_RM and ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$. We choose 8169 AGNs in sample 3 to illustrate this influence. On average, the corrected M_RM becomes larger by 1 order of magnitude than M_RM, and the corrected ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$ decreases by about 10 times (see Figure 3). The substantial increase of M_RM will significantly impact the black hole mass function of these SDSS AGNs, e.g., leading to more AGNs with higher masses. The substantial decrease of ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$ loosens the requirement for the accretion rate of the accretion disk and might make the distinction between high- and low-accreting sources less obvious. If L_bol/L_Edd ≥ 0.1, i.e., $\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}\geqslant 0.42$, for high-accreting sources, the percent of AGNs with $\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}\geqslant 0.42$ is 31.4%, but the percent of AGNs with $\mathrm{log}({\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}/f)\geqslant 0.42$ is only 0.3% (see Figure 3). The percent of high-accreting sources is decreased by about 100 times due to the f-correction. In a sense, the f-correction blurs the distinction between high- and low-accreting sources.

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The virial factor of Hβ is consistent with that of Hα for 8169 AGNs in sample 3 (see Figure 4). The Hα lags are consistent with or (slightly) larger than the Hβ lags for RM AGNs (e.g., Kaspi et al. 2000; Bentz et al. 2010; Grier et al. 2017). Because the Hα optical depth is larger than the Hβ optical depth, the optical depth effects may result in the larger Hα lags that cause the Hα emission line to seemingly appear at larger distances than Hβ (see Bentz et al. 2010), even though the Hβ and Hα broad emission lines are from the same region. Thus, it seems that r_BLR(Hβ) ≈ r_BLR(Hα). For broad emission lines with different r_BLR, there will be $f\propto {r}_{\mathrm{BLR}}^{\alpha }$ (α > 0) as F_r is considered and the BLR clouds are in the virialized motion for a given AGN (Liu et al. 2017). The Hβ and Hα BLRs have similar virialized kinematics for type 1 AGNs in SDSS DR16 (Rakić 2022). If r_BLR(Hβ) ≈ r_BLR(Hα) and $f\,\propto \,{r}_{\mathrm{BLR}}^{\alpha }$, it will be expected that f(Hβ) is on the whole consistent with f(Hα) for AGNs in sample 3, as shown in Figure 4.

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Quasars at z ≳ 6 can probe the formation and growth of SMBHs in the Universe within the first billion years after the Big Bang. The quasars, with M_RM ≳ 10⁹ M_⊙ at z ≳ 7 and M_RM ≳ 10¹⁰ M_⊙ at z ≳ 6, make the formation and growth of SMBHs ever more challenging (e.g., Wu et al. 2015; Fan et al. 2023). These SMBHs will need a combination of massive early black hole seeds with highly efficient and sustained accretion (e.g., Fan et al. 2023). However, the single-epoch spectrum method had been widely used to estimate the M_RM of the high-z quasars (e.g., Willott et al. 2010; Wu et al. 2015; Wang et al. 2019; Eilers et al. 2023), and this may result in underestimation of M_RM, overestimation of L_bol/L_Edd, and significant influence on the formation and growth of SMBHs in the early Universe. Based on Equation (4), we will use $\mathrm{log}f=0.8+0.8\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$ to estimate f and study its influence for quasars at z ≳ 6.

There are 113 quasars at 6 ≲ z ≲ 8 with reliable Mg ii–based black hole mass estimates (Fan et al. 2023). These 113 quasars have 〈f〉 = 78 (f = 12–189), $\langle \mathrm{log}({M}_{\mathrm{RM}}/{M}_{\odot })\rangle =9.0$, $\langle \mathrm{log}({{fM}}_{\mathrm{RM}}/{M}_{\odot })\rangle =10.9$, and 〈L_bol/L_Edd/f〉 = 0.01 (see Table7). The f-correction makes M_RM increase by 1–2 orders of magnitude. Also, the substantially reduced L_bol/L_Edd/f =0.007–0.014 will make these 113 quasars accrete at well below the Eddington limit, although likely in the radiatively efficient regime via a geometrically thin, optically thick accretion disk (Shakura & Sunyaev 1973). Based on Equation (7) in Fan et al. (2023), ${M}_{\bullet }(t)\propto \exp (t)$, the growth times of SMBHs in these quasars will increase by a factor of 2.5–5.2 due to the f-correction. Thus, the black hole seeds do not seem to have enough time to grow up for SMBHs in quasars at z ≳ 6, and this gives more strong constraints on the formation and growth of the black hole seeds. Thus, the f-correction will make it more difficult to explain the formation and growth of SMBHs at z ≳ 6; e.g., larger masses of SMBHs need more massive early black hole seeds and/or longer growth times. Bogdán et al. (2024) found evidence for the heavy-seed origin of early SMBHs from a z ≈ 10 X-ray quasar. Our results of corrected masses support heavy-seed origin scenarios of early SMBHs.

Table 7. Quasars at z ≳ 6

Name	Redshift	$\mathrm{log}\tfrac{{M}_{\mathrm{RM}}}{{M}_{\odot }}$	$\mathrm{log}\tfrac{{L}_{3000}}{\mathrm{erg}\,{{\rm{s}}}^{-1}}$	$\tfrac{{L}_{\mathrm{bol}}}{{L}_{\mathrm{Edd}}}$	f	$\mathrm{log}\tfrac{{{fM}}_{\mathrm{RM}}}{{M}_{\odot }}$	$\tfrac{{L}_{\mathrm{bol}}}{{{fL}}_{\mathrm{Edd}}}$
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
J000825.77−062604.42	5.930	8.72	46.32	1.626	127.4	10.82	0.013
J002031.47−365341.82	6.834	9.23	46.42	0.633	59.8	11.01	0.011
J002429.77+391318.97	6.621	8.43	46.18	2.293	167.7	10.66	0.014

Note. L_bol = 5.15L₃₀₀₀, where L₃₀₀₀ is the UV quasar continuum luminosity at rest-frame wavelength 3000 Å (Fan et al. 2023). f is estimated by $\mathrm{log}f=0.8+0.8\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$, where ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$ is estimated by L_bol/L_Edd/η and η = 0.038.

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The highest-redshift quasar, J031343.84−180636.40, at z = 7.6423, has $\mathrm{log}({M}_{\mathrm{RM}}/{M}_{\odot })=9.2$, $\mathrm{log}({{fM}}_{\mathrm{RM}}/{M}_{\odot })=11.0$, and L_bol/L_Edd/f = 0.01. Quasar J0100+2802, the most luminous quasar known at z > 6, has $\mathrm{log}({M}_{\mathrm{RM}}/{M}_{\odot })\sim 10$ and L_bol/L_Edd ∼ 0.8 (Wu et al. 2015). So, J0100+2802 has $\mathrm{log}({{fM}}_{\mathrm{RM}}/{M}_{\odot })\sim 12$ and L_bol/L_Edd/f ∼ 0.01. Quasar J140821.67+025733.2, at z = 2.055, has a black hole mass of 10^11.3 M_⊙ with an uncertainty of 0.4 dex (Kozłowski 2017). It seems reasonable that J0100+2802 has $\mathrm{log}({{fM}}_{\mathrm{RM}}/{M}_{\odot })\sim 12$ with an uncertainty of 0.4. Recently, Kokorev et al. (2023) found an AGN at z = 8.50 with an Hβ-based mass of $\mathrm{log}({M}_{\mathrm{RM}}/{M}_{\odot })\,=8.2$ and L_bol/L_Edd ∼ 0.33. We have $\mathrm{log}({{fM}}_{\mathrm{RM}}/{M}_{\odot })\sim 9.7$ and L_bol/L_Edd/f ∼ 0.01 for this AGN. In addition, samples 1–3 each have 〈L_bol/L_Edd/f〉 = 0.01. It is interesting that 〈L_bol/L_Edd/f〉 =0.01 exists for quasars/AGNs at z < 0.35 and ≳6, and perhaps it is a coincidence.

It is not determined whether or not the SMBHs coevolve with their host galaxies (e.g., Kormendy & Ho 2013), especially the SMBHs in AGNs with high accretion rates. Coevolution had been supported by the M_•–σ_* relations of local quiescent galaxies. For 31 nearby galaxies, Tremaine et al. (2002) obtained $\mathrm{log}({M}_{\bullet }/{M}_{\odot })=8.13+4.02\mathrm{log}({\sigma }_{* }/{\sigma }_{0})$ with σ₀ = 200 km s⁻¹. McConnell & Ma (2013) presented a revised scaling relation of $\mathrm{log}({M}_{\bullet }/{M}_{\odot })=8.32\,+5.64\mathrm{log}({\sigma }_{* }/{\sigma }_{0})$ for dynamical measurements of M_• at the centers of 72 nearby galaxies. For 72 nearby quiescent galaxies with dynamical measurements of M_•, Woo et al. (2013) obtained $\mathrm{log}({M}_{\bullet }/{M}_{\odot })=8.37+5.31\mathrm{log}({\sigma }_{* }/{\sigma }_{0})$. For 19 local luminous AGNs at z < 0.01, Caglar et al. (2020) obtained$\mathrm{log}({M}_{\bullet }/{M}_{\odot })=8.14+3.38\mathrm{log}({\sigma }_{* }/{\sigma }_{0})$.

We collect σ_* from Table 1 in Woo et al. (2015) for AGNs in our samples and M_RM, σ_*, L₅₁₀₀, etc. for other SDSS quasars from Table 1 in Shen et al. (2015b). There are 62 AGNs and 88 quasars collected (see Tables 8 and 9). The 62 AGNs are at z = 0.013–0.100 with 〈z〉 = 0.063, which is beyond the local Universe. The 88 quasars are at z = 0.116–0.997 with 〈z〉 = 0.581, which is well beyond the local Universe. The significant difference of redshift might influence whether these 150 sources follow the same M_•–σ_* relationship as the local sources. The (M_RM, σ_*) data of these 150 soures do not roughly follow the four local M_•–σ_* relations, and the deviation of the 88 quasars is more obvious (see Figure 5). When studying the coevolution of SMBHs, the local M_•–σ_* relation is basically equivalent to the local black hole–galaxy bulge mass relation. Quasars at z ∼ 6 are above the local mass relation (e.g., Fan et al. 2023). So, the quasars at z ∼ 6 should be above the local M_•–σ_* relation. Thus, these z ≳ 6 quasars using fM_RM might be above the local M_•–σ_* relations.

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Table 8. 62 SDSS AGNs in M_•–σ_* Map Research

Name	Redshift	$\mathrm{log}\tfrac{{M}_{\mathrm{RM}}}{{M}_{\odot }}$	$\tfrac{{L}_{\mathrm{bol}}}{{L}_{\mathrm{Edd}}}$	f	$\tfrac{{\sigma }_{* }}{\mathrm{km}\,{{\rm{s}}}^{-1}}$	$\mathrm{log}\tfrac{{{fM}}_{\mathrm{RM}}}{{M}_{\odot }}$
(1)	(2)	(3)	(4)	(5)	(6)	(7)
J010409.16+000843.7	0.071	6.78	0.065	6.4	66 ± 16	7.59
J030417.78+002827.4	0.045	6.54	0.147	25.9	88 ± 8	7.95
J073106.87+392644.7	0.048	6.39	0.120	29.8	72 ± 14	7.86

Note. The σ_* of the 62 AGNs in our samples are taken from Table 1 in Woo et al. (2015).

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Table 9. 88 SDSS Quasars in M_•–σ_* Map Research

Name	Redshift	$\mathrm{log}\tfrac{{M}_{\mathrm{RM}}}{{M}_{\odot }}$	$\mathrm{log}\tfrac{{L}_{5100}}{\mathrm{erg}\,{{\rm{s}}}^{-1}}$	$\tfrac{{L}_{\mathrm{bol}}}{{L}_{\mathrm{Edd}}}$	f	$\tfrac{{\sigma }_{* }}{\mathrm{km}\,{{\rm{s}}}^{-1}}$	$\mathrm{log}\tfrac{{{fM}}_{\mathrm{RM}}}{{M}_{\odot }}$
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)
141359.51+531049.3	0.8982	7.94	44.11	0.117	15.6	121 ± 31	9.13
141324.28+530527.0	0.4559	8.36	43.91	0.028	4.9	191 ± 4	9.05
141323.27+531034.3	0.8492	8.93	44.28	0.017	3.4	166 ± 20	9.46

Note. The 88 SDSS quasars are taken from Table 1 in Shen et al. (2015b). L_bol = 9.8L₅₁₀₀. f is estimated by $\mathrm{log}f=0.8+0.8\mathrm{log}{\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$, where ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$ is estimated by L_bol/L_Edd/η and η = 0.038.

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The local luminous AGNs in Caglar et al. (2020) have 〈L_bol/L_Edd〉 = 0.07, which corresponds to f = 10, indicating that the M_•–σ_* relation of Caglar et al. (2020) should be corrected by moving vertically upward by an order of magnitude in the M_•–σ_* map. Though the (fM_RM, σ_*) data of these 150 sources deviate from (are above) these local M_•–σ_* relations, they roughly follow the corrected M_•–σ_* relation (see Figure 5). This deviation implies requirements of more massive black hole seeds, longer growth times, larger AGN duty cycles, and/or higher mass accretion rates in the long-term accretion history for them. Also, it seems that the agreement of the (fM_RM, σ_*) data with the corrected M_•–σ_* relation is better than the agreement of the (M_RM, σ_*) data with the local M_•–σ_* relations (see Figure 5). These results might shed light on the possible redshift evolution in the M_•–σ_* relationship. The formation and growth of the local SMBHs and host galaxies might be different from those of the SMBHs in higher-redshift AGNs/quasars and host galaxies.

The redward shift z_g can also be estimated by the λ_b and λ_n of the Hβ and Hα lines (see Equation (3)). Because of the absence of the uncertainty of λ_n for Hβ in Table 2 of Liu et al. (2019), z_g is estimated by the λ_b and λ_n of Hα for 7552 AGNs in sample 2, z_g(Hα)(b-n). z_g(Hα)(b-n) is roughly consistent with the [O iii] λ5007–based z_g(Hα) (see Figure 6). Considering the uncertainties of z_g(Hα) and z_g(Hα)(b-n) (see columns (3) and (4) in Table 2), they are consistent with each other. Thus, the results of z_g(Hα) are reliable. Based on z_g(Hα) and z_g(Hα)(b-n), the virial factors of f(Hα) and f(Hα)(b-n) are estimated and compared to test their reliabilities. Figure 6 shows that f(Hα) and f(Hα)(b-n) are basically consistent. Considering the uncertainties, which have a mean of 2.0 and median of 0.8 for f(Hα) and a mean of 1.6 and median of 0.7 for f(Hα)(b-n) (see columns (8) and (9) in Table 2), f(Hα) is consistent with f(Hα)(b-n). Thus, the selection of [O iii] λ5007 as a reference to estimate z_g in Equation (3) will not influence our results.

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It is very difficult to get a real individual value of η to estimate ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$ for a large sample of AGNs, because η is closely related to the difficultly measured spin of a black hole. Usually, the Eddington ratio is regarded as a proxy of the accretion rate of the black hole. Even though these correlations of ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$ with f are likely influenced by the unknown individual value of η, there are still correlations of the Eddington ratio with f, because a difference of only 0.038 exists between ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$ and L_bol/L_Edd in Tables 1–3. Davis & Laor (2011) found a strong correlation of $\eta =0.089{M}_{8}^{0.52}$ for a sample of 80 Palomar–Green quasars, where M₈ is the black hole mass in units of 10⁸ M_⊙ and η was estimated from the mass accretion rate and L_bol. This empirical relation is used to estimate η in order to test the influence of using η = 0.038 on these correlations of ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$ with f. Correlation analyses are made for ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$ and f in Figure 1, with ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$ to be reestimated by L_bol/L_Edd in Tables 1–3 and the estimated η. There are still correlations very similar to those found in Figure 1 when using these new dimensionless accretion rates (see Figure 7 and Table 4). Thus, the ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$–f correlations found in this work do not result from using the fixed value of η = 0.038.

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Equation (2) can give for v_FWHM, f, and z_g

$\begin{eqnarray}&&\mathrm{log}{\left(\displaystyle \frac{{v}_{\mathrm{FWHM}}}{c}\right)}^{2}=-\mathrm{log}\left(\displaystyle \frac{3}{2}f\right)+\mathrm{log}{z}_{{\rm{g}}},\end{eqnarray} \tag{ 5 }$

which is similar to Equation (6) in Mediavilla et al. (2018). Mediavilla et al. (2018) found a tight correlation between the widths and redward shifts of the Fe iii λ λ2039–2113 blend for lensed quasars, which supports the gravitational interpretation of the Fe iii λ λ2039–2113 redward shifts. A series of lines based on Equation (4) with different f are compared to the observational data points (see Figure 8). From top to bottom, the corresponding f increases. Because of the codependence among the Eddington ratio, dimensionless accretion rate, and v_FWHM, the large ranges of the former two quantities may lead to the large span in the direction roughly perpendicular to these lines (see Figure 8). These lines with f = 1–100 recover the observational data in Figure 8, and this indicates the gravitational origin of z_g. At the same time, the internal physical processes, e.g., the microturbulence, within the BLR cloud can broaden and smooth the line profles (Bottorff & Ferland 2000). Also, the turbulence velocity of the BLR cloud can influence the widths of the line profiles. These turbulence processes will influence v_FWHM and then f for different AGNs. The combination of the column density of the BLR cloud, the metallicity of the BLR cloud, internal physical processes within the BLR cloud, etc., may decrease these correlations in Figure 1 (e.g., Liu et al. 2022).

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There are the various outflows at accretion disk, BLR, NLR, and kiloparsec scales, driven by F_r from AGNs (Dyda & Proga 2018; Kang & Woo 2018; Dannen et al. 2019; Mas-Ribas & Mauland 2019; Nomura et al. 2020; Meena et al. 2021; Singha et al. 2021). Thus, F_r is prevalent and may contribute to the force budget for inflow, e.g., F_r decelerates inflow (Ferland et al. 2009). RM observations of PG 0026+129 indicate a decelerating inflow if z_g originates from inflow. If the decelerating inflow is prevalent, z_g will increase with the increasing r_BLR/r_g, but this expectation is not consistent with the negative trend found in Figure 2. Thus, the inflow seems not to be the origin of z_g. In RM observations, the asymmetric lag maps and shifts of broad emission lines for AGNs usually differ from the theoretical expectation that inflow will generate the redward-shifted broad emission lines with the blueward asymmetric lag maps (e.g., Denney et al. 2010; Zhang et al. 2019; Hu et al. 2020; Feng et al. 2021a, 2021b). This kind of broad emission line may originate from an elliptical disklike BLR (Kovačević et al. 2020; Feng et al. 2021a). Therefore, the redward-shifted broad emission lines in AGNs do not necessarily originate from inflow.

Mejía-Restrepo et al. (2018) determined the virial factor in a smaller set of sources using a different method than proposed here and found a relation whereby f ∝ 1/v_FWHM, which is attributed to inclination effects, but without excluding the possibility of radiation pressure effects over a wide luminosity range. Their sources have $\mathrm{log}[{v}_{\mathrm{FWHM}}/(\mathrm{km}\,{{\rm{s}}}^{-1})]$ ≈ 3.2–4.0, which are much narrower than the $\mathrm{log}[{v}_{\mathrm{FWHM}}/(\mathrm{km}\,{{\rm{s}}}^{-1})]$≈ 2.7–4.4 in our samples. Also, their sources have $\mathrm{log}({M}_{\mathrm{RM}}/{M}_{\odot })\approx$ 7.5–9.7 and $\mathrm{log}[{L}_{5100}/(\mathrm{erg}\,{{\rm{s}}}^{-1})]$ = 44.3–46.2, which are much narrower than the $\mathrm{log}({M}_{\mathrm{RM}}/{M}_{\odot })$ ≈5.2–9.7 and $\mathrm{log}[{L}_{5100}/(\mathrm{erg}\,{{\rm{s}}}^{-1})]$ = 40.6–45.6 in our samples, respectively. There are positive correlations between z_g and v_FWHM for our samples, ${z}_{{\rm{g}}}\propto {v}_{\mathrm{FWHM}}^{1.5}$ (see Figure 9). Based on ${z}_{{\rm{g}}}\propto {v}_{\mathrm{FWHM}}^{1.5}$ and Equation (2), with v_FWHM partly contributed by inclination effects, we have $f\propto 1/{v}_{\mathrm{FWHM}}^{0.5}$, which is qualitatively consistent with but shallower than f ∝ 1/v_FWHM. This discrepancy might be generated by our consideration of radiation pressure and the estimation of M_• using standard thin accretion disk models for sources with the narrower parameter coverage (Mejía-Restrepo et al. 2018). In this sense, the results and interpretations promoted here are consistent with Mejía-Restrepo et al. (2018).

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The AGNs with high accretion rates show shorter time lags by factors of a few compared to the predictions from the r_BLR–L₅₁₀₀ relationship (Du et al. 2015). Du & Wang (2019) found that accretion rate is the main driver for the shortened lags and established a new scaling relation,

$\begin{eqnarray}&&\mathrm{log}{r}_{\mathrm{BLR}}({\rm{H}}\beta )=1.65+0.45\mathrm{log}{L}_{44}-0.35{R}_{\mathrm{Fe}\,{\rm\small{II}}},\end{eqnarray} \tag{ 6 }$

where r_BLR(Hβ) is r_BLR in units of light-days for Hβ, L₄₄ = L₅₁₀₀/(10⁴⁴ erg s⁻¹), and R_{Fe II} is the line ratio of Fe ii to Hβ. Replacing ${r}_{\mathrm{BLR}}=33.65{L}_{44}^{0.533}$ with Equation (6), the mass of the black hole is given by

$\begin{eqnarray}\begin{array}{rcl}\mathrm{log}{M}_{\mathrm{RM}}({R}_{\mathrm{Fe}\,{\rm\small{II}}}) & = & \mathrm{log}{M}_{\mathrm{RM}}-0.083\mathrm{log}{L}_{44}\\ & & -0.35{R}_{\mathrm{Fe}\,{\rm\small{II}}}+0.123,\end{array}\end{eqnarray} \tag{ 7 }$

which is used to estimate the dimensionless accretion rate, ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}({R}_{\mathrm{Fe}\,{\rm\small{II}}})$. Samples 1 and 3 are used to investigate the influence of Equation (6) on the f–${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$ relation. R_{Fe II} is estimated by equivalent widths of Hβ and Fe ii taken from Table 2 of Liu et al. (2019) for 5997 AGNs in sample 1 and 5365 AGNs in sample 3. First, ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}({R}_{\mathrm{Fe}\,{\rm\small{II}}})$ is overall consistent with the original ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$ (see Figure 10). Second, f is well correlated with ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}({R}_{\mathrm{Fe}\,{\rm\small{II}}})$ (see Figure 10), and Equation (6) has a slight impact on the f–${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$ relation. Also, r_BLR(R_{Fe II} )/r_g(R_{Fe II} ) is estimated, and there exists the anticorrelation trend between z_g and r_BLR(R_{Fe II} )/r_g(R_{Fe II} )/〈f〉 (see Figure 11), same as in Figure 2. The potential effect of $\mathop{{\mathscr{M}}}\limits^{\,.}$, especially at the high mass accretion rate end (Du et al. 2015), does not lead to qualitatively different results of r_BLR/r_g.

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Based on the assumption of a gravitational origin for the redward shifts of the broad emission lines Hβ and Hα and their widths and redward shifts for more than 8000 SDSS DR7 AGNs with z < 0.35, we measured the virial factor in M_RM, estimated by the RM method and/or the relevant secondary methods. The measured virial factor contains the overall effect of F_r from accretion disk radiation and the geometric effect of the BLR. Our findings can be summarized as follows.

• 1.  
  There are positive correlations of f with ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$ and L_bol/L_Edd, which are a combined effect of several physical mechanisms, such as the Doppler effect, the gravitational redshift, the gravity of the black hole, the radiation pressure force, etc. f spans a large range, and f > 1 for >96% of AGNs in samples 1–3. The f-correction makes the percent of high-accreting AGNs decrease by about 100 times and blurs the distinction between high- and low-accreting sources.
• 2.  
  z_g is anticorrelated with r_BLR/r_g. z_g and r_BLR/r_g/〈f〉 marginally follow the 1:1 line. A series of lines with different f basically reproduce the v_FWHM–z_g distribution for the broad Hβ and Hα. These results suggest that the redward shifts of the broad Hβ and Hα are governed by the gravity of the central SMBHs.
• 3.  
  For quasars at z ≳ 6, the f-correction changes them from near-Eddington accreting sources to low-accreting sources, likely in the radiatively efficient regime via a geometrically thin, optically thick accretion disk. The f-corrected masses indicate that quasars at z ≳ 6 have more massive early black hole seeds and longer growth times, supporting the heavy-seed origin scenarios of early SMBHs. These results will make it more challenging to explain the formation and growth of SMBHs at z ≳ 6.
• 4.  
  A total of 62 AGNs and 88 quasars beyond the local Universe do not follow the local M_•–σ_* relations. After the f-correction, these 150 sources are above the local M_•–σ_* relations, but they roughly follow the f-corrected M_•–σ_* relation of the local luminous AGNs in Caglar et al. (2020). These results might shed light on possible redshift evolution in the M_•–σ_* relationship.

Our results show that radiation pressure force should be considered in estimating the virial masses of SMBHs. The usually used values of f should be corrected for high-accreting AGNs, especially quasars at z ≳ 6. The f-correction to M_RM will make the coevolution (or not) of SMBHs and host galaxies more complex for local and higher-redshift sources. Positive correlations of f with ${\mathop{{\mathscr{M}}}\limits^{\,.}}_{{f}_{{\rm{g}}}=1}$ and L_bol/L_Edd need to be further tested by the redward-shifted broad emission lines of the RM AGNs without the signatures of inflow and outflow in the BLR, which can be picked out by the velocity-resolved time lag maps.

We are grateful to the anonymous referee for constructive comments and suggestions that significantly improved this manuscript. We are thankful for the financial support of the National Key R&D Program of China (grant No. 2021YFA1600404), the National Natural Science Foundation of China (grants Nos. 12373018, 12303022, 12203096, 12063005, and 11991051), Yunnan Fundamental Research Projects (grant Nos. 202301AT070358 and 202301AT070339), Yunnan Postdoctoral Research Foundation Funding Project, Special Research Assistant Funding Project of Chinese Academy of Sciences, and the science research grants from the China Manned Space Project with grant No. CMS-CSST-2021-A06. We acknowledge the Program for Innovative Research Team (in Science and Technology) in University of Yunnan Province (IRTSTYN).