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

Under the hypothesis of gravitational redshift induced by the central supermassive black hole and based on line widths and shifts of redward-shifted Hβ and Hα broad emission lines for more than 8000 Sloan Digital Sky Survey DR7 active galactic nuclei (AGNs), we measure the virial factor in determining supermassive black hole masses. The virial factor had been believed to be independent of accretion radiation pressure on gas clouds in broad-line regions (BLRs) and only dependent on the inclination effects of BLRs. The virial factor measured spans a very large range. For the vast majority of AGNs (>96%) in our samples, the virial factor is larger than the f = 1 usually used in the literature. The f-correction makes the percent of high-accreting AGNs decrease by about 100 times. There are positive correlations of f with the dimensionless accretion rate and Eddington ratio. The redward shifts of Hβ and Hα are mainly of gravitational origin, confirmed by a negative correlation between the redward shift and the dimensionless radius of the BLR. Our results show that radiation pressure force is a significant contributor to the measured virial factor, containing the inclination effects of the BLR. The usually used values of f should be corrected for high-accreting AGNs, especially high-redshift quasars. The f-correction increases their masses by 1–2 orders of magnitude, which will make it more challenging to explain the formation and growth of supermassive black holes at high redshifts.


INTRODUCTION
Black hole mass, M • , is an important fundamental parameter of black hole.Reliable measurement of M • always is a key issue of black hole related researches.For active galactic nuclei (AGNs), the reverber-ation mapping (RM) method or the relevant secondary methods based on single-epoch spectra were widely used to measure M • by a virial mass M RM = f v 2 FWHM r BLR /G when clouds in broad-line region (BLR) are in virialized motion, where f is the virial factor, v FWHM is full width at half maximum of broad emission line, r BLR is radius of 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 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 source, and τ ob is the observed time lag of the broadline 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 -L 5100 relation for Hβ emission line of the RM AGNs, where L 5100 is 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 had been carried out (e.g., King et al. 2015;Shen et al. 2015aShen et al. ,b, 2016;;Grier et al. 2017;Hoormann et al. 2019;Shen et al. 2019).Non-survey RM observation researches had been made for more than 100 AGNs over the last several decades (e.g., Kaspi & Netzer 1999;Kaspi et al. 2000;Peterson et al. 2005;Bentz et al. 2006;Kaspi et al. 2007;Bentz et al. 2010;Denney et al. 2010;Barth et al. 2011;Haas et al. 2011;Pozo Nuñez et al. 2012;Du et al. 2014;Pei et al. 2014;Wang et al. 2014;Barth et al. 2015;Du et al. 2015;Hu et al. 2015;Bentz et al. 2016;Du et al. 2016;Lu et al. 2016;Pei et al. 2017;Du et al. 2018a,b;Xiao et al. 2018a,b;Zhang et al. 2019;Hu et al. 2020;Bentz et al. 2021;Feng et al. 2021a,b;Hu et al. 2021;Li et al. 2021;Lu et al. 2021;Bentz et al. 2022;Li et al. 2022;Bentz et al. 2023).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 with σ * to be stellar velocity dispersion of 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 BLR, and the issues related to masses of 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), coevolution (or not) of SMBHs and host galaxies (e.g., Tremaine et al. 2002;Kormendy & Ho 2003;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,b) 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 DR 5 quasars with redward shifted Hβ and Fe II broad emission lines, Liu et al. (2022) made further efforts of researching f and f σ .Fe IIIλλ 2039-2113 UV line blend comes from an inner region of 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 BLR are yet unclear for AGNs (e.g., Wang et al. 2017).Thus, it is unclear of the origin of redward shifts of broad emission lines.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 redward shift of the Fe III blend is preferred than 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 DR 7 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 DR 16, Rakić (2022) showed that the BLR gas seems to be virialized.The velocity-resolved lag maps of Hβ broad emission line for Mrk 50 and SBS 1518+593 show characteristic of Keplerian disk 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 accretion disk has significant influences on the stability and dynamics of clouds in BLR (e.g., Marconi et al. 2008;Netzer & Marziani 2010;Krause et al. 2011Krause et al. , 2012;;Naddaf et al. 2021).The dynamics of clouds can determine the three-dimensional geometry of BLR (Naddaf et al. 2021).However, radiation pressure was not considered in estimating M RM , and the virial factor had been believed to be only from the geometric effect of BLR.Lu et al. (2016) found that the BLR of NGC 5548 could be jointly controlled by the radiation pressure force from accretion disk and the gravity of the central black hole.Krause et al. (2011) found that stable orbits of clouds in 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 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, out of Table 2 in Liu et al. (2019), will be adopted to investigate the virial factor, 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 method.Section 3 describes sample selection.Section 4 presents analysis and results.Section 5 is potential influence on quasars at z 6. Section 6 is potential influ-ence on M • − σ * map of AGNs.Section 7 presents discussion, and Section 8 is conclusion.Throughout this paper, we assume a standard cosmology with H 0 = 70 km s −1 Mpc −1 , Ω M = 0.3, and Ω Λ = 0.7 (Spergel et al. 2007).

METHOD
A BLR cloud is subject to gravity of 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, many unknown physical parameters are likely various for different AGNs.Thus, a new method was proposed to measure f and then M RM , avoiding to use the averages of the virial factor or the numerical calculation of F r (Liu et al. 2017(Liu et al. , 2022)).
The virial factor formula in Liu et al. (2022) was derived from the Schwarzschild metric for clouds in the virialized motion 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 • /c 2 ), we have (2) Mediavilla & Jiménez-Vicente (2021) pinpointed that the observed redward shift of the Fe IIIλλ 2039-2113 emission line blend in quasars originates from the gravity of black hole, while these Fe IIIλλ 2039-2113 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 not only determined by the kinematics of BLR, but also determined by the gravity of black hole.The virialized assumption in measuring M RM will ensure that the line position should be governed by the gravity of 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. (2017Liu et al. ( , 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 4 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 , 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 14584 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 wavebands, 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 broad emission lines Hβ and Hα are defined as

SAMPLE SELECTION
where λ b is the centroid wavelength of 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 narrow emission line corrected by z SDSS , and λ 0 is the vacuum wavelength of spectrum line (λ 0 = 4862.68for Hβ, λ 0 = 6564.61for Hα, and λ 0 = 5008.24for [O III]λ5007)1 .
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 standard of estimating z g for the broad Hβ and Hα, the [O III]λ5007 line is used in Equation (3).First, one of choice criteria is AGN's flag = 0, which means no emission line with multiple peaks (Liu et al. 2019), because that the multiple peaks of 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 broad emission lines are redward within 1σ uncertainties.Because the empirical r BLR -L 5100 relation is established for broad emission line Hβ, the relevant researches on the virial factor are made with the broad Hβ and Hα in this work.9185 AGNs are selected out of the 14584 AGNs as Sample 1 only for the broad Hβ.9271 AGNs are selected out of the 14584 AGNs as Sample 2 only for the broad Hα.The cross-identified AGNs in Samples 1 and 2 are used as Sample 3 that contains 8169 AGNs with 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), L 5100 , M RM , L bol /L Edd , and the dimensionless accretion rate Ṁ fg=1 .The bolometric luminosity L bol was estimated in Liu et al. (2019) using L bol = 9.8L 5100 (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).Ṁ fg=1 = L bol /L Edd /η, where η is the efficiency of converting rest-mass energy to radiation.Hereafter, in addition to special statement, we adopt η = 0.038 (Du et al. 2015).
For our selected AGNs, L 5100 spans four orders of magnitude, M RM spans more than four orders of magnitude, and L bol /L Edd spans more than three orders of magnitude.These parameters cover at least one order of magnitude wider than those in Liu et al. (2022).The measured values of f span more than three orders of magnitude, which cover at least one order of magnitude wider than those in Liu et al. (2022).These much wider parameters can ensure that this work is feasible.
-7 -Table 1.The Relevant Parameters for 9185 AGNs in SDSS DR7 for Sample 1   2 of Liu et al. (2019) or converted from the relevant quantities in Table 2 of Liu et al. (2019).-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).(This table is available in its entirety in machine-readable form.)    2 of Liu et al. (2019) or converted from the relevant quantities in Table 2 of Liu et al. (2019).(This table is available in its entirety in machine-readable form.)-9 -    2 of Liu et al. (2019) or converted from the relevant quantities in Table 2 of Liu et al. (2019).-999 and † are same as in Table 1.(This table is available in its entirety in machine-readable form.)

ANALYSIS AND RESULTS
In order to study the correlation between f , Ṁ fg=1 , L 5100 , 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 correlation between f and Ṁ fg=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 pvalue 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 Ṁ fg=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 ∆a and ∆b, but it does not take into account the obervational errors of 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 10 4 times to generate the distributions of a MC , b MC , ∆a MC , and ∆b 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 ∆a where the p-values of the hypothesis test are < 10 −40 , and in the fittings, the uncertainty of log Ṁ fg=1 is taken to be 0.4 determined by the uncertainty of 0.4 dex usually used in M RM .Equations ( 4a)-(4d) correspond to the best fits to data sets ( f , Ṁ fg=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 ∝ Ṁ 0.8−0.9fg=1 , and then f ∝ (L bol /L Edd ) 0.8−0.9 .Since f may be affected by F r , it is possible that f is correlated with L 5100 .In fact, there are weak correlations between f and L 5100 (see Tables 4-5).Also, the dependence of f and Ṁ fg=1 on v FWHM may result in a false correlation.Thus, the partial correlation analysis is needed to test the log f -log Ṁ fg=1 correlation when excluding the influence of v FWHM and/or L 5100 .Based on the Pearson's correlation coefficients in Table 5, the 1st order partial correlation analysis gives a confidence level of > 99.99% for the log f -log Ṁ fg=1 correlation when excluding the dependence on L 5100 or v FWHM (see Table 6).The 2nd order partial correlation analysis gives a confidence level of > 99.99% for the log f -log Ṁ fg=1 correlation when excluding the dependence on v FWHM and L 5100 , except for the broad Hα in Sample 3 at a confidence level of 99.84% (see Table 6).Thus, the positive correlation exists between f and Ṁ fg=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 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.
In order to test the gravitational origin of the redward shift of broad emission line, we compare z g to r BLR /r g , the dimensionless radius of BLR in units of r g .The Spearman's rank correlation test shows 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 that M RM can not 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% AGNs in Sample 1, > 97% AGNs in Sample 2, and for > 96% 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).
Since f > 1 for most of SDSS AGNs in Samples 1-3, the f correction might result in substantial influence on M RM and Ṁ fg=1 .We choose 8169 AGNs in Sample 3 to illustrate this influence.On average, the corrected M RM becomes larger by one order of magnitude than M RM , and the corrected Ṁ fg=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 Ṁ fg=1 loosens the requirement for accretion rate of accretion disk, and might make the distinction between high-and low-accreting sources less obvious.If L bol /L Edd ≥ 0.1, i.e., log Ṁ fg=1 ≥ 0.42, for high-accreting sources, the percent of AGNs with log Ṁ fg=1 ≥ 0.42 is 31.4%,but the percent of AGNs with log( Ṁ fg=1 / f ) ≥ 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.
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 the 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 ∝ r α BLR (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 DR 16 (Rakić 2022).If r BLR (Hβ) ≈ r BLR (Hα) and f ∝ r α BLR , 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. r BLR /r g corrected by f = 12.5 for Sample 2. Panel (c): Hβ shift z g vs. r BLR /r g corrected by f = 13.9 for Sample 3. Panel (d): Hα shift z g vs. r BLR /r g corrected by f = 12.7 for Sample 3. The Spearman test shows negative correlations between these two physical quantities.Note.-Based on the Pearson's correlation coefficient r in Table 5, the partial correlation coefficient rp and the p-value Pp of the hypothesis test are estimated using the Website for Statistical Computation (http://vassarstats.net/index.html).Orders 1 and 2 denote the 1st and 2nd order partial correlation coefficients, respectively.Ṁfg=1 = Lbol/LEdd/η and η = 0.038.

POTENTIAL INFLUENCE ON QUASARS AT z 6
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 10 9 M ⊙ at z 7 and with M RM 10 10 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 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 log f = 0.8 + 0.8 log Ṁ fg=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), log(M RM /M ⊙ ) = 9.0, log( f M RM /M ⊙ ) = 10.9, and L bol /L Edd / f = 0.01 (see Table 7).The f correction makes M RM increase by one-two orders of magnitude.Also, substantially reduced L bol /L Edd / f = 0.007-0.014will make these 113 quasars accreting 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 • (t) ∝ exp(t), the growth times of SMBHs in these quasars will increase by a factor of 2.5-5.2due to the f correction.Thus, the black hole seeds don't 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. (2023) found evidence for 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.
We collect σ * from Table 1 in Woo et al. (2015) for AGNs in our samples, and M RM , σ * , L 5100 , etc for other SDSS quasars from Table 1 in Shen et al. (2015b).There are 62 AGNs and 88 quasars collected (see Tables 8-9).These 62 AGNs are at z = 0.013-0.100with z = 0.063, which are beyond the local Universe.These 88 quasars are at z = 0.116-0.997with z = 0.581, which are 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 these four local M • − σ * relations, and the deviation of these 88 quasars is more obvious (see Figure 5).As study the coevolution of SMBHs with host galaxies, 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, these quasars at z ∼ 6 should be above the local M • − σ * relation.Thus, these z 6 quasars using f M RM might be above these local M • − σ * relations.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 ( f M 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 long-term accretion history for them.Also, it seems that agreement of the ( f M RM ,σ * ) data with the corrected M • − σ * relation is better than agreement of the (M RM ,σ * ) data with these local M • − σ * relations (see Figure 5).These results might shed light on 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.

DISCUSSION
The redward shift z g can also be estimated by 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 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 [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 3-4 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 are 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 a median of 0.8 for f (Hα) and a mean of 1.6 and a median of  -Lbol = 5.15L3000, and L3000 is the UV quasar continuum luminosity at rest-frame wavelength 3000 Å (Fan et al. 2023).f is estimated by log f = 0.8 + 0.8 log Ṁfg=1, where Ṁfg=1 is estimated by Lbol/LEdd/η and η = 0.038.(This table is available in its entirety in machine-readable form.)   1 in Shen et al. (2015b).Lbol = 9.8L5100.f is estimated by log f = 0.8 + 0.8 log Ṁfg=1, where Ṁfg=1 is estimated by Lbol/LEdd/η and η = 0.038.(This table is available in its entirety in machine-readable form.)0.7 for f (Hα)(b-n) (see columns 8-9 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.
It is very difficult to get real individual value of η to estimate Ṁ fg=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 accretion rate of black hole.Even though these correlations of Ṁ fg=1 with f are likely influenced by the unknown individual value of η, there are still correlations of the Eddington ratio with f , because only a difference of 0.038 exists between Ṁ fg=1 and L bol /L Edd in Tables 1-3.Davis & Laor (2011) found a strong correlation of η = 0.089M 0.52 8 for a sample of 80 Palomar-Green quasars, where M 8 is the black hole mass in units of 10 8 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 Ṁ fg=1 with f .Correlation analyses are made for Ṁ fg=1 and f in Figure 1 with Ṁ fg=1 to be re-estimated 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, these Ṁ fg=1f correlations found in this work do not result from using the fixed value of η = 0.038.
Equation ( 2) can give for v FWHM , f , and 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 IIIλλ 2039-2113 blend for lensed quasars, which supports the gravitational interpretation of the Fe IIIλλ 2039-2113 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 micro-turbulence, 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, 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).
There are the various outflows at accretion disk scales, the BLR scales, the NLR scales and the kpc scales, driven by F r from AGNs (Kang & Woo 2018;Dyda & Proga 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,b).This kind of broad emission lines 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 log[v FWHM /(km s −1 )] ≈ 3.2-4.0,which are much narrower than log[v FWHM /(km s −1 )] ≈ 2.7-4.4 in our samples.Also, their sources have log(M RM /M ⊙ ) ≈ 7.5-9.7 and log[L 5100 /(erg s −1 )] = 44.3-46.2,which are much narrower than log(M RM /M ⊙ ) ≈ 5.2-9.7 and log[L 5100 /(erg s −1 )] = 40.6-45.6 in our samples, respectively.There are positive correlations between z g and v FWHM for our samples, z g ∝ v 1.5 FWHM (see Figure 9).Based on z g ∝ v 1.5 FWHM and Equation (2) with v FWHM partly contributed from inclination effects, we have f ∝ 1/v 0.5 FWHM , 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, these results and interpretations promoted here are consistent with Mejía-Restrepo et al. (2018).
The AGNs with high-accretion rates show shorter time lags by factors of a few compared to the predictions from the r BLR -L 5100 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: where r BLR (Hβ) is r BLR in units of light days for Hβ, L 44 = L 5100 /(10 44 erg s −1 ), and R FeII is the line ratio of Fe II to Hβ. Replacing r BLR = 33.65L 0.533 44 with Equation ( 6), the mass of black hole is given by log which is used to estimate the dimensionless accretion rate, Ṁ fg=1 (R FeII ).Samples 1 and 3 are used to investigate the influence of Equation ( 6) on the f -Ṁ fg=1 relation.R FeII 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, Ṁ fg=1 (R FeII ) is overall consistent with the original Ṁ fg=1 (see Figure 10).Second, f is well correlated with Ṁ fg=1 (R FeII ) (see Figure 10), and Equation ( 6) has a slight impact on the f -Ṁ fg=1 relation.Also, r BLR (R FeII )/r g (R FeII ) is estimated, and there exits the anti-correlation trend between z g and r BLR (R FeII )/r g (R FeII )/ f (see Figure 11), same as in Figure 2. The potential effect of Ṁ , especially at the high mass accretion rate end (Du et al. 2015), do not lead to qualitatively different results of r BLR /r g .

CONCLUSION
Based on the assumption of a gravitational origin for the redward shifts of 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 BLR.Our findings can be summarized as follows: 1.There are positive correlations of f with Ṁ fg=1 and L bol /L Edd , which are a combined effect of several physical mechanisms, such as the Doppler effects, the gravitational redshift, the gravity of black hole, the radiation pressure force, etc. f spans a large range, and f > 1 for >96% 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 anti-correlated 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 makes them from the close Eddington accreting sources become 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 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.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 the local sources and the higher redshift sources.Positive correlations of f with Ṁ fg=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 BLR, which can be picked out by the velocity-resolved time lag maps.

Fig. 1 .
Fig. 1.f vs. Ṁ fg=1 .Panel (a): for Hβ of 9185 AGNs in Sample 1. Panel (b): for Hα of 9271 AGNs in Sample 2. Panel (c): for Hβ of 8169 AGNs in Sample 3. Panel (d): for Hα of 8169 AGNs in Sample 3. The dashed green line denotes f g = 1 for v FWHM .The dashed red line denotes the best bisector linear fit.The blue solid line denotes the 95% confidence ellipse.∆ f is the fitting residuals.

Fig. 3 .Fig. 4 .
Fig. 3.-The f correction effects on mass and accretion rate for 8169 AGNs in Sample 3. The blue lines are y = 0.42 and x = 0.42.
Fig. 5.-M • − σ * map for 62 AGNs in our samples (solid circles), and 88 quasars in Shen et al. (2015b) (open squares).The black symbols correspond to M RM , and the colourful symbols are f M RM .The bule dashed line is the Tremaine et al. (2002) relation for nearby inactive galaxies.The olive dashed line is the Woo et al. (2013) relation for nearby quiescent galaxies.The magenta dashed line is the McConnell & Ma (2013) relation for 72 nearby galaxies.The cyan dashed line is the Caglar et al. (2020) relation for local luminous AGNs.The cyan dash-dotted line is the Caglar et al. (2020) relation moved vertically upward by an order of magnitude.

4. 62
AGNs and 88 quasars, beyond the local Universe, do not follow these local M • − σ * relations.After the f correction, these 150 sources are above these local M • − σ * relations, but they roughly follow the f -corrected M • − σ * relation of these local luminous AGNs in Caglar et al. (2020).These results might shed light on possible redshift evolution in the M • − σ * relationship.

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

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

Table 6 .
Partial correlation analysis results

Table 8 .
62 SDSS AGNs in M • − σ * map research Note.σ* of 62 AGNs in our samples are taken from Table 1 in Woo et al. (2015).(This table is available in its entirety in machine-readable form.)