A LOCAL BASELINE OF THE BLACK HOLE MASS SCALING RELATIONS FOR ACTIVE GALAXIES. III. THE M_BH–σ RELATION

In Paper I, we described in detail an image analysis code "Surface Photometry and Structural Modeling of Imaging Data (SPASMOID)" designed to allow for simultaneous fitting of multi-filter images with arbitrary constraints between the parameters in each band (Bennert et al. 2011a, 2011b). This joint multi-wavelength analysis enables a much more powerful disentanglement of the nuclear and host-galaxy components than using single-band imaging alone (e.g., using GALFIT, Peng et al. 2002). The bluer bands provide a robust measurement of the normalization of the nuclear flux while the redder data exploit the more favorable contrast between the AGN and the host galaxy to constrain the morphological structure of the latter. The approach of simultaneously using structural and photometric information is most successful for imaging of AGN hosts given the presence of a bright AGN point source. SPASMOID's reliance on a Markov Chain Monte Carlo (MCMC) technique also provides realistic uncertainties and the ability to explore covariances between various model parameters.

We use SPASMOID to perform surface-brightness photometry on the SDSS images, simultaneously fitting the AGN by a point-spread function (PSF) and the host galaxy by a combination of spheroid (Sérsic with free index n, in a range between 0.7 and 4.7), and if present, disk (exponential profile), and bar (Sérsic with index n = 0.5, i.e., a Gaussian). The results given in Table 1 correspond to the maximum a posteriori (MAP) values. Note that this approach differs slightly from Paper I, in which we fitted the spheroid with a single de Vaucouleurs (1948) profile. We thus here included all 25 objects from Paper I again and ran SPASMOID on the full sample of 103 objects. From the final sample, 11 objects were omitted due to either image defects, bright nearby stars complicating the fit, or no reliable fit achieved.

We use the Sérsic index to distinguish between classical bulges and pseudo-bulges (see Section 5.1). The spheroid radius is used to determine the stellar velocity dispersion within that effective radius (see Section 3.2). The PSF g'-band magnitude is corrected for Galactic extinction (subtracting the SDSS DR7 "extinction_g''" column), and then extrapolated to 5100 Å, assuming a power law of the form ${f}_{\nu }$ ∝ ${\nu }^{\alpha }$ with α = −0.5. The resulting AGN luminosity free of host-galaxy contribution (except potential dust attenuation) is used for ${M}_{\mathrm{BH}}$ measurements (see Section 3.3). In the subsequent papers of the series (V. N. Bennert et al. 2015, in preparation), we will discuss luminosity and stellar masses of the different components when deriving the remaining ${M}_{\mathrm{BH}}$-scaling relations.

From the full sample of 103 objects, the spectra of 21 objects did not yield a robust measurement of the stellar kinematics, due to dominating AGN flux and high redshift (12 objects) or problems with the instrument (9 objects), so our final kinematic sample consists of 82 objects (see Paper II, Table 2). While the exclusion of 10% of objects with faint galaxies compared to the AGN can in principle introduce a systematic effect in our sample, we consider this effect negligible, given the overall sample size.

For the majority of objects, broad nuclear Fe ii emission (∼5150–5350 Å) is present and interferes with the measurements of both σ in the MgIb range and broad Hβ width. Thus, for those objects, a set of IZw1 templates (varying width and strength) and a featureless AGN continuum were fitted simultaneously and subtracted.

Stellar-velocity dispersion σ was measured from three different spectra regions: around CaH&K$\lambda \lambda 3969,3934$ (hereafter CaHK), around the Mg Ib $\lambda \lambda \lambda 5167,5173,5184$ (hereafter MgIb) lines and around Ca ii $\lambda \lambda \lambda 8498,8542,8662$ (hereafter CaT). σ measurements were obtained from a Python-based code described in detail in Papers I and II. In short, it is based on the algorithm by van der Marel (1994), fitting a linear combination of Gaussian-broadened (30–500 km s⁻¹) template spectra (G and K giants of various temperatures as well as spectra of A0 and F2 giants from the Indo-US survey) and a polynomial continuum using a MCMC routine (with the best derived σ measurements corresponding to the MAP values). Telluric and AGN emission lines were masked and thus excluded from the fit.

In Paper II, σ measurements were derived for both aperture and spatially resolved spectra, i.e., as a function of distance from the center. The extracted spatially resolved spectra were used to determine the velocity dispersion within the spheroid effective radius ${\sigma }_{\mathrm{spat},\ \mathrm{reff}}$, free from broadening due to a rotating disk component. (Note that this assumes that the spheroid component dominates the velocity dispersion within the spheroid effective radius, and that contributions from bar and disk are negligible in comparison.) To do so, we calculate the velocity dispersion within the spheroid effective radius as determined from the surface photometry:

$$\begin{eqnarray}&&{\sigma }_{\mathrm{spat},\mathrm{reff}}^{2}=\displaystyle \frac{{\displaystyle \int }_{-{R}_{\mathrm{eff}}}^{{R}_{\mathrm{eff}}}({\sigma }_{\mathrm{spat}}^{2}(r)+{v}_{\mathrm{spat}}^{2}(r))\cdot I(r)\cdot r\cdot {dr}}{{\displaystyle \int }_{-{R}_{\mathrm{eff}}}^{{R}_{\mathrm{eff}}}I(r)\cdot r\cdot {dr}}\end{eqnarray} \tag{ 1 }$$

with I(r) = $I(\mathrm{reff})\cdot \mathrm{exp}(-{\kappa }_{n}\cdot [{(r/{r}_{\mathrm{reff}})}^{1/n}-1])$ the surface brightness of the spheroid fitted as a Sérsic profile. Here, ${v}_{\mathrm{spat}}$ is the rotational component of the spheroid. We approximated ${\kappa }_{n}$ = 1.9992n-0.3271 (valid for 0.5 $\;\lt \;n\;\lt$ 10, Capaccioli et al. 1989; Prugniel & Simien 1997). (Note that n, ${r}_{\mathrm{reff}}$, and $I(\mathrm{reff})$ are taken from the image analysis.) Since stellar velocity dispersions were measured for spectra on both sides of the center, along the major axis, we integrate from "$-{R}_{\mathrm{eff}}$" to "$+{{\rm{R}}}_{\mathrm{eff}}$." A spline function is used to interpolate over the appropriate radial range, since the ${\sigma }_{\mathrm{spat}}$ measurements are discrete. In Section 5.2, we discuss other σ definitions used in the literature and compare them with the fiducial value used throughout this Paper from Equation (1).

We measure the second moment of the broad Hβ emission line from the central blue Keck/LRIS spectrum (1 square-arcsecond in size),using the IDL-based code implemented by Park et al. (2015), allowing for a multi-component spectral decomposition of the Hβ region. Here, we briefly summarize the procedure. First, we model and subtract the observed continuum by simultaneously fitting the pseudo-continuum, which consists of the AGN featureless power-law continuum, the AGN Fe ii emission template from Boroson & Green (1992), and the host-galaxy starlight templates from the Indo-US spectral library (Valdes et al. 2004), in the emission-line free windows of 4430–4770 Å and 5080–5450 Å. Second, we model the continuum-subtracted Hβ emission-line region by simultaneously fitting Gauss–Hermite series (van der Marel & Franx 1993; Woo et al. 2006; McGill et al. 2008) to the broad and narrow Hβ emission lines and the [O iii] $\lambda \lambda 4959,5007$ narrow emission lines and fitting Gaussian functions to the broad and narrow He ii $\lambda 4686$ emission lines (when blended with the broad Hβ component; see Section 3.1 of Park et al. 2015 for details). The final fits are shown in Figure 3.

From the resulting fit, the second moment of the broad Hβ component (${\sigma }_{{\rm{H}}\beta }$) is combined with the 5100 Å AGN luminosity derived from surface photometryto estimate ${M}_{\mathrm{BH}}$:

$$\begin{eqnarray}&&\mathrm{log}{M}_{\mathrm{BH}}=0.71+6.849+2\mathrm{log}\left(\displaystyle \frac{{\sigma }_{{{\rm{H}}}_{\beta }}}{1000\;\mathrm{km}\;{{\rm{s}}}^{-1}}\right)\\ &&+\;0.549\mathrm{log}\left(\displaystyle \frac{\lambda {L}_{5100}}{{10}^{44}\;\mathrm{erg}\;{{\rm{s}}}^{-1}}\right).\end{eqnarray} \tag{ 2 }$$

This equation is derived from adopting the most recent BLR radius-luminosity relation (Bentz et al. 2013, Table 14, Clean2+ExtCorr) and a virial factor of $\mathrm{log}f=0.71$ (Park et al. 2012; Woo et al. 2013). The results are given in Table 1 for a sample of 79 objects (see the next paragraph). We assume a nominal uncertainty of the BH masses measured via the virial method of 0.4 dex (Vestergaard & Peterson 2006).

Note that from the full sample of 103 objects presented in Paper II, six objects showed only a broad Hα line in the SDSS spectrum and no broad Hβ line in either the SDSS or Keck spectra. While we can still estimate ${M}_{\mathrm{BH}}$ from the broad Hα line in the SDSS spectrum, we decided to exclude them from the sample, for consistent ${M}_{\mathrm{BH}}$ measurements. An additional eight objects showed broad Hα and Hβ lines in the SDSS spectrum, but did not reveal any broad Hβ line in the Keck spectrum. We excluded these objects here as well, and our ${M}_{\mathrm{BH}}$ sample is thus comprised of 79 objects. However, we will discuss them individually in an upcoming paper (V. N. Bennert et al. 2015, in preparation). This upcoming paper will also include a direct comparison between the SDSS spectrum and the Keck spectrum, to study any broad-line variability. 57 objects are included in the BH mass function study by Greene & Ho (2007) with BH masses derived from the broad Hα line and luminosity in the SDSS spectra, with an overall good agreement in the mass measurements.

We visually inspected the multi-filter SDSS images as well as the fits and residuals to determine the best host-galaxy decomposition. The majority of the host galaxies are classified as Sa or later (51/66 = 77%) and was fitted either by a spheroid+disk decomposition (40) or spheroid+disk+bar (11). For the remaining 15 objects, a spheroid only fit was deemed sufficient. The high fraction of spiral galaxies is typical for a sample of (mostly radio-quiet) Seyfert galaxies (e.g., Hunt & Malkan 1999 and references therein). This is consistent with the majority of objects (∼60%) showing rotation curves with a maximum velocity between 100 and 200 km s⁻¹. 6% of the sample (4/66) are merging or interacting galaxies. This is lower than for our high-redshift Seyfert galaxies (∼30% at $z\simeq 0.4-0.6$, Park et al. 2015) and more comparable to quiescent galaxies in the local universe.

Of the 79 objects in our sample, 75 are included in the morphological classification by Galaxy Zoo (Lintott et al. 2011), but for only 28 did the vote reach the necessary 80% mark to flag the morphology as either spiral or elliptical. For those, our classification agrees in the majority of cases (82%) with the rest being classified as ellipticals by Lintott et al. (2011) while we classified them as spirals. However, we consider our classification as more robust since it also takes into accounts the fits and residuals, especially given the AGN central point source, which could lead to an overestimation of the presence of a bulge by Galaxy Zoo.

For the spiral galaxies in our sample, a little more than half of spheroid components are fitted by a Sérsic index $\lt 2$ (28 objects = 55%, corresponding to 43% of the total sample). For the rest (23 objects), the spheroid component is fitted by a Sérsic index $\geqslant 2$. On average, the Sérsic index for the spiral galaxies is 2.2 ± 1.6, for $n\geqslant 2$ bulges 3.8 ± 0.8, and for $n\lt 2$ bulges 0.9 ± 0.3.

If we take solely the Sérsic index as an indicator for the existence of a classical vs. pseudo-bulge, half of our spiral galaxies have pseudo-bulges; we consider them candidate pseudo-bulges. The host-galaxies of the reverberation-mapped AGNs have a similar distribution: Ho & Kim (2014) classify 75% as spiral galaxies with roughly half having classical bulges and half having pseudo-bulges. In a bulge+disk decomposition of galaxies in SDSS, roughly 25% of galaxies with sufficient image quality to study their bulge profile shape (a total of ∼53,000) have pseudo-bulges, if we use the same criterion with Sérsic index $\lt 2$ (Simard et al. 2011, their Figure 15, when excluding bars with n = 0.5).

However, we further follow the guidelines by Kormendy & Ho (2013) to distinguish between classical bulges and pseudo-bulges, applying the following four criteria that we can probe with our data: (i) Sérsic index $n\lt 2$ for pseudo-bulges, $n\geqslant 2$ for classical bulges; (ii) bulge-to-total luminosity ratios (in all four SDSS bands) B/T > 0.5 for classical bulges; (iii) ${v}_{\mathrm{max},\mathrm{reff}}$/${\sigma }_{\mathrm{center}}$ > 1 for pseudo-bulges, < 1 for classical bulges (here, ${v}_{\mathrm{max},\mathrm{reff}}$ is the maximum velocity at the effective radius of the bulge and ${\sigma }_{\mathrm{center}}$ is the stellar velocity dispersion in the center); (iv) the presence of a bar in face-on galaxies as an indicator of a pseudo-bulge. To be conservative, we only classify objects as having a pseudo-bulge for which at least three of the above four criteria are met. That leaves us with a total of eight spiral galaxies with a definite pseudo-bulge.

The measurement of the stellar velocity dispersion profiles and rotation curves is described in detail in Paper II. In this paper, with the addition of the surface photometry parameters, we have all the necessary information to investigate the systematic uncertainties and biases related to the definition of the stellar velocity dispersion.

With this goal in mind, we carry out a systematic comparison between different definitions of the velocity dispersion parameter taken from the literature and our fiducial measurement (${\sigma }_{\mathrm{spat},\mathrm{reff}}$, Equation (1)). Specifically we compute velocity dispersions as average of the second moment of the velocity field, by varying the size of the aperture, by considering the difference between correcting and not correcting the velocity rotation curve for inclination, and by considering the effects of contamination by nuclear light. While not 100% exhaustive, our list of definitions includes most of the choices adopted in the literature. For example, Ferrarese & Merritt (2000) use the second moment, integrating over one quarter of the effective radius of the galaxy and correcting the velocity for inclination. Neither Gültekin et al. (2009), McConnell & Ma (2013), nor Kormendy & Ho (2013) correct the velocity for inclination. However, Gültekin et al. (2009) and McConnell & Ma (2013) use the effective radius of the galaxy, while Kormendy & Ho (2013) use half of the effective radius of the galaxy. McConnell & Ma (2013) additionally discuss the choice of the minimum radius and find that setting it to zero (as done by e.g., Gültekin et al. 2009) can result in σ values smaller by 10%–15%, since it includes signal from within the BH gravitational sphere of influence. Thus, they instead set the minimum radius to the latter. However, for our data, we are not resolving the BH gravitational sphere of influence. Finally, many σ measurements in the literature are derived from aperture spectra, such as SDSS fiber spectra or spectra of distant galaxies integrated over different aperture sizes, making a direct comparison difficult. The reverberation-mapped AGN comparison sample falls into this category (Woo et al. 2015).

Table 2 lists the results for different possible comparisons, and Figure 1 shows three examples. Here, ${\sigma }_{\mathrm{spat},0.5\mathrm{reff}}$ (${\sigma }_{\mathrm{spat},0.25\mathrm{reff}}$) integrates out to half (one quarter) of the effective bulge radius. ${\sigma }_{\mathrm{spat},\ \mathrm{reff},\ \mathrm{sin}i}$ additionally corrects the velocity for inclination, as estimated from the disk:

$$\begin{eqnarray}&&{\sigma }_{\mathrm{spat},\ \mathrm{reff},\ \mathrm{sin}i}^{2}=\\ &&\displaystyle \frac{{\displaystyle \int }_{-{R}_{\mathrm{eff}}}^{{R}_{\mathrm{eff}}}({\sigma }_{\mathrm{spat}}^{2}(r)+{v}_{\mathrm{spat}}^{2}/\mathrm{sin}{(i)}^{2}(r))\cdot I(r)\cdot r\cdot {dr}}{{\displaystyle \int }_{-{R}_{\mathrm{eff}}}^{{R}_{\mathrm{eff}}}I(r)\cdot r\cdot {dr}}.\end{eqnarray} \tag{ 3 }$$

Again, we adopt smaller integration radii in ${\sigma }_{\mathrm{spat},\ 0.5\ \mathrm{reff},\ \mathrm{sin}i}$ and ${\sigma }_{\mathrm{spat},\ 0.25\ \mathrm{reff},\ \mathrm{sin}i}$ (half and one quarter of the effective bulge radius, respectively). ${\sigma }_{\mathrm{spat},\ \mathrm{reff},\ \mathrm{galaxy}}$ integrates out to the effective radius of the galaxy instead, or alternatively half/one quarter of that (${\sigma }_{\mathrm{spat},\ 0.5\ \mathrm{reff},\mathrm{galaxy}}$; ${\sigma }_{\mathrm{spat},\ 0.25\ \mathrm{reff},\mathrm{galaxy}}$). It follows that ${\sigma }_{\mathrm{spat},\ \mathrm{reff},\ \mathrm{galaxy},\ \mathrm{sin}i}$ corrects the velocity for inclination, considering different integration limits in ${\sigma }_{\mathrm{spat},\ 0.5\ \mathrm{reff},\ \mathrm{galaxy},\ \mathrm{sin}i}$ and ${\sigma }_{\mathrm{spat},\ 0.25\ \mathrm{reff},\ \mathrm{galaxy},\ \mathrm{sin}i}$. Finally, ${\sigma }_{\mathrm{spat},\ \mathrm{SDSS}}$ integrates within the 15 radius of the SDSS fiber. (Note that in fact our ${\sigma }_{\mathrm{ap},\ \mathrm{SDSS}}$ corresponds to a rectangular region with 15 radius and 1'' width, given the width of the long slit used.)

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Table 2.  Comparison between Different Definitions for the Stellar Velocity Dispersion

Ratio	Mean
(1)	(2)
${\sigma }_{\mathrm{spat},\ 0.5\ \mathrm{reff}}$/${\sigma }_{\mathrm{spat},\ \mathrm{reff}}$	0.98 ± 0.01
${\sigma }_{\mathrm{spat},\ 0.25\ \mathrm{reff}}$/${\sigma }_{\mathrm{spat},\ \mathrm{reff}}$	0.99 ± 0.02
${\sigma }_{\mathrm{spat},\ \mathrm{reff},\ \mathrm{sin}i}$/${\sigma }_{\mathrm{spat},\ \mathrm{reff}}$ a	1.31 ± 0.09
${\sigma }_{\mathrm{spat},\ 0.5\ \mathrm{reff},\ \mathrm{sin}i}{\text{}}/{\sigma }_{\mathrm{spat},\ \mathrm{reff}}$	1.14 ± 0.06
${\sigma }_{\mathrm{spat},\ 0.25\ \mathrm{reff},\ \mathrm{sin}i}{\text{}}/{\sigma }_{\mathrm{spat},\ \mathrm{reff}}$	1.05 ± 0.04
${\sigma }_{\mathrm{spat},\ \mathrm{reff},\mathrm{galaxy}}{\text{}}/{\sigma }_{\mathrm{spat},\ \mathrm{reff}}$ a	0.99 ± 0.01
${\sigma }_{\mathrm{spat},\ 0.5\ \mathrm{reff},\mathrm{galaxy}}{\text{}}/{\sigma }_{\mathrm{spat},\ \mathrm{reff}}$	0.96 ± 0.01
${\sigma }_{\mathrm{spat},\ 0.25\ \mathrm{reff},\mathrm{galaxy}}{\text{}}/{\sigma }_{\mathrm{spat},\ \mathrm{reff}}$	0.98 ± 0.02
${\sigma }_{\mathrm{spat},\ \mathrm{reff},\mathrm{galaxy},\ \mathrm{sin}i}{\text{}}/{\sigma }_{\mathrm{spat},\ \mathrm{reff}}$	1.43 ± 0.09
${\sigma }_{\mathrm{spat},\ 0.5\ \mathrm{reff},\mathrm{galaxy},\ \mathrm{sin}i}{\text{}}/{\sigma }_{\mathrm{spat},\ \mathrm{reff}}$	1.18 ± 0.06
${\sigma }_{\mathrm{spat},\ 0.25\ \mathrm{reff},\mathrm{galaxy},\ \mathrm{sin}i}{\text{}}/{\sigma }_{\mathrm{spat},\ \mathrm{reff}}$	1.07 ± 0.04
${\sigma }_{\mathrm{spat},\ \mathrm{SDSS}}{\text{}}/{\sigma }_{\mathrm{spat},\ \mathrm{reff}}$ a	0.97 ± 0.01
${\sigma }_{\mathrm{ap},\ \mathrm{reff}}{\text{}}/{\sigma }_{\mathrm{spat},\mathrm{reff}}$	1.01 ± 0.02

Notes. Comparison between the different definitions for the stellar velocity dispersion used in the literature. Column (1): ratio between a given definition of $\sigma$ (see text for details) and the fiducial ${\sigma }_{\mathrm{spat},\ \mathrm{reff}}$ used throughout the paper (Equation (1)). Column (2): mean and uncertainty of the ratio.

^aShown in Figure 1.

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The largest (average) effect on the derived σ is the correction of the velocity for inclination, which can result in σ measurements by on average 31% ± 9% larger (see Figure 1) (43% ± 9% in case of the galaxy effective radius), with individual objects as much as doubled. This discrepancy is reduced by considering smaller integration limits, since rotation is negligible at the center of most objects. However, such large differences are mainly due to galaxies seen close to face on, for which the velocities are highly uncertain and sometimes inflated beyond reasonably physical limits, resulting in large outliers. For example, if we impose arbitrarily that $v/\mathrm{sin}(i)\lt \sigma$ (i.e., rotation not dominant), the average ratio falls to 1.06 ± 0.01. Or if instead less stringent we impose that $v/\mathrm{sin}(i)\lt 2\sigma$ (e.g., pseudo-bulges may be rotation dominated), the average ratio is 1.14 ± 0.02. Based on this source of uncertainty, we emphasize that the large difference should not be taken at face value, but more as an indication that for face-on galaxies we simply do not know the contribution of the rotational support. The intrinsic uncertainty in inclination correction for face-on objects is compounded by the fact that we do not have a good estimate of the inclination of the bulge. In the definitions of σ above labeled as "sini," we used the inclination of the disk component as a proxy. Given these caveats, we decided to not correct the velocities for inclination in our fiducial measurement, ${\sigma }_{\mathrm{spat},\mathrm{reff}}$ (Equation (1)), which is a common practice in the literature, including the comparison samples considered here. However, we caution that σ might be underestimated in some cases.

When increasing the radius to the galaxy effective radius, the σ measurement decreases on average by 1% ± 1% (see Figure 1). This is expected, since most objects show a decreasing stellar velocity dispersion with distance from the center (Paper II).

However, there is another effect that might counterbalance this trend: choosing a larger radius also includes more and more rotational velocities, in particular those of the disk. Depending on the viewing angle, disk rotation can lead to over-estimating σ, if seen edge-on, or under-estimating σ, if seen face-on, given that the disk is kinematically cold (e.g., Woo et al. 2006, 2013, Papers I and II). This effect can be potentially important, given the variety of host-galaxy morphologies in both local (e.g., Malkan et al. 1998; Hunt & Malkan 2004; Kim et al. 2008; Bennert et al. 2009, 2011b) and distant AGNs (Bennert et al. 2010, 2011b; Park et al. 2015). When considering the different morphologies and inclinations in our comparison between ${\sigma }_{\mathrm{spat},\mathrm{reff}}$ and ${\sigma }_{\mathrm{spat},\mathrm{reff},\mathrm{galaxy}}$, we confirm the expected trend.

Bellovary et al. (2014) use cosmological smoothed particle hydrodynamics (SPH) simulations of five disk galaxies and find that the line of sight effect due to galaxy orientation can affect σ by 30% with face-on views resulting in systematically lower velocity dispersion measurements and edge-on orientations leading to higher values due to a contamination of rotating disk stars. Both fiber-based SDSS data (e.g., Greene & Ho 2006; Shen et al. 2008) as well as aperture spectra for distant galaxies (e.g., our studies on the evolution of the ${M}_{\mathrm{BH}}$–σ relation; Woo et al. 2006, 2008; Treu et al. 2007) can suffer from the added uncertainty of the effect of the disk on σ. The evolutionary studies rely on active galaxies, by necessity, and given the distance of the galaxies, stellar velocity dispersion measurements are challenging. Aperture spectra typically include the central few kpc of the galaxy (for a typical aperture of 1'' = 5–7 kpc for the redshift range of z = 0.36–0.57 covered by, e.g., Woo et al. 2008), and can be contaminated by the disk kinematics, especially for Seyfert-1 galaxies chosen for these studies due to their relatively weak AGN power-law continuum. However, the evolutionary trend found by these studies is that of distant spheroids having on average smaller velocity dispersions than local ones. This effect cannot be explained by disk kinematics, since we would expect the opposite from rotational support. Thus, the offset of the distant AGNs from the local ${M}_{\mathrm{BH}}$–σ scaling relations is, if anything, underestimated.

Choosing a generic radius of $1\buildrel{\prime\prime}\over{.} 5$ as the SDSS fiber results in an underestimation of σ by 3% ± 1% (Figure 1). Note, however, that this difference is very sensitive to the distance of the particular objects, and the ratio between aperture size to bulge (or galaxy) effective radius. For our sample, the effective bulge radius (on average 2.9 ± 0.3) is close to the SDSS fiber size, so the difference is small.

In all the σ definitions mentioned above, we measured σ (and the velocity) as a function of distance from the center and integrated over it later out to the effective bulge radius (according to Equation (1)). However, this is not the same as directly integrating over the entire spectrum (out to the effective radius) and then measuring σ, since the latter will always include AGN power-law and emission lines, while in the former approach the central spectrum is sometimes excluded due to AGN contamination (Paper II). Thus, we also compare our fiducial stellar velocity dispersion ${\sigma }_{\mathrm{spat},\mathrm{reff}}$ with the corresponding value derived from aperture spectra which on average overestimates σ by 1% ± 2%.

To conclude, these comparisons show that the choice of the exact definition of σ can have a non-negligible effect (up to 40%) and care needs to be taken, especially when comparing to other values in the literature. It is important to understand the effects of inclination, the rotational contribution of the disk, and the AGN contribution in the center, on the measurement of stellar velocity dispersion. We consider the value used in this paper, ${\sigma }_{\mathrm{spat},\mathrm{reff}}$ (Equation (1)), the stellar velocity dispersion within the bulge effective radius derived from spatially resolved σ and velocity measurements, the most robust measurement since it excludes contributions of disk rotation and AGN emission. Moreover, distinguishing between bulge and disk in the context of the BH mass scaling relations is especially important if BHs correlate only with the bulge component and not the disk, as suggested by recent studies (Kormendy et al. 2011; Kormendy & Ho 2013).

In Figure 2, we show the resulting ${M}_{\mathrm{BH}}$–$\sigma$ relation, as well as the offset from the fiducial relation. Overall, our sample follows the same ${M}_{\mathrm{BH}}$–$\sigma$ relation as that of reverberation-mapped AGNs as well as that of quiescent galaxies. Our sample covers a small dynamical range in BH mass (6.7 $\lt \mathrm{log}$ ${M}_{\mathrm{BH}}$ < 8.2), mainly due to the fact that we selected low-luminous Seyfert galaxies with lower mass BHs to enable $\sigma$ measurements. Considering the uncertainties of ${M}_{\mathrm{BH}}$ of 0.4 dex, we cannot independently determine the slope of the local relationship. The sample size is, however, sufficient to determine the zero point and scatter of the distribution around the local relationship, assuming a choice of the slope. Thus, when fitting a linear relation to the data of the form

$$\begin{eqnarray}&&\mathrm{log}({M}_{\mathrm{BH}}/{M}_{\odot })=\alpha +\beta \mathrm{log}(\sigma /200\;\mathrm{km}\;{{\rm{s}}}^{-1})\end{eqnarray} \tag{ 4 }$$

we keep the value of $\beta$ fixed to the corresponding relationships of quiescent galaxies (5.64 for McConnell & Ma 2013 and 4.38 for Kormendy & Ho 2013) or reverberation mapped AGNs (Woo et al. 2015, 3.97). The results are summarized in Table 3. Both zero point and intrinsic scatter are comparable to that of the quiescent galaxies, within the uncertainties, consistent with the hypothesis that the ${M}_{\mathrm{BH}}$ estimates we adopt do not introduce significant uncertainty in addition to the estimated one. Furthermore, biases based on different selection functions (reverberation mapped AGNs selected based on variability; quiescent galaxies selected based on the ability to resolve the BH gravitational sphere of influence; our sample selected based on Hβ line width) can be considered negligible.

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Table 3.  Fits to the Local ${M}_{\mathrm{BH}}$–σ Relations

Sample	$\alpha$	$\beta$	Scatter	Offset	Reference
(1)	(2)	(3)	(4)	(5)	(6)
Quiescent galaxies (72)	8.32±0.05	5.64±0.32	0.38	⋯	McConnell & Ma (2013) a
Quiescent galaxies (51)	8.49±0.05	4.38±0.29	0.29	⋯	Kormendy & Ho (2013)
Reverberation-mapped AGNs (29)	8.16±0.18	3.97±0.56	0.41±0.05		Woo et al. (2015)
AGNs (66)	8.38±0.08	5.64 (fixed)	0.43±0.09	−0.01±0.01	this paper
AGNs (66)	8.20±0.06	4.38 (fixed)	0.25±0.10	0.06±0.01	this paper
AGNs (66)	8.14±0.06	3.97 (fixed)	0.19±0.10	0.02±0.01	this paper

Notes. Fits to the ${M}_{\mathrm{BH}}$–σ relation, $\mathrm{log}({M}_{\mathrm{BH}}/{M}_{\odot })=\alpha +\beta \mathrm{log}(\sigma /200\;\mathrm{km}\;{{\rm{s}}}^{-1})$ Column (1): sample and sample size in parenthesis. Column (2): mean and uncertainty on the best fit intercept. Column (3): mean and uncertainty on the best fit slope. Column (4): mean and uncertainty on the best fit intrinsic scatter. Column (4): mean and uncertainty of offset from fiducial relation of either McConnell & Ma (2013) (slope fixed to 5.64), Kormendy & Ho (2013) (slope fixed to 4.38), or Woo et al. (2015) (slope fixed to 3.97). Column (5): references for fit. Note that the quoted literature uses FITEXY with a uniform prior on the intrinsic scatter, so our fits assume the same.

^aRelation plotted as dashed lines in Figure 2 and used as fiducial relation when calculating residuals.

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When probing dependencies on host-galaxy morphology, we find that barred galaxies (comprising 17% of the sample) do not lie preferentially off the ${M}_{\mathrm{BH}}$–σrelation, in agreement with studies by Graham (2008; Bentz et al. 2009; Beifiori et al. 2012) (see, however, Graham & Li 2009). Also, neither merging galaxies (6%) nor pseudo-bulges (12%) form particular outliers from the relation. This statement also holds true when considering all candidate pseudo-bulges (i.e., objects with spheroid with Sérsic index $n\lt 2$; a total of 43% of the sample). In fact, candidate pseudo-bulges show an even smaller scatter in the relation.

While the latter is in agreement with some studies (e.g., Kormendy 2001; Gu et al. 2009), more recent studies suggest the opposite (Hu 2008; Greene et al. 2010; Kormendy & Bender 2011; Kormendy & Ho 2013; Ho & Kim 2014). Pseudo‐bulges, characterized by nearly exponential light profiles, ongoing star formation or starbursts, and nuclear bars, are believed to have evolved secularly through dissipative processes rather than mergers (Courteau et al. 1996; Kormendy & Kennicutt 2004), unlike their classical counterparts.

Given the sample of Seyfert-1 galaxies comprised of a majority of late-type galaxies and the small fraction of mergers, our results are consistent with secular evolution, driven by disk or bar instabilities and/or minor mergers, growing both BHs through accretion and spheroids through a re-distribution of mass from disk to bulge (e.g., Croton 2006; Jahnke et al. 2009; Parry et al. 2009; Bennert et al. 2010, 2011b; Cisternas et al. 2011; Schramm & Silverman 2013). We can only speculate here that the smaller scatter exhibited by pseudo-bulges might in fact be explained by a synchronizing effect that secular evolution has on the growth of BHs and bulges, growing both simultaneously at a small but steady rate. Major mergers, on the contrary, believed to create classical bulges (and elliptical galaxies) are a more stochastic and dramatic phenomenon with episodes of strong BH and bulge growth that can be out of sync due to the different time scales involved for growing BH and bulge in a major merger (e.g., Hopkins 2012). We cannot directly probe the latter with our data—as already mentioned, galaxies with obvious signs of interactions and mergers do not form particular outliers from the relation, but this is based on a very small sample statistics of 6% mergers.

However, we conclude with a note of caution: For one, when splitting our sample into sub-samples (such as classical versus pseudo-bulges), the results suffer from small sample statistics. Second, our morphological classification relies on ground-based SDSS images with inherent limitations such as limited depth and resolution. For example, the kpc-scale of bars and bulges is comparable to the spatial resolution of the SDSS images (a $\sim 1\buildrel{\prime\prime}\over{.} 5$ PSF corresponds to 1.5 kpc at z = 0.05), significant given the presence of the bright point source in AGNs. The seeing in the ground-based SDSS images might also cause the host galaxies to appear rounder than they actually are. Likewise, faint tidal features indicative of merger events might have been missed in these shallow images. Deeper and higher spatial-resolution images are required to further test the connection between bars and bulges and BHs.