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A&A
Volume 691, November 2024
Article Number A114
Number of page(s) 24
Section Extragalactic astronomy
DOI https://doi.org/10.1051/0004-6361/202348353
Published online 01 November 2024

© The Authors 2024

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1. Introduction

The discovery of the relations between host galaxy bulge properties and the mass of central supermassive black holes (BHs; Kormendy & Richstone 1995; Magorrian et al. 1998; Ferrarese & Merritt 2000; Gebhardt et al. 2000a) has been pivotal in extragalactic astronomy during the last two decades. Nowadays, it is widely accepted that all galaxies with a massive bulge host a supermassive BH, with the BHs being a key element in regulating galaxy formation and evolution, so that the growth of both components is intimately related (Kormendy & Ho 2013, see also Greene et al. 2020). The empirical relations are key ingredients in numerical, theoretical, and semi-analytic models to reproduce the galaxy population properties (Croton et al. 2006; Schaye et al. 2015; Sijacki et al. 2015; Volonteri et al. 2016; Thomas et al. 2019; Li et al. 2020).

Among the relations between BH mass (M•) and host galaxy bulge properties, the BH mass-bulge mass and the BH mass-bulge velocity dispersion (σe)1 correlations stand out due to their equally small scatter (∼0.29 dex; Kormendy & Ho 2013). Several studies have attempted to find correlations with other galaxy properties (e.g., Saglia et al. 2016; de Nicola et al. 2019), but the uncertainties in the quantities prevent definitive conclusions from being reached. In all cases, the difficulty lies in measuring M• and σe for a sufficiently large galaxy population. In the very nearby universe, both quantities can be directly computed. The BH masses can be measured from dynamical analysis for massive BHs (see review by Kormendy & Ho 2013), while spatially resolved galaxy spectra are needed to measure σe (e.g., Jørgensen et al. 1995; Cappellari et al. 2006). Beyond the nearby universe, instrumental resolution limits dynamical measurements of BH mass. We have to rely on indirect measurements that can only be applied for active galactic nucleus (AGN) host galaxies. Based on the response of the broad-line region (BLR) gas to the variable AGN continuum radiation, the reverberation mapping (RM) technique (Blandford & McKee 1982; Peterson et al. 2004; Kaspi et al. 2005; Bentz et al. 2013; Shen et al. 2015; Bentz & Manne-Nicholas 2018; Lira et al. 2018; Hu et al. 2021; Kaspi et al. 2021) has proven useful in estimating BH masses. This is possible because nearby RM AGNs follow a M• − σe relation that is roughly parallel to that of inactive systems, offering a calibration method (Gebhardt et al. 2000b; Ferrarese et al. 2001; Nelson et al. 2004; Ho & Kim 2014). The extension of this technique, the single-epoch virial BH mass estimate (Vestergaard & Peterson 2006; Greene & Ho 2005b; Ho & Kim 2015), has allowed statistical studies of BH masses to be carried out for representative samples of the active galaxy population. The advent of high-resolution spectro-astrometric measurements with interferometry (e.g., Gravity Collaboration 2018, 2023) promises a new option for measuring M• in active systems.

Studying AGNs in the context of M• − σe is still a difficult task, mainly because quantifying σe for such systems is challenging. Stellar velocity dispersion (σ*) measurements are susceptible to the underlying stellar population properties, and their relative contribution to the observed luminosity weighted galaxy spectrum. This is probably more significant for AGNs, which have complex stellar populations because of the prevalence of ongoing star formation (Jarvis et al. 2020; Shangguan et al. 2020a; Torbaniuk et al. 2021; Xie et al. 2021; Zhuang & Ho 2022; Molina et al. 2023a). In addition, the rich emission-line spectrum of type 1 AGNs blends and confuses with the stellar absorption features of the host galaxy. Moreover, the AGN continuum strongly dilutes the host galaxy spectrum. These problems are specially concerning when measuring σ* from stellar features in the optical (Greene & Ho 2006a). Observing in the near-infrared (near-IR) is a plausible alternative (e.g., Woo et al. 2013); however, the AGN glare also limits the modeling of the stellar features at those wavelengths, even when adaptive optics-assisted observations are employed (Dasyra et al. 2007; Watson et al. 2008; Grier et al. 2013). Another option is to model narrow emission lines, such as [O III]λ5007, to adopt the gas kinematics as a surrogate for σ* (e.g., Nelson 2000; Shields et al. 2003). Nevertheless, those estimates are prone to large uncertainties (Nelson 2000; Onken et al. 2004; Bonning et al. 2005; Greene & Ho 2005a; Bennert et al. 2018). To avoid these important complications, the best strategy is to target the Calcium near-IR triplet (CaT) absorption lines (e.g., Ferrarese et al. 2001; Barth et al. 2002; Nelson et al. 2004; Onken et al. 2004; Greene & Ho 2006b; Woo et al. 2010) at 8498, 8542, and 8662 Å, the “gold standard” for deriving stellar velocity dispersion in active galaxies (Greene & Ho 2006a). The key advantages of using the CaT for measuring σ* (or σe) is that these stellar features are largely insensitive to the underlying stellar population properties (Dressler 1984), and they are located in a spectral window that is mostly free of strong AGN emission lines. However, observing the CaT for large samples of active galaxies has been prohibitive because the absorption features can be strongly affected by sky emission, and they quickly get redshifted out of the spectral window that is accessible with optical spectrographs, only allowing σ* to be characterized for low-redshift sources.

For this work, we used seeing-limited and ground-layer adaptive optics-aided Very Large Telescope (VLT) Multi Unit Spectroscopic Explorer (MUSE) observations to model the CaT and estimate the bulge velocity dispersion for the host galaxies of local type 1 AGNs and quasars. Our sample was built from the Palomar-Green (PG) quasar survey (Boroson & Green 1992) and the Close AGN Reference Survey (CARS; Husemann et al. 2022), with all of these at z ≲ 0.1 (Figure 1). We analyzed these systems in the context of the M• − σe relation, finding no difference between the active and inactive galaxies. Section 2 summarizes the properties of the active galaxy sample and outlines the observations. Section 3 details the procedures used to derive M• and σe. Section 4 presents the PG quasars and CARS AGNs in the context of the M• − σe relation, and examines possible deviations from this relation in terms of the Eddington ratio. We discuss and summarize our findings in Section 5. Hereinafter, we refer to the central stellar spheroid component of the host galaxy simply as the “bulge”, which pertains to both classical and pseudo bulges (Kormendy & Kennicutt 2004), unless explicitly stated. We adopted a ΛCDM cosmology with Ωm = 0.308, ΩΛ = 0.692, and H0 = 67.8 km s−1 Mpc−1 (Planck Collaboration XIII 2016).

thumbnail Fig. 1.

AGN monochromatic luminosity at 5100 Å as a function of the BH mass. The error bar in the bottom right corner represents the typical uncertainty of the quantities. The dashed lines correspond to fixed Eddington ratios. The contours show the distribution of the broad-lined AGNs taken from the 7th data release of the Sloan Digital Sky Survey (SDSS; Abolfathi et al. 2018) at z < 0.35 presented by Liu et al. (2019).

2. Sample and observations

We use VLT-MUSE wide-field-mode integral field unit (IFU) observations for our sample, which consists of 42 local (z  <  0.1; Table A.1 in Appendix A) type 1 AGNs, nine PG quasars (Boroson & Green 1992) and 33 CARS AGNs (Husemann et al. 2022). All observations were conducted as part of multiple European Southern Observatory (ESO) programs [094.B−0345(A), 095.B−0015(A), 097.B−0080(A), 099.B−0242(B), 099.B−0294(A), 0101.B−0368(B), 0103.B−0496, 0104.B−0151(A), and 106.21C7.002], spanning June 2015 to March 2021. In a single snapshot, MUSE provides a wide field-of-view of approximately 1′×1′, with a 0 . ″ 2 × 0 . ″ 2 $ 0{{\overset{\prime\prime}{.}}}2 \times 0{{\overset{\prime\prime}{.}}}2 $ pixel size. It generates ∼90 000 spectra per pointing. The wavelength coverage of MUSE ranges from ∼4700 to ∼9350 Å, with a wavelength sampling of 1.25 Å channel−1 at a mean resolution of R ≈ 3000 (full width at half maximum FWHM = 2.65 Å). We analyze archival MUSE data cubes obtained from the ESO science portal2. These data cubes correspond to a combination of multiple observing blocks, typically ranging from two to four per target. In the particular case of PG 0050+124, we used the “MUSE-DEEP” data cube3, which is a combination of 15 single observing blocks aimed at maximizing the signal contrast (Weilbacher et al. 2020). Most of the MUSE observations were conducted under natural-seeing conditions, with typical values ∼ 0 . ″ 5 − 1 . ″ 4 $ {\sim} 0{{\overset{\prime\prime}{.}}}5{-}1{{\overset{\prime\prime}{.}}}4 $ (Table A.2). Only four observations were assisted by the ground-layer adaptive optics module. In these cases, the MUSE spectra were masked at the 5840–5940 Å wavelength range surrounding the Na I Dλλ5890, 5896 emission by the data reduction pipeline.

We further applied the Zurich Atmosphere Purge (ZAP) sky-subtraction tool (version “2.1.dev”; Soto et al. 2016) to remove residual sky features. We set CFWIDTHSP = 5, while keeping the other parameters at their default values. Additionally, we masked all spectra at wavelengths where strong sky-line residuals are present in the preprocessed data cubes (5578.5, 5894.6, 6301.7, 6362.5, and 7640 Å). To account for instrumental resolution, we adopt the MUSE line-spread function parameterization of Guérou et al. (2017) to correct the line widths.

2.1. PG quasars

We use the available MUSE data for nine local quasar host galaxies previously presented in Molina et al. (2022). These systems are member of the broader sample of 87 z < 0.5 quasars (Boroson & Green 1992) belonging to the PG survey of optical/ultraviolet color-selected quasars (Schmidt & Green 1983). These targets have substantial multiwavelength data across the entire electromagnetic spectrum, ranging from X-ray (Reeves & Turner 2000; Bianchi et al. 2009) to optical (Boroson & Green 1992; Ho & Kim 2009), mid-IR (Shi et al. 2014; Xie et al. 2021; Xie & Ho 2022), far-IR (Petric et al. 2015; Shangguan et al. 2018; Zhuang et al. 2018), mm (Shangguan et al. 2020b,a; Molina et al. 2021), and radio (Kellermann et al. 1989, 1994; Silpa et al. 2020, 2023) wavelengths. High-resolution ( ∼ 0 . ″ 1 $ {\sim} 0{{\overset{\prime\prime}{.}}}1 $) Hubble Space Telescope (HST) imaging in the optical and near-IR are available for many of the host galaxies, securing accurate characterization of the host galaxy morphology (Kim et al. 2008; Zhang et al. 2016; Kim & Ho 2019; Zhao et al. 2021). Stellar masses (M*) were computed from the multiband host galaxy images after applying the mass-to-light ratios of Bell & de Jong (2001, see Zhao et al. 2021 for more details). We estimate bolometric luminosities (Lbol) from the AGN monochromatic luminosity at 5100 Å, λ L λ ( 5100 Å ) $ \lambda L_\lambda(5100\,\mathrm{\AA}) $, by adopting the conversion of Richards et al. (2006), L bol = 10 λ L λ ( 5100 Å ) $ L_{\mathrm{bol}} = 10\, \lambda L_\lambda (5100\, \mathrm{\AA}) $. Our PG quasar subsample is characterized by ⟨Lbol⟩ = 1045.6 erg s−1, ⟨M•⟩ = 108.3 M⊙, ⟨M*⟩ = 1010.8 M⊙, and ⟨z⟩ = 0.060.

2.2. CARS AGNs

The CARS sample was selected from the broader Hamburg/ESO survey (HES) of ultraviolet-excess sources covering an area of ∼ 9000 deg 2 $ {\sim} 9000\, \rm deg^2 $ in the southern hemisphere. The type 1 AGNs were confirmed through follow-up spectroscopy (Wisotzki et al. 2000; Schulze et al. 2009). Specifically, CARS corresponds to a representative survey of 41 systems randomly selected from the broader subsample of 99 HES objects at z ≲ 0.06 (Bertram et al. 2007). CARS probes the bright tail of the AGN luminosity function in the local universe (Schulze et al. 2009). The CARS host galaxies have been observed at multiple wavelengths, including the mm (Bertram et al. 2007) and radio (König et al. 2009). The panchromatic spectral energy distribution decomposition and stellar mass estimation for the host galaxies are presented in detail in Smirnova-Pinchukova et al. (2022). Available IFU data are presented in Husemann et al. (2022) for 41 targets, 37 of these observed by MUSE. From the MUSE dataset, we discard two targets that do not correspond to type 1 AGNs, but instead starbursts with extremely blue continuum and broad-line components tracing starburst-driven outflows (Husemann et al. 2022). Another two targets overlap with the PG quasar sample and were also discarded. Hence, we analyze 33 CARS systems (Table A.1). These CARS host galaxies are characterized by ⟨z⟩ = 0.042, ⟨Lbol⟩ = 1044.5 erg s−1, ⟨M•⟩ = 107.9 M⊙, and ⟨M*⟩ = 1010.6 M⊙.

3. Methods

Our aim is to study local type 1 AGNs and quasars in the context of the M•–σe relation. To characterize σe, we focus on modeling the CaT. We minimize the effect of the AGN emission in diluting the stellar spectrum features by using annular (elliptical) apertures when extracting the spectra. For each source, we measure σ* as close as possible to the bulge half-light radius (Re). We develop an annular aperture correction recipe to estimate σe from σ*. We further correct for systematics associated with MUSE instrumental artifacts to ensure accurate estimates. Additionally, we control for AGN emission diluting the stellar continuum and the signal-to-noise (S/N) of the stellar features.

3.1. Host galaxy spectra extraction

We begin by characterizing the projected geometry of the host galaxies on the sky. We collapse the MUSE data cubes across the spectral axis to derive “white-light” images. For each white-light image, we use the BACKGROUND2D task from PHOTUTILS (Bradley et al. 2022) to compute the background level. Then, we smooth the white-light image with a 1″-wide Gaussian kernel and apply DETECT_SOURCES to build a segmentation map. During this process, we mask sources with S/N < 5. We use the segmentation map to isolate the host galaxy from any other sources in the unsmoothed white-light image, and we apply SOURCECATALOG to derive the system position angle and ellipticity. The AGN location is assumed to coincide with the center of the host galaxy. We use the geometric parameters to construct a series of concentric annuli with thickness equal to the point-spread function (PSF) FWHM. The spectra are bounded to a window of ±1000 km s−1 to enclose the redshifted CaT4, with a wavelength upper limit set to 9300 Å in the observer frame to avoid the red spectral limit of MUSE. The spectra are masked whenever sky lines are present. These sky lines are identified in the corresponding variance spectra, analogously extracted from the MUSE variance data cubes. Nonetheless, we recall that the sky-line features have been minimized by applying the ZAP tool on the data cubes.

3.2. MUSE instrumental feature correction

We identify a MUSE instrumental feature at 9060–9180 Å in the observer frame (Figure 2), which must be corrected. The key characteristic of this instrumental feature is its consistent presence at the same wavelength range whenever MUSE observes a relatively bright source, such as a bright field star or the AGN. We speculate that this instrumental feature is caused by minor variations in the instrument sensitivity at specific wavelengths. We employ a “flat field-like” procedure to correct for this feature. We note that this instrumental feature overlaps with at least one CaT absorption line for AGNs at 0.046 < z < 0.080.

thumbnail Fig. 2.

Example of MUSE instrumental feature correction. The shaded regions highlight the masked wavelengths that encompass the CaT. The spectra have been vertically shifted to improve figure visualization.

We characterize the shape of the MUSE instrumental feature by using field stars (identified by Gaia) within the MUSE field-of-view. Out of all data cubes, we find that the spectra of 29 bright stars exhibit instrumental features. Stars are seen as point-like sources in the MUSE white-light images, so we model them using a Moffat (1969) function. We extract the stellar spectrum by spatially collapsing the data cube over a circular aperture equal to twice the Moffat model FWHM. Each stellar spectrum is fitted within the 4750–9280 Å wavelength range using the penalized pixel-fitting (pPXF; Cappellari & Emsellem 2004) routine. We identify at least ten field stars with good-quality spectrum models, but we only use the four5 that deliver the best removal of the MUSE instrumental feature. We use the stellar spectra to obtain an instrumental feature template from the data minus model residuals. Then, we flux-density normalize these templates before proceeding with the MUSE instrumental feature correction. We found that the normalization value for the instrumental feature template slightly varies for each MUSE observation (see Appendix B for more details). Therefore, we determine a specific normalization value for each case. For each target, we fit the spectra extracted from all annuli over the 9030–9200 Å wavelength range using a power-law continuum model plus the instrumental feature template multiplied by a scaling factor as free parameter. During the fit, we mask a spectral window of ±300 km s−1 around each CaT absorption line, if present6. We compare the best-fit scaling factor with the local median spectrum flux density (computed over 9030–9200 Å), finding that the ratio between these two quantities remains approximately constant for each annulus-extracted spectrum (Figure 3). Thus, for each target, we calculate the normalization constant by averaging the ratio over all annuli-extracted spectra. Finally, we divide each annulus-extracted spectrum by its corresponding normalized instrumental feature template. Considering the ten field stars with good-quality spectrum models for building different instrumental feature templates, we estimate systematic uncertainty of 0.08 dex induced by template (star) selection.

thumbnail Fig. 3.

Scaling factor for the instrumental feature template as a function of the median flux density for the annuli-extracted spectra over the 9030–9200 Å wavelength range (observer frame). The model corresponds to a linear function adjusted to the data. The bottom panel shows the model residuals. We only show the data for one target. We present the rest of the sample in Appendix B.

3.3. CaT modeling and stellar velocity dispersion measurements

We employ pPXF with input stellar templates taken from the INDO-U.S. stellar spectral library of Valdes et al. (2004) to model the spectra. The stellar templates cover the 3460–9464 Å wavelength range at a uniform spectral resolution of FWHM = 1.35 Å (Beifiori et al. 2011). We use a combination of late-type (F, G, K, and M) red giant (luminosity class III) stars of near-solar metallicity, plus an A-type dwarf (luminosity class V) star. We broaden the stellar templates in velocity space, taking into account their spectral resolution difference (in quadrature) with respect to the width of the MUSE line-spread function. We note that our results are not particularly sensitive to our choice of stellar template library, with an uncertainty of 0.04 dex associated with adopting other options for stellar spectral library (e.g., Vazdekis et al. 2012), or a simpler weighted linear combination of A and K stellar templates (Kong & Ho 2018). We avoid employing high-order moments (e.g., h3 and h4) when modeling the CaT because of the modest absorption line S/N and data spectral resolution. Even though the contrast between the AGN and its host galaxy is minimized at the CaT wavelengths, the featureless spectrum of the AGN continuum can still dominate the observed emission. The wing of the O Iλ8446 broad emission line can also interfere in the bluer end of the spectral range considered here (Caglar et al. 2020). To model such possible spectral subcomponents, we include an additive polynomial of order 0 and a multiplicative polynomial of order 3 in the pPXF setup. The inclusion of additive and multiplicative polynomials during the fit also helps to correct imperfect sky subtraction or scattered light with the former, and inaccuracies in spectral calibration or mismatches in dust reddening correction with the latter (Cappellari 2017). We caution that the inclusion of an additive polynomial tends to downweight the young (age ≲1 Gyr) stellar templates during the fit; however, the Ca II absorption line widths are largely insensitive to stellar population properties (Dressler 1984). Besides σ*, deriving other parameters for the host galaxy stellar component is beyond the scope of this work.

For many of the high-luminosity AGNs the CaT lines close to the nucleus are significantly diluted by the underlying nonstellar continuum, to the extent that the stellar features are undetectable in the innermost spectrum extracted from the MUSE data cube. This allow us to consider this spectrum as an effective AGN template that can be used for modeling the AGN emission for the rest of the annuli-extracted spectra. Here, our main assumption is that the AGN is seen as a point source whose spectrum has been blurred by the MUSE PSF across the data cube. For those targets (see Table 1), we add this empirical AGN template to the pool of spectra models, but as an additional component in pPXF to avoid the AGN template broadening and shift in velocity space. For the stellar component we keep the pPXF setup as detailed above. Thus, we model the AGN emission and the CaT simultaneously. We note that the additive and multiplicative polynomials affect all the templates when using pPXF (see Equation (13) of Cappellari 2023 for more details). Figure 4 shows an example for PG 2130+099, where the AGN component is subtracted from the observed spectrum, and the residuals clearly show the CaT stellar features.

thumbnail Fig. 4.

Example of the AGN emission subtraction procedure. The shaded regions highlight the masked spectral windows encompassing the CaT. The AGN template has been vertically shifted to improve figure visualization. The residuals correspond to the observed spectrum minus AGN template.

Table 1.

Measured and derived quantities.

For each host galaxy, we select σ* from the model spectrum taken at an aperture radius as close as possible to the bulge half-light radius, following the host galaxy bulge properties derived from HST image modeling for the PG quasars (Zhao et al. 2021) and MUSE white-light images for the CARS AGNs (Husemann et al. 2022). This radial constrain minimizes the aperture correction factor needed to convert σ* to σe (see Section 3.4). For our sources, the Re values are ∼1 kpc on average. For the selected spectra, we also compute the equivalent width (EW) and S/N of the Ca IIλ8542 absorption line. We choose Ca IIλ8542 as reference considering that the correct modeling of this stellar feature is enough to accurately quantify σ* from the CaT region (Harris et al. 2012). We present the selected spectra and their corresponding models in Figure A.1.

We use Monte Carlo resampling to derive the uncertainties in σ* (Cappellari 2023). For each spectrum, the noise level (root-mean-square) is determined from the CaT model residuals. Then, we add simulated noise to the corresponding spectrum, assuming a normal distribution, and repeat the CaT fit. We iterate 1000 times to obtain a probability distribution for the best-fit parameters to estimate the 1σ uncertainties from the 16th and 84th percentiles. Table 1 provides the adopted aperture radius of the modeled spectrum and the σ* values for each host galaxy. We emphasize that this procedure also accounts for imperfect removal of the MUSE instrumental feature (Section 3.2), which is reflected in the spectrum model residuals (e.g., see HE 0224−2834 and/or HE 2128−0221 in Figure A.1).

3.4. Stellar velocity dispersion aperture correction

For each AGN host galaxy, we use σ* to estimate σe. We follow the practice of the SAURON/ATLAS3D team for estimating σe, which derives the effective velocity dispersion from the luminosity weighted spectrum within Re (e.g., Emsellem et al. 2007). The option of using spatially resolved velocity fields to derive σe (e.g., Gültekin et al. 2009) cannot be applied due to data limitation. However, both methods provide consistent σe estimates for local massive systems with classical bulges (Figure 11 of Kormendy & Ho 2013). We provide further tests showing agreement between both procedures in Appendix C).

To estimate σe from σ*, we must apply an aperture correction that accounts for the radial gradient of σ* in galaxies (Jørgensen et al. 1995; Cappellari et al. 2006). We note that the aperture correction factors provided in the literature cannot be applied to our case because they were developed for spectra extracted using circular apertures rather than annular apertures. Here, we develop a correction recipe for estimating σe from σ* when using annular apertures. By assuming that the bulge surface brightness profile can be well described by a Sérsic model, an isotropic velocity dispersion (for simplicity), and a constant mass-to-light ratio, we find

F ap = ∑ k = 0 2 A k ( n ) ( R / R e ) k , $$ \begin{aligned} F_{\rm ap} = \sum ^2_{k = 0} A_k(n) (R/R_e)^k, \end{aligned} $$(1)

where Fap corresponds to the correction factor by which σ* must be divided to obtain σe. The polynomials Ak correspond to

A k ( n ) = A k 0 ′ + A k 1 ′ n − 1 + A k 2 ′ n + A k 3 ′ n 2 , $$ \begin{aligned} A_k(n) = A^{\prime }_{k0} + A^{\prime }_{k1} n^{-1} + A^{\prime }_{k2} n + A^{\prime }_{k3} n^2, \\ \end{aligned} $$(2)

where the coefficients A′kj depend on the central bulge stellar mass profile, modeled by the Sérsic index n. Equation (1) is flexible enough to account for the blurring effects of the MUSE PSF, parameterized in terms of the ratio ξ = PSF FWHM/Re. We provide the coefficients A′kj in Table 2. Equation (1) is within 2% for annular apertures located between 0.5 to 2.5 Re, bulge Sérsic indexes ranging from n = 0.5 to 8, and ξ = 0 − 2.0. This range of ξ covers the typical properties of our MUSE observations (Table A.2). Appendix D gives more details about how we derived this numerical recipe. The 1σ error for the aperture correction factor can be estimated from the uncertainties of the bulge surface brightness Sérsic profile parameters. However, these uncertainties are often underestimated due to systematics associated with non-axisymmetric components present in galaxies. Thus, we adopt a more conservative approach to compute the aperture correction factor uncertainty. For each target, we perform a Monte Carlo simulation where we vary Re and n to compute Fap using Eq. (1). The bulge half-light radius values are varied following a Gaussian distribution with width equal to the observation PSF FWHM (0 . ″ $ \overset{\prime \prime }{.} $2 for HST observations; Zhao et al. 2021). For the Sérsic index, we assume values varying between 0.5 and 8 with no prior information. We iterate 1000 times to obtain a probability distribution for Fap and compute the 1σ error from its standard deviation. We measure a typical aperture correction factor relative uncertainty of 11% and 8% for PG quasars and CARS AGNs, respectively.

Table 2.

Coefficient values for computing the annuli aperture correction.

Figure 5 shows the correction factors applied to our sample, categorized into PG quasars and CARS AGNs. The annular aperture correction factors are ∼0.6 − 1.2. On average, we find that the aperture correction factors are more significant for the PG quasar sample. This can be attributed to two factors: (1) the PG quasars tend to be at higher redshifts (⟨z⟩ = 0.060) compared with the CARS AGNs (⟨z⟩ = 0.042), resulting in spectra extracted farther away from the bulge Re due to their smaller projected sizes on the sky; and (2) the combination of the PSF blur and high AGN luminosity in the PG quasars limiting the extraction of σ* close to the bulge Re. We note that the average observation seeing conditions for both AGN samples are similar ( ∼ 0 . ″ 83 ± 0 . ″ 26 $ {\sim} 0{{\overset{\prime\prime}{.}}}83 \pm 0 {{\overset{\prime\prime}{.}}}26 $).

thumbnail Fig. 5.

Annular aperture correction factor, Fap ≡ σ*/σe, for both AGN samples.

3.5. CaT EW and S/N-associated systematics

The scattered light from the nucleus in type 1 AGNs dilutes the host galaxy stellar spectrum by a featureless continuum. This spectrum dilution leads to a decrease in the EW and S/N for the stellar features (e.g., Alexandroff et al. 2013), introducing an additional source of uncertainty when deriving σ*. Here, we investigate the effect of the AGN dilution for our measurements.

We conduct a series of Monte Carlo simulations using MUSE observations of inactive galaxies taken from the AMUSING++ survey (López-Cobá et al. 2020). After removing Seyfert galaxies, mergers, low-quality data cubes (Ca IIλ8542 line S/N < 5), and matching the galaxies without AGN based on redshift (0.015 < z < 0.09), stellar mass (109.5–1011.5 M⊙), and star formation rate (0.32–32 M⊙ yr−1), we use 75 MUSE observations of 9 E/S0, 14 spiral, 7 peculiar, and 45 unclassified galaxies. We extracted the spectra following the procedure outlined in Section 3.1, but applying circular apertures instead of annular apertures since we do not need to consider any nuclear emission. We mimic the effects of dilution by AGN emission on the line EW by adding a constant continuum and its associated Poisson noise until the recovered Ca IIλ8542 EW reduces to 0.2 Å. In a second test, we study the accuracy of the absorption-line width recovery against spectrum quality. This is done by adding Poisson noise to the inactive galaxy spectra until the resulting S/N of the Ca IIλ8542 line is ∼1. In both tests we encompass the Ca IIλ8542 line EW and S/N values measured for the type 1 AGNs presented in this work7. Figure 6 summarizes the results from both simulations. Although the Ca IIλ8542 line EW is significantly reduced owing to its dilution, the uncertainty of the σ* measurements remains controlled within ∼0.1 dex for reasonable AGN-host galaxy contrast. This is because the added Poisson noise is not high enough to critically reduce the S/N of the absorption lines. We highlight this in our second test which emphasizes the stronger link between σ* recovery and the S/N of the stellar absorption feature. Our test results are in qualitative agreement with similar reports by Caglar et al. (2020). Note that our analysis does not include the effect of CaT being influenced by blending with AGN emission lines, which are largely absent in this spectral region. The effect of CaT S/N on the uncertainty of σ* is already considered in our Monte Carlo resampling routine.

thumbnail Fig. 6.

Systematic uncertainty for the stellar velocity dispersion as a function of the (a) EW and (b) S/N of Ca IIλ8542. The dashed line represents the equality between the observed σ* and the value obtained after degrading the spectra quality; dot-dashed and dotted lines represent the 0.1 and 0.2 dex scatter, respectively. At the bottom of each panel, the normalized histograms show the distribution of EW or S/N values obtained for the PG quasars, CARS AGNs, and the inactive galaxy subsample taken from the AMUSING++ survey.

3.6. BH mass

We estimate the BH masses following Ho & Kim (2015), who, building upon the work of Greene & Ho (2005b), present recalibrated single-epoch virial mass estimators based on the updated virial coefficients for classical bulges and pseudo bulges (Ho & Kim 2014) and the BLR size and continuum luminosity relation of Bentz et al. (2013). The BH masses (Table A.1) are computed as

log M ∙ ( H β ) = log [ ( FWHM ( H β ) 1000 km s − 1 ) 2 ( λ L λ ( 5100 Å ) 10 44 erg s − 1 ) 0.533 ] + a , $$ \begin{aligned} \log M_\bullet (\mathrm{H}\beta ) = \log \left[ \left( \frac{\mathrm{FWHM}(\mathrm{H}\beta )}{1000\,\mathrm{km\,s}^{-1}} \right)^2 \left( \frac{\lambda L_\lambda (5100\, {\AA })}{10^{44}\,\mathrm{erg\,s}^{-1}} \right)^{0.533} \right] + a, \end{aligned} $$(3)

where a = 7.03 ± 0.02 for classical bulges and a = 6.62 ± 0.04 for pseudo bulges. The zero-point difference implies that host galaxies presenting pseudo bulges have 0.41 dex lower BH mass for fixed Hβ linewidth and AGN luminosity. The M• uncertainties are conservatively estimated as the sum in quadrature of the scatter of the M• − σe relation used to calibrate the BH mass prescription (0.29 dex for classical bulges and ellipticals, and 0.46 dex for pseudo bulges; Ho & Kim 2014) and the scatter of the BLR-size relation (0.19 dex; Bentz et al. 2013). Thus, we adopt M• uncertainties of 0.35 dex and 0.50 dex for classical bulges and pseudo bulges, respectively. These estimates bracket the typical BH mass uncertainty of traditional single-epoch M• prescriptions (e.g., ∼0.43 dex; Vestergaard & Peterson 2006). We classify the bulge type following Ho & Kim (2015), with classical bulges presenting n > 2 (but see Gao et al. 2020).

4. Results

In total, we successfully detect the CaT for 40 out of 42 host galaxies among the PG quasars and CARS AGNs. We model the CaT stellar features at a median distance of ∼1.1 ± 0.5 kpc away from the host galaxy nucleus (Table 1), corresponding to ∼0.8 times the bulge Re. We consider the host galaxy morphology when constructing the annular apertures, minimizing the effect of host galaxy inclination when measuring σ*. Figure A.1 provides a qualitative view for the spectral fits. Large residuals in the CaT models often correspond to sky lines that were masked during the fitting process (e.g., HE 0429−0247), and/or inaccurate removal of MUSE instrumental features (e.g., HE 0224−2834 and HE 2128−0221). The latter are considered when estimating the uncertainties for the line widths. We detect the Ca IIλ8542 line with high significance (S/N ≥ 10) in 16 host galaxies, with low significance (10 > S/N ≥ 5) in 18 cases, and with poor significance (S/N < 5) in six host galaxies. We note that systems at higher redshifts tend to have lower S/N detection. We are unable to detect the CaT in two CARS AGNs because reliable spectra extraction for the host galaxies was not possible. These host galaxies are found too compact, with the AGN radiation dominating the observed emission. After discarding targets with S/N < 5 detection, we obtain a total of 34 host galaxies with reliable σe measurements.

4.1. Stellar velocity dispersion estimates

We provide the stellar velocity measurements in Table 1. We find that σ* values range from 60 to 230 km s−1, with median values of 152 km s−1 for PG quasars and 116 km s−1 for CARS AGNs. The typical σ* uncertainties are ∼10%±10% (∼14 km s−1), although 8 sources have larger 1σ errors due to low-S/N detection of the CaT (e.g., HE 0224−2834, HE 1017−0305, HE 1353−1917, PG 1426+015). We note that the MUSE instrumental resolution at 9000 Å (∼36 km s−1; Guérou et al. 2017) is relatively low compared with the measured σ* values. Velocity dispersion values close to or below the instrumental resolution may have significant scatter or be overestimated (Scott et al. 2018). For instance, Koss et al. (2022b) exclude any σ* measurements within 20% of the instrumental resolution limit for their sample of local type 2 quasars. In our case, all of our targets are above this limit, with HE 2018−0221 presenting the lower σ* value, 67% higher than the MUSE spectral resolution.

From the literature, we find few σ* measurements for our targets that can be used to make rough comparisons. Here, we report any σ* measurement irrespective of the stellar feature observed. Dasyra et al. (2007) presented near-IR H−band spectra modeling (mainly using CO stellar features; their Figure 2) for PG 0050+124 (σ* = 188 ± 36 km s−1), PG 1126−041 (σ* = 194 ± 29 km s−1), PG 1426+015 (σ* = 185 ± 67 km s−1), and PG 2130+099 (σ* = 156 ± 18 km s−1). Those spectra were obtained by using slits with widths of ∼ 1 . ″ 0 − 1 . ″ 4 $ {\sim} 1{{\overset{\prime\prime}{.}}}0-1{{\overset{\prime\prime}{.}}}4 $, but avoiding the nuclear zones ( ≲ 0 . ″ 3 − 0 . ″ 4 $ {\lesssim}0{{\overset{\prime\prime}{.}}}3-0{{\overset{\prime\prime}{.}}}4 $). Grier et al. (2013) presented adaptive optics-assisted Gemini near-IR IFU measurements, also taken in the H-band, for PG 1426+015 (σ* = 211 ± 15 km s−1) and PG 2130+099 (σ* = 167 ± 19 km s−1). Those measurements were extracted over annular apertures avoiding the AGN emission; they set annulus inner radius ∼ 0 . ″ 2 − 0 . ″ 4 $ {\sim} 0{{\overset{\prime\prime}{.}}}2{-}0{{\overset{\prime\prime}{.}}}4 $ and outer radius ∼ 0 . ″ 6 − 1 . ″ 7 $ {\sim} 0{{\overset{\prime\prime}{.}}}6{-}1{{\overset{\prime\prime}{.}}}7 $. Bennert et al. (2011) use Keck/LRIS long-slit spectroscopy to isolate the host galaxy emission and measure σ* for HE 0203−0031 and HE 0119−0118. Based on CaT modeling and apertures equal to the bulge Re, they report σ* = 200 ± 9 km s−1 and 89 ± 10 km s−1, respectively. Busch et al. (2015) targeted HE 1029−1931 with seeing-limited SINFONI observations in the K band, reporting σ* = 104 ± 20 km s−1, using an aperture ∼ 0 . ″ 6 $ {\sim} 0{{\overset{\prime\prime}{.}}}6 $. Finally, for HE 1237−0504, Caglar et al. (2020) estimate σ* = 145 ± 4 km s−1 by modeling the CO(2−0) absorption features observed with SINFONI (adaptive optics-assisted). Although different procedures and modeled stellar features make direct comparisons difficult, all of these literature measurements of σ* are consistent with ours (Table 1). Using the adaptive optics observations as the main reference for duplicate literature values, we find a mean σ* ratio (literature divided by ours) of 1.04 with a standard deviation of 0.17.

Figure 7 shows the distribution for our σ* measurements. In contrast with the PG quasars, for the CARS sample we find more systems with σ* ≲ 150 km s−1, which is consistent with the CARS survey covering less luminous AGNs with less massive BHs in less massive host galaxies and at lower redshifts (Section 2). Additionally, we compare with the σ* distributions for two other AGN samples: local z ≤ 0.1 type 1 AGNs (Shen et al. 2008) from the SDSS and type 2 AGNs from the BAT AGN Spectroscopic Survey (BASS; Koss et al. 2022a). We only select the BASS type 2 AGNs with CaT-based σ* values, corresponding to host galaxies at z ≲ 0.065 (Koss et al. 2022b). Both samples provide a broader reference for type 1 AGNs, but it should be noted that their measurements may be less precise than ours for classical bulges and pseudo bulges because of aperture effects. For example, the SDSS data were obtained using 3″-wide fibers. Despite this limitation, the BASS and SDSS samples still serve as a valuable comparison, especially considering the small correction needed for aperture effects. In the joint sample of PG quasar and CARS AGN host galaxies, we find a range of σ* values similar to those estimated for the SDSS z ≤ 0.1 sample. The BASS type 2 AGNs tend to present a higher fraction of host galaxies with larger σ*. This type 1-type 2 AGN dichotomy has been already noted by Koss et al. (2022b), and can be linked to the differences in the properties of their host galaxies, with the type 2 AGN host galaxies likely being more massive (see also Koss et al. 2011).

thumbnail Fig. 7.

Distribution of the stellar velocity dispersion for the active galaxies. Literature estimates consider SDSS z ≤ 0.1 type 1 AGNs (Shen et al. 2008) and BASS type 2 Seyferts (including type 1.8 and 1.9 Seyferts; Koss et al. 2022b). We caution that the literature values were derived from spectra obtained using different aperture observation setups.

After applying the aperture correction factors (Figure 5), we report σe values in the range of 60 − 250 km s−1, with median values of 185 km s−1 for the PG quasars and 108 km s−1 for the CARS sample. The typical σe uncertainties ∼16%±6% (≲25 km s−1).

4.2. The M• − σe relation

Figure 8 shows the relation between BH mass and σe for the active galaxies. We primarily sample the parameter space at M• ≲ 108 M⊙, where we detect a large scatter (∼0.6 dex) when considering the relation of Kormendy & Ho (2013) as reference. The CARS AGNs mainly span M• − σe at M• ≲ 107.5 M⊙, while the PG quasars complement CARS for higher BH masses. The lack of AGN data at higher BH masses is mainly due to the redshift upper limit for our sample (z ≲ 0.1). By performing a multivariate Cramér test (Baringhaus & Franz 2004), using the Kormendy & Ho (2013) sample as reference for M• ≲ 108 M⊙, we find no difference between the active and inactive galaxy samples (p-value = 0.69). The p-value value decreases to 0.42 when considering a BH mass upper limit of 108.5 M⊙, but this is largely due to poor AGN sample statistics at M• ≳ 108 M⊙.

thumbnail Fig. 8.

M• − σe relation for local type 1 AGNs and inactive galaxies. Colored open circles correspond to three AGNs for which the median sample aperture correction factor was adopted to estimate σe. The data for inactive galaxies are taken from Kormendy & Ho (2013), as well as the best-fit relation (their Equation (7)) and scatter.

Our BH mass estimates rely on the BLR size-luminosity relation of Bentz et al. (2013). Woo et al. (2024) recently suggested a slightly shallower trend for this relation. They report typical BLR sizes being smaller for the more luminous AGNs [ λ L λ ( 5100 Å ) ≳ 10 44 $ \lambda L_\lambda (5100\, {\AA}) \gtrsim 10^{44} $ erg s−1], suggesting less massive BHs because M• scales linearly with BLR size. We explore this possibility by reestimating the BH masses following the BLR size-luminosity relation of Woo et al. (2024) and Ho & Kim (2015). For AGNs with λ L λ ( 5100 Å ) ≳ 10 44 $ \lambda L_\lambda (5100\, {\AA}) \gtrsim 10^{44} $ erg s−1 (eight systems; six PG quasars) the BH masses decrease by ≳0.15 dex. HE 1248−1356 is the only system for which the BH mass increases by ≳0.15 dex due to its low AGN luminosity [ λ L λ ( 5100 Å ) ≲ 10 42 $ \lambda L_\lambda (5100\, {\AA}) \lesssim 10^{42} $ erg s−1]. For all other AGNs, the BH masses vary by ≲0.15 dex, well below the typical M• uncertainty range. Figure 9 presents our sample with updated BH masses (M•W24). A multivariate Cramér test suggest no difference between the active and inactive galaxy samples (p-value = 0.49), in agreement with our previous report.

Considering the 1σ lower limit of the M• − σe relation (0.29 dex scatter) and the 1σ uncertainties of our M• and σe estimates, Figure 8 suggests four out of eight PG quasars and five out of 26 CARS AGNs lie below the M• − σe relationship for ellipticals and classical bulges (Kormendy & Ho 2013), largely following the galaxies with pseudo bulges (Saglia et al. 2016). Finding some systems below M• − σe might not be surprising, as we adopted the single-epoch BH mass prescription of Ho & Kim (2015), which systematically offsets the systems with pseudo bulge −0.41 dex below the classical M• − σe relation. To check whether this assumption drives our results, in Figure 10 we show M• − σe differentiated by bulge type. For pseudo bulges, we adopt the best-fit reported by Ho & Kim (2014, ∼0.46 dex scatter). They modeled the data collated in Kormendy & Ho (2013) keeping the slope fixed to that of the M• − σe relation of classical bulges and ellipticals during the fitting process. We find that five out of nine of the AGN hosts located below M• − σe for classical bulges and ellipticals are systems with pseudo bulges, suggesting that the adopted BH mass prescription may be effectively producing such trend. However, we report four AGN hosts with classical bulges below M• − σe (e.g., HE 0949−0122, PG 2130+099). Literature data also show some systems with classical bulges following such trend. These AGN hosts are immune to the BH mass systematic described above. For example, by adopting the BH mass prescription of Vestergaard & Peterson (2006), which does not differentiate AGN host galaxies by bulge type, to reestimate M•, all the systems with pseudo bulges shift upward in the M• − σe plane (Figure 11), as expected. Only two AGN hosts with pseudo bulge remain below the M• − σe relation for classical bulges and ellipticals after considering their BH mass uncertainty (0.43 dex; Vestergaard & Peterson 2006). However, the four AGN host galaxies with classical bulges continue to line below the M• − σe relation. We observe similar trends for the literature data. We note that a multivariate Cramér test indicates that our sample does not differ from that of inactive galaxies taken from Kormendy & Ho(2013, p-value  < 0.01) after adopting the BH mass prescription of Vestergaard & Peterson (2006).

Figure 8 also shows two systems lying above the M• − σe relation, HE 0119−0118 and PG 1211+143. When looking carefully, we find that HE 0119−0118 has a barred host galaxy (Husemann et al. 2022), and PG 1211+143 is compact (Zhao et al. 2021). It is well-known that measuring bulge properties in barred galaxies is difficult (Gao & Ho 2017, and references therein). We conjecture similar issues for PG 1211+143, where the host galaxy morphology and AGN emission critically undermine accurate host galaxy image decomposition. Therefore, we cannot be certain of the accuracy of the offset with respect to M• − σe for these two systems. It is possible that those measurements also reflect biased estimations of σe due to poor host galaxy bulge characterization, in cases where the bulge is too compact and the region from where we extract σ* corresponds to that of a bar or disk. We caution that the results presented in this section depend, in part, on the adopted M• scaling relationship (e.g., Shankar et al. 2019) and the criteria used to classify bulge type.

thumbnail Fig. 9.

M• − σe relation for local type 1 AGNs, as in Figure 8, but updating the BH mass estimates following the BLR size-luminosity relation of Woo et al. (2024).

thumbnail Fig. 10.

M• − σe relation for local type 1 AGNs, as in Figure 8, highlighting the best-fit relations for classical bulges (Kormendy & Ho 2013) and pseudo bulges (Ho & Kim 2014). We differentiate the systems by bulge type. The shaded regions represent the intrinsic scatter for both relations, 0.29 dex and 0.46 dex, respectively. The dashed line indicates the extrapolation of the best-fit relation for pseudo bulges at high BH masses. We also show the RM AGNs presented in Ho & Kim (2014, 2015) and the local AGN sample of Bennert et al. (2021).

thumbnail Fig. 11.

M• − σe relation for local type 1 AGNs, as in Figure 10, but adopting the single-epoch BH mass prescription of Vestergaard & Peterson (2006). The M• uncertainty is 0.43 dex.

Another source of uncertainty arises from estimating σe (and σ*) itself. We have estimated σe from a spectrum obtained by collapsing the host galaxy emission over an aperture. This method differs from that usually employed in observations that spatially resolve the galaxy kinematics (e.g., Gültekin et al. 2009; Kormendy & Ho 2013). Differential rotation, which is minimized by analyzing kinematic fields, may artificially broaden the spectrum obtained from an aperture, effectively increasing the estimated σe. While this effect may be minor for bulges, it may be significant for pseudo bulges due to their higher rotation support. Greene & Ho (2005a) suggest that rotational broadening may be small. Bennert et al. (2011) find an average increase ∼10% for σe due to rotational broadening, but they note that rotational broadening could be more significant for edge-on systems (up to ∼40%, implying ∼0.6 dex difference in BH mass offset). Note that, in the context of elliptical galaxies σe is computed by co-adding spectra within apertures (e.g., Cappellari et al. 2006) even though they also typically rotate (Cappellari 2016). We emphasize that we are comparing our sample with the inactive galaxy sample of Kormendy & Ho (2013), whose σe measurements were obtained from spatially resolved kinematics. Kormendy & Ho (2013) compared their estimates with aperture-based values provided by the SAURON/ATLAS3D team for massive galaxies with classical bulges, and found no major discrepancies (see their Figure 11b). We test the consistency of both procedures for systems with bulge and pseudo bulges by analyzing mock data in Appendix C. The mock data are build with galaxy properties encompassing those of our targets (Tables A.1 and 1). We find good agreement between both methods in estimating σe, with aperture-based values overestimating those obtained from spatially resolved kinematic maps by ∼2%±2% for σe ≳ 75 km s−1. However, we caution that our test suggests a significant overestimation of σe by the aperture-based method when the linewidths of the absorption features are comparable to the spectral resolution of the observations (see Appendix C, for more details).

One remaining uncertainty is whether to account for inclination effects when computing σe. Bennert et al. (2015) show that correcting σe for inclination may increase the values by up to ∼40%. However, as Bennert et al. (2015) note, not correcting σe for inclination is the common practice in the literature, meaning that any potential systematic bias would affect all galaxy samples, not just those hosting AGNs. Nevertheless, we find no correlation between BH mass offset from the M• − σe relation and galaxy inclination for our sample.

4.3. Deviations from the M• − σe relation

Literature studies suggest that active galaxies with more efficiently accreting BHs tend to deviate from M• − σe, with the Eddington ratio8 being inversely correlated with the BH mass offset (e.g., Shen et al. 2008; Ho & Kim 2014). The CARS AGNs and the PG quasars contain local AGNs with high Lbol/LEdd, making our sample ideal for testing departures from the M• − σe relation. The PG quasars mainly sample Lbol/LEdd ≳ 0.1, while the CARS AGNs are spread all over the range Lbol/LEdd = 0.01 − 1, although the majority of these systems present Lbol/LEdd ≲ 0.1. We compute the BH mass offset from M• − σe, using as reference Equation (7) of Kormendy & Ho (2013):

Δ log ( M ∙ / M ⊙ ) = log ( M ∙ M ⊙ ) − 4.38 log ( σ e 200 km s − 1 ) + 8.49 . $$ \begin{aligned} \Delta \log \,(M_\bullet / M_\odot ) = \log \left(\frac{M_\bullet }{M_\odot }\right) - 4.38\,\log \left(\frac{\sigma _e}{200\,\mathrm{km}\,\mathrm{s}^{-1}}\right) + 8.49. \end{aligned} $$(4)

Figure 12 shows the BH mass offset as a function of Eddington ratio. We find no clear trends. Dividing the data into two bins of Eddington ratio, AGNs with Lbol/LEdd ≤ 0.1 present a median M• offset of −0.15, with a scatter of 0.52 dex. Objects with Lbol/LEdd > 0.1 show a median BH mass departure equal to −0.53, with a scatter of 0.65 dex. We compute Kendall τ correlation coefficients for both AGN samples and the joint sample (PG quasars plus CARS AGNs). In this analysis, we include the 1σ uncertainties of the quantities involved. We consider a correlation to be valid only if the probability of it occurring by chance is ≤0.05. For the PG quasars, we compute r = 0 . 06 − 0.22 + 0.22 $ r = 0.06^{+0.22}_{-0.22} $ with a p-value of 0.92, while for the CARS AGNs we estimate r = − 0 . 04 − 0.10 + 0.09 $ r = -0.04^{+0.09}_{-0.10} $ with a p-value of 0.79. For the joint sample, we compute r = − 0 . 14 − 0.08 + 0.08 $ r = -0.14^{+0.08}_{-0.08} $ with a p-value of 0.24. No statistically significant correlation is seen. We find similar results when only considering host galaxies with CaT S/N > 10. Bearing in mind that galaxies with pseudo bulges are offset from M• − σe and exhibit larger scatter (Kormendy & Ho 2013), we further distinguish host galaxies with classical or pseudo bulges. Figure 13 presents the data color-coded by bulge type, with the classical bulges labeled following two common classification schemes: the traditional Sérsic index > 2 threshold (Kormendy & Kennicutt 2004; Fisher & Drory 2008), and B/T > 0.1 (Gao et al. 2020; Quilley & de Lapparent 2023). For the first case, we find no correlation for either bulge type (classical bulges: r = − 0 . 13 − 0.15 + 0.18 $ r=-0.13^{+0.18}_{-0.15} $, p = 0.65; pseudo bulges: r = − 0 . 15 − 0.11 + 0.11 $ r=-0.15^{+0.11}_{-0.11} $, p = 0.31). For the second case, the limited statistics preclude us from computing a correlation coefficient for pseudo bulges. For galaxies hosting classical bulges, we find no significant correlation ( r = − 0 . 16 − 0.09 + 0.10 $ r=-0.16^{+0.10}_{-0.09} $, p = 0.22) after correcting LEdd, M•, and BH mass offset considering our adopted bulge-type dependent BH mass prescription (Ho & Kim 2015). However, we caution that using the B/T ratio for classifying bulge type in AGN host galaxies is highly misleading, as it is known that the central spheroid can be overluminous from recent star formation activity (Kim & Ho 2019). Furthermore, bulge model uncertainties are pernicious in the presence of a bright AGN glare. While we have some confidence in the AGN-host decomposition of the PG quasars, which were based on HST images (Zhao et al. 2021), the image analysis of the CARS AGNs was limited to MUSE white-light images (Husemann et al. 2022). We lack the statistics necessary to study subtle trends with respect to host galaxy morphology.

thumbnail Fig. 12.

BH mass offset from the M• − σe relation of Kormendy & Ho (2013, their Equation (7)) as a function of the Eddington ratio. The error bars in the bottom right corner represent the BH mass and Eddington ratio 1σ uncertainties. We detail the correlation coefficient r and p-value for the PG quasars, CARS AGNs, and the joint sample.

thumbnail Fig. 13.

Similar to Figure 12, but color-coding the data by the host galaxy bulge type, which is differentiated by (a) Sérsic index n and (b) the B/T ratio. We correct the BH mass-related quantities according to the bulge-type classification following the BH mass prescription of Ho & Kim (2015).

We complement our sample by including the less-luminous RM AGNs presented in Ho & Kim (2014, 2015) and the local AGNs observed by Bennert et al.(2021, Figure 14). When combining the three data sets, the correlation coefficient slightly reduces to r = − 0 . 12 − 0.05 + 0.05 $ r=-0.12^{+0.05}_{-0.05} $ with a p-value of 0.05. However, we note that the low p-value is mainly driven by the AGNs with Eddington ratio < 0.01 (only four objects). By excluding these AGN hosts we obtain r = − 0 . 09 − 0.05 + 0.04 $ r = -0.09^{+0.04}_{-0.05} $ and p = 0.15. The p-value is not low enough to imply a significant inverse correlation between BH mass offset and Lbol/LEdd.

thumbnail Fig. 14.

Similar to Figure 12, but we add the RM AGNs presented in Ho & Kim (2014, 2015), the local AGN sample of Bennert et al. (2021), and the SDSS z ≤ 0.1 type 1 AGNs provided by Shen et al. (2008).

5. Discussion and summary

Early studies commonly reported that the AGN host galaxies tend to follow a different M• − σe relation compared with that of inactive galaxies (e.g., Greene & Ho 2006a; Shen et al. 2008; Ho & Kim 2014). This was interpreted as a natural consequence of bright AGNs being preferentially hosted in late-type galaxies, considering that these systems tend to present pseudo bulges (Kormendy & Kennicutt 2004) and pseudo bulge hosts lie below the M• − σe relation of classical bulges and ellipticals (Hu 2008; Greene et al. 2010; Kormendy & Ho 2013; Saglia et al. 2016; de Nicola et al. 2019). However, opposite reports have also been presented (e.g., Nelson et al. 2004; Woo et al. 2010, 2013; Bennert et al. 2011, 2015; Caglar et al. 2020). By analyzing spatially resolved σ* profile data, Bennert et al. (2021) suggest that AGN host galaxies with with a pseudo bulge follow the M• − σe relation of classical bulges and ellipticals. Our measurements for the type 1 AGNs and quasars agree with the classical bulge–pseudo bulge dichotomy for M• − σe. Many of the PG quasars are located below the relation of Kormendy & Ho (2013) for classical bulges, in the regime of pseudo bulges, even though their central spheroids are classical bulges. Similar systems are also present in the sample of Bennert et al. (2021, Figure 10). It is unlikely that underestimated BH masses is the sole cause in producing such trends. We have used the BH mass prescription of Ho & Kim (2015), which was explicitly calibrated for RM AGNs (Ho & Kim 2014) using the M• − σe relation of inactive galaxies (Kormendy & Ho 2013). The difference in zero-point applied for host galaxies with classical bulges and pseudo bulges (Equation 3) can only account for a M• offset up to 0.41 dex for systems with a pseudo bulge. Still some AGN hosts with a pseudo bulge deviate ≳1 dex from M• − σe (Figure 13a). However, we find average BH mass offsets from M• − σe of ∼ − 0.25 ± 0.18 dex for AGN hosts with a classical bulge and −0.31 ± 0.11 dex for those with a pseudo bulge, implying that any claim is on weak grounds. If we adopt the traditional single-epoch BH mass prescription of Vestergaard & Peterson (2006), which does not distinguish by bulge type, many of the AGN hosts with a pseudo bulge shift upward, closer to M• − σe. Only a minor fraction (16%) of AGN hosts remain below M• − σe (Figure 11). However, in such a case, our AGN sample distribution on the M• − σe plane would become significantly different from that of inactive galaxies taken from Kormendy & Ho (2013, multivariate Cramér test p-value  < 0.01). It is unclear why both samples may depart from each other, such that most of the disk-like AGN hosts may present overmassive BHs (or pseudo bulges with a lower σe) compared with the inactive spirals. BH masses could be underestimated if the BLR has a significant amount of dust, with ∼0.3 − 0.4 dex offsets reported for less luminous AGNs (Caglar et al. 2020). However, such BH mass offsets are not enough to explain our data, as mentioned above. Adopting the recent BLR-size luminosity relation of Woo et al. (2024) instead of that of Bentz et al. (2013) strengthens our finding, as Woo et al. (2024)’s BLR-size relation suggests less massive BHs for the more luminous AGNs. We note that the CaT is largely insensitive to the underlying stellar population properties (Dressler 1984), suggesting that our findings should be robust against recent star formation activity that might induce systematic bias in AGN hosts. These findings are consistent with reports on active galaxies in the context of the BH mass-bulge mass relation, where some active galaxies have a systematically lower M• than inactive galaxies for both cases when (Molina et al. 2023a) or when not differentiating (Kim et al. 2008; Zhao et al. 2021; Ding et al. 2022) by bulge type when estimating BH masses.

In our search for more subtle trends, we did not find any correlation between the Eddington ratio and offset from the M• − σe relation, even after complementing our sample with the less-luminous RM AGN data. However, we cannot discount the possibility that the large uncertainties involved when estimating both the BH mass offset from M• − σe and the Eddington ratio (∼0.3 − 0.4 dex) may be washing out any potential weak correlation, further concealed by our small sample size. This is suggested in Figure 14 by the SDSS type 1 AGN data, which can be interpreted as a rough reference for the population of type 1 AGNs covering a representative range of Eddington ratios at low redshifts (Shen et al. 2008)910. The SDSS data show an inverse correlation between the M• offset and Lbol/LEdd ( r = − 0 . 31 − 0.04 + 0.05 $ r=-0.31^{+0.05}_{-0.04} $, p < 0.01). It is worth noting that sample selection effects are also concerning. On the one hand, as discussed in Shen et al. (2008), an inverse correlation between Lbol/LEdd and the BH mass offset cannot be explained by the interdependence between Lbol, LEdd, and M•, but it can be qualitatively produced if σe and λ L λ ( 5100 Å ) $ \lambda L_\lambda (5100\, {\AA}) $ are positively correlated. We ascertain that this is the case for the AGN sample analyzed here ( r = 0 . 25 − 0.05 + 0.05 $ r = 0.25^{+0.05}_{-0.05} $, p = 0.03). On the other hand, applying a luminosity threshold for selecting active galaxies, as was done for the PG quasars (Boroson & Green 1992) and CARS AGNs (Schulze et al. 2009), biases the samples toward AGNs with a high BH mass when considering the single-epoch virial mass estimate (Shen & Kelly 2010), making it harder to detect any residual correlation between the BH mass offset from M• − σe and the Eddington ratio. The effects of these potential contending biases in producing the observed trend are, admittedly, uncertain.

There have been reports that AGNs with high Lbol/LEdd being systematically below the BH mass–host stellar mass relation (Volonteri et al. 2015; Shankar et al. 2019). Zhuang & Ho (2023) suggest that the level of BH accretion and star formation of an active galaxy is related to its position on the M• − M* plane, with BH growth outpacing the host galaxy stellar growth when the BH is undermassive. They show that most of the local AGN hosts below the M• − M* relation are late-type systems, with plenty of gas reservoirs as suggested by their blue color. We may be observing a similar trend in the M• − σe plane as well. If the BHs are growing more rapidly than the host galaxy bulges in local AGNs and quasars with a high Eddington ratio, then they may culminate on the M• − σe relation for classical bulges and ellipticals. Molina et al. (2023a) note that in the more gas-rich PG quasars, the BHs can increase their mass significantly while the bulges may have already finished their mass buildup, unless the host galaxies undergo merging. The amount of mass that BHs can accrete may be rooted at the given formation stage of the host galaxy central stellar spheroid (e.g., Merritt & Poon 2004; Miralda-Escudé & Kollmeier 2005; Anglés-Alcázar et al. 2017a), perhaps further modulated by a bursty nuclear stellar feedback episode (Anglés-Alcázar et al. 2017b), or self-limited by feedback from the central engine surrounding the BH (King 2010; King & Pounds 2015). Menci et al. (2023) suggest that the more efficiently accreting BHs tend to lie below the M• − σe for classical bulges and ellipticals due to the large gas fuel supply available within the host galaxies and the relatively weak AGN feedback in the high Eddington ratio regime. We can neither rule out nor probe this possibility, but we note that Molina et al. (2023b) report a weak correlation between the Eddington ratio and molecular gas fraction for z ≲ 0.5 luminous AGNs, including the PG quasars and CARS AGNs.

To summarize, this study used archival MUSE observations for 42 local (z ≲ 0.1) type 1 AGNs and quasars taken from the CARS and PG quasar surveys. Our main goal was to measure the bulge stellar velocity dispersion from the CaT stellar features and investigate the location of the host galaxies in the M• − σe relationship. We used annular apertures to extract the spectra and mitigate the effect of the AGN emission in diluting the host galaxy stellar features. Novel aperture corrections were developed to estimate accurate σe in these systems. Our main results are as follows:

  1. The active galaxies have stellar velocity dispersion in the range σ* = 60 − 230 km s−1. After correcting for aperture effects, bulge morphology, and observations’ beam-smearing, we estimated σe values in the range 60 − 250 km s−1.

  2. By assuming a BH mass prescription that differentiates between classical bulges and pseudo bulges, we find that the CARS AGNs and PG quasars span over the M• – σe plane, with no statistical difference with respect to the inactive galaxies. The type 1 AGNs tend to preferentially follow the M• – σe relation of elliptical and classical bulges, with ∼25% AGN host galaxies located in the M• – σe regime of pseudo bulges. We find two host galaxies above the local M• – σe relation, but systematics associated with galaxy morphology are likely hindering their σe estimates. If we do not differentiate by bulge type when estimating BH masses, the fraction of AGN hosts found below M• – σe mildly reduces to ∼16%, and the sample distribution AGNs on the M• – σe plane becomes inconsistent with that of the inactive galaxies.

  3. We do not find any correlation between the BH mass offset from M• – σe and the Eddington ratio, even after complementing our sample with the less luminous reverberation-mapped AGNs and AGN host galaxies with spatially resolved σ* profiles. However, we caution that this may be due to the large measurement uncertainties plus the small sample size.

1

We refer to σe as the stellar velocity dispersion measured inside Re, the bulge effective radius.

4

For PG 1426+015 we use a window of width ±1500 km s−1 to better characterize the continuum.

5

These stars correspond to Gaia DR3 3195905922532342656, 2684962933533311488, 2504238273149858816, and 2366885700060472576, which are consistent with being K-type, but were not previously selected by stellar classification.

6

For PG 1011−040 we use a window of width ±600 km s−1 because of the presence of calcium in emission (Persson 1988).

7

Even though spectrum quality is commonly quantified by the S/N of the stellar continuum, we prefer to use the S/N of the Ca IIλ8542 line as reference because AGN continuum subtraction is not needed for all sources (see Table 1).

8

We compute the Eddington ratio as Lbol/LEdd, where LEdd = 1.26 × 1038(M•/M⊙) erg s−1.

9

We only consider the σe estimates based on CaT modeling. The data were corrected by aperture effects; however, the bulge morphology was roughly approximated (see Shen et al. 2008, for more details).

10

The BH masses and Eddington ratios of the SDSS sample were updated following Ho & Kim (2015) for classical bulges.

Acknowledgments

We thank the anonymous referee for constructive comments and suggestions. LCH was supported by the National Science Foundation of China (11721303, 11991052, 12011540375, 12233001), the National Key R&D Program of China (2022YFF0503401), and the China Manned Space Project (CMS-CSST-2021-A04, CMS-CSST-2021-A06). K. K. acknowledges support from the Knut and Alice Wallenberg Foundation. This research has made use of the services of the ESO Science Archive Facility, and based on observations collected at the European Organization for Astronomical Research in the southern hemisphere under ESO programme IDs 094.B−0345(A), 095.B−0015(A), 097.B−0080(A), 099.B-0242(B), 099.B-0294(A), 0101.B−0368(B), 0103.B−0496(B) and 0104.B−0151(A), 106.21C7.002.

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Appendix A: Additional material

Table A.1.

Basic parameters of the sample.

thumbnail Fig. A.1.

Extracted spectrum at the CaT wavelength range for each host galaxy. For clarity, the model residuals have been shifted to the flux density level indicated by the dotted line. The gray shaded regions represent the spectral windows that are masked to avoid prominent sky-line features detected in the corresponding variance spectra.

thumbnail Fig. A.1.

Continued.

Table A.2.

MUSE observational setup.

Appendix B: MUSE instrumental feature characterization

For each target, we characterize and correct the MUSE data for an instrumental feature seen at 9060–9180 Å in the observer frame. We derive an instrumental feature template from field stars that are also observed by MUSE. For each target, the annuli-extracted spectra are modeled using an instrumental feature template multiplied by a scaling factor plus a power-law continuum component. A template flux normalization is determined by calculating the average of the ratio between the template scaling factor and local spectrum continuum level across all annular apertures. Figure B.1 shows the scaling factor as a function of the local continuum level for the PG quasar and CARS AGN samples. For all targets, the scaling factor is nearly proportional to the local continuum level, with minor variations corresponding to less accurate fits. By fitting a linear function to the data of each host galaxy, we recover slopes close to unity (as detailed for one case in Figure 3), in agreement with our supposition that the observed spectrum features are related to instrument response to incoming flux. The zero point variation among the sources indicates that the flux density normalization of the instrumental feature varies for each observation.

thumbnail Fig. B.1.

Scaling factor for the instrumental feature template as a function of the annuli-extracted spectra median flux density over 9030–9200 Å (observer frame) for (a) PG quasars and (b) CARS AGNs. The solid lines connect the data points corresponding to each observation.

Appendix C: Luminosity weighted σe

We estimate σe from a spectrum obtained by spatially collapsing the data cube over a given aperture. That is, we estimate σe from spatially unresolved data by adopting the convention of the SAURON/ATLAS3D team (Emsellem et al. 2007)

G e = ∫ 0 R e I ( R ) G ( R ) R d R ∫ 0 R e I ( R ) R d R , $$ \begin{aligned} G_e = \frac{\int _0^{R_e} I(R) G(R)\, R\,dR}{\int _0^{R_e} \, I(R)\, R\, dR}, \end{aligned} $$(C.1)

thumbnail Fig. C.1.

Example of a galaxy model analyzed in our test. (a) Intrinsic surface brightness distribution of the bulge. (b) Galaxy bulge plus disk surface brightness distribution convolved by PSF. (b) “RMS” velocity ( V RMS ≡ σ V 2 + V 2 ) $ (V_{\mathrm{RMS}} \equiv \sqrt{\sigma_V^2 + V^2} \,) $ map. (c) Line-of-sight velocity field. (d) Velocity dispersion map. In all panels, the magenta dashed curve shows the bulge half-light radius. We apply the color scale in panels (a) and (b). The galaxy kinematics obtained from JAMPY (Cappellari 2008).

where I(R) is the surface brightness and Ge is the luminosity weighted mean of a quantity G(R) within 1 Re. This approach differs from that commonly employed in M• − σe relation studies, as these usually use spatially resolved observations to compute the luminosity weighted σe as follows (e.g., Gültekin et al. 2009)

σ e 2 = ∫ 0 R e ( σ V 2 + V 2 ) I ( R ) d R ∫ 0 R e I ( R ) d R , $$ \begin{aligned} \sigma _e^2 = \frac{\int _0^{R_e} (\sigma _V^2 + V^2) I(R)\,dR}{\int _0^{R_e} I(R)\,dR}, \end{aligned} $$(C.2)

where V is the rotational component, and σV is the velocity dispersion. Kormendy & Ho (2013) show that both procedures provide consistent σe values for a local set of galaxies, independent of their bulge type (their Figure 11). In one of their tests, Bennert et al. (2015) show that their σe measurements are consistent with SDSS-based estimates, with the later based on aperture-extracted spectra.

Here, we test if both procedures used to estimate σe provide consistent results for mock data with properties similar to our sample. We model a galaxy as disk plus bulge, with each surface brightness component described by a Sérsic profile when projected on the sky. After setting the galaxy profile, we compute the second velocity moment map using JAMPY (Figure C.1 Cappellari 2008). We consider a PSF FWHM = 1″. From this map, we estimate effective velocity dispersion within the bulge Re following the common practice (σe2D; Gültekin et al. 2009). To build a three-dimensional data cube, we adopt a K-star spectrum template taken from the INDO-U.S. stellar spectral library of Valdes et al. (2004), and the two-dimensional surface brightness and kinematic maps for our galaxy model. The kinematic maps set the spectra Doppler shift and broadening at each data cube pixel that we must apply for. We convolve the spectra by the MUSE line spread function, accounting for the stellar template spectral resolution (FWHM = 1.35 Å; Beifiori et al. 2011). Finally, we collapse the data cube over a circular aperture equal to the bulge Re to extract the spectrum and obtain the effective velocity dispersion (σe1D). We repeat this process 500 times, varying the galaxy parameters randomly following uniform distributions with parameter ranges that encompass the observed values for our sample (Table A.1 and 1). Specifically, we consider galaxies with total mass between 9.6–11.2 M⊙ and bulge-to-total ratio in the 0.01–0.97 range. Sérsic index values in the 0.5–8 range for the bulge and 0.5–3 range for the disk. We set bulge effective radius in the range of 0 . ″ $ {{\overset{\prime\prime}{.}}} $5–5″, and the disk size can be 1.1–5 times larger. The bulge and disk can have different observed inclination on the sky. For both cases, the minor-to-major axis ratio range is 0.3–0.95, that is inclination angles between 0° (face-on) and 88° (edge-on). The axis ratio lower limit is set considering an intrinsic component axis-ratio of 0.2. Figure C.2 shows our test results. We find a good agreement between both estimates; the mean trend shows that σe1D agrees with σe2D by ≲10% for σe2D ≳ 50 km s−1. For σe2D ≳ 75 km s−1, σe1D overestimates σe2D by merely ∼2%±2%. Such factor is negligible considering the typical uncertainty of our measurements (∼10%; Section 4.1). For σe2D ≲ 50 km s−1, we find that σe1D increasingly overestimates σe2D for lower values. However, both quantities remain closely correlated, with no significant dependence on the properties of the simulated systems. The overestimation of σe2D by σe1D for σe2D ≲ 50 km s−1 seems to be related to measuring velocity dispersion values close to or below the spectral resolution of the data (Scott et al. 2018), which is ∼36 km s−1 for MUSE at 9000 Å $ 9000\,\mathrm{\AA} $ (Guérou et al. 2017) and our mock data. Given the measured σ* values for our sample (Table 1), this analysis suggests that we can safely compare our estimates with the spatially resolved σ* measurements provided in the literature.

thumbnail Fig. C.2.

Relative difference between spatially resolved and aperture-based σe measurements. The orange circles represent the mean trend over 25 km s−1-wide bins. The error bars correspond to the data scatter. The shaded region represents the spectrum resolution of MUSE in terms of line width (∼ FWHM/2.355).

Appendix D: Annular aperture correction factor

Aperture correction factors must be applied for estimating σe from projected (line-of-sight) second velocity moment (σlos) measurements. We derive aperture correction factors under the assumption that the host galaxy bulge surface brightness emission on the sky is well-described by a Sérsic (1963) radial profile

I ( R ) = I 0 exp ( − b n ( R / R e ) 1 / n ) , $$ \begin{aligned} I(R) = I_0 \exp (-b_n (R/R_e)^{1/n}), \end{aligned} $$(D.1)

where I0 is the central intensity and bn is the constant that sets Re as the half-light radius. By further assuming an isotropic velocity dispersion, σlos can be calculated as (Baes & Ciotti 2019)

σ los 2 ( R ) = 2 I ( R ) ∫ R ∞ ν ( r ) M ( r ) r 2 − R 2 d r r 2 , $$ \begin{aligned} \sigma _{\rm los}^2(R) = \frac{2}{I(R)} \int _R^\infty \frac{\nu (r) M(r) \sqrt{r^2-R^2} dr}{r^2}, \end{aligned} $$(D.2)

where ν(r) and M(r) are the three-dimensional luminosity density and mass profiles, respectively. The ν(r) and M(r) profiles are given by (Binney & Tremaine 2008)

ν ( r ) = − 1 π ∫ R ∞ d I ( R ) d R dR r 2 − R 2 , M ( r ) = 4 π ( M L ) ∫ 0 r ν ( r ′ ) r ′ 2 d r ′ , $$ \begin{aligned} \begin{array}{l} \nu (r) = \frac{-1}{\pi } \int _R^\infty \frac{\mathrm{d}I(R)}{\mathrm{d}R} \frac{dR}{\sqrt{r^2-R^2}}, \\ M(r) = 4 \pi \left( \frac{M}{L} \right) \int _0^r \nu (r^{\prime })r^{\prime 2} dr^{\prime }, \end{array} \end{aligned} $$(D.3)

where M/L is the mass-to-light ratio, assumed to be constant in this work. The ν(r) and M(r) profiles can be computed in terms of the Fox H function (Mathai et al. 2009) following Equations (17) and (25) of Baes & Ciotti(2019, see also Baes & van Hese 2011). For completeness, we also provide the formula for computing the projected (luminosity weighted) second velocity moment when adopting a circular aperture:

σ los 2 ( < R ) = ∫ 0 R I ( R ′ ) σ los 2 ( R ′ ) R ′ d R ′ ∫ 0 R I ( R ′ ) R ′ d R ′ . $$ \begin{aligned} \sigma _{\rm los}^2( < R) = \frac{\int _0^R I(R^{\prime }) \sigma _{\rm los}^2(R^{\prime }) R^{\prime } dR^{\prime }}{\int _0^R I(R^{\prime }) R^{\prime } dR^{\prime }}. \end{aligned} $$(D.4)

The PSF effect is considered following Emsellem et al.(1994, see also Cappellari 2008). We compute the PSF-convolved line-of-sight second velocity moment ( σ ∼ los $ \widetilde{\sigma}_{\mathrm{los}} $) as

σ ∼ los 2 ( R ) = ( I σ los 2 ) ( R ) ∗ PSF ( R ) I ( R ) ∗ PSF ( R ) , $$ \begin{aligned} \widetilde{\sigma }_{\rm los}^2(R) = \frac{(I \sigma _{\rm los}^2)(R)*\mathrm{PSF}(R)}{I(R) *\mathrm{PSF}(R)}, \end{aligned} $$(D.5)

where the PSF follows a Moffat (1969) profile with β ≈ 2.2 for MUSE data (Bacon et al. 2017; Guérou et al. 2017).

thumbnail Fig. D.1.

Aperture correction as a function of (a) the circular aperture size and (b) annular aperture location. The circular aperture correction values are normalized to the effective velocity dispersion at Re/2 (σe/2), following Cappellari et al. (2006). We also highlight the empirical mean circular aperture correction provided by Cappellari et al. (2006) for local E/S0 galaxies, with the shaded region representing its scatter. The numerical approximation for the annular aperture correction factors corresponds to that of Equation 1, using the input coefficients presented in Table 2.

Figure D.1 shows the annular and circular aperture correction factors as a function of radius, normalized to Re, and for different values of the Sérsic index n in the case of no PSF convolution. When using circular apertures, the correction factor is small (∼10%), but non-negligible. The analytical formula accurately reproduces the empirical correction found by Cappellari et al. (2006) for local E/S0 galaxies. When considering annular apertures, the second velocity moment correction factor can be as high as 50%, depending on the Sérsic index and/or reasonable values for annular distance from Re. Observing bulges with large Sérsic index (e.g., classical bulges) generally requires applying more significant corrections for σlos to estimate σe at a fixed radius. Note that at large annular radii additional systematics could influence the estimation of σe (e.g., the kinematics of the disk). Figure D.1 highlights the difference between adopting annular and circular apertures for estimating σe and the importance of measuring σlos as close as possible to Re.

Figure D.2 shows the effect of PSF convolution on the recovery of σe. The overall effect is a decrease of the annular aperture correction factor that must be applied when the PSF FWHM increases relative to Re, as expected due to the spreading of the central emission. This conclusion applies for values of Sérsic index beyond the n = 1 and 4 cases presented here. Figure D.3 presents the annular aperture correction factors as function of radius, normalized to Re, for different Sérsic index n, and various ratios of ξ≡ PSF FWHM/Re. The annular aperture correction factor becomes less sensitive to the bulge Sérsic index for large ξ values, indicating that determining the underlying surface brightness profile is less critical.

thumbnail Fig. D.2.

PSF convolution effect on the annular aperture correction factor for different values of PSF FWHM, normalized to Sérsic model Re. We only show results for two Sérsic surface brightness profile cases (a) n = 1 and (b) n = 4.

thumbnail Fig. D.3.

Annular aperture correction factor as a function aperture radial location. Panels (a) to (h) show particular cases of PSF FWHM values normalized to Sérsic model Re. The numerical approximation for annular aperture correction factor corresponds to that of Equation 1, adopting the input coefficients presented in Table 2.

Given that computing Fap is considerably time intensive, we develop a cheaper numerical parameterization that can be used once the bulge profile and observation PSF have been characterized. The numerical parameterization is detailed in Equations 1 and 2, with input coefficients given in Table 2. These coefficients were derived by fitting Fap in a radius range from 0.5 to 2.5 Re in steps of 0.1 Re, n = 0.5 − 8 varying in bins 0.5 wide, and ξ = 0 − 2.0 sampled in steps of 0.25. This numerical recipe is accurate up to 2%, sufficient given the larger uncertainties expected from modeling AGN host galaxy bulges. We note that, with the exception of the case of no PSF blurring (ξ = 0), our numerical recipe in only valid when the observation PSF is well-described by a Moffat profile. However, the procedure outlined here can be adopted for different representations of the PSF.

All Tables

Table 1.

Measured and derived quantities.

In the text
Table 2.

Coefficient values for computing the annuli aperture correction.

In the text
Table A.1.

Basic parameters of the sample.

In the text
Table A.2.

MUSE observational setup.

In the text

All Figures

thumbnail Fig. 1.

AGN monochromatic luminosity at 5100 Å as a function of the BH mass. The error bar in the bottom right corner represents the typical uncertainty of the quantities. The dashed lines correspond to fixed Eddington ratios. The contours show the distribution of the broad-lined AGNs taken from the 7th data release of the Sloan Digital Sky Survey (SDSS; Abolfathi et al. 2018) at z < 0.35 presented by Liu et al. (2019).
In the text
thumbnail Fig. 2.

Example of MUSE instrumental feature correction. The shaded regions highlight the masked wavelengths that encompass the CaT. The spectra have been vertically shifted to improve figure visualization.
In the text
thumbnail Fig. 3.

Scaling factor for the instrumental feature template as a function of the median flux density for the annuli-extracted spectra over the 9030–9200 Å wavelength range (observer frame). The model corresponds to a linear function adjusted to the data. The bottom panel shows the model residuals. We only show the data for one target. We present the rest of the sample in Appendix B.
In the text
thumbnail Fig. 4.

Example of the AGN emission subtraction procedure. The shaded regions highlight the masked spectral windows encompassing the CaT. The AGN template has been vertically shifted to improve figure visualization. The residuals correspond to the observed spectrum minus AGN template.
In the text
thumbnail Fig. 5.

Annular aperture correction factor, Fap ≡ σ*/σe, for both AGN samples.
In the text
thumbnail Fig. 6.

Systematic uncertainty for the stellar velocity dispersion as a function of the (a) EW and (b) S/N of Ca IIλ8542. The dashed line represents the equality between the observed σ* and the value obtained after degrading the spectra quality; dot-dashed and dotted lines represent the 0.1 and 0.2 dex scatter, respectively. At the bottom of each panel, the normalized histograms show the distribution of EW or S/N values obtained for the PG quasars, CARS AGNs, and the inactive galaxy subsample taken from the AMUSING++ survey.
In the text
thumbnail Fig. 7.

Distribution of the stellar velocity dispersion for the active galaxies. Literature estimates consider SDSS z ≤ 0.1 type 1 AGNs (Shen et al. 2008) and BASS type 2 Seyferts (including type 1.8 and 1.9 Seyferts; Koss et al. 2022b). We caution that the literature values were derived from spectra obtained using different aperture observation setups.
In the text
thumbnail Fig. 8.

M• − σe relation for local type 1 AGNs and inactive galaxies. Colored open circles correspond to three AGNs for which the median sample aperture correction factor was adopted to estimate σe. The data for inactive galaxies are taken from Kormendy & Ho (2013), as well as the best-fit relation (their Equation (7)) and scatter.
In the text
thumbnail Fig. 9.

M• − σe relation for local type 1 AGNs, as in Figure 8, but updating the BH mass estimates following the BLR size-luminosity relation of Woo et al. (2024).
In the text
thumbnail Fig. 10.

M• − σe relation for local type 1 AGNs, as in Figure 8, highlighting the best-fit relations for classical bulges (Kormendy & Ho 2013) and pseudo bulges (Ho & Kim 2014). We differentiate the systems by bulge type. The shaded regions represent the intrinsic scatter for both relations, 0.29 dex and 0.46 dex, respectively. The dashed line indicates the extrapolation of the best-fit relation for pseudo bulges at high BH masses. We also show the RM AGNs presented in Ho & Kim (2014, 2015) and the local AGN sample of Bennert et al. (2021).
In the text
thumbnail Fig. 11.

M• − σe relation for local type 1 AGNs, as in Figure 10, but adopting the single-epoch BH mass prescription of Vestergaard & Peterson (2006). The M• uncertainty is 0.43 dex.
In the text
thumbnail Fig. 12.

BH mass offset from the M• − σe relation of Kormendy & Ho (2013, their Equation (7)) as a function of the Eddington ratio. The error bars in the bottom right corner represent the BH mass and Eddington ratio 1σ uncertainties. We detail the correlation coefficient r and p-value for the PG quasars, CARS AGNs, and the joint sample.
In the text
thumbnail Fig. 13.

Similar to Figure 12, but color-coding the data by the host galaxy bulge type, which is differentiated by (a) Sérsic index n and (b) the B/T ratio. We correct the BH mass-related quantities according to the bulge-type classification following the BH mass prescription of Ho & Kim (2015).
In the text
thumbnail Fig. 14.

Similar to Figure 12, but we add the RM AGNs presented in Ho & Kim (2014, 2015), the local AGN sample of Bennert et al. (2021), and the SDSS z ≤ 0.1 type 1 AGNs provided by Shen et al. (2008).
In the text
thumbnail Fig. A.1.

Extracted spectrum at the CaT wavelength range for each host galaxy. For clarity, the model residuals have been shifted to the flux density level indicated by the dotted line. The gray shaded regions represent the spectral windows that are masked to avoid prominent sky-line features detected in the corresponding variance spectra.
In the text
thumbnail Fig. A.1.

Continued.
In the text
thumbnail Fig. B.1.

Scaling factor for the instrumental feature template as a function of the annuli-extracted spectra median flux density over 9030–9200 Å (observer frame) for (a) PG quasars and (b) CARS AGNs. The solid lines connect the data points corresponding to each observation.
In the text
thumbnail Fig. C.1.

Example of a galaxy model analyzed in our test. (a) Intrinsic surface brightness distribution of the bulge. (b) Galaxy bulge plus disk surface brightness distribution convolved by PSF. (b) “RMS” velocity ( V RMS ≡ σ V 2 + V 2 ) $ (V_{\mathrm{RMS}} \equiv \sqrt{\sigma_V^2 + V^2} \,) $ map. (c) Line-of-sight velocity field. (d) Velocity dispersion map. In all panels, the magenta dashed curve shows the bulge half-light radius. We apply the color scale in panels (a) and (b). The galaxy kinematics obtained from JAMPY (Cappellari 2008).
In the text
thumbnail Fig. C.2.

Relative difference between spatially resolved and aperture-based σe measurements. The orange circles represent the mean trend over 25 km s−1-wide bins. The error bars correspond to the data scatter. The shaded region represents the spectrum resolution of MUSE in terms of line width (∼ FWHM/2.355).
In the text
thumbnail Fig. D.1.

Aperture correction as a function of (a) the circular aperture size and (b) annular aperture location. The circular aperture correction values are normalized to the effective velocity dispersion at Re/2 (σe/2), following Cappellari et al. (2006). We also highlight the empirical mean circular aperture correction provided by Cappellari et al. (2006) for local E/S0 galaxies, with the shaded region representing its scatter. The numerical approximation for the annular aperture correction factors corresponds to that of Equation 1, using the input coefficients presented in Table 2.
In the text
thumbnail Fig. D.2.

PSF convolution effect on the annular aperture correction factor for different values of PSF FWHM, normalized to Sérsic model Re. We only show results for two Sérsic surface brightness profile cases (a) n = 1 and (b) n = 4.
In the text
thumbnail Fig. D.3.

Annular aperture correction factor as a function aperture radial location. Panels (a) to (h) show particular cases of PSF FWHM values normalized to Sérsic model Re. The numerical approximation for annular aperture correction factor corresponds to that of Equation 1, adopting the input coefficients presented in Table 2.
In the text

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