Defining the (Black Hole)-Spheroid Connection with the Discovery of Morphology-Dependent Substructure in the $M_{\rm BH}$–$\rm n_{sph}$ and $M_{\rm BH}$–$\rm R_{e,sph}$ Diagrams: New Tests for Advanced Theories and Realistic Simulations

It is widely known that the mass of the black hole (BH) residing at the centre of most galaxies is correlated with both the host spheroid’s stellar mass ($M_{\rm*,sph}$) and its central stellar velocity dispersion ($\sigma$). At the same time, bulgeless galaxies, for example, NGC 2478, NGC 4395, and NGC 6926, have also been observed to house massive BHs (e.g. Secrest et al., 2013; Simmons et al., 2013; den Brok et al., 2015; Davis et al., 2019, and references therein), and one of the tightest scaling relations is between black hole mass ($M_{\rm BH}$) and the winding/pitch angle of the spiral arms in spiral galaxies (Seigar et al., 2008; Berrier et al., 2013; Davis et al., 2017). Additional correlations exist between $M_{\rm BH}$ and disk stellar mass (Davis et al., 2018), disk rotation, and dark matter halo mass (Ferrarese, 2002; Baes et al., 2003; Volonteri et al., 2011; Davis et al., 2019). Collectively, this goes beyond the notion of a single primary (causal) relation for all galaxies plus secondary (indirect/consequential) relations, and reveals a greater level of complexity. Indeed, the markedly different $M_{\rm BH}$–$M_{\rm*,gal}$ and $M_{\rm BH}$–$M_{\rm*,sph}$ relations for early-type galaxies (ETGs, comprised of E-, ES¹¹1ES-type represents ellicular galaxies which have an intermediate-scale stellar disk confined to within their spheroid (Liller, 1966; Graham, 2019a).-, and S0-types) and late-type galaxies (LTGs), i.e. spiral (Sp) galaxies (Davis et al., 2018, 2019; Sahu et al., 2019a), undoubtedly reflects the different physical processes, gas supply history, net angular momentum, involved in building these systems.

The review of the BH scaling relations by Graham (2016) highlighted the need to achieve internal consistency among the various scaling relations, in particular between the $M_{\rm BH}$–$\sigma$, $M_{\rm BH}$–$M_{\rm*,sph}$, and $\sigma$–$M_{\rm*,sph}$ relations. This followed Graham (2012) who reported on a near-linear and super-quadratic $M_{\rm BH}$–$M_{\rm*,sph}$ relation, respectively, for spheroids with a Sérsic or core-Sérsic²²2Core-Sérsic galaxies have a deficit of light at their centre; hence, their central (bulge) light profile is described using a shallow power-law followed by a Sérsic (1963) function beyond the core (Graham et al., 2003b). This population was first discussed by King & Minkowski (1966, 1972). light profile (see also Graham & Scott (2013) and Scott et al. (2013)). Savorgnan et al. (2016) subsequently discovered an improved division due to ETGs and LTGs (none of LTGs have core-Sérsic bulge profiles) in the $M_{\rm BH}$–$M_{\rm*,sph}$ diagram, and in the $M_{\rm BH}$–($\rm L_{gal}$, galaxy luminosity) diagram. This was also later reported by van den Bosch (2016). Savorgnan et al. (2016) coined the notion of a red and blue sequence when two tracks, due to ETGs and LTGs, are evident in a BH mass scaling diagram (see also Terrazas et al., 2016; Dullo et al., 2020). Sahu et al. (2019a) additionally found that the $M_{\rm BH}$–$M_{\rm*,sph}$ relation for ETGs with a disk (ES and S0) and ETGs without a disk (E-type) is roughly quadratic, while the two relations are offset by more than an order of magnitude in the $M_{\rm BH}$-direction. This has since been found in a recent simulation by Marshall et al. (2020). Clearly, it is not simply the amount of stellar mass that matters, but also how it was accumulated and is now distributed. In this vein, we explore the relationship that the BH mass has with the size and shape (centrally concentrated or diffused) of the surrounding bulge/spheroid— terms that we use interchangeably—and as a function of the morphology of the host galaxy.

The above mentioned developments represent a key advance in our understanding of the coevolution of galaxies and black holes. It built upon works such as Wandel (1999), Salucci et al. (2000), Laor (2001), and Graham (2012) and voided the notion (Dressler, 1989; Kormendy & Richstone, 1995; Magorrian et al., 1998) that the black hole mass simply co-evolved linearly with the spheroid mass. The recognition of a more nuanced situation is perhaps not surprising given the variety of accretion/merger histories, and resulting structures among galaxies. For example, core-Sérsic galaxies, thought to be built from the dry merger of galaxies with pre-existing black holes (Begelman, 1984), appear to follow a steeper relation in the $M_{\rm BH}$–$\sigma$ diagram (Sahu et al., 2019b), see also Terrazas et al. (2016, their Figure 3a) and Bogdán et al. (2018, their Figure 5).

Based on the low intrinsic scatter about the $M_{\rm BH}$–$\sigma$ relation (Ferrarese & Merritt, 2000; Gebhardt et al., 2000), some studies have concluded that it is the most fundamental relation between black hole mass and galaxy (e.g. van den Bosch, 2016; de Nicola et al., 2019). However, it may be a premature conclusion without considering the correlations between BH mass and other basic galaxy properties, or allowing for the morphology-dependence and thus (formation physics)-dependence of galaxies. Moreover, it overlooks that the $M_{BH}$–pitch angle ($\phi$) relation has the least total scatter at 0.43 dex (Davis et al., 2017) compared to 0.51 dex in the latest $M_{\rm BH}$–$\sigma$ diagram (Sahu et al., 2019b).

Establishing if a, and which, single relation is the most fundamental, i.e., the primary relation, and how it depends upon morphology is important for understanding the co-evolution of galaxies and BHs. The secondary scaling relations — not to be confused with the morphology dependent substructure which reveals an additional parameter/factor modulating the co-joined growth of galaxies and BHs³³3We could re-frame these results by constructing a simplified ‘fundamental plane’, i.e. a 3-parameter equation involving $M_{\rm BH}$, $\sigma$ (or $M_{*}$) and morphological type (even if just a binary parameter). This would effectively unite the morphology-dependent $M_{\rm BH}$–$\sigma$ ($M_{\rm BH}$–$M_{*}$) relations and reduce the scatter about the two-parameter relations which ignore the morphological type. We will pursue this in future work. — are, however, also important. They can still be used, for example, to predict BH masses or to check on the accuracy of computer simulations e.g., CLUES (Yepes et al., 2009), Magneticum (Dolag, 2015), Bolshoi (Klypin et al., 2011), EAGLE (Schaye et al., 2015), Illustris (Vogelsberger et al., 2014), IllustrisTNG (Pillepich et al., 2018), FIRE (Hopkins et al., 2018), and SIMBA (Davé et al., 2019), which are trying to produce realistic galaxies⁴⁴4Simulations lacking primary information about the spheroid can still be tested against the non-linear, morphology-dependent, $M_{\rm BH}$–$M_{\rm*,gal}$ relations (Davis et al., 2018; Sahu et al., 2019a).. These empirical relations help to decipher the physics behind the effect of the central supermassive black hole on the host spheroid/galaxy properties and vice versa. How such black hole feedback drives galaxy evolution is the challenge yet to be fully answered (Choi et al., 2018; Ruszkowski et al., 2019; Terrazas et al., 2020; Martín-Navarro et al., 2020).

Here, we will expand upon the previous efforts in establishing the $M_{\rm BH}$–$\rm n_{sph}$ relation (e.g., Graham et al., 2003a; Graham & Driver, 2007a; Vika et al., 2012; Beifiori et al., 2012; Savorgnan et al., 2013; Savorgnan, 2016), the $M_{\rm*,sph}$–$\rm n_{sph}$ relation (e.g., Andredakis et al., 1995; Jerjen et al., 2000; Graham & Guzmán, 2003; Ferrarese et al., 2006; Savorgnan, 2016), the $M_{\rm*,sph}$–$\rm R_{e,sph}$ relation (e.g., Sérsic, 1968a; Graham & Worley, 2008; Lange et al., 2015), and the $M_{\rm BH}$–$\rm R_{e,sph}$ relation (e.g., de Nicola et al., 2019) using our extensive sample of 83 ETGs and 40 LTGs with careful (individual, not automated) multi-component decompositions. Importantly, we explore potential substructures due to galaxy sub-morphologies, i.e., Sérsic versus core-Sérsic galaxies, barred versus non-barred galaxies, galaxies with a stellar disk versus galaxies without a stellar disk, and ETGs versus LTGs. We also investigate the relation between $M_{\rm BH}$ and the central concentration index (Graham et al., 2001a), which is known to vary monotonically with the Sérsic index (Trujillo et al., 2001; Graham et al., 2001b).

As with the $M_{\rm BH}$–$M_{\rm*,sph}$ relation, the $M_{\rm BH}$–$\rm n_{sph}$ and $M_{\rm BH}$–$\rm R_{e,sph}$ relations can be applied to large surveys of galaxies (e.g., Casura et al., 2019) — if their bulge Sérsic parameters are reliable — to estimate their black hole masses and further construct the black hole mass function (BHMF). The BHMF holds interesting information for cosmologists, e.g., to estimate the mass density of the Universe contained in BHs (e.g. Fukugita & Peebles, 2004; Graham & Driver, 2007b), to map the growth of BHs and constrain theoretical models of BH evolution (e.g. Kelly & Merloni, 2012). In addition, the latest BHMF, which takes part in calculating the black hole merger rate (Chen et al., 2019; Volonteri et al., 2020), will help improve the prediction for the amplitude and time until detection of the long-wavelength (micro to nano Hertz) gravitation wave background — as generated from merging supermassive black holes — using pulsar timing arrays (Siemens et al., 2013; Shannon et al., 2015; Sesana et al., 2016) and using the upcoming Laser Interferometer Space Antenna (LISA, Danzmann, 2017; Baker et al., 2019).

Section 2 details the galaxy sample and parameters which we used for our investigation, and the regression routines applied to obtain the correlations. Various correlations we observed, including their dependencies on galaxy morphology, are described in the subsections of Section 3. In sub-section 3.1, we present the scaling relations observed between the spheroid stellar mass and spheroid Sérsic index. Sub-section 3.2 presents the expected correlation between black hole mass and the bulge Sérsic index by combing the correlation observed between spheroid stellar mass and spheroid Sérsic index with our latest correlation between black hole mass and spheroid stellar mass. It then presents the observed correlations between black hole mass and the bulge Sérsic index based on our data-set. We also show the relationship between the Sérsic index and the central light concentration, and we present the correlation observed between the black hole mass and the central concentration index. In sub-section 3.3, we present the correlations observed between the spheroid stellar mass and the effective spheroid half-light radius. Here, we also explore the correlations of the spheroid stellar mass with the spheroid radii containing $\rm 10\%$ and $\rm 90\%$ of the light of the spheroid. Sub-section 3.4 provides the expected correlation that the spheroid half-light radius might have with the black hole mass, before presenting the observed correlations between the black hole mass and the spheroid effective half-light radius, along with the correlations between the black hole mass and the spheroid radii containing $\rm 10\%$ and $\rm 90\%$ of spheroid’s light. These subsections additionally provide a discussion and some explanation for the correlations that we find. Finally, Section 4 presents a summary of our main results.

The Sérsic (1963, 1968a) function is nowadays used to describe the light profiles of elliptical galaxies (E) and, when present, the spheroidal component of galaxies with a disk (ES/S0/Sp). A review of the Sérsic function, and its many associated expressions, can be found in Graham & Driver (2005). Briefly, the intensity of a Sérsic light profile can be expressed as a function of the projected galactic radius (R), such that {IEEEeqnarray}rCl I(R)= I_e exp[-b_n { (RRe )^1/n-1 }], where $\rm I_{e}$, $\rm R_{e}$, and $\rm n$ are profile parameters. The term $\rm I_{e}$ is the effective intensity at the effective radius $\rm R_{e}$, which bounds 50% of the total light in the associated 2D image. Graham (2019b) provides a detailed review of this popular radius and addresses the misconceptions about its physical significance. The surface brightness at this effective radius ($\rm\mu_{e}$) is related to $\rm I_{e}$ through $\rm\mu_{e}\equiv-2.5\log(I_{e})$.

The Sérsic index n (also known as the shape parameter), describes the curvature of the light profile, such that a Sérsic light profile with a higher Sérsic index is steeper at the centre and has a shallower distribution at larger radius, whereas, a profile with a smaller Sérsic index is shallower at the centre followed by a steeper drop at outer radii (see Figure 2 in Graham, 2019b). Thus, the Sérsic index traces the central concentration of the light within the spheroid (Trujillo et al., 2001; Graham et al., 2001b, their Figure 2); and also the inner gradient of the gravitational potential⁵⁵5This holds when dark matter is negligible, and there is no significant stellar mass-to-light ratio gradient. (Terzić & Graham, 2005, their Figures 2 and 3). The value of the term $\rm b_{n}$ in Equation 2 depends on n, and is obtained by solving $\Gamma(\rm 2n)=2\gamma(\rm 2n,b_{n})$, where $\Gamma$ denotes the gamma function and $\gamma$ is the incomplete gamma function. It can also be approximated by $\rm b_{n}\approx 1.9992\,n-0.3271$ for $0.5<n<10$ (Capaccioli, 1989).

In this work, we have used a sample of 123 galaxies with directly-measured black hole masses, for whom the Sérsic model parameters ($\rm n,R_{e},and\,\mu_{e}$) describing their spheroid’s surface brightness distribution were obtained by a careful multi-component decomposition of the galaxy’s light. These parameters are collectively taken from Savorgnan & Graham (2016), Davis et al. (2019), and Sahu et al. (2019a). These studies performed a 2-dimensional (2D) isophotal analysis, first extracting a 2D luminosity model using Isofit and Cmodel (Ciambur, 2015) to capture the radial gradients in the ellipticity, position angle, and Fourier harmonic coefficients describing the isophote’s deviations from a pure ellipse, and then performing a multi-component decomposition using the isophotal-averaged 1D surface-brightness profile along the major and geometric-mean⁶⁶6The geometric-mean axis, which is also known as the “equivalent axis”, is the radius of the circularized form of the elliptical isophote with major axis radius $\rm R_{maj}$ and minor axis radius $\rm R_{min}$, which conserves the same amount of flux. This results in the equivalent axis radius ($R_{\rm eq}$) being the geometric mean of $\rm R_{maj}$ and $\rm R_{min}$ ($\rm R_{eq}=\sqrt{\rm R_{maj}*R_{min}}$), which is also represented as $\rm R_{geom}$ (for more details see the appendix section in Ciambur, 2015). axis of the galaxies. For this purpose, they used the software Profiler (Ciambur, 2016), which is inbuilt with many functions for specific galaxy components, including the Sérsic function for galactic spheroids. The major and geometric-mean axis were modelled independently (see Section 3 in Sahu et al., 2019a, for more details).

Table 3 in our appendix lists both the major-axis bulge parameters ($\rm n_{maj},\,R_{e,maj},\,\mu_{e,maj}$), and the equivalent-axis bulge parameters ($\rm n_{eq},\,R_{e,eq},\,\mu_{e,eq}$), plus the morphologies, and the bulge masses ($M_{*,\rm sph}$) taken from Savorgnan & Graham (2016), Davis et al. (2019), and Sahu et al. (2019a), along with the distances and the directly-measured black hole masses of the galaxies. To show the consistency between the structural decomposition of the major- and equivalent-axis surface brightness profiles, we have plotted $\rm\mu_{e,sph,maj}$ versus $\rm\mu_{e,sph,eq}$ in Figure 1. The 1$\sigma$ scatter in this diagram is $\rm 0.58\,mag\,arcsec^{-2}$ which corresponds to a 1$\sigma$ scatter in $\rm R_{e}$ of $\approx$30% given that the Sérsic model’s surface brightness profile has slopes of $\sim$1.8 to $\sim$2.1 (for $\rm n=1$ to 10) at $\rm R=R_{\rm e}$, where $\rm{\rm d}\mu(R)/{\rm dR}|_{R_{e}}=2.5b_{n}/(\ln(10)\,n\,R_{e})\approx[2.17-0.36/n]/Re$. Table 3 also provides the radial concentration index (C: see Section 3) and the physical (arcsec to kpc) size scale of the galaxies⁷⁷7The physical scale is calculated using the python version of Edward (Ned) L. Wright’s cosmological calculator (Wright, 2006), written by James Schombert, assuming the cosmological parameters $\rm H_{0}=67.4\,(km\,s^{-1})/Mpc$, $\Omega_{m}$=0.315, and $\Omega_{v}$=0.685 (Planck Collaboration et al., 2018).. The morphologies of these galaxies are based on the multi-component decompositions found in Savorgnan & Graham (2016), Davis et al. (2019), and Sahu et al. (2019a).

The black hole masses used here have been obtained from various sources in the literature. Their original sources are listed in Savorgnan et al. (2016) and Sahu et al. (2019a) for the ETGs, and in Davis et al. (2019) for the LTGs. These black hole masses have been directly-measured using either the stellar dynamical modelling, gas dynamical modelling, megamaser kinematics, proper motions (Sgr $A^{*}$), or the latest direct imaging method (M87*). As the distances to the galaxies have been revised over time, the BH masses have also been updated to keep pace with this, and thereby provide a consistent analysis with the arcsecond-to-kpc and apparent-to-absolute magnitude conversions.

Our total sample is comprised of 123 galaxies, of which 83 are ETGs, and 40 are LTGs. We have used the Bivariate Correlated Errors and Intrinsic Scatter (BCES) regression (Akritas & Bershady, 1996) to obtain the symmetric (bisector) best-fit lines for all our correlations. The BCES⁸⁸8We used the Python module from (Nemmen et al., 2012), which is available at https://github.com/rsnemmen/BCES regression considers the measurement errors in both variables and allows for intrinsic scatter in the data. It is a modified form of the ordinary least square (OLS) regression. It calculates the OLS($Y|X$) line by minimizing the scatter in the Y-direction, and the OLS($X|Y$) line by minimizing the scatter in the X-direction. The BCES(Bisector) line symmetrically bisects the OLS($Y|X$) and OLS($X|Y$) lines. We prefer to use the bisector line as it offers equal treatment to the quantities plotted on the X-and Y-axes. Additionally, we also checked the consistency of our correlations by employing the modified-FITEXY (MPFITEXY) regression (Press et al., 1992; Tremaine et al., 2002; Williams et al., 2010; Markwardt, 2012), where we had to bisect the best-fit lines obtained from the forward MPFITEXY($Y|X$) and inverse MPFITEXY($X|Y$) regressions to obtain the symmetric fit to our data (see Novak et al., 2006, for more details about the MPFITEXY regression).

For our investigation, we adopt a $20\%$ uncertainty for the Sérsic bulge parameter n. Various factors which can contribute to the uncertainty in the measurement of the Sérsic bulge parameters include: inappropriate sky subtraction; incomplete masking; inaccurate point-spread function (PSF) for the telescope; uncertainties in the identification of components; especially the nuclear (bar/disk/ring/star cluster) or faint components during the multi-component decomposition of the galaxy luminosity. Thus, it is challenging to quantify the uncertainty in the bulge parameters for every galaxy individually.

In past studies, various measures have been taken to quantify realistic errors on the bulge/galaxy Sérsic index. For example, Caon et al. (1993a) noted a typical error of $\sim 25\%$ corresponding to a $25\%$ variation in the (observed - fitted) residual, while some studies (e.g. Graham & Driver, 2007a; Savorgnan et al., 2013) adopted a constant uncertainty of $\sim 20\%$, and others employed Monte Carlo simulations (e.g. Beifiori et al. (2012) obtaining up to a $\sim 15\%$ error-bar). Others varied the sky subtraction by $\pm 1\sigma$ to estimate error-bars (Vika et al., 2012), some used mean/median errors based on a broader comparison with published parameters from other studies (Graham & Worley, 2008; Laurikainen et al., 2010) producing up to $\sim 30\%$ uncertainty, whereas Savorgnan (2016) used $20\%$, $42\%$, and $52\%$ uncertainties, respectively, for their grade 1, grade 2, and grade 3 galaxies following Savorgnan & Graham (2016, their Section 4.2). As Savorgnan & Graham (2016) noted, their generous uncertainties arose when comparing published parameters based upon an array of differing decompositions for the same galaxy. For example, sometimes a single Sérsic component had been fit while other times the image analysis additionally included, as separate components, a disk and sometimes also a bar.

Given that our sky-background intensities are measured carefully (Sahu et al., 2019a, see their Figure 1 and Section 2.2 ), and that our parameters are obtained from multi-component decompositions, we have ruled out our two major sources of systematic errors (i.e. over/under-estimation of the sky and failing to account for a biasing component), and as such we adopt a 20% uncertainty for n, and a 30% uncertainty for $\rm R_{e}$ based on the $1\sigma$ scatter in $\rm\mu_{e}$ for our galaxy sample as already described in this section. We do, however, test and confirm that our scaling relations are not significantly dependent upon this. Our results are stable (no change in slope or intercept at the $1\sigma$ uncertainty level) upon using an uncertainty up to 30% in $\rm n$ and 40% in $\rm R_{e}$. Furthermore, we also performed all the correlations using the major subsample of our total sample for whom the spheroid parameters are derived using $\rm 3.6\,\mu m$ images (see Table 3), and the correlations are found to be consistent with the correlations obtained using the total sample within the $\pm 1\sigma$ uncertainty bounds of the slopes and intercepts.

During our linear regressions, we have excluded certain potentially biasing galaxies, which are either stripped galaxies (NGC 4342 and NGC 4486B), a single galaxy with $M_{\rm BH}<10^{5}M_{\odot}$ (NGC 404), or more than $2\sigma$ outliers (NGC 1300, NGC 3377, NGC 3998, NGC 4945, NGC 5419) in any of the correlations presented here. NGC 4342 and NGC 4486B are stripped of their stellar mass due to the gravitational pull of their massive companion galaxies NGC 4365 (Blom et al., 2014) and NGC 4486 (Batcheldor et al., 2010), respectively. Hence, NGC 4342 and NGC 4486B can bias the black hole scaling relations as they have smaller $\rm n$ or $\rm R_{e}$ than they would have had if they weren’t stripped of their mass. NGC 404, the only galaxy in our sample with a BH mass below $10^{6}M_{\odot}$, can bias the best-fit lines due to its location at the end of the distribution and thus its elevated torque strength. The galaxies NGC 3377, NGC 3998, NGC 4945, and NGC 5419 in the $M_{\rm BH}$–$\rm n$ diagram, and NGC 1300 in the $M_{\rm BH}$–$\rm R_{e}$ diagram, are more than $\pm 2\sigma_{\rm rms}$ outliers from the corresponding best-fit lines and slightly alter their slopes⁹⁹9Including these galaxies in the regressions changes the slopes by $\sim 1\sigma$ uncertainty level of current slopes.. Hence, these galaxies are better excluded in all our regressions to obtain robust correlations. These eight excluded galaxies are indicated in all the plots. This exclusion leaves us with a reduced sample of 115 galaxies.

The stellar masses of our galactic spheroids ($M_{\rm*,sph}$) are derived from the luminosities measured using the Sérsic model (for the bulge) fit to the equivalent- (or geometric-mean) axis light profile, parameterized by $\rm n_{sph,eq}$, $\rm R_{e,sph,eq}$, and $\rm I_{e,sph,eq}$. Therefore, it is expected to find some correlation between $M_{\rm*,sph}$ and the Sérsic index, and also between $M_{\rm*,sph}$ and the effective half-light radius. The issue of parameter coupling potentially explaining the trends between the Sérsic parameters and the luminosity was explored and dismissed using model-independent measures of both luminosity and size (Caon et al., 1993b; Trujillo et al., 2001), implying the observed correlation between luminosity versus Sérsic properties (n and $\rm R_{e}$) are indeed real. Moreover, the errors in n and $\rm R_{e}$ adopted here are not big enough for parameter coupling in the fitting process to explain the observed trends.

We have used the largest sample of galaxies to date with directly-measured black hole masses, and carefully measured bulge parameters obtained from multi-component decomposition of their galaxy light in our previous studies (Savorgnan & Graham, 2016; Davis et al., 2019; Sahu et al., 2019a). Using this extensive data-set, we have investigated the correlations between black hole mass ($M_{\rm BH}$) and the bulge Sérsic index ($\rm n_{sph}$), bulge central light concentration index (C), and the bulge effective half-light radius ($\rm R_{e,sph}$).

For our sample, we also investigated the correlations between bulge mass ($M_{\rm*,sph}$) and both the bulge Sérsic index and bulge half-light radius. We then combined these with the latest $M_{\rm BH}$–$M_{*,sph}$ relations to predict and check upon the observed correlations of $M_{\rm BH}$ with $\rm n_{sph}$ and $\rm R_{e,sph}$.

In all of the relations we investigated, we explored the possibility of substructure due to various subcategories of galaxy morphology, i.e., Sérsic versus core-Sérsic galaxies, galaxies with a stellar disk versus galaxies without a stellar disk, barred versus non-barred galaxies, and ETGs versus LTGs.

Parameters for all the correlations presented in this paper are separately listed in Table 1 and Table 2. The slope of the correlations that we obtained for $M_{\rm BH}$ or $M_{\rm*,sph}$ with the major-axis bulge parameters ($\rm n_{sph,maj}$ and $\rm R_{e,sph,maj}$) are consistent with the slope from the corresponding correlations of $M_{\rm BH}$ or $M_{*,sph}$ with the equivalent-axis bulge parameters ($\rm n_{sph,eq}$ and $\rm R_{e,sph,eq}$).

Our prime results can be summarized as follows,

• •

  ETGs and LTGs follow two different $M_{\rm*,sph}$–$\rm n_{sph}$ relations (see Figure 2), with slopes equal to $3.27\pm 0.25$ and $1.31\pm 0.22$, and total rms scatter equal to $\Delta_{\rm rms|sph}$=0.46 dex and 0.32 dex, respectively (Equations 2 and 2), in the $M_{\rm*,sph}$–$\rm n_{sph,maj}$ diagram. As the Sérsic index is a measure of the central concentration of a bulge’s light, these different slopes for the $M_{\rm*,sph}$–$\rm n_{sph}$ relation suggest distinct mechanisms for the evolution of spheroid mass and central light (or stellar mass) concentration in ETGs and LTGs.

• •

  In the $M_{\rm BH}$–$\rm n_{sph}$ diagram, ETGs and LTGs seem to follow two different relations with $M_{\rm BH}\propto\rm n_{sph,maj}^{3.95\pm 0.34}$ and $M_{\rm BH}\propto\rm n_{sph,maj}^{2.85\pm 0.31}$ with $\Delta_{\rm rms|BH}$ = 0.65 dex and 0.67 dex, respectively (Figure 4, Equations 3 and 3).

• •

  In the diagram showing the black hole mass versus the spheroid central concentration index, C(1/3), we again find two (slightly) different relations due to ETGs and LTGs (Figure 7, Table 2), analogous to the $M_{\rm BH}$–$\rm n_{sph}$ diagram. The slopes for the $M_{\rm BH}$–$\rm C(1/3)$ relations are $8.94\pm 0.86$ and $6.88\pm 0.97$ with $\Delta_{\rm rms|BH}$= 0.72 dex and 0.68 dex, respectively, for ETGs and LTGs.

• •

  We find a tight near-linear relation between $M_{\rm*,sph}$ and $\rm R_{e,sph}$ for our range of data (Figures 8 and 9). Both ETGs and LTGs define the log-linear relation $M_{\rm*,sph}\propto\rm R_{e,sph,maj}^{1.08\pm 0.04}$ (Equation 8) with $\Delta_{\rm rms|sph}$=0.27 dex. An extended view of the $M_{\rm*,gal}$–$\rm R_{e,gal}$ relation for ETGs is curved (Graham, 2019b), and our $M_{\rm*,sph}$–$\rm R_{e,sph}$ relation, somewhat dominated by massive spheroids, agrees with the quasi-linear part of the curve at high-masses where E-type galaxies dominate.

• •

  ETGs and LTGs define two different relations between black hole mass and bulge $\rm R_{e}$ (Figure 11), such that $M_{\rm BH}$–$\rm R_{e,sph,maj}^{(1.26\pm 0.08)}$ and $M_{\rm BH}$–$\rm R_{e,sph,maj}^{(2.33\pm 0.31)}$ for ETGs and LTGs, with $\Delta_{\rm rms|BH}$=0.58 dex and 0.62 dex, respectively (Equation 3.4 and 3.4). This is analogous to the substructure in the $M_{\rm BH}$–$M_{\rm*,sph}$ diagram due to ETGs and LTGs (Sahu et al., 2019a).

• •

  In the $M_{\rm BH}$–$\rm R_{e,sph}$ diagram, ETGs with a disk (ES, S0) and ETGs without a disk (E) follow two different, almost parallel, relations with slopes $\sim 2\pm 0.2$ (Figure 12), which are steeper than the above single-regression $M_{\rm BH}$–$\rm R_{e,sph}$ relation for all ETGs (see Tables 1 and 2 for parameters) and offset by a factor of $\sim$10 in the vertical $M_{\rm BH}$-direction. This is again analogous to the offset observed between the $M_{\rm BH}$–$M_{\rm*,sph}$ relations followed by ETGs with and without a disk (Sahu et al., 2019a). Given $M_{\rm*,sph}$ depends on $\rm R_{e,sph}$ via $M_{\rm*,sph}=(M/L)2\pi R_{e}^{2}\langle I\rangle_{e}$, we find that the offset in the $M_{\rm BH}$–$M_{\rm*,sph}$ diagram originates from the offset between ETGs with and without a disk in the $M_{\rm BH}$–$\rm R_{e,sph}$ diagram. The reason behind the offset is the smaller spheroid half-light radius of ETGs with a disk relative to that of elliptical (purely spheroidal) galaxies.

• •

  In the $M_{\rm*,sph}$–$\rm R_{z,sph}$ and $M_{\rm BH}$–$\rm R_{z,sph}$ diagrams for z=$10\%$ and $90\%$ (see Figures 10 and 14), we recover the same substructures as the $M_{\rm BH}$–$\rm R_{e,sph}$ and $M_{\rm*,sph}$–$\rm R_{e,sph}$ relations mentioned above, with the slopes of correlations gradually decreasing with increasing z (see Table 2 for parameters).