Splitting the lentils: Clues to galaxy/black hole coevolution from the discovery of offset relations for non-dusty versus dusty (wet-merger-built) lenticular galaxies in the $M_{\rm bh}$-$M_{\rm *,spheroid}$ diagram

This work advances the (galaxy morphology)-dependent (black hole mass, $M_{\rm bh}$)-(spheroid/galaxy stellar mass, $M_*$) scaling relations by introducing `dust bins' for lenticular (S0) galaxies. Doing so has led to the discovery of $M_{\rm bh}$-$M_{\rm *,sph}$ and $M_{\rm bh}$-$M_{\rm *,gal}$ relations for dusty S0 galaxies - built by major wet mergers and comprising half the S0 sample - offset from the distribution of dust-poor S0 galaxies. The situation is reminiscent of how major dry mergers of massive S0 galaxies have created an offset population of ellicular and elliptical galaxies. For a given $M_{\rm bh}$, the dust-rich S0 galaxies have 3 to 4 times higher $M_{\rm *,sph}$ than the dust-poor S0 galaxies, and the steep distributions of both populations in the $M_{\rm bh}$-$M_{\rm *,sph}$ diagram bracket the $M_{\rm bh} \propto M_{\rm *,sph}^{2.27+/-0.48}$ relation defined by the spiral galaxies, themselves renovated through minor mergers. The new relations offer refined means to estimate $M_{\rm bh}$ in other galaxies and should aid with: (i) constructing (galaxy morphology)-dependent black hole mass functions; (ii) estimating the masses of black holes associated with tidal disruption events; (iii) better quantifying evolution in the scaling relations via improved comparisons with high-$z$ data by alleviating the pickle of apples versus oranges; (iv) mergers and long-wavelength gravitational wave science; (v) simulations of galaxy/black hole coevolution and semi-analytic works involving galaxy speciation; plus (vi) facilitating improved extrapolations into the intermediate-mass black hole landscape. The role of the galaxy's environment is also discussed, and many potential projects that can further explore the morphological divisions are mentioned.


INTRODUCTION
Stars create metals and dust, polluting or rather fertilising galaxies. Today, there is typically a couple of hundred times more total-gas mass than dust mass, at least in local late-type galaxies (LTGs) where the mass ratio ranges from 10 to 1000, and 1.5 log(M HI /M dust ) 3 dex (Casasola et al. 2020, 2022, their Fig. 2 and3, and Table 4). In an ensemble of early-type galaxies (ETGs) detected by the Herschel Observatory, the average dust-tostellar mass ratio has been reported as log(M dust /M * ) = −4.3 dex by Smith et al. (2012). Despite the low fraction of dust, when not diffusely distributed (e.g. Knapp et al. 1989;Goudfrooij & de Jong 1995), it can be easy to spot in silhouette against the uniform screen E-mail: AGraham@swin.edu.au of an ETG's optical light (e.g. Sadler & Gerhard 1985;Veron-Cetty & Veron 1988). It can appear as expansive dust lanes patchy clouds, and tangled serpentine shapes, formed in dense molecular clouds, where the dust has condensed into larger particles with (typically CO and N) ice mantles and polycyclic aromatic hydrocarbons, built in part from the larger reservoir of dust, carbon and silicate grains, and atomic and ionic metals in the diffuse interstellar medium.
The ETGs can be built through collisions. Violent mergers which erase the orbital angular momentum of the progenitor system form elliptical (E) galaxies. When the bulk of the orbital angular momentum is not cancelled, major collisions will produce lenticular (S0) 1 disc galaxies (e.g. Naab & Burkert 2003), such also made with other galaxy types, and the larger sample contains ∼100 galaxies with directly measured black hole masses. A handful of exclusions are described in Section 2.2, also see Section 3.3, along with a recognition that some of the apparent outlying systems may hold clues to an even more intricate understanding of the evolution of galaxies. Given the use of bulge stellar masses obtained from bulge luminosities (Section 2.1), Section 2.3 offers some additional insight into how bulges are identified. It also discusses an emerging trend in which bulges are giving up ground to other galaxy components, in a few cases leading to ambiguity over whether a galaxy may be better considered bulgeless. Circumventing potential dependence on the galaxy decompositions, the M bh -M * ,gal diagram is additionally used in Section 3.1, along with the M bh -M * ,sph and M bh -R e,sph diagrams, to reveal the division between dusty and non-dusty S0 galaxies.
The literature was searched for evidence of mergers and accretion in the 40 S0 galaxies, and the results are reported throughout and summarised in Table 2. Recognising that the dusty S0 galaxies are the product of major wet mergers, and given their location on the high M * ,sph /M bh side of the distribution of disc galaxies in the M bh -M * ,sph diagram, Section 3.2 briefly reviews the location of the S galaxies positioned between the visibly dust-rich and dust-poor S0 galaxies. Section 3.3 further reports on systems, i.e., spheroid/galaxy/black hole pairings, which do not conform to the general trends for one reason or another; and may be worthy of further consideration in future work. Having derived M bh -M * ,sph relations for dusty and non-dusty S0 galaxies in Section 3.1, the latter is used in Section 3.4 to derive the typical bulge-to-total (B/T ) stellar mass ratio associated with a (dry merger)-induced jump from the non-dusty S0 galaxy M bh -M * ,sph relation to the E galaxy M bh -M * ,sph relation, refining the calculation in Graham (2022) that was based on a single M bh -M * ,sph relation for all S0 galaxies. A discussion of several topics is provided in Section 4, including the concept that some S0 galaxies are faded S (and faded S0) galaxies, the role of environment, AGN feedback, the benefits of refined black hole mass estimates, and more. Rather than reiterating the new relations for the S0 galaxies, some of the many impacts and potential future projects they enable are listed in Section 5.
Finally, one likely ingredient for understanding what is occurring may sound obvious, but the presence of gas reveals an environment which permits its existence. The dusty S0 galaxies have grown with neither gas-stripping nor a central 'Benson Burner' 4 curtailing gas-cooling and preventing star formation. Therefore, a cursory investigation into the (cluster, group, or field) environment has been made here. While this did not prove fruitful, it may be a helpful reference for future work and is included in an Appendix.

DATA SET
Graham & Sahu (2022a) describe a sample of 104 galaxies with directly measured black hole masses and multicomponent decompositions of images taken at 3.6 µm by the Spitzer Space Telescope and analysed by , Davis et al. (2019a), Sahu et al. (2019), and . The galaxies' Figure 1. Examples of the four 'dust bins' (dust code: N = strong No; n = weak no, only nuclear dust ring/disc; y = weak yes; Y = strong Yes) applied in Table 2. Upper-left: (N)o visible dust. NGC 2778, inner ∼2.2×2.2 kpc (HST Obs. ID 6099). Upper-middle: (n)uclear dust disc only. NGC 5845, inner ∼4.4×4.4 kpc with 0.9x0.9 kpc inset (Obs. ID 6099). Upper-right: (y)es dust but not much. NGC 5813, inner ∼2.9×2.9 kpc (Obs. ID 5454). Lower-left: (Y)es plenty of dust. NGC 524, inner ∼3.5×3.5 kpc (Obs. ID 5999). Lower-middle: (Y)es widespread dust. NGC 1316, ∼14.2×14.2 kpc (Obs. ID 9409). Lower-right: (Y)es plenty of dust. NGC 6861, inner ∼2.6×2.6 kpc (Obs. ID 5999). Images are central cutouts (not the whole galaxy) from F814W/F555W WFPC2/PC images available at the HLA, except for NGC 1316, which is also from the HLA but is an F814W/F435W ACS/WFC image with conjoined twin CCDs. black hole masses were taken from the literature, as indicated in the above papers. As noted in Davis et al. (2019a, their Section 2.1) and Sahu et al. (2019, their Table 4), the black hole masses are routinely adjusted to reflect their updated distances. The uncertainty on these black hole masses increased to incorporate the (random error) uncertainty in those distances   Table 1). The distances were obtained from a range of methods, thereby helping to erase the potential for an unknown systematic error in the distance -introduced by individual distance indicator methods -that might produce a bulk systematic bias in the masses of the black holes (and galaxies). 5 Having noted the above, many of the ETG distances were based on their surface brightness fluctuations (Tonry et al. 2001;Blakeslee et al. 2002Blakeslee et al. , 2009, which did recently experience a systematic correction of 1 per cent due to a reduced distance modulus for the Large Magellanic Cloud (Pietrzyński et al. 2019). As such, 5 In late 2022, Martin Bureau reminded the author that the community tend to overlook the two (random and systematic) uncertainties in the distance when reporting the uncertainty on the black hole mass. These should be considered in addition to the typically reported random error on the black hole mass measurement, which the author notes often does not include an allowance for systematic error in its measurement from, for example, issues around spatial resolution or radially varying stellar populations due to galaxy components. their black hole masses shifted by 1 per cent. As done by Kelson et al. (1996) and Lelli et al. (2022), for example, systematic errors can be separated from random errors. With the existence of (galaxy morphology)-dependent scaling relations, coupled with the use of the same distance indicator technique for galaxies of the same morphological type, there might be an erroneous, bulk systematic shift to the values of M bh for each galaxy type. This unknown systematic error can readily be attached as an additional uncertainty on the intercept of the (galaxy morphology)-specific M bh -(stellar velocity dispersion, σ) relations if the same distance indicator method was used for the galaxies that define each relation. However, things are more convoluted when dealing with the M bh -M * relations given that systematic errors in the distance also impact M * . This leads to correlated errors in M bh and M * , coming from both the random and systematic uncertainty in the distance. Nonetheless, this impact will be insignificant relative to the much larger factors of a few difference in M bh /M * , at fixed M * , that are now observed between the different morphological types. Therefore, the issue of systematic errors in the distance is left for a more thorough treatment elsewhere.
of 73 ETGs, of which 35 can be thought of as E galaxies 6 and 38 can be thought of as S0 galaxies 7 , plus 31 LTGs, i.e., spiral galaxies, of which two are bulgeless (NGC 4395 and NGC 6926) and two are reclassified here (and previously elsewhere) as S0 galaxies (NGC 2974 and NGC 4594). For convenience, masses and mor-6 Technically, ten of these 35 are ellicular (ES,e) galaxies with intermediate-scale discs . 7 Technically, four of these 38 are ellicular (ES,b) galaxies without largescale discs ).
phologies are provided in Table 1 for the S0 galaxies investigated here. To be as inclusive as possible, Davis et al. (2019a) considered NGC 2974 andNGC 4594 (Sombrero) to be S galaxies. However, as noted above, they are now treated as S0 galaxies, thereby increasing the S0 count from 38 to 40. NGC 4594 is classified in the literature as both an unbarred Sa galaxy and an S0 galaxy. With a B/T flux ratio of 0.6, it had the highest spheroid mass of all the 31 supposed S galaxies. 8 The shell galaxy NGC 2974 (aka NGC 2652: Tal et al. 2009) is often considered an S0 galaxy. It has a weak inner bar and a nuclear, gaseous, two-armed spiral structure within the inner 200 pc. It is thought to have accreted material and been built up by mergers with smaller galaxies. It has 0.7 × 10 9 M of HI (Kim et al. 1988;Weijmans et al. 2008) and dominates the isolated NGC 2974 Group of five galaxies.
Hubble Space Telescope (HST) images, primarily 'level 4' colour images available from the Hubble Legacy Archive (HLA) 9 of the 40 S0 galaxies, were inspected for signs of dust. The Mikulski Archive for Space Telescopes (MAST) 10 was also used. The S0 galaxies were assigned to one of four 'dust bins', denoted as follows: • N for (N)o visible sign of dust; • n for (n)ot much other than a (n)uclear dust ring/disc and otherwise dust-free appearance; • y for (y)es a little; and • Y for a stronger (Y)es, with obvious signs of widespread dust beyond the nucleus.
Examples of the four dust bins are shown in Fig. 1. The outcome of the visual inspection is reported in Table 2, along with a reference if dust was seen or if merger activity had previously been reported.
Some dusty S0 galaxies are well known to be young mergers (e.g., NGC 5128), while others are old mergers which have since curtailed, if not ceased, their star formation (e.g., NGC 1316). It is expected that within the 'down-sizing' scenario, the lower-mass dusty S0 galaxies will, on average, have been built most recently (Cairós & González-Pérez 2020;Zhang et al. 2020). What is of relevance here is that they all experienced a major wet merger in the past, evolving them to higher masses.
2.1 Dust and the estimated M * /L 3.6 ratios This section explores how dust might contribute to an elevated derivation of the stellar masses in the dusty S0 galaxies. Indeed, dust can redden optical colours, such as B − V, thereby increasing the (B − V colour)-dependent stellar mass-to-light ratios. Based on the models of Into & Portinari (2013) and the Kroupa (2002) IMF, (Graham & Sahu 2022a, their Eq. 4) present the following M/L expression for fluxes at 3.6 microns: From the 0.8 B−V 1.0 colours of the S0 and ES galaxies (Graham & Sahu 2022a, their Fig. 1), none have unusually red colours. However, perhaps some dusty S0 mergers have formed a younger population in sufficient numbers that a more appropriate (dust-free) colour would be more blue. If the dust-free colour ranged from 0.65 B−V 0.9, as seen for the LTGs, then the M/L ratios would be 0.4 M * /L 3.6 0.7 rather than 0.6 M * /L 3.6 0.9, with the latter based on the observed 0.8 B − V 1.0. Dropping from a mid-point M * /L 3.6 value of 0.75 to a midpoint of M * /L 3.6 = 0.55 would result in a ∼27 per cent reduction in the mass-to-light ratio, resulting in a ∼0.135 dex decrease to the stellar mass of the dusty, possibly intrinsically bluer, S0 galaxies. However, the applicability of this adjustment is debatable given that the M/L formula was constructed using realistic dusty models for galaxies with a range of morphologies. Dust can also brighten the observed flux at 3.6 µm, which, although still dominated by the stellar continuum, can contain the thermal glow of warm dust. For this reason, Querejeta et al. (2015) used a 25 per cent lower M * /L 3.6 ratio for LTGs than for ETGs. Therefore, one could argue for an additional ∼0.125 dex reduction to the stellar masses of (some of) the dusty S0 galaxies (if ongoing star formation is heating a substantial amount of dust), giving a total 0.26 dex (≈82 per cent) reduction when combined with the reduction mentioned above based on dust reddening. As we shall see later, this is insufficient to explain the upcoming offset between the dust-rich and dust-poor S0 galaxies.

Exclusions
Readers not interested in the details of the sample can skip to Section 3. In what follows is a description of previously and currently excluded galaxies by the author and the reasons for the exclusion. Section 2.3 then provides a somewhat esoteric but relevant discussion about the bulge masses of a few S and S0 galaxies in an arguably undermining, albeit insufficiently so, manner not typically seen in black hole scaling relation papers.
Identifying outlying data points is important for a couple of reasons. Their inclusion in regression analyses can yield a biased relation relative to that defined by the bulk of the remaining population, especially if the outliers are located towards the end of a distribution. Their offset may be due to measurement error, or it may signal that some additional process, such as stripping, has modified the system. As such, removing the outlying data is appropriate and also serves as a means of flagging it for future study. Suppose a relation is curved or bent when including more data beyond the current sampling. In that case, removing the apparent outlier(s) at the extremity of what currently appears to be a linear (or log-linear) distribution may enable the recovery of a more optimal approximation over that fitted data range.

Previous exclusions
Kormendy & Ho (2013) used 17 and excluded 26 bulges from their sample of 43 disc galaxies, declaring the bulk of the exclusions to be pseudobulges which do not obey the near-linear M bh -M * ,sph relation. Among what they thought were elliptical galaxies, they additionally excluded classical bulges in known mergers, galaxies with depleted cores, and galaxies with big black holes, basically systems which did not follow the near-linear M bh -M * ,sph relation.
The current work instead builds on the (galaxy morphology type)-dependent M bh -M * ,sph and M bh -M * ,gal relations most recently presented in Graham (2022), where some galaxies were also ex-cluded from the Bayesian regression analyses performed there. 11 To maintain sample consistency in Graham (2022), if a galaxy was flagged as an outlier in one diagram, such as the M * ,sph -σ diagram, it was excluded from the regression in all diagrams. These excluded galaxies are shown in Fig. 2, except for the two bulgeless galaxies (NGC 4395 and NGC 6926) and one galaxy with no black hole mass (NGC 5055) 12 , and they are discussed below.
Previously, five of the initial 104 galaxies (NGC 1194, NGC 1316, NGC 2960 13 were a priori excluded due to their well-recognised wet merger status. Although the community, and author, had tended to discard such galaxies as (unrelaxed) deviant systems that should be shunned from the scaling diagrams, these four S0 galaxies plus one S galaxy (NGC 2960) have yielded vital insight into understanding the (evolution of the) S0 galaxy population. These five galaxies are marked with pink hexagons in Fig. 2. An additional disturbed S galaxy (Circinus: Elmouttie et al. 1998) -marked in Fig. 2 with a black square around a blue star -was also excluded from the past regression analyses for the same reason.
Three galaxies stood out as outliers from the morphologydependent M * ,sph -σ relations (Graham 2022). The outliers included one E galaxy (NGC 4291), one dusty S0 galaxy (NGC 2787), and one S galaxy (NGC 4945), and may represent measurement error. As discussed in Appendix B of Graham (2022), NGC 4291 may be an S0 galaxy rather than an E galaxy and therefore have an incorrectly high spheroid mass assigned to it. NGC 4945 may also have an erroneous spheroid mass (and spheroid Sérsic index), while only NGC 2787 remains something of a mystery. These three galaxies have been over-plotted with a black dot in Fig. 2.
This leaves four more past exclusions. 14 NGC 404 (S0) is located at the low-mass end of the diagram, and as such, it could have excessive weight in the regression analyses, torquing the fitted line towards it. It may, however, be in the correct location. The bulgeless galaxy NGC 4395 resides close to it in the M bh -M * ,gal diagram (Fig. 2). Second, NGC 4342 is a stripped S0 galaxy, having had its stellar mass reduced by an unknown amount. The ES,e ellicular galaxy NGC 3377 (Nykytyuk 2015), also known as a discy fast-rotating elliptical galaxy, was excluded due to its biasing nature in the M bh -M * ,gal diagram (Graham 2022, his Fig. 2). 15 Finally, NGC 7457 (S0) has an unusual kinematic structure, including cylindrical rotation about its major axis, signalling that it too experienced a past merger (Sil'chenko et al. 2002;Naab & Burkert 2003;Molaeinezhad et al. 2019).
In Figure 2, the M bh -M * ,gal and M bh -M * ,sph relations from Graham (2022), based on these exclusions, are shown for the E, S0 and S galaxies.  Graham (2022). Black hole mass versus galaxy stellar mass. Five points enclosed in a black square (three S0, the S galaxy Circinus S, and the ES.e galaxy NGC 3377 mentioned in Section 2.2.1), plus three points over-plotted with a black circle (the S0 galaxy NGC 2787, the E galaxy NGC 4291, and the S galaxy NGC 4945), were excluded from the Bayesian regression analyses performed in Graham (2022), as were the bulgeless galaxies NGC 4395 and NGC 6926, denoted by the empty black squares in the left panel. The regressions shown here were for the E, ES,e and two core-Sérsic galaxies (red line), the S0 and ES,b galaxies (cyan line), and the S galaxies (blue line). Right: Similar, but using the spheroid stellar mass rather than the galaxy stellar mass. The regressions sample the same systems in both panels. For plotting purposes only, NGC 2974 (aka NGC 2652) and NGC 4594 (M104) have been reclassified here from S to S0.

Current exclusions
Given the inclusion here of the above five mergers (NGC 1194, NGC 1316, NGC 2960 and the current focus on S0 (including ES,b) galaxies, only four S0 galaxies from the above list are excluded: NGC 404 (low-mass extremity); NGC 2787 (currently unexplained outlier); NGC 4342 (stripped); and NGC 7457 (merger with unusual kinematics). They are marked in all relevant diagrams.
In addition, based on the appearance of dust, as reported in Table 2, one dusty (NGC 3489) and one relatively dust-free S0 galaxy (NGC 1332) are flagged and excluded from the regression analyses. These two systems are not thought to be in error but reflect that the merging process is not a perfectly regimented sequence. That is, a small fraction of real outliers exist. For example, the S0 galaxies NGC 1023, NGC 4762, and NGC 7332 are not dusty but are, or may be, post-mergers (see the comments in Table 2). These and a few other (included) systems are discussed further in Section 3.3. While NGC 4945 -the only previously excluded spiral galaxy -does not have an undue influence on the mass-mass diagrams presented here and is therefore included, the spiral galaxy NGC 1300 was flagged as an outlier in  and has been excluded here (see Fig. 3) in deriving slightly revised S galaxy scaling relations.
These galaxies are partly noted because, like the stripped galaxy NGC 4342, they may offer further insight into the processes which mould galaxies. They may, however, turn out to have measurement errors or instead reflect true scatter in the evolutionary sequence.

Unknown exclusions and selection biases
Disc-dominated galaxies with small bulges, including low surface brightness disc galaxies with large disc scalelengths (e.g. Hamabe 1982; Graham & de Blok 2001), may be under-represented. Indeed, several LTGs were excluded from the parent sample established in  because they needed to be analysed using Hubble Space Telescope data rather than Spitzer Space Telescope data to resolve their smaller bulges. However, their inclusion by Davis et al. (2018) reveals a consistent trend in the M bh -M * ,gal and M bh -M * ,gal diagram. Nonetheless, one should be mindful of potential selection bias (Disney 1976) in the M bh -M * ,gal diagram.
The previous concern of a sample selection bias between galaxies with directly measured black hole masses and those without (Shankar et al. 2016) arose from the use of an inconsistent stellar mass-to-light ratio between the samples imaged in different bands. In the M * ,gal -σ diagram, ETGs with directly measured black hole masses do not appear offset from those without a directly measured black hole mass ).

What is a bulge?
Quantifying a bulge once meant performing a two-component bulge+disc decomposition. However, closer inspection of nearby galaxies has revealed the presence of many components that, when accounted for, can whittle away at what was once considered the bulge. This subsection is primarily included to raise awareness of this issue, which previously received some impetus from the threecomponent bulge+bar+disc fits in Laurikainen et al. (2005). However, multicomponent fits with R 1/n bulges were pioneered by Prieto et al. (1997) and Prieto et al. (2001).
The bulge masses used here come from multicomponent decompositions in which additional components, such as bars, ansae, rings, nuclear star clusters, nuclear discs, nuclear bars, and more, were identified in images and sometimes spectra. In many instances, the components were already known in the literature. There is usually little ambiguity over the bulge, but sometimes, there is scope for question. In pushing the frontiers of what is feasible, a couple of examples are discussed below.
While NGC 2787 (log σ[km s −1 ] = 2.28) has a light profile -shown in  -which is similar to that of NGC 4371 (log σ[km s −1 ] = 2.11) -shown in Sahu et al. (2019) -the stellar mass of NGC 2787's spheroid and galaxy is ∼4 times smaller. Curiously, Gadotti et al. (2015) reveal how the bulge component of NGC 4371 might be smaller and, indeed, how it could be whittled away to nothing. NGC 4371, like NGC 4429, is a 'figureof-eight' galaxy, with its X-shaped 'pseudobulge' structure lining up with its partial ring (likely the ansae of a now dispersed bar) to form a 'figure-of-eight' pattern. While Sahu et al. (2019) fit a Sérsic bulge plus a (low-n Sérsic) barlens component for the pseudobulge in NGC 4371, Gadotti et al. (2015) fit a point-source, a nucleus (<0 .8), plus an "inner disc" (likely dominating from 1 to 20 ) for the pseudobulge component. Gadotti et al. (2015) noted that their pseudobulge component additionally encompassed unmodelled light from both a 3 .2 disc-like feature 16 (light which contributed to the bulge model of Sahu et al. 2019) and a 10 ring (which contributed to the barlens model of Sahu et al. 2019). Although this does not impact the M bh -M * ,gal diagram explored here, it raises the question, What is a bulge? This can influence the M bh -M * ,sph diagram if nearby galaxies or galaxies with new, better spatial resolution are decomposed ad infinitum. It also raises the question as to which component(s) dominate the stellar velocity dispersion, σ, used in M bh -σ diagrams.
There is also the curious case of the giant irregular galaxy NGC 6926, which, at 86 Mpc, is the most distant S galaxy in the present sample. 17 It is thought to be disturbed by its dwarf ETG neighbour NGC 6929 and has a nuclear, molecular disc thought to be edge-on and thus some 18 to 32 degrees misaligned with the inclination of the main stellar disc (Sato et al. 2005;Davis et al. 2019a). Despite its large stellar mass, this S galaxy is bulgeless. 18 NGC 4699 is another galaxy with an unexpectedly low bulge-tototal stellar mass ratio, of just 0.10, for a massive spiral galaxy .
While systems with ambiguous bulges are thought to represent a minority of the sample, Fig. 3 presents not only the M bh -M * ,sph diagram (and M bh -R e,sph diagram) but also the M bh -M * ,gal diagram for the S0 galaxies, along with the S galaxies. The 'dust bin' of each S0 galaxy is indicated there. Given the focus on the S0 galaxies, the 16 Could this be the inner part of the bar? 17 Aside from the S galaxy merger NGC 2960 at 73 Mpc, and UGC 3789 at 51 Mpc, all of the sample's S galaxies are located within 40 Mpc. 18 NGC 6926 was modelled as having a pseudobulge rather than a classical bulge (Davis et al. 2019a).
S galaxies are greyed out but retained for reference. The 'dust bin' to which each S0 galaxy was assigned can also be seen in Table 2. It is apparent (from the M bh -M * ,gal diagram) that the division among the S0 galaxies is not due to the decomposition process.

Splitting the lenticular galaxies
Compared to S galaxies, S0 galaxies exhibit a considerable amount of scatter in the M bh -M * ,gal , and M bh -M * ,sph diagrams (Fig. 2). An exploration of this scatter was the motivation for this paper.
As noted in Section 2, a visual inspection was performed on the sample of S0 galaxies. Getting one's hands dirty and a little more intimate with the galaxies proved highly informative. Doing so, coupled with recourse to the literature, a general division emerged between the S0 galaxies living on the right-hand and lefthand side of the S0 galaxy distribution in the M bh -M * ,gal and M bh -M * ,sph diagrams, evident in Fig. 3. By and large, those S0 galaxies on the left-hand side do not display much evidence for dust, a tracer indicative of an environment suitable for hosting cold gas. In contrast, most of those on the right-hand side abound in dust and tend to be previously-recognised wet mergers (Table 2). Furthermore, after completing this work, it was discovered that the notion of two S0 galaxy subtypes was realised by van den Bergh (1990) via an exploration of the galaxy luminosity functions and ellipticity histograms, although the S0 population remained somewhat enigmatic (van den Bergh 2009). Fig. 1 shows an illustrative 'facebook' for some S0 galaxies in the different dust bins. Additional examples of dusty S0 galaxies can be seen in Fig. B1 of Krajnović et al. (2013) and Dustpedia 19 (Davies et al. 2017;Clark et al. 2018). Upon commencing this project, only four of the sample's S0 galaxies had been flagged as recognised mergers. Appearing as low M bh /M * ,sph outliers from past near-linear M bh -M * ,sph relations, they were previously identified as mergers and dismissed as presumably unrelaxed systems. However, they need not be dismissed, nor are they rare; they are simply the lower M bh /M * ,sph ratio members of a more extensive distribution of dusty S0 galaxies built by substantial wet mergers.

New S0 galaxy M bh -M * scaling relations
Given the trends in Fig. 3, the previous M bh -M * ,sph relations for S0 and S galaxies from Graham (2022) are revisited in Fig. 4, where separate relations for the dust-poor and dust-rich S0 galaxies are reported, along with a slightly revised relation for the S galaxies due to the changes mentioned in Section 2. What can be seen in Fig. 4 is tantamount to a minor breakthrough in understanding the connection between galaxies. It had been a mystery why the S galaxies followed a steeper relation than the ETGs Davis et al. 2019a, Fig. 2) and how it arose that the S galaxies defined a steep M bh -M * ,sph relation between the S0 and E galaxies (Sahu et al. 2019). Fig. 3 and 4 reveal, for the first time, that the dust-poor and dust-rich S0 galaxies also follow steep M bh -M * ,sph relations, which sandwich the S galaxies. The dust-poor S0 galaxies define the relation   (Emsellem et al. 1996). NGC 4596 Y, n Inner dust disc (Kent 1990;Gerssen et al. 1999). NGC 5018 † ‡ Y, ... Dust lanes, merger, shells, gas-bridge to NGC 5022 (Hilker & Kissler-Patig 1996;Tal et al. 2009;Spavone et al. 2018). NGC 5128 † ‡ Y, ... Dusty gas-rich merger (Malin et al. 1983;Israel 1998). NGC 5252 Y, ... Merger, misaligned dust lanes (Morse et al. 1998;Keel et al. 2015;Kim et al. 2015). NGC 5813 ‡ y, n Widespread dust, old merger, core-Sérsic galaxy, misaligned kinematic axis, X-ray halo (Tomita et al. 2000;Krajnović et al. 2015). NGC 6861 Y, ... Dusty gas disc; interacting with NGC 6868, and located near IC 4943 (Machacek et al. 2010).
Column 1: Same as Table 1. Column 2: Is there dust visible in the HST optical image: (Y)es plenty; (y)es but not a lot; (n)ot really, just a (n)uclear dust disc/ring; and (N)o, none. The second entry, after the comma, is the "Dust feature" classification from Krajnović et al. (2011): n = no dust; d = dusty disc.
Column 3: Comments and references showing the dust and/or noting accretion/merger activity.
at least for 7 < log(M bh /M ) < 9 dex. An expression with a smaller uncertainty at a different intercept is provided in the caption to Fig. 4. At a given black hole mass, the dust-rich S0 galaxies have a spheroid mass that is, on average, ∼0.54 dex greater than that of the dust-poor S0 galaxies in the sample. The Bayesian regression hints at a steeper slope for the dust-rich S0 galaxies; however, the uncertainty is sufficiently large that this remains inconclusive. Nonetheless, the intercept of the dust-rich S0 galaxy M bh -M * ,sph relation (for 15 galaxies excluding NGC 3489) appears notably different in noting that a different normalisation point, at log(M * ,sph /M ) = 10.70 dex, has been used in Eq. 3.
Building on , which revealed how dry S0 mergers build E galaxies -by folding in disc stars - Fig. 5 is a helpful schematic to illustrate how wet mergers involving spiral galaxies can build dusty S0 galaxies.
Unlike the M bh -M * ,sph diagram (Fig. 4), the M bh -M * ,gal diagram ( Fig. 6) offers less clear relations for the S0 galaxies. Nonetheless, the general offset nature of the dust-poor and dustrich S0 galaxies is evident. Here, the (wet merger)-built, dust-rich S0 galaxies appear as a high-mass extension to the S galaxies. The dust-poor S0 galaxies have notably lower M * ,gal /M bh ratios for a given black hole mass. M bh -M * ,gal relations are provided in the caption to Fig. 6. The size of the offset between the different types of disc galaxy (S, dust-poor S0, and dust-rich S0) are substantial and suggest that considerable gains could be made by revisiting the (M bh -M * ,sph )-and/or (M bh -M * ,sph )-based derivation of the virial f - factors used to convert AGN virial masses into black hole masses (Bentz et al. 2009;Bennert et al. 2021).
The above division between the S0 galaxies is strong but could be better because of some discrepant systems This is, however, expected given the partly stochastic nature of mergers and the changing neighbourhood where galaxies can find themselves. This is explored further in Section 3.3.

Potential dust bias
As seen in Fig. 3, the effects of dust on the adopted stellar massto-light ratio (Section 2.1) are insufficient to fully explain the offset between the S0 galaxies with and without visible signs of dust. However, it could account for up to half of the offset if both of the ∼0.13 dex offsets (in log M * ,gal ) mentioned in Section 2.1 were to hold.
Of the four previously well-recognised S0 mergers (NGC 1194, NGC 1316, NGC 5018 andNGC 5128), all appear to have typical V−[3.6] colours around 3.6 mag, except for NGC 5018, whose spheroid could be 0.25-0.50 mag too bright at 3.6 µm   Fig. A1). Nonetheless, the offset behaviour in some, if not all, dusty S0 galaxies -relative to the dust-poor S0 galaxies -in the M bh -M * ,gal diagram appears to be due to their merger origin noted in Table 2. This may reflect an S+S merger, an S+(dust-poor S0) merger, or a gas-rich but initially dust-poor S0+S0 merger 20 if it subsequently produces stars that yield a dusty interstellar medium.

Spiral galaxies
The M bh -M * ,sph relation for the 25 S galaxies (derived excluding NGC 1300 and the Milky Way) can be written as in good agreement with (Davis et al. 2019a). Sahu et al. (2019) show that the S galaxy M bh -M * ,gal relation predominantly resides to the right of the S0 galaxy M bh -M * ,gal relation. This can be seen in Figure 2 and might be explained by ongoing star formation in S galaxies while S0 galaxies are, in effect, left behind. This concept also allows for a certain amount of AGN growth in the S galaxies, so long as it does not out-pace the stellar growth by too much. However, as we learned in Section 3.1, this picture is incomplete, in that the S0 galaxy population is comprised of systems that may have either been left behind or advanced to higher stellar masses and M * /M bh ratios through major wet mergers. The S galaxies now appear as an intermittent population (Fig. 4). Those galaxies located to the high M * ,sph /M bh side of the S galaxy distribution have experienced substantial mergers. The S galaxies have probably experienced less substantial mergers.
There are five S galaxies on this high M * /M bh envelope with bulges more massive than log(M * ,sph /M ) = 10.35 dex (see Fig. 3). One of these is the Sa? merger NGC 2960, marked with a pink hexagon in Fig. 2. The four other galaxies (NGC 1097, NGC 1398, NGC 4501, and NGC 2273 are mentioned below. NGC 1097 is a dusty, barred spiral galaxy undergoing substantial star formation, likely induced by interaction with its dwarf neighbour ESO 416-19, aka NGC 1097A (Ondrechen et al. 1989).
NGC 1398 is the brightest galaxy of the NGC 1398 Group (10 members) located within the Eridanus Cluster associated with the larger Fornax Cluster. Dust lanes associated with the bar are evident, as are dust lanes crossing nearly perpendicular to the bar. This double-ringed galaxy has the biggest galaxy stellar mass of all the spiral galaxies in the sample. Despite this, a cautionary flag is raised as to whether its bulge mass has been over-estimated due to the potential presence of an (unmodelled) barlens. 21 NGC 4501 (aka M88) is a dusty star-forming spiral galaxy with a nascent HI tail pointing away from M87 (Cayatte et al. 1990) due to ram-pressure stripping in the Virgo cluster (Vollmer et al. 2008). The spheroid mass may have been overestimated due to a possible (unmodelled) upturn in the inner disc light. A slightly similar situation may exist with M33 (not included here), which was initially reported to have a bulge (Boulesteix et al. 1979;Bothun 1992) that was subsequently diminished in stature (e.g., Kent 1987;Minniti et al. 1993).
Finally, NGC 2273 has a UV-bright star-forming ring structure reminiscent of that in NGC 2974. Both galaxies have a bulgeto-total stellar mass ratio of one-third. Furthermore, NGC 2273 has a nuclear bar and stellar spiral structure within the main bar (Ferruit et al. 2000;Petitpas & Wilson 2002), somewhat similar to NGC 2974, with its nuclear gas spiral within its weak main bar. NGC 2273 has roughly 10 9 M of HI and is the brightest galaxy in the NGC 2273 Group of three, also referred to as the 'Lyon Groups of Galaxies' LGG 137 (Garcia 1993). NGC 2273 is suspected of having interacted with UGC 03530, aka NGC 2273B (Byrd et al. 1987). In contrast with NGC 2974, the spheroid in NGC 2273 has an effective half-light radius which is 2.4 times (0.38 dex) smaller, a velocity dispersion which is 0.22 dex smaller, and a black hole mass which is 19 times (1.28 dex) smaller.
This subsection's divergence was intended to see if the tabulated spiral galaxies with reportedly massive spheroids may have had their formation aided by (expectedly minor) 22 mergers and interactions. Because most spiral galaxies contain dust, with the more massive spirals being dustier (van den Bergh & Pierce 1990), alternative signatures of interactions, accretion, or mergers were searched for in the literature and reported above. Further research is, however, required on this front, given that many spiral galaxies, perhaps including those not numbered in Fig. 3, may also show signs of interactions. Nonetheless, removing NGC 1398 and NGC 4501, the above small sample of (5-2=3) 'spiral' galaxies includes one recognised spiral merger (NGC 2960), the spiral galaxy NGC 2273 thought to have undergone a past interaction and having features remarkably similar to that seen in NGC 2974, and the 21 Barlenses, not to be confused with inner discs, have been associated with bar buckling events which produce boxy/X/(peanut shell)-shaped 'pseudobulges' (Bardeen 1975;Hohl 1975;Combes & Sanders 1981;Athanassoula 2005;Laurikainen et al. 2011;. 22 Major mergers would destroy the spiral pattern and build S0 galaxies.
previously interacting spiral galaxy NGC 1097. These galaxies are identified in the lower-middle panel of Fig. 3.
Considering S galaxies with spheroid stellar masses below log(M * ,sph /M ) = 10.35 dex, an additional four spiral galaxies are labelled in Fig. 3 because of how they define the continuation of an apparent high M * ,sph /M bh envelope for the S galaxies. 23 These include NGC 4826 (aka the Black Eye Galaxy), with copious amounts of dust and gas counter-rotating to the stellar disc (Braun et al. 1992;Rubin 1994), likely brought in by a smaller, gasrich galaxy. The others are the Milky Way and NGC 3079, which are somewhat similar to each other, plus NGC 1320 (Martini et al. 2003), which could perhaps be called the 'Eye Liner Galaxy' with a reasonably prominent dust-arm visible (on just one side of the galaxy) in the HST/WFPC2/F791W-F547M image available at the HLA.

Interesting and/or nonconforming galaxies
As noted in Section 2.2, looking for galaxies which do not conform with the general trend may be insightful. While some systems might represent measurement error, others may reflect true scatter and thus further insight into the evolutionary paths of galaxies. An attempt has been made here to provide additional information on interesting or nonconforming galaxies if not already discussed in Section 2. This subsection is considered additional to the main result and can be bypassed by readers not after every detail.
There is a cluster of five dust-poor S0 galaxies (NGC 1023, NGC 3384, NGC 4371, NGC 4762, and NGC 7332) in the lowerright of the distribution in the M bh -M * ,gal diagram (Fig. 6, righthand side). They overlap with the spiral galaxies and the lowmass extrapolation of the M bh -M * ,gal relation for dust-rich S0 galaxies. While they are not major merger remnants, four of these five show signs of accretion or minor mergers. NGC 1023 and NGC 7332 have chemically decoupled stellar nuclei (Sil'chenko 1999). NGC 7332 has a 2.5±0.5 Gyr old nucleus, a gas disc that is counter-rotating with respect to the stellar disc (Fisher et al. 1994) and a small HI mass of 4 × 10 6 M (Serra et al. 2012). It is unclear if the nuclear disc was solely built by accretion or if the bar had a hand to play (Cole et al. 2014). NGC 1023 has a 7 Gyr old nucleus within an older bulge and a reasonably thin/flat stellar disc. It has a disturbed large-scale HI disc (Allsopp 1979;Sancisi et al. 1984) of 2 × 10 9 M (Serra et al. 2012), giving M HI /M * ,gal = 0.026, plus outer kinematics (Noordermeer et al. 2008;Bettoni et al. 2012) and a nuclear, stellar disc (Dalla Bontá et al. 2014) suggestive of a past interaction or small merger event. Despite the HI gas mass, no dust is visible, presumably still diffusely distributed or tied up in the gas phase metallicity.
Another galaxy in the grouping of five is NGC 3384, which may be interacting with a supergiant intergalactic HI ring in the Leo I Group (Sil'chenko et al. 2003). This galaxy contains little noticeable dust in the optical images (Tomita et al. 2000) but has been detected in CO (J=2-1 transition), yielding an H 2 mass of (3.5 ± 0.8) × 10 6 M (Welch & Sage 2003), and it has a global HI mass of ∼2×10 7 M (Serra et al. 2012). NGC 4371, with the smallest black hole mass of the five, stands out for not displaying any obvious signs of past/ongoing interaction, accretion, or minor  Fig. 7 in . Here, two steep M bh -M * ,sph relations for visibly dust-poor and (wet-merger-built) dust-rich lenticular galaxies are shown to bracket the steep relation for spiral galaxies (Eq. 4), which previously appeared as an unusually steep relation relative to that of the (dust-poor and dust-rich) lenticular galaxies and the (mergerbuilt) elliptical galaxies (Fig. 2). merger activity, as seen in the other four (growing) galaxies. However, it has a second nuclear star-forming ring with a major-axis radius of ∼10 (∼0.8 kpc) (Comerón et al. 2010). Finally, NGC 4762 has a thick asymmetric gas disc and mild asymmetry in its stellar structure (Wakamatsu & Hamabe 1984;Wozniak 1994). It has a warped S-shaped outer stellar disc and a blue lens possibly built from star formation associated with a neighbourly interaction or the consumption of a smaller galaxy (Wakamatsu & Hamabe 1984;Wozniak 1995). In passing, it is noted that NGC 3115 (in sample) and NGC 5866 (not in sample) may represent more dusty analogues of NGC 4762, having more substantial thick stellar discs (Burstein 1979). The dusty S0 galaxy NGC 5252, another known merger, represents an even more evolved state of affairs, in which the stellar disc has thickened considerably, but the edge-on thin disc is still apparent. Curiously, NGC 4762 has a rather low spheroid-to-galaxy stellar mass ratio for a merger, at just 0.08, but perhaps this is reflective of what some minor mergers produce.
In addition to NGC 3115, mentioned above, there is another dusty ES,b galaxy in the sample. It is NGC 6861, shown in Fig. 1, and dubbed here 'The Speaker' due to its strong dust rings over the inner kpc, which make it resemble an audio speaker. NGC 6861 is interacting with NGC 6868 in the Telescopium galaxy group (Machacek et al. 2010). It is interesting because it is a reasonably compact massive galaxy (R e,gal ∼ 2.5 kpc, M * ,gal ∼ 10 11 M ) reminiscent of relic red nuggets. Nevertheless, it also appears to have grown/accreted an intermediate-scale disc suggestive of a wet merger event that might have contributed to its spheroid's development.
Also near the top of the M bh -M * ,gal relation for dust-rich S0 galaxies (Fig. 6) are the two core-Sérsic S0 galaxies: NGC 524 and NGC 5813 (Krajnović et al. 2009;Richings et al. 2011;Dullo & Graham 2014;Krajnović et al. 2015). Core-Sérsic galaxies are thought to have been built from mergers in which a binary massive black hole scoured away the central stellar phase-space of what may have been a Sérsic (1963) R 1/n light profile (Begelman et al. 1980;Ebisuzaki et al. 1991;Graham 2004;Merritt 2013). Additional core-Sérsic S0 galaxies can be found in Dullo & Graham (2013). As mentioned in Section 1, the core-Sérsic galaxy NGC 5813 is immersed in a group-sized X-ray halo which has shut down star formation. While NGC 5813 is an old merger (Hopp et al. 1995), it is less clear when NGC 524 experienced its merger event. NGC 524 is immersed in a smaller, galaxy-sized X-ray halo (Osmond & Ponman 2004) that has not yet removed all the dust and cold gas from the centre of NGC 524 (Sil'chenko 2000). These two core-Sérsic S0 galaxies are interesting in that depleted cores are generally considered to be a sign of dry mergers (e.g. Merritt et al. 2007; Sesana et al. 2008b) due to the efficient gravitational drag on black holes from gas clouds and/or a circumbinary disc sparing the ejection of hypervelocity stars from the core region of a wet merger (Merritt 2006;Sesana et al. 2006;Mayer et al. 2007;Kelley et al. 2017). However, the dust in these two galaxies reveals a more complex history, affecting accretion and feedback (Liao et al. 2022).
Returning to the low-mass end, another system which stood out is the S0 galaxy NGC 7457, already mentioned in Section 2.2.1 as a merger remnant revealed through its kinematics. Apart from a faint non-nuclear dust ring, it appears dust-poor. It has been reported to have an H 2 mass of (3.3 ± 1.0) × 10 6 M , comparable to the 5.5 × 10 6 M of H 2 gas reported by Welch & Sage (2003) in the gas-rich dwarf galaxy NGC 404, which has an order of magnitude less stellar mass. Like NGC 404, NGC 7457 resides in the lower-left of the M bh -M * ,gal diagram. NGC 7457 has a young ∼2 Gyr old nucleus and experienced star-formation 2-3 Gyr ago, giving rise to new globular clusters (Chomiuk et al. 2008) and perhaps the now faint dust ring. However, it is a dust-poor post-merger residing among the low-mass S0 galaxies. Although not S0 galaxies, NGC 3377 (ES,e) and Circinus (S), mentioned in Section 2, also appear to be post-merger systems residing among the low-mass S0 galaxies. These three galaxies are enclosed with black squares in Fig. 2.

Updating the typical B/T ratio in dry S0 mergers
Unlike wet mergers, where the progenitor galaxies' B/T stellar mass ratios are harder to establish given the likelihood of star formation during the merger, this is not a problem for dry mergers. The dry major-merger-induced-jump shown by Graham (2022, section 3.2, final paragraph), to convert two equal S0 galaxies into an E galaxy, was based on an M bh -M * ,sph relation defined by all of the S0 galaxies. This jump from the S0 galaxy M bh -M * ,sph relation to the E galaxy M bh -M * ,sph relation was shown to equate to the merger of two S0 galaxies with a bulge-to-total stellar mass ratio of 0.37. Removing the offset dust-rich S0 galaxies and focussing on the dust-poor S0 galaxies, they define a steeper relation in the M bh -M * ,sph diagram, which can be seen offset to smaller M * ,sph at M bh ∼ 10 8 M (Fig. 4). As a consequence, a dry merger induced jump from the new dust-poor S0 galaxy M bh -M * ,sph relation to the unchanged E galaxy M bh -M * ,sph relation is still associated with the same doubling of the black hole mass. However, it now requires a greater increase in the spheroid mass. This means that more disc stars from the two progenitor galaxies are required to create the new, bigger spheroid, i.e., the elliptical galaxy. The merger of two S0 galaxies with log(M bh /M ) = 7.7 dex, and log(M * ,sph /M ) = 10.01 dex, on the dust-poor S0 galaxy M bh -M * ,sph relation (shown in the right-hand panel of Fig. 4) required to build an E galaxy with log(M bh /M ) = 8.0 dex on the E galaxy M bh -M * ,sph relation (Fig. 2), requires an increase in spheroid mass of 0.84 dex. This balance is achieved by folding in the disc and bulge stars of two S0 galaxies with B/T = 0.29. This ratio is in line with observations of S0 galaxy B/T ratios (Graham & Worley 2008).  Figure 7 shows the size-(stellar mass) diagram for the bulges of the disc galaxies studied here. The ES,b systems such as NGC 6861 -which are bulge-dominated and possibly largely unevolved relics from z ∼ 2.5 -have the same sizes and stellar masses as the dusty S0 galaxy bulges thought to have been built/bolstered by wet mergers at lower redshifts. This is not necessarily a problem. The M * ,sph -R e,sph trend seen in Fig. 7 and explored in Graham (2022) and Hon et al. (2022b) is expected given the M * ,sph -σ relation for S0 galaxies, coupled with the virial theorem approximation M * ,sph ∝ σ 2 R e,sph . Fig. 7 reveals that this seems Of note here is that one may have a compact massive spheroid, with, say, log(M * ,sph /M ) > 10.6-11 dex and R e,sph > 1-2 kpc, that was recently (z < 1-2) built by a wet merger. Such a spheroid is not a 'relic' from the early Universe. This is not to say that ES,b galaxies which accreted an intermediate-scale disc are not relics, only that the evolutionary pool appears to be muddied at 10.6 log(M * ,sph /M ) 11.2 dex by major wet mergers that build new spheroids in dusty S0 galaxies. Presumably, from Fig. 6, the compact bulges of the S0 galaxies in Hon et al. (2022a), with 10 < log(M * ,sph /M ) < 10.6 dex, were found in dust-poor galaxies. As seen in Fig. 2, the spheroidal component of dust-poor ES,e and true E galaxies tend to have distinctly bigger sizes (R e,sph > 2 kpc). As such, they would not have been counted among the potential relic red nuggets. Those systems do not contain a preserved nugget but have been transformed by major mergers ).

Mergers: shaking the dominance of AGN feedback
After reviewing the literature, it is apparent that the dusty S0 galaxies are known merger products, while the relatively dust-poor S0 galaxies are not recognised as such -although some are reported to have experienced minor accretion or disturbances (Table 2). Fig. 5 qualitatively illustrates how wet (gas-rich) mergers evolve the dust-poor S0 galaxies across to the dust-rich S0 galaxy sequence in the M bh -M * ,sph diagram, quantitatively shown in Fig. 4 but without the evolutionary tracks. For a quarter of a century, AGN feedback has been heralded as the driving force behind the black hole scaling relations (e.g. Silk & Rees 1998;Fabian 2012). However, mergers also have a considerable yet often understated role. Indeed, mergers have a dominant role in creating the E (and ES,e) galaxies, for which the bulk of the progenitor's disc stars are folded into the E galaxy Graham 2022).
Collisions aside, stars tend to form in discs rather than in bulges. Therefore, to fully understand the M bh -M * ,sph relations requires an element in which some disc stars eventually become bulge stars. Gas-fuelled AGN and disc stellar growth need not be automatically accompanied by bulge stellar growth. Indeed, if AGN feedback regulated the star formation, one might expect to see stronger M bh -M * ,disc and M bh -M * ,gal relations than the M bh -M * ,sph relations. Furthermore, quasar feedback is not isotropic but occurs in bi-directional polar jets, limiting its ability to quench star formation within the disc plane. It is, therefore, perhaps not surprising that mergers, moving disc stars into bulges, is an important ingredient of galaxy/black hole evolution.

Black hole mass functions
The ability to make refined predictions of the central black hole mass in other galaxies benefits various research programmes. This includes work on the local black hole mass function (Salucci et al. 1999;Vika et al. 2009;Shankar 2013) and the masses of active black holes in distant quasars (Vestergaard & Osmer 2009;Willott et al. 2010). Initially, a single black hole scaling relation tended to be applied to all galaxies, irrespective of their morphology. Knowing the galaxy or spheroid luminosity function of, say, the LTGs and the ETGs, or their distribution of Sérsic indices or velocity dispersions, one could construct the LTG and ETG black hole mass function (e.g. Vika et al. 2009). However, one can now do better than this, given that the black hole mass is related in different ways to the spheroid stellar mass of ETGs of different morphology, reflecting their merger history. For a given spheroid stellar mass, the black hole mass of an ETG can differ by an order of magnitude or more. This adjustment will considerably impact the black hole mass function, especially at low masses, due to the steepness of the M bh -M * ,sph relations. Keel et al. (2015) present eight dusty Seyfert 2 S0 and S galaxies, including the dusty S0 galaxy NGC 5252, with its suspected binary SMBH (Kim et al. 2015;Yang et al. 2017). These eight galaxies are merger remnants showing a fading AGN. Other examples of galaxies with a directly measured black hole mass but with possible evidence for a binary SMBH are the spiral galaxy NGC 4151 (Bon et al. 2012) and the dusty S0 galaxy NGC 1194 (Vasylenko et al. 2015;Fedorova et al. 2016). It seems reasonable to speculate that, if built up by mergers, these eight Seyfert galaxies should have black hole masses, which place them on the right-hand side of the S0 galaxy M bh -M * ,gal relation (Fig. 6, left-hand side). That is, they are predicted here to have M bh /M * ,gal ratios an order of magnitude smaller than typically observed in dust-poor S0 galaxies, as is observed with NGC 1194, NGC 4151, and NGC 5252. This has nothing to do with pseudobulges built from secular evolution; quite the opposite, it is because their bulges have been built, in part, by mergers which folded in some pre-existing disc stars and thereby lowered the M bh /M * ,sph ratio. The dust-rich S0 galaxy M bh -M * ,gal relation, which somewhat overlaps with the S galaxy M bh -M * ,gal relation (Fig. 6, right-hand side), has been used to predict these eight galaxies' central black hole mass (Table 3).

Gravitational waves
The benefits of refined black hole mass predictions from (galaxy morphology)-dependent black hole scaling relations can also be extended to efforts to predict the gravitational wave background Table 3. Predicted M bh for Seyfert galaxies from Keel et al. (2015).   (Luo et al. 2016;Mei et al. 2021) aiming to detect gravitational waves in the sub-milliHertz to 1 Hz range. The refined relations should also have importance for current endeavours to detect the gravitational wave background arising from numerous coalescing massive black holes and monochromatic signals from individual binaries. By better populating (mock or real) catalogues of galaxies with more accurate black hole masses and with a better knowledge of which (real) galaxy types experienced a major merger, it enables improved predictions of massive black hole binaries and thus an improved knowledge of the collective ocean of gravitational waves that they generate. This, in turn, benefits our expectations for the local influence of these waves on the arrival time of pulsed emission from pulsars located in an array around us (Foster & Backer 1990;Sesana et al. 2008a;Burke-Spolaor et al. 2019). Ongoing experiments to indirectly detect these waves include the Parkes Pulsar Timing Array (PPTA: Manchester et al. 2013;Shannon et al. 2013;Goncharov et al. 2022), the European Pulsar Timing Array (EPTA: Lentati et al. 2015;Chen et al. 2021), the North American Nanohertz Observatory for Gravitational Waves (NANOGrav: Arzoumanian et al. 2021;Middleton et al. 2021), plus newcomers such as the Indian Pulsar Timing Array (Tarafdar et al. 2022), and the South African MeerTime Pulsar Timing Array (Spiewak et al. 2022). It is thus a large global enterprise. The new (galaxy morphology)-dependent black hole scaling relations, with their order of magnitude differences in M bh /M * ,sph ratio, is valuable knowledge in one's toolkit. Knowing which galaxies have formed from major merger events, which were gas-poor or gas-rich mergers (impacting the binary BH merger timescale), what 24 www.lisamission.org progenitor galaxies likely built the merger product (giving insight on the initial BH masses), and what the expected final black hole mass is based on the galaxy morphology, offers significant advantages over the past use of a single near-linear 'red sequence'. These advances should help with the interpretation, and perhaps even the discovery, of nanoHertz to 1 Hertz gravitational waves.

Evolution of the scaling relations
Studies of possible evolution in the M bh -M * ,sph and M bh -M * ,gal relations with look back time to higher redshifts (e.g. Treu et al. 2007;Merloni et al. 2010;Sun et al. 2015) will also benefit from a greater awareness of (galaxy morphology)-dependent relations. For instance, the scaling relation for quasars in distant disc galaxies is likely to be different from the relation defined by a local sample dominated by E galaxies. A measured offset in these two relations (high-z AGN in disc galaxies and low-z E galaxies) may reflect the apple versus orange sample selection. Indeed, the local S and E galaxy M bh -M * ,sph relations differ. This is obviously not due to evolution over different look-back times because both samples have the same cosmological time stamp. As the local benchmark is now better understood, with a suite of (galaxy morphology)-dependent scaling relations, and as it becomes better calibrated over time, improvements in measurements of the evolution in the scaling relations can be made. Due to its spatial resolution, aperture size, and infrared capabilities, the James Webb Space Telescope (JWST Gardner et al. 2006) is expected to aid such endeavours.

Tidal disruption events
Better estimates can also be made for the masses of black holes which induce flares from their nearby stars, torn open by the gravitational differential across them as they approach the event horizon and experience a 'tidal disruption event' (TDE: Coughlin et al. 2017;Shu et al. 2020), which fuels an accretion disc. For example, the extragalactic transient Swift J164449.3+573451 (aka Sw J1644+57) is a TDE associated with a black hole estimated to have a mass < 2 × 10 7 M (Bloom et al. 2011;Levan et al. 2011). The appendix of Burrows et al. (2011) notes that this TDE occurred at the centre of a spiral galaxy and that the galaxy luminosity is log(L H /L ,H ) = 9.58 dex. Adopting even a high H-band M/L ratio of 2, the spiral galaxy M bh -M * ,gal relation seen in Fig. 6 predicts the discovery (not previously announced) of an intermediate-mass black hole (IMBH) with M bh = 0.5×10 5 M . This is more than two orders of magnitude smaller than the published upper limit obtained using the old near-linear 'red sequence' for bulges. For comparison with another IMBH in a spiral galaxy, it is noted that LEDA 87300 has a galaxy stellar mass of 2.4 × 10 9 M , and its central AGN has an estimated black hole mass of 0.3 × 10 5 M (Baldassare et al. 2015;Graham et al. 2016).

S0 galaxies at low masses
It would be interesting to examine dust-poor S0 galaxies with M * ,gal 10 10 M (M * ,sph 0.3 × 10 10 M ). Figure 3 suggests that they may have M bh 10 7 M , which is currently restricted mainly to the purview of the S galaxies. Dwarf S galaxies are, however, rare, while dwarf S0 galaxies are abundant. It may therefore be that many IMBHs are waiting to be discovered among the low-mass dwarf S0 galaxies. Some will inevitably be the delivery vehicle for massive black holes into S galaxies (e.g. Graham et al. 2021).
Given the merger-driven evolution to higher masses across the M bh -M * ,sph diagram, the future creation of dusty S0 galaxies with M bh ≈ 10 6 M and M sph ≈ 10 10 M is also predicted here. Such S0 galaxies, with M bh /M sph ∼ 10 −4 , are absent in the current data set. however, they are an anticipated member of the galaxy population -unless the increasing rate of cosmological expansion prevents their formation (Eingorn et al. 2013).

Specific SFRs and SFHs across the M bh -M * ,sph diagram
Coupled with an awareness that the elliptical galaxies built from dry mergers are offset to the right of the dust-poor S0 galaxies in the M bh -M * ,gal diagram, it is noted here that a star formation history (SFH) gradient likely exist across the face of the M bh -M * ,gal diagram, from the dust-poor S0 galaxies to dusty S and S0 galaxies, and then on to the E galaxies ). An additional mass-dependent trend of drying-out and reduced specific star-formation rate (SFR), and SFH, may also exist along the currently observed S and dusty S0 galaxy M bh -M * ,gal relations as things proceed to higher masses (Fig. 6). While showing these trends would not be a discovery, it would be a value-added figure explained by the underlying distribution of galaxy morphology.

Environment
The presence or absence of cold gas, and a galaxy's ability to attain cold gas, can gravely affect its evolution. Many mechanisms can remove or prevent the presence of cold gas and thus curtail star formation and quasar activity, effectively halting a galaxy in its evolutionary track across the galaxy/black hole scaling diagrams.
In LTGs -used here as a reference given their dusty nature -metals in the gas phase of their interstellar medium (ISM) can amount to a few (2-4) times the dust mass (Casasola et al. 2022). As the gas cools in the dense ISM, these metals adhere to and grow mantles around the refractory dust cores/grains produced during star formation. This build-up of the dust helps with the formation of molecular gas clouds by shielding them from ultraviolet rays capable of dissociating molecules like H 2 and C0, which are both a signpost of dust (Alatalo et al. 2013;Kokusho et al. 2017Kokusho et al. , 2019 and stepping stone toward the gravitational collapse of the gas clouds to form stars. Therefore, retaining dusty gas -comprised of graphite, silicon carbide, aluminium oxide and other molecules, which condensed out of the cooling winds of AGB stars or the decompressing gas of supernova (e.g. Clayton & Nittler 2004) -can help raise a further generation of stars.
While falling into a galaxy cluster can result in the removal of gas and prevent the acquisition of new gas (Gunn & Gott 1972;Larson et al. 1980), quasi-isolation or membership of a small galaxy group may result in the ongoing accretion of gas clouds which replenish this fuel supply without destroying a galaxy's disc (Ferguson & Mackey 2016). Small galaxy groups are also more conducive than clusters to galaxy collisions. These collisions can shock and compress gas clouds, causing a burst of star formation at a rate beyond that in regular spiral (S) galaxies (Schweizer 2005), and the dust content of the ISM is known to increase with the specific star formation rate (da Cunha et al. 2010;De Looze et al. 2020). Perhaps together with dust from at least one S galaxy progenitor, the merger-induced dusty starbursts set the dust-poor and dust-rich S0 galaxies apart. Shocks can also reduce or remove the orbital angular momentum of gas clouds, causing the gas to fall toward the centre of the merger product and trigger a starburst (Sanders et al. 1988). The creation of tidally-induced bars may also torque the gas clouds and drive them (someway) inward, where further star formation may occur. 25 All this is to say that the environment matters. However, a galaxy's environment has also taken something of a back seat to the assumed dominance of AGN feedback in shaping the coevolution of galaxies and black holes.
The observation that quasars reside in lower density regions than radio galaxies (Lietzen et al. 2011) meshes with the notion that once galaxies are in a (cold gas)-stripping environment, their growth shuts down (Constantin et al. 2008). Radio mode feedback (from a 'Benson Burner') essentially maintains the status quo in galaxies with 'radio mode' AGN. For these galaxies, subsequent evolution across the black hole scaling diagrams no longer occurs through AGN accretion or star formation but via mergers (Graham & Sahu 2022a,b), building bigger E galaxies and brightest cluster galaxies (BCG).
It might be tempting to speculate that the trend observed for the dust to reside in the more massive S0 galaxies reflects their ability to (gravitationally) hold on to it, while the lower-mass S0 galaxies cannot. However, although feasible, the equally lowermass spiral galaxies can hold onto their dust, with many maintaining a Seyfert at their centre. Therefore, there must be more afoot than simply the stellar mass. The environment may come into play, with many lower-mass S0 galaxies ram-pressure stripped of gas due to their residence in, or passage into, a hot X-ray gas cloud. Although, contradicting matters is that some visibly dust-poor S0 galaxies contain hydrogen gas (e.g., Kokusho et al. 2017, and references therein). Spiral galaxies are, however, known to have a preference for existing in the field and group environment.
To probe this plausible but speculative idea -of a (dust and gas)-stripped galaxy -the broad environment of the lenticular galaxies was tracked down, specifically, if they are isolated or belong to a group or a cluster. This information has been recorded in the Appendix but, on its own, appears not to offer clarity. This may, in part, be due to the randomness of mergers in small groups or whether the group has entered a cluster's X-ray halo -which has not been explored.

Faded S galaxies
There is some suggestion in Fig. 3 that the more massive S0 galaxies (M * ,gal 7 × 10 10 M ; M * ,sph 2.5 × 10 10 M ) retain their dusty gas, at least prior to the onset of a hot X-ray halo which may heat the gas and destroy the dust. If so, then perhaps the five (less massive) dust-poor S0 galaxies (NGC: 1023;3384;4371;4762;and 7332) tracking the lower-end of the spiral galaxy M bh -M * ,gal relation 26 were built by a wet merger and should, in an evolutionary sense, be grouped with the dusty S0 galaxies. Perhaps the dusty clue to their past has been swept away. Their proximity to the spiral merger NGC 2960 -the pink hexagon with the lowest mass in Fig. 6 -also adds credence to the notion (speculation) that these S0 galaxies might be faded S galaxies, or faded S0 galaxies, 'frozen-in' (into their resting place) in the M bh -M * ,sph diagram. 25 Nuclear discs and nuclear bars (aka secondary bars, e.g., Pfenniger & Norman 1990) may funnel gas into AGN (Shlosman et al. 1989;Mulchaey et al. 1997). The abundance of nuclear stellar discs in S0 galaxies (e.g. Jaffe et al. 1994;Rest et al. 2001;Balcells et al. 2007) may be a sign/means, even if a relic, of AGN fuelling. Counter-rotating nuclear discs are, of course, also a sign of past accretion events. 26 Four of the above five have bulges to the right of the S galaxy M bh -M * ,sph relation (Fig. 4).
The offset between the dusty S0 galaxies, the S galaxies, and the dust-poor S0 galaxies may blend well with the idea that S galaxies might fade into S0 galaxies once they lose their cold gas (Larson et al. 1980;Bekki et al. 2002;Barway et al. 2009). However, the average ∼0.3 dex difference in the galaxy stellar mass, at fixed black hole mass, between the S galaxy and the dust-poor S0 galaxy M bh -M * ,gal relations may pose too large a stumbling block for this scenario. One alternative is dramatic black hole growth from a circumnuclear gas disc with relatively little global star formation, effectively pumping the S galaxies up onto the dust-poor S0 galaxy relation in the M bh -M * ,gal diagram.
As noted above, rather than evolve from the S galaxy relation to the dust-poor S0 galaxy relation, the tidal-stripping of gas may cause the S galaxies to freeze in the M bh -M * ,gal diagram and not undergo further movement. In contrast, those with an ongoing (cold gas)-supply continue the march to higher stellar and black hole masses. The dust-poor S0 galaxies with high M bh /M * ,sph ratios on the upper left-hand side of the mass-mass scaling diagram may then be something of a relic population 27 , frozen-in long ago and revealing how the (cold gas)-rich M bh -M * ,gal relation looked in the past before it became today's dust-poor S0 galaxy relation.

Non-exponential discs
Simulations have shown that when discs are present, stars from infalling satellite galaxies have a tendency to align with and heat the pre-existing disc (e.g. Johnston et al. 2017). As the captured gas cools, modulo any ongoing gravitational perturbations, it settles to the mid-plane (e.g., NGC 4233, NGC4710, NGC 5866) and forms a new generation of stars (Fall & Efstathiou 1980). This captured material roughly follows an exponential light profile, given the observed light profiles of disc galaxies. However, it may contribute to some anti-truncated discs by disproportionately adding material to the pre-existing disc at either large or small radii, thereby producing a double-exponential disc light profile (Kazantzidis et al. 2009). It is noted that other mechanisms may also be capable of doing this, such as bars which can enhance the, or arguably create an additional, inner disc (Bureau & Athanassoula 2005). Plus, the cluster environment appears to reduce the stellar disc scalelengths (Gutiérrez et al. 2004) and may cause disc truncations (van der Kruit & Searle 1981). Given the smaller stellar masses of the dust-poor S0 galaxies relative to the dust-rich S0 galaxies (Fig. 6), they will naturally have smaller disc sizes. It might be insightful to check for disc truncations and anti-truncations among the dust-poor and dust-rich S0 galaxies. Such an investigation will be left for elsewhere.

Blue ETGs and star formation
Studies of dust in ETGs (e.g. Smith et al. 2012) will benefit from a division of not just E versus S0 galaxies but dust-poor versus dustrich S0 galaxies. This division is likely related to the discovery of a population of blue spheroids/ETGs (Ellis et al. 2005;Driver et al. 2007), half of which have moderate to high star-formation rates (Kannappan et al. 2009;Schawinski et al. 2009;Mahajan et al. 2018). The low-mass blue ETGs have already been called out as major merger remnants fading onto the red sequence (Kannappan et al. 2009). As the star-formation in these wet-merger-built systems ceases, their emission lines first disappear while they remain blue before they become dusty red ETGs, like NGC 1316. Therefore, it is not simply a matter of star-forming versus nonstarforming ETGs, or blue versus red ETGs: both blue ETGs and dust-rich red ETGs are expected to be offset from the dust-poor S0 galaxies that did not experience the continuation of wet mergers that built-up the dust-rich S0 galaxies.
As shown here, a deeper insight into a galaxy's evolution can come from knowledge of its morphological type (E, ES, S0), M bh /M * ,sph and M bh /M * ,gal ratio, and visible signs of dust. For instance, Schawinski et al. (2009) found that 5 per cent of their ETG sample is blue, reflecting current/recent star formation (in the absence of substantial dust reddening). By looking at the dust rather than the colour, it is found here that roughly half of the S0 galaxies experienced a major wet merger. This is a ten-fold increase in systems that experienced star formation from a major wet merger, larger in part because it does not only capture systems currently/recently undergoing star formation and in part because hidden, dust-reddened star-forming systems are captured. An additional reason is evident when comparing with Driver et al. (2006), which rewrote the notion that all ETGs are 'red and dead'. They found a one-third blue fraction of ETGs. The difference is because Driver et al. (2006) used a sample of galaxies extending >3 mag fainter than the L * or brighter sample of Schawinski et al. (2009). In descending to these fainter luminosities, it is noted that although dwarf spiral galaxies are rare, some dwarf S0 galaxies do contain weak spiral structures (e.g. Jerjen et al. 2000;Barazza et al. 2002;. There is also a population of blue compact dwarf (BCD) galaxies, comprised of an old underlying disc coloured blue by accretion-induced star formation (e.g. Bekki 2008;Amorín et al. 2009;Ju et al. 2022).  introduced the notion of a blue and red sequence in the M bh -M * ,sph and M bh -M * ,gal diagram, with the LTGs defining a steep relation and the ensemble of ETGs defining a near-linear relation. However, it is now understood that the red sequence was artificial. With dusty S0, dust-poor S0, and E galaxies all following a steep relation, one can add a third parameter (replacing morphology) to the M bh -M * ,sph diagram to show the starforming galaxies, which includes not only S galaxies but some of the younger S0 merger products like NGC 5128. A galaxy's star formation rate can be estimated in many ways. One is from the blackbody glow at infrared wavelengths of the heated dust. Terrazas et al. (2016) used this approach, and interpreted the trend in the M bh -M * ,gal diagram as strong evidence of AGN feedback, such that a low specific black hole mass, i.e., lower M bh /M * , ratio, was less capable of quenching star formation or, conversely, that higher M bh /M * ,gal ratios in (some of) the ETGs were a signature of AGN suppression of star formation. However, the interpretation in the current paper is that the systems with higher star formation rates primarily have these because of external refuelling events dictating their morphology and shaping their distribution in the M bh -M * ,gal diagram.
A variant of this theme for visualising the blue/red sequence of  was used by Dullo et al. (2020, their Fig. 3), who showed the NUV-[3.6] colour as the third parameter. Young et al. (2014) had previously used the NUV-K colour for the ATLAS 3D galaxies to explore how it relates to their cold gas content. While Young et al. (2014) concluded that S0 galaxies are quenched S galaxies that underwent bulge growth due to cold gas accretion and modest star formation in some, it seems plausible that the dusty S0 galaxies studied by Young et al. (2014) are instead the result of an S0-building wet merger involving at least one S galaxy.
Counter-rotating gas discs, counter-rotating stellar discs, and polar discs (e.g., Sil'chenko & Afanasiev 2004) have long been recognised as signatures of accretion or mergers (e.g., Bertola et al. 1992;Kuijken et al. 1996). Past events can also reveal themselves as chemically distinct features (e.g., Sil'chenko 2000). Regarding the counter-rotating inner dust disc of NGC 3032 (not in our sample), Young et al. (2008) suggest that the "molecular gas was captured through cold accretion from the intergalactic medium or in an interaction or a minor merger with a gas-rich neighbor". They add that, "Perhaps it is even a remnant of a major merger which formed the present galaxy. The high degree of regularity in the gas kinematics and stellar morphology suggests that this event did not occur recently." Several galaxies in our sample appear to fit this bill -albeit with co-rotating discs, which are a more probable occurrence -such as NGC 4459, a likely merger with a high bulge-to-total stellar mass ratio of ∼0.6, and NGC 4526, which was also studied by Young et al. (2008).

ALMA's window
Long-baseline submillimetre-to-radio interferometry currently enables better spatial resolution than single aperture mirrors, yielding measurements of smaller and/or more distant black holes (e.g. Combes et al. 2019;Nguyen et al. 2021;Smith et al. 2021). Such measurements do, however, require specific conditions, such as a cool, stable gas disc or ring that does not dominate the mass budget within the sampled volume around the black hole. This has proved most successful regarding maser emission, with some two dozen and counting black hole masses measured this way (e.g. Miyoshi et al. 1995;Kuo et al. 2020, and references therein). Galaxies with such favourable conditions 28 are overwhelmingly spiral galaxies, with a few dusty S0 mergers (e.g., NGC 1194 and NGC 2960) also in the mix. The spiral galaxies define steep M bh -M * relations Davis et al. 2019a, plus Fig. 4 and 6), with only the most massive overlapping with, and thus appearing to conform with, the near-linear M bh -M * ,sph relation dominated by restricted samples of ETGs (no 'over-massive' black holes, no obvious dusty mergers, not many low-mass dust-poor S0s which act to steepen the slope. This does not mean that the bulk of the maser sample does not follow black hole scaling relations but instead reveals the inadequacy and limitation of the old near-linear relation. The tendency for the more massive S0 galaxies to contain visible signs of dust (Fig. 3) meshes well with the observation that massive S0 galaxies tend to have carbon-monoxide (CO) discs (Alatalo et al. 2013). Given the Atacama Large Millimeter/submillimeter Array's (ALMA's) ability to detect the emission from nuclear CO discs, ALMA's window into the black hole scaling diagrams may not offer a full view of the black hole landscape. For instance, if the occurrence of nuclear molecular discs predominantly coincides with the presence of visible dust and cold gas on large scales, then observing programmes to improve the black hole scaling relations may be limited to dusty galaxies, thereby offering a somewhat limited view and potentially curtailing the interpretation of the data and the scaling relations. For example, the dust-poor S0 galaxies and the (merger-built) elliptical galaxies, which occupy the lowerleft and upper-right, respectively, of the distribution in the M bh -M * ,gal diagram ( Fig. 2 and 4), might be relatively sparsely sampled relative to the subset of dusty S0 and S galaxies. That is, ob-serving campaigns with ALMA might be restricted to sampling a limited/biased distribution of systems across the Jeans-Lundmark-Hubble galaxy sequence (Jeans 1919;Lundmark 1925;Hubble 1926;Lundmark 1927;Jeans 1928;Graham 2019, and references therein). Although, it remains to be seen how many types of galaxy ALMA will be able to properly sample.
Moreover, coupled with an awareness of the morphologydependent black hole scaling relations, submillimetre-to-radio interferometers are not only refining previous black hole mass measurements and bolstering statistics with increased numbers, but they should result in further insight into the unfolding story of coevolution via mergers, fuelling (star formation and AGN), and galactic speciation. Studies of CO and other molecules, such as hydrogen cyanide (HCN), in the discs of high-z galaxies offer exciting pathways to witness further the growth of black holes and the metamorphosis of their host galaxies (e.g. Riechers 2007;Kakkad et al. 2017;Farina et al. 2022;Tripodi et al. 2022, and references therein). Awareness of the morphology-dependent black hole scaling relations will be necessary for the comparison with local galaxies and interpretation.

FUTURE WORK
The discovery of interlaced M bh -M * ,sph relations (from dust-poor S0 galaxies, S galaxies, dusty S0 galaxies to E galaxies) invites further quantitative probing. The following list mentions many potential research endeavours which could be pursued to help shore up, refine, or modify the new picture of galaxy/black hole coevolution involving 'punctuated equilibrium' jumps through the different galaxy types. There may additionally be a kind of 'gradualism', or 'secular evolution', along the relations for the different morphological types, e.g., Sd toward Sa and M bh /M * ,sph growth among the dusty S0 galaxies if the accreted gas from the merger grows the black hole at a faster rate than star formation (Colberg & Di Matteo 2008;Bonoli et al. 2009;Dubois et al. 2012;Seymour et al. 2012).
The following list is also intended to help indicate the breadth of galaxy research pertaining to, and stemming from, the black hole scaling relations.
• The location of a disc galaxy in the M bh -M * ,sph scaling diagrams could be explored as a function of the (cold gas)-to-stellar mass ratio, normalised to the stellar mass: atomic HI or molecular CO and H 2 gas.
• Consider the specific dust mass as a third parameter. Recognition of the dusty S0 galaxies as something of an intermediatestellar-mass population between dust-poor S0 galaxies and E galaxies may bring crucial insight for studies trying to connect dust mass with other properties of ETGs (Smith et al. 2012).
• Check on the presence of hot X-ray gas. In particular, are the low-mass S0s located within such an environment, acting to freeze them in time, modulo cluster-related tidal effects snapping at their heels and eating away at them?
• The dominant gas-removal mechanism for many individual galaxies could be explored, as was carefully done for the dwarf galaxy DDO 113 by Garling et al. (2020).
• The presence of truncated or anti-truncated stellar discs could be explored to see if they predominantly occur in a particular part of the diagram.
• The star formation history could be examined (e.g. Chilingarian et al. 2007;Weisz et al. 2011), adding this information as a (coloured) third parameter in the scaling diagram. Another option includes showing the amount of Hα luminosity from star formation.
• Beyond the M bh -M * ,gal , M bh -M * ,sph , and M bh -R e,sph diagrams seen here, it may be insightful to explore the location of visibly dusty (wet merger)-built S0 galaxies and dust-poor S0 galaxies in the M bh -σ, M bh -n, and M bh -ρ diagrams (Merritt 2000;Saglia et al. 2016;Sahu et al. 2022a).
• Explore if the mass scaling relation between nuclear star clusters and their host spheroid, discovered two decades ago (Balcells et al. 2003;, also displays a dependence on galaxy morphology. This may reveal insight into their coevolution. • Investigate whether the (nuclear star cluster mass)-(black hole mass) relation, first presented at a conference in China in August of 2014 (Graham 2016) and more recently refined by Graham (2020), has any dependence on the host galaxy properties.
• Pursue the interaction and accretion history of all the spiral galaxies in the sample.
• Investigate the abundance of bars (and nuclear bars) in dustpoor and dust-rich S0 galaxies, and S galaxies, and also along their M bh -M * ,sph sequences.
• The S0 galaxies built from gas-rich mergers (of, say, an S0 and an S galaxy or two S galaxies) may display evidence of this merger in their globular cluster systems (GCSs). This is evident in the Milky Way, where detailed information has revealed at least three past mergers (Kruijssen et al. 2019), and it may be evident in the dusty S0 galaxies as bimodal and trimodal colour distributions of globular clusters (Forbes et al. 1997;Zepf & Ashman 1999;Brodie & Strader 2006). In contrast, the dust-poor S0 galaxies may have a more simple GCS. Building on Pota et al. (2013) and González-Lópezlira et al. (2022), a programme to measure the colour histograms of the GCSs of galaxies with directly measured black masses may be insightful, or even comparing the histograms in just the dust-poor versus dust-rich S0 galaxies could be telling.
• Improved predictions can now be made about where intermediate-mass black holes (IMBHs) may reside. Given the near-absence of dwarf S galaxies, the super-quadratic scaling relation for the dust-poor S0 galaxies appears to offer the most promise for predicting the masses of IMBHs in dwarf galaxies, perhaps even providing a direct bridge to the lower end of the IMBH range 10 2 < M bh /M < 10 5 (Abbott et al. 2022).
• Check if simulations focussed on S galaxies reproduce the steep M bh -M * ,gal relation and the steep M bh -V rot and M bh -M dark−matter relations (Davis et al. 2019b, and references therein).

ACKNOWLEDGEMENTS
The author is grateful for discussions with Anita Pappas and David Brown, Billanook College. Part of this research was conducted within the Australian Research Council's Centre of Excellence for Gravitational Wave Discovery (OzGrav) through project number CE170100004. This work has used the NASA/IPAC Infrared Science Archive (IRSA) and the NASA/IPAC Extragalactic Database (NED), funded by NASA and operated by the California Institute of Technology. Based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the Mikulski Archive for Space Telescopes and the Hubble Legacy Archive. This research has also used the NASA/SAO Astrophysics Data System Bibliographic Services and the Rstan package available at https: //mc-stan.org/.

DATA AVAILABILITY
The imaging data underlying this article are available in the NASA/IPAC Infrared Science Archive. The derived spheroid and galaxy stellar masses are tabulated in . wet mergers, have even larger spheroid stellar masses for the same black hole mass. These (galaxy morphology)-dependent M bh -M * ,sph relations imply a gas dependence in the scaling diagrams that has yet to receive much attention and may, in part, be controlled by the environment (e.g. Shin et al. 2012;Yang et al. 2018).
Indeed, some galaxy environments are less conducive to wet mergers than others. Clusters, for example, may 'dry out' galaxies, removing their gas. Furthermore, the high speeds of galaxies in clusters reduce the chances of galaxy collisions. Collectively, this could leave a population of dust-poor S0 galaxies unlikely to acquire new gas and evolve. Unless the BCG assimilates them, they would, in a sense, be frozen in time, albeit containing an ageing stellar population.
Hot haloes of X-ray emitting gas have likely halted star formation and quasar activity in elliptical galaxies (Benson et al. 2003;Hickox et al. 2009;, and references therein). The NGC 5813 subgroup associated with the N5846 Group in the sprawling Virgo cluster is one such example, with a 'Benson Burner' at the heart of the S0 galaxy NGC 5813 keeping the gas hot (Randall et al. 2015). This mechanism may also contribute to the tendency for some larger spiral galaxies to be deficient in atomic hydrogen (e.g. Valluri & Jog 1991). In addition to thermal evaporation (Cowie & Songaila 1977), cluster-sized X-ray haloes can lead to ram-pressure stripping of cold HI gas (Gunn & Gott 1972;Davies & Lewis 1973;Giovanelli & Haynes 1985;Yagi et al. 2010;Wang et al. 2021). Group-sized X-ray bright haloes can also remove gas from a galaxy (Forman et al. 1979) and thus strangle the supply of cold gas, which may have otherwise arisen from the cooling of hot gas (Kawata & Mulchaey 2008;Peng et al. 2015). Coldgas removal mechanisms appear to also operate in compact groups and some loose groups (e.g. Verdes-Montenegro et al. 2001;Omar & Dwarakanath 2005;Sengupta et al. 2007;Kolcu et al. 2022). The affected galaxies are observed to have reduced HI content and reduced HI disc sizes. Reduced Hα sizes have also been observed (Vaughan et al. 2020), and reduced stellar disc sizes in galaxy clusters were discovered two decades ago (Gutiérrez et al. 2004), likely also a result of gravitational tides (Roche 1850) from repeated fleeting encounters with each other (Haynes et al. 1990;Moore et al. 1998;Wang et al. 2022), thereby building up the intracluster light. Table A1 provides the S0 galaxy environments gleaned from the literature. Much of the group membership can be found in Garcia (1993), later refined by (Osmond & Ponman 2004;Makarov & Karachentsev 2011). No apparent trends have been found between the dust-rich and dust-poor S0 galaxies. This paper has been typeset from a T E X/L A T E X file prepared by the author. Column 1: Galaxy name. Column 2: Is there dust visible in the HST optical image: (Y)es plenty, (y)es but not much, only a small (n)uclear dust disc/ring, or (N)othing obvious. Column 3: Environment. Common knowledge supplemented primarily by the Lyon Groups of Galaxies (LGG) catalogue (Garcia 1993) and the group catalogue of Makarov & Karachentsev (2011), accessed via the Extragalactic Distance Database (http://edd.ifa.hawaii.edu / Tully et al. 2009).