# The Carnegie-Irvine Galaxy Survey<EMAIL_ADDRESS>Bulges in Stellar Mass-based Scaling
Relations

Hua Gao (高桦) Kavli Institute for Astronomy and Astrophysics, Peking
University, Beijing 100871, China  Kavli Institute for the Physics and
Mathematics of the Universe (WPI), The University of Tokyo Institutes for
Advanced Study, The University of Tokyo, Kashiwa, Chiba 277-8583, Japan
Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu
HI 96822, USA  Luis C. Ho Kavli Institute for Astronomy and Astrophysics,
Peking University, Beijing 100871, China  Department of Astronomy, School of
Physics, Peking University, Beijing 100871, China  Zhao-Yu Li Department of
Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China Hua Gao
<EMAIL_ADDRESS>
###### Abstract

We measure optical colors for the bulges of 312 disk galaxies from the
Carnegie-Irvine Galaxy Survey and convert their previously available $R$-band
structural parameters to stellar mass parameters. We also measure their
average stellar mass surface density in the central 1 kpc ($\Sigma_{1}$).
Comparing the mass-based Kormendy relation with the original one based on
flux, we find that the majority of the classifications into classical and
pseudo bulges, as well as their overall statistical properties, remain
essentially unchanged. While the bulge type classifications of the Kormendy
relation are robust against stellar population effects, the mass-based
classification criteria do produce better agreement between bulge structural
properties and their stellar populations. Moreover, the mass-based Kormendy
relation reveals a population of ultra-dense bulges akin to high-$z$ compact
early-type galaxies, which are otherwise hidden in the original Kormendy
relation. These bulges are probably relics of spheroids assembled in the early
Universe, although for some we cannot rule out some contribution from secular
growth. We confirm previous studies that $\Sigma_{1}$ correlates well with
bulge surface densities.

Galaxy bulges (578), Galaxy structure (622), Galaxy photometry (611), Disk
galaxies (391), Elliptical galaxies (456), Galaxy evolution (594)

##  1 Introduction

First established in the 1980s (Djorgovski & Davis, 1987; Dressler et al.,
1987), the fundamental plane is a tight relation between observables in size,
luminosity, and kinematics of early-type galaxies that unifies its two earlier
discovered projections, the Faber–Jackson (1976) relation between luminosity
and stellar velocity dispersion and the Kormendy (1977) relation between
effective radius and surafce brightness. Besides its original usage as a
distance indicator, it has provided much insight into the formation and
evolution of early-type galaxies (Dressler et al., 1987; Faber et al., 1987;
Djorgovski et al., 1988; Lynden-Bell et al., 1988; D’Onofrio et al., 2013,
2017). Rooted in the virial theorem, the fundamental plane and its projections
have achieved notable success because they provide an empirical foundation to
distinguish stellar systems of different physical nature. For instance, the
fundamental plane has helped establish that spheroidal galaxies are physically
distinct from giant ellipticals (Kormendy, 1985, 1987; Bender et al., 1992;
Binggeli, 1994), and that a dichotomy exists between slow-rotator and fast-
rotator ellipticals (Faber et al., 1997). Recently, the Kormendy relation has
played an increasingly important role as a diagnostic tool to distinguish
classical bulges from pseudo bulges (Gadotti, 2009; Neumann et al., 2017; Gao
et al., 2020). Classical bulges share similar observational properties with
ellipticals (Kormendy & Kennicutt, 2004; Fisher & Drory, 2008; Gao et al.,
2020) and therefore are often regarded as end products of violent processes
such as galaxy mergers and coalescence of stellar clumps in high-$z$ disks
(e.g., Toomre, 1977; Bournaud, 2016; Tacchella et al., 2019). Pseudo bulges,
by contrast, are basically miniature disks masquerading as bulges in the
center of a galaxy (Kormendy, 1993; Kormendy & Kennicutt, 2004; Fisher &
Drory, 2008; Fisher et al., 2009; Gao et al., 2020). They originate from
redistributed disk material by secular processes driven by non-axisymmetric
mass distributions such as bars and spirals (Combes & Sanders, 1981; Kormendy,
1981, 1982; Pfenniger & Norman, 1990; Sellwood & Wilkinson, 1993; Kormendy &
Kennicutt, 2004; Sellwood, 2014).

Our previous study showed that classical and pseudo bulges selected by the
Kormendy relation have physical properties consistent with what we expect from
their distinctive formation paths (Gao et al., 2020). We performed
multicomponent decomposition of $R$-band images of galaxies from the Carnegie-
Irvine Galaxy Survey (CGS; Ho et al., 2011) to measure accurate bulge
parameters, and presented the statistics of classical and pseudo bulges based
on classifications made using the Kormendy relation. Pseudo bulges generally
are less concentrated and less prominent, and they have characteristically
fainter surface brightness than classical bulges. Classical bulges exhibit an
apparent ellipticity distribution skewed toward lower values compared with
pseudo bulges, hinting that classical bulges are intrinsically rounder. In
contrast with previous studies, we found that the distribution of the Sérsic
(1968) index of bulges is not bimodal, and that the two kinds of bulges cannot
be well-separated by the widely adopted criterion of $n=2$ (e.g., Fisher &
Drory, 2008, 2016). We also found many fewer pseudo bulges in early-type disk
galaxies, which helps alleviate the tension presented by the overbundance of
massive, pure disk galaxies in a hierarchically clustered, $\Lambda$CDM
Universe (e.g., Kormendy & Fisher, 2008; Kormendy et al., 2010; Peebles &
Nusser, 2010).

Despite the above success, we note that relying on single-band measurements
has potential drawbacks. The influence of stellar population must be explored.
Systematic variations in mass-to-light ratio ($M/L$) have been reported to
cause the tilt of the fundamental plane with respect to the virial plane
(e.g., Djorgovski & Davis, 1987; Renzini & Ciotti, 1993; Kroupa, 2001;
Cappellari et al., 2012). Cappellari et al. (2013a) found that replacing
luminosity with dynamical mass brought the relation closer to the virial plane
and reduced the scatter. This is also true for the case of stellar mass (Hyde
& Bernardi, 2009). It is well known that pseudo bulges have younger stellar
populations than classical bulges (e.g., Fisher et al., 2009; Sánchez-
Blázquez, 2016), and some classical bulges may also have residual star
formation (Luo et al., 2020). In light of these considerations, we expect that
replacing surface brightness with stellar mass surface density in the Kormendy
relation would provide a more fundamental, physical parameter space with which
to investigate bulges.

The structural properties of galaxies are closely related to their level of
star formation activity. Early-type galaxies, being centrally concentrated,
are also predominantly quiescent in the local Universe. Various structural
parameters have been explored as discriminators of star-forming and passive
galaxies, including Sérsic index, surface stellar mass density, and stellar
mass divided by the half-light radius (e.g., Kauffmann et al., 2003, 2006;
Bell, 2008; Franx et al., 2008; Wuyts et al., 2011; Bell et al., 2012).
Inspired by these studies, Cheung et al. (2012) first introduced the parameter
$\Sigma_{1}$, the average stellar mass surface density in the central 1 kpc,
to trace the star formation status of galaxies, comparing its efficacy
relative to other structural parameters. Subsequently, $\Sigma_{1}$ enjoyed
wide application, especially in the context of scaling relations to study
galaxy quenching and structural transformation (Fang et al., 2013; Woo &
Ellison, 2019; Woo et al., 2017). By design, this parameter closely traces
bulge properties, as typical bulge size is $\sim 1\,\mathrm{kpc}$ (Fisher &
Drory, 2008; Gao et al., 2020). The increase of $\Sigma_{1}$ as galaxies
evolve from the blue cloud to the red sequence is often associated with
compaction processes such as violent disk instabilities and mergers (Zolotov
et al., 2015; Tacchella et al., 2016), during which bulges also grow.
Therefore, it is quite natural to extend its application to study bulges (Luo
et al., 2020; Yesuf et al., 2020). The classifications of bulge type based on
$\Sigma_{1}$ agree with those based on the Kormendy relation. Most
importantly, $\Sigma_{1}$ is simple and economical to measure compared with
detailed bulge structural parameters that require adequate spatial resolution
and time-consuming, often highly uncertain bulge-to-disk decompositions.
Nevertheless, some information would be lost from averaging the central mass
distribution, and contamination from other structural components is
inevitable.

The success of working with stellar mass in studying galaxy evolution
motivates us to extend the measurements of the $R$-band bulge structural
parameters of 320 CGS disk galaxies (Gao et al., 2019). This contribution
focuses on measuring the stellar mass parameters of these objects based on
their optical colors. We then use these measurements to construct a revised
mass-based Kormendy relation of bulges and ellipticals and compare
classifications of bulge types with those obtained with the original Kormendy
relation (Gao et al., 2020). We will also evaluate the feasibility of using
$\Sigma_{1}$ as an additional diagnostic to study bulge and galaxy evolution.

Bulge colors can be measured through multiband bulge-to-disk decompositions,
as conducted, for instance, in the MegaMorph project (e.g., Bamford et al.,
2011; Vika et al., 2014; Kennedy et al., 2016). Their approach can increase
the signal-to-noise ratio of the data and self-consistently account for the
flux contribution from other components. Due to the large sample size, their
surface brightness models are relatively simple for practical reasons. The
multicomponent models of the CGS sample are much more detailed and complicated
than those adopted in large data sets (Gao et al., 2018, 2019), rendering it
extremely computationally expensive and unfeasible to perform multiband
fitting for our study. Moreover, MegaMorph’s assumption that the structural
parameters vary as polynomial functions of wavelength lacks physical basis.
Instead, we adopt the pragmatic strategy of estimating the bulge color using
aperture photometry in different bands, defining the aperture size to include
only the region where the bulge surface brightness dominates. Although we do
not remove the disk rigorously, its contamination can be minimized by proper
choice of aperture size.

This study uses the same sample as that presented in Gao et al. (2019). Sample
selection and data quality are described in Section 2. Section 3 discusses the
methodology to measure bulge color and stellar mass. Results and discussions
are given in Sections 4 and 5. We summarize the paper in Section 6.

##  2 Sample and Data

Our sample, introduced in Gao et al. (2019, 2020), is a subset of CGS (Ho et
al. 2011), which comprises bright ($B_{T}\leq 12.9\,\mathrm{mag}$) galaxies in
the southern hemisphere without any selection on morphology, size, or
environment. High-quality images in $B$, $V$, $R$, and $I$, along with
isophotal parameters, were presented in Li et al. (2011). The images have a
field-of-view of $8\farcm 9\times 8\farcm 9$, a median seeing of $\sim
1\arcsec$, and a median surface brightness sensitivity of 26.9 mag arcsec-2 in
$B$ and 26.4 mag arcsec-2 in $R$. The median luminosity distance of the sample
is 26 Mpc, almost all galaxies are within 80 Mpc (Gao et al., 2019), and the
median seeing ($1\arcsec$) of R-band images corresponds to $\sim
0.1\,\mathrm{kpc}$ and $\sim 0.4\,\mathrm{kpc}$ at these distances,
respectively. Note that the typical bulge half-light radius is $\sim$1 kpc,
which is well resolved in the CGS images. This study makes use only of the $B$
and $R$ image products.

In previous papers of this series, we performed two-dimensional bulge-to-disk
decompositions of the disk (lenticular and spiral) galaxies of the CGS sample
in the $R$ band (Gao et al., 2018, 2019) and studied the statistical
properties of classical and pseudo bulges using the measurements (Gao et al.,
2020). The bulge-to-disk decompositions were performed on a subset of CGS
galaxies selected by morphological type index $-3\leq T\leq 9.5$ and
inclination angle $i\leq 70\arcdeg$. We complemented the sample with a handful
of misclassified ellipticals and removed objects whose decomposition was
unsuccessful, arriving at a final sample of 320 galaxies. The decomposition
strategy, detailed in Section 3 of Gao et al. (2019), considered nuclei,
nuclear/inner lenses, inner rings, bars, and disk breaks. Our modeling
strategy treated nuclear rings and nuclear bars as part of the bulge
component, while other features such as spiral arms, outer lenses, and outer
rings were omitted from the fits because they are not crucial for accurate
bulge measurements according to the experiments in Gao & Ho (2017). Active
galaxy nuclei and nuclear star clusters are modeled as point sources, in order
to remove their contamination to other galaxy components. The error budget of
the bulge parameters includes the uncertainties from sky measurements and
model assumptions.

This study builds on the 320 disk galaxies with accurate bulge structural
parameters. We succeed in deriving optical $B-R$ colors for the bulges of all
but eight of the galaxies. We measure $\Sigma_{1}$ for the 312 disk galaxies,
as well as an additional 83 CGS ellipticals serving as a reference sample. The
structural parameters of the ellipticals were already presented in Gao et al.
(2020).

##  3 Methodology

Based on the two-dimensional best-fit models of the galaxies from Gao et al.
(2019), to estimate bulge colors we define a “bulge aperture” as the set of
image pixels where the flux of the bulge dominates over that of other
components. In order to secure a reasonably large aperture to estimate bulge
colors for most of the galaxies, we select pixels having a bulge flux
threshold at least 25% brighter than all the other components combined. Next,
we remove pixels that are affected by dust lanes and foreground sources. Gao
et al. (2019) already paid careful attention to mask manually central dust
lanes when performing bulge-to-disk decompositions. However, we do not expect
that the dust lanes were perfectly removed. Therefore, we take extra
precaution to further minimize dust extinction effects by performing
$1.5\sigma$ clipping of pixels that show irregular colors compared with other
pixels at the same galactic radius. We further remove pixels whose flux
contains non-zero contribution from active galactic nuclei or nuclear star
clusters, if present, in order to minimize their impact on estimating bulge
colors. Finally, we remove pixels inside 3 times the seeing disk; for galaxies
with small bulges, we remove pixels inside one seeing disk to preserve
sufficient pixels to estimate their colors. Such galaxies are mostly late-type
spirals with pseudo bulges, whose centers are often plagued by severe dust
extinction, which further exacerbates the situation. Eventually, we omit eight
galaxies from the sample of Gao et al. (2019) because there are not enough
pixels left to ensure robust color measurement.

We calculate an integrated $B-R$ color over the above-selected pixels in the
$B$ and $R$ images and treat it as the bulge color. The $R$-band stellar mass-
to-light ratio follows from

$\log\left(M/L\right)_{R}=0.683(B-R)-0.523,$ (1)

with a typical error of $\sim 0.1\,\mathrm{dex}$ (Bell et al., 2003). We then
derive stellar mass and average stellar mass surface density within the
effective radius $r_{e}$ ($\langle\Sigma_{e}\rangle$) of the bulges. We
perform the same calculations for the ellipticals, which already have
available colors (Li et al., 2011) and structural parameters (Gao et al.,
2020). The uncertainties of $B-R$ include photometric errors in both bands,
uncertainties in defining bulge aperture introduced by the bulge structural
parameters, and dispersion of colors in the bulge aperture caused by possible
radial gradients. The latter may reflect intrinsic color gradients and/or
contamination from other structural components (e.g., disk). It is reassuring
that for most cases the photometric errors dominate the error budget. We
propagate the uncertainties of $B-R$, bulge magnitudes, and Equation (1) to
the uncertainties of the stellar mass parameters. The mean uncertainties of
bulge stellar masses and average effective stellar mass surface density are
0.19 dex and 0.18 dex, respectively (Table 1).

Table 1: Colors and Stellar Mass Parameters

Name | $B-R$ | $\log M_{\star}$ | $\log\langle\Sigma_{e}\rangle$ | $\log\Sigma_{1}$
---|---|---|---|---
| (mag) | ($M_{\sun}$) | ($M_{\sun}\,\mathrm{kpc}^{-2}$) | ($M_{\sun}\,\mathrm{kpc}^{-2}$)
(1) | (2) | (3) | (4) | (5)
ESO 027–G001 | 0.89±0.23 | 8.66±0.21 | 9.60±0.20 | 8.87±0.19
ESO 121–G026 | 1.47±0.10 | 9.92±0.14 | 9.82±0.13 | 9.63±0.12
ESO 137–G034 | 1.39±0.03 | 10.40±0.12 | 9.68±0.11 | 9.77±0.10
ESO 138–G010 | 1.31±0.24 | 9.29±0.30 | 8.78±0.27 | 8.94±0.19
ESO 185–G054 | 1.44±0.02 | 11.69±0.12 | 8.55±0.13 | 10.13±0.10
ESO 186–G062 | 1.20±0.26 | 9.51±0.26 | 8.19±0.27 | 8.60±0.19
ESO 213–G011 | 1.38±0.09 | 9.74±0.16 | 8.96±0.15 | 9.28±0.12
ESO 221–G026 | 1.05±0.23 | 9.93±0.27 | 9.64±0.25 | 9.44±0.19
ESO 221–G032 | 0.97±0.06 | 9.09±0.16 | 9.82±0.14 | 9.06±0.11
ESO 269–G057 | 1.48±0.23 | 10.58±0.21 | 10.09±0.20 | 10.03±0.19
ESO 271–G010 | 0.87±0.06 | 8.37±0.13 | 8.31±0.12 | 8.60±0.11

Note. — Col. (1): Galaxy name. Col. (2): Color of bulge or elliptical; data
for ellipticals are from Li et al. (2011). Col. (3): Stellar mass of bulge or
elliptical. Col. (4): Mean stellar mass surface density within the effective
radius of bulge or elliptical. Col. (5): Mean stellar mass surface density
within a radius of 1 kpc.

(Table 1 is published in its entirety in machine-readable format. A portion is
shown here for guidance regarding its form and content.)

Measurements of $\Sigma_{1}$ of disk and elliptical galaxies are
straightforward. Given the isophotal analysis provided by Li et al. (2011), we
integrate the $B$-band and $R$-band surface brightness profiles from the
galaxy center outward to calculate the $B-R$ color and $\Sigma_{1}$, using
Equation (1) to estimate $M/L$. As with the measurements of stellar mass
parameters, we propagate the uncertainties from the photometric errors,
interpolation errors in surface brightness profiles, and errors in Equation
(1) into the uncertainties of $\Sigma_{1}$. The mean uncertainty of
$\Sigma_{1}$ is 0.15 dex (Table 1). Note that we exclude the hosts of known
active galactic nuclei from our sample when measuring $\Sigma_{1}$.

Figure 1: Color–magnitude diagram of bulges. From top to bottom, the symbols
are color-coded according to the bulge-to-total ratio $B/T$, Sérsic index $n$,
and residual average effective surface brightness
$\Delta\langle\mu_{e}\rangle$. The left panels show the original data, and the
right panels are the LOESS-smoothed version. The symbols surrounded by large
open squares are ultra-dense bulges (Section 4.2.3). The stars in the left
panels mark three outliers (NGC 4947, NGC 5188, and NGC 7689), which are
discussed in the main text. The black dashed line in the bottom panel is the
demarcation between classical and pseudo bulges classified using the Kormendy
relation, obtained, for illustration purposes only, by fitting the data with
$\Delta\langle\mu_{e}\rangle\approx 0$ (yellowish symbols).  Figure 2: The
Kormendy relation of ellipticals and bulges (labeled E and B, respectively),
shown in the traditional flux-based units in the $R$ band (left) and stellar
mass units (right). The dashed and dotted lines represent the best-fit
relation and its $3\sigma$ boundary of ellipticals (filled black squares). The
filled circles in the left panel are bulges color-coded based on their $B-R$
color. In the right panel, the filled blue circles and open red circles are
pseudo bulges (labeled PB) and classical bulges (labeled CB), respectively,
classified using the flux-based Kormendy relation; the filled magenta circles
are ultra-dense bulges (labeled UDB; Section 4.2.3); the smaller
greenish/blueish squares are high-$z$ ETGs color-coded according to their
redshift and LOESS-smoothed. Representative error bars are given in the lower-
left corner.

##  4 Results

###  4.1 Strategies for Classifying Bulges

Figure 1 presents the color–magnitude diagram of bulges, color-coded,
respectively, using the bulge-to-total flux ratio ($B/T$), Sérsic index ($n$),
and residual average effective surface brightness relative to the lower
$3\sigma$ boundary of the Kormendy relation ($\Delta\langle\mu_{e}\rangle$;
Figure 2). The structural parameters are taken from Gao et al. (2020) and were
measured in the $R$ band. To minimize the effects of object overlay and aid
visual inspection of the average trends, we smoothed the data by locally
weighted regression (LOESS; Cleveland & Devlin, 1988).111We adopt the
implementation of the two-dimensional LOESS method by Cappellari et al.
(2013b), who kindly provide the code in http://purl.org/cappellari/idl. Three
extremely red ($B-R>2$ mag) but small bulges (NGC 4947, NGC 5188, and NGC
7689) are readily recognized as outliers to the majority (stars in Figure 1).
We are confident that NGC 5188 has an unreliable color, because its center is
complicated by intricate dust features that are beyond the capacity of manual
masking and pixel clipping. NGC 4947 has well-defined dust lanes that were
masked during measuring its bulge color, but we cannot exclude the possibility
that its exceptionally red color is caused by continuous dust extinction.
Nothing particularly suspicious stands out about the bulge color of NGC 7689.
We do not remove these three sources from the analysis, as they do not affect
the interpretation.

Comparison of the three color–magnitude diagrams shows that the Kormendy
relation is superior in classifying bulges compared with single structural
parameters such as $B/T$ or $n$, because only $\Delta\mu_{e}$ presents a
systematic trend from red, luminous systems, characteristic of typical
classical bulges, to blue, faint ones, as expected of pseudo bulges. Notably,
$n$ gives the worst performance, showing no discernible trends with either
bulge color or luminosity. The bulge-to-total ratio does a better job,
although the levels of constant $B/T$ do not arrange bulges into a sequence
that resembles the transition we expect from classical to pseudo bulges. These
results echo the findings of Gao et al. (2020), who conclude that $n$ is not
an effective indicator of bulge properties, whereas the Kormendy relation is
more robust. Pseudo bulges selected according to the Kormendy relation exhibit
a wider range of colors than classical bulges. Indeed, Figure 1 reveals a
population of pseudo bulges with colors as red as classical bulges
($B-R\approx 1.5$ mag). This suggests that incorporating stellar population
information into the Kormendy relation may be able to improve the
classification further. Figure 2 illustrates how bulge colors are distributed
in the Kormendy relation (left panel). We expect that some of the red pseudo
bulges (having large $M/L$) close to the lower $3\sigma$ boundary will be
classified as classical bulges once we replace surface brightness with mass
surface density, and vice versa.

###  4.2 The Mass-based Kormendy Relation

Figure 2 directly contrasts the traditional $R$-band Kormendy relation (Gao et
al., 2020) with our new stellar mass-based version222Unlike Gao et al. (2020),
the effective radius here has been circularized by
$r_{e,c}=r_{e}\sqrt{1-\epsilon}$, with $\epsilon$ the apparent ellipticity of
the best-fit bulge model., which replaces the average effective surface
brightness ($\langle\mu_{e}\rangle$) with the average effective stellar mass
surface density ($\log\langle\Sigma_{e}\rangle$). In addition to CGS
ellipticals and bulges, we also overlay for comparison the sample of massive,
high-redshift ($z\approx 1.5\textrm{--}2.5$) early-type galaxies (ETGs), whose
structural parameters were compiled by Huang et al. (2013) from the literature
(Scarlata et al., 2007; Scoville et al., 2007; Tasca et al., 2009; Conselice
et al., 2011; Damjanov et al., 2011; Grogin et al., 2011; Papovich et al.,
2012; Szomoru et al., 2012). As in Gao et al. (2020), we use the best-fit
relation of the ellipticals to separate classical and pseudo bulges:

$\displaystyle\log(\langle\Sigma_{e}\rangle/M_{\sun}\,\mathrm{kpc}^{-2})$
$\displaystyle=$ $\displaystyle\left(-0.78\pm
0.07\right)\log(r_{e,c}/\mathrm{kpc})$ (2) $\displaystyle+\left(9.50\pm
0.05\right),$

which has a scatter of 0.17 dex in $\log\langle\Sigma_{e}\rangle$. We define
pseudo bulges as those with $\log\langle\Sigma_{e}\rangle$ smaller than the
lower $3\sigma$ boundary of the above relation,

$\log(\langle\Sigma_{e}\rangle/M_{\sun}\,\mathrm{kpc}^{-2})<-0.78\log(r_{e,c}/\mathrm{kpc})+8.99.$
(3)

The rest are classical bulges.

As expected, in Figure 2 we notice bulges classified as one type using the
original Kormendy relation cross over to the other category once we account
for the effects of $M/L$. Among the original sample of 94 pseudo bulges, 15
are reclassified as classical bulges according to the mass-based Kormendy
relation; as for classical bulges, we relabel 12 of 218 as pseudo bulges. The
consistency rate is $91\%$. Thus, as discussed in the next section, the
overall statistics of the two bulge populations are not significantly changed.

####  4.2.1 Better Agreement between Structural and Stellar Population
Properties of Bulges

Bulge classifications based on structural and stellar population properties
often yield inconsistent results. This is already seen in Figure 1 and
described in Section 4.1. Some pseudo bulges appear to have colors comparable
to those of typical classical bulges. Figure 3 gives another view of the bulge
color–magnitude diagram, now with the bulges color-coded by their distance to
the lower $3\sigma$ boundary of the mass-based Kormendy relation
($\Delta\langle\Sigma_{e}\rangle$). Comparison with Figure 1 shows that the
average demarcation line of classical and pseudo bulges tilts counter-
clockwise when the classifications are made using the mass-based relation
(Figure 3). This allows more red bulges to enter the domain of classical
bulges, thereby alleviating the tension between bulge structural and stellar
population properties.

Figure 3: Color–magnitude diagram of bulges. The symbols are color-coded
according to (top) $\Delta\log\langle\Sigma_{e}\rangle$ and (bottom)
$\Delta\log\Sigma_{1}$. The left panels show the original data, and the right
panels are the LOESS-smoothed version. The black dashed lines in the right
panels are the demarcation between classical and pseudo bulges classified
using (top) the mass-based Kormendy relation and (bottom) the
$M_{\star}$–$\Sigma_{1}$ relation (Section 4.3), obtained, for illustration
purposes only, by fitting the data with
$\Delta\log\langle\Sigma_{e}\rangle\approx 0$ and $\Delta\log\Sigma_{1}\approx
0$, respectively (yellowish symbols).

####  4.2.2 Updated Statistical Properties of Bulges

As mentioned in Section 1, based on the classifications made using the
$R$-band Kormendy relation, pseudo bulges generally have less concentrated
(lower $n$) light profiles, are less prominent (smaller $B/T$), and have
fainter average surface brightness (larger $\langle\mu_{e}\rangle$) than
classical bulges. Classical bulges show an apparent ellipticity distribution
skewed toward lower values compared to pseudo bulges, since they are
intrinsically rounder objects. We also found that the Sérsic index
distribution is not bimodal, and that the two kinds of bulges cannot be well
separated at $n\approx 2$. Bulges classified using the Kormendy relation
naturally show well-separated $\langle\mu_{e}\rangle$ distributions for
classical and pseudo bulges. Our decomposition generally yielded less
conspicuous bulges (smaller $B/T$) than previously measured, for both
classical and pseudo bulges, on account of our more detailed and more accurate
treatment of galactic subcomponents (Gao & Ho, 2017).

In light of the new classification criterion presented in this study, Figure 4
reassesses the overall statistical properties of the structural parameters of
classical and pseudo bulges classified using the original Kormendy relation
(dot-dashed histograms) and the new mass-based version (solid histograms). The
revision in classification criteria does not significantly alter the overall
statistics of the two populations, apart from the tendency for the bulge-to-
total mass ratios of the two types to be better segregated than the bulge-to-
total flux ratios, now forming a nearly bimodal distribution. And the two
types now perhaps show a somewhat more noticeable separation in terms of
Sérsic index. Overall, however, the conclusions in Gao et al. (2020) largely
still hold. This reinforces our previous conclusion that the Kormendy relation
is already a robust tool to classify the two populations of bulges. Switching
to stellar mass hardly affects the overall statistics.

Figure 4: Comparison of structural parameters of classical (reddish) and
pseudo (blueish) bulges classified using the original (dot-dashed histograms;
Gao et al. 2020) and the new (solid histograms) mass-based Kormendy relation.
From left to right, top to bottom, the panels display distributions of
$R$-band bulge-to-total flux ratio, bulge-to-total mass ratio, average
effective surface brightness ($\langle\mu_{e}\rangle$), Sérsic index ($n$),
apparent ellipticity ($\epsilon$), and effective radius ($r_{e}$) of the
bulge.  Figure 5: Correlation between $\Sigma_{1}$ and $M_{\star}$ of CGS
ellipticals and disk galaxies. Black filled squares are ellipticals; filled
circles are disk galaxies. The hosts of classical bulges are in red, and the
hosts of pseudo bulges, as selected by Equation (3), are in blue; the hosts of
ultra-dense classical bulges defined by Equation (4) are in magenta. The
contours show the distribution of all galaxies. The red sequence is readily
recognizable but the blue cloud is sparsely sampled. The black dashed line is
the structural valley from Luo et al. (2020). Figure 6: Correlation between
$\Delta\log\Sigma_{1}$ and bulge $\Delta\langle\mu_{e}\rangle$ (left) and
bulge $\Delta\log\langle\Sigma_{e}\rangle$ (right) for disk galaxies. The
Pearson correlation coefficient ($r$) and Spearman’s rank correlation
coefficient ($\rho$) are given in the lower-right corner. Consistency is
defined as the fraction of bulges in the first and third quadrants.

####  4.2.3 Ultra-dense Bulges

We draw attention to a population of classical bulges located above the upper
$3\sigma$ boundary of the best-fit relation of the CGS ellipticals (magenta
filled circles; Figure 2b). The threshold

$\log(\langle\Sigma_{e}\rangle/M_{\sun}\,\mathrm{kpc}^{-2})>-0.78\log(r_{e,c}/\mathrm{kpc})+10.01$
(4)

selects 19 such ultra-dense bulges. This population is more prominent in the
mass-based relation than in the original Kormendy relation, where only one
such case stands out (Figure 2a). Ultra-dense bulges have mass surface
densities higher than those of $z\approx 0$ ellipticals but are comparable to
those of $z\gtrsim 1.5$ ETGs. As shown in Figure 1, these objects are not
necessarily the most prominent (large $B/T$ or bulge luminosity) or the most
concentrated (large $n$) bulges, but they are the reddest at a given bulge
luminosity. As discussed in Section 5.2, ultra-dense bulges may be well-
preserved high-$z$ red nuggets or bulges with an early assembly history
(Graham et al., 2015; Gao et al., 2018).

###  4.3 Correlation between $\Sigma_{1}$ and Bulge Parameters

Luo et al. (2020) investigated the feasibility of using $\Sigma_{1}$ to
classify bulges, and concluded that this parameter can produce bulge
classifications consistent with those derived from the Kormendy relation. They
also stressed the importance of removing the mass dependence in $\Sigma_{1}$
when separating the two bulge populations. Our ability to revisit these issues
is somewhat hampered by the under-representation of low-mass galaxies in CGS,
as well as the overall small-number statistics of the survey, which preclude
us from robustly defining the structural valley and measuring
$\Delta\Sigma_{1}$ for the sample in a self-consistent manner, as shown in
Figure 5333The figure omits a few outliers for better visualization.. For the
sake of convenience, we adopt the formulation of the structural valley from
Luo et al. (2020) and calculate
$\Delta\log\Sigma_{1}=\log\Sigma_{1}+0.275(\log M_{\star})^{2}-6.445\log
M_{\star}+28.059$.

Figure 6 illustrates that $\Delta\log\Sigma_{1}$ correlates well with the
other bulge type indicators, $\Delta\langle\mu_{e}\rangle$ and
$\Delta\log\langle\Sigma_{e}\rangle$, with the latter showing a somewhat
tighter correlation than the former. In both cases, the scatter is larger at
the low-density end. Nevertheless, $\Delta\log\Sigma_{1}$ yields greater than
80% consistency in bulge classification compared to the other two criteria.

We examine how $\Delta\log\Sigma_{1}$ is distributed in the bulge
color–magnitude diagram. The average separation line between classical and
pseudo bulges classified using $\Delta\log\Sigma_{1}$ (Figure 3) is more or
less consistent with that delineated using the mass-based Kormendy relation,
$\Delta\log\langle\Sigma_{e}\rangle$. Although the average trend is similar,
$\Delta\log\Sigma_{1}$ leads to greater tension between the structural and
stellar population properties of bulges. This is evident when comparing the
left panels of Figure 3 in the faint end of bulges: there are more bulges with
positive $\Delta\log\Sigma_{1}$ in the lower panel than those with positive
$\Delta\log\langle\Sigma_{e}\rangle$ in the upper panel, both of which are
outliers to the systematic trend. Namely, there would be more blue and faint
classical bulges if the classifications were based on $\Sigma_{1}$. This is to
be expected. After all, $\Sigma_{1}$ is a galaxy-wide parameter, whereas
$\langle\Sigma_{e}\rangle$ is tailored specifically for the bulge component.

##  5 Discussion

###  5.1 Effective Criteria for Bulge Classification

Neumann et al. (2017) found that classifications made using only the Kormendy
relation are consistent with those obtained using multiple single-band
photometric or stellar kinematic criteria. In our previous work, this was the
major motivation for promoting the usage of the Kormendy relation to classify
bulges (Gao et al., 2020). The more accurate bulge-to-disk decomposition
afforded by the detailed two-dimensional analysis of CGS images revealed that
the distribution of bulge Sérsic indices is, in fact, not bimodal, casting
doubt on the common practice of classifying bulges by the boundary of $n=2$
(e.g., Fisher & Drory 2008). Also, classical and pseudo bulges overlap
significantly in $B/T$ (Gao et al., 2020; cf. Figure 4). In Section 4.1, we
showed that in the color–magnitude diagram of bulges
$\Delta\langle\mu_{e}\rangle$ on average effectively sorts the bulge
population into a smooth, monotonic sequence from red, luminous systems to
blue, faint systems (Figure 1). The performance of $B/T$ is inferior, and $n$
fares the worst. This reinforces our previous conclusion that the Kormendy
relation is more effective than any single photometric structural parameter in
classifying bulges. Converting the Kormendy relation to a parameter space
based on stellar mass offered us an additional insight: the bulk of the bulge
classifications remained unchanged, and, so too, the overall statistics of the
two bulge types. We conclude that bulge classifications based on the Kormendy
relation are robust against stellar population effects.

Still, working with stellar mass brings other advantages that should not be
overlooked. Classifying bulges using the stellar mass-based Kormendy relation
leads to better agreement between the structural properties and stellar
populations of classical and pseudo bulges. This is evident by inspecting
Figures 1 and 3: the mass-based parameters recognize more optically red bulges
as classical bulges. This is, of course, expected. The selection criteria aim
to pick denser bulges as classical bulges, and red bulges have higher $M/L$.
Moreover, our mass-based method evidently identifies a much larger population
of ultra-dense bulges (19 vs. 1; Section 4.2.3), intriguing objects whose
origin will be discussed more extensively in Section 5.2. Lastly, the
$\Sigma_{1}$ parameter is a cheap but effective parameter for bulge studies.
We confirm previous studies that using $\Sigma_{1}$ to classify bulges not
only yields good consistency with both the original and the mass-based
Kormendy relation, but that $\Sigma_{1}$ also correlates with the two bulge
indicators ($\Delta\langle\mu_{e}\rangle$ and
$\Delta\langle\Sigma_{e}\rangle$) very well. In other words, they can be used
to predict each other. A tremendous asset is that measuring $\Sigma_{1}$ does
not require bulge-to-disk decomposition whatsoever, and it also places less
stringent demand on image resolution.

Since our sample is not complete at low masses, we lack sufficient dynamic
range to fully explore the bimodality of bulges (e.g., Figure 5) and the
benefits of working with stellar mass. Given the prevalence of multiband
images, however, we recommend working with the more physically meaningful mass
parameters to study bulges whenever possible.

###  5.2 Ultra-dense Bulges

In Section 4.2.3, we designated a subsample of 19 classical bulges as ultra-
dense bulges, whose $\langle\Sigma_{e}\rangle$ are upper $3\sigma$ outliers of
the best-fit mass-based Kormendy relation of the CGS ellipticals. Here, we
summarize their properties. While they have the highest $\Sigma_{1}$ of all
galaxies (Figure 5), as expected from their definition, ultra-dense bulges are
not necessarily the most concentrated or most luminous members among the
classical bulges (Figure 1). The similarity of their
$\langle\Sigma_{e}\rangle$ with high-$z$ ($\gtrsim 1.5$) ETGs, coupled with
their very red optical colors at fixed bulge luminosity, hints of an early
formation epoch. With a median total stellar mass of $10^{11.1}\,M_{\sun}$,
their hosts are the most massive among all disk galaxies, as massive as the
ellipticals in CGS (median stellar mass of $10^{11.2}\,M_{\sun}$). Two-thirds
of them have early Hubble types (S0–Sa). The $BVI$ three-color CGS images of
the host galaxies are displayed in Figure 7.

Figure 7: Atlas of the galaxies that host ultra-dense bulges. Their stellar
masses and Hubble types from HyperLeda (Paturel et al., 2003) are labeled in
the lower-left and upper-right corner of each panel, respectively.

The above-mentioned characteristics of ultra-dense bulges are reminiscent of
previously discovered relic high-$z$ massive compact galaxies (Trujillo et
al., 2014; Ferré-Mateu et al., 2017; Yıldırım et al., 2017). These high-$z$
survivors, fortunate enough to have escaped the violence of hierarchical
merging, remain as dense and compact as high-$z$ ETGs at $z\approx 0$, while
most of the other massive compact galaxies have experienced significant size
growth through dry, minor (but see Davari et al. 2017) mergers to become
present-day massive ellipticals (e.g., Bezanson et al., 2009; Huang et al.,
2013, 2016; Zhu et al., 2021). The ultra-dense bulges highlighted here, with
median $\log\left(M_{\star}/M_{\sun}\right)=10.3$ and median
$R_{e}=0.4\,\mathrm{kpc}$, bear a close resemblance to the population of
similarly massive, dense bulges found lurking in the centers of nearby disk
galaxies (Graham et al., 2015; de la Rosa et al., 2016; Costantin et al.,
2020). Costantin et al. (2021, 2022) argued that this subset of bulges
descended directly from highly dissipative structures associated with an early
population of galaxies formed at $z\gtrsim 3$.

By analogy, can we hypothesize that the ultra-dense bulges are relic
progenitors of the other classical bulges? It is not likely the case, because
their hosts are more massive and have higher $\Sigma_{1}$ than many of the
other hosts of classical bulges. Despite some effects that could reduce it,
$\Sigma_{1}$ generally increases as galaxies evolve. Assuming that the ultra-
dense bulges are relic high-$z$ ETGs, $\Sigma_{1}$ of their hosts should
barely increase, yet their $\Sigma_{1}$ are still higher at $z\approx 0$ than
the other classical bulge hosts. Thus, it is safe to conclude that the
progenitors of the ultra-dense bulge hosts and the majority of the other
classical bulge hosts are different in terms of stellar masses and/or central
densities, at least already at the formation epoch of the ultra-dense bulges.
Thus, the CGS ultra-dense bulges are probably relic high-$z$ ETGs, but their
characteristics are not representative of the progenitors of most present-day
classical bulges. A sample that is more complete in the lower stellar mass
regime may help to clarify what the progenitors of other classical bulges look
like.

Having tried to link the ultra-dense bulges to structures that were assembled
quickly at early times, it is also worthwhile to mention other possibilities.
We note that six of the galaxies that host ultra-dense bulges were recognized
to have buckled bars by Li et al. (2017; NGC 1291, 2811, 3271, 3885, 4699, and
4727). In addition, NGC 6893 seems to host a barlens, although it not was
recognized by Li et al. (2017). The stellar mass range of the ultra-dense
bulge hosts ($\gtrsim 10^{11}\,M_{\sun}$) corresponds to the range where the
fraction of buckled bars reaches its maximum ($\sim 40\%$) in the CGS disk
galaxy sample (Li et al., 2017). When a bar gets buckled, its stellar mass in
the thickened part will be modeled as part of the photometric bulge in our
decomposition strategy, thereby potentially increasing the measured average
mass surface density within the half-light radius of the bulge. Moreover, the
resultant bulge is expected to have an old stellar population, because the
stellar population of bars in early-type disks is often old (Phillips, 1996),
the buckling process does not involve new star formation (Combes et al.,
1990), and buckled bars could subsequently reduce gas inflow to the galaxy
center (Fragkoudi et al., 2016). In light of these factors, the bar buckling
may be a viable mechanism for forming the red ultra-dense bulges found in this
study. We also find that some galaxies (NGC 1068, 1291, 3271, 4699, and 6753;
Gao et al. 2019) host nuclear bars and/or rings. The presence of these
features implies the presence of an underlying kinematically cold structure,
namely a nuclear disk, which is a product of secular processes. However, de
Lorenzo-Cáceres et al. (2019a) claimed, on the basis of structural properties
and colors of their bulges, that most galaxies that have both a nuclear and a
main bar host classical bulges. Moreover, the stellar population of nuclear
disks is not necessarily young (e.g., NGC 1291; de Lorenzo-Cáceres et al.
2019b; Bittner et al. 2021).

To summarize: both early, rapid collapse and protracted secular evolution can
contribute to the formation of ultra-dense bulges. If secular processes do
play a role, it is difficult to understand why their influence is not more
widespread in other bulges. Additional data and analysis more refined than
that attempted here will be required to disentangle the contribution of
various mechanisms.

##  6 Summary

We measured optical colors of bulges for a sample of 312 CGS disk galaxies and
converted our previous measurements of bulge structural parameters from flux
to stellar mass units. We constructed the mass-based Kormendy relation of
bulges and ellipticals and applied it to classify bulge types. Comparison of
the classifications based on the original Kormendy relation and the mass-based
one showed that the classifications of bulge types are consistent (to better
than $90\%$) and the overall statistics of classical and pseudo bulges are
barely changed. We conclude that the Kormendy relation as a classification
tool is robust against stellar population effects. However, working with mass
allows for better agreement between bulge structural properties and stellar
populations, as it allows more bulges with old stellar population to be
classified as classical bulges. It also helps to reveal a more significant
population of ultra-dense bulges, which are probably relic primordial bulges
as dense as high-$z$ ETGs, although secular processes may also contribute to
their growth. We confirmed previous studies that $\Sigma_{1}$ is well
correlated with bulge surface densities.

This work was supported by the National Science Foundation of China (11721303,
11991052, 12011540375, 12122301) and the China Manned Space Project (CMS-
CSST-2021-A04, CMS-CSST-2021-A06). ZYL is supported by a Shanghai Natural
Science Research Grant (21ZR1430600), and by the “111” project of the Ministry
of Education under grant No. B20019. Kavli IPMU was established by World
Premier International Research Center Initiative (WPI), MEXT, Japan. We thank
Aaron Barth for helpful suggestions and for his contributions to the Carnegie-
Irvine Galaxy Survey. HG thanks Hassen Yesuf for useful discussions. We thank
the referee for helpful comments that improved the manuscript.

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