# Morphological asymmetries of quasar host galaxies with Subaru Hyper Suprime-
Cam

Shenli Tang,1,2,3,4 John D. Silverman,2,3,5 Hassen M. Yesuf,2,6 Xuheng Ding,2
Junyao Li,7 Connor Bottrell,2Andy Goulding,8 Kiyoaki Christopher Omori,9
Yoshiki Toba,10,11,12 and Toshihiro Kawaguchi13
1Department of Physics, School of Science, The University of Tokyo, 7-3-1
Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
2Kavli Institute for the Physics and Mathematics of the Universe (WPI), The
University of Tokyo, Kashiwa, Chiba 277-8583, Japan
3Center for Data-Driven Discovery, Kavli IPMU (WPI), UTIAS, The University of
Tokyo, Kashiwa, Chiba 277-8583, Japan
4Institute for Cosmic Ray Research, The University of Tokyo, 5-1-5
Kashiwanoha, Kashiwa, Chiba 277-8582, Japan
5Department of Astronomy, School of Science, The University of Tokyo, 7-3-1
Hongo, Bunkyo, Tokyo 113-0033, Japan
6Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing
100871, China
7Department of Astronomy, University of Illinois at Urbana-Champaign, Urbana,
IL 61801, USA
8Department of Astrophysical Sciences, Princeton University, Princeton, NJ
08540, USA
9Division of Particle and Astrophysical Science, Nagoya University, Furo-cho,
Chikusa-ku, Nagoya 464–8602, Japan
10National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo
181-8588, Japan
11Academia Sinica Institute of Astronomy and Astrophysics, 11F of Astronomy-
Mathematics Building, AS/NTU, No.1, Section 4, Roosevelt Road, Taipei 10617,
Taiwan
12Research Center for Space and Cosmic Evolution, Ehime University, 2-5
Bunkyo-cho, Matsuyama, Ehime 790-8577, Japan
13Department of Economics, Management and Information Science, Onomichi City
University, Hisayamada 1600-2, Onomichi, Hiroshima 722- 8506, Japan E-mail:
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

How does the host galaxy morphology influence a central quasar or vice versa?
We address this question by measuring the asymmetries of 2424 SDSS quasar
hosts at $0.2<z<0.8$ using broad-band ($grizy$) images from the Hyper Suprime-
Cam Subaru Strategic Program. Control galaxies (without quasars) are selected
by matching the redshifts and stellar masses of the quasar hosts. A two-step
pipeline is run to decompose the PSF and Sérsic components, and then measure
asymmetry indices ($A_{\rm CAS}$, $A_{\rm outer}$, and $A_{\rm shape}$) of
each quasar host and control galaxy. We find a mild correlation between host
asymmetry and AGN bolometric luminosity ($L_{\rm bol}$) for the full sample
(spearman correlation of 0.37) while a stronger trend is evident at the
highest luminosities ($L_{\rm bol}>45$). This then manifests itself into
quasar hosts being more asymmetric, on average, when they harbor a more
massive and highly accreting black hole. The merger fraction also positively
correlates with $L_{\rm bol}$ and reaches up to 35% for the most luminous.
Compared to control galaxies, quasar hosts are marginally more asymmetric
(excess of 0.017 in median at 9.4$\sigma$ level) and the merger fractions are
similar ($\sim 16.5\%$). We quantify the dependence of asymmetry on optical
band which demonstrates that mergers are more likely to be identified with the
bluer bands and the correlation between $L_{\rm bol}$ and asymmetry is also
stronger in such bands. We stress that the band dependence, indicative of a
changing stellar population, is an important factor in considering the
influence of mergers on AGN activity.

###### keywords:

quasars: supermassive black holes – galaxies: active – galaxies: interactions
– galaxies: evolution – methods: observational – methods: statistical

††pubyear: 2023††pagerange: Morphological asymmetries of quasar host galaxies
with Subaru Hyper Suprime-Cam–C

##  1 Introduction

Measurements of morphology and structural parameters of type-1 quasar hosts
are hampered due to the central bright point source. However, these
measurements are essential to understand the role that supermassive black
holes (SMBHs) play in galaxy evolution. It is suggested that the merger of
galaxies may drive gas inflow onto the central SMBH, thus igniting a luminous
quasar (active galactic nuclei (AGNs) with $L_{\text{bol}}\gtrsim 10^{43}$ erg
$\mathrm{s}^{-1}$) (Hernquist, 1989; Barnes & Hernquist, 1991; Di Matteo et
al., 2005; Capelo et al., 2015), and at the same time, build up the tight
correlations between the BH mass and stellar velocity dispersion (Ferrarese &
Merritt, 2000; Gebhardt et al., 2000; Merritt & Ferrarese, 2001) and bulge
mass (Laor, 2001; McLure & Dunlop, 2002; Marconi & Hunt, 2003; Häring & Rix,
2004). In this scenario, it is expected that AGNs show more disturbed features
than inactive galaxies.

A traditional way to test this scenario is to select a control inactive galaxy
sample and visually classify the mergers in both populations (Cisternas et
al., 2010; Kocevski et al., 2011; Weston et al., 2016; Fan et al., 2016;
Mechtley et al., 2016; Ricci et al., 2017; Donley et al., 2018; Ellison et
al., 2019; Marian et al., 2019, 2020). However, visual classification is
subjective and difficult to reproduce. Additionally, quantitative and
systematic approaches to merger classification are necessary when handling
large data volumes from current and future astronomical imaging surveys.

Conselice (2003) developed an asymmetry index (A) to classify major mergers
and showed that ULIRGS tend to have asymmetry indices larger than 0.35, which
was further used to identify major mergers out to $z\sim 3$ in HDF. The A
index is measured by rotating the image of a galaxy by $180^{\circ}$ and
subtract it from the original image. After being normalized by the total flux
of the galaxy, this value can well describe the asymmetry of a galaxy
(Elmegreen & Elmegreen, 1982; Conselice, 1997). Besides mergers, asymmetry can
also be used as an indicator of star-formation activity. (e.g., Conselice,
2003, 2014; Yesuf et al., 2021). Since blue star-forming galaxies tend to be
more asymmetric than red quiescent galaxies.

Regarding quasars, only a few studies measured A index of of their host
galaxies due to the difficulty of separating the contribution from the quasar
light from its host and minimizing its effect on the asymmetry measurements
(Zhao et al., 2022). Consequently, most studies have focused on Type-2 AGNs
(Gabor et al., 2009; Boehm et al., 2013; Lackner et al., 2014; Glikman et al.,
2015; Goulding et al., 2018; Zhao et al., 2022), and some mixed the two
populations with X-ray selection (Kocevski et al., 2011; Cotini et al., 2013;
Villforth et al., 2014; Villarroel et al., 2017).

In particular, Goulding et al. (2018) studied the morphology of 2552 WISE-
selected AGNs at $z<0.9$ in Subaru HSC-SSP footprint. The merger fraction was
then estimated using a Random Forest Classifier based on the morphological
parameters including asymmetry. They found that the AGN fraction in merger
samples are a factor of $2\sim 7$ higher than in non-interacting galaxies, and
the merger fraction of AGNs is enhanced in the most luminous population. Their
results strongly suggest a positive connection between the mergers and AGNs.
However, they excluded Type-1 AGNs from their samples due to the stronger
contamination from the point sources. Studies have pointed out that Type-1
AGNs have intrinsic differences with Type-2 AGNs in their physical properties
(Satyapal et al., 2014; Zakamska et al., 2016; Villarroel et al., 2017; Kong &
Ho, 2018; Zhuang & Ho, 2020), such as Eddington ratio and star formation rate
(SFR). Also, how heavily the AGNs are obscured may also make a factor of 2
difference in merger fraction (Kocevski et al., 2015), but currently tested
with only $\sim 150$ sources. Therefore, a dedicated effort focusing on Type-1
AGNs with comparable statistics to the Goulding et al. (2018) study is
required to fill in the gap. Furthermore, Type-1 AGNs will allow us to
investigate the dependence of merger features on BH mass and Eddington ratio.

While there is growing consensus that mergers are not the main driver of BH
accretion, there remains some discrepancies on how much of an excess in the
merger fraction of quasars exists compared to inactive galaxies. X-ray studies
typically find no excess of the AGN merger fraction over inactive galaxies at
$0.3\lesssim z\lesssim 2.5$ (Gabor et al., 2009; Kocevski et al., 2011; Boehm
et al., 2013; Villforth et al., 2014; Villarroel et al., 2017); however,
Silverman et al. (2011) find a $2\times$ enhancement at $0.2<z<1.0$ and Cotini
et al. (2013) report a $\sim 5\times$ excess in the local Universe. Optical
selected Type-1 AGNs show observational evidence for both positive (Ellison et
al., 2011; Marian et al., 2020) and negative (Cisternas et al., 2010; Marian
et al., 2019; Zhao et al., 2022) excess for the merger fraction. The variation
in AGN merger fractions, as exhibited in the aforementioned studies, is likely
attributed to a number of factors including luminosity evolution (Treister et
al., 2012; Weigel et al., 2018; Kim et al., 2021), redshift evolution
(Steinborn et al., 2018), and merger-stage dependence (Ricci et al., 2017;
Goulding et al., 2018).

In this work, we present another approach to lessen the tension between
various measurements by considering the properties of the stellar population
of AGN host galaxies. We measure the asymmetry indices from five broad-band
images, which allows us to compare asymmetry as a function of wavelength which
differs due to the contributions from different stellar populations. In
addition, we study correlations between physical properties of our quasars and
asymmetries. Such correlations could be triggered by the dependence of merger
fractions on those properties as mentioned above.

Here, we use 2424 Sloan Digital Sky Survey (SDSS York et al., 2000) Type-1
quasars at $0.2<z<0.8$ selected from the DR14 quasar catalog (Myers et al.,
2015; Pâris et al., 2018), with measured properties (sizes, Sérsic indices) of
their quasar hosts from Li et al. (2021a) based on Subaru imaging from the
HSC-SSP (Miyazaki et al., 2018, s21a_wide). High-quality optical imaging
enables parametric morphological decompositions to be more accurate, and
improve measurements of galaxy asymmetries, as shown by Bottrell et al. (2019)
using deep and shallow images from SDSS Stripe 82. The wide coverage of HSC-
SSP allows us to build the largest statistical sample of Type-1 quasar hosts
for non-parametric asymmetry measurements. A common practice in previous works
is to measure quasar host asymmetry using the Galfit (Peng et al., 2002)
package to decompose the quasar image and remove the point source. Then the
asymmetry is calculated on the $pure$ quasar host image (Zhao et al., 2022,
e.g.,). In this work, we follow this routine, while making use of a more
recently developed decomposition tool GaLight (Ding et al., 2021), and
morphology measurement tool statmorph (Rodriguez-Gomez et al., 2019).

This paper is arranged as following: section 2 describes the sample selection.
section 3 presents the methodology of measurements. Results are shown in
section 4, and further discussion are made in section 5. We adopt $\Lambda
CDM$ cosmology with $\Omega_{\Lambda}=0.7$, $\Omega_{\mathrm{m}}=0.3$, and
$H_{0}=70\mathrm{~{}km}\mathrm{~{}s}^{-1}\mathrm{Mpc}^{-1}$ across the paper.

Figure 1: Left: $L_{\rm bol}$ as a function of redshift for Type-1 quasars
used in this work (red points in the main plot and red histograms in the
appended plots) and the entire SDSS DR14 quasar population in the same
redshift range (blue contours in the main plot and blue empty histograms in
the appended plots). The green cumulative histograms show the HSC quasar
distribution. Right: Similar to the left, but for Eddington ratio and $M_{\rm
BH}$.

##  2 Sample selection

The optical imaging used in this study is from the HSC-SSP data release 4
(internal). It covers $\sim 1100$ deg2 with five broad bands ($grizy$
Kawanomoto et al., 2018) to a full $5\sigma$ depth reaching 26.5, 26.5, 26.2,
25.2, 24.4 mag with median seeing of 0.79, 0.75, 0.61, 0.68, 0.68 arcseconds
respectively (Aihara et al., 2022). The data were reduced with the HSC
Pipeline (Bosch et al., 2018).

###  2.1 Type-1 quasars

The type-1 quasars used in the Li et al. (2021a) study were originally
selected from SDSS DR14 quasar catalog (Myers et al., 2015; Pâris et al.,
2018) and cross-matched with HSC-SSP PDR2 (Aihara et al., 2019). The authors
studied the host galaxy size-mass relation and $M_{\rm BH}$ \- $M_{*}$ mass
relation of these quasars (Li et al., 2021b). The final sample contains 2424
quasars at $0.2<z<0.8$. The lower limit was set to avoid fine structures such
as spiral arms that could not be well modeled in the parametric fitting. We
keep this lower limit because we are interested in the correlations between
those parameters and asymmetry. While at $z>0.8$, the host galaxies are
difficult to detect in HSC due to the surface brightness dimming (Ishino et
al., 2020). The quasar host galaxies are magnitude-limited ($i_{mag}<23$) and
have 90% completeness at lower limits of $10^{9.3-10.3}M_{\odot}$ for
z=0.3-0.7. This selection results in 13.5% of the sources being rejected by
these limits. An upper limit of $10^{11.5}M_{\odot}$ was set because galaxies
above this threshold are very likely undetected in blue bands thus may have
inaccurate stellar mass measurements. Such a cut may lead to a biased fraction
towards star-forming hosts among the quasars, but this number is minimal
($\sim 1\%$). They also applied some quality checks to the subtraction and
fitting results as detailed in Section 5 and Table 2 of their paper. In this
work, we updated the imaging from PDR2 to DR4 s21a_wide, because the sky
subtraction algorithm was optimized over the recent data release. In
particular, the global sky extraction was improved to deal with a bias that
might be caused by dead CCDs (see Figure 7 of Aihara et al., 2022). In
addition, local sky subtraction was implemented to mitigate the impact of
extended wings of bright objects. We note that a complementary paper (Yesuf et
al. in prep) will examine the asymmetries of a large sample of BLAGNs at
$z<0.35$ imaged by HSC.

The BH properties and redshifts of these quasars are estimated by Rakshit et
al. (2020) based on SDSS spectral properties. Specifically, BH masses ($M_{\rm
BH}$) are measured from the single epoch virial method using H$\beta$ and
$L_{5100}$ (Vestergaard & Peterson, 2006):

$M_{\mathrm{BH}}=10^{6.91}L^{0.5}_{5100,44}\left(\frac{\mathrm{FWHM}_{H\beta}}{1000\mathrm{~{}km}\mathrm{~{}s}^{-1}}\right)^{2}M_{\odot}$
(1)

where $L_{5100,44}$ is the monochromatic luminosity at rest-frame 5100 Å in
units of $10^{44}\ \rm{erg\ s^{-1}}$ measured by power-law continuum fitting
to the SDSS spectra. The bolometric luminosity $L_{\rm bol}$ is calculated
using the bolometric correction factor in Richards et al. (2006):

$L_{\mathrm{bol}}=9.26\times L_{5100}$ (2)

Then the Eddington ratio is given by:

$\lambda_{\rm edd}=\log(L_{\mathrm{bol}}/L_{\mathrm{edd}})$ (3)

where $L_{\mathrm{edd}}$ is the Eddington luminosity estimated from $M_{\rm
BH}$:

$L_{\mathrm{Edd}}\cong 1.3\times
10^{38}\left(M_{\mathrm{BH}}/M_{\odot}\right)\mathrm{erg}\mathrm{s}^{-1}$ (4)

We show the distribution of the 2424 quasars used in this work in $L_{\rm
bol}$ as versus redshift and $\lambda_{\rm edd}$ versus $M_{\rm BH}$ diagrams
(Figure 1). Compared to the whole SDSS DR14 quasar population, our samples are
slighted shifted to low $L_{\rm bol}$, thus the low $M_{\rm BH}$ and low
$\lambda_{\rm edd}$ region. There is also a small decrement in density ($\sim
30\%$) at $z>0.75$. Both of the above biases are caused by the failure in
detecting the host galaxies of these populations, i.e., relatively bright and
high-z quasars being rejected in Li et al. (2021a) catalog.

###  2.2 Control Galaxies

Control galaxies are selected from the catalog of Kawinwanichakij et al.
(2021), which includes $\sim$ 1.8 million galaxies in HSC PDR2 footprints with
single Sérsic fits. We also updated the images to DR4 data for the same reason
as mentioned above. Kawinwanichakij et al. (2021) controlled the quality of
galaxy images by setting the following parameters in the HSC-SSP database:

  1. 1.

(g&r)_inputcount_value $\geq$ 4

  2. 2.

(i&z&y)_inputcount_value $\geq$ 6

  3. 3.

i_cmodel_mag brighter than 24.5 AB

  4. 4.

i_cmodel_magerr $\leq$ 0.1 AB

  5. 5.

i_extendedness_value = 1

  6. 6.

PSFMAG-CMODELMAG $>$ 0.2

In addition, we increased the threshold of the 6th criterion from the original
value of 0.145 to 0.2 to be more strict with the removal of point sources. We
find that some close double stars could be included when using the lower
value. To assure that our control sample does not include AGNs, we matched our
control galaxies with the latest Million Quasars (Milliquas) catalog v7.8c
(Flesch, 2021), which includes 1.4 million type-1, type-2 quasars selected
from radio, optical, and X-ray surveys, and the HSC quasar catalog, which is
based on HSC broad band color selections (Goulding et al. in prep.). This
rejects 0.02% and 0.04% sources from our control galaxies, respectively.
Stellar mass and photo-$z$ of these galaxies, published in Nishizawa et al.
(2020), are estimated using a template fitting-code mizuki (Tanaka, 2015;
Tanaka et al., 2018). The code uses a set of theoretical templates generated
with the Bruzual & Charlot (2003) stellar population synthesis code to fit the
CModel magnitudes of the sources. The results are calibrated to spectroscopic
redshifts achieved from other surveys that overlap with the HSC-SSP survey.
Stellar mass is estimated as the median value derived from the PDF
marginalized over all the other parameters including SFR and dust attenuation.

Based on these measurements, we apply the KD-Tree method to select control
galaxies matching our quasars in redshift and stellar mass. KD-Tree is a
binary space partitioning method that can efficiently find neighborhood data
points in k-dimensional space (Fuchs et al., 1980). We set the leaf size to 20
in sklearn.neighbors.KDTree and select 3 inactive galaxies for every quasar.
We allow duplicated selection of galaxies, such galaxies are counted multiple
times in our results. In total, 7060 galaxies are selected, with 212
duplicates. We show the redshift and stellar mass distribution of these
quasars and control galaxies in Figure 2.

The photo-$z$ measurements of the control galaxies were qualified by
Kawinwanichakij et al. (2021) as shown in their Appendix A, Figure 12. They
compared HSC-SSP PDR2 mizuki photo-$z$ with available spec-$z$ measurements
from other surveys such as zCOSMOS (Lilly et al., 2009), VIPERS (Garilli et
al., 2014) and PRIMUS (Cool et al., 2013). They reported that the median
offset between the spec-$z$ and photo-$z$ is 0.001. While $\sim 7\%$ of the
sample has catastrophic failures ($|\Delta z|/\left(1+z_{\text{spec
}}\right)>0.15$), most of which happen at $z>1.0$ or $z<0.2$. Between this
redshift range, the photo-$z$ is well-constrained thanks to the 4000 Å break,
in which the failure fraction reduces to 3%. We expect this fraction to be
lower for our control galaxies as they are relatively bright sources (typical
$i$-band magnitudes $\sim 20.7$ compared to the lower cut of 24.5 in
Kawinwanichakij et al. (2021)). On the other hand, the authors expect a
typical scatter $\sim 0.25$ dex for the mizuki stellar mass measurements. They
applied correction of $0.08-0.11$ dex for the “out-shining” effect of star-
forming galaxies (Sorba & Sawicki, 2015). The uncertainties on the stellar
mass of quasar hosts were estimated by Li et al. (2021a) via comparison to
simulated quasar images based on HST images in CANDELS-COSMOS field. The
values weakly depend on masses and are typically between 0.1-0.3 dex.

Figure 2: The distribution of stellar mass and redshift for quasar hosts (red
dots and red histograms) and control galaxies (blue contours and blue empty
histograms). The peaked redshifts of control galaxies at certain values are
artifacts of the binning and precision of photo-$z$. Green curves are the
cumulative distribution for the quasars hosts.

##  3 Methodology

We measure the asymmetry of the quasar host galaxies and a matched control
sample. For the former, this requires us to generate an image of the hosts
free of quasar emission. Thus, we decompose the optical images into separate
source components (i.e., quasar and host emission; Section 3.1). As detailed
below, two images are constructed, one with the nearby companions and the
other without. This allows us to assess the level of asymmetric features
either external or internal to quasar hosts (Section 3.3). The procedures for
control galaxies are similar with details provided in Appendix B.

###  3.1 Quasar subtraction through 2D decomposition

We use GaLight (Ding et al., 2020, 2021) to perform two-dimensional
decomposition of quasar and host emission. GaLight is developed under Python3
environment and incorporates structures from Lenstronomy (Birrer et al.,
2021). For example, the particle swarm optimization (PSO; Kennedy & Eberhart,
1995) algorithm used in GaLight offers improvements in the gradient descent
algorithms used in other analysis tools which can become trapped in local
minimal and dependent on the initial input values. GaLight models the imaging
in multiple steps. First, the cutout size of the image frame is determined
that includes any extended emission (e.g., tidal tails) of nearby objects, if
present, and minimizes computing time by limiting the total number of pixels.
Square cutouts, centered on the target, have dimensions ranging from 84 to 196
pixels ($14-33\arcsec$ in HSC) in steps of 28 pixels.

We then fit all galaxies detected in the cutout image using Sérsic profiles.
We set $supersampling\\_factor$ to 3 instead of 1 to allow a higher resolution
sub-pixel sampling of surface brightness. The quasar component is represented
by an additional point spread function (PSF) model which is generated by HSC
Pipeline at the same position as the quasar (Bosch et al., 2018). Lastly,
GaLight uses the PSO algorithm with 200 iterations to minimize the overall
$\chi^{2}$ for the model parameters including Sérsic radius, Sérsic index,
ellipticity, and central positions of the sources. In this work, we fixed the
Sérsic radius and Sérsic index to the same as measured in Li et al. (2021a) to
better reproduce the host magnitudes. Thus, their SED fitting results (e.g.,
stellar mass and rest frame U-V color) can be directly used in this work.

To demonstrate the model fitting procedure, we show in Figure 3 an example
(SDSS J000219.11+002232.9) which is a Type-1 quasar hosted by a galaxy with a
stellar mass of $10^{10.6}M_{\odot}$ at $z=0.547$. Based on the raw image
(data panel) and GaLight modeled image (model panel), we first remove quasar
emission, i.e., the point source component, from the raw image. Thus, the
quasar host and companions are left, we refer this to the “host+comp" frame.
Because we do not know whether the companions are physically related to the
quasar hosts, we make another frame that subtracts the best fit Sérsic models
of all of those companions from the “host+comp" frame. More importantly, this
enables us to assess whether any quasar response to host asymmetries depends
on scale from the host to its nearby neighbors. We refer to this frame as the
“host only" frame. We will carry out our measurements on both frames.

Figure 3: Example 2D decomposition of SDSS J022111.26-022919.3. Top panels
from left to right are (1) raw $i$-band HSC image of the quasar; (2) best-fit
GaLight model; (3) data minus point source model, we refer to it as
“host+comp" frame; (4) data minus point source minus all the other companions
except the quasar host, we refer to it as “host only" frame. The bottom panel
proves the wellness of fitting via comparing the 1D surface brightness profile
(as a function of distance from the image center) between model (blue solid
line) and data (open circles). The model consists of the point source (orange
solid line) and all galaxy components including the quasar host and companions
(green solid line). The lower panel presents the residual of data minus model.

###  3.2 Parametric quasar host properties

Besides the Sérsic parameters from Li et al. (2021a), we also use their rest-
frame U-V color and $M_{*}$ results estimated by using the SED fitting code
CIGALE (Boquien et al., 2019) based on five-band HSC photometry. They tested
that the extrapolation of their SED fitting results to MIR wavelengths agree
well with the Wide-field Infrared Survey Explorer (WISE; Wright et al. (2010))
detections at 3.4 µm after removing the quasar contribution. To qualify the
reliability of their results, they selected 1141 CANDELS galaxies in the
COSMOS field that are imaged by both HSC and HST. They made mock quasar images
via adding PSF models to the HSC images of the CANDELS galaxies to examine how
well the decomposition method can recover the real host flux. They reported
that the underestimation level is only $\sim$ 7%. Then they compared HSC- and
CANDELS-based measurements on $M_{*}$ and rest-frame U-V color. They found a
small offset of $\sim$ 0.1 dex towards higher $M_{*}$ in HSC, and the median
difference of rest-frame U-V color is around 0.03. They also separated the
quasar host types into star-forming galaxies and quiescent galaxies based on a
boundary line drawn using the Support Vector Machine algorithm:

$U-V=0.16\times\log M_{*}+0.16$ (5)

Assuming the CANDELS-based UVJ classification is the ground truth, they
estimated that the above classification method can successfully identify 89%
of the massive star-forming hosts with log$M_{*}>10.0$ based on HSC
photometry. We apply the same UVJ cut to divide the control non-AGN host
galaxies.

###  3.3 Morphological measurements

We use statmorph, which is a python-based code developed by Rodriguez-Gomez et
al. (2019) for estimating both parametric and non-parametric morphological
parameters of galaxies. This tool has been tuned based on simulated images
from IllustrisTNG and observational data from PanSTARRS at low redshift
($z\sim 0.05$). It has been used up to $z\sim 4$ (Pearson et al., 2019). The
routine requires a cutout image of the galaxy, a segmentation map that
indicates the pixels assigned to the galaxy of interest, and a mask image to
reject pixels with most of the flux from other sources. For each of our
“host+comp" and “host only" images, we generate the segmentation maps and mask
images (Section 3.3.1) that are then input to statmorph to calculate the
asymmetry values (Section 3.3.2).

Figure 4: Example production of the segmentation map and mask image (see text
for full details) for SDSS J022111.26-022919.3. (a) HSC $i$-band image. (b)
Combination of a low nsigma “cool" mode detection (colored segments) and high
nsigma “hot" mode detection. (c) All remaining “cool" segments from the
previous step are combined to make the final segmentation map. (d) The other
“cool" segments are combined to create the mask file.  Figure 5: Asymmetry
measurements of one typical quasar: SDSS J022111.26-022919.3, which is located
at $z=0.528$ and with stellar mass of $10^{11.1}M_{\odot}$. For every quasar,
we make two image frames: 1st row for the “host+comp" frame, and the 2nd row
for the “host only" frame. We measure three asymmetry indices for each:
$A_{\rm CAS}$, $A_{\rm outer}$, and $A_{\rm shape}$ for each column. Colored
images are shown based on all five broad-band HSC data. For $A_{\rm CAS}$
column (panels a,d), we plot the segmentation map envelope with solid lines
and 1.5$R_{P}$ aperture with dashed circles. For $A_{\rm outer}$ column
(panels b,e), we plot the half-light radius $R_{\rm half}$ apertures with
solid ellipses, and maximum radius $R_{\rm max}$ apertures with dashed
ellipses. For $A_{\rm shape}$ column (panels c,f), we show the segmentation
map of the object for all five bands, overplotted with the asymmetry center
(dots), and $R_{\rm max}$ apertures in dashed circles. For all panels, colors
of the contours refer to the bands as labeled in panel (a), from which 5
asymmetry values are measured for every panel.

####  3.3.1 Source deblending and masking

We use SExtractor (Bertin, E. & Arnouts, S., 1996) for source extraction and
deblending that is available in python through the photutils package (Bradley
et al., 2022). The three main parameters of concern are as follow.

  1. 1.

nsigma: Detection threshold for a single pixel in units of the standard
deviation of the background noise level.

  2. 2.

npixels: Minimum number of connected pixels required to be considered a single
source

  3. 3.

contrast: The flux fraction to deblend a local peak as a separate object from
another.

Other parameters are set to default values. However, an optimal separation of
the target quasar and close companions often requires a case-by-case tuning of
these parameters which is not feasible for a large sample. Sazonova et al.
(2021) followed the work of Galametz et al. (2013) and implemented a so-called
“hot+cool" method that can be effectively applied to a large and diverse set
of galaxy images. The basic idea of this method is to use a low nsigma "cool
mode" detection to include faint structures such as tidal tails and then apply
a high nsigma "hot mode" detection to remove nearby contaminating sources. In
Figure 4, we demonstrate our implementation of this “hot+cool" method using
the $i$-band image of SDSS J022111.26-022919.3, an SDSS quasar at $z=0.53$.
The steps are described in detail as follows:

Step 1: “cool" mode. Using the HSC science image (Figure 4a), we first perform
a “cool" mode detection with a $1\sigma$ threshold, 5 pixels minimum area, and
$1e-6$ contrast fraction (the same as Sazonova et al. (2021)) to find all
faint sources within the field of view, as shown by the colored segments in
Figure 4b.

Step 2: Masking unassociated neighbors. We identify the central source and use
its kron radius $R_{k}$ (Kron, 1980) given in the SourceCatalog as determined
by photutils. We then make an elliptical aperture using $5\times R_{k}$ to
include the faint features of the galaxy as shown by the red dashed ellipse in
Figure 4b. We mask the detected objects outside $5\times R_{k}$. However, we
retain pixels for nearby sources if they have a cool segment with at least
$0.01\times A_{k}$ (area within kron aperture) pixels within the aperture of
the main target. As demonstrated in Figure 4b, two faint components (green and
pink connected segments) pass this step.

Step 3: “hot" mode. We then apply a “hot" mode detection routine using a
$2.5\sigma$ threshold and no deblending. The segments resulting from this
“hot" mode are plotted as white contours in Figure 4b. To save computational
time, the “hot" mode detection is only applied to sources left by Step 2.

Step 4: Masking close contaminants. The purpose of the “hot" mode detection is
to check the deblending results based on the “cool" mode, which may
inadvertently split a galaxy into multiple components. Specifically, for each
“cool” segment (colored regions), if it covers over 80% of any “hot” segment
(white contours), we consider it as a separated bright peak that is then
masked. In Figure 4b, the pink “cool" segment completely covers its “hot"
segment thus determined to be a contaminant. While the “hot" segment
counterpart of the green “cool" segment covers only a fraction of the large
“hot" segment thus it’s considered an important (i.e., possibly interacting)
component of the central source. Also, if the area of a “hot" segment more
than doubles that of the central galaxy, we mask its “cool" segment.
Considering the brightness and redshift of our Type-1 quasars, they are
unlikely to be secondary members of a minor merger. Such large “hot" segments
are more likely to be foreground galaxies or saturated stars.

Step 5: Generate final segmentation map and object mask. Finally, we combine
the “cool" segments from Step 4 to produce the final segmentation map (Figure
4c), and all the other “cool" segments as objects to be masked (Figure 4d)

We apply this procedure to the “host+comp" images of our Type-1 quasars. The
colored solid contours in Figure 5a denote the final segmentation map output
from the pipeline. For the “host only" images, we only perform a simple “cool"
mode detection with $1\sigma$ threshold and $5$ pixels minimum area, without
any deblending, to include everything remaining in the frame (Figure 5d). This
is because all neighbors have been removed during the image-fitting procedure
described in Section 3.1.

####  3.3.2 Asymmetry measurements

The CAS (concentration, asymmetry, smoothness) system is a commonly used non-
parametric means to characterize the structure of galaxies (Conselice, 2014).
In this work, we focus on the asymmetry values with three definitions: $A_{\rm
CAS}$, $A_{\rm outer}$, and $A_{\rm shape}$. All can be generally written in
the following form as defined in Conselice et al. (2000):

$A=\min\left(\frac{\Sigma\left|I_{0}-I_{180}\right|}{\Sigma\left|I_{0}\right|}\right)-\min\left(\frac{\Sigma\left|B_{0}-B_{180}\right|}{\Sigma\left|I_{0}\right|}\right)$
(6)

For $A_{\rm CAS}$, given the segmentation map of a source (solid envelopes in
Figure 5 left column), $I_{0}$ is the image pixel values within 1.5 Petrosian
radius $R_{P}$ ($\eta=0.2$, Petrosian (1976), dashed circles in Figure 5 left
column). $I_{180}$ is rotated from $I_{0}$ by $180^{\circ}$, which is then
subtracted from $I_{0}$. The total absolute residual values are normalized by
the total values of the original $I_{0}$. This constitutes the first term,
namely, object asymmetry. However, this value could be biased due to
underlying background gradients (Conselice et al., 2000). Therefore, the
second term, background asymmetry (we refer to it as $A_{\rm bkg}$) is
subtracted from the first term to get the intrinsic object asymmetry.

statmorph measures the background asymmetry with a so-called "sky box" method.
With a maximum box size of 32$\times$ 32 pixels, it iterates over the whole
image to find an empty area that includes no pixels from detected objects. If
there is no such area with the maximum size, the length of the box side is
reduced by a factor of 2 and iterates until an empty area is found. We measure
the sky asymmetry on the images of the quasars with the size of
$90\arcsec\times 90\arcsec$ before any model subtractions or cutout using
GaLight. This value is then applied to our measurements on "host+comps" and
“host only" frames. Further proof and refinement of the sky asymmetry
measurements are described in Appendix A.

As a caveat on the traditional $A_{\rm CAS}$ measurement for bulge-dominated
galaxies, the asymmetry index could be underestimated because $R_{P}$ is
effectively smaller thus causing the outskirt features to be ignored. To
compensate for this, Wen et al. (2014) suggested a new asymmetry index, namely
the outer asymmetry $A_{\rm outer}$ that is calculated in a similar way to
$A_{\rm CAS}$. It replaces the 1.5$R_{P}$ circular aperture with an elliptical
annulus from the half-light radius ($R_{\rm half}$, solid ellipse in Figure
5b) that contains half of the total flux within $R_{\rm max}$ and a maximum
radius of the target emission ($R_{\rm max}$, dashed ellipse in Figure 5b).
The elongation and orientation of the ellipse are calculated based on the
segmented source image. By doing so, the central region of the galaxy is
masked, and the asymmetry value will be more sensitive to the outskirt of the
galaxy.

As another modification, Pawlik et al. (2016) defined the shape asymmetry
($A_{\rm shape}$) to increase the sensitivity to faint features by converting
all pixel fluxes of the galaxy to a binary mask (Figure 5cfi shadowed regions,
all five bands are overlapping). Thus, $A_{\rm shape}$ carries information
only about the shape of the segmentation map, as shown in Figure 5c. It is
calculated within a circular aperture with radius $R_{\rm max}$ (Figure 5cfi
dashed circle), and no background asymmetry subtraction is required.

In total, we calculate 30 asymmetry values for every Type-1 quasar, combining
five photometric bands, two image frames (“host+comp" and “host only"), and
three asymmetry indices.

Figure 6: Ratio of our asymmetry measurements as a function of redshift which
indicates the level of reliability. A target being assessed with statmorph
$flag=0$ and successfully passing the entire mask and deblending pipeline are
considered a reliable measurement. The dashed histograms are the total number
of objects in every redshift bin (left y-axis), and the corresponding ratios
are shown in colored broken lines for each band (right y-axis).  Figure 7:
Quality assessment (i.e., asymmetry recovery tests) measured as an offset
($\Delta$) of $A_{\rm CAS}$ between asymmetry values before and after
decomposing the simulated quasar images in each of the five bands separately.
The x-axis refers to the host-to-total flux ratio ($f_{\rm gal}$). The
contours show the entire sample, which are $1\sigma$, $2\sigma$, and $3\sigma$
envelopes of binned 2d histogram distributions (i.e., 99.7% data points are
located within the outermost contours). Similar contours in plots hereafter
will use the same definition. The colored dots show the subset after applying
statmorph flags. The dashed black lines mark the zero offsets. We find the
flags can effectively reduce the cases that might have quasar subtraction
issues at low $f_{\rm gal}$. Figure 8: Comparisons between $A_{\rm CAS}$,
$A_{\rm outer}$ (left column), and $A_{\rm shape}$ (right column), for
“host+comp" frames (top row) and “host only" frames (bottom row) in the
$i$-band.  The black solid lines indicate a 1:1 relation, and the grey
shadowed regions indicate $\pm 0.25$ offsets from that. We give the Pearson
correlation coefficient $\rho$ in the upper left corner of each panel.  Figure
9: $A_{\rm CAS}^{i}$ as a function of quasar and host galaxy properties.
Contours show 1, 2 and $3\sigma$ envelopes of the 2D distributions. Median
values are based on binning either according to their physical property (shown
on the x-axis, green open circles) or the $A_{\rm CAS}^{i}$ values (y-axis,
orange open circles). Error bars represent the standard deviation of each
binned subset. We highlight the top-left panel that shows the strongest
correlation with asymmetry and has an upturn at $\log~{}L_{\rm bol}=45$.

###  3.4 Quality assessment

We assess the reliability of our asymmetry measurements for both the
“host+comp" and “host only" science frames separately. Reasons for inaccurate
measurements could be multi-fold which originate mainly from the statmorph
flags Rodriguez-Gomez et al. (2019), and a few from the mask and deblending
pipeline (Section 3.3.1, minor). Issues can include quasar hosts undetected in
“cool" mode, residuals of the subtraction may lead to an improper Petrosian
radius, or the background not being properly assessed. We show the ratio of
the measurements that are free of these issues as a function of redshift in
Figure 6, as a means to gauge the level of reliability in the asymmetry
measurements. The dashed histograms indicate the total number of objects in
each redshift bin, and the colored broken lines are the reliability ratios in
each band. Generally, the ratio of reliable measurement is highest in
$i$-band, while it drops rapidly towards higher redshift and in bluer bands.
This is because the $i$-band wavelength coverage (695–845 nm) is less affected
by sky background noise and galactic extinction. The survey was designed to
take $i$-band images under better seeings and darker nights (Aihara et al.,
2017). These factors make the quality of $i$-band images better than the other
four, thus making the reliable ratio of measurements higher in this band. In
total, the numbers of targets with reliable measurements for the five bands
are: $g$, 886; $r$, 1488; $i$, 1987; $z$, 1939; $y$, 1759, respectively. We
also checked the dependence of the reliability ratio on other properties, we
find that it is almost independent of $M_{\rm BH}$ and $\lambda_{\rm edd}$
(the difference between each bin is within 0.1 in $i$ band). While the ratio
is higher when $M_{*}$ is larger, as the host galaxy is easier to be detected,
the difference between the most and least massive bins is $\sim 0.2$.

We then use simulated quasar images to assess a possible over-subtraction
issue of quasar hosts and its impact on the asymmetry measurements. Li et al.
(2021a) created a set of 1141 simulated quasar images by inserting a model HSC
PSF (mock quasar) into HSC images of real inactive CANDELS galaxies matched to
the redshift and stellar mass of the quasar hosts. The flux of the mock quasar
is set to the actual $i$-band value for the matched HSC quasar. It is then
scaled to other bands using a power law function in wavelength. By comparing
the original galaxy magnitudes and those after adding and subtracting the mock
quasar, they found that the decomposition process results in only a $\sim 7\%$
over-subtraction affecting the host-to-total flux ratio ($f_{\rm gal}$). We
reran our asymmetry measurement pipeline on both the original and processed
galaxy images and compared the results in terms of offsets of in $A_{\rm CAS}$
($\Delta A_{\rm CAS}=A_{\rm original}-A_{\rm processed}$). As shown by the
contours in Fig. 7, the entire sample has $1\sigma$ scatter of $\sim 0.05$ for
$\Delta A_{\rm CAS}$. There is no strong bias on whether the subtraction leads
to systematically larger or smaller asymmetry values. Even so, the data are
more scattered at low $f_{\rm gal}$, i.e., luminous quasars. The impact of the
over-subtraction is stronger in bluer bands as the quasars are brighter in
those bands. We then apply the statmorph flags and only keep the reliable
measurements (colored dots in Fig. 7) that result in the scatter being
significantly reduced. Taking the $i$-band as an example, if we define
$|\Delta A_{\rm CAS}|>0.1$ as a significant offset, then only $\sim 4.5\%$ of
the sources are affected by subtraction issues, and $\sim 50\%$ of those are
flagged out by statmorph. This is because the strange residuals of bad-
subtracted cases would more likely lead to a failure of statmorph, e.g., in
finding a proper asymmetry center. Therefore, we suggest that for our reliably
measured quasar hosts, only $\sim 2\%$ are affected by significant subtraction
issues. For the following main results of this work, we only use these
reliable measurements (a test of keeping the statmorph-rejected sources show
consistent results, see Appendix C).

For the reliable measurements, we further compare different asymmetry indices
for both the “host only" and “host+comp" frames in Figure 8. The left column
compares $A_{\rm CAS}$ with $A_{\rm outer}$, and right column compares $A_{\rm
CAS}$ with $A_{\rm shape}$. The $i$-band data are shown in $1\sigma$,
$2\sigma$, and $3\sigma$ contours. The black solid lines indicate 1:1
relations in each panel. Grey shadowed regions indicate $\pm 0.25$ offsets
from the 1:1 relations. We show the Pearson correlation coefficient $\rho$
between the asymmetry indices at the left upper side of each panel. For all
bands and both frames, $\rho$ between $A_{\rm CAS}$ and $A_{\rm outer}$ is
$0.84-0.9$, and is $0.37-0.56$ between $A_{\rm CAS}$ and $A_{\rm shape}$.
Compared to “host+comp" measurements, the removal of companion sources
generally reduces all asymmetry values, making $A_{\rm CAS}$ and $A_{\rm
outer}$ values more constrained to be below 0.5 in “host only" measurements.
$A_{\rm shape}$ has larger scatter than $A_{\rm outer}$, but still has a
moderate correlation with $A_{\rm CAS}$.

##  4 Results

###  4.1 Correlations between asymmetry and quasar properties

We first consider the dependence of asymmetry on quasar and host properties
including $L_{\rm bol}$, $M_{\rm BH}$, $\lambda_{\rm edd}$, redshift, stellar
mass $M_{*}$, rest-frame U-V color. We show the distribution of $A_{\rm
CAS}^{i}$ measured on “host only" frames as a function of these properties in
Figure 9. Here, the use of “host only" frames reduces the influence of
projection effects (i.e., foreground and background objects) on the asymmetry
measurements. We bin the measurements according to their physical properties
with fixed intervals (0.5 dex for $L_{\rm bol}$, $M_{\rm BH}$, and
$\lambda_{\rm edd}$; 0.3 dex for $M_{*}$; 0.4 mag for rest-frame U-V; 0.1 for
redshift). For each bin, median values are shown as green open circles with
standard deviation represented by the error bars. We find that $A_{\rm
CAS}^{i}$ has moderate positive correlations with $L_{\rm bol}$, $M_{\rm BH}$,
$\lambda_{\rm edd}$, $M_{*}$, and a weak negative correlation with rest-frame
U-V color of the host galaxy. There is no clear correlation with redshift.
While not shown, we also tested with the observed host galaxy color ($g-r$);
it shows a weak negative correlation with $A_{\rm CAS}^{i}$, similar to the
rest frame $U-V$ color. We highlight the $L_{\rm bol}$-A correlation turns out
to be the strongest among these parameters, which will be discussed in more
detail in the following sections. Here, we first point out a turning point at
$L_{\rm bol}\sim 45$, beyond which the correlation appears to strengthen.

In addition, we bin the data by $A_{\rm CAS}^{i}$ (in intervals of 0.1, orange
open circles in the top left panel. $A_{\rm CAS}$ is likely related to longer
term activity of a quasar host galaxy, thus binning by $A_{\rm CAS}$ might
smooth out short term variations. In this way, we find that more asymmetric
hosts have slightly more luminous quasar activity. This is similar to what we
find by binning along the x-axis values.

Figure 10: Median asymmetry of quasar hosts in bins of BH mass and Eddington
ratio. In each cell, the top value shows the number of objects per bin, the
lower number shows the median asymmetry value of these objects. Only cells
with more than five objects are kept. The value of the asymmetry is indicated
by the scale bar on the right. We report a trend from lower left to upper
right that when $M_{\rm BH}$ and $\lambda_{edd}$ increase, the host galaxy of
quasars is slightly more asymmetric on average.  Figure 11: Merger fractions
of quasar hosts as a function of $L_{\rm bol}$ (left) and redshift (right) in
all five bands (Q_grizy). The error bars in the merger fraction are estimated
from the Poisson errors of mergers. The x-values of the points are shifted a
bit for clarity of the data in each band. The $i$-band data for quasar hosts
in the left panel is fitted with a linear relation (solid line, Equation 7)
and a power law (dashed line, Equation 8). The merger fraction for the control
galaxy sample in the $i$-band is shown as the grey line with shadowed Poisson
error.

We further quantify the typical asymmetries of our quasar sample by the
properties of its BH and host galaxy. We first bin our quasars in $M_{\rm BH}$
from $10^{6.5}$ to $10^{9.5}\mathrm{M_{\odot}}$ and $\lambda_{edd}$ from -3 to
0 both with intervals of 0.5 dex (Figure 10). We only keep bins that include
more than five objects. The counts are shown by the upper number in each bin.
Then we calculated the median asymmetry values as noted by the lower number in
each bin. The darker color of the cell indicates a larger median asymmetry
value. We see a trend from the lower left to the upper right. Therefore, we
report that the quasar hosts are more asymmetric when they harbor a more
massive and more actively-accreting BH.

Figure 12: Probability density distribution of $A_{\rm CAS}^{i}$ of quasar
hosts (solid blue histograms) and control galaxies (dashed red histograms).
From left to right, the columns show different samples: all hosts, star-
forming hosts, and quiescent hosts. The top row is based on the “host+comp"
frames while the bottom row uses the “host only" frames.

###  4.2 Merger fraction

We then convert the asymmetry values to merger fractions and study its
correlations with $L_{\rm bol}$ and redshift. Following Conselice (2003), we
classify objects with either $A_{\rm CAS}>0.35$ or $A_{\rm outer}>0.35$ as
major mergers. We only use “host+comp" frames for this purpose to minimize the
risk of removing genuine companions. However, this estimate is an upper limit
because it includes projected nearby contaminants. We show the merger fraction
as a function of $L_{\rm bol}$ and redshift in all five bands (Figure 11). In
this section, we only focus on the $i$-band results, i.e., the green points.
The quasars are binned in a similar way as in Figure 9, but per 0.4 dex for
$L_{\rm bol}$ based on the distribution of the samples. The error bars on the
x-axis indicate the standard deviation of $L_{\rm bol}$ and redshift in each
bin. The uncertainties on the merger fractions are the assumed Poisson error.

We find that the merger fractions of quasar hosts are overall low (16.2% in
$i$-band) and have little dependence on redshift (spearman correlation
coefficient $\rho=0.2$ with $p$-value=0.7). Therefore, most of the quasars are
not triggered by mergers at $z<0.8$. Following Treister et al. (2012), we fit
a linear relation (black solid line) and a power law (black dashed line) to
the $i$-band merger fractions of quasar hosts as functions of $L_{\rm bol}$.
The fitting was performed with scipy.optimize.curve_fit package using least
squares optimization. The best fit result for linear relation is:

$\operatorname{frac}(L)=\frac{\log\left(L_{\mathrm{bol}}\right)-43.4^{+0.4}_{-0.4}}{9.9^{+2.4}_{-2.4}}$
(7)

and for power law:

$\operatorname{frac}(L)=\left(\frac{L_{\mathrm{bol}}}{4.51^{+3.4}_{-3.4}\times
10^{47}\mathrm{erg}\ \mathrm{s}^{-1}}\right)^{0.31^{+0.05}_{-0.05}}$ (8)

The fitting of Treister et al. (2012) was based on multiple studies, thus
could be affected by a lack of uniformity in sample selection and
classification criteria. Here our evolution is based on one single sample set
and one single criterion of classification. Nevertheless, the zero point of
our fitted linear relation is consistent with Treister et al. (2012) within
the error ($43.4\pm 0.4$ versus 43.2), which indicates an expected threshold
of $L_{\rm bol}$, below which the major merger fraction of AGNs is negligible.
Above this threshold, the power law slope is close to that previously reported
($0.31\pm 0.05$ versus 0.4), indicating a similar increasing tendency of
merger fraction with increasing $L_{\rm bol}$. The difference between the
exact values of merger fractions in our work and their work is likely caused
by the different classification criteria, which leads to the different scaling
of our fitted functions.

###  4.3 Asymmetry of Quasar hosts and control galaxies

We compare the $i$-band CAS asymmetry ($A_{\rm CAS}^{i}$) of quasar hosts and
inactive galaxies in Figure 12 for both “host+comp" and “host only" frames.
The left column shows the probability density distribution of $A_{\rm
CAS}^{i}$ for the entire population of quasars and galaxies. The comparisons
between star-forming and quiescent types are shown in the middle and right
columns. Overall, “host+comp" frames have longer tails towards higher
asymmetries than “host only" frames in all sub-panels, indicating that the
highest asymmetries are due to the companions. Considering the star-forming
galaxies only, the median excess of quasar host asymmetry over control
galaxies is below 0.01. While for quiescent galaxies, quasar hosts are slight
biased toward higher asymmetry. The entire sample of quasar hosts has an
excess of 0.017 in median over the control galaxies for the “host only"
frames. The statistical error of the excess (i.e., $\sigma/\sqrt{N}$ of
$A_{Q}-A_{G}$) is 0.0018, thus this excess has a significance level of
$9.4\sigma$. For the “host+comp" frames, this excess is $0.014\pm 0.004$.

Next, we bin our quasars with respect to $M_{*}$ from $10^{9.6}$ to
$10^{11.4}M_{\odot}$ per 0.3 dex, and rest-frame U-V color of the host galaxy
from 0.7 to 2.5 per 0.3 mag. We compare these values to that of control
galaxies with the same binning. The results, shown in Figure 13, indicate
generally similar trends in both panels that the asymmetry values increase
with increasing stellar mass and bluer color. Considering the typical
uncertainty level of $A_{\rm CAS}^{i}$ to be $\sim$0.03 (see Appendix A for
the details on the uncertainties of asymmetry measurements), we find that the
presence of a quasar leads to a slight excess of asymmetry as shown in some
low mass bins and at the red and high-mass end. A simple interpretation could
be that less massive galaxies are more easily affected by AGN feedback which
may redistribute gas and hence influence the location of subsequent star
formation. Alternatively, if quasars trigger star-forming activity, which lead
to asymmetries, it may be a relatively stronger effect in systems that
originally had low star-forming activities (quiescent and red galaxies).

Figure 13: Similar to Fig. 10, median asymmetry but now binned with $M_{*}$
from $10^{9.6}$ to $10^{11.4}M_{\odot}$ per 0.3 dex, and rest-frame U-V host
color from 0.7 to 2.5 per 0.3 mag. The left (right) panel is for quasar hosts
(control galaxies).

###  4.4 Band dependencies

Figure 14: Similar to Fig. 9, $A_{\rm CAS}^{i}$ as a function of quasar and
host galaxy properties but now shown for all five bands. For clarity, we only
show the median values, not all of the individual values, and the y-axis is in
log scale. The vertical bars indicate the error on the mean
($\sigma/\sqrt{N}$) for each bin.

As mentioned above, we find that our results depend on the optical band under
consideration. We first revisit the correlations, shown in Figure 9, for all
five optical bands (Figure 14). We calculate the Spearman correlation
coefficients between $A_{\rm CAS}$ and the other parameters. As an example, we
find that correlation coefficients, between $A_{\rm CAS}$ and $L_{\rm bol}$,
reduce from $\sim 0.35$ in bluer bands ($g$, $r$, and $i$) to 0.26 in $z$-band
and 0.17 in $y$-band. In addition, for each panel in Figure 14, the data
points for the bluer bands are mostly located above those from the redder
bands. As a remark, the $g$-band results are not likely to be greatly affected
by residuals from the model point source subtraction even with a lower host
fraction in $g$-band and poorer seeing conditions (see Sec. 3.4). The mean and
standard deviation of host fraction for the reliable objects in $grizy$ bands
are: $0.26\pm 0.22$, $0.37\pm 0.22$, $0.45\pm 0.21$, $0.49\pm 0.20$, and
$0.51\pm 0.20$, respectively.

We find a similar trend in Figure 11 that the correlations between merger
fraction and $L_{\rm bol}$ become weaker in the redder bands. The results in
$gri$ bands are roughly consistent, while elevated from the $zy$ bands. The
merger fraction of control galaxies are also lower than in $zy$ bands than in
$i$-band, which are $14.2\%\pm 0.5\%$ and $10.4\%\pm 0.4\%$, respectively.
These two values are generally higher than the quasar hosts even in the
luminous cases. We suggest a possible explanation that, if the quasars are
triggered by major mergers, they are more likely gas-rich (“wet") mergers
(Hopkins et al., 2008), which are easier to be found in bluer bands than in
redder bands.

Figure 15: Comparison of empirical Cumulative Distribution Functions (eCDF) of
$A_{\rm CAS}$ for the $r$ (left panel), $i$ (middle panel), and $z$ (right
panel) bands. The sample is separated into four categories (colored curves;
see text for details). The Kolmogorov–Smirnov test (K-S test) statistic $D$
and $p$ values are provided in each panel. $D_{\rm sf}$ and $p_{\rm sf}$ are
measured between Q_sf and G_sf, while $D_{\rm qs}$ and $p_{\rm qs}$ are
measured between Q_qs and G_qs.

In Figure 15, we further illustrate the information shown in Figure 12 by
using empirical Cumulative Distribution Functions (eCDF) in $r$, $i$, and $z$
bands. The curves represent the different samples: blue=star-forming quasar
hosts (Q_sf), orange=quiescent quasar hosts (Q_qs), green=star-forming
galaxies (G_sf), and red=quiescent galaxies (G_qs). We show the
Kolmogorov–Smirnov test (k-s test) statistic $D$ and $p$ value calculated
using scipy.stat.ks_2samp between quasar hosts and inactive galaxies separated
by their types. The k-s $D$ statistic indicates the maximum vertical distance
between the two eCDFs.

We see a single trend that the $p$ value increases from blue to red and the
$D$ value decreases, especially for star-forming hosts. The null hypothesis
for $p$ value here is that the two populations are identical in asymmetry,
thus larger $p$ value means higher probability that the two populations have
the same distribution. Because we keep two digits, $p<0.01$ will be shown as
0. The $D$ statistic indicates the maximum distance between the two eCDFs
which can also be visually identified by the solid and dashed curves that are
approaching each other from left to right. As in Section 4.3, we calculate the
excess median quasar host asymmetry over inactive galaxies for the entire
sample in $r$, $i$, and $z$ bands, which are $0.030\pm 0.002$, $0.017\pm
0.002$, $0.006\pm 0.002$, respectively. Therefore, we find that the answer to
whether quasar hosts are more asymmetric than inactive galaxies has an optical
band dependence. The reason for this effect will be discussed in detail in
Section 5.3

##  5 Discussion

###  5.1 The $L_{\rm bol}-A$correlation

Considering the correlations between asymmetry and quasar properties (Section
4.1), we start our discussion on the $i$-band results first and consider the
other bands later. We calculate the Spearman correlation coefficient
$\rho_{s}$ between two of the seven parameters. The results are shown as a
matrix in Figure 16. Values in each cell indicate $\rho_{s}$ between the x
parameter and y parameter. A lighter color refers to a stronger positive
correlation while a darker color indicates a stronger negative correlation.
The parameters given in the bottom row are of primary interest which show
their dependence of $A_{\rm CAS}^{i}$. We find that $A_{\rm CAS}^{i}$ is
subtly related to $M_{\rm BH}$ and $\lambda_{\rm edd}$, while more strongly
dependent on $L_{\rm bol}$ ($\rho_{s}=0.37$). Besides the BH properties, the
host properties ($M_{*}$ and rest-frame U-V color) play a lesser role on
$A_{\rm CAS}^{i}$. There is almost no correlation between redshift and $A_{\rm
CAS}^{i}$, as also shown in Figure 9. For cells in the other rows with
$|\rho_{s}|>0.3$ (Figure 16), the redshift dependence of $L_{\rm bol}$
correlation between $M_{*}$ and rest-frame U-V color traces the SFR-stellar
mass relation (Williams et al., 2009). The correlations between $M_{*}$ and BH
properties describe the BH-host co-evolution scenario (Kormendy & Ho, 2013),
and agrees with Suh et al. (2019), who reported 0.44 and 0.24 Pearson
correlation coefficients between $L_{\rm bol}^{\rm X-ray}$ and $M_{*}$ for
AGNs at $z<0.5$ and $0.5<z<1.0$ AGNs. The correlations between the three BH
properties are driven by Equations 3 and 4. Considering the mutual dependence
between the above-mentioned parameters, we also calculated the partial
Spearman correlation coefficient with python package pingouin.partial_corr
between $L_{\rm bol}$ and $A_{\rm CAS}^{i}$. The effect of $M_{\rm
BH}$111Either $M_{\rm BH}$ or $\lambda_{\rm edd}$ could be controlled, but not
both. Otherwise $L_{\rm bol}$ will be fixed, $M_{*}$, rest U-V, and redshift
are controlled. We find that the partial correlation coefficient remains at
0.31, with a 95% confidence level between $0.27\sim 0.35$. Therefore, we claim
that there is a weak independent correlation between quasar $L_{\rm bol}$ and
quasar host asymmetry.

Figure 16: Spearman correlation coefficient matrix for seven parameters. We
claim that $L_{\rm bol}$ has the strongest correlation with quasar host
asymmetry.

We then separate the samples into faint and luminous quasars with a threshold
of $L_{\rm bol}=10^{45}\mathrm{erg\ s^{-1}}$ and re-investigate the $L_{\rm
bol}-A$ correlation for both subcategories in all five bands using three
asymmetry indices. The threshold is decided by the evidence shown in Figure 9
(first panel), in which the median asymmetry (green curve) turns up at $\sim
45$. We show the 95% confidence levels of the $L_{\rm bol}-A$ partial
correlation Spearman coefficients as colored bars in Figure 17. The number of
reliable sources in each subcategory is shown in grey. We find that the
results for the three asymmetry indices are consistent. The faint subcategory
tends to have weaker $L_{\rm bol}-A$ correlation than the luminous sample in
all five bands. This indicates that, only when quasars are bright enough,
their luminosity could have some correlation with their host. On the other
hand, we also see a trend that this correlation is stronger in bluer bands
than in redder bands, a reproduction of the results presented in Figure 14. It
could still be explained by the stellar population scenario mentioned above.
While this analysis uses "host only" measurements, the results are similar for
"host+comp" frames.

Figure 17: Confidence levels (95%) on the Spearman partial correlation
coefficients between $L_{\rm bol}$ and $A_{\rm CAS}$ (left), $A_{\rm outer}$
(middle), $A_{\rm shape}$ (right). Results for the $grizy$ bands are shown as
different colored bars, followed with the numbers of reliable sources in each.
Quasars are separated into faint and luminous subcategories with a threshold
of $L_{\rm bol}=10^{45}\mathrm{ergs^{-1}}$. The effects of $M_{\rm BH}$,
$M_{*}$, rest frame U-V color, and redshift are controlled for this partial
correlation.

Based on these findings, the $L_{\rm bol}-A$ correlation can be interpreted in
multiple ways. First, it may mean that high asymmetries cause an increase in
$L_{bol}$, consistent with the widely explored merger-trigged AGN accretion
scenario. Early simulation studies (e.g., Hernquist, 1989; Barnes & Hernquist,
1991; Di Matteo et al., 2005) showed that galaxy mergers may trigger gas
inflow and ignite the central SMBH. This scenario is partly supported by
recent observations showing that more luminous AGNs tend to have larger merger
fractions (Treister et al., 2012; Glikman et al., 2015; Hong et al., 2015; Fan
et al., 2016; Kim et al., 2017; Alonso et al., 2018; Goulding et al., 2018;
Ellison et al., 2019; Gao et al., 2020; Kim et al., 2021). With correlations
between $L_{\rm bol}$, $M_{\rm BH}$, and $\lambda_{edd}$, it is also natural
to see an enhanced merger fraction when the BH is more massive (Hong et al.,
2015) or has a higher accretion rate (Marian et al., 2020). However, given the
existing evidence against the correlation between BH properties and merger
fraction (Kocevski et al., 2011; Villforth et al., 2014; Mechtley et al.,
2016; Villforth et al., 2017; Hewlett et al., 2017), and the overall low
fraction of mergers among AGNs (Gabor et al., 2009; Cisternas et al., 2010;
Silverman et al., 2011), it is still unclear whether the merger-trigged AGN
scenario is enough to explain the correlations described above. As a test, we
removed all the mergers identified by $A_{\rm CAS}^{i}>0.35$ in “host+comp"
frames, and re-calculated $\rho_{s}$. Interestingly, $\rho_{s}$ between
$A_{\rm CAS}^{i}$ and $L_{\rm bol}$ remains almost unchanged (0.37). This
agrees with Figure 15 which shows that, for the population with $A_{\rm CAS}$
below 0.35 in $gri$ bands, quasar hosts are also more asymmetric than control
galaxies. Therefore, there is something other than major mergers making the
quasar hosts more asymmetric than the control galaxies, and it is correlated
to $L_{\rm bol}$. It is still possible to be merger-related (i.e., minor or
late-stage mergers) because the $A>0.35$ criteria mainly selects major
mergers. Recent semi-analytic models such as $v^{2}GC$ (Shirakata et al.,
2019) showed that minor mergers could trigger $30\%\sim 90\%$ of AGNs at $z<1$
from the brightest population to the faintest (also see Fanidakis et al.
(2012); Menci et al. (2014); Griffin et al. (2019)). Future work with Gini and
$M_{20}$ measurements (Lotz et al., 2004) on the same sample set will help us
to test this scenario.

Alternatively, it can be $L_{\rm bol}$ that causes increased asymmetries
($A_{\rm CAS}^{i}$). It is known that morphological parameters trace the star
formation activity to some level (Conselice et al. (2000); Conselice (2003),
also discussed). The star-forming main sequence (SFMS, SFR-stellar mass
relation) has been reported to have a scatter of $\sim 0.3$ dex (Daddi et al.,
2007; Speagle et al., 2014). This means besides the stellar mass, some other
properties of the galaxies are related to the SFR. Using $\sim 3700$ SDSS
Stripe 82 star-forming galaxies (SFGs), Yesuf et al. (2021) applied the mutual
information framework to study the relevance of several structural parameters
on the offset of specific SFR from SFMS. They found that asymmetry provides
the most information on the offset residual, thus is the second important
parameter (next to $M_{*}$) to the SFR of galaxies. This is supported by the
negative correlation between the color and asymmetry that we see in Figure 16,
indicating bluer quasar hosts are more asymmetric. Therefore, here we roughly
translate the $L_{\rm bol}-A$symmetry correlation (after removing the effect
of $M_{*}$) to the positive AGN feedback scenario, in which the BH activity
may trigger SFR (Ishibashi & Fabian, 2012; Silk, 2013). This scenario may also
explain our findings in Section 5.3. The correlation is stronger in bluer
bands because it is the younger stellar population that directly relates to
the AGN activity, while the older population is less affected. And if the AGN
is more powerful (luminous), it will have stronger feedback to form stars.
Especially in quiescent quasar hosts, the newly-born stars are separated from
the old population, while in star-forming quasar hosts, they could be mixed up
with pre-quasar phase star-forming activities (Figures 12 and 15). Also, if
the quasar host is less massive, it would be easier for the quasar to affect
it (Figure 13). Studies have directly tested this connection using different
SFR estimators including far IR luminosity, SED fitting, and [O ii]-[O iii]
emission lines (Hickox et al., 2014; Lanzuisi et al., 2017; Zhuang & Ho,
2020). However, our HSC data does not cover far IR and five-band optical data
is not reliable enough to fit a good SED model for SFR estimation. The Rakshit
et al. (2020) catalog does not include the [O ii] emission line. We leave the
estimation of SFR of our quasar hosts for a future work.

Lastly, the correlation between $L_{\rm bol}$ and $A_{\rm CAS}^{i}$
(considering it here as an indicator of SFR) could be driven by a third
parameter that is mutually related to both and not considered in Figure 16.
One possibility is the molecular gas mass $M_{\mathrm{H}_{2}}$. It is
suggested that the growth of BH and star formation activity could be fueled by
the same cold gas reservoir (Springel et al., 2005; Hopkins & Hernquist,
2006). For example, Shangguan et al. (2020) studied CO lines of 40 Palomar-
Green (PG) quasars at $z<0.3$ observed by ALMA. They estimated the molecular
gas mass of the quasar hosts according to the CO-to-$\mathrm{H_{2}}$
conversion factor ($\rm\alpha_{CO}$). They found the correlation between IR
luminosity $L_{\rm IR}$, which is used as an indicator of SFR, and AGN
luminosity $\lambda L_{\lambda}(5100\text{\AA})$ disappears after removing the
dependence of $L_{\rm IR}$ on $L_{\rm CO}$, an indicator of
$M_{\mathrm{H}_{2}}$. Latter work done by Zhuang et al. (2021) investigated
453 SDSS DR7 quasars at $0.3<z<0.35$. They estimated the molecular gas of
quasar hosts using a new method developed by Yesuf & Ho (2019) based on dust
extinction and gas-phase metallicity. They also found that the correlation
between BH activity (they used BHAR) and SFR have mutual dependence on
$M_{\mathrm{H}_{2}}$. And even after removing the effect of
$M_{\mathrm{H}_{2}}$, a weak correlation still exists. Yesuf & Ho (2020) also
applied the same method and found that the molecular gas fraction and the
Eddington ratio have moderately strong correlations in AGN-dominated galaxies.
However, we cannot test this gas-driven scenario with either method. We do not
have CO line information about our quasars, and the Yesuf & Ho (2019) method
is also not applicable for the whole sample. Because the dust extinction was
traced by $\mathrm{H\alpha/H\beta}$, while not all of our quasar spectra cover
H$\alpha$. Yesuf et al. (in preparation) is performing a test on this method
using $\sim 2000$ HSC Type-1 AGNs at $z<0.35$.

In summary, there are at least three possible scenarios that can explain the
$L_{\rm bol}-A$symmetry correlation. Nevertheless, it is difficult to
distinguish between them with our current information in hand. More detailed
classification of mergers, and more reliable measurements of SFR and gas mass
are required in future work.

###  5.2 Merger rates

Considering the best-fit relations in Figure 11, the intersection between the
merger fractions of quasars and control galaxies is located at $L_{\rm
bol}\sim 45$, similar to the turning point we reported in Figure 9. Below this
luminosity, the merger fractions of quasars are lower than that of control
galaxies. The merger fraction of the entire control galaxy sample in $i$-band
is $16.8\pm 0.5\%$ (see Sec. 4.3 for asymmetry values of control galaxies),
which is shown as the shaded grey line in Figure 11. We suggest a possible
explanation that low luminosity quasars are preferentially formed by minor
mergers (Shirakata et al., 2019), or secular processes such as the influence
of bars (Crenshaw et al., 2003; Ohta et al., 2007) and disk instabilities
(Hirschmann et al., 2012), or found as fading quasars in merger remnants
(Bennert et al., 2008). Therefore, low luminosity quasars are a biased
population that is less likely to be classified as mergers using the asymmetry
values, which is only sensitive to major mergers (Conselice, 2014). On the
other hand, luminous quasars ($L_{\rm bol}>45$) have an excess of mergers up
to $\sim 15\%$ compared to the control galaxies. Referring to examples shown
in Figure 18, seven of the eight most asymmetric cases have $L_{\rm bol}>45$.
Our findings that luminous quasars are more likely to be in mergers reflects
the high $L_{\rm bol}$ values for those within the high asymmetry tail of the
distribution as indicated by the orange data points in Figure 9. Note that the
most luminous bin in Figure 11 is roughly the second most luminous bin in
Figure 9, we did not show the merger fraction for the most luminous quasars
with $L_{\rm bol}>46$ here due to the poor statistics (error bars covering
$\sim 20\%$). Therefore, the claim here that luminous quasars are more likely
to be mergers does not conflict with the claim in Section 4.1 that most of the
most luminous quasars are not mergers. It’s worth noting that our best-fit
relation (Fig. 11) reaches 0.35 at most thus never approaches unity.

Figure 18: $i$-band images after removing point source and companions for 8
most luminous cases (upper two rows) and 8 most asymmetric cases (bottom two
rows). Each source is titled with its SDSS name and spectroscopic redshift.
$A_{\rm CAS}$, $A_{\rm outer}$, and $A_{\rm shape}$ values are shown for each
source. We find that the most luminous quasars and most asymmetric quasar
hosts are two different populations. The asymmetry values of the former are
mainly contributed by intrinsic processes and typically around 0.2, while that
of the later are mainly contributed by extrinsic processes and typically above
0.3.

###  5.3 Interpretation of optical band dependencies and its impact on
determining merger fraction

Figure 19: Examples of four “host only" Type-1 quasar hosts (SDSS name and
redshift are titled on top of each column), whose CAS asymmetries were
measured in $giy$ bands from top to bottom. We show that the variation of CAS
asymmetry values over bands could be caused by band-dependent, faint tidal
tails, bad quasar/AGN subtraction, the band-dependent prevalence of young
stellar populations, and blurred secondary galaxies.

Previous works have not reached a clear consensus on whether the merger rate
of quasar hosts is higher than inactive galaxies over a broad range of
redshift and across the demographics of the AGN population. For example,
Marian et al. (2020) finds an enhancement of the merger fraction of AGNs at
the highest Eddington ratios at $z<2$. As well, Goulding et al. (2018) finds a
clear excess of mergers associated with luminous obscured AGNs in the HSC-SSP
survey. However, Villforth et al. (2017) reports that even the most luminous
AGNs do not have an enhanced rate of disturbance compared to inactive
galaxies. Therefore, the tension is not yet fully resolved. Beyond the
luminosity dependence that we have shown in Figure 11, we suggest that the
band dependence could be another factor to reconcile the differences seen in
the literature.

To demonstrate the dependence on a given optical band, we show four cases of
“host only" quasar images whose $A_{\rm CAS}$ values differ in blue and red
bands (Figure 19). For SDSS J011040.14-002434.2, the excess of $A_{\rm
CAS}^{i}$ is mainly driven by the better image quality in this band (Section
3.4). In the second case, SDSS J010922.75-002621.1, we find the excess in $gr$
band could be artificial due to the subtraction residual in the center. In a
third case, SDSS J154603.79+421816.3, the excess in bluer band is caused by
better-resolved spiral arms and star-forming regions. In a fourth case, SDSS
J230648.66+004106.4 is probably a minor merger. The secondary galaxy is
evident in bluer bands, while blurred with the main galaxy in $y$-band.

Correspondingly, we consider four general reasons for the excess asymmetry in
the bluer bands. First, it could be a technical bias related to the image
quality in different bands. Specifically, the S/N and resolution are two main
factors that may affect the asymmetry measurements. In Appendix A, we show how
we correct the S/N issue according to a simulation work by Thorp et al.
(2021). While not correcting for resolution-based issues, we see in Figure 23
that the excess of asymmetry in bluer bands still exists even in smaller
spatial bins. The values in Figure 19 have already been corrected for S/N.
Second, it could be a systematic effect caused by our 2D image decomposition
method using GaLight. However, the excess still exists as seen from a check of
the results based on the “host+comp" frames which include the companions.
Considering the subtraction of point sources, because the residual mainly
affects the central region, $A_{\rm outer}$ can mitigate this effect to some
level. And since $A_{\rm shape}$ flattens all the values, it is less sensitive
to the poor residuals. Therefore, we check $A_{\rm outer}$ and $A_{\rm shape}$
and find they also have excesses in the bluer bands. Third, it could be caused
by the stellar population. The bluer bands capture light from a younger
stellar population, while the redder bands are more sensitive to the older
stars. In that sense, the excess in bluer bands would indicate that the
younger stellar population is more unevenly distributed as compared to the
older stellar population. This has already been pointed out by early works
(Conselice, 1997; Conselice et al., 2000), in which they refer to it as
"flocculent" asymmetry. Fourth, it could be a systematic bias in identifying
mergers in different bands. For example, when the secondary galaxy or tidal
features are dominated by young stars that are newly triggered by the merger,
the system will more likely be identified as a merger. This is somewhat
entangled with the "flocculent" asymmetry caused by star-formation activity
but mainly driven by the dynamics. Spectroscopic selection of mergers such as
being applied in Ellison et al. (2013, 2019) will yield purer merger samples
to deal with the systematic effects. Considering that most of our samples are
non-mergers, and they usually do not have significant issues with subtraction
residuals (otherwise rejected), we suggest that "flocculent" asymmetry to be
the main cause of the statistical excess of asymmetry in the bluer bands.

We then compared our results with previous works that have studied the merger
rate of Type-1 AGNs at various luminosities (Table 1). None of these works
discussed the "flocculent" asymmetry because most of them only used one
photometric band. Here, we can explore whether the asymmetry is affected by
the stellar population by comparing results at a common rest-frame wavelength
(Column 6). All of these works compared the merger fractions between AGNs and
control inactive galaxies, which are listed in Columns 7 and 8. Because the
definition of mergers vary in different studies, instead of the fraction
itself, comparison between $f_{\rm merger}^{Q}$ and $f_{\rm merger}^{G}$ are
more meaningful. In that sense, we find these works follow such a pattern:
when considering stellar populations at $\lambda_{\rm eff}^{\rm
rest}<5200\text{\AA}$ (our HSC study in $gri$ bands, Gabor et al. (2009);
Cisternas et al. (2010); Koss et al. (2010); Boehm et al. (2013); Mechtley et
al. (2016); Marian et al. (2019, 2020)), they tend to find more or less an
excess of $f_{\rm merger}^{Q}$ over $f_{\rm merger}^{G}$. When sensitive to
$5200\text{\AA}<\lambda_{\rm eff}^{\rm rest}<9000\text{\AA}$ (This work in
$zy$ band, Cotini et al. (2013); Ellison et al. (2019); Zhao et al. (2022)),
the results are mixed. At $\lambda_{\rm eff}^{\rm rest}>9000\text{\AA}$
(Villforth et al., 2014, 2017), so far no excess of $f_{\rm merger}^{Q}$ is
reported. Rest-frame wavelengths around 5200 Å are roughly the turning point
in the spectra of G and K stars (Yan et al., 2019). Its blue ward emission is
contributed by younger O, B, A, and F stars. Up to 9000 Å, it is contributed
by all types of stars. While beyond 9000 Å, the M stars dominate the spectra.
We reemphasize our point mentioned in Section 4.4 and Figure 11 that, if a
quasar is formed in major merger, such a merger is more likely to be gas-rich
(Hopkins et al., 2008), thus have young stellar population. Such a merger is
easier to be classified either with quantification method or visual inspection
in bluer bands than in redder bands. We do not deny a possible intrinsic
redshift evolution of the asymmetry at higher redshift, since we only showed a
flattened curve at $z<0.8$ (Figure 14). Also, we clarify that our results are
only for Type-1 quasars.

Table 1: Merger fractions of quasar hosts and normal galaxies

works | $N_{\rm AGN}$ | observed band | $z_{\rm range}$ | $<z>$ | $\lambda_{\rm eff}^{\rm rest}(\text{\AA})$ | log$L_{\rm bol}$ | $f_{\rm merger}^{Q}$ | $f_{\rm merger}^{G}$ | classifier
---|---|---|---|---|---|---|---|---|---
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10)
This work | 886 | HSC_g | $0.2-0.8$ | 0.49 | 3194 | $43.1-46.4$ | $15.8\%\pm 1.4\%$ | $14.4\%\pm 0.7\%$ | Asymmetry
This work | 1488 | HSC_r | $0.2-0.8$ | 0.54 | 3984 | $43.1-46.4$ | $16.0\%\pm 1.0\%$ | $14.7\%\pm 0.6\%$ | Asymmetry
This work | 1987 | HSC_i | $0.2-0.8$ | 0.60 | 4801 | $43.1-46.4$ | $16.2\%\pm 0.9\%$ | $16.8\%\pm 0.5\%$ | Asymmetry
This work | 1939 | HSC_z | $0.2-0.8$ | 0.60 | 5567 | $43.1-46.4$ | $10.3\%\pm 0.7\%$ | $14.2\%\pm 0.5\%$ | Asymmetry
This work | 1759 | HSC_y | $0.2-0.8$ | 0.58 | 6177 | $43.1-46.4$ | $8.0\%\pm 0.7\%$ | $10.4\%\pm 0.4\%$ | Asymmetry
Gabor et al. (2009) | 19 | ACS_F814W | $0.3-1.0$ | $\sim 0.7$ | 4690 | $43.3-45.3$ | 68% a |  $\sim 25\%$ a | Asymmetry
Cisternas et al. (2010) | 83 | ACS_F814W | $0.3-1.0$ | 0.8 | 4429 | $43.6-46.3$ | $15.0\%\pm 8.8\%$ | $12.6\%\pm 6.5\%$ | Visual
Koss et al. (2010) | 72 | SDSS_gri | $<0.05$ | 0.03 | 4536 b | $\sim 44$ | 18% | 1% | Visual
Kocevski et al. (2011) | $72^{*}$ | WFC3_F160W c | $1.5-2.5$ | $\sim 2.0$ | 5093 | $43.3-45.3$ | $16.7_{-3.5}^{+5.3}\%$ | $15.5_{-2.2}^{+2.8}\%$ | Visual
Boehm et al. (2013) | 21 | ACS_F606W | $0.5-1.1$ | 0.71 | 3397 | $43.3-45.3$ |  $0\%-65.5\%$ d | $20.2\%-52.6\%$ | Both
Cotini et al. (2013) | $59^{*}$ | SDSS_r | $0.003-0.03$ | $\sim 0.02$ | 6021 | $\sim 44$ | $20_{-5}^{+7}\%$ | $4_{-1.2}^{+1.7}\%$ | Both
Villforth et al. (2014) | $60^{*}$ | WFC3_F160W | $0.5-0.8$ | $\sim 0.65$ | 9259 | $42.3-45.8$ | $<6\%$ | $\sim 10\%$ | Both
Mechtley et al. (2016) | 19 | WFC3_F160W | $1.9-2.1$ | 2.0 | 5093 | $46.7-47.7$ | $39\%\pm 11\%$ | $30\%\pm 5\%$ | Visual
Villforth et al. (2017) | $20^{*}$ | WFC3_F160W | $0.5-0.7$ | 0.62 | 9431 | $45.4-47.2$ | $\sim 25\%$ | $\sim 25\%$ | Both
Ellison et al. (2019) | 1124 | CFIS_r | $0-0.25$ | $\sim 0.1$ | 5760 | $41.8-46.3$ |  $7.7\%-30.9\%$ e | $2.9\%-16.0\%$ | Visual
Marian et al. (2019) | 21 | WFC3_F160W | $1.8-2.2$ | 1.95 | 5179 | $46.5-47.2$ | $24\%\pm 9\%$ | $19\%\pm 4\%$ | Visual
Marian et al. (2020) | 17 | FORS2_B | $0.09-0.19$ | 0.15 | 3687 | $45.3-46.9$ |  $41\%\pm 12\%$ f | $8\%\pm 6\%$ | Visual
Marian et al. (2020) | 17 | FORS2_V | $0.09-0.19$ | 0.15 | 4723 | $45.3-46.9$ | $41\%\pm 12\%$ | $8\%\pm 5\%$ | Visual
Zhao et al. (2022) | 245 | PS1_i | $0.04-0.15$ | 0.09 | 6883 | $43.4-43.8$ | 3.7% | 3.2% | Asymmetry

List of works that have been working on the comparison between merger
fractions of Type-1 AGN host and inactive galaxies.
Column (2): Number of Type-1 AGNs used in the work. If the Type-1 and Type-2
were not separated (usually selected from X-ray), we count the total number
and mark it with a *.
Column (3): The bands from which the imaging was taken and classified. Column
(4): Redshift range of the AGNs.
Column (5): The median redshift of the samples. If it is not provided in the
work, we roughly estimate it from the distribution plots.
Column (6): The effective wavelength midpoint of the band divided by $1+z_{\rm
median}$. Information on the bands is taken from the SVO filter profile
service.
Column (7): Range of bolometric luminosities of the AGNs. When only X-ray
luminosity is provided in the work, we convert it to $L_{\rm bol}$ with a
typical correction factor $L_{\rm bol}/L_{\rm X(2-10keV)}=20$ according to
Lusso et al. (2012) or $L_{\rm bol}/L_{\rm X(14-195keV)}=8$ according to Koss
et al. (2017). Some works also use $L_{\rm[OIII]}$, we convert it with a
correction factor $L_{\rm bol}/L_{\rm[OIII]}=600$ according to Kauffmann &
Heckman (2009).
Column (8-9): Merger fraction of AGNs (or Quasars) and matched inactive
galaxies. For this work, the uncertainties are the Poisson errors, i.e.,
$\sqrt{N_{\rm merger}}/N_{\rm total}$.
Column (10): Methods being used to classify mergers. "Both" means both visual
and quantitative methods were used.
a: We only count for their Type-1 AGNs (class "X1") with good fittings. The
mergers are classified as $A>0.35$, although not used in their work. They did
not provide the precise merger fraction of the control galaxies, or the data.
So we roughly counted $f_{\rm merger}^{G}$ from their Figure 10 X-ray panel.
However, this comparison might not be fair since the Type-1 AGNs are a subset
of their X-ray AGNs, while the control galaxies are matched to all X-ray AGNs.
b: They used composite gri images for classification, we see from Figure 1
that the interacting features are mostly blue. Thus, we note the $\lambda_{\rm
eff}^{\rm rest}$ for SDSS_g band here. c: They also had F125W band in their
data, but was not used in the results.
d: According to their Table 1 and 2. The lower limit assumes only their major
mergers are real, while the upper limit adds all their unclear cases as well.
We take the average values of their 4 criteria as the number of mergers.
e: According to their Table 1, the lower limit counts the interacting pairs
only, while the upper limit includes the post-mergers.
f: According to their Figure 3, cut-off rank = 10. The same for the V band
below.

##  6 Conclusions

We have measured asymmetry indices of 2424 quasar host galaxies at $0.2<z<0.8$
from the SDSS DR14 catalog using all five broad-band optical images from the
HSC SSP. This represents the largest sample of Type-1 quasar hosts with
asymmetry measurements. We determine the dependence of asymmetry on intrinsic
quasar and host properties including systematic differences between optical
bands. Control galaxies are selected by matching the redshift and stellar mass
of the quasar hosts. The GaLight image decomposition tool is used to subtract
the point source from the quasar images. We use two types of quasar host
images, either with or without the nearby companions removed based on model
fitting. Both are then analyzed using statmorph to measure the $A_{\rm CAS}$,
$A_{\rm outer}$, and $A_{\rm shape}$ values. Simulations with mock quasars do
not show a bias in the asymmetry measurements caused by the quasar
subtraction. After applying the statmorph flags, we keep $\sim 80\%$ of the
total sample in $i$-band ($37\%-80\%$ in other bands) of which only $\sim 2\%$
of these sources are affected by significant subtraction issues (Section 3.4).
Based on these sources, we find the following:

  * •

The asymmetry indices of quasar hosts are correlated with $L_{\rm bol}$
primarily due to a broad tail of higher asymmetries at higher quasar
luminosities (Sec. 4.1 and 4.2). There is a clear upturn in the median
asymmetries binned in luminosity, hence merger rates, at $L_{\rm Bol}=45$.
This correlation is stronger in the bluer bands.

  * •

Quasar hosts are more asymmetric when they harbor more massive and more active
black holes (i.e., higher Eddington rate), and when their host galaxies are
more massive and bluer (Section 4.1).

  * •

Type-1 quasar hosts are _slightly_ more asymmetric than inactive galaxies with
a median difference of $\sim$ 0.02 in $i$-band with a significance of
9.4$\sigma$. The overall merger fractions of quasar hosts and inactive
galaxies are both around $\sim 8.0-17.1\%$.

  * •

An optical band (wavelength) dependence is an essential factor in (1) the
excess of asymmetry in quasar hosts over control galaxies, (2) the
correlations between the physical properties, and (3) merger ratios. Bluer
bands ($gri$) generally have larger asymmetry values than the redder bands
($yz$). Such an effect is stronger for quasar hosts than the inactive
galaxies. In bluer bands, the differences between quasar hosts and inactive
galaxies is larger, and the correlations between quasar properties and
asymmetry also become stronger. While all the correlations are weaker in
redder bands (Section 4.4).

  * •

Bringing our results to common rest-frame wavelengths and comparing to
previous works, we find that the discrepancies on whether quasar hosts have
merger fractions higher than inactive galaxies could be explained partly by
the stellar populations of the quasar host galaxies. At rest-frame wavelengths
bluer than 5200 Å, $f_{\rm merger}^{Q}$ is more or less larger than $f_{\rm
merger}^{G}$. When the rest wavelengths are between 5200 Å and 9000 Å, the
results are mixed. Beyond 9000 Å, no excess of $f_{\rm merger}^{Q}$ is
reported (Section 5.3).

## Acknowledgements

The Hyper Suprime-Cam (HSC) collaboration includes the astronomical
communities of Japan and Taiwan, and Princeton University. The HSC
instrumentation and software were developed by the National Astronomical
Observatory of Japan (NAOJ), the Kavli Institute for the Physics and
Mathematics of the Universe (Kavli IPMU), the University of Tokyo, the High
Energy Accelerator Research Organization (KEK), the Academia Sinica Institute
for Astronomy and Astrophysics in Taiwan (ASIAA), and Princeton University.
Funding was contributed by the FIRST program from the Japanese Cabinet Office,
the Ministry of Education, Culture, Sports, Science and Technology (MEXT), the
Japan Society for the Promotion of Science (JSPS), Japan Science and
Technology Agency (JST), the Toray Science Foundation, NAOJ, Kavli IPMU, KEK,
ASIAA, and Princeton University.

This paper is based [in part] on data collected at the Subaru Telescope and
retrieved from the HSC data archive system, which is operated by Subaru
Telescope and Astronomy Data Center (ADC) at NAOJ. Data analysis was in part
carried out with the cooperation of Center for Computational Astrophysics
(CfCA) at NAOJ. We are honored and grateful for the opportunity of observing
the Universe from Maunakea, which has the cultural, historical and natural
significance in Hawaii.

This paper makes use of software developed for Vera C. Rubin Observatory. We
thank the Rubin Observatory for making their code available as free software
at http://pipelines.lsst.io/.

The Pan-STARRS1 Surveys (PS1) and the PS1 public science archive have been
made possible through contributions by the Institute for Astronomy, the
University of Hawaii, the Pan-STARRS Project Office, the Max Planck Society
and its participating institutes, the Max Planck Institute for Astronomy,
Heidelberg, and the Max Planck Institute for Extraterrestrial Physics,
Garching, The Johns Hopkins University, Durham University, the University of
Edinburgh, the Queen’s University Belfast, the Harvard-Smithsonian Center for
Astrophysics, the Las Cumbres Observatory Global Telescope Network
Incorporated, the National Central University of Taiwan, the Space Telescope
Science Institute, the National Aeronautics and Space Administration under
grant No. NNX08AR22G issued through the Planetary Science Division of the NASA
Science Mission Directorate, the National Science Foundation grant No.
AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the
Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation.

We thank the anonymous reviewers for their insightful comments on an earlier
draft of this work. S.T. thanks Mallory D. Thorp for her advice on the
application of corrections to asymmetry values. J.D.S. is supported by the
JSPS KAKENHI Grant Number JP22H01262, and the World Premier International
Research Center Initiative (WPI Initiative), MEXT, Japan. H. Yesuf was
supported by JSPS KAKENHI Grant Number JP22K14072 and the Research Fund for
International Young Scientists of NSFC (11950410492). X.D. is supported by
JSPS KAKENHI Grant Number JP22K14071. C.B. gratefully acknowledges generous
support from the Natural Sciences and Engineering Research Council of Canada
through their post-doctoral fellowship program.

Software: numpy (Harris et al., 2020); matplotlib (Hunter, 2007); pandas
(pandas development team, 2020); seaborn (Waskom, 2021); scikit-learn
(Pedregosa et al., 2011); TOPCAT (Taylor, 2011); GaLight (Ding et al., 2021);
photutils (Bradley et al., 2022); statmorph (Rodriguez-Gomez et al., 2019).

Database: SVO filter profile service, http://svo2.cab.inta-csic.es/theory/fps/

## Data Availability

Table 2: Asymmetry measurements for quasar hosts, complete version is available online  name | host_A_i | host_oA_i | host_sA_i | host_s2n_i | host_res_i | host_flag_i | …
---|---|---|---|---|---|---|---
000002.50+021818.9 | 0.129 | 0.334 | 0.233 | 9.187 | 1.56 | 0 | …
000010.97+005653.3 | 0.031 | 0.038 | 0.069 | 16.818 | 0.77 | 0 | …
000017.74+020027.9 | 0.065 | 0.143 | 0.445 | 29.709 | 1.67 | 0 | …
000017.88+002612.6 | 0.074 | 0.117 | 0.132 | 14.253 | 1.42 | 0 | …
000022.05+011742.5 | 0.047 | 0.275 | 0.567 | 61.347 | 1.11 | 0 | …
… | … | … | … | … | … | … | …
Table 3: Asymmetry measurements for control galaxies, complete version is available online  name | host_A_i | host_oA_i | host_sA_i | host_s2n_i | host_res_i | host_flag_i | …
---|---|---|---|---|---|---|---
43128210455553649 | 0.091 | 0.151 | 0.268 | 5.559 | 1.14 | 1 | …
42072271270998533 | 0.036 | 0.025 | 0.146 | 5.29 | 1.21 | 1 | …
44191880581244422 | 0.038 | 0.086 | 0.236 | 7.529 | 1.04 | 1 | …
40972617909296239 | 0.029 | 0.042 | 0.102 | 10.603 | 0.94 | 0 | …
40985945192811690 | 0.084 | 0.127 | 0.153 | 4.13 | 1.4 | 0 | …
… | … | … | … | … | … | … | …

The measurements of asymmetries for each source, including the quasar hosts
and control inactive galaxies are available online. The entire catalog
included the physical parameters of the quasars, we refer the readers to Table
1 of Li et al. (2021a) for the meaning of those lines. The asymmetry
measurement results are appended from the 36th column to the 100th column.
Basically, the columns are named as
“{sersic/host}_{A/oA/sA/s2n/res/flag}_{g/r/i/z/y}", where “sersic" stands for
the measurement for “host+comp" frames, “host" stands for that of “host only"
frames. “A", “oA", and “sA" represent the CAS asymmetry, outer asymmetry, and
shape asymmetry, respectively. “s2n" stands for the signal-to-noise ratio per
pixel of the host galaxies measured by statmorph, “res" stands for the
resolution of the galaxy estimated with half-light radius and FWHM. These two
parameters are used in Appendix A to qualify the corrections of asymmetry
values using simulation. All the asymmetry values in our data table have
applied the corrections. Finally, “flag" stands for the statmorph flag, 0=good
and 1=bad (Section 3.4). The empty cells in the data table are the failures at
the stage of deblending and masking (Section 3.3.1). These make up $2\times
6\times 5=60$ columns, a short example for “host only" frames in $i$ band is
shown in Table 2. We also added five columns for the sky background asymmetry
measured for each source, which is named “sky_{g/r/i/z/y}". These are the
$A_{\rm bkg}/1.2$ values mentioned in Section A.

We keep the same format for the measurements of control inactive galaxies,
with a short example shown in Table 3. In the complete version, we include the
photo-z and stellar mass measurements from Kawinwanichakij et al. (2021). The
asymmetry measurement results are appended from the 9th column to the 73th
column.

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##  Appendix A Refinement of asymmetry measurements

Figure 20: Proof of the "sky box" method to measure the sky asymmetry. Left
panel: the five-band colored raw image of the quasar. The sky boxes are
selected at empty spaces where there is no source detected. The sky asymmetry
values per pixel are labeled. Right panel: $i$-band mask for the cutout image
generated from a simple “cool" mode detection on the raw data.  Figure 21:
Distribution of background asymmetry in all five bands ($A_{\rm
bkg}^{\\{g/r/i/z/y\\}}/1.2$) for the quasar hosts that have reliable
measurements. Mean ($\mu$) and standard deviation ($\sigma$) are shown for
each band. We take $\sqrt{2}\sigma$ as the typical uncertainty levels of our
asymmetry measurements on sources.

We first demonstrate the “sky box" method in statmorph that we applied in
Section 3.3.2. Figure 20 (left panel) is a five-band colored image of quasar
generated in the same way as Figure 5. With a raw image with size of
$90\arcsec\times 90\arcsec$, the mask for sky asymmetry measurements (Figure
20 right panel) is made from a simple “cool" mode detection (Section 3.3.1)
without any deblending. Thus, all the sources that have at least 5 continuous
pixels above $1\sigma$ threshold are masked. We make this mask for all five
bands independently and the sky box is also drawn independently (colored
squares in Figure 20 left panel). They do not necessarily take the same area
of the sky. We show the sky asymmetry values per pixel on the lower right
side.

In Figure 21, we show the distribution of background asymmetry measurements
for all reliable samples. The standard deviation $\sigma$ can be considered as
the typical uncertainty value for the background in each band. We can assume
that the object also has the same level of uncertainty. Then according to
function 6, the uncertainty of the source measurement can be combined as
$\sqrt{2}\sigma$. Generally, redder bands have higher sky asymmetry values and
larger standard deviation than bluer bands, as the background noise increase
from optical to NIR.

Thorp et al. (2021) pointed out that this method may over-subtract the sky
asymmetry and lead to an underestimation of the real asymmetry value of the
source. This is because the second item of Equation 6 takes the absolute value
of $B_{0}-B_{180}$, even a random noise field will have a positive value. This
also explains why the noisier redder bands have larger sky asymmetry values.
Thorp et al. (2021) randomly selected 1000 galaxies from the IllustrisTNG
simulation (Pillepich et al., 2018) with $M_{*}\geq 10^{9}M_{\odot}$ at $z=0$.
They added observational effects including the resolution and noise to the
simulated galaxies and studied how well statmorph can reproduce the intrinsic
asymmetry values. Their first main finding is that, instead of subtracting the
whole $A_{\rm bkg}$ (they refer to it as $A_{\rm noise}$), dividing the value
by 1.2 better recovers the intrinsic asymmetry. We tested with our $A_{\rm
CAS}^{i}$ measurements on “host only" images via subtracting the whole $A_{\rm
bkg}$, or $A_{\rm bkg}/1.2$, or $A_{\rm bkg}/1.4$ (Figure 22). We do not know
the intrinsic asymmetry of our data, but since this modification aims to
correct the bias caused by S/N, we would expect a successful correction to
flatten the asymmetry values concerning S/N (similar to their Figure 4). In
that sense, we roughly reproduced their curves at S/N$<$80, in which the
subtraction of $A_{\rm bkg}/1.2$ has the stablest $A_{\rm CAS}$ over the whole
S/N range for all five bands. While subtracting $A_{\rm bkg}/1.4$ and $A_{\rm
bkg}$ tend to overestimate and underestimate $A_{\rm CAS}$ at S/N$<$15\. Also,
subtracting $A_{\rm bkg}/1.2$ instead of $A_{\rm bkg}$ significantly reduced
the fraction of negative $A_{\rm CAS}$ from $19\%\sim 68\%$ to $3\%\sim 23\%$
in these bands.

Figure 22: Comparison of $A_{\rm CAS}$ measured on “host only" image via
subtracting $A_{\rm bkg}$ (blue points), $A_{\rm bkg}/1.2$ (blue points), or
$A_{\rm bkg}/1.4$ (blue points). The data are binned in S/N with a range of
four for $g$ and $y$ bands, a range of five for $r$, $i$, and $z$ bands.
Median values with mean $\sigma$ uncertainties are shown for each bin.  Figure
23: The noise-corrected $A_{\rm CAS}$ of quasar hosts binned in resolution
($R_{\text{1/2}}/\mathrm{FWHM}$). Median values are shown with mean $\sigma$
errors in all five bands. A clear trend exists that $A_{\rm CAS}$ increases
with resolution, the reason of which could either be technical or physical. We
do not apply the resolution correction to our results given its strong
degeneracy with S/N. Nevertheless, the offset between the bands at almost all
resolution bins in this plot indicates that the band dependence of $A_{\rm
CAS}$ should attribute to some other reasons, at least not only the
resolution.

Besides S/N, the low resolution may also smooth out an image and lead to
underestimated asymmetry values. To address this, we calculated the resolution
of each quasar host galaxy in each band as
$\mathrm{resolution}=R_{\text{1/2}}/\mathrm{FWHM}$ where $R_{\text{1/2}}$ is
the circular half-light radius calculated by
statmorph.source_morphology.rhalf_circ, and FWHM is calculated using
galight.tools.measure_tools.measure_FWHM on the input HSC PSF images. Figure
23 shows the median $A_{\rm CAS}$ in resolution bins for all five bands. There
is a clear trend that the values decrease towards low resolution. However, it
is unclear whether this trend is purely caused by resolution since it can be
entangled with S/N. Also, it could be physical effect since disturbed galaxies
are more likely to have larger $R_{\text{1/2}}$. Thorp et al. (2021) also
performed a fitting function to correct for the resolution issue, but the
scatter increases significantly compared to the results with only noise
corrections. Therefore, we decided to adopt the $A_{\rm bkg}/1.2$ noise
correction and not the resolution correction to our results. In the mean time,
we claim from this figure that the band differences are independent of
resolution thus supportive of the results given in Section 4.4.

##  Appendix B Control Galaxy measurements

For the inactive galaxies (control sample), we proceed with the image analysis
as done for the quasar images as described in Section 3. Here, the difference
is that there is no point source to subtract. All components within the image
are modeled with Sérsic profiles at the same depth (200 iterations) for the
PSO algorithm. Then these components, with the exception of the central
galaxy, are removed to produce the “host only" frame. While the control
galaxies are not "hosting" a quasar, we keep the name as used for the quasar
frames. As well, the processes of making segmentation masks and measuring
asymmetry are the same as done for the quasars. Figure 25 shows an example of
the statmorph measurements on one of the galaxies.

Figure 24: Similar to Figure 3, but for control galaxy, HSC ID
40972373096155148, RA=133.87344, Dec=-0.08921. In case of control galaxies,
data (first panel) play the role of "host+comp" frame.  Figure 25: Similar to
Figure 5, but for the control galaxy in Figure 24.

We then select reliable measurements from the control galaxies in the same
manner as for the quasars. The corresponding numbers of reliable objects in
every band are g, 2363; r, 4288; i, 6428; z, 6200; y, 5199. Figure 26 shows
the counts in each redshift bin and the corresponding reliability ratios in
each band, which generally shows the same trends as the quasar hosts.

Figure 26: Similar to Figure 6, but for control galaxies.

##  Appendix C Robustness of the correlations

Figure 27: Similar to Figure 9, but include the sources rejected by statmorph,
i.e., flag=1. We found the results (trends) are consistent in both cases.

We test the robustness of the reported correlations from two aspects. In
Section 3.4, we rejected sources that are flagged by statmorph. It is
difficult to figure out the exact reason of each rejection since only one flag
value is provided. We visually checked some of the rejected sources and did
not find significant issues in most of the cases. Therefore, we put those
sources back and reproduced Figure 9 as Figure 27. This increases the reliable
source number from 1987 to 2409. Now the rejection is only made by our
deblending and mask pipeline. For the rejected 15 sources, 10 are rejected due
to bad subtraction residuals, which lead to a failure in source detection; 3
are rejected due to contamination from large foreground sources; 1 is rejected
due to artifact; 1 is rejected due to no-detection of the host. None of these
rejected sources has merger features, and the median bolometric luminosity is
45.4, similar to the median of entire sample within $1\sigma$ ($45.2\pm 0.4$).
Therefore, our rejections do not bias to specific type of objects. In Figure
27, we find consistent trend as in Figure 9, thus as a proof that our results
are robust with different rejection criteria.

Figure 28: Distribution of the spearman correlation coefficients between
$A_{\rm CAS}^{i}$ and other parameters over 1000 time Monte Carlo simulations
for all the quasars that have reliable measurements of $A_{\rm CAS}^{i}$. The
16th, 50th, and 84th percentiles are labeled by the dashed vertical lines and
the numbers on top of each panel.

We then tested how the uncertainty of the parameters affect the correlations
reported in Figure 16 using Monte Carlo simulations. We assumed gaussian
distribution of the six parameters: $A_{\rm CAS}^{i}$, $L_{\rm bol}$, $M_{\rm
BH}$, $\lambda_{\rm edd}$, $M_{*}$ and rest-frame U-V color. The distribution
in Figure 21 can be used to estimate the uncertainty level of $A_{\rm CAS}$ as
$\sqrt{2}\sigma$ combining the object term and background term. For $i$-band,
we take it as $\sqrt{2}\times 0.02=0.03$. The uncertainty on $L_{\rm bol}$ is
mainly attributed to the bolometric correction (BC) factor. Richards et al.
(2006) reported a $\sigma$ of $\sim$ 2 for $\mathrm{BC}_{5100\text{\AA}}$,
which converts to $\sim$ 0.1 in log $L_{\rm bol}$ according to Equation 2. The
uncertainty level of $L_{5100}$ is $\sim$ 0.01 according to the Rakshit et al.
(2020) catalog, thus we ignore it and only considered the uncertainty on BC.
The error of $M_{\rm BH}$ is provided in the Rakshit et al. (2020) catalog,
which is mainly caused by the spectral fitting and inherent systematics of the
viral method. We use these $M_{\rm BH}$ errors as the $\sigma$ for each of our
quasar. We then estimated the uncertainty level of $\lambda_{\rm edd}$ via
combining the $\sigma$ of $L_{\rm bol}$ and $M_{\rm BH}$ in Equation 3. The
errors ($\sigma$) on $M_{*}$ is provided by Li et al. (2021a), which is a
result of SED fitting using CIGALE. Li et al. (2021a) also compared their rest
frame U-V color measurements to the values of CANDELS galaxies, and found a
median difference of 0.03. We used this as the $\sigma$ on the U-V color.

We perform this simulation for all quasars that have a valid $A_{\rm CAS}^{i}$
measurements. For each iteration, we calculated the spearman correlation
coefficients between $A_{\rm CAS}^{i}$ and the other parameters. The
distribution of the coefficients in 1000 iterations is shown in Figure 28 with
their 16th, 50th, and 84th percentile values labelled as dashed lines. We find
that adding these uncertainties reduces the correlation coefficient that we
have reported in Figure 16 by $\sim 20\%$, which does not affect our main
results.