#  Two rest-frame wavelength measurements of galaxy sizes at $z<1$: the
evolutionary effects of emerging bulges and quenched newcomers

Angelo George,1 Ivana Damjanov,1 Marcin Sawicki,1 Stéphane Arnouts,2 Guillaume
Desprez,1 Stephen Gwyn,3 Vincent Picouet,2 Simon Birrer4,5 and John
Silverman6,7,8
1Institute for Computational Astrophysics and Department of Astronomy &
Physics, Saint Mary’s University, 923 Robie Street, Halifax, NS B3H 3C3,
Canada
2Aix-Marseille University, CNRS, CNES, LAM, Marseille, France
3NRC Herzberg Astronomy and Astrophysics, 5071 West Saanich Road, Victoria, BC
V9E 2E7, Canada
4Department of Physics, Kavli Institute for Particle Astrophysics and
Cosmology, Stanford University, Stanford, CA 94305, USA
5SLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA
6Kavli Institute for the Physics and Mathematics of the Universe, The
University of Tokyo, Kashiwa, 277-8583, Japan
7Center for Data-Driven Discovery, Kavli IPMU (WPI), UTIAS, The University of
Tokyo, Kashiwa, Chiba 277-8583, Japan
8Department of Astronomy, School of Science, The University of Tokyo, 7-3-1
Hongo, Bunkyo, Tokyo 113-0033 Japan
E-mail<EMAIL_ADDRESS>Research ChairCanada Research Chair

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We analyze the size evolution of $16000$ star-forming galaxies (SFGs) and
$5000$ quiescent galaxies (QGs) with mass $M_{*}>10^{9.5}$M⊙ at $0.1<z<0.9$
from the COSMOS field using deep CLAUDS+HSC imaging in two rest-frame
wavelengths, $3000$ Å (UV light) and $5000$ Å (visible light). With half-light
radius ($R_{e}$) as proxy for size, SFGs at characteristic mass $M_{0}=5\times
10^{10}$M⊙ grow by $20\%$ ($30\%$) in UV (visible) light since $z\sim 1$ and
the strength of their size evolution increases with stellar mass. After
accounting for mass growth due to star formation, we estimate that SFGs grow
by $75\%$ in all stellar mass bins and in both rest-frame wavelengths. Redder
SFGs are more massive, smaller and more concentrated than bluer SFGs and the
fraction of red SFGs increases with time. These results point to the emergence
of bulges as the dominant mechanism for the average size growth of SFGs. We
find two threshold values for the stellar mass density within central $1$ kpc
($\Sigma_{1}$): all SFGs with $\log\Sigma_{1}\gtrsim 9$ are red and only QGs
have $\log\Sigma_{1}\gtrsim 9.7$. The size of $M_{*}=M_{0}$ QGs grows by
$50\%$ ($110\%$) in the UV (visible) light. Up to $\sim 20\%$ of this increase
in size of massive QGs is due to newcomers (recently quenched galaxies).
However, newcomers cannot explain the observed pace in the size growth of QGs;
that trend has to be dominated by processes affecting individual galaxies,
such as minor mergers and accretion.

###### keywords:

galaxies: general – galaxies: evolution – galaxies: structure – galaxies:
bulges – galaxies: photometry

††pubyear: 2024††pagerange: Two rest-frame wavelength measurements of galaxy
sizes at $z<1$: the evolutionary effects of emerging bulges and quenched
newcomers–D

##  1 Introduction

The assembly history of galaxies in the aging universe depends on their
evolutionary stage at the cosmic time (redshift) of observations. Galaxy
transformations over time include changes both in the average properties of
their stellar population (e.g., age and metallicity) and in galaxy structure
(i.e., size and shape). By observing evolutionary trends in morphological
properties of galaxies in large (spectro-)photometric surveys (e.g., SDSS,
York et al. 2000; Abazajian et al. 2009; Aihara et al. 2011; HSC-SSP Aihara et
al. 2018, 2019; DESI, Dey et al. 2019), we can probe the assembly histories of
different galaxy populations.

A suite of studies conducted over the last two decades show that for both SFGs
and QGs galaxy size depends on stellar mass. (e.g., Shen et al., 2003; Guo et
al., 2009; Williams et al., 2010; Newman et al., 2012; Mosleh et al., 2012;
van der Wel et al., 2014; Lange et al., 2015; Faisst et al., 2017; Roy et al.,
2018; Mowla et al., 2019b; Matharu et al., 2019, 2020; Kawinwanichakij et al.,
2021; Barone et al., 2022; Mercier et al., 2022; Damjanov et al., 2023, and
many others). Both galaxy populations show an expected positive trend in size
with stellar mass, in agreement with theoretical studies (e.g., Price et al.,
2017; Genel et al., 2018; Rosito et al., 2019; de Graaff et al., 2022).

However, the parameters of the relation between galaxy size and stellar mass
(also called the size-mass relation; SMR) differ between SFGs and QGs. The
SFGs show a weak relation between galaxy size and stellar mass with
$R_{e}\varpropto M_{*}^{\sim 0.2}$, where $R_{e}$ is the effective radius
(radius that includes $50\%$ of galaxy light) in kpc and $M_{*}$ is the
stellar mass in M⊙ (e.g., Shen et al., 2003; Williams et al., 2010; van der
Wel et al., 2014; Mowla et al., 2019b; Kawinwanichakij et al., 2021; Barone et
al., 2022). On the other hand, QGs exhibit a more complex SMR. Below the pivot
mass ($\sim 3\times 10^{10}$M⊙), QGs show a weak trend with $R_{e}\varpropto
M_{*}^{-0.1}$, resembling the equivalent relation for SFGs (e.g., Morishita et
al., 2017; Kawinwanichakij et al., 2021). In sharp contrast, the exponent of
the power-law SMR for QGs above the pivot mass increases significantly
($R_{e}\varpropto M_{*}^{\sim 0.7}$; e.g., Shen et al., 2003; Williams et al.,
2010; Lange et al., 2015; Huang et al., 2017; Mowla et al., 2019b; Mosleh et
al., 2020; Kawinwanichakij et al., 2021; Nedkova et al., 2021; Damjanov et
al., 2023).

Mowla et al. (2019a) and Kawinwanichakij et al. (2021) show that the pivot
mass of the SMR of QGs matches the pivot point of the stellar mass-halo mass
(SMHM) relation, a relation between galaxies and their host dark matter halos
(e.g., Kereš et al., 2005, 2009; Bower et al., 2006; Croton et al., 2006;
Behroozi et al., 2010; Huang et al., 2017; Golden-Marx & Miller, 2019; Golden-
Marx et al., 2022; Gabrielpillai et al., 2022; Zaritsky & Behroozi, 2023). The
SMHM relation describes how efficiently baryons accreted into the halo are
cooled and converted into stars. The pivot point of the SMHM relation is the
stellar mass that corresponds to the maximum efficiency of this conversion
(e.g., Behroozi et al., 2010; Leauthaud et al., 2012; Moster et al., 2013;
Rodríguez-Puebla et al., 2017). According to Mowla et al. (2019a) and
Kawinwanichakij et al. (2021), because the pivot points of SMR and SMHM
relation are similar, the pivot point of the SMR indicates the stellar mass at
which in-situ galaxy growth via star formation is replaced by the ex-situ
driven growth via mergers and accretion.

In addition, the pace at which galaxy sizes evolve also differ between massive
($M_{*}\gtrsim 3\times 10^{10}$M⊙) SFGs and QGs. On average, the evolution in
galaxy sizes (i.e., in $R_{e}$) with cosmic time is stronger for QGs than for
SFGs (e.g., van der Wel et al., 2014; Kawinwanichakij et al., 2021). This
difference suggests that the observed evolution in galaxy morphology can probe
physical processes that affect galaxy growth before and after the star
formation in them ceases.

The strength of SFG size evolution varies with the observed redshift interval.
The observational studies that cover a wider redshift range ($0<z<3$) show a
strong evolution of sizes (e.g., van der Wel et al., 2014; Mowla et al.,
2019b). However, several studies show weak to no evolution since $z\sim 1$
(e.g., Lilly et al., 1998; Ravindranath et al., 2004; Barden et al., 2005;
Kawinwanichakij et al., 2021) whereas studies that focus only at higher
redshifts ($z>2$) show very strong evolution in SFG sizes with redshift (e.g.;
Oesch et al., 2010).The range of results for different redshift intervals
indicates that the size evolution of SFGs may be slowing down with cosmic
time.

There are several reasons why the size growth of SGFs could slow down with
decreasing redshift. One obvious reason is the availability of cold gas. As
the average density of gas decreases with time, there is less gas (star-
forming material) in galaxy environment that can be accreted. The lack of
star-forming material leads to reduced levels of cosmic star formation at
redshift $z\lesssim 1$ (Madau & Dickinson, 2014). If the size growth of SFG is
driven predominantly by the in-situ star formation, the reduced amount of
available gaseous material will affect the observed pace of their size growth.

Another reason for slower size growth of SFGs at $z<1$ is the emergence of
centrally concentrated bulges in galaxies (Martig et al., 2009; Sachdeva et
al., 2017; Hashemizadeh et al., 2022). Older galaxies tend to have a
significant bulge component in their center compared to younger galaxies. In
massive galaxies at $z<1$, these bulges contain older and less massive red
stars that dominate the light in visible and near infra-red wavelength
regimes. In contrast, the light in outer disks is dominated by the
contribution from young stars that emit strongly in UV (e.g., Breda et al.,
2020). Therefore, the overall size of a galaxy appears smaller and more
centrally concentrated in the rest-frame $5000$ Å where the light from old
stars dominates. In studies that target this redder rest-frame wavelength, the
observed slower pace of size evolution with decreasing redshift can indicate
the emergence of concentrated red bulges.

Galaxies undergo morphological transformations when they become quiescent. A
quenched galaxy typically has smaller size than a star-forming one. If an SFG
is a disk-dominated system and the quenching process does not affect the
stellar distribution within it, the stellar disk fades away when the star-
formation ceases (e.g., Christlein & Zabludoff, 2004; Carollo et al., 2016;
Matharu et al., 2020). As galaxy disks are more extended than their more
centrally concentrated bulges/spheroids, the half-light radius of a galaxy
decreases when the disk fades. These transformations can be probed further by
analyzing galaxy light profile components (e.g., bulges and disks) separately,
investigating mass complete spatially resolved spectroscopic datasets, and
measuring galaxy dynamical properties (e.g., Hudson et al., 2010; Belli et
al., 2017, 2019; Newman et al., 2018).

QGs, on the other hand, evolve and grow in size through various physical
processes, including major mergers, minor mergers, accretion and adiabatic
expansion (e.g., Naab et al., 2007, 2009; Fan et al., 2008, 2010; van Dokkum
et al., 2010; Trujillo et al., 2011; Cimatti et al., 2012; López-Sanjuan et
al., 2012; Huertas-Company et al., 2013; Zahid et al., 2019; Damjanov et al.,
2023). Additionally, newly quenched galaxies being added to the quiescent
population (newcomers) affect the overall size distribution and the average
size growth of QGs (progenitor bias; e.g., van Dokkum & Franx, 2001; Saglia et
al., 2010; Carollo et al., 2013; Genel et al., 2018; Damjanov et al., 2023).
The estimated level of influence that the newcomers have on the average size
evolution of the quiescent population range from minimal (e.g., Zanella et
al., 2016; Damjanov et al., 2023) to substantial (e.g., Carollo et al., 2013;
Fagioli et al., 2016). However, apart from the progenitor bias that affect the
bulk of the QG population, a majority of studies tracing morphological
evolution of QGs indicate that minor mergers are the major driver of size
growth for this galaxy population. (Newman et al., 2012; Beifiori et al.,
2014; Ownsworth et al., 2014; Belli et al., 2015; Buitrago et al., 2017; Zahid
et al., 2019; Hamadouche et al., 2022; Damjanov et al., 2023). These galaxy
encounters affect the light in different rest-frame wavelengths by depositing
younger and/or metal-poor stellar material at galaxy outskirts (e.g., Suess et
al., 2020).

So far, the large scale studies have been focused on a single rest-frame
wavelength (typically $5000$ Å). Galaxy light profiles at this wavelength are
dominated by the light from older stars that contribute most of the stellar
mass of a galaxy (e.g., van der Wel et al., 2014; Mowla et al., 2019b;
Kawinwanichakij et al., 2021). By studying galaxy morphology in ultraviolet
(UV) light, we can probe the distribution of recently formed stars in galaxies
because the shorter wavelengths ($2000<\lambda<4000$ Å) are dominated by the
light from young stars (e.g., Bruzual & Charlot, 2003). In addition,
simultaneous analysis of galaxy morphology at longer wavelengths
($\lambda>4000$ Å) enables us to probe the difference in spatial distribution
of young and old stars.

Figure 1: CLAUDS+HSC images of an SFG (centre) with mass $\log M=10.53$ at
$z=0.67$ ($\alpha$: 150.17897, $\delta$: 2.03213) in u, g, r, i and z bands.
Image cutouts are of dimensions $\sim 33.8^{\prime\prime}\times
33.8^{\prime\prime}$ and have the same intensity stretch. A
$10^{\prime\prime}$ ($72.3$ kpc) scale is shown for reference in the middle
panel.

CFHT CLAUDS (Sawicki et al., 2019) and Subaru HSC-SSP (Aihara et al., 2018,
2019) surveys together provide the ideal dataset for studies of galaxy
morphology in rest-frame UV and visible lights at $z<1$. The CLAUDS+HSC survey
includes deep images of the sky over an overlapping area of $18.6$ deg2 in $6$
bands ($Ugrizy$) that range from UV to near infrared (IR). In this pilot
study, we use a subset of this dataset (the central $\sim 1.6$ deg2 region of
the COSMOS field) to analyse the SMR and size evolution of $\sim 16000$ SFGs
and $\sim 5200$ QGs at $0.1<z<0.9$ in two rest-frame wavelengths: $3000$ Å and
$5000$ Å. We will expand the study to the entire survey volume in the followup
analysis.

The plan of this paper is as follows. Section 2 provides an overview of
CLAUDS+HSC survey, the data and auxiliary data products we use, sample
selection and classification of galaxies into SFGs and QGs. In Section 3, we
present details of the methodology for 2D galaxy light profile fitting
(Section 3.1). In Section 3 we also provide the results of the simulations we
perform to estimate the robustness of the fitting pipeline and the systematic
uncertainties present in our morphological measurements (Section 3.2). In
addition, we describe how we estimate the rest-frame Sérsic parameters in
Section 3.3. We validate our size measurements by comparing them with those
from the literature (Section 3.4). In Section 4, we describe the analytic fits
to the SMR and present the results of SMR fitting and size evolution for SFGs
and QGs in two rest-frame wavelengths. We discuss the implications of our
findings in Section 5. We summarize the major conclusions based on our
analysis in Section 6.

Throughout this study, we use $M$ to denote stellar mass in solar units
($M_{\star}/$M⊙) unless otherwise stated. Additionally, we use the AB
magnitude system and adopt a standard cosmology with $\Omega_{M}=0.3$,
$\Omega_{\Lambda}=0.7$ and $h=0.7$ throughout this paper.

##  2 Data and Sample Selection

###  2.1 CLAUDS+HSC Survey

We use the ultra deep multi-band ($u+griz$) imaging data from the central
$1.6$ deg2 region of the COSMOS field from CFHT Large Area $U$-band Deep
Survey111https://www.clauds.net/ (CLAUDS; Sawicki et al., 2019) and the Hyper
Suprime-Cam Subaru Strategic Program222https://hsc.mtk.nao.ac.jp/ssp/ (HSC-
SSP; Aihara et al., 2018, 2019). In total, the combined deep CLAUDS+HSC
dataset covers an overlapping area of $18.6$ deg2. In u and i bands, the
median depth is $27.1$ mag (measured at $5\sigma$ level in $2^{\prime\prime}$
apertures). The depth of the dataset is optimal for the analysis of massive
galaxy morphology out to their faint outskirts. The CLAUDS+HSC bands also have
similar sensitivities, except for y-band where the limiting magnitude is
$25.6$ mag. Figure 1 shows an example galaxy from our dataset in u, g, r, i
and z bands.

Recently, Kawinwanichakij et al. (2021) conducted an extensive morphological
study using HSC-SSP data only (i.e., without the u-band images that are
available only for the Deep and UltraDeep layers of the HSC-SSP). The u-band
data are important for constraining photometric redshifts and internal galaxy
properties. For example, u-band photometry is essential for bracketing the
Balmer and $4000$ Å breaks in order to estimate photometric redshifts
(photo-$z$’s) accurately for galaxies at intermediate redshifts ($0<z\lesssim
0.7$, Sawicki et al., 2019). At these intermediate redshifts, u-band data also
play a vital role in constraining star formation rates (SFRs) of galaxies.
With the addition of u-band data from CLAUDS to the Deep and UltraDeep layer
of HSC-SSP, we are able to minimize the contamination when separating SFGs and
QGs at $z<1$ and to analyze structural evolution of the two populations in a
unique way.

Although our final goal is to analyze the entire $18.6$ deg2 of the survey
region, we limit this pilot study to the COSMOS/UltraVISTA field, covering
$\sim 1.6$ deg2 on the sky. We target this smaller area because it is a widely
studied survey region in the sky. Choosing a widely studied field is
beneficial as we can compare the results of our work with those of previous
and parallel studies. This approach ensures quality control of certain steps
implemented in this project and enables us to draw scientific conclusions
regarding the physical processes driving the observed trends with more
certainty. In addition, the COSMOS field is rich in ancillary data, which we
utilize in this study for mass and redshift estimates. In the next stages, we
will expand the study to the entire CLAUDS+HSC data to improve statistics and
analyze the impact of environment on size evolution of galaxies.

Figure 2: A sample PSF generated for u-band using PSFEx. The left panel shows
the image of a star in u-band. The middle panel shows the PSF image generated
for the same location in the sky. The last panel shows the residual image
obtained by subtracting the PSF from the stellar image.

###  2.2 Point Spread Function

Atmospheric turbulence (atmospheric seeing) dominates the blurring of
astronomical images in ground-based observations. A point spread function
(PSF) describes how a point source in the sky (a star) looks like in the image
due to this blurring. Hence, to study the intrinsic shapes of galaxies, we
must remove the effect of PSF from galaxy images.

The HSC Pipeline uses the PSFEx algorithm333http://ascl.net/1301.001 (Bertin,
2011) to characterize the PSF of the HSC images. The PSFex adaptation in the
HSC Pipeline models PSF from the images of unsaturated stars in an iterative
way to remove contamination from neighbouring objects (Bosch et al., 2018).
PSF models at any position within the footprint of the survey are available
through the PSF picker444https://hsc-release.mtk.nao.ac.jp/psf utility in the
Public Data Release by HSC-SSP.

Although PSFs for every sky coordinate covered by the HSC-SSP are available
via the PSF picker, fetching a PSF model for every galaxy would be
computationally expensive. Hence, we divide every HSC image patch555Each image
patch consists of $4200\times 4200$ pixels where each image pixel covers $\sim
0.168^{\prime}\times 0.168^{\prime}$ of the sky. The pixel and patch sizes are
the same for both CLAUDS and HSC images. into $36$ sub-regions of dimensions
$117.6^{\prime\prime}\times 117.6^{\prime\prime}$. For all galaxies within a
sub-region, we use the PSF image at the center of this region.

The PSF images for CLAUDS u-band data are not available externally and thus we
model PSFs for these images using PSFEx. Although the software selects the
bright unsaturated stars using the catalogue generated by SExtractor, we
refine the automated star selection by identifying a region from peak in the
surface brightness versus magnitude relation where the unsaturated stars lie.
PSEFx then uses these stars to model PSFs for the 36 sub-regions in each image
patch.

Figure 2 shows a sample PSF in u-band we generate using PSFEx. The third panel
shows the residual after PSF model (second panel) is removed from an image of
a star in u-band (first panel). The residual image shows no visible structure,
confirming that the u-band PSFs we generated are reliable.

The PSF changes with the band of observations and location in the sky. Among
the HSC bands, the i-band has the best seeing with full width at half maximum
(FWHM) of the PSF ranges between $0.5^{\prime\prime}$ and
$0.7^{\prime\prime}$. The FWHM in u-band is also in the same range. The median
FWHM in other bands is $\sim 0.7^{\prime\prime}$.

###  2.3 Other Data Products

We make use of the COSMOS2020 (Weaver et al., 2022), the latest photometric
catalog from Cosmic Evolution Survey (COSMOS, Scoville et al. (2007)). This
catalog is based on new imaging and incorporates CFHT CLAUDS u-band data, HSC-
SSP PDR2 grizy data, VISTA VIRCAM YJHKs-band data of UltraVISTA DR4 (McCracken
et al., 2012) and Spitzer/ IRAC channel $1$, $2$, $3$, and $4$ data of the
Cosmic Dawn Survey (Collaboration et al., 2022). The catalogue covers an area
of $2$ deg2 and includes multiwavelength photometric measurements and derived
properties for $1.7$ million sources. From the catalogue, we use the
photometric redshifts, stellar masses and star formation rates derived using
LePhare (Ilbert et al., 2006) and rest-frame UVJ colours from Eazy code
(Brammer et al., 2008).

Figure 3: Classification of galaxies into SFGs (blue) and QGs (red) in the UVJ
diagram using Equation 1. The upper and lower panels show the classification
in two redshift bins. Rest-frame UVJ colours are taken from Weaver et al.
(2022). Figure 4: An example of galaxy profile modelling using Galfit. Panels
from left to right show the input cutout image of an SFG ($z=0.81$; $\log
M=9.56$) in i-band (A), masked image (B), best-fit models (C) and residual
image (D). The fitting of the target galaxy (center) is performed
simultaneously with a neighbouring bright galaxy on its bottom right. The
best-fit parameters of the target galaxy is given in panel D. We also show a
$10^{\prime\prime}$ ($77.6$ kpc) scale in panel A for reference.

###  2.4 Sample Selection and Galaxy Classification

We limit the redshift range of our work to $0.1<z<0.9$ because we cannot
constrain the stellar mass of galaxies well beyond $z\sim 0.9$ by using
CLAUDS+HSC data alone. This problem arises due to the limitation in the
available wavelength coverage to bracket several spectral features like the
$4000$ Å break. Although this pilot study utilises COSMOS2020 catalogue with
wider wavelength coverage (Section 2.3), our future studies will cover the
complete CLAUDS+HSC deep survey region and thus we will be relying on the data
from $Ugrizy$ bands alone.

Furthermore, we select only massive galaxies for this study. The mass
completeness limits varies with redshift as well as with the level of galaxy
star formation activity. The SFGs at lower redshifts have lower stellar mass
completeness limit than the QGs at higher redshifts. To have a uniform mass
completeness limit for the entire dataset, we limit our study to galaxies more
massive than $\log M>9.5$ (Chen et al. 2024, in prep).

We classify selected galaxies into SFGs and QGs using UVJ colour-colour
diagrams (e.g., Wuyts et al., 2007; Williams et al., 2009; Brammer et al.,
2011; Muzzin et al., 2013; Carnall et al., 2018) by applying a selection
criterion from Whitaker et al. (2011),

$\begin{split}&(U-V)>0.88\times(V-J)+0.69\ [\mathrm{for}\ z<0.5],\\\
&(U-V)>0.88\times(V-J)+0.59\ [\mathrm{for}\ z>0.5],\end{split}$ (1)

where $U-V$ and $U-J$ are rest-frame colours. Figure 3 shows the UVJ
classification of galaxies in two redshift regimes. The sample we analyze in
this study contains in total $\sim 16000$ SFGs (blue points) and $\sim 5200$
QGs (red points).

##  3 Methodology

Figure 5: An example for profile modelling of simulated galaxies using Galfit
in i-band. The panels are same as in Figure 4 but for a simulated galaxy
instead for a real one.

###  3.1 Galaxy Profile Fitting using Galfit

We model galaxy light profiles with a single Sérsic profile,

$\log\left(\frac{\Sigma(R)}{\Sigma_{e}}\right)=-b_{n}\left[\left(\frac{R}{R_{e}}\right)^{1/n}-1\right],$
(2)

where $R_{e}$ is the effective radius, $\Sigma_{e}$ is the surface brightness
at $R_{e}$ and $n$ is the Sérsic index, which defines the concentration of the
profile. The coefficient $b_{n}$ is not a free parameter; it is a function of
$n$ and it ensures that the region within $R_{e}$ encloses half of the total
luminosity of the galaxy. Although galaxies usually have several internal
components such as disks and bulges, we can fit the overall light profile of a
galaxy using a single Sérsic function to obtain its global features (size,
light concentration, axis ratio, etc.; van der Wel et al., 2012).

In this work we use Galfit, a two-dimensional ($2$D) parametric light profile
fitting software tool to fit galaxy profiles (Peng et al., 2002, 2010a). The
software extends the one dimensional Sérsic profile from Equation 2 to $2$D
space for direct fitting of galaxy images. Based on the simulations we perform
(and describe in Section 3.2), we introduce magnitude cut in each band
(u$<26$; g$<25.5$; r, i, z$<25$) that corresponds to $\sim 75\%$ success rate
for our fitting pipeline. We do not apply any limit based on other galaxy
structural parameters such as size, Séric index or axis ratio, because they do
not affect the robustness of our fitting pipeline (see Appendix A).
Additionally, we do not fit y-band data because the simulations show that our
pipeline does not produce accurate measurements of the structural parameters
in that band even for fairly bright galaxies (e.g., $24<y<25$; see Section
3.2.3).

At the start of the fitting procedure, we make an image cutout centred around
the target galaxy. We ensure that the size of the cutout is sufficiently large
for good fitting by adopting a minimum cutout size of $25\times R_{eS}$, where
$R_{eS}$ is the SExtractor effective radius from the COSMOS2020 catalogue
(this radius does not take into account the effects of seeing). The cutout
size does not exceed $84^{\prime\prime}$. Our simulations (Section 3.2.2) show
that the fitting of galaxy profiles using the cutout size of $25\times R_{eS}$
performs significantly better than when using cutout sizes $<20\times R_{eS}$
and that the performance level plateaus around $20-25\times R_{eS}$. Because
the HSC PSF images have dimensions of 43 pixels per side, we ensure that the
image cutout is at least 45 pixels per side, which is also the dimension of
the convolution box for PSF.

We flag neighbouring galaxies for simultaneous fitting if their projected
distance from the target galaxy is
$<2\times(R_{eS}$(target)$+R_{eS}$(neighbour)$)$, and the magnitude difference
is $m_{\text{neighbour}}-m_{\text{target}}<0.5$. If the target galaxy is
fainter than the $23$ mag, we further restrict the magnitude range for
neighbour selection to ${m_{\text{neighbour}}-m_{\text{target}}<0.1}$.

We then mask out all other fainter galaxies and bright objects in the cutout
image. We use a watershed segmentation technique implemented in the Photoutils
package for Python for this purpose. The watershed algorithm treats pixel
values in the image as an inverted local topography where the centroids of
bright objects are at the local minima of this topography. The algorithm
generates a segmentation map for the light sources in the image cutout. The
algorithm then deblends all overlapping regions and separates the objects in
into regions (‘watersheds’). Finally, it creates a file containing pixel
coordinates belonging to the ‘watersheds’ of all objects to be masked out.

The next step is to estimate the initial parameters for galaxy fitting. We use
magnitudes from COSMOS2020 catalogue as the input magnitude value for Galfit
in the respective filters. We roughly estimate the initial values of the other
Sérsic parameters (size, Sérsic index, axis ratio) using
PetroFit666https://github.com/PetroFit/petrofit/releases/tag/v0.4.1, a galaxy
light profile fitting Python package, and provide a random angle for the
position angle. We estimate the mean sky background using the areas outside
the watershed segments and provide it as the initial parameter. However, if a
galaxy is fainter than $24.5$ mag, we fix the sky brightness at this mean sky
value while running Galfit. In addition, the software requires
$\sigma$-images, that contain the pixel level uncertainties in the data, and
the PSF images.

A Galfit Séric model is a fuction of several parameters: centroid pixel
coordinates ($x_{0}$, $y_{0}$), magnitude ($m_{tot}$), $R_{e}$, $n$, axis
ratio ($b/a$), and position angle ($\theta_{PA}$). We fit galaxy profiles
using Galfit in two steps, a procedure similar to the one used by Matharu et
al. (2019). In the first step, we keep all parameters free to vary and obtain
refined values of galaxy initial position parameters: centroid pixel
coordinates ($x_{0}$, $y_{0}$), $b/a$ and $\theta_{PA}$. We provide the best-
fit parameters from the first step as the input for the second step, where we
force three position parameters to remain constant (within uncertainties) and
keep the remaining three structural parameters ($m_{tot}$, $R_{e}$ and $n$)
free. We also constrain the parameter space for both steps to ensure that the
software reaches global minimum. These constraints are $10\leq m_{tot}\leq
30$, $0.1\leq R_{e}\leq 50$ pix, $0.1\leq n\leq 20$ and $0.02\leq b/a\leq 1$.
In addition, we do not allow the Galfit-derived magnitude to differ from the
magnitude obtained using SExtractor by more than $2$ mags.

Figure 4 illustrates the steps in fitting of a galaxy light profile. Panel A
shows the image cutout, panel B shows the masking using watershed algorithm
and panel C the best fit Sérsic models of the target and a neighbouring
galaxy. Panel D provides the residual image obtained by subtracting panel C
from panel A. From the residual image, it is clear that the fitting pipeline
finds Sérsic profile that best matches the observed galaxy (i.e., Panel D
shows no features in the region of fitted galaxy). Since we use only a single
Sérsic profile for a galaxy, in some cases structural features (such as spiral
arms) are visible in the residual image (see the residuals for the
neighbouring galaxy in panel D).

###  3.2 Simulations

Figure 6: The relative difference in galaxy sizes in i-band from simulations
with different cutout sizes. The contours in each panel show the distribution
of the relative difference in size: $\Re_{d}=$ (input - output)/input
(Equation 3). The red curve in each panel shows the median relative difference
in sizes in bins of input galaxy sizes. We perform five sets of simulations,
changing the cutout size in relation to the effective radii of simulated
galaxies: $5R_{e}$, $10R_{e}$, $15R_{e}$, $20R_{e}$ and $25R_{e}$. In the last
panel, we give the median value of the offset in $\Re_{d}(R_{e})$ for each
cutout size.

####  3.2.1 Pipeline

We perform simulations to test the robustness of our galaxy profile fitting
procedure (Section 3.1) and to estimate the systematic uncertainties present
in the measurements. In these simulations, we generate artificial galaxies and
plant them in real images. To ensure that the simulated galaxies reflect the
properties of real galaxies, we use the structural parameters of galaxies in
the COSMOS field from the Zurich Structure and Morphology
Catalogue777https://irsa.ipac.caltech.edu/data/COSMOS/gator_docs/cosmos_morph_zurich_colDescriptions.html
(ZSMC; Sargent et al., 2007; Scarlata et al., 2007). ZSMC contains Hubble
Space Telescope (HST)/ACS F814W-based structural properties of galaxies
(Sargent et al., 2007), derived from a single Sérsic profile fitting using
GIM2D (Simard, 1998). We first estimate a multivariate probability
distribution function (PDF) of GIM2D-derived size ($R_{e}$), Sérsic index
($n$) and axis ratio ($q$) along with u, g, r, i, z and y magnitudes from the
CLAUDS+HSC-SSP catalogue (Desprez et al., 2023) using multi-dimensional kernel
density estimation. As there are only a few galaxies with
$R_{e}>1.5^{\prime\prime}$ in ZSMC, we include additional artificial galaxies
with these larger sizes. We then randomly draw parameters from this PDF to
simulate galaxy 2D profiles. While doing so, we make sure that we do not draw
a parameter set from the multivariate PDF where any of the parameters has a
non-physical value (for example, $R_{e}<0$ pixels).

In the next step, we model each artificial galaxy with its randomly chosen
parameter set from the PDF using Galfit. We also convolve these artificial
galaxies with PSF at the position where we plant them in the image. We then
plant these galaxies in real images with randomly chosen position angles
($\theta_{PA}$). Random values of $\theta_{PA}$ ensure that galaxies are
placed in the images with orientations similar to those of real galaxies. We
also make sure that the centroid of the simulated galaxy does not fall within
the effective radius of another real galaxy in the image.

We fit each planted galaxy using the same procedure we use with the real data
(Section 3.1). Figure 5 shows an example of a simulated galaxy and its best-
fitting model obtained by the pipeline. We perform the planting and fitting
procedures using one artificial galaxy at a time to avoid overcrowding of
images. One of the caveats of our simulation is that artificial galaxies have
single Sérsic profiles, which is not true for most of the real galaxies.
Nevertheless, this estimate is sufficient for the present study as we fit real
galaxy profiles with a single-profile model.

####  3.2.2 Cutout image size

Using simulations we first explore the impact of the image cutout size on the
fitting results. We perform several sets of simulations with varying cutout
sizes based on the size of the target galaxy: $5R_{e}$, $10R_{e}$, $15R_{e}$,
$20R_{e}$ and $25R_{e}$.

We define relative difference for a given parameter $x$ as

$\Re_{d}(x)=\frac{x\,\text{(input)}-x\,\text{(output)}}{x\,\text{(output)}}.$
(3)

We then analyse the distribution of $\Re_{d}(R_{e})$ as a function of galaxy
size in bins of various cutout sizes (Figure 6). Clearly, the smallest cutout
size ($5R_{e}$; first panel) performs the worst irrespective of the intrinsic
size of the target galaxy. In this case, our pipeline yields sizes that are,
based on the median relative difference, $\sim 20\%$ smaller than input values
(red line in the first panel of Figure 6). In addition the scatter in relative
differences is significantly higher than for any other cutout sizes (blue
contours). This error in size measurements is mainly due to two reasons: (1)
galaxy profiles are extended beyond $5R_{e}$; (2) there are a few pixels in
the cutout to effectively estimate the sky background.

The offset in the size measurements decreases with increasing cutout sizes.
With the $25R_{e}$ cutout (last populated panel of Figure 6), the median
relative offset (red line) shows that the best-fit $R_{e}$ is equivalent to
the real (input) $R_{e}$ irrespective of the input galaxy size. In addition,
the scatter in relative galaxy size differences is slightly higher for
$20R_{e}$ than for $25R_{e}$ cutout size. Since this scatter represents the
systematic uncertainty in galaxy size measurements, we consider a cutout size
of $\sim 25R_{e}$ as the best among the five cutout sizes chosen for the
simulation and adopt that cutout size for our fitting procedure.

However, unlike simulated galaxies, we do not know the intrinsic sizes of
galaxies in real data apriori. Therefore, for real galaxies, we use size
estimations from SExtractor to determine the required sizes of image cutouts.

####  3.2.3 Robustness

Figure 7: Simulated success rate (Equation 4) as a function of magnitudes in
CLAUDS+HSC bands. The u-band data performs the best while the y-band data
performs the poorest (see text for details). Hence, we do not use y-band data
in this work. Figure 8: Relative difference in Sérsic parameters in i-band
from simulations. The contours in each panel show the distribution of the
relative difference in parameters: the top, middle and bottom rows show the
relative differences in size, Sérsic index and axis ratios respectively. From
left to right, the columns show the relative differences in parameters as
functions of Sérsic parameters: magnitude, output size, output Sérsic index
and output axis ratio. Note that we have used the relative difference here as
$\Re_{d}=$ (input - output)/input (Equation 3). In each panel, the red curve
represents the median of the relative difference.

To investigate the robustness of the pipeline, we define the simulated success
rate as

$\text{Success rate}=1-\frac{\text{Number of failures}}{\text{Total number of
simulated galaxies}}.$ (4)

We consider a fit as failed if any of the parameters lies at the limits of the
allowed parameter space. We show this ratio in all the six CLAUDS+HSC bands in
Figure 7.

We find that the success rate is a function of magnitude in the fitted band
with limiting magnitude ranging from $\sim 25$ to $26$ mag for a success rate
of $0.75$ in u, g, r, i and z bands. However, we do not find any significant
dependence of the success rate on any other Sérsic parameter (see Appendix A).
Hence, we limit our analysis based on the magnitude of galaxies in five
CLAUDS+HSC bands.

We also note that success rate in the y-band is the lowest among the
CLAUDS+HSC bands. It is because the y-band is shallow and has poor seeing
compared to other HSC bands. Hence, the fits based on the y-band profiles are
more likely to fail for even relatively bright galaxies. Therefore, we do not
use galaxy morphology in the y-band for the analysis of galaxy morphological
evolution in this work. This does not impact our ability to probe the light
profiles of galaxies in two rest-frame wavelengths. For galaxies in our
highest redshift bin ($0.75<z<0.9$, Section 4), we use z-band images to model
light profiles at the redder rest-frame wavelength ($5000$ Å).

Figure 9: Violin plots showing the distribution of the relative difference in
galaxy sizes $\Re_{d}(R_{e})$ in bins of i-band magnitude ($x$-axis), its
measured size (rows) and measured Sérsic index (columns). The shape of the
violin plots represents the distribution of $\Re_{d}(R_{e})$ in each $3$-D
parameter bin, and its $16^{\text{th}}$, $50^{\text{th}}$ and $84^{\text{th}}$
percentiles are shown as horizontal lines. The distributions are approximately
Gaussian with their medians around $0$. However, the distributions differ from
Gaussian and their median values are significantly different from $0$ for
galaxies with higher Sérsic indices or larger sizes that represent the tail of
the distribution of galaxy structural properties.

####  3.2.4 Uncertainty Estimation

An important reason for running these simulations is to estimate the
systematic uncertainties in our measurements. To do this, we analyze the
distribution of $\Re_{d}(x)$ as a function of magnitude, measured size and
measured Sérsic index. The first column of Figure 8 shows $\Re_{d}(x)$ as a
function of i-band magnitude. The median relative difference (red curve) in
size, Sérsic index or axis ratio does not exhibit any significant systematic
offset as a function of magnitude.

If the measured size of a galaxy is large ($R_{e}>0.8^{\prime\prime}$; the
first row in the second column of Figure 8), the measurement tends to be
slightly larger than the input size. This offset increases with the measured
size of the galaxy (offset is above $10\%$ if $R_{e}>1.5^{\prime\prime}$). We
also tend to overestimate the Sérsic index of a galaxy if its measured size is
greater than $1.5^{\prime\prime}$ (second panel in the middle row). These
offsets are not significant, and galaxies with large sizes are extremely rare.
The median relative differences in $R_{e}$ and $n$ also show offsets if output
Sérsic index is greater than $4$ (red curves in the first and second row of
the third column in Figure 8). These high Sérsic index galaxies could have
complex formation history and are least likely to be described by a single
Sérsic index. Furthermore, the ability of our algorithm to estimate structural
parameters do not depend on the $b/a$ of galaxies (third column). Therefore,
we do not introduce any offset to measured parameters in this study.

In summary, the simulations show that we are able to recover median values for
structural parameters in all bands. We do not find any significant systematic
offset between the measured $R_{e}$, $n$ or $b/a$ and their input values
irrespective of the brightness, size, Sérsic index and axis ratio for most of
our model galaxies ($R_{e}\lesssim 1^{\prime\prime}$ and/or $n\lesssim 4$,
Figure 8). The median difference between input and output parameters (red
curves) in the simulations is always close to $0$. We also find that the size
measurements are more robust than the measurements of Sérsic index, as
reflected in the scatters (blue contours) of relative differences of $R_{e}$
and $n$. The scatter in the distribution of $\Re_{d}(R_{e})$ is significantly
smaller than the scatter in $\Re_{d}(n)$. Finally, we are able to recover the
size of the simulated galaxies even if they are very small
($R_{e}<0.2^{\prime\prime}$).

We incorporate the $\Re_{d}$ scatter ( blue contours in Figure 8) into the
measurement uncertainty as systematic uncertainty. We do not estimate the
systematic uncertainty in a given parameter as a function of each Sérsic
parameter separately due to covariance between them. We instead analyze the
scatter in the relative differences in multi-dimensional bins of Sérsic
parameters. As we do not find any dependence of the $\Re_{d}$ scatter on the
measured axis ratio, we do not include $b/a$ in our uncertainty analysis. The
violin plots in Figure 9 show the distribution of $\Re_{d}(R_{e})$ in 3-D bins
of $m$, $R_{e}$ and $n$. Except for galaxies with high $R_{e}$ and $n$
($R_{e}>0.7^{\prime\prime}$ and $n>3$; bottom right panels), the distribution
is approximately Gaussian. Hence, we can use the standard deviation (std) of
the distributions in Figure 9 as a quantitative measure of the systematic
uncertainties. We consider the $x\times\text{std}(\Re_{d}(x))$ to be the
systematic uncertainty in the parameter $x$, $\sigma_{sys}(x)$. We then
estimate the total uncertainty in a measured parameter to be
$\sigma^{2}=\sigma_{sys}^{2}+\sigma_{ran}^{2}$, where $\sigma_{ran}$ is the
random uncertainty reported by Galfit for a given galaxy. We note that in most
cases the $\sigma_{sys}$ is significantly larger than $\sigma_{ran}$.

###  3.3 Size Measurements in Two Rest-frame Wavelengths

The goal of this work is to perform morphological analysis of galaxies in two
rest-frame wavelengths. We choose rest-frame $3000$ Å (UV light) and $5000$ Å
(visible light) because they bracket the $4000$ Å break in galaxy spectrum
(Bruzual & Charlot, 2003; Conroy, 2013). Because UV light is dominated by the
light from young ($<1$ Gyr) massive stars, this light traces the regions of
recent star formation activity in galaxies. In contrast, light from the old
($>1$ Gyr) low mass stars contributes significantly to the visible wavelength
region of galaxy spectrum. Since low-mass stars represent the bulk of galaxy
stellar mass, the visible light traces closely the stellar mass distribution
within galaxies.

Earlier studies generally focus on the rest-frame $5000$ Å that traces the
stellar mass distribution in galaxies. Furthermore, though many of the studies
have multi-wavelength dataset, they often fit galaxy profiles in a single band
alone. They first fit a subset of galaxies in multiple bands and estimate
colour gradients (i.e., how galaxy size decreases with increasing wavelength).
They then use this colour gradient estimation and correct the size
measurements of all galaxies in a selected band to obtain size measurement in
any desired rest-frame wavelength (e.g., van der Wel et al., 2014;
Kawinwanichakij et al., 2021).

In contrast, we fit the light profiles of all galaxies in five bands and use
this multi-band structural information to estimate the Sérsic parameters in
two rest-frame wavelength regimes. We first estimate the characteristic
redshifts at which effective wavelengths of CLAUDS+HSC bands cover these rest-
frame wavelengths. As an example, the rest-frame $5000$ Å corresponds to the
effective (observed) wavelengths of r and i bands for galaxies at the
characteristic redshift of $z\sim 0.235$ and $z\sim 0.5422$, respectively.

For a galaxy at redshift $z$, which falls between the characteristic redshifts
$z_{a}$ and $z_{b}$ corresponding to bands $a$ and $b$, we estimate the rest-
frame Sérsic parameter as

$x_{\text{RF}}=w_{a}x_{a}+w_{b}x_{b},$ (5)

where $x_{a}$ and $x_{b}$ are Sérsic parameter measurements in bands $a$ and
$b$ respectively. We add weights $w_{a}$ and $w_{b}$ as

$w_{a}=\left|{\frac{{z-z_{a}}}{{z_{b}-z_{a}}}}\right|\\\ \text{and}\\\
w_{b}=\left|{\frac{{z-z_{b}}}{{z_{b}-z_{a}}}}\right|.$ (6)

If a galaxy is missing Sérsic parameter measurement in one of the bands
(either $a$ or $b$) due to failed fit or magnitude limit, we use the
measurement from the available single band.

###  3.4 Validation via Comparison with Literature

Figure 10: Comparison between size measurements in our study and those in the
literature. Left panels show the comparison with the size measurements by van
der Wel et al. (2014, v+14) and the right panels show the comparison with
Kawinwanichakij et al. (2021, K+21), both corrected to rest-frame $5000$ Å. We
use measurements in r, i and z bands to bracket the rest-frame $5000$ Å
depending on the redshift of the galaxy (see Section 3.3). Our size
measurements are on $y$-axis and those from the literature are on $x$-axis.
The blue line shows the median for SFGs and the maroon line is the median
relation for QGs. The shaded region indicate the uncertainties in these median
sizes. The black line represents the $1:1$ relation. The lower panels show the
relative difference in sizes measured as a function of their measurements in
the literature. We estimate the relative difference as
$\Re_{d}=(R_{e}$(literature) - $R_{e}$(this work$)/$ $R_{e}$(literature).
Similar to the upper panels, the median values with their uncertainties are
also shown. There is a good agreement between the measurements in this study
and those in the literature.

A number of studies use HST to perform the structural analysis of galaxy light
profiles. Since we use ground-based data, our images have (on average) $\sim
5$ times larger PSF than HST images. Hence, it is important to investigate how
well our ground-based measurements perform when compared to the measurements
based on higher-resolution HST data.

We compare our size measurements with those by van der Wel et al. (2012) based
on HST Wide Field Camera 3 (WFC3/IR) images (the two left panels in Figure
10). We perform this comparison on a galaxy-by-galaxy basis because the study
by van der Wel et al. (2012) also contains the Sérsic profile measurements of
galaxies in the COSMOS field. Following the prescription described in that
study, we correct HST size measurements to the constant rest-frame wavelength
of $5000$ Å. In our measurements, we use weighted average of the measurements
in two bands that are closest to the rest-frame $5000$ Å at a given redshift
(Section 3.3).

Equivalent to our fitting approach, the measurements by van der Wel et al.
(2012) also come from a single Sérsic profile fitting using Galfit. Because of
this similarity in the approaches, a comparison between two size measurements
is an independent test of the accuracy of galaxy size measurements based on
the ground-based data. A direct comparison on a galaxy-by-galaxy basis shows
that the galaxy sizes in our study tend to be on average $\sim 5\%$ larger
than the measurements based on HST data for both SFGs and QGs (bottom panel in
the first column in Figure 10). The fact that ground-based data have worse PSF
than the HST data at least partially contributes to the observed offset. At
the same time, our data are more sensitive to the low-surface brightness
regions than the HST data and therefore our images better incorporate
information from outer regions of galaxies. Since the offset is not very large
(within uncertainties of individual measurements) and we do not find any trend
in this offset with galaxy size, we infer that the size measurements are
robust when obtained from the ground-based data with good seeing.

Kawinwanichakij et al. (2021) use the same ground-based images as we do (HSC-
SSP). In the right panel of Figure 10 we compare the two ground-based size
measurements, using the size estimates in rest-frame $5000$ Å for the
comparison (the only estimates avaiable in Kawinwanichakij et al., 2021).
Although in general we find a good agreement between these two size
measurements, there is a significant offset at smaller sizes
($R_{e}<0.3^{\prime\prime}$): our size measurements are systematically larger
than Kawinwanichakij et al. (2021). The offset increases with decreasing
galaxy sizes and it is more prominent for SFGs than QGs.

We speculate that the differences between our size measurements and those of
Kawinwanichakij et al. (2021) arise from a number of differences between the
two fitting approaches. Kawinwanichakij et al. (2021) use Lenstronomy (Birrer
et al., 2015; Birrer & Amara, 2018) whereas we use Galfit. In contrast to
their single-band size measurements corrected to the rest-frame, we estimate
the rest-frame $5000$ Å sizes by combining multi-band data that correspond to
this rest-frame wavelength directly, without any corrections. In addition,
compared to their cutout images, we select significantly larger cutout images
to model the extended profiles of the galaxies. Our simulations show that a
large-fitting region is essential to measuring accurately the overall galaxy
sizes (Section 3.2.2).

##  4 Results

Figure 11: Fitting smoothly broken double power law to the QGs and single
power law to the SFGs at rest-frame $5000$ Å in 5 redshift bins. The
individual SFGs are shown as blue points; red points represent QGs. The median
uncertainties in size measurements for SFGs and QGs are given in the top left
corner of each panel. The best-fitting power law relations for SFGs and QGs
are shown in navy blue and maroon, respectively, with the uncertainties from
Bayesian posterior plotted as shaded region.

cccccccc[hvlines]

1-8Rest-frame $3000$ Å

$z$ median $z$ $\alpha$ $\beta$ $R_{p}$ $\log M_{p}$ $\sigma_{\log{R_{e}}}$
$R_{0}$

$0.1-0.3$ $0.22\pm 0.01$ $0.05\pm 0.14$ $0.61\pm 0.11$ $2.26\pm 0.33$
$10.16\pm 0.17$ $0.21\pm 0.01$ $4.52\pm 0.06$

$0.3-0.45$ $0.35\pm 0.00$ $0.19\pm 0.06$ $0.65\pm 0.08$ $3.27\pm 0.41$
$10.52\pm 0.12$ $0.23\pm 0.01$ $4.07\pm 0.02$

$0.45-0.6$ $0.53\pm 0.00$ $0.20\pm 0.12$ $0.60\pm 0.07$ $3.18\pm 0.56$
$10.52\pm 0.16$ $0.27\pm 0.01$ $3.91\pm 0.03$

$0.6-0.75$ $0.68\pm 0.00$ $0.12\pm 0.08$ $0.71\pm 0.04$ $2.65\pm 0.20$
$10.49\pm 0.07$ $0.26\pm 0.01$ $3.48\pm 0.01$

$0.75-0.9$ $0.83\pm 0.00$ $0.04\pm 0.12$ $0.59\pm 0.06$ $3.35\pm 0.32$
$10.57\pm 0.10$ $0.28\pm 0.01$ $3.82\pm 0.01$

1-8Rest-frame $5000$ Å

$z$ median $z$ $\alpha$ $\beta$ $R_{p}$ $\log M_{p}$ $\sigma_{\log{R_{e}}}$
$R_{0}$

$0.1-0.3$ $0.22\pm 0.01$ $0.01\pm 0.12$ $0.75\pm 0.14$ $2.08\pm 0.28$
$10.25\pm 0.15$ $0.22\pm 0.01$ $4.15\pm 0.05$

$0.3-0.45$ $0.36\pm 0.00$ $0.15\pm 0.04$ $0.71\pm 0.05$ $2.38\pm 0.19$
$10.47\pm 0.07$ $0.21\pm 0.00$ $3.28\pm 0.01$

$0.45-0.6$ $0.53\pm 0.00$ $0.17\pm 0.06$ $0.73\pm 0.05$ $2.18\pm 0.22$
$10.47\pm 0.09$ $0.24\pm 0.01$ $3.02\pm 0.01$

$0.6-0.75$ $0.68\pm 0.00$ $0.04\pm 0.05$ $0.70\pm 0.04$ $1.87\pm 0.12$
$10.38\pm 0.06$ $0.24\pm 0.00$ $2.87\pm 0.01$

$0.75-0.9$ $0.84\pm 0.00$ $0.02\pm 0.06$ $0.70\pm 0.04$ $1.83\pm 0.12$
$10.43\pm 0.07$ $0.26\pm 0.00$ $2.60\pm 0.00$

Table 1: The best-fit parameters of double power law SMR for QGs in two rest-
frame wavelengths: $3000$ Å and $5000$ Å (Equation 8). Here $R_{0}$ denotes
the characteristic size of QGs at $5\times 10^{10}$M⊙. Uncertainties smaller
than $0.005$ are rounded as $0.00$. Both $R_{p}$ and $R_{0}$ are in kpc and
$M_{p}$ is in M⊙.

cccccc[hvlines]

1-6Rest-frame $3000$ Å

Galaxy $z$ median $z$ $\alpha$ $R_{0}$ $\sigma_{\log{R_{0}}}$

5-1SFGs 0.1-0.3 $0.23\pm 0.00$ $0.22\pm 0.03$ $6.35\pm 0.30$ $0.22\pm 0.01$

0.3-0.45 $0.36\pm 0.00$ $0.28\pm 0.01$ $6.49\pm 0.15$ $0.24\pm 0.00$

0.45-0.6 $0.53\pm 0.00$ $0.22\pm 0.01$ $6.02\pm 0.14$ $0.25\pm 0.00$

0.6-0.75 $0.69\pm 0.00$ $0.20\pm 0.01$ $5.95\pm 0.09$ $0.24\pm 0.00$

0.75-0.9 $0.84\pm 0.00$ $0.19\pm 0.01$ $5.83\pm 0.08$ $0.24\pm 0.00$

5-1QGs 0.1-0.3 $0.23\pm 0.01$ $0.58\pm 0.08$ $4.57\pm 0.23$ $0.20\pm 0.01$

0.3-0.45 $0.37\pm 0.00$ $0.61\pm 0.05$ $4.14\pm 0.13$ $0.21\pm 0.01$

0.45-0.6 $0.53\pm 0.00$ $0.59\pm 0.06$ $3.94\pm 0.17$ $0.25\pm 0.01$

0.6-0.75 $0.69\pm 0.00$ $0.70\pm 0.03$ $3.48\pm 0.09$ $0.25\pm 0.01$

0.75-0.9 $0.84\pm 0.00$ $0.59\pm 0.04$ $3.77\pm 0.13$ $0.26\pm 0.01$

1-6Rest-frame $5000$ Å

Galaxy $z$ median $z$ $\alpha$ $R_{0}$ $\sigma_{\log{R_{0}}}$

5-1SFGs 0.1-0.3 $0.23\pm 0.00$ $0.20\pm 0.02$ $5.76\pm 0.23$ $0.21\pm 0.01$

0.3-0.45 $0.36\pm 0.00$ $0.24\pm 0.01$ $5.74\pm 0.11$ $0.21\pm 0.00$

0.45-0.6 $0.53\pm 0.00$ $0.19\pm 0.01$ $5.32\pm 0.09$ $0.22\pm 0.00$

0.6-0.75 $0.69\pm 0.00$ $0.18\pm 0.01$ $5.35\pm 0.07$ $0.22\pm 0.00$

0.75-0.9 $0.84\pm 0.00$ $0.17\pm 0.01$ $5.01\pm 0.06$ $0.22\pm 0.00$

5-1QGs 0.1-0.3 $0.25\pm 0.01$ $0.73\pm 0.08$ $4.18\pm 0.19$ $0.21\pm 0.01$

0.3-0.45 $0.37\pm 0.00$ $0.67\pm 0.04$ $3.36\pm 0.08$ $0.21\pm 0.01$

0.45-0.6 $0.53\pm 0.00$ $0.69\pm 0.04$ $3.10\pm 0.09$ $0.23\pm 0.01$

0.6-0.75 $0.69\pm 0.00$ $0.68\pm 0.03$ $2.90\pm 0.06$ $0.24\pm 0.01$

0.75-0.9 $0.84\pm 0.00$ $0.67\pm 0.03$ $2.64\pm 0.05$ $0.24\pm 0.01$

Table 2: The best-fitting parameters from the single power law SMR for SFGs
and QGs in two rest-frame wavelengths (Equation 7). Uncertainties smaller than
$0.005$ are rounded as $0.00$. The unit of $R_{0}$ is in kpc.

###  4.1 Size-Stellar Mass Relation

Observations show that galaxies exhibit a relation between their stellar mass
and size(SMR) at least up to $z\sim 3$ (e.g., Shen et al., 2003; Trujillo et
al., 2004; Guo et al., 2009; Williams et al., 2010; Newman et al., 2012; van
der Wel et al., 2014; Lange et al., 2015; Faisst et al., 2017; Roy et al.,
2018; Mowla et al., 2019b; Matharu et al., 2019, 2020). The form of the SMR
differs between the SFGs and QGs (e.g., van der Wel et al., 2014; Mowla et
al., 2019b; Kawinwanichakij et al., 2021).

SFGs exhibit a linear relation between galaxy size and stellar mass in log-log
space:

$\log R_{e}=\log R_{0}+\alpha\log\left(\frac{M}{M_{0}}\right),$ (7)

where $\alpha$ is the slope and $R_{0}$ is the characteristic size at the
fiducial mass, $M_{0}$ (e.g., van der Wel et al., 2014; Kawinwanichakij et
al., 2021). In line with the literature, we adopt $M_{0}=5\times 10^{10}$M⊙.
Figure 11 illustrates the SMR at rest-frame $5000$ Å for SFGs in our sample
divided into five redshift bins (with data points in blue and the best fit
relation as blue-shaded line). In Appendix B, we show the SMR for our SFGs at
rest-frame $3000$ Å.

However, QGs have a more complex SMR where the slope changes at the pivot mass
$\sim 3\times 10^{10}$M⊙ (Lange et al., 2015; Mowla et al., 2019a; Mosleh et
al., 2020; Kawinwanichakij et al., 2021; Nedkova et al., 2021; Damjanov et
al., 2023). Hence, to explore the trend in size with stellar mass for QGs, we
first fit a smoothly broken double power law,

$R(M)=R_{p}\left(\frac{M}{M_{p}}\right)^{\alpha}\left[\frac{1}{2}{\left(1+\frac{M}{M_{p}}\right)^{\delta}}\right]^{\frac{\beta-\alpha}{\delta}},$
(8)

where $M_{p}$ is the pivot stellar mass at which the slope changes, $R_{p}$ is
the effective radius at the pivot stellar mass, $\alpha$ is the slope of the
SMR at the low-mass end, $\beta$ is the slope at the high-mass end, and
$\delta$ is the smoothing factor. Following Mowla et al. (2019a), we adopt
$\delta=6$ to reduce degeneracy between $\delta$ and the slopes.

We fit the SMR for both SFGs and QGs using
Dynesty888https://zenodo.org/record/7832419, a Python package designed to fit
the data by implementing Bayesian inference (Speagle, 2020). We follow the
fitting procedure from van der Wel et al. (2014, see their Section 3). The
differences between our approach and the method of van der Wel et al. (2014)
are the use of the double power law for QGs and how total likelihood is
estimated in our study. van der Wel et al. (2014) introduce a $10\%$
contamination in the dataset due to misclassification of SFGs and QGs. Since
we use UVJ classification where the separation criteria varies with redshift,
we do not account for contamination in this study. Additionally, they also
allow for $1\%$ outliers (objects that should not be in the data) in the
likelihood estimation.

Our likelihood function is of the form

$\displaystyle\mathcal{L}=\ln{(wp)},$ (9)

where $w$ is a weighting factor and $p$ is the probability of observing the
given size for a galaxy of mass $M_{*}$ as described by van der Wel et al.
(2014, their Equation 4). The weighting factor, $w$, is inversely proportional
to the galaxy number density at a given stellar mass taken from the stellar
mass functions of Muzzin et al. (2013). By applying this weight, we ensure
that each stellar mass range contributes equally to the SMR fit.

We first fit the double power-law to the SMR of QGs. Figure 11 shows (in red)
the distribution of galaxies in this parameter space and the best-fit
relations in five different redshift bins in the rest-frame $5000$ Å (see
Appendix B for the results in rest-frame $3000$ Å). It is clear that QGs have
shallower SMR slopes at stellar masses below the pivot mass compared to those
of more massive QGs. The SMR for the low mass QGs is horizontal within
$2\sigma$ (i.e., $\alpha\sim 0$). These shallow slopes of low mass QGs
($M<M_{p}$) are also consistent, within their uncertainties, with the results
of Kawinwanichakij et al. (2021) and Nedkova et al. (2021). However, the QGs
show a steep SMR above the pivot mass, $M_{p}(z)$. This pivot mass is around
$\log M\sim 10.5$, except in the lowest redshift bin where it drops to $\sim
10.3$. In addition, the high-mass end slopes $\beta$ do not change
significantly across the full redshift interval. Table 4 gives the best fit
parameters of the double power law fitting of QGs in two rest-frame
wavelengths.

Once we estimate the pivot point, we fit the SMR for the QGs with stellar
masses larger than $M_{p}(z)$ and SFGs that are more massive than
$10^{9.5}M_{\odot}$ (mass completeness limit; Section 2.4) using the single-
power law from Equation 7. We use the Equation 7 for QGs to be able to
directly compare our findings with the results from the literature (Section
4.2). This approach captures the nature of SMR for QGs when we limit our
analysis to very massive QGs (above the pivot point)999We note that although
we fit the QGs in two ways (double power law given in Table 4 and single power
law in Table 4), the results for massive QGs are very similar.. The SMR
fitting procedure is the same as above. We fit the SMR for SFGs and QGs
separately without considering any cross-contamination between the two
populations. Unlike van der Wel et al. (2014), we use moving UVJ separation
line (with $z$) which further reduces contamination (Whitaker et al., 2011).
The best-fit parameters are given in Table 4.

Figure 12: The best-fitting single power-law SMR for SFGs and massive QGs in
two rest-frame wavelengths ($3000$ Å and $5000$ Å) in five redshift bins. The
shaded regions represent the fit uncertainties estimated from the Bayesian
posterior for SFGs and QGs respectively. The SMR fits for SFGs and QGs in the
shorter wavelength are plotted in cyan and red, respectively. The fits for
SFGs and QGs in the longer wavelengths are shown in navy blue and maroon,
respectively.

Figure 12 shows the resulting SMR in the rest-frame $5000$ Å for SFGs and QGs
(in navy blue and maroon, respectively) separated into five redshift bins.
This figure includes the best-fit SMRs for SFGs from Figure 11. For QGs, the
Figure 12 shows the best-fitting single power-law for massive QGs above the
pivot point ($M>M_{p}$).

The results are mostly consistent with previous works. Except at very high
masses, the sizes of SFGs are larger than those of QGs at a given stellar mass
as shown previously by Shen et al. (2003), Williams et al. (2010), van der Wel
et al. (2014), Mosleh et al. (2020), Kawinwanichakij et al. (2021) and others.
At very high masses ($\log M\gtrsim 11.2$), we find that the average size of
QGs becomes comparable to that of SFGs. The slopes of the SMR for SFGs
($\alpha\sim 0.2$) and massive QGs ($\alpha\sim 0.7$) are similar to those in
the literature (van der Wel et al., 2014; Mowla et al., 2019b; Kawinwanichakij
et al., 2021; Nedkova et al., 2021).

Figure 12 also shows the resulting best-fit SMRs in the rest-frame $3000$ Å
for both SFGs and QGs (in cyan and red, respectively; see Figure 28 for the
distribution of individual data points in this parameter space). Except for
the clear offset in the zero-points, the results in the rest-frame $3000$ Å
are also very similar to the results in the rest-frame $5000$ Å. For stellar
masses $\log M\lesssim 11.2$, the SFGs are, on average, larger than QGs. The
slopes of the SFGs are comparable between the two wavelengths ($\alpha\sim
0.2$). At the same time, the SMR for QGs is systematically $\sim 10\%$ steeper
in the red light than in the blue light, although this difference is within
$2\sigma$ uncertainty on the slope estimates. We return to the difference in
the SMR zero points for two rest-frame wavelengths in Section 4.3.

###  4.2 Median Size Evolution in the Rest-frame Visible Light

Figure 13: Evolution of the median galaxy sizes at fixed stellar mass in rest-
frame $5000$ Å for galaxies segregated in four mass bins. Mass bins are noted
in the top right corner of each panel. SFG population is represented by blue
colour and the QG population is shown in red. The median sizes in five
redshift bins are shown with stars for SFGs and with circles for QGs along
with their bootstrapped error bars. The size evolution is represented by a
power-law, $A(1+z)^{\beta}$ (solid curves with uncertainties from Bayesian
posterior as shaded region). For comparison, we show the results from van der
Wel et al. (2014, v+14) as well. Their best fitting evolution curves for SFGs
and QGs at rest-frame $5000$ Å are shown in blue and red hatched regions
respectively. In general, there is a good agreement between our results and
those of v+14.  Figure 14: Comparison of the size evolution at fixed stellar
mass for SFGs and QGs we trace in four stellar mass bins with the results of
Mowla et al. (2019b, M+19) and Kawinwanichakij et al. (2021, K+21). The size
evolution reported by M+19 and K+21 are shown as hatched regions. Rest of the
details in the figure are the same as in Figure 13. In general, there is a
good agreement between our results and the literature. Figure 15: Comparison
of the median size evolution for SFGs and QGs at fixed mass between rest-
frames UV ($3000$ Å) and visible ($5000$ Å) light. As in Figure 13, panels
show the median size evolution in various mass bins as denoted in the top
right corner of each panel. Cyan and navy blue represent SFG population in
rest-frames $3000$ Å and $5000$ Å respectively. Similarly, red and maroon
colours denote QG population in rest-frame $3000$ Å and $5000$ Å respectively.
The best fitting power-law functions (solid curves) for the size evolution of
SFGs and QGs are shown with their uncertainties (shaded regions). Galaxies are
more extended in shorter wavelength than in longer wavelength.

We further employ galaxy size measurements to explore the trend in median size
with redshift for SFGs and QGs segregated by stellar mass. First, we use
median galaxy sizes to explore galaxy size evolution in the rest-frame
wavelength of $5000$ Å.

Figure 13 shows the median sizes of SFGs and QGs in this rest-frame wavelength
as a function of redshift in four different stellar mass bins. Within each
bin, the stellar mass range is fixed across the full redshift interval. In
addition to SFGs having larger sizes than QGs, median sizes of both galaxy
populations at a given fixed stellar mass generally grow with cosmic time.

We fit the redshift evolution in size of SFGs and QGs with a power law,

$R_{e}=R_{e}^{0}(1+z)^{\beta},$ (10)

where $R_{e}^{0}$ is $R_{e}$ at $z=0$. When traced in the rest-frame visible
light, the pace of size evolution differs between SFGs and QGs (blue vs. red
solid lines in Figure 13). In all mass bins below $\log M\sim 11$, sizes of
QGs evolve faster than those of SFGs. In the mass range $10.5<\log M<11$ and
over the redshift interval $0.1<z<0.9$, the median sizes of QGs grow as
$R_{e}\varpropto(1+z)^{-1.0}$ whereas the sizes of SFGs grow as
$R_{e}\varpropto(1+z)^{-0.5}$. For example, the average sizes of $\log M\sim
10.75$ QGs grow by $\sim 73\%$ in the time interval of 6.1 Gyrs. In contrast,
similarly massive SFG grows in size by only $\sim 31\%$ over the same time
interval.

Furthermore, for both galaxy populations, the pace at which galaxies grow is
faster for more massive systems. The growth rate
$\beta=\frac{d\log{(R_{e}/\mathrm{kpc})}}{d\log(1+z)}$ for SFGs ranges from
$0.10\pm 0.18$ for the least massive to $-2.07\pm 0.45$ for the most massive
SFGs (blue lines in Figure 13). Over the same mass range, the growth rate for
QGs changes from $\beta=-0.55\pm 0.4$ to $\beta=-1.94\pm 0.64$ (red lines in
Figure 13). QGs grow significantly in size at $z<1$ even at lower masses
($\log M<10.5$). In contrast, the size evolution of SFGs at these low masses
is, within uncertainties, consistent with no growth.

We note that these growth rates are the median trends for galaxy populations,
not individual galaxies. Furthermore, for now we do not consider the fact that
stellar mass of SFGs also grows through star formation. Finally, here we do
not account for the effect of progenitor bias among QG population. We estimate
the amplitude of the effects that mass growth and progenitor bias have on the
evolution in average size of SFGs and QGs in Section 5.

Our findings are in good agreement with results reported in the literature. In
Figure 13, we compare our results (solid curves and shaded regions) with those
from van der Wel et al. (2014, hatched region) at rest-frame $5000$ Å. Like
us, they have fitted galaxy images using Galfit. However, their fits are based
on galaxy images taken with WFC3 onboard HST, which has at least 4 times
better resolution than our ground-based HSC images (see Section 3.4).
Furthermore, we note that their best-fitting relation between median size and
redshift spans the redshift interval $0<z<3$. Finally, they only have two
redshift bins at $z<1$ while we have a finer sampling of this redshift range
with $5$ bins. Despite these differences, we find an excellent agreement
between our trends and results reported in van der Wel et al. (2014).

In Figure 14 we compare our results (solid lines with shaded regions) with
Mowla et al. (2019b, backward hatched regions) and Kawinwanichakij et al.
(2021, forward hatched regions). Mowla et al. (2019b) follow a similar
methodology as ours: they fit the galaxy light profiles using Galfit. However,
their measurements are based on the same HST imaging as the ones reported in
van der Wel et al. (2014) and thus do not suffer from atmospheric seeing
effects. Furthermore, they measure galaxy size growth over the redshift
interval $0<z<3$, with only two redshift bins below $z\sim 1$. At the same
time, Kawinwanichakij et al. (2021) use a ground-based HSC dataset similar to
ours, but follow a different methodology (see Section 3.4 for details).

Except for the marginal difference for SFGs in the lowest mass bin (first
panel of Figure 14), the agreement between our results and the results of
Mowla et al. (2019b) is excellent. With our dataset that covers larger area of
the COSMOS field (Section 2.1), we are able to obtain a narrower confidence
interval (shaded regions in Figure 14).

In general, our median size evolution estimates for QGs are in agreement with
Kawinwanichakij et al. (2021), but we do find some discrepancy between the
measurements for SFGs. Overall, the median SFG size measurements of
Kawinwanichakij et al. (2021, blue forward hatched regions in Figure 14) are
smaller than our measurements and those of Mowla et al. (2019b). This offset
could could be a combination of several effects. As we discuss in Section 3.4,
the two ground-based studies (this study and that of Kawinwanichakij et al.
(2021)) use different methodology to fit galaxy light profiles. In addition,
we select significantly larger cutout images to model the extended galaxy
profiles. Furthermore, we have much better estimation of photo-$z$’s and UVJ
based SFG-QG classification thanks to $U+grizy+JHK$ data from COSMOS2020,
whilst Kawinwanichakij et al. (2021) use only HSC $grizy$ data to segregate
galaxies into SFGs and QGs. On the other hand, our study is somewhat affected
by cosmic variance because our dataset covers almost two orders of magnitude
smaller area on the sky (1.6 deg2 of the COSMOS field vs $100$ deg2 of the
HSC-SSP analyzed in Kawinwanichakij et al. 2021). Using the Equation $3$ from
Driver & Robotham (2010), we estimate that the cosmic variance will decrease
by a factor of $4$ when we expand to the full CLAUDS+HSC survey area (from
$12\%$ to $3\%$).

The slower pace of evolution for SFGs compared to QGs for galaxies with $\log
M<11$ in the rest-frame $5000$ Å is consistent with the suite of previous
studies (e.g., Lilly et al., 1998; Ravindranath et al., 2004; Barden et al.,
2005; van der Wel et al., 2014; Mowla et al., 2019b; Kawinwanichakij et al.,
2021). Additionally, similar to our findings, van der Wel et al. (2014), Mowla
et al. (2019b) and Kawinwanichakij et al. (2021) find that the rate of median
size evolution for both QGs and SFGs depends on galaxy stellar mass. This
comparison confirms that the resolution of the HSC+CLAUDS imaging dataset is
well suited for detailed probes of galaxy size evolution up to $z\sim 1$.

###  4.3 Median Size Evolution in Rest-frame UV Light

In addition to the rest-frame visible light, we also investigate the size
evolution of galaxies in rest-frame UV ($3000$ Å). The comparison of galaxy
sizes in two rest-frame wavelengths using a large dataset is unique to our
work. These two rest-frame wavelengths ($3000$ and $5000$ Å) represent light
coming from young and old stars (see Section 3.3). Hence, comparisons of
galaxy morphology in these two rest-frame wavelengths can advance our
understanding of the connection between galaxy structural change (i.e., size
growth) and the evolution in their stellar content.

We combine the median evolutionary trends in galaxy size at fixed stellar mass
in rest-frame UV and visible light for both SFGs and QGs in Figure 15. The
median sizes of SFGs and QGs in UV are represented by cyan crosses and red
diamonds respectively and those in visible light are in blue stars and maroon
circles. We also show the best-fit size evolution trends (solid curves) along
with uncertainties (shaded area).

On average, we find that the two galaxy popoulations are larger in rest-frame
$3000$ Å than in $5000$ Å. For example, the median size of SFGs with stellar
mass $\log M\sim 10.5$ at $z\sim 0.5$ is $12\%$ larger in rest-frame blue
light than in red light (cyan and blue curves in the second panel of Figure
15). The sizes of QGs display even larger difference when measured in two
rest-frame wavelengths. At the same mass and redshift ($\log M\sim 10.5$,
$z\sim 0.5$) QGs are, on average, $33\%$ more extended in the observations at
shorter rest-frame wavelength ($3000$ Å) than in $5000$ Å (red and maroon
curves in the second panel of Figure 15).

Several evolutionary trends in galaxy size at the rest-frame wavelegth of
$3000$ Å are similar to those we observe at the rest-frame wavelength of
$5000$ Å. Firstly, the median UV-based sizes of both SFGs and QGs evolve with
cosmic time. At stellar mass range of $10.8<\log M<11$, the median UV-based
sizes of SFGs grow by $\sim 50\%$ over the span of $\sim 6.1$ Gyrs ($z=0.9$ to
$z=0.1$) whilst sizes of similarly massive QGs grow by $21\%$ in the same
rest-frame regime and during the same cosmic period. However, the pace of
evolution in rest-frame $3000$ Å is slower than in $5000$ Å. For comparison,
in the rest-frame visible light SFGs of similar mass grow by $90\%$ over the
same redshift interval; based on their visible light profiles, similarly
massive QGs grow by $78\%$ since $z\sim 0.9$.

Finally, the trend in the pace of size evolution with galaxy mass in the rest-
frame $3000$ Å is similar to the trend we observe at $5000$ Å except for QGs
in the most massive bin ($\log M>11$). SFGs in the most massive bin grow by
$\sim 165\%$ whilst at lowest mass bin we do not find any significant evidence
for size evolution of either SFG or QG population. However, unlike the rest-
frame visible light, the most massive QGs in rest-frame UV do not show any
significant signs of growth in median size since $z\sim 1$. Nevertheless, we
note that our sample size is small at these higher mass end and the
uncertainties are large.

cccc[hvlines]

1-4Rest-frame $3000$ Å

Galaxy $\log M$ $R_{e}^{0}$ $\beta$

4-1SFGs $9.7$ $3.62\pm 0.39$ $0.03\pm 0.24$

$10.2$ $4.95\pm 0.27$ $-0.11\pm 0.12$

$10.7$ $6.83\pm 0.40$ $-0.26\pm 0.13$

$11.2$ $9.57\pm 1.17$ $-0.45\pm 0.27$

2-1QGs $10.7$ $5.06\pm 0.64$ $-0.59\pm 0.28$

$11.2$ $9.68\pm 0.58$ $-0.44\pm 0.14$

1-4Rest-frame $5000$ Å

Galaxy $\log M$ $R_{e}^{0}$ $\beta$

4-1SFGs $9.7$ $3.59\pm 0.30$ $-0.08\pm 0.19$

$10.2$ $4.75\pm 0.28$ $-0.21\pm 0.13$

$10.7$ $6.28\pm 0.38$ $-0.35\pm 0.14$

$11.2$ $8.37\pm 0.76$ $-0.50\pm 0.20$

2-1QGs $10.7$ $5.07\pm 0.58$ $-1.10\pm 0.26$

$11.2$ $11.95\pm 1.96$ $-1.24\pm 0.36$

Table 3: The best-fit parameters for the redshift evolution of characteristic
sizes at fixed stellar mass for SFGs and QGs in two rest-frame wavelengths
($R_{e}=R_{e}^{0}(1+z)^{\beta}$). The best-fit curves are displayed in Figures
16 and 21 for SFGs and QGs respectively.

cccc[hvlines]

1-4SFGs

Restframe $\lambda$ $\log M_{(z=0.9)}$ $R_{e}^{0}$ $\beta$

4-1$3000$ Å $9.7$ $7.00\pm 0.44$ $-0.91\pm 0.14$

$10.2$ $8.06\pm 0.63$ $-0.81\pm 0.17$

$10.7$ $9.72\pm 1.15$ $-0.78\pm 0.27$

$11.2$ $12.21\pm 2.03$ $-0.80\pm 0.37$

4-1$5000$ Å $9.7$ $6.42\pm 0.45$ $-0.91\pm 0.16$

$10.2$ $7.25\pm 0.55$ $-0.82\pm 0.17$

$10.7$ $8.52\pm 0.80$ $-0.79\pm 0.21$

$11.2$ $10.37\pm 1.27$ $-0.82\pm 0.28$

Table 4: The best-fit parameters of redshift evolution of characteristic sizes
of SFGs with evolving mass as described in Section 5.1.2. The corresponding
evolutionary curves are displayed in Figure 17.

##  5 Discussion

So far we have used the median sizes in several mass bin to trace the size
evolution of galaxies in our dataset. Although this approach provides direct
comparison with the results based on median size measurements from other
studies, it does not formally take into account the uncertainties present in
the individual size and mass measurements. At the same time, the SMR fitting
procedure incorporates these measurement uncertainties and applies weighting
based on galaxy stellar mass functions (Section 4.1). Therefore, we decide to
use the size estimates from the best-fitting SMRs for the interpretation of
our results. Our conclusions remain the same when we replace the estimates
from the best-fit SMRs with the measured median sizes in bins of galaxy
stellar mass.

In addition, we also compare this size evolution in two rest-frame
wavelengths: $3000$ Å and $5000$ Å. The analysis of the size growth in two
rest-frame wavelengths enables us to trace the distribution of young and old
stellar populations in galaxies. We list the parameters of the best-fit
function for the redshift evolution of SFG and QG sizes in both wavelengths in
Table 4.3.

Figure 16: Evolution of characteristic sizes of SFGs at a given fixed stellar
mass. Panels show the size evolution of SFGs for a range of characteristic
masses (as labeled). The SFG sizes at rest-frame $5000$ Å are shown in navy
blue and at rest-frame $3000$ Å in cyan. The sizes at shorter wavelength are
larger than those at longer wavelengths. The size evolution is fit by a power-
law $R_{e}=R_{e}^{0}(1+z)^{\beta}$, and the best fitting parameters are given
in Table 4.3. The solid curves show the evolutionary trends as described by
these power-law functions with shaded regions showing the $1\sigma$ confidence
intervals.

###  5.1 Size Evolution of Star-Forming Galaxies

####  5.1.1 Overview of the Observed Trends

Figure 16 shows the size evolution of SFGs in two rest-frame wavelengths for
four fixed characteristic stellar masses since $z\sim 1$. As expected, for the
evolution in rest-frame $5000$ Å (visible light, navy blue symbols and solid
lines with shaded area in Figure 16) we find similar trends as in Section 4.2.
The size growth of SFGs is mass dependent: the higher the mass of SFGs, the
faster they grow in size with cosmic time. At $\log M=9.7$, SFGs grow only
$\sim 5\%$ in size but at $\log M=11.2$, they grow by $31\%$ in $6$ Gyrs from
$z\sim 0.9$ to $z\sim 0.1$.

At the characteristic mass of $5\times 10^{10}\text{M}_{\odot}$, van der Wel
et al. (2014) report the power-law fit (Equation 10) to the size evolution of
SFGs with $\beta=-0.75$ whereas we find $\beta=-0.35\pm 0.14$. When probing
the trends in near-IR-based size with redshift for $\log M>10.8$ galaxies
Williams et al. (2010) also find a steeper size evolution for SFGs
($\beta=-0.77\pm 0.08$). We note that both Williams et al. (2010) and van der
Wel et al. (2014) studies cover wider redshift ranges: $0.5<z<2$ and $0<z<3$
respectively. Thus, their observed evolution is affected by SFGs at $z>1$.
Furthermore, the flatter distribution of median SFG sizes below $z\sim 1$ from
Figure 4 of Williams et al. (2010) and the agreement between our size
measurements and those of van der Wel et al. (2014) suggest that the size
evolution of SFGs weakens at $z<1$. Moreover, Oesch et al. (2010) and Mosleh
et al. (2012) report faster evolution of SFG sizes at $z>2$. At the same time,
studies that focus at $z<1$ (e.g., Lilly et al., 1998; Ravindranath et al.,
2004; Barden et al., 2005; Kawinwanichakij et al., 2021) report either a slow
or no evolution of the average size of SFGs. Thus, our analysis confirms that
the size evolution of SFGs slows down with decreasing redshift and that this
change in pace occurs at $z\sim 1$.

As in Figure 15, Figure 16 also includes the size evolution of SFGs in the
rest-frame $3000$ Å (cyan symbols and solid lines with shaded area). In
general, we find that the sizes of SFGs are larger at shorter wavelength than
at the longer one, as we indicate in Section 4.3. For example, at $z=0.5$,
SFGs appear $\sim 5\%-17\%$ larger in rest-frame UV than in visible light
depending on their stellar mass. Since the shorter wavelength probes the
distribution of younger stellar population, the larger size in $3000$ Å may
imply extended (less concentrated) star-forming regions in those star-forming
systems. A smaller size at longer wavelength suggests that the older
population is more centrally concentrated, in line with the inside-out
quenching scenario (Tacchella et al., 2015, 2018; Lin et al., 2019).

The difference between the two rest-frame wavelengths also depends on galaxy
stellar mass. The sizes of SFGs at low mass ($\log M<10$) are almost the same
at both wavelengths, and the differences in the sizes between the wavelength
regimes increase with their stellar mass. At $z=0.5$, an average SFG with
stellar mass $\log M=9.7$ has very similar size (within uncertainties) in both
red and blue light. However, a massive SFG ($\log M=11.2$) is $\sim 17\%$
larger in blue than in red light.

We interpret the lack of difference in two rest-frame sizes of low-mass SFGs
as a sign that low-mass galaxies tend to have their younger and older stars
well mixed within their light profiles, without any strong signals of either
inside-out or outside-in quenching. However, as the galaxies grow in mass and
size, the bulges also grow in the centres of these galaxies. These bulges may
eventually suppress star formation by stabilizing the gas within them (e.g.,
Martig et al., 2009; Saintonge et al., 2012; Sachdeva et al., 2017;
Hashemizadeh et al., 2022). As the star formation ceases in the centre, the
peak of star formation activity moves to the outskirts of disks. Thus, the
massive SFGs tend to look bigger in rest-frame UV than in visible light.

At fixed stellar mass, we find that the pace of SFG size evolution is slightly
slower at $3000$ Å than at $5000$ Å. For example, at $\log M=10.7$, SFGs grow
by $21\%$ ($\beta=-0.35$) in red light and by $15\%$ ($\beta=-0.26$) in blue
light between $z\sim 0.9$ and $z\sim 0.1$. The direction towards faster growth
in red light is systematic across all stellar masses of SFGs, but we also note
that, within uncertainty, this difference is consistent with the same pace of
size evolution in two rest-frame wavelengths.

The slower pace of evolution in blue light than in red light could also
indicate the role of bulge growth in the evolution of SFGs. Bulges become more
and more significant in the overall galaxy light profile with time (e.g.,
Sachdeva et al., 2017). As the bulges grow, this growth is reflected in the
size evolution of SFGs at longer wavelength where we see a slightly faster
pace of size evolution.

There are two scenarios for bulge growth that we can consider: (1) both the
bulge and the disk grow in SFGs, and (2) bulge grows into the disk and pushes
the peak star-forming regions we can observe at $3000$ Å further out. In the
first scenario, the slightly faster pace of SFG size evolution in longer
wavelength probing older stellar population could suggest that the growth of
bulges is faster than the growth of disks at this redshift range. In the
second scenario, since only bulges grow significantly, the size growth in
visible light is faster as well. Although pushing the peak star forming
regions outward will make the sizes in UV bigger, growth in visible light is
still stronger because it encompasses the light from stars in both the growing
bulge and the disk. However, we note that we only fit a single Sérsic profile
to observed galaxy light profiles. We will explore the two scenarios in more
detail in the follow-up study by performing bulge$+$disk decomposition.

Figure 17: Evolution of characteristic sizes of SFGs assuming evolving stellar
mass we calculate based on Equation 12 at $z<1$. Each panel shows the size
evolution for a galaxy that has labeled stellar mass at $z\sim 0.9$. This
characteristic mass then evolves along the redshift space in each panel via
in-situ star formation. We adopt the SFR values of star formation main
sequence from Speagle et al. (2014) to estimate this stellar mass growth. We
then adopt galaxy size considering increased mass and observed SMR at each
subsequent redshift point. As in Figure 16, the navy blue stars denote the
size in rest-frame $5000$ Å and the cyan crosses denote those in rest-frame
$3000$ Å, along with their bootstrapped error bars. The best fitting power-
laws with their uncertainties are represented by solid curves and shaded
areas. The parameters of the best-fit power-laws are given in Table 4.3. By
selecting galaxies with increased stellar mass at lower redshifts we attempt
to analyze the size evolution of SFGs by connecting progenitors to their
descendants.

####  5.1.2 The Effects of SFG Mass Growth

The earlier studies in general look at the galaxy size evolution at a constant
stellar mass across the redshift range that they probe (e.g., van der Wel et
al., 2014; Mowla et al., 2019b; Kawinwanichakij et al., 2021, and many
others). So far, we have also analyzed the size evolution in this manner.
However, SFGs are actively forming stars and growing in both size and mass
with cosmic time. To study the true size evolution of SFGs, it is important to
consider their mass growth.

We use a toy model to trace the size evolution of galaxies having progenitors
with similar mass at $z\sim 0.9$ by incorporating mass growth via star-
formation. We assume that the SFGs remain on the star-forming main sequence
since $z\sim 1$ and that the mass growth is only due to in situ star formation
in galaxies. Although galaxies quench and leave the SFG population over cosmic
time interval we probe, in the toy model we also assume that this quenching is
random and does not affect any sub-population of SFGs specifically.

We adopt the star-forming main sequence relation from Speagle et al. (2014),

$\log\mathrm{SFR}(M,t)=(0.84-0.026t)\log M+0.11t-6.51,$ (11)

where $t$ is the age of the universe in Gyr and SFR is in
$\hbox{M${}_{\odot}$}/$yr. For SFGs with a specific mass at $z=0.9$, we
estimate their mass at various cosmic times between $z=0.9$ and $z=0.1$ as

$M(t_{2})=M(t_{1})+\int_{t_{1}}^{t_{2}}10^{\log\mathrm{SFR}(M,t)}dt,$ (12)

where $t_{1}$ is the age of the Universe at $z=0.9$ and $t_{2}$ is its age at
the target redshift. We then estimate the characteristic sizes of these SFGs
at lower redshifts by taking their increased stellar mass as the input for the
SMR (Equation 7) with redshift-dependent parameters from Table 4. Finally, we
fit the size evolution of SFGs growing both in mass and size at two different
rest-frame wavelengths. Figure 17 illustrates the results and Table 4.3 lists
the best-fit parameters of the evolutionary trends for the SFGs with growing
stellar mass.

When considering stellar mass growth, for every initial stellar mass (i.e.,
stellar mass at $z=0.9$) we find that the size growth is faster than the
growth at constant mass. In rest-frame visible light, SFGs with initial
stellar mass of $\log M=9.7$ grow in size by $\sim 65\%$ when we consider mass
growth (navy blue curve in the first panel of Figure 17) against only $\sim
5\%$ growth when the mass growth is ignored in the redshift range we probe
(navy blue curve in the first panel of Figure 16). The observed weak/no
evolution at the lowest mass bin in Figure 16 ($\beta=-0.08\pm 0.19$ in $5000$
Å) seems to be due to the selection of different galaxies at different
redshifts.

At the same time, the SFGs with initial stellar mass $\log M=11.2$ grow in
size by $\sim 55\%$ and $\sim 32\%$ with and without stellar mass growth,
respectively (last panels in Figures 17 and 16). Thus the difference in size
growth with or without simultaneous stellar mass growth decreases with stellar
mass. The difference we observe demonstrates that the size evolution of
individual galaxies is faster than what is being suggested in the literature
because size growth of SFGs cannot be separated from the increase in their
stellar mass.

Additionally, we do not find any differential size evolution with stellar mass
when we consider mass growth. Our analysis shows that the sizes of all massive
SFGs ($\log M>9.5$) grow, within uncertainties, at a constant pace
($\beta\sim-0.8$ to $-0.9$) (Figure 17 and Table 4.3).

Furthermore, the differences in the rate of size growth between two wavelength
regimes that we report in Section 5.1.1 disappear when we include stellar mass
growth. Both cyan and navy blue curves represent power-law functions with very
similar exponents in all panels of Figure 17. However, throughout the redshift
range we probe, the SFGs remain more extended in the shorter wavelength than
in the longer wavelength in all initial stellar mass bins.

####  5.1.3 Slowing down of Size Evolution for SFGs

In combination with the observed rates of size growth at $z>1$ from the
literature, our results for the star-forming population point to slowing down
of SFG size evolution with cosmic time (5.1.1). To explore this further, we
separate SFGs into two groups based on $U-V$ colour: red SFGs ($U-V>1$) and
blue SFGs ($U-V\leq 1$). In Figure 18 we show the characteristics of these two
populations in size-stellar mass parameter space for sizes measured at the
rest-frame $5000$ Å. The figure shows that (1) blue SFGs (blue squares) cover
a smaller range of masses (up to $10^{10.5}$M⊙) than red SFGs (purple
diamonds); (2) at the same stellar mass, blue SFGs are around $20\%$ larger
than red SFGs.

The lack of massive blue SFGs may be related to the extent of the star
formation episodes. Blue SFGs are forming stars at high rates and thus are
still "catching up" with the SFGs that are in slightly more advanced stage of
the evolution (but still on the main sequence). As the SFGs evolve, they
become redder in colour because low-mass stars become dominant light source. A
fraction of these stars are in central bulges. The presence of bulges in red
SFGs makes them more centrally concentrated in the rest-frame $5000$ Å than
blue SFGs of the same mass. Hence, on average, these red SFGs have smaller
sizes in visible light than blue SFGs.

Figure 18: The median rest-frame $5000$ Å sizes of red SFGs ($U-V>1$) and blue
SFGs ($U-V\leq 1$) in ten equally populated mass bins. The red SFGs are shown
as indigo diamonds and blue SFGs as blue squares. All errors are bootstrapped.
Blue SFGs tend to be less massive than red SFGs and thus mass bins are
concentrated at lower masses for blue SFGs. At a given mass, blue SFGs are
$20\%$ larger than red SFGs. Figure 19: The redshift evolution of the SFG red
fraction: fraction of red SFGs in the complete SFG population along with
bootstrapped error bars (diamonds with solid line). Conversely, the fraction
of blue SFGs decrease with cosmic time. We find that the fraction of red SFGs
increases with cosmic time. Since the majority of blue SFGs have mass $\log
M<10.3$ (see Figure 18), we also show the red SFG fraction in this mass range
(crosses with dashed line) to demonstrate that the trend remains the same.
This fractional increase in the smaller and redder SFGs with cosmic time could
explain the slowing pace of the characteristic size evolution of SFGs since
$z\sim 1$. Figure 20: The distribution of Sérsic indices of blue SFGs (blue
filled histograms) and red SFGs (purple unfilled histograms). Galaxies with
$n>1.5$ are often considered to have a significant bulge component whilst
galaxies with $n>2.5$ are considered to bulge dominated. Among bulge-dominated
SFGs, $83\%$ are red. We also show the p-value from the K-S test in the figure
legend.

We also find that the fraction of red galaxies in the SFG population is
steadily increasing with cosmic time across the redshift range we explore
(Figure 19). At $0.75<z<0.9$, red SFGs constitute $\sim 64\%$ of the SFG
sample (purple diamonds with solid curve). This percentage increases to $\sim
80\%$ in the lowest redshift bin. Conversely, the fraction of blue SFGs
decreases with redshift because the sum of fractions of red and blue SFGs
equals to $1$ in each redshift bin.

Since blue SFGs have low masses, we investigate how the fraction of red SFGs
change with redshift for $\log M<10.3$ (crosses with dashed curve in Figure
19). We find a similar trend: the fraction of red SFGs increases with
decreasing redshift. Low-mass red SFGs constitute $\sim 48\%$ of the low-mass
SFG sample in the highest redshift bin but this percentage increases to $\sim
82\%$ in the lowest redshift bin. This increase in the fraction of smaller red
galaxies in the SF population could explain the observed slower pace of sizes
growth for SFGs at $z<1$ with respect to higher redshifts.

Based on the observational studies, galaxies with Sérsic index $n\gtrsim 1.5$
are generally considered to have a significant bulge component and those with
$n\gtrsim 2.5$ are considered bulge-dominated or spheroidal galaxies (e.g.,
Dutton, 2009; Sachdeva, 2013). Figure 20 shows that in our sample red SFGs
dominate both the $n>1.5$ (galaxies with significant bulge component) and
$n>2.5$ (bulge-dominated galaxies) subsamples with 81% and 83%, respectively.
The average Sérsic indices of red SFGs are higher than their blue
counterparts. The median $n$ of red SFGs ($n=1.14$) is $33\%$ higher than blue
SFGs ($n=0.86$). We perform a Kolmogorov–Smirnov test (K-S test, Kolmogorov-
Smirnov et al., 1933; Smirnov, 1948) to check whether the two distributions
differ statistically from each other. An extremely small p-value implies that
the two samples do not originate from the same parent (sub)population.

Even when we remove massive red SFGs and restrict the analysis of Sérsic
indices to $\log M<10.3$, we find similar trends as in Figure 20. Among low-
mass SFGs with significant bulge component, $\sim 69\%$ are red SFGs; among
bulge dominated SFGs, this percentage is $\sim 72\%$.

To summarize, we see that blue SFGs are more extended but lower in mass than
red SFGs. The fraction of red SFGs in the parent SFG population is increasing
steadily with cosmic time. Red SFGs tend to be more concentrated and host
significant bulge component in them compared to their bluer SFG counterparts.
Together with the difference we see between sizes of SFGs at shorter and
longer rest-frame wavelengths (Figures 12, 15, 16, and 17) and the scenarios
that can explain it (Section 5.1.1), this analysis suggests that the emergence
of bulges is driving the observed average size evolution of SFGs.

Several physical processes may be responsible for the emergence and growth of
bulges in galaxies. Bulge growth can happen internally (secular evolution) or
it can be induced externally.

In secular evolution, bulge formation and growth is often a result of
perturbations in the galaxy disks due to internal substructures such as spiral
arms and bars (Kormendy & Kennicutt, 2004; Athanassoula, 2005; Kormendy, 2008;
Gadotti, 2009; Sellwood, 2014). This secular growth of bulges (also called
pseudo-bulges) is global to all disk galaxies. The pseudo-bulges are
characterised by disk like profiles ($n\sim 1$) and thus do not increase
galaxy Sérsic index.

In contrast, bulge formation and growth induced externally (e.g., through
mergers) by displacing the stars and gas in the disks towards galaxy centres,
can contribute to the increase in galaxy Sérsic index (Naab et al., 2006;
Gadotti, 2009; Hopkins et al., 2010; Oser et al., 2010; Tacchella et al.,
2019). Bulges formed in this manner (classical bulges) have light profiles and
properties similar to elliptical galaxies. These bulges have higher fraction
of old red stars than in the disks and appear redder in colour (e.g., Breda et
al., 2020). They generally have centrally concentrated surface brightness
distributions, which are reflected in their higher $n$.

Several processes can cause a compaction towards galaxy centre resulting in
central bulge formation and growth. Those mechanisms include galaxy wet
mergers (Zolotov et al., 2015; Inoue et al., 2016) and collisions of counter-
rotating gas streams that feed the galaxy (Danovich et al., 2015). In both
cases, dissipative gaseous accretion promotes bursts of star formation and
growth of stellar mass towards galaxy centre (e.g., Dekel et al., 2009;
Lapiner et al., 2023). The absence of further gas inflow results in gas
depletion and eventual quenching of the core regions (e.g., Ceverino et al.,
2010; Zolotov et al., 2015; Tacchella et al., 2016). Thus, the bulges in SFGs
tend to have high central mass concentrations and appear redder in colour as
they age. We note, however, that wet compaction events are more common at
earlier cosmic times than within the redshift interval we probe here.

Figure 21: Evolution of median sizes of QGs. Panels show the size evolution of
SFGs at two different characteristic masses. The QG sizes at rest-frame $3000$
Å is shown in red and at rest-frame $5000$ Å in maroon. The sizes at shorter
wavelength is larger than those at longer wavelengths. The size evolution is
fit using a power-law: $A(1+z)^{\beta}$ and the best fitting parameters are
given in Table 4.3. The solid curves show the best-fit evolutionary curves
with shaded regions showing their $1\sigma$ confidence intervals.

Studies based on cosmological hydrodynamic simulations (e.g., IllustrisTNG)
predict that the emergence of centrally concentrated spheroidal components is
a common phenomena in SFGs (e.g., Tacchella et al., 2019). Observational
studies show that the overall Sérsic index and bulge-to-total ratios increase
with cosmic time for SFGs which agrees with the predictions of simulations
that the bulges are growing in SFGs (e.g., Lang et al., 2014). In light of
these results, the emergence of classical bulges must be responsible for the
high $n$ tail of the Sérsic index distribution for SFGs in our sample.

####  5.1.4 Limitations

Effects of age and metallicity on galaxy light profiles cannot be disentangled
without spatially-resolved spectroscopic data. Observed larger galaxy sizes in
shorter wavelength can result from either young or low-metallicity stars in
their outskirts as both stellar population properties contribute to galaxy
colours (e.g., Gustafsson, 1989; Streich et al., 2014). If light profiles of
our $\log M>9.5$ SFGs in bluer rest-frame wavelength are at least in part
affected by the presence of low-metallicity stars, measured larger sizes in
$3000$ Å can point towards accretion (in addition to star-formation) in the
outskirts of these galaxies. Low mass galaxies generally have lower
metallicities than massive ones (e.g., Tremonti et al., 2004; Foster et al.,
2012; Zahid et al., 2014; Ma et al., 2016). Thus the accretion of low-mass
satellites can lower the average metallicity and increase the surface
brightness of galaxy outskirts in bluer light.

The presence of dust can also cause observed differences between rest-frame
$3000$ Å and $5000$ Å. The rest-frame $3000$ Å is more impacted by dust
absorption than $5000$ Å. Some studies show that the dust attenuation is
stronger in bulges than in disks (e.g., Driver et al., 2007). This radial
dependence of dust reddening could impact the fitting of Sérsic profile at
$3000$ Å. As the contribution of unobscured blue light from the outskirts
dominates the overall light profile of a galaxy, its $R_{e}$ is overestimated
and $n$ is underestimated. However, we find that the red SFGs tend to be
smaller and have higher Sérsic indices than blue SFGs in the rest-frame $5000$
Å where galaxy light is less affected by dust compared to the rest-frame
$3000$ Å (Figures 18 and 20). Hence, we conclude that the observed differences
between red and blue SFGs in our study are not driven by dust reddening alone.

With this study, we do not perform detailed analysis of galaxy components
(bulges v/s disks). This is a necessary next step in testing the bulge growth
scenario for the size evolution of SFGs. In the follow-up study we plan to
perform bulge-disk decomposition and estimate bulge-to-total (B/T) ratios for
all CLAUDS+HSC galaxies using a multi-component model fitting approach.

###  5.2 Size Evolution of Quiescent Galaxies

####  5.2.1 Overview of the Observed Trends

As in Section 5.1, we use characteristic sizes estimated from the SMR to trace
the evolution in size for QG population in two rest-frame waveleghts (Figure
21). We limit the analysis to two different characteristic masses ($\log
M=10.7,11.2$) that are above the pivot point of the SMR (Section 4.1) for
quiescent population.

As expected, the results we show in Figure 21 are in line with our findings in
Section 4.3. Similar to SFGs, we find that the sizes of QGs are larger in
rest-frame $3000$ Å than in $5000$ Å (red and maroon curves in Figure 21). At
$z\sim 0.5$, QGs with mass $\log M=10.7$ are $\sim 23\%$ larger in size in
shorter wavelength than in longer wavelength. This offset in size decreases
with increase in stellar mass. At the same redshift, QGs with mass $\log M\sim
11.2$ are $\sim 12\%$ more extended in the rest-frame UV light than in visible
light.

With respect to SFGs in our sample, QGs exhibit much stronger size evolution
in rest-frame $5000$ Å than in $3000$ Å. For example, in the rest-frame $3000$
Å, QGs with mass $\log M\sim 10.7$ grow in size by $\sim 38\%$ in the rest-
frame UV and by $\sim 82\%$ in the rest-frame visible light over the span of 6
Gyrs that we probe. For very massive QGs ($\log M\sim 11.2$), this growth is
by $\sim 27\%$ and $\sim 97\%$, respectively, for sizes measured in rest-frame
UV and visible light.

Furthermore, the average size of a QG is significantly smaller than an SFG at
a given mass and redshift (Figure 15). Comparison of the last two panels of
Figure 16 with Figure 21 shows that at $z=0.5$ and rest-frame wavelength
$5000$ Å, $\log M=10.7$ SFG is $\sim 70\%$ larger than QG of the same stellar
mass. However, SFGs and QGs with $\log M=11.2$ have similar sizes. The trends
in the bluer wavelength are also similar to those in the redder wavelength.
This is interpreted to be due to the fading of disk when a galaxy fully
quenches (e.g., Christlein & Zabludoff, 2004; Carollo et al., 2016; Matharu et
al., 2020; Estrada-Carpenter et al., 2023). Since the disks are more extended
than their more centrally concentrated bulges/spheroids, the overall size of a
galaxy shrinks when the disk completely fades.

Alternatively, the smaller sizes of QGs relative to SFGs of comparable stellar
mass could be due to the preferential growth of central regions over the outer
ones during and after quenching. We can investigate the effects of both disk
fading and growth of bulges through bulge$+$disk decomposition in two rest-
frame wavelengths. When the star formation in galaxy disk is quenched, the
disk become fainter, especially at UV wavelengths (although it does not
disappear completely). The surface brightness of the disk diminishes
significantly in the rest-frame UV regime and less so at longer (rest-frame
optical) wavelengths. However, this fading should not affect the central
regions (galaxy bulges). We will explore this effect in our future work
through bulge$+$disk decomposition.

Finally, the QGs tend to grow in size at a more rapid pace compared to SFGs
even though they do not actively form stars. This observation is in line with
the literature (e.g., Conselice, 2014; van der Wel et al., 2014; Straatman et
al., 2015; Damjanov et al., 2019; Kawinwanichakij et al., 2021). This strong
size evolution could be driven by several physical processes affecting
individual QGs, such as major mergers (Conselice et al., 2003, 2022; Bluck et
al., 2012; Kaviraj et al., 2014; Man et al., 2016; Mantha et al., 2018; Huško
et al., 2022, 2023), minor mergers and accretion (Khochfar & Silk, 2006; Naab
et al., 2009; Bluck et al., 2012; Newman et al., 2012; Fagioli et al., 2016;
Sawicki et al., 2020; Conselice et al., 2022; Huško et al., 2023; Suess et
al., 2023), and adiabatic expansion (Boily & Kroupa, 2003; Goodwin & Bastian,
2006; Baumgardt & Kroupa, 2007; Fan et al., 2008, 2010; Damjanov et al., 2009;
Ragone-Figueroa & Granato, 2011; Lapi et al., 2018). In addition, progenitor
bias due to newcomers affects the average QG size evolution (Franx et al.,
2008; Carollo et al., 2013; Fagioli et al., 2016; Damjanov et al., 2019,
2023).

Galaxy mergers can contribute significantly to the size evolution of QGs.
Although major dry mergers can yield an increase in size of QGs proportional
to the increase in mass (eg., Boylan-Kolchin et al., 2006; Naab et al., 2007),
the average number of major mergers per galaxy since $z\sim 1$ is low (e.g.,
Bundy et al., 2009; López-Sanjuan et al., 2010; Kaviraj et al., 2014; Thibert,
2018). However, both observational studies (e.g., Bezanson et al., 2009; van
Dokkum et al., 2010; Trujillo et al., 2011; Fagioli et al., 2016; Zahid et
al., 2019) and theoretical/simulation based studies (e.g., Khochfar & Silk,
2006; Naab et al., 2009; Oser et al., 2010, 2012; Hilz et al., 2012) show that
minor mergers and accretion can support size growth in QGs. Theoretical
studies (e.g., Naab et al., 2009) and observational studies that incorporate
these theoretical predictions (e.g., Bezanson et al., 2009; Zahid et al.,
2019) indicate that a series of minor mergers, with the fractional growth in
$R_{e}\propto M_{*}^{2}$ (where $M_{*}$ is the fractional growth in stellar
mass) can explain QG size growth since $z\sim 2-3$.

The fact that the QGs have larger sizes in shorter wavelength (red curves in
Figure 21) than the longer one (maroon curves in Figure 21) could indicate
growth through minor mergers and accretion (e.g., Bezanson et al., 2009; Naab
et al., 2009; Suess et al., 2020, 2023). As these are passive galaxies, they
are not actively forming stars, but the bulk of the young and/or low-
metallicity stars traced by shorter wavelength could be added ex-situ or
formed from ex-situ added gas through accretion and minor mergers. These
materials are added to the outskirts puffing up galaxy light profile.

On the other hand, the longer wavelength profiles are dominated by the in-situ
old stars which are concentrated towards galaxy centre. As time progresses,
the ex-situ stars can also migrate inward and add to the galaxy profile in the
longer wavelength as they age (e.g., Boecker et al., 2023). Since the longer
wavelength probes newly added old ex-situ stars along with the already
existing in-situ aged stellar population in galaxies, we see the stronger
evolution in rest-frame $5000$ Å.

Figure 22: Size Evolution of QGs in two rest-frame wavelengths when progenitor
bias is removed. The size evolution curves of the entire QG population are
shown in gray (dotted and solid curves for rest-frame UV and visible light
respectively). The best-fit parameters are given in Table 5.2.2. Other details
are same as for Figure 21.

####  5.2.2 Removing the Effect of Progenitor Bias

Newly quenched galaxies (newcomers) can affect the distribution of QG sizes at
a given mass because they are, on average, more extended than the rest of the
QG population (progenitor bias; Saglia et al., 2010; Damjanov et al., 2023).
Progenitor bias should not contribute significantly to the size evolution of
QGs. For example, Damjanov et al. (2023) find that the newcomers contributed
only $11\%$ to the growth of QGs at $z<0.6$. Nonetheless, it is important to
remove the impact of progenitor bias from the QG population to understand how
aging QGs may evolve with time. Here we account for the effect of progenitor
bias by removing the most extended galaxies and re-fitting the “corrected" QG
SMR in five redshift bins that cover the $0.1<z<0.9$ interval.

We assume that in total $\sim 10\%$ of the QG population above the SMR pivot
point in each redshift bin are newcomers (Moutard et al., 2016). However, the
probability for an SFG to quench increases with stellar mass and is given by

$\begin{split}p_{Q}&\varpropto\left(1-e^{-\frac{M}{M^{*}}}\right),\\\
&=c\left(1-e^{-\frac{M}{M^{*}}}\right),\end{split}$ (13)

where $M$ is the stellar mass of the SFG in solar masses, $c$ is the
normalization constant and $M^{*}$ is the characteristic mass from stellar
mass (Schechter) function for SFGs in M⊙ (Peng et al., 2010b). $M^{*}$ varies
with redshift and we adopt the values from Muzzin et al. (2013). In addition,
the number densities of SFGs and QGs also vary with stellar mass and redshift
(e.g., Muzzin et al., 2013).

To incorporate this information into a toy model of QG size growth, we first
divide massive QGs ($M>M_{p}$) in each redshift bin into $10$ equal frequency
bins of stellar mass. Then we determine the fraction of galaxies to be removed
from each mass bin as

$f_{pb}=p_{Q}\frac{n_{Q}(M)}{n_{T}(M)},$ (14)

where $n_{Q}(M)$ and $n_{T}(M)$ are the number densities of QGs and all
galaxies respectively. We adopt the number densities from Muzzin et al.
(2013). We use the normalization constant $c$ in the equation for $p_{Q}$
(Equation 13) to keep the percentage of the newly quenched galaxies to be
$\sim 10\%$ . We adopt $c=3/5$ which removes $\sim 8-13\%$ of the most
extended galaxies in each redshift bin above the SMR pivot point (see also
Appendix C). Across the stellar mass bins, this percentage varies from $\sim
0\%$ to $\sim 20\%$ of the QG population in that bin. We then perform the SMR
fitting and examine the evolution of characteristic sizes of remaining (aging)
QGs at two different stellar masses in $3000$ Å and $5000$ Å (Figure 22; the
best-fit parameters of the size evolution are given in Table 5.2.2).

cccc[hvlines]

1-4QGs

Restframe $\lambda$ $\log M_{(z=0.9)}$ $R_{e}^{0}$ $\beta$

2-1$3000$ Å $10.7$ $4.58\pm 0.60$ $-0.69\pm 0.28$

$11.2$ $8.28\pm 0.70$ $-0.27\pm 0.19$

2-1$5000$ Å $10.7$ $4.48\pm 0.59$ $-1.13\pm 0.29$

$11.2$ $9.45\pm 0.70$ $-0.91\pm 0.17$

Table 5: The parameters of the best-fit power-law (Equation 10) to the
redshift evolution of characteristic QG sizes after the removal of the the
progenitor bias (Section5.2).

When we remove the supposed newcomers from the QG sample, the average
characteristic sizes of QGs become smaller, as expected. The decrease in the
average size of QGs with stellar mass $\log M=10.7$ is similar across the
redshift range at both wavelength regimes ($11-15\%$). However, at high
stellar mass ($\log M=11.2$), this decrease ranges from $3\%$ ($5\%$) at
$z=0.9$ to $18\%$ ($13\%$) at $z=0.1$ in rest-frame visible (UV) light.

We do not find any significant difference in the pace of size growth for QGs
with mass $\log M=10.7$ when we account for progenitor bias in both wavelength
regimes (compare coloured curves against black curves in the first panel of
Figure 22). However, there are indications that the size evolution becomes
weaker when we remove the progenitor bias for QGs with mass $\log M=11.2$
(second panel of Figure 22). Nonetheless, the difference is within
uncertainties for both UV and visible light. Thus, the progenitor bias only
causes a significant vertical offset in the redshift evolution of average QG
sizes without strong signals of change in the exponent of the evolutionary
trends for stellar masses higher than the pivot mass ($\log M\sim 10.5$,
Section 4.1).

Progenitor bias can affect the size evolution of QGs in two ways. First, the
sizes of SFGs are larger than QGs at the same stellar mass and the size of
SFGs increase with time (e.g., Figures 11 and 15). Thus, the newcomers are
larger than the those quenched earlier in time. This causes a vertical shift
in the size evolution curve by increasing the average size of QGs at all
redshifts. We see this effect in our sample (the black curves are higher than
coloured curves in Figure 22).

Second, disks of galaxies fade when they quench (Christlein & Zabludoff, 2004;
Carollo et al., 2016; Estrada-Carpenter et al., 2023). Bulge becomes more
prominent in a galaxy light profile resulting in smaller effective sizes of
galaxies. Since the disk fading affects light profiles in rest-frame $3000$ Å
more than $5000$ Å and the quenched fraction increases with cosmic time, the
progenitor bias effect should slow down the pace of size evolution in the
shorter wavelength (red curves in Figure 22). We expect to see a strengthening
of the size evolution for QGs in rest-frame UV light after we remove the
progenitor bias. Instead, we find that the pace of the size evolution is
unaffected indicating that the process of disk fading does not affect the
average size evolution of massive QGs.

Thus, we conclude that the effects of progenitor bias on the massive QG
population is mainly on the vertical offset of the evolutionary trend. The
strength of the redshift evolution in size for massive QGs is determined by
other physical processes such as minor mergers and accretion adding low-mass
and/or low-metallicity stellar population (Section 5.1.4) to galaxy outskirts.

###  5.3 Evolution in Central Stellar Mass Surface Density

Central stellar mass surface density, the projected stellar mass surface
density within central $1$ kpc radius, is another physical parameter that is
widely used in the analysis of galaxy evolution (e.g., Cheung et al., 2012;
Fang et al., 2013; Barro et al., 2013, 2014, 2017; Suess et al., 2021, 2020;
Xu & Peng, 2021; Ji et al., 2022; Ji & Giavalisco, 2022, 2023; Estrada-
Carpenter et al., 2023). Stellar mass surface density is derived using both
$R_{e}$ and $n$ and thus it does not include uncertainties due to covariance
between $R_{e}$ and $n$ making it a more robust parameter than these two
parameters individually (Ji et al., 2020, 2022).

We calculate $\Sigma_{1}$ in units of $\hbox{M${}_{\odot}$}/\text{kpc}^{2}$
and fit the relation between $\Sigma_{1}$ and galaxy stellar mass in Appendix
D. We first fit the $\Sigma_{1}$-stellar mass relation in log-log space for
all SFGs and QGs (Figure 31) and also for SFGs split into blue and red
subpopulations (Figure 32). One of the parameters of the fits we perform in
Appendix D is the characteristic $\Sigma_{1}$ value101010Throughout this work,
we use its logarithmic value for analysis, i.e.,
$\log(\Sigma_{1}/\hbox{M${}_{\odot}$}/\text{kpc}^{2})$. for galaxies with
stellar mass of $5\times 10^{10}$ M⊙. Figure 23 shows the redshift evolution
of this characteristic value for SFGs (dark blue stars) and QGs (maroon filled
circles).

Figure 23: The evolution of characteristic $\Sigma_{1}$ for galaxies with mass
$5\times 10^{10}$ M⊙. We show these values for all SFGs (dark blue stars) and
QGs (maroon dots) along with their uncertainties. We also show the evolution
in $\Sigma_{1}$ in two subpopulations of SFGs - blue (blue diamonds) and red
(purple squares). We compare our results with the estimated characteristic
$\Sigma_{1}$ values at higher redshift for SFGs (blue crosses) and QGs (maroon
open circles) from Barro et al. (2017).

The QGs have $\sim 2$ ($3$) times higher $\Sigma_{1}$ than SFGs at $0.1<z<0.3$
($0.75<z<0.9$). As the majority of the SFGs are red (Figure 19), the average
$\Sigma_{1}$ of red SFGs (purple diamonds) are almost identical to those of
all SFGs. However, blue SFGs (blue squares) have significantly lower
$\Sigma_{1}$ values than red SFGs, except at $0.1<z<0.3$ where the sample size
is small. Red SFGs have $\sim 1.5$ ($2.7$) times higher central stellar mass
density than blue SFGs at $0.3<z<0.45$ ($0.75<z<0.9$). This difference is in
line with SFGs growing their central regions and increasing $\Sigma_{1}$
during their evolution towards the red sequence (as also reported by Guo et
al., 2021; Quilley & de Lapparent, 2022).

We compare our results with the evolution in $\Sigma_{1}$ for SFG and QG at
$0.5<z<3$ (Barro et al., 2017, maroon open circles (QGs) and blue crosses
(SFGs) in Figure 23). High-redshift $\Sigma_{1}$ are estimated based on
spatially resolved SED fits derived from surface brightness profiles in nine
HST bands. We do not have spatially resolved SED fits for $z<1$ galaxies. By
incorporating spatially resolved SEDs, Barro et al. (2017) are potentially
able to incorporate the effects of colour gradients in the $\Sigma_{1}$
calculation. However, these gradients in both SFGs and QGs do not evolve at
$z<1$ significantly (Suess et al., 2019). Hence, the fact that we consider
only global rather than spatially resolved colour gradients does not affect
the difference in the evolutionary trends for $\Sigma_{1}$ between two
studies.

Figure 23 shows decreasing trend in $\Sigma_{1}$ of SFGs at $z>1$ with cosmic
time. At $z<1$, we find that the SFGs with mass $5\times 10^{10}$M⊙ have, on
average, a constant $\Sigma_{1}$ value. We interpret this lack of evolution in
$\Sigma_{1}$ for SFGs in our sample as another aspect of prominent bulge
growth in them. The growth of centrally concentrated bulges increases the
$\Sigma_{1}$ values at lower redshift, resulting in flattening of the observed
evolutionary trend.

The characteristic $\Sigma_{1}$ values for QGs also decrease with time since
$z\sim 3$. Potential mechanisms driving this decrease in $\Sigma_{1}$ include
minor mergers (resulting in much larger size than stellar mass increase; e.g.,
Bezanson et al., 2009; Hopkins et al., 2009; Naab & Ostriker, 2009; van Dokkum
et al., 2010; Fagioli et al., 2016; Zahid et al., 2019), adiabatic expansion
due to mass loss and the rearrangement of stellar orbits in shallower
gravitational potential (e.g., Damjanov et al., 2009, 2011; van Dokkum et al.,
2010, 2014; Lapi et al., 2018), and the influx of larger newcomers (Section
5.2.2; Carollo et al., 2013; Damjanov et al., 2019, 2023; Ji & Giavalisco,
2022).

However, QGs at $z<1$ show that the pace of redshift evolution in $\Sigma_{1}$
slows down at $z<0.9$. For example, $\Sigma_{1}$ of QGs decreases by $\sim
10\%$ from $z\sim 0.7$ to $z\sim 0.2$ (or over 4 Gyrs of cosmic time), while
the decline is somewhat more dramatic at higher $z$ ($20\%$ between the
redshift bins $1.5<z<2$ and $1<z<1.5$, corresponding to $\sim 2$ Gyrs of
cosmic time). This weakening of $\Sigma_{1}$ evolution reflects the lack of
evolution in $\Sigma_{1}$ for SFGs (especially red SFGs shown in purple
diamonds) of the same mass. This similarity in trends between SFGs and QGs is
driven partly by the quenching processes that transfer a fraction of SFGs in
one redshift bin to the the QG population in the next lower redshift bin.

Figure 24: The relation between specific star formation rate (sSFR) and
$\Sigma_{1}$ in four redshift bins for SFGs (red, blue and all) and QGs. We
divide each galaxy groups into $10$ equally populated bins of $\Sigma_{1}$ and
estimate median sSFR. The errorbars correspond to the bootstrapped errors for
the medians.

Earlier studies show that galaxy star formation activity is correlated with
their central stellar mass surface density (e.g., Barro et al., 2017; Whitaker
et al., 2017; Suess et al., 2021). These studies demonstrate that galaxies
follow an "L-shaped track" in the specific star formation rate (sSFR) vs
$\Sigma_{1}$ parameter space. We explore this relation in Figure 24, where we
show median sSFR values for both SFGs (dark blue stars) and QGs (maroon filled
circles) in ten equally populated $\Sigma_{1}$ bins. The sSFR values for SFGs
are roughly stable at $\log\Sigma_{1}\lesssim 9$. After that point, the sSFR
starts to decrease with the increase in $\Sigma_{1}$. As the SFGs build up
their central regions, their star-formation activity declines. However, we
note that the observed correlation between $\Sigma_{1}$ and the sSFR may not
necessarily imply a causal relation between the two parameters since a third,
hidden parameter may be involved (as also noted by, e.g., Lilly & Carollo,
2016; Chen et al., 2020; Ji & Giavalisco, 2022, 2023).

When we divide the SFG population into red and blue SFGs (purple diamonds and
blue squares, respectively, in Figure 24), red SFGs show somewhat lower sSFR
at any given $\Sigma_{1}$. Furthermore, there is a maximum $\Sigma_{1}$ above
which only red SFGs exist.

The sSFR values for blue SFGs do not change with $\Sigma_{1}$, indicating that
these young SFGs are actively forming stars at all galactocentric radii. The
sSFR for red SFGs declines with $\Sigma_{1}$ above the threshold value where
blue SFGs are absent. This trend suggests that by probing the red SFG
subpopulation, we trace how these galaxies build up their central bulges and
undergo inside-out quenching.

The threshold $\Sigma_{1}$ value above which all SFGs are red is
$\log\Sigma_{1}\sim 9$. This threshold value does not evolve with redshift. In
line with the values reported by Whitaker et al. (2017), our result suggests
that as SFGs build up sufficient central stellar mass density, quenching
process becomes prominent above this universal minimum value.

In addition, we find that there is a threshold $\Sigma_{1}$
($\log\Sigma_{1}\sim 9.7$) beyond which a galaxy cannot sustain star
formation. This finding broadly agrees with previous results of Whitaker et
al. (2017), Suess et al. (2021) and Estrada-Carpenter et al. (2023).

Thus, we observe the minimum value for $\Sigma_{1}$ that corresponds to the
point when quenching becomes prominent globally (throughout the galaxy) and
the maximum value above which all galaxies are quiescent. Between these two
threshold values of $\Sigma_{1}$, red bulges continue to grow in SFGs as
indicated by the negative correlation between $\Sigma_{1}$ and sSFR (Figure
24).

The range of $\Sigma_{1}$ values that red SFGs can have once they drop below
sSFR for the blue subpopulation suggests that the process we trace is the slow
process of bulge growth that drives this decline in sSFR. A number of studies
support the emergence/significant growth of centrally concentrated bulges in
galaxies before and while they transition through the green valley and quench
(Barro et al., 2017; Wu et al., 2020; Guo et al., 2021; Quilley & de
Lapparent, 2022; Ji & Giavalisco, 2023). Therefore, we conclude that the bulge
growth is the major driving force behind the evolution of SFGs towards the red
sequence at $z<0.9$.

##  6 Conclusions

We measure morphological parameters for 21,200 massive ($\log M>9.5$) star-
forming and quiescent galaxies (SFGs and QGs) in the redshift range
$0.1<z<0.9$ and in two rest-frame wavelengths, $3000$ Å and $5000$ Å. We use
the multi-band images from the central region of the COSMOS field in the
CLAUDS+HSC Survey and ancillary data from COSMOS2020 catalogue for the
estimates of galaxy redshifts and stellar masses. We measure galaxy sizes by
fitting a single-component Sérsic profile to their light profiles using Galfit
software.

We fit the size-mass relation (SMR) for SFGs and QGs separately in five
redshift bins in the two rest-frame wavelengths (Equations 7 and 8. We then
analyse the redshift evolution of galaxy sizes by fitting a power-law
(Equation 10).

The main conclusions of our study are:

  1. 1.

Both SFGs and QGs on average appear more extended in blue light (rest-frame
$3000$ Å) than in red light (rest-frame $5000$ Å) at all redshifts and masses
(Figures 12, 15, 16 and 21). We find that the pace of evolution at fixed
stellar mass is stronger in the rest-frame visible light than in the rest-
frame UV light and this difference is more prominent for QGs (Figure 21). In
addition, the strength of the evolution in both wavelengths in general
increases with the stellar mass of galaxies (Figures 15 and 16).

  2. 2.

When we incorporate mass growth for SFGs assuming the evolving star formation
main sequence, the pace of the SFG size evolution (i.e., the exponent of the
best-fitting power-law) becomes independent of both galaxy stellar mass and
the rest-frame wavelength (Figure 17).

  3. 3.

The strength of the size evolution at constant stellar mass for SFGs is weaker
than the strength reported in studies that cover wider redshift ranges. We
explore this slowing down of the size evolution since $z\sim 1$ by dividing
the SFG population into two groups: blue SFGs ($U-V\leq 1$) and red SFGs
($U-V>1$). We find that red SFGs are smaller and more massive than blue SFGs
and that their fraction increases by $25\%$ during the $6$ billion years of
cosmic time that we explore. In addition, red SFGs constitute the majority
($83\%$) of the bulge-dominated SFGs ($n>2.5$). We conclude that the emergence
of bulges is a primary driver of size evolution for SFGs.

  4. 4.

We use a a toy model to remove the effect of newcomers (progenitor bias) from
the observed trend in size evolution for QGs. The removal of the largest QGs
changes the characteristic size of QGs at $z=0$ ($R_{e}^{0}$ in Equation 10).
This change in $R_{e}^{0}$ is systematically negative for all masses and in
both rest-frame wavelengths. Our analysis shows that newcomers increase the
average size of QGs by up to $\sim 20\%$ since $z\sim 1$. In contrast, the
pace of QG size evolution is not affected systematically by this effect. We
conclude that the strength of the evolution is not determined by disk fading
during the process of quenching but rather by physical processes affecting
individual galaxies after they become quiescent (e.g., mergers and accretion).

  5. 5.

We find that the redshift evolution of $\Sigma_{1}$ stops (slows down) for
SFGs (QGs) at $z<1$. This change in the evolutionary pace compared to higher
$z$ is consistent with the growth of bulges in SFGs and the influx of
newcomers to the QG population. We also find that all SFGs with
$\log\Sigma_{1}\gtrsim 9$ are red ($U-V>1$) and that all galaxies with
$\log\Sigma_{1}\gtrsim 9.7$ are quiescent. Finally, the sSFR of red SFGs
within these two threshold values of $\Sigma_{1}$ declines with $\Sigma_{1}$,
suggesting that the growth of bulges drives the evolution of SFGs towards the
QG population.

To explore further the role of bulges in the average size growth of the SFG
population and the impact of various physical processes on the QG size
evolution, we will perform bulge$+$disk decomposition. We plan to implement
this procedure in the future by fitting a double-component Sérsic profile to
galaxy light profiles. Furthermore, here we do not investigate the impact of
environment on the galaxy size evolution due to the small volume we probe. To
take into account environmental effects, we will expand our analysis based
both on a single- and double-component Sérsic profile to the entire $18.6$
deg2 of the CLAUDS+HSC survey. Our pilot study demonstrates the power of deep
ground-based photometric surveys and galaxy light profiles fitting in multiple
rest-frame wavelengths to probe in detail the strength of galaxy structural
evolution at $z<1$ and constrain its driving mechanisms.

## Acknowledgements

We thank the anonymous referee for their thoughtful suggestions that helped us
improve the clarity and completeness of the manuscript. We gratefully
acknowledge the contributions of Anneya Golob, Bobby Sorba, Harrison
Souchereau and Ikuru Iwata. We thank the extra-galactic research group at
Saint Mary’s University, Canada, especially Vicente Estrada-Carpenter,
Lingjian Chen and Devin Williams for their valuable inputs and support. We
also thank Lalitwadee Kawinwanichakij, Dan Taranu, Connor Bottrell, Arjen van
der Wel, Chien Peng, Lamiya Mowla and Jasleen Matharu for helpful discussions.
The research of A.G., I.D. and M.S. is supported by the National Sciences and
Engineering Council (NSERC) of Canada.

These data were obtained and processed as part of the CFHT Large Area U-band
Deep Survey (CLAUDS), which is a collaboration between astronomers from
Canada, France, and China described in Sawicki et al. (2019). CLAUDS is based
on observations obtained with MegaPrime/ MegaCam, a joint project of CFHT and
CEA/DAPNIA, at the CFHT which is operated by the National Research Council
(NRC) of Canada, the Institut National des Science de l’Univers of the Centre
National de la Recherche Scientifique (CNRS) of France, and the University of
Hawaii. CLAUDS uses data obtained in part through the Telescope Access Program
(TAP), which has been funded by the National Astronomical Observatories,
Chinese Academy of Sciences, and the Special Fund for Astronomy from the
Ministry of Finance of China. CLAUDS uses data products from TERAPIX and the
Canadian Astronomy Data Centre (CADC) and was carried out using resources from
Compute Canada and Canadian Advanced Network For Astrophysical Research
(CANFAR).

The Hyper Suprime-Cam (HSC) collaboration includes the astronomical
communities of Japan and Taiwan, and Princeton University, USA. 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