# Reconstructing the Assembly of Massive Galaxies. II. Galaxies Develop
Massive and Dense Stellar Cores as They Evolve and Head Toward Quiescence at
Cosmic Noon

Zhiyuan Ji University of Massachusetts Amherst, 710 North Pleasant Street,
Amherst, MA 01003-9305, USA  Mauro Giavalisco University of Massachusetts
Amherst, 710 North Pleasant Street, Amherst, MA 01003-9305, USA

###### Abstract

We use the SED-fitting code Prospector to reconstruct the nonparametric star
formation history (SFH) of massive ($\log M_{*}>10.3$) star-forming galaxies
(SFGs) and quiescent galaxies (QGs) at redshift $z_{\rm{obs}}$ $\sim 2$ to
investigate the joint evolution of star-formation activity and structural
properties. Compared to extended SFGs, compact SFGs are more likely to have
experienced multiple star-formation episodes, with the fractional mass formed
during the older ($\geq 1$ Gyr) episode being larger, suggesting that high-
redshift SFGs assembled their central regions earlier and then kept growing in
central mass as they become more compact. The SFH of compact QGs does not
significantly differ from the average for this category, and shows an early
burst followed by a gradual decline of the star formation rate. The SFH of
extended QGs, however, is similar to that of post-starburst galaxies and their
morphology is also frequently disturbed. Knowledge of the SFH also enables us
to empirically reconstruct the structural evolution of individual galaxies.
While the progenitor effect is clearly observed and accounted for in our
analysis, it alone is insufficient to explain the observed structural
evolution. We show that, as they evolve from star-forming phase to quiescence,
galaxies grow massive dense stellar cores. Quenching begins at the center and
then propagates outward to the rest of the structure. We discuss possible
physical scenarios for the observed evolution and find that our empirical
constraints are in good quantitative agreement with the model predictions from
dissipative accretion of gas to the center followed by massive starbursts
before final quiescence (wet compaction).

Galaxy formation(595); Galaxy evolution(594); Galaxy structure(622); High-
redshift galaxies(734); Galaxy quenching (2040)

††software: GALFIT (Peng et al., 2002, 2010a), Prospector (Leja et al., 2017;
Johnson et al., 2021), FSPS (Conroy et al., 2009; Conroy & Gunn, 2010), MIST
(Choi et al., 2016; Dotter, 2016), MILES (Falcón-Barroso et al., 2011)

##  1 Introduction

Observations over a wide range of cosmic time, from the present day up to the
so-called epoch of “cosmic noon” at $1<z<4$ (Madau & Dickinson, 2014), have
shown that galaxies are characterized by a bimodality of colors at UV through
NIR wavelengths (e.g. see Baldry et al. 2004; Bell et al. 2004; Brammer et al.
2009; Williams et al. 2009, among many other). Blue galaxies on average are
actively forming stars, and have higher star formation rates (SFRs) than red
galaxies which generally have substantially lower levels of star formation
activity, leading to a natural classification into the star-forming galaxies
(SFGs) and the quiescent galaxies (QGs). Understanding the transition from
SFGs to QGs, a process referred to as galaxy quenching, remains a key missing
piece toward a complete picture of galaxy evolution.

Broadly, two general categories of quenching mechanisms – mass quenching and
environmental quenching – have been identified (Peng et al., 2010b).
Environmental quenching is associated with the external environment of a
galaxy, and it might be able to alter the structural evolution of galaxies
(e.g., Valentinuzzi et al., 2010; Papovich et al., 2012; Cappellari, 2013;
Huertas-Company et al., 2013; Strazzullo et al., 2013; Newman et al., 2014;
Matharu et al., 2019). Observations generally show that the environmental
effect on galaxy evolution primarily happens in relatively low-mass (stellar
mass $\log M_{*}<10$), low-redshift ($z<1.5$) galaxies in dense environments
such as groups/clusters of galaxies (Peng et al., 2010b; Guo et al., 2017;
Kawinwanichakij et al., 2017; Ji et al., 2018), which, however, are not the
subject of the investigation presented here. In this work, we will
specifically focus on massive galaxies at cosmic noon epoch, defined here as
having stellar mass $\log M_{*}\geq 10.3$ and in the redshift range of $1<z<4$
(see section 2). Observations show that at this epoch massive galaxies begin
to quench in large numbers (e.g., Muzzin et al., 2013) and suggest that the
quenching should be mainly due to physical processes internal to the galaxies
themselves, i.e., mass quenching (see a recent review by Man & Belli 2018 and
references therein).

Strong correlations are observed between the star-formation activity and the
structural properties of galaxies, such as size, light profile and central
mass density. It is now well-established that these correlations persist at
least out to $z\sim 3$. These include the emergence of the Hubble Sequence at
$z>2$ (e.g. Franx et al., 2008; Kriek et al., 2009; Wuyts et al., 2011), and
the dependence of the mass-size relation and Sérsic index $n$ on specific star
formation rate (sSFR, e.g. Williams et al. 2010; Wuyts et al. 2011; Newman et
al. 2012; Barro et al. 2013; Patel et al. 2013; van der Wel et al. 2014;
Shibuya et al. 2015). On average, QGs have much smaller sizes, steeper light
profiles (i.e. larger $n$) and larger central mass densities than SFGs of
similar mass and redshift. The mass-size relation of QGs also is substantially
steeper than that of SFGs. Phenomenologically, these correlations could be
interpreted as evidence of a causal link between the structural
transformations and the quenching of galaxies, but because the relative timing
sequence of these two events remains empirically unconstrained, whether the
causality is real remains unknown. In fact, some evidence suggests that at
$z\sim 2$ galaxies quench first and then secularly transform their structures
later. This is based on the observations that (1) a substantial amount of
massive QGs at redshift $z=2\sim 3$ appear to have a disk-like morphology
(e.g. McGrath et al., 2008; Bundy et al., 2010; van der Wel et al., 2011;
Bruce et al., 2012; Chang et al., 2013), and (2) the spatially resolved
spectra of a handful of strongly lensed QGs show disk-like kinematics, namely
systems that are predominantly supported by rotation although with a larger
velocity dispersion than modern disks (Newman et al., 2015; Toft et al., 2017;
Newman et al., 2018).

Imaging observations, especially those with high-angular resolution taken with
the Hubble Space Telescope (HST), have unveiled what appears to be a threshold
of central stellar-mass surface density ($\Sigma$) above which galaxies are
found to be predominately quiescent. This observational finding seemingly
suggests that the presence of dense central core in massive galaxies might be
a necessary condition for, or a consequence of quenching (e.g. Kauffmann et
al., 2003; Franx et al., 2008; Cheung et al., 2012; Fang et al., 2013; Lang et
al., 2014; Whitaker et al., 2017; Barro et al., 2017; Lee et al., 2018), and
this consideration motivated a number of theoretical studies to explore
possible physical processes that would result in such a phenomenology. For
example, one such mechanism is actually a class of processes, generically
referred to as “wet compaction” whose main feature is highly dissipative gas
accretion toward the central regions of a galaxy which, in turn, promotes
enhanced activity of star formation at a higher rate relative to the average
SFR (Dekel et al., 2009). Specifically, the larger gas mass fraction of
galaxies at high redshifts compared to local ones (Tacconi et al. 2020 and
references therein) makes the early galactic disks gravitationally unstable on
the scale of $\lesssim 1$ kpc (Toomre, 1964). The disks can thus fragment into
star-forming clumps (e.g. Genzel et al., 2008; Guo et al., 2015) which then
migrate toward the center of galaxies, trigger central starbursts and form
dense central cores. In the absence of further gas inflow, these lead to the
central gas depletion and finally the quenching of central star formation
(Ceverino et al., 2010; Dekel & Burkert, 2014; Wellons et al., 2015; Zolotov
et al., 2015; Tacchella et al., 2016). Simulations suggests that any mechanism
that can cause a drastic dissipative loss of angular momentum would lead to
hydrodynamical instabilities and the compaction, such as wet major mergers
(Zolotov et al., 2015; Inoue et al., 2016) or collisions of counter-rotating
streams that feed the galaxy (Danovich et al., 2015).

An apparent correlation between quiescence and central density, including the
the apparent $\Sigma$ threshold, however, does not necessarily imply any
physical causality between the two properties, because it can also be the
result of the “progenitor effect” (Lilly & Carollo, 2016; Ji & Giavalisco,
2022). In an expanding universe, bound systems should keep memory of the
cosmic density at the time when the halo collapses, implying that halos formed
earlier should have higher densities than those formed later (Mo et al.,
2010). Therefore, at a fixed epoch QGs are expected to be smaller and denser
than SFGs, as the star-forming progenitors of what are being observed as QGs
have necessarily formed earlier. The implication is that the apparent
correlations between structural and star-formation properties reflect the
collective effects of the interplay among different physical mechanisms, e.g.,
the progenitor effect vs. the physical compaction. This makes the empirical
investigation of the relative contribution from each of them critical for
understanding the structural transformations of galaxies and their
relationships with quenching.

A substantial body of existing observational studies has investigated the
redshift evolution of the mass-size and mass-$\Sigma$ relations of SFGs and
QGs. Both relations contain, in an integral form, information of the
structural evolution. This means that a certain galaxy population, observed at
redshift $z_{\rm{obs}}$, includes galaxies that have formed at all different
epochs of $z>$ $z_{\rm{obs}}$. The progenitor effect, therefore, is inherently
mixed with whatever other physical mechanism (e.g., wet compaction) is at
play. This makes the interpretation of the observations model-dependent (see
e.g., section 3.3 of Barro et al. 2017).

To account for the progenitor effect on the apparent structural evolution, the
key is to identify galaxies that formed at different epochs. In this way we
can then avoid aggregating all galaxies together. Some early attempts achieved
this by selecting galaxies with a constant number density (e.g., van Dokkum et
al., 2010; Patel et al., 2013). This is based on the idea of selecting dark
matter halos of the same mass, which at high redshifts would be roughly
coeval. In our view, a further step forward is to utilize the star-formation
history (SFH) of galaxies, i.e. their full stellar-mass assembly histories,
and to select galaxies following a common evolutionary path, namely formed at
a similar epoch, have a similar stellar mass and are observed at a similar
evolutionary stage. In this way any form of the progenitor effect is naturally
taken into account. We can then statistically reconstruct the structural
evolution of individual galaxies. And by looking at how the average properties
of galaxies that formed at the same time and are observed at the same
evolutionary stage change as a function of time, we can empirically constrain
the physics behind the structural transformations of galaxies.

With the modelling of Spectral Energy Distributions (SEDs) growing in accuracy
and sophistication, and with the availability of high-quality data that cover
truly panchromatic swathes of wavelengths, significant progress has been made
to reconstruct the SFH of galaxies at high redshifts, both in parametric (e.g.
Maraston et al., 2010; Papovich et al., 2011; Ciesla et al., 2016; Lee et al.,
2018; Carnall et al., 2018, 2019) and nonparametric forms (e.g. Ocvirk et al.,
2006; Tojeiro et al., 2007; Pacifici et al., 2012; Leja et al., 2017, 2019;
Belli et al., 2019; Iyer et al., 2019). The flexibility of the nonparametric
form, in particular, is critical for not only an unbiased inference of
physical parameters such as stellar mass, SFR and stellar age (e.g Leja et
al., 2019; Lower et al., 2020), but also for the fidelity of the
reconstruction of the SFH itself, which can have arbitrarily complex forms
generally not captured by parametric models (Leja et al., 2019; Johnson et
al., 2021; Tacchella et al., 2022).

In the first paper of this series (Ji & Giavalisco 2022, hereafter Paper I),
we have utilized the fully Bayesian SED fitting code Prospector (Leja et al.,
2017; Johnson et al., 2021) to reconstruct the nonparametric SFH of a sample
of massive QGs at $z\sim 2$ to quantify the progenitor effect. We found that
the progenitor effect is strong in QGs with
$\log_{10}(M_{*}/M_{\sun})=10.3\sim 11$, while much milder in more massive
QGs, implying that the post-quenching mass and size growths of the latter
happen via mergers, which reduce the memory of the time of their formation in
their structural properties. What remains to be explored is if, in addition to
the progenitor effect, the central regions of galaxies grow denser and more
massive relative to the outskirts as they evolve toward quenching. In this
second paper, we expand our investigation to all massive galaxies at $z\sim
2$, selected from the CANDELS/COSMOS and GOODS fields, regardless of their
star-formation activity. We use the SFHs to empirically and quantitatively
estimate the contributions from different physical mechanisms to the apparent
correlations between star-formation and structural properties. Throughout this
paper, we adopt the AB magnitude system and the $\Lambda$CDM cosmology with
parameters $\Omega_{m}=0.3$, $\Omega_{\Lambda}=0.7$ and
$\rm{h=H_{0}/(100kms^{-1}Mpc^{-1})=0.7}$.

##  2 The sample

The parent sample considered in this work is extracted from the portions of
the COSMOS, GOODS-South and GOODS-North fields observed by the CANDELS program
(Grogin et al., 2011; Koekemoer et al., 2011). The original CANDELS
photometric catalogs in each field, which already include HST, Spitzer and
ground-based broadband data, and have been augmented by additional narrow- and
medium-band data from deep ground-based imaging.

The photometric catalogs used in this work are the same as those in Paper I.
In brief, for the CANDELS/COSMOS field, we use the photometric catalog from
Nayyeri et al. (2017) which includes the intermediate- and narrow-band optical
photometry from Subaru (Taniguchi et al., 2015) and medium-band NIR photometry
from Mayall NEWFIRM (Whitaker et al., 2011). For the GOODS-North field, we use
the photometric catalog from Barro et al. (2019) which includes twenty-five
medium-band photometry at the optical wavelengths acquired during the SHARDS
survey (Perez-Gonzalez et al., 2013). Finally, for the GOODS-South field, we
use the latest photometric catalog from the ASTRODEEP project (Merlin et al.
2021). This expands the original Guo et al. (2013) catalog in the GOODS-South
with eighteen new medium-band photometric data obtained with the Subaru
SuprimeCAM (Cardamone et al., 2010), and another five NIR photometric bands
observed with the Magellan Baade FourStar during the ZFOURGE survey (Straatman
et al., 2016). In all cases the photometry has been aperture-matched where low
angular-resolution images were de-blended based on positional priors from high
angular-resolution ones. Overall, densely sampled ($\approx$ 40 bands), deep
photometry covering the spectral range from the rest-frame UV to NIR for
galaxies at $z\sim 1-4$, is available in all the three fields considered here.

Starting from the CANDELS photometric catalog, we first impose a cut on the
integrated isophotal signal-to-noise ratio, S/N $\geq 10$, in the HST/WFC3
$H_{160}$ band to ensure good-quality photometry and morphology. We then
remove galaxies that have clear evidence of AGN activity using the AGNFlag in
the aforementioned catalogs. Because the focus of this work is on massive
galaxies at cosmic noon, we then select galaxies with redshift $1.2<$
$z_{\rm{obs}}$ $<4$ and stellar mass $\log_{10}(M_{*}/M_{\odot})>10$ using the
existing measurements of Nayyeri et al. (2017) for the COSMOS field and of Lee
et al. (2018) for the GOODS fields. This initial cut on stellar mass also
ensures high stellar-mass completeness in all fields considered in this work,
namely $>80\%$, see Ji et al. 2018 for the CANDELS/GOODS fields and Nayyeri et
al. 2017 for the CANDELS/COSMOS field. In principle, these cuts on
$z_{\rm{obs}}$ and $M_{*}$ would not be needed, because we could use the
results from our new SED modeling with Prospector (section 3.1) from the
upstart. In practice, however, running Prospector is computationally very
intensive111In our case, typical running time for a single galaxy is $\approx
10-20$ hours, making it impractical to run it on the entire CANDELS catalogs
for the primary selection. Thus, we resort to first make the cuts on
$z_{\rm{obs}}$ and $M_{*}$ from previous measurements, and then run Prospector
to derive the SFH of a substantially smaller sample.

Following Paper I, to ensure the quality of the adopted de-blending procedures
of low-resolution images and hence the reliability of the photometry, we have
visually inspected the galaxies of our sample in all bands in order to select
galaxies with high-quality photometry (see Figure 2 of Paper I). We retain in
the sample only galaxies with GALFITFlag = 0 from van der Wel et al. (2012) to
ensure that morphological measurements are reliable (section 3.2). Finally,
using the more refined measures of the stellar mass from our Prospector
fitting (section 3.1), we adjust the stellar-mass cut to
$\log_{10}(M_{*}/M_{\sun})>10.3$ such that the environmental effect should
only play a relatively minor role in the evolution and quenching of galaxies
(e.g., Ji et al., 2018). Our final sample contains 1317 galaxies in total,
with 492 located in the COSMOS, 448 in the GOODS-South and 377 in the GOODS-
North. In Figure 1, we show the distribution of the final sample in the rest-
frame UVJ diagram (Williams et al., 2009).

Figure 1: The rest-frame UVJ-color diagram. Background grey contours show the
distribution of galaxies in the CANDELS/COSMOS field, with
$\log_{10}(M_{*}/M_{\sun})>10$, $1.2<$ $z_{\rm{obs}}$ $<4$ and S/N $>10$ in
the $H_{160}$ band. Black solid lines show the SFG-QG separation boundary of
Schreiber et al. (2015). The final sample of this work (section 2) is shown as
individual points that are color-coded according to sSFR derived from our new
SED fitting with Prospector (section 3.1).

##  3 Measurements

###  3.1 SED Fitting with Prospector

We have derived the integrated properties of the stellar populations of the
galaxies in our sample by modeling their SEDs with the Prospector package
(Leja et al., 2017; Johnson et al., 2021).

Prospector is built upon a fully Bayesian framework, and is ideally positioned
to exploit the vast improvement in quality, quantity and wavelength coverage
of modern (mostly) photometric data accumulated over the last two decades in
legacy fields, such as the three fields that are subjects of this work, i.e.
the COSMOS and the two GOODS fields. Packages such as Prospector now make it
possible to fit the SED of high-redshift galaxies with advanced, complex
stellar population synthesis models that provide robust constraints on the
stellar age of galaxies. One key feature of Prospector is that it allows
flexible parameterizations of galaxies’ SFHs, including a piece-wise,
nonparametric form which we have elected to adopt in this work.

Adopting a nonparametric SFH means that the assembly history of galaxies can
be arbitrarily complex and yet the code yields an unbiased inference of the
properties of stellar populations. Despite the reconstruction of SFHs is still
a relatively new development, several recent studies have tested its
robustness with Prospector using synthetic observations generated from
cosmological simulations (Leja et al., 2019; Johnson et al., 2021; Tacchella
et al., 2022; Ji & Giavalisco, 2022). These tests have demonstrated that, at
least in the case of the SFHs of the synthetic galaxies encountered in the
simulations, Prospector is capable of recovering the nonparametric SFHs with
high fidelity, if high-quality, panchromatic data coverage of the SED is
available.

In this work, we assume the same model priors and settings of Prospector as we
did in Paper I, and we refer readers to that paper for a detailed description
and the quality tests that we have already conducted. Here we only briefly
summarize the key assumptions. For the basic setup of the fitting procedure,
we adopt the Flexible Stellar Population Synthesis (FSPS) code (Conroy et al.,
2009; Conroy & Gunn, 2010) where the stellar isochrone libraries MIST (Choi et
al., 2016; Dotter, 2016) and the stellar spectral libraries MILES (Falcón-
Barroso et al., 2011) are used. We assume the Kroupa 2001 initial mass
function (IMF), the Calzetti et al. 2000 dust attenuation law and fit the
V-band dust optical depth with a uniform prior $\tau_{V}\in(0,2)$, and the
Byler et al. 2017 nebular emission model. During the SED modeling, we assume
that all stars in a galaxy have the same metallicity $Z_{*}$, and fit it with
a uniform prior in the logarithmic space
$\log_{10}(Z_{*}/Z_{\sun})\in(-2,0.19)$, where $Z_{\sun}=0.0142$ is the solar
metallicity, and the upper limit of the prior is chosen because this is the
highest value of metallicity that the MILES library has. We do not keep track
of the time evolution of the metallicity. Finally, we also fit the redshift
$z_{\rm{obs}}$ as a free parameter but we use a strong prior in the form of a
normal distribution centered at the best available redshift value from the
CANDELS catalogs, either photometric or spectroscopic redshift when available,
and with a width equal to the corresponding redshift uncertainty, i.e. either
the spectroscopic redshift error bar or the width of the posterior
distribution function of photometric redshifts.

Following recent studies (Leja et al., 2017, 2019, 2022; Estrada-Carpenter et
al., 2020; Tacchella et al., 2022; Ji & Giavalisco, 2022), we model the
nonparametric SFH with a piece-wise step function with nine bins of lookback
time. Specifically, the first and second bins are fixed to be 0 - 30 and 30 -
100 Myr, the last bin is assumed to be 0.9$\rm{t_{H}}$ \- $\rm{t_{H}}$ where
$\rm{t_{H}}$ is the Hubble Time of $z_{\rm{obs}}$, and the remaining six bins
are evenly spaced in logarithmic lookback time intervals between 100 Myr and
0.9$\rm{t_{H}}$. Also, as done in Paper I, our fiducial measures throughout
the paper have been obtained assuming the nonparametric SFH with the Dirichlet
prior that (1) results in a symmetric prior on stellar age and sSFR, and a
constant SFH with $\rm{SFR(t)=M_{*}/t_{H}}$; (2) has been tested with nearby
galaxies across all different types (Leja et al., 2017, 2018); and (3) has
been validated with simulated observations of synthetic galaxies from
cosmological simulations (Leja et al., 2019; Ji & Giavalisco, 2022). However,
we stress that the choice of the prior for the nonparametric SFH still remains
somewhat arbitrary and the validity of each assumption still needs to be
further investigated. For example, some other recent work (e.g Estrada-
Carpenter et al., 2020; Tacchella et al., 2022) adopted a so-called Continuity
prior that is very similar to the Dirichlet prior, except it strongly
disfavours sudden changes of SFRs in adjacent time bins, such as those
resulting from powerful bursts of star formation. Which prior should be used
for high-redshift galaxies is awaiting to be tested with future spectroscopic
data that can provide independent measures of parameters such as stellar ages
and timescales of star formation and of quenching. It is very likely that we
will need a more complex prior than the aforementioned two to better capture
different episodes of galaxies’ assembly histories (Suess et al., 2022), and
the choice of such priors might also depend on galaxy types. Here, without
further constraints from other independent measurements, we decide to use the
well-tested Dirichlet prior as our fiducial one. But we have also run another
set of Prospector fittings for the entire sample with the Continuity prior, to
check whether the choice of prior introduces significant differences in our
results and conclusions. As Appendix A shows, we find that the choice of the
nonparametric SFH prior leaves the key conclusions of this work unchanged. In
the following all the results presented, therefore, are the ones obtained
using the Dirichlet prior.

Finally, once we have obtained the SFHs for the individual galaxy in our
sample, we use a number of metrics, which we have introduced in Paper I, to
characterize the overall shape of the SFHs, including different relevant
timescales. We refer readers to Paper I for the definitions and
characteristics of the metrics. Here we only briefly outline the ones
discussed in the main text of this work, i.e.

  * •

Mass-weighted age $t_{\rm{age}}$

  * •

Formation redshift $z_{\rm{form}}$: the redshift corresponds to when the time
interval between $z_{\rm{form}}$ and $z_{\rm{obs}}$ equals to $t_{\rm{age}}$

  * •

Asymmetry $\tau_{\rm{SF}}$/$\tau_{\rm{Q}}$ (see equation 5 in Paper I): the
ratio between the mass-weighted time widths of the period between
$t_{\rm{age}}$ $\rm{<t<t_{H}}$ ($\tau_{\rm{SF}}$) divided by that of the
period between $0<t<$ $t_{\rm{age}}$ ($\tau_{\rm{Q}}$)

####  3.1.1 Comparing with Existing Measurements

We first compare our measurements of $M_{*}$, SFR and sSFR with previous ones.
Specifically, for the galaxies in the CANDELS/COSMOS field we compare our
measures with those done by Nayyeri et al. (2017), who assumed a parametric,
exponentially declining SFH, i.e. $\rm{SFR(t)\propto e^{-t/\tau}}$. Their SFRs
at $z_{\rm{obs}}$ were estimated by combining the UV and IR luminosities, i.e.
$\rm{SFR_{UV}^{Obs}+SFR_{IR}}$, which is the sum of the observed far-UV
emission and the reprocessed thermal dust emission in the IR, after
calibration onto a scale of SFR (Kennicutt, 1998). For the galaxies in the
CANDELS/GOODS fields we compare our measures with those done by Lee et al.
(2018), who treated the functional form of the SFH as a free parameter, chosen
from a set of five parametric analytical models. Their SFRs at $z_{obs}$ were
also estimated using $\rm{SFR_{UV}^{Obs}+SFR_{IR}}$ when a galaxy is detected
in and/or at wavelengths longer than the 24$\micron$ Spitzer/MIPS images, or
using the SED-derived SFRs otherwise. In Figure 2 we show the comparisons in
the CANDELS/COSMOS and GOODS fields, respectively. We also run the Pearson
correlation tests between our Prospector measurements and the previous ones
for each parameter. In all cases we find a rather tight correlation between
the two sets of measurements, with a Pearson correlation coefficient $\gtrsim
0.6$. Yet, systematic deviations are also clearly seen.

As Figure 2 shows, on average our Prospector fitting returns a $0.3-0.4$ dex
larger $M_{*}$ than the previous measures. This systematically larger $M_{*}$
has been extensively discussed and now well understood (Leja et al., 2019,
2022). Using nonparametric SFHs returns older stellar ages, and hence larger
stellar masses, than using parametric ones, because they can easily
accommodate a larger fraction of older stellar populations for a given SED
(Carnall et al., 2019; Leja et al., 2019). Evidence that this systematically
larger $M_{*}$ is actually more accurate than that found when using parametric
forms for the SFH has also been found, including (1) the evolution of galaxy
stellar-mass function inferred using nonparametric SFHs during SED fitting is
more consistent with direct observations (Leja et al., 2020); (2) tests using
synthetic observations generated from cosmological simulations show that
$M_{*}$ derived from the SED fitting with nonparametric SFHs is unbiased, and
in much better agreement with the intrinsic value than that derived with
parametric SFHs (Lower et al., 2020).

Regarding the SFR measures, our Prospector fitting procedure returns values
that are $0.3-0.8$ dex smaller than the previous measures, in quantitative
agreement with the findings of Leja et al. (2020) and Leja et al. (2022).
These authors have shown that this is primarily due to older, evolved stellar
populations in massive galaxies. Specifically, the older stellar populations
can contribute to the IR flux via dust heating, which consequentially leads to
the overestimated $\rm{SFR_{IR}}$. This systematics is particularly important
in low-sSFR galaxies (Conroy, 2013), and it becomes increasingly important at
higher redshifts when the evolved stellar populations were more luminous just
as they were younger. For QGs, their SFR and sSFR measures thus are arguably
more accurate via panchromatic SED fitting than via
$\rm{SFR_{UV}^{Obs}+SFR_{IR}}$, considering that SED fitting takes into
account the full shape of SEDs and self-consistently models stellar
populations with different ages. In fact, this point has been demonstrated
using mock observations of galaxies from semi-analytic models (e.g., see
Figure 6 of Lee et al. 2018). This is also supported by the fact that we find
a larger Pearson correlation coefficient, i.e. a stronger correlation, when
comparing our new SFR and sSFR measures with those from Lee et al. (2018) than
with those from Nayyeri et al. (2017). As Figure 2 shows, the correlation
improvement is mainly caused by galaxies with low sSFRs, because Lee et al.
(2018) used SED-derived SFRs for galaxies without MIR/FIR detections and most
of the low-sSFR galaxies are not detected in the MIR/FIR bands.

Figure 2: The comparisons between our Prospector fitting results (y-axis) and
previous measurements (x-axis). Galaxies in the CANDELS/GOODS fields are shown
in orange, while those in the CANDELS/COSMOS field are shown in black. In each
panel, the red dashed line marks the one-to-one relation. In the legend we
label the median and standard deviation of the differences between the two
measurements, i.e. the Prospector one minus the previous one, and the
coefficient of Pearson correlation test between the two measurements. Our
measurements are in great qualitative agreement with previous ones, although
systematic offsets are also clearly observed between the two measurements (see
section 3.1.1 for a detailed discussion).

We then proceed to compare the results from our fitting procedure with the
star-forming main sequence measured by Leja et al. (2022), who also used
Prospector. Instead of separating galaxy populations according to their star-
formation properties prior to measuring the star-forming main sequence, Leja
et al. (2022) utilized the normalizing flow technique to model the density
distribution of full galaxy populations, from which they measured the redshift
evolution of the ridge (i.e. mode) and the mean of the density distribution in
the $M_{*}$-SFR parameter space, i.e. the star-forming main sequence. In this
way they demonstrated that systematic errors of the star-forming main
sequence, which are introduced by the selection methods of SFGs, can be
effectively mitigated. More importantly, Leja et al. (2022) showed that within
this framework they were able to substantially mitigate, if not resolve, a
long-standing systematic offset of the star-forming main sequence between the
observations and the predictions from cosmological simulations (see their
section 7.1). In Figure 3, we compare the distribution of our sample with the
density ridge222We have checked that using the mean density of Leja et al.
(2022) as the main sequence does not change any of our conclusions of this
work. (their equation 9) of Leja et al. (2022). Because the star-forming main
sequence evolves with redshift, for a better comparison, instead of plotting
Leja et al. (2022)’s star-forming main sequence at the median redshift of our
entire sample, we first bin galaxies in our sample according to their $M_{*}$,
and then within individual $M_{*}$ ranges we plot the star-forming main
sequence at the median redshift value of that given $M_{*}$ bin (blue solid
line in Figure 3). Our measurement is in great agreement with that of Leja et
al. (2022). In Figure 3 we also plot the subsample of galaxies which are
classified as QGs with the UVJ technique. They are well below the star-forming
main sequence, again showing the consistency between the SED fitting and the
UVJ selection that we already saw in Figure 1.

Figure 3: The plot of star-forming main sequence. Grey contours show the
distribution of the entire sample in this study. Red contours show the
distribution of a subsample that is classified as QGs using the UVJ technique.
All distributions are estimated with a Gaussian kernel. The blue solid line
marks the star-forming main sequence of Leja et al. (2022) who also used
Prospector for their SED fittings.

The other way to compare our measurements with Leja et al. (2022)’s is through
the distribution of $\rm{\Delta MS}$ (Elbaz et al., 2011), i.e. the distance
to the star-forming main sequence that is defined as

$\rm{\Delta MS=\frac{SFR}{SFR_{MS}}}$ (1)

where $\rm{SFR_{MS}}$ is the star formation rate of a galaxy on the star-
forming main sequence of Leja et al. (2022). We show the comparison in Figure
4 where we also fit a Gaussian distribution to the galaxies with $\rm{\Delta
MS}$ $\geq 0.1$. While this study and that work both use Prospector, we note
that our fiducial SED model is not the same as Leja et al. (2022)’s, because
they used the Continuity prior for the nonparametric SFH reconstructions and
assumed a different dust attenuation law. Yet, the Gaussian fit shows on
average $\rm{\Delta MS}$ $\approx 1$, showing no systematic shifts between our
star-forming main sequence and Leja et al. (2022)’s. The Gaussian fit also
shows that the width (i.e. dispersion) of the star-forming main sequence is
$0.3-0.4$ dex, which is broadly consistent with Leja et al. (2022) and other
recent studies (e.g. Whitaker et al., 2012; Speagle et al., 2014; Schreiber et
al., 2015; Lee et al., 2018).

Figure 4: The distribution of $\rm{\Delta MS}$, i.e. the distance to the star-
forming main sequence of Leja et al. (2022). The entire sample is divided into
two groups using the median stellar mass. The red solid line shows the
Gaussian fit to the distribution of galaxies with $\rm{\Delta MS}$ $\geq 0.1$,
the best-fit center ($\mu$) and width ($\sigma$) are shown in the plot. The
$\mu($$\rm{\Delta MS}$) being consistent with one shows the consistency
between our star-forming main sequence and Leja et al. (2022)’s.

####  3.1.2 Examples of Results of the SED Fitting Procedure

In Figure 5 we show the SED fitting results, together with the HST images, of
four examples of massive galaxies at $z_{\rm{obs}}$ $\sim 2$ with
$\log_{10}(M_{*}/M_{\sun})\sim 11$ in our sample whose SFHs and morphological
properties are illustrative of the variety that we have encountered in this
study. The first row shows a QG with sSFR $<0.01$ Gyr-1, and whose morphology
is extended, with $R_{\rm{e}}$ $\approx$ 4.5 kpc in the $H_{160}$ band. The
SFH shows that the galaxy is a post-starburst one, because it experienced a
recent burst of star formation $\approx 0.1$ Gyr prior to $z_{\rm{obs}}$ and
it has a relatively low-level of ongoing SFR, which in this case is the SFR
within 30 Myr, i.e the first lookback time bin of the nonparametric SFH.
Interestingly, its HST images in bluer bands, e.g. the $I_{814}$ band that
probes rest-frame $\sim 2800$Å, show features consistent with a merging event,
fully in line with it being a post-starburst galaxy. The second row shows a
very compact QG with $R_{\rm{e}}$ $\approx$ 0.7 kpc, whose SFH shows a
relatively gradual decline over the past 1 Gyr. The third and fourth rows show
an extended SFG and a compact SFG, respectively. While in the $H_{160}$ band
the size ($R_{\rm{e}}$) of the former is about two times larger than the
latter, the images of both galaxies show a very compact central component in
the $I_{814}$ band, and their reconstructed SFHs suggest they are experiencing
an on-going burst of star formation. These four galaxies are representative of
the types of correlations between the morphological properties and SFHs at the
core of this study, and we will discuss and quantify them for the entire
sample in the remaining of this paper.

Figure 5: Examples of four different kinds of galaxies in this study, from top
to bottom, namely an extended QG, a compact QG, an extended SFG and a compact
SFG. The left column shows the RGB images made with multi-band HST images.
Basic information of individual examples is listed in each panel. The middle
column shows the best-fit SED derived with Prospector, where in the bottom
inset we show the relative error between data and model-predicted photometry.
The right column shows the reconstructed nonparametric SFH.

###  3.2 Morphological Measurements

Similarly to Paper I, the morphological measures of the sample galaxies are
taken from van der Wel et al. (2012), where the light profile of galaxies was
modelled with a 2-dimensional Sérsic function using Galfit (Peng et al.,
2010a). We take the measurements in the $H_{160}$ band, because it better
probes the stellar morphology for galaxies at $z\sim 2$ as it is the reddest
HST imaging available in the three fields. To secure good quality, we follow
the recommendation of van der Wel et al. (2012) to only use the measurements
with GALFITFlag = 0. This means that the galaxies with unreliable single
Sérsic fitting results are excluded, including when (1) the Galfit-derived
total flux significantly deviates from that derived with SExtractor (Bertin &
Arnouts, 1996), (2) at least one parameter reaches either bounds of the range
of allowed values set prior to the fitting (e.g. $0.2<n<8$) and (3) the Sérsic
fitting is unable to converge.

Morphological parameters considered in this work include the

  * •

Sérsic index $n$

  * •

circularized effective/half-light radius $R_{\rm{e}}$
$=R_{e,maj}\times\sqrt{b/a}$, where $R_{e,maj}$ is the effective semi-major
axis and $b/a$ is the axis ratio

  * •

stellar-mass surface density within the effective radius $\Sigma_{e}$
$=M_{*}/(2\pi R_{e}^{2})$

  * •

stellar-mass surface density within the central radius of 1kpc,
$\Sigma_{\rm{1kpc}}$ (Cheung et al., 2012). As shown in Ji et al. (2022),
because $\Sigma_{\rm{1kpc}}$ is derived as the combination of $R_{\rm{e}}$ and
$n$, this helps to reduce the uncertainty stemming from the strong covariance
between the two parameters, making $\Sigma_{\rm{1kpc}}$ a more robust
parameter to quantify galaxy compactness than $n$ or $R_{\rm{e}}$
individually.

  * •

fractional mass within the central radius of 1 kpc, $\rm{M_{1kpc}/M_{*}}$ (Ji
et al., 2022). Similar to the stellar-mass surface density,
$\rm{M_{1kpc}/M_{*}}$ also is a good metric of the compactness of galaxies.
Because $\Sigma$ is strongly correlated with $M_{*}$, it is hard to interpret
correlations of any property with $\Sigma$, because they could be driven by
$M_{*}$ rather than the compactness. Because the dependence of
$\rm{M_{1kpc}/M_{*}}$ on $M_{*}$ is very weak (Ji et al., 2022), using it can
provide a more direct view on the links between compactness and other physical
properties.

##  4 Results

###  4.1 The Progenitor Effect

To begin, we investigate the progenitor effect on the apparent structural
evolution of galaxies by studying the correlations between structural
properties and $z_{\rm{form}}$. We first divide the entire sample into UVJ-
selected SFG and QG subsamples, and then divide each subsample using its
median $M_{*}$, motivated by Paper I where we found that among QGs the
strength of the progenitor effect depends on $M_{*}$. In Figure 6, we show the
relationships of $z_{\rm{form}}$ with $R_{\rm{e}}$, $\Sigma_{\rm{1kpc}}$,
$\Sigma_{e}$ and $\rm{M_{1kpc}/M_{*}}$, respectively. For each relationship we
also fit a power-law function, i.e. (1+$z_{\rm{form}}$)-β, which is plotted as
a dashed line in Figure 6. To estimate the uncertainty of best-fit power-law
relations, similarly to what we did in Paper I, we bootstrap the sample 1000
times, during each time we resample the value of each measurement using a
normal distribution with a width equal to the measurement uncertainty. The
best-fit power-law relation and the corresponding uncertainty are labeled in
the legend of each panel of Figure 6.

As Figure 6 shows, all considered structural properties depend on the
formation epoch ($z_{\rm{form}}$) of galaxies, which, in essence, is the
progenitor effect. Regarding the relationship between $R_{\rm{e}}$ and
$z_{\rm{form}}$, galaxies formed earlier, i.e. having larger $z_{\rm{form}}$,
tend to have smaller sizes. A clear dependence on $M_{*}$ is found for QGs
where the relationship is steeper for lower-mass QGs than for higher-mass
ones. This is most likely because of the increasing importance of repeated
minor merging events that take place in more massive galaxies after quenching.
The miner mergers drive the after-quenching size growth that flattens the
$R_{\rm{e}}$-$z_{\rm{form}}$ relationship of more massive QGs, as we have
already discussed in detail in Paper I. Compared to QGs, the $R_{\rm{e}}$ of
SFGs decreases with (1+$z_{\rm{form}}$) seemingly at a slower rate.
Quantitatively interpreting this relationship of SFGs requires the full
knowledge333Because rejuvenation of star formation apparently is rare, we
consider the whole history of mass assembly of any galaxy as the period that
starts from the Big Bang and ends at the time when a galaxy becomes quiescent.
of their mass assembly histories, which we do not have because (1) by the time
of $z_{\rm{obs}}$ SFGs are still undergoing active star formation and have not
yet completed assembling their masses, and (2) we cannot predict their SFHs
after $z_{\rm{obs}}$. Nevertheless, the decreasing trend of $R_{\rm{e}}$ with
$z_{\rm{form}}$ shows, even for SFGs, that the progenitor effect also plays a
role in their apparent size evolution.

Because the measurements of $\Sigma_{\rm{1kpc}}$, $\Sigma_{e}$ and
$\rm{M_{1kpc}/M_{*}}$ directly depend on $R_{\rm{e}}$ (section 3.2), it is
naturally expected that these metrics of compactness also depend on
$z_{\rm{form}}$, a relationship that we indeed observe and show in the left
three columns of Figure 6. Regardless of star-formation properties at
$z_{\rm{obs}}$, galaxies that formed earlier, i.e. having larger
$z_{\rm{form}}$, tend to have a more compact morphology, i.e. have larger
$\Sigma_{\rm{1kpc}}$, $\Sigma_{e}$ and $\rm{M_{1kpc}/M_{*}}$.

Unlike using $z_{\rm{obs}}$ to select samples, which results in galaxies
formed at different epochs of $z\geq$ $z_{\rm{obs}}$ being grouped together,
using $z_{\rm{form}}$ naturally separates galaxies formed at different epochs,
which automatically allows one to investigate, and account for, the progenitor
effect. Figure 6 shows that the progenitor effect at least in part contributes
to the apparent evolutions of the size and of the compactness metrics
considered in this work with $z_{\rm{obs}}$, if galaxies are mixed together
with no accounting for their formation epochs. As Figure 6 shows, variations
up to 50% can be observed in $R_{\rm{e}}$ and up to 1 full dex in
$\Sigma_{\rm{1kpc}}$, depending on the stellar-mass range, can be accounted
for solely because of the progenitor effect. This highlights the importance of
accounting for its contribution before attempting any interpretation or
modeling of the apparent structural evolution in terms of different physical
mechanisms.

Figure 6: The relationships between the morphological properties of galaxies
and their $z_{\rm{form}}$. The top and bottom rows show the results of UVJ-
selected SFGs and QGs, respectively. We remind that the y-axis’ scale of the
two rows are different. Each galaxy sample is further divided using its median
stellar mass. In each panel, the contours show the density distributions
estimated using a Gaussian kernel, the points with error bars show the medians
and standard deviations of the corresponding morphological property of
galaxies in the individual bins of $z_{\rm{form}}$, and the dashed lines show
the results of power-law fittings where the best-fit relations are labeled in
the panel legend. The plots clearly show the dependence of galaxies’
morphology on their formation time, i.e. the progenitor effect.

###  4.2 Correlations Between the Structural and Star-formation Properties of
Galaxies Formed at a Similar Epoch

Taking advantage of the reconstructed SFHs, we can group galaxies according to
their formation epochs. Once the formation epoch is fixed, the progenitor
effect should then be essentially eliminated from the apparent evolution. If
correlations between star-formation and structural properties are still
observed, then these will provide strong empirical constraints on physical
mechanisms of the observed structural evolution. In what follows, we use
$z_{\rm{form}}$ as a proxy for the formation epoch of galaxies. Specifically,
we divide the entire sample into four subsamples using quartiles of
$z_{\rm{form}}$, and study the relationships of the compactness metrics of
galaxies with $\rm{\Delta MS}$ (equation 1). The results are plotted in Figure
7.

In all bins of $z_{\rm{form}}$, the majority of galaxies are distributed
following a “pistol”-shape pattern, both in the diagram of $\rm{\Delta MS}$
vs. $\Sigma_{\rm{1kpc}}$ (the top row of Figure 7), and in the diagram of
$\rm{\Delta MS}$ vs. $\rm{M_{1kpc}/M_{*}}$ (the middle row of Figure 7). At a
fixed $z_{\rm{form}}$, galaxies with smaller $\rm{\Delta MS}$, i.e. being more
quiescent, have larger $\Sigma_{\rm{1kpc}}$ and $\rm{M_{1kpc}/M_{*}}$, i.e.
being more compact, while at the same time the distributions of
$\Sigma_{\rm{1kpc}}$ and $\rm{M_{1kpc}/M_{*}}$ also become narrower. In the
diagram of $\rm{\Delta MS}$ vs. $\Sigma_{e}$ (the bottom row of Figure 7), a
similar trend is also observed for the QGs, which in general have larger
$\Sigma_{e}$ whose distribution also is narrower than that of the SFGs,
although the knee of the pistol pattern is not as obviously seen as that in
the diagrams of $\Sigma_{\rm{1kpc}}$ and of $\rm{M_{1kpc}/M_{*}}$.

A similar pistol pattern between the star-formation properties and the central
density of galaxies have been observed in recent literature (e.g. Barro et
al., 2013, 2017; Lee et al., 2018). However, in those studies the pistol
pattern was produced without regard to the formation epoch of galaxies,
meaning that the quantitative details and overall significance of the observed
patterns were at least in part due to the progenitor effect. In this work we
refine the characterization of the evolutionary trends in the sense that we
utilize the information extracted from the reconstructed SFHs to eliminate the
contribution from the progenitor effect to the observed correlations. As
Figure 7 shows, the pistol pattern, although different in shape, is still
observed after the elimination of the progenitor effect, demonstrating the
existence of a physical link between the effective radius, central mass
density and compactness of galaxies and their star-formation properties.
Overall as galaxies quench, their effective radii become smaller, central mass
densities increase and thus their compactness also increases.

Figure 7:  The relationships between $\rm{\Delta MS}$ (equation 1) and galaxy
compactness metrics $\Sigma_{\rm{1kpc}}$ (the top row), $\rm{M_{1kpc}/M_{*}}$
(the middle row) and $\Sigma_{e}$ (the bottom row). To mitigate the progenitor
effect, we use the quartiles of the distribution of $z_{\rm{form}}$ to divide
the entire sample into four subsamples, which are shown in individual columns,
such that galaxies shown in the same panel formed at a similar epoch. We
remind that the scale of y-axis is reversed, so galaxies being more quiescent,
i.e. having smaller $\rm{\Delta MS}$, appear in upper regions of the plots.
Each galaxy is color-coded according to its size $R_{\rm{e}}$. Background grey
contours show the density distribution estimated using a Gaussian kernel. The
dependence of the compactness metrics adopted here on $\rm{\Delta MS}$ is
still observed even after we mitigate the progenitor effect by binning the
sample by $z_{\rm{form}}$, implying a physical link between the compactness of
galaxies and their star-formation properties.

###  4.3 Automated Morphological Classification: the Dawn of Spheroids?

While on average galaxies with smaller $\rm{\Delta MS}$ are more compact,
Figure 7 shows that there are also substantial numbers of galaxies with
suppressed star formation, i.e. $\rm{\Delta MS}$ $<0.1$, that have a extended
morphology. Morphological metrics derived from the single Sérsic fitting, such
as the ones considered above, have been extensively adopted to characterize
the overall shape of the light distribution of galaxies. However, those
metrics entirely ignore substructures such as clumps and nonaxisymmetric
features like bars and tidal features that contain key information about the
evolutionary mechanism of galaxies’ structures. In fact, those substructures
are commonly observed not only in SFGs at high redshifts (e.g. Elmegreen et
al., 2007; Förster Schreiber et al., 2011; Guo et al., 2012) but also in some
QGs as well (Giavalisco & Ji in preparation). Although in the nearby universe
visual classifications are effective in identifying them (e.g. Lintott et al.,
2008), at high redshifts identifying the substructures are very challenging
because of (1) the cosmological dimming of surface brightness and (2) the
limitations imposed by the available angular resolution that even with HST,
and now JWST, are such that the substructures are still hard to visually
identify. Rapidly developing techniques based on machine learning and
artificial intelligence hopefully can greatly help improve the morphological
classification of galaxies in the early universe. For example, utilizing deep
learning techniques, Huertas-Company et al. (2015) trained the Convolutional
Neural Networks to classify the morphological types of CANDELS’ galaxies
brighter than 24.5 magnitude (AB) in the $H_{160}$ band. The quality of their
machine-based classifications is very high, namely they have no bias (with a
scatter of $\sim 10\%$) compared to those done by human classifiers and the
fraction of mis-classifications is better than $1\%$.

We cross match our sample with the catalog of Huertas-Company et al., and then
plot the successfully-matched subsample in Figure 8. Overall, a clear trend
can be observed such that the morphology of galaxies changes from disk-like to
spheroidal/bulge-like as they become more compact and quench. With the
progenitor effect eliminated, this apparent morphological transformation of
galaxies is closer to the intrinsic structural transformations that galaxies
undergo as they evolve and quench in time. Thus, morphologically spheroidal
structures (no dynamical information is included in this discussion), whether
bulges or elliptical galaxies, emerge as galaxies suppress their star
formation and terminate the assembly of the bulk of their stellar masses. We
shall return later on this point from a different perspective.

Regarding galaxies with suppressed star formation in particular, the
morphology of the extended ones is not only more disky, but also more
disturbed compared to the compact ones. In particular, we use the median
values of $\Sigma_{\rm{1kpc}}$, $\Sigma_{e}$ and $\rm{M_{1kpc}/M_{*}}$ of SFGs
to divide the sample galaxies into compact and extended ones. The key result
is that, regardless of the metrics adopted to classify galaxies’ compactness,
the probability of a QG to have a disturbed morphology is $\sim 3-4$ times
higher when it is extended relative to when it is compact. Specifically, for
galaxies with $\rm{\Delta MS}$ $<0.1$, the fraction of those having a
disturbed morphology, namely being classified as an irregular disk or a merger
according to Huertas-Company et al.’s criteria (see their section 6 for
details), is $68.6\pm 12.9\%$ (48/70) for those with
$\log_{10}$$\Sigma_{\rm{1kpc}}$ $<9.5$ compared to $18.2\pm 0.3\%$ (42/231)
for those with $\log_{10}$$\Sigma_{\rm{1kpc}}$ $>9.5$. If we use
$\rm{M_{1kpc}/M_{*}}$ $=0.1$ as the dividing line between extended and compact
galaxies, the fraction is $71.7\pm 16.4\%$ (33/46) for the former compared to
$22.4\pm 0.3\%$ (57/255) for the latter. Similarly, if we use
$\log_{10}\,$$\Sigma_{e}$ $=9.2$ instead, the fraction is $64.3\pm 12.3\%$
(45/70) for the extended ones compared to $19.5\pm 0.3\%$ (45/231) for the
compact ones. This distinct morphological difference within the population of
QGs already indicates the possibility that they reflect markedly different
formation and evolutionary paths. We are going to elaborate this point more in
the next section.

Figure 8: Similar to Figure 7, but for the subsample of galaxies that are
successfully matched to the catalog of Huertas-Company et al. (2015) where the
Convolutional Neural Networks were used to classify the morphological type of
galaxies, including a purely bulge- (red), a purely disk- (blue), a
bulge+disk- (green), an irregular disk- (cyan) and a merger- (orange) type
morphologies. The morphology of galaxies changes from disk-like to bulge-like
as they become more compact and quench.

###  4.4 The Relationship Between the Morphology and the Assembly History of
Galaxies

So far, we have only considered the formation epoch ($z_{\rm{form}}$) of
galaxies as a key piece of information stemming from the knowledge of SFHs. In
this section we take the full shape of reconstructed SFHs into account to
further study the relationship between the morphological and star-formation
properties. To better illustrate our findings, we divide the entire sample
into four galaxy groups, namely (1) extended quiescent galaxies (eQGs); (2)
compact quiescent galaxies (cQGs); (3) extended star-forming galaxies (eSFGs)
and (4) compact star-forming galaxies (cSFGs).

We have verified that the results presented below are not sensitive to the
exact criteria adopted to select the four galaxy groups. Specifically, to
separate SFGs and QGs, we have tested our results by both using the UVJ
technique and using the distance from the star-forming main sequence provided
by the dividing line of $\rm{\Delta MS}$ $=0.1$. To separate compact and
extended galaxies, we have tested our results by using both the fixed dividing
line of $\log_{10}$$\Sigma_{\rm{1kpc}}$ $=9.5$ and also the varying dividing
line of the median $\Sigma_{\rm{1kpc}}$ of individual $z_{\rm{form}}$ bins. We
have not observed any substantial changes in our results. Our results also
appear to be insensitive to the exact choice of the compactness metrics
adopted to separate compact and extended galaxies, as we demonstrate in
Appendix B using $\rm{M_{1kpc}/M_{*}}$ and $\Sigma_{e}$. In what follows, we
show the results of using $\rm{\Delta MS}$ $=0.1$ to separate star-forming and
quiescent galaxies, and using $\log_{10}$$\Sigma_{\rm{1kpc}}$ $=9.5$ to
separate compact and extended galaxies.

We first measure the average SFHs. Similarly to what we did in the previous
sections, we first bin the galaxies according to their $z_{\rm{form}}$. In
each bin of $z_{\rm{form}}$, instead of calculating the median/mean of best-
fit SFHs of the individual galaxy groups, we stack the posteriors derived from
the Prospector fittings, hence the full probability density distributions of
the individual fittings are taken into account. Because in this work we only
consider galaxies within a relatively narrow range of stellar mass, i.e.
$\log_{10}(M_{*}/M_{\sun})=10.3\sim 11.7$, to emphasize the shape of SFHs,
before stacking we normalize each SFH by $M_{*}$ observed at $z_{\rm{obs}}$ so
that

$\int_{0}^{t_{H}}\widetilde{\rm{SFR}}(t)\,dt=1,$ (2)

where $\widetilde{\rm{SFR}}(t)=\rm{SFR}(t)/M_{*}$. We then calculate the
median and standard deviation (1$\sigma$) from the stacked posteriors, and
finally obtain the average SFHs.

In Figure 9, we show the average SFHs of eQGs (magenta), cQGs (red), eSFGs
(cyan) and cSFGs (blue), respectively. Clear differences in the shape of the
SFH of each subgroup are clearly observed. While SFGs have either an overall
flat or rising SFH, QGs show a clear decline of recent SFR, as expected. The
novel aspect is the dependence of SFHs on morphology: while cQGs underwent an
early phase of intense star formation followed by a gradual decline and
eventual quiescence, eQGs followed the opposite trend, namely, a gradual rise
of star formation toward a relatively recent peak followed by a rapid decrease
toward quiescence. Among SFGs, the differences in the average SFH as a
function of morphology seem to be more subtle, but we observe that the overall
shape of their average SFH is similar to that of the eQGs minus the decrease
toward quiescence. We also observe that the rapid rise toward the very early
peak seems to be absent among SFGs, suggesting that this type of SFH is
typical of the early type galaxies but becomes rare or absent in galaxies that
are still forming stars later on in the cosmic evolution, even among cSFGs. In
the following, we will study in detail the relationships between the SFH and
the compactness of QGs (section 4.4.1), and of SFGs (section 4.4.2),
respectively.

####  4.4.1 The Case of QGs

Clear trends are observed between the assembly history and the morphology of
QGs. As Figure 9 shows, the stacked SFH of eQGs peaks at a much later time,
i.e. smaller lookback time, of $\lesssim 0.5$ Gyr compared to the cQGs whose
stacked SFH peaks at $\gtrsim 1$ Gyr ago. Thus, while the stacked SFH of cQGs
is very similar to that of typical QGs, which have assembled most mass early
on followed by a gradual decline of their SFRs, the stacked SFH of eQGs is
very similar to that of post-starburst galaxies, which shows a recent (a short
lookback time from $z_{\rm{obs}}$) burst of star formation followed by a rapid
decline of SFR.

In Figure 10 we compare the distributions of $t_{\rm{age}}$ and of
$\tau_{\rm{SF}}$/$\tau_{\rm{Q}}$. The eQGs have smaller $t_{\rm{age}}$ (i.e.
younger) and larger $\tau_{\rm{SF}}$/$\tau_{\rm{Q}}$ compared to the cQGs,
which are consistent with the shape of the stacked SFHs. We also run both the
Kolmogorov–Smirnov and the Anderson-Darling tests on the null hypothesis that
the distributions of the two QG groups are the same. We can reject the null
hypothesis with a $\gtrsim 3\sigma$ confidence level, except in the largest
$z_{\rm{form}}$ bin where we can reject it with an $\sim 2.5\sigma$ confidence
level. Thus, compared to the eQGs, in general the cQGs have both a larger
mass-weighted age and a shorter star-formation timescale relative to the
quenching timescale.

Taken together with the higher frequency of finding eQGs with a disturbed
morphology, as we have already observed in Figure 8 and discussed in section
4.2, the finding of eQGs’ average SFH being consistent with that of post-
starburst galaxies supports a scenario that some eQGs might be merging
transients, namely, they are merger remnants observed at a stage when the SFR
rapidly declines after a merger-induced recent starburst. The stacked SFH of
eQGs suggests that the decline of SFR happened $\lesssim 0.5$ Gyr ago, which
is in broad agreement with studies of galaxy mergers based on hydrodynamical
simulations (e.g. Springel et al., 2005; Hopkins et al., 2008). A complete
merging event very likely includes multiple episodes of starburst, and the
whole process can help galaxies to form dense central cores because strong
gravitational torques induced by the merging galaxy can effectively drive gas
to the center and then trigger central star formation (e.g. Sanders et al.,
1988; Hopkins et al., 2010). Because the eQGs do not seem to have built up
their central cores by the time of observation444It is possible that some of
the eQGs might have highly dust-obscured dense cores. Unfortunately, the
sensitivity of existing MIR/FIR observations in the three fields is relatively
low. This possibility can be tested with future observations such as the
upcoming JWST surveys., if they indeed come from mergers, then they likely
have just finished one earlier episode of merger-induced starburst, meaning
that their star formation can possibly rejuvenate. Interestingly, $23\pm 3\%$
QGs in our sample are the eQGs, and this fraction is quantitatively consistent
with the rejuvenation rate measured in recent spectroscopic studies of QGs at
lower redshift $z=0.5\sim 1$ (e.g. Belli et al., 2017; Chauke et al., 2019),
although we note that using different criteria for the rejuvenation event can
result in quantitatively different rates (e.g., section 6.5 of Tacchella et
al., 2022).

Figure 9: The stacked SFHs of QGs (the top row) and SFGs (the bottom row),
where the solid line and shaded region correspond to the median and 1$\sigma$
range derived from the stacked posteriors. Because each SFH has been
normalized before stacking, the y-axis has an arbitrary scale (see section 4.4
for details). Different columns show galaxies formed at different epochs.
Galaxies are also divided according to their compactness. These end up with
four groups of galaxies, namely extended/compact QGs (e/c QGs) and
extended/compact SFGs (e/c SFGs). The selection criteria of the individual
groups, as well as the color scheme, are illustrated in the right-most panel.
The differences in stacked SFHs between eQGs and cQGs are significant, while
they are subtle between eSFGs and cSFGs. Figure 10: The distributions of the
mass-weighted age ($t_{\rm{age}}$, the top row) and the Asymmetry of SFHs
($\tau_{\rm{SF}}$/$\tau_{\rm{Q}}$, the bottom row) of QGs. Similar to Figure
9, here we refer to galaxies with $\rm{\Delta MS}$ $<0.1$ as the QGs.
Different columns show galaxies formed at different epochs. The magenta and
red solid lines show the distributions of eQGs and cQGs, respectively. The
median of each distribution is labeled using a downward arrow with the
corresponding color. For reference, in every panel the distribution of the
entire QG sample is shown as the filled area in light red. Also labeled are
the $p$-values of the Kolmogorov–Smirnov (KS) and the Anderson-Darling (AD)
tests on the null hypothesis that the distributions of eQGs and cQGs are
identical. The histograms again show the significant differences between the
stacked SFHs of eQGs and of cQGs.

####  4.4.2 The Case of SFGs

Although on average the SFHs of eSFGs and of cSFGs are very similar (Figure
9), a correlation indeed is found between the central stellar-mass surface
density of SFGs and a more detailed SFH classification that we describe in
what follows. Observations of nearby elliptical galaxies555Given the stellar
mass range the majority of the SFGs considered in this study have likely
become massive elliptical galaxies at $z=0$. suggest that their masses were
assembled via multiple episodes that are responsible for the formation of
different structures (e.g. Huang et al., 2013). Motivated by this, we visually
inspect individual SFHs to identify the SFGs with multiple, prominent episodes
of star formation, i.e. two or more peaks that are clearly shown in a
reconstructed SFH. As Figure 11 illustrates, both the best-fit SFH and the
corresponding uncertainty are taken into account during the process of visual
classifications. We then study the relationship between $\Sigma_{\rm{1kpc}}$
and the fraction of SFGs with multiple star-formation episodes, i.e.

$\mathcal{F}_{\rm{multi-SF}}=\frac{N_{\rm{multi-SF}}}{N_{\rm{tot}}},$ (3)

where $N_{\rm{tot}}$ is the total number of SFGs.

Figure 11: Examples of individual reconstructed SFHs to illustrate our visual
selection of SFGs with more than one prominent episode of star formation. The
black solid line shows the best-fit reconstructed SFH, and the dashed line
shows the 1$\sigma$ uncertainty. To highlight the older age bins, unlike
previous figures about SFHs, we remind that we use the linear scale along the
x-axis here. The upper panel shows an ideal case where two episodes of star
formation are clearly seen. The middle panel shows a possible case where the
uncertainty of the reconstructed SFH makes it hard to judge the robustness of
the $\sim 2.5$ Gyr-old star-formation episode, although the best-fit SFH
indeed suggests a two-epoch formation. The bottom panel shows an unlikely case
where the SFR monotonically rises since the Big Bang.

Before proceeding to discuss the results, we clarify in more detail our visual
classification procedure, and also we caution about possible systematics
affecting $\mathcal{F}_{\rm{multi-SF}}$. First, given the limitation imposed
by the data, the SFH reconstruction in this work is done with the integrated
photometry and those refer to the galaxy as a whole. Also, the time resolution
of the SFH reconstruction remains relatively low compared to the timescale of
starbursts, i.e. Gyr vs. tens or hundreds of Myr. The combination of these two
effects means that every star-formation feature, such as a peak or a burst,
identified in a reconstructed SFH might in fact consist of more than one
independent star-formation events that happen at approximately the same epoch
but are not physically associated, e.g., two or more episodes of star
formation in different, physically uncorrelated HII regions. As a result, for
each galaxy we are unable to identify all independent star-formation events,
which however is not our goal. Our goal simply is to find SFGs that by the
time of $z_{\rm{obs}}$ have had more than one enhanced and distinct epoch of
mass assembly (as opposed to independent star-formation events). Because we
only have nine bins of look-back time (section 3.1), for nearly all cases of
our SFGs the “multiple star-formation episodes” really means two well-
separated episodes, in the form of peaks of SFR enhancement, a younger one and
an older one with a typical dividing line of 1 Gyr. Because all SFGs, by
definition, include the younger, on-going star-formation episode in their
SFHs, what really is measured by $\mathcal{F}_{\rm{multi-SF}}$ can be
considered as the fraction of SFGs with the clear presence of older stellar
populations created in previous episodes of star formation. Second, ambiguous
cases certainly exist such as the “possible” example illustrated in Figure 11
where the uncertainty of reconstructed SFHs makes it hard to tell the
robustness of individual star-formation episodes. While the measures presented
below only include the “secure” cases, we have checked that, qualitatively,
our findings do not change if the “possible” cases are also included. Finally,
although for each galaxy we have $\approx 40$ bands photometry, most of these
cover the spectral range bluer than the rest-frame V-band for galaxies at
$z\sim 2$. In this wavelength range young, bright stars outshine older stars,
meaning that either a large mass fraction of older stellar populations is
present or a comparatively high S/N of the photometric data (especially in the
rest-frame optical and NIR) is required to robustly recover the older stellar
populations in the reconstructed SFHs (e.g. Papovich et al., 2001). Therefore,
even for the “unlikely” cases illustrated in Figure 11 it is still possible
that the galaxies in fact have older stellar populations which, however,
cannot be recovered by our Prospector fitting because of wavelength coverage
and data quality at the moment. This can lead to under-estimated
$\mathcal{F}_{\rm{multi-SF}}$.

The key result of this investigation is shown in Figure 12. We find that
$\mathcal{F}_{\rm{multi-SF}}$ generally increases with $\Sigma_{\rm{1kpc}}$,
implying that the probability of finding SFGs that have a sizeable stellar
mass in older stellar populations increases as their central stellar mass
densities grow. The grey line in Figure 12 illustrates the same point in a
different way by plotting the probability of finding SFGs that have a Sérsic
index $n$ between 1.5 and 3, i.e. whose light profiles in the $H_{160}$ band
indicate their morphology likely is a combination of a dense central core plus
an extended outskirt, as a function of $\Sigma_{\rm{1kpc}}$. We have checked
that this trend does not change if we use a slightly different range of $n$,
e.g., between 1.5 and 2.5. We stress that the two trends of
$\Sigma_{\rm{1kpc}}$ that we have just presented, one with
$\mathcal{F}_{\rm{multi-SF}}$ and the other with the shape of light profiles
rely on entirely independent observables, the former based on the SFH and the
latter based on morphology. Yet we find that the two trends depict a
consistent picture, which on the one hand points to the substantial robustness
of the SFH reconstructions, and on the other to the power of incorporating the
SFHs to study the structural evolution of galaxies.

Figure 12: The relationship between $\Sigma_{\rm{1kpc}}$ and
$\mathcal{F}_{\rm{multi-SF}}$, i.e. the fraction of SFGs (i.e. $\rm{\Delta
MS}$ $\geq 0.1$) showing multiple, prominent star-formation episodes in their
reconstructed SFHs. We remind that in this plot only the “secure” cases (see
Figure 11 for illustration) are included, but the overall increasing trend
does not change if the “possible” cases are also included. In the top-left
corner we list in detail the number of galaxies in each bin of
$\Sigma_{\rm{1kpc}}$. Also plotted is the fraction of SFGs having Sérsic index
$n=1.5\sim 3$, which is shown as the black dashed line and the grey shaded
region that marks the 1$\sigma$ range. All uncertainties in the plot are
estimated using the Poisson errors. This plot shows that the probability of
finding SFGs that have a sizeable stellar mass in older stellar populations
increases as the central stellar-mass surface density grows (see section 4.4.2
for a detailed discussion).

For the SFGs with multiple episodes of star formation, we have further studied
the fractional mass of stellar populations assembled earlier, namely the mass
of the stellar populations older than $\rm{\frac{1}{3}t_{H}}$ divided by the
total stellar mass observed at $z_{\rm{obs}}$. The choice of
$\rm{\frac{1}{3}t_{H}}$ is based on results from hydrodynamical simulations,
which suggest that the characteristic timescale of galaxy compaction via
dissipative processes at the redshift considered here is roughly a constant
fraction ($0.3\sim 0.4\times$) of the Hubble Time (see section 5.2 of Zolotov
et al., 2015). We have also checked our results by using the fixed 1 Gyr, and
found no substantial changes in our results. Figure 13 shows the cumulative
distribution of the fractional mass of older stellar populations in bins of
$\Sigma_{\rm{1kpc}}$. The fractional mass increases with $\Sigma_{\rm{1kpc}}$.
Taken together with the strong, increasing trend of $\mathcal{F}_{\rm{multi-
SF}}$ with $\Sigma_{\rm{1kpc}}$, this provides strong evidence that massive
galaxies at cosmic noon assembled first their central regions which kept
growing both in mass and density as the outer regions developed. These make
the galaxies appear more compact and nucleated, and with a shrinking effective
radius. This picture is broadly consistent with conclusions reached by using
independent observables such as the radial profile of SFR obtained from
spatially-resolved maps of H$\alpha$ emission (e.g. Nelson et al., 2012,
2016).

Figure 13: The cumulative distribution of fractional mass of the older stellar
populations of the SFGs with multiple, prominent star-formation episodes. Here
the older stellar populations are referred to those with age larger than one
third of the Hubble Time at $z_{\rm{obs}}$, i.e. $\frac{1}{3}\rm{t_{H}}$. For
reference, $\frac{1}{3}\rm{t_{H}}$ corresponds to $\approx 1$ Gyr at $z=2$.
Galaxies with different $\Sigma_{\rm{1kpc}}$ are colored-coded using the same
color scheme as Figure 12 which is also labeled in the upper-left.

##  5 Discussion

Galaxies form and evolve accompanied by changes of their structural properties
which, if their evolutionary tracks can be reconstructed, contain key
empirical information on the physical drivers of structural evolution. Unlike
cosmological simulations where the evolutionary path of galaxies and the
contributions from different physical processes are known at any time, the
morphology and other properties of real galaxies, however, can only be known
at the time of observation, $z_{\rm{obs}}$. Here we propose the following
methodology to statistically reconstruct the structural evolution of
individual galaxies using their reconstructed SFHs. Because our method relies
on good morphological measures which are only available at $z\lesssim 3$ at
the moment, only galaxies with $z_{\rm{form}}$ $\leq 3$ are included to the
following analysis.

###  5.1 The Method

The method is straightforward, and it is built upon two simple basic
assumptions. Thanks to the HST’s high angular resolution, over the last two
decades significant progress has been made in measuring the morphology of
massive galaxies of all spectral types up to $z\sim 3$ (van der Wel et al.,
2014; Shibuya et al., 2015; Mowla et al., 2019, just to name a few). A common
result from these studies is that, at least for massive galaxies, structural
properties are strongly correlated with $M_{*}$ (e.g. mass-size relation, Shen
et al. 2003; van der Wel et al. 2014), star-formation activity (e.g. SFR,
Salim et al. 2007; Elbaz et al. 2007; Whitaker et al. 2012) and redshift. At
least to first order, it is thus reasonable for us to assume that the
structure of a galaxy is determined once its redshift, $M_{*}$ and SFR are
known. But we also have to keep in mind substantial, non-negligible intrinsic
scatters in such relationships exist. For example the intrinsic scatter of
galaxy’s mass-size relation has been found to be $\sim 0.2$ dex at $0<z<3$
(van der Wel et al., 2014). This suggests the possibility that additional
parameters also determine the structure of galaxies. Nonetheless, because
little information on how additional parameters drive the scatter is
available, here we ignore this issue entirely. The second assumption of the
method is that existing galaxy samples observed at any given redshift are
representative of the population of massive ($>10^{10}M_{\sun}$) galaxies at
that redshift. We do not see any strong evidence that this assumption is
grossly violated at the redshifts discussed here. We remind, however, that
given the relatively small area covered by the three legacy fields considered
in this study, the results could still be biased if significant cosmic
variance exists, which remains to be tested with future larger-area surveys.

The method includes three steps. First, for any given galaxy observed at
$z_{\rm{obs}}$ we reconstruct its SFH and we use it to measure $z_{\rm{form}}$
and from this we obtain the galaxy’s stellar mass and star-formation rate at
$z_{\rm{form}}$, i.e. $M_{*}^{\rm{z_{form}}}$ and SFR$\rm{{}^{z_{form}}}$.
This first step requires accurate multi-band photometry (section 2) which is
critical for the robust reconstruction of the SFH and thus of the evolutionary
track of individual galaxies in the plane of SFR vs. $M_{*}$. About this
point, we remind that the use of Prospector with nonparametric SFHs (Leja et
al., 2020) has essentially eliminated the long-standing tension between the
observed cosmic star-formation rate density and the cosmic stellar-mass
density, where the time integral of the former used to over-predict the latter
by $\approx 60\%$ (Madau & Dickinson, 2014). In Figure 14 we demonstrate this
point in a more direct way using our SFH reconstructions. In particular, we
use the SFHs of the UVJ-selected QGs in our sample to predict their
distribution in the SFR-$M_{*}$ diagram at $z_{\rm{form}}$, and then compare
the distribution with the star-forming main sequence measured at that
redshift. Similarly to what we did in section 3.1.1, the star-forming main
sequence used for the comparison is the one from Leja et al. (2022). As Figure
14 shows, there is excellent agreement between the prediction from the
reconstructed SFHs and the observed star-forming main sequence, demonstrating
that Prospector is able to robustly reconstruct the evolutionary tracks of
galaxies in the $M_{*}$ vs. SFR plane.

Figure 14: Predicting the star-forming main sequence using the reconstructed
SFHs of the UVJ-selected QGs. The individual evolutionary tracks are plotted
as the light grey lines. The red triangles show the $M_{*}$ and SFR of the
galaxies at $z_{\rm{obs}}$. The black circles show the $M_{*}$ and SFR of the
galaxies at $z_{\rm{form}}$ as inferred by their reconstructed SFHs. The blue
dashed line marks the star-forming main sequence of Leja et al. (2022)
measured at $z_{\rm{form}}$. Great agreement is seen between the measured and
the predicted star-forming main sequence.

The second step is to select from the existing samples those galaxies whose
($z_{\rm{obs}}$, $M_{*}$, SFR) are similar to the values of ($z_{\rm{form}}$,
$M_{*}^{\rm{z_{form}}}$, SFR$\rm{{}^{z_{form}}}$). We utilize the posteriors
from individual Prospector fittings, and select all galaxies whose
($z_{\rm{obs}}$, $M_{*}$, SFR) are within the 2$\sigma$ posterior contours of
($z_{\rm{form}}$, $M_{*}^{\rm{z_{form}}}$, SFR$\rm{{}^{z_{form}}}$). We have
also tested our results using 1$\sigma$ contours, and found no substantial
changes.

We remind that using different SED fitting procedures generally results in
systematic shifts in the measures of physical properties of galaxies (section
3.1.1). Here we use the entire sample of this work as the default, because all
physical properties are measured consistently, namely using the same SED
fitting procedure and under the same SED assumptions. However, our selection
criteria are very strict (section 2) because of our emphasis on the robustness
of SFH reconstructions, which requires high-quality, densely sampled
photometric data. Because all we need for this step are ($z_{\rm{obs}}$,
$M_{*}$, SFR), which compared to the measurement of SFHs are less sensitive to
the data quality, it is possible to use an enlarged sample to increase the
statistics at the potential cost of introducing systematics stemming from
blending measures derived from different SED fitting procedures. Specifically,
we have tested the robustness of our results using the enlarged samples of the
CANDELS/COSMOS and GOODS fields where all galaxies with reliable Galfit
fitting results have been included. Because running Prospector is
computationally quite expensive and most galaxies in the enlarged sample do
not have available Prospector fitting results, we decide to empirically
correct the existing measurements of $M_{*}$ and SFR from the enlarged CANDELS
catalogs using the median systematic shifts found between previous
measurements and the ones we obtained with Prospector (section 3.1.1 and
Figure 2). To ensure a uniform correction, for all galaxies in the enlarged
sample we use the SFRs estimated using the UV+IR ladder of Barro et al. (2019)
and then correct them by -0.7 dex (the middle panel of Figure 2), and the
median stellar mass of the CANDELS catalogs using the Hodges–Lehmann method
(Santini et al., 2015) and then correct them by +0.3 dex (the left panel of
Figure 2). As one can see in the inset of Figure 15, using the enlarged sample
does not substantially change our results. In the following, we therefore
choose the results from the sample of our own selection as the fiducial ones,
although we do also report and compare the results from the enlarged sample in
the following discussions.

The final step of our method is to compute the weighted average of the
structural properties of the galaxies selected from the step two, where we use
the corresponding posterior as the weights, and use them as the inferred
structural properties of a galaxy at its formation epoch $z_{\rm{form}}$.
Accordingly, the uncertainties of the inferred structural properties are
computed as the weighted standard deviation of the selected galaxies. By
comparing the inferred properties at $z_{\rm{form}}$ with the observed ones at
$z_{\rm{obs}}$, we can get the evolutionary track of structural properties.

Finally, before proceeding to discuss the results, we highlight the important
differences of our method from earlier studies. Previous works have attempted
to reconstruct the evolutionary trajectories of galaxies in planes defined by
observables that include morphological (structural) properties, $M_{*}$ and
star-formation properties (e.g. Barro et al., 2013, 2017), with the ultimate
goal of identifying the underlying physics that drives specific evolutionary
phases. The method described above of reconstructing the evolutionary tracks
provide another such attempt. What is different here, and provides a crucial
advantage, is that we are able to statistically estimate the true tracks of
the individual galaxies from the time of their formation, estimated by
$z_{\rm{form}}$, to that of their observation, i.e. $z_{\rm{obs}}$, thanks to
the reconstruction of SFHs without mixing together galaxies born at different
times. This difference from previous works is key and an important step
forward in the sense that we can avoid bundling together galaxies that formed
at different epochs and experienced different assembly histories. This means
that we can mitigate, in fact eliminate, any form of the progenitor effect
solely based on empirical quantities. And the population averages, which
define the general trends, are naturally obtained from the individual galaxies
as they evolve, and are not predictions from models.

###  5.2 The Structural Evolution of Galaxies Since $z_{\rm{form}}$

Figure 15: The inferred structural changes since $z_{\rm{form}}$ of individual
galaxies using the evolution of $R_{\rm{e}}$ (upper-left), $\Sigma_{e}$
(upper-right), $\Sigma_{\rm{1kpc}}$ (bottom-left) and $\rm{M_{1kpc}/M_{*}}$
(bottom-right) that we reconstruct with the SFHs (see section 5.1 for detail).
Galaxies are color coded according to $\rm{\Delta MS}$. Individual galaxies
are plotted as the light-colored, small dots. The large dots with thick error
bars show the medians and 1$\sigma$ ranges of galaxies within the individual
bins of $\rm{\Delta MS}$. In the inset of each panel, we plot the probability
density distribution of the corresponding morphological property, where the
solid line shows our fiducial results and the dashed line shows the results of
using an enlarged sample for our reconstruction of evolution of the structural
property (section 5.1). Also labeled in each panel are the Spearman
coefficient $\rho$ and the Pearson coefficient $r$ of the correlation tests
between the corresponding structural change and the $\rm{\Delta MS}$. The
inferred structural changes clearly depend on star-formation properties of
galaxies.

Figure 15 shows the change of structural parameters, namely the observed
structural properties at $z_{\rm{obs}}$ divided by the inferred ones at
$z_{\rm{form}}$, as a function of the star-formation property of individual
galaxies. Specifically, the structural changes considered are: $\rm{\Delta
logR_{e}}$, $\rm{\Delta log\Sigma_{e}}$, $\rm{\Delta log\Sigma_{1kpc}}$ and
$\rm{\Delta log(M_{1kpc}/M_{*})}$. We list the inferred median changes of
these structural properties as a function of $\rm{\Delta MS}$ in Table 1,
where we also include the results from the enlarged sample as well. Also
listed in Table 1 are the uncertainties of the median values that we estimate
by (1) bootstrapping the sample 1000 times and (2) during each bootstrapping
using a normal distribution to resample the value of inferred structural
properties at $z_{\rm{form}}$ with the estimated uncertainties that we
described in the final step in section 5.1.

$\rm{\Delta MS}$ | $\rm{\Delta log\,R_{e}}$ | $\rm{\Delta log\,\Sigma_{e}}$ | $\rm{\Delta log\,\Sigma_{1kpc}}$ | $\rm{\Delta log\,(M_{1kpc}/M_{*})}$
---|---|---|---|---
$>1$ |  $0.11\pm 0.03$ ($0.17\pm 0.03$) |  $-0.07\pm 0.04$ ($0.08\pm 0.05$) |  $-0.01\pm 0.04$ ($0.21\pm 0.03$) |  $-0.27\pm 0.04$ ($-0.40\pm 0.03$)
$0.1-1$ |  $0.02\pm 0.02$ ($0.07\pm 0.02$) |  $0.15\pm 0.05$ ($0.22\pm 0.05$) |  $0.24\pm 0.04$ ($0.36\pm 0.03$) |  $-0.08\pm 0.03$ ($-0.19\pm 0.03$)
$0.01-0.1$ |  $-0.19\pm 0.05$ ($-0.11\pm 0.04$) |  $0.55\pm 0.09$ ($0.65\pm 0.07$) |  $0.48\pm 0.05$ ($0.65\pm 0.05$) |  $0.14\pm 0.05$ ($0.05\pm 0.04$)
$<0.01$ |  $-0.34\pm 0.04$ ($-0.28\pm 0.04$) |  $0.91\pm 0.08$ ($0.93\pm 0.07$) |  $0.72\pm 0.05$ ($0.76\pm 0.04$) |  $0.29\pm 0.04$ ($0.21\pm 0.04$)
Table 1: The median inferred structural changes since $z_{\rm{form}}$ for
galaxies with different $\rm{\Delta MS}$ (also see Figure 15). The values in
the parentheses are the results of using the enlarged sample for the
reconstruction of structural evolution (section 5.1). The errors represent the
uncertainties of the median.

A clear trend, with an absolute correlation coefficient $\sim 0.5$ of both the
Spearman’s rank and the Pearson tests, of the structural evolution since
$z_{\rm{form}}$ with $\rm{\Delta MS}$ is observed for all parameters
considered. The $\rm{\Delta logR_{e}}$ increases with the $\rm{\Delta MS}$.
Galaxies observed to have on-going star formation (i.e. large $\rm{\Delta
MS}$) grew in size since $z_{\rm{form}}$. For galaxies with suppressed star
formation we see that $R_{\rm{e}}$ decreases and the decrease continuously
becomes more prominent as $\rm{\Delta MS}$ becomes smaller, i.e. the galaxies
become more quiescent: galaxies do shrink as they approach quiescence, at
least when the size is measured by $R_{\rm{e}}$.

In general, $\rm{\Delta log\Sigma_{e}}$, $\rm{\Delta log\Sigma_{1kpc}}$ and
$\rm{\Delta log(M_{1kpc}/M_{*})}$ decrease with increasing $\rm{\Delta MS}$.
One the one hand, for galaxies above the star-forming main sequence, i.e.
$\rm{\Delta MS}$ $>1$, $\Sigma_{e}$ does not significantly evolve since
$z_{\rm{form}}$. Depending on the exact sample selection (the step two
described in section 5.1), $\Sigma_{\rm{1kpc}}$ either remains constant using
the reference sample adopted in this work, or it exhibits a very mild increase
of $\approx 0.2$ dex if using the enlarged sample. On the other hand, since
$z_{\rm{form}}$, for galaxies with very little or no on-going star formation
by the time of $z_{\rm{obs}}$, $\Sigma_{e}$ and $\Sigma_{\rm{1kpc}}$
significantly increase by $\approx 0.9$ dex and $\approx 0.7$ dex,
respectively, regardless of the adopted sample. More importantly, as Figure 16
shows, we find that the inferred evolution of $\Sigma_{\rm{1kpc}}$ is larger
than that of $\Sigma_{e}$ for galaxies with $\rm{\Delta MS}$ $>0.1$, while for
those with $\rm{\Delta MS}$ $<0.1$ the opposite is found, namely the
$\rm{\Delta log\Sigma_{e}}$ is larger than the $\rm{\Delta log\Sigma_{1kpc}}$
and the difference, $\rm{\Delta log\Sigma_{1kpc}-\Delta log\Sigma_{e}}$,
becomes more negative as the galaxies become more quiescent.

The findings above have two important implications. First, galaxies grow in
mass preferentially in the central regions as they are still in the star-
forming phase, which explains $\rm{\Delta log\Sigma_{1kpc}\gtrsim\Delta
log\Sigma_{e}}$ that we find for $\rm{\Delta MS}$ $>0.1$ galaxies. This is
consistent with what we found earlier in section 4.4.2, namely that compact
SFGs are not only more likely to have a sizeable presence of older stellar
populations (Figure 12) but also their fractional mass is larger (Figure 13)
than extended SFGs. We notice this conclusion is also consistent with that
reached by Barro et al. (2017) (see their section 4) which, however, is based
upon totally different assumptions. Barro et al. showed that the scatter of
the relationship between $\Sigma$ and $M_{*}$ is smaller than the expected
growth in mass within the redshift intervals covered in their analysis.
Because of this, they argued that it was reasonable to assume the observed
$\Sigma$-$M_{*}$ relationship as a proxy for the evolutionary track of SFGs.
Because the observed slope of the $\Sigma_{\rm{1kpc}}$-$M_{*}$ relationship is
steeper than that of the $\Sigma_{e}$-$M_{*}$ relationship, Barro et al.
concluded that high-redshift galaxies preferentially build up their centeral
regions within 1 kpc. Our approach also remains entirely empirical, and
includes the substantial improvement of getting rid of the key assumption of
Barro et al. (2017). Our approach uses the SFHs to statistically reconstruct
the structural evolution of individual galaxies, including their central
densities, from the data alone. It also avoids the progenitor effect stemming
by avoiding to bundle together galaxies with different formation histories,
which adds a spurious contribution to the intrinsic structural evolution, as
we already discussed in section 4.1.

The second important implication is that the growth in central stellar-mass
surface density after galaxies go through their last major episode of star
formation, i.e. after they go through quenching, is slower within the central
radius of 1 kpc than within $R_{\rm{e}}$. We interpret this as a consequence
of two effects. First, as they are still in the star-forming phase galaxies
continuously grow their central stellar mass densities. When the quenching
starts, the central density, e.g. $\Sigma_{\rm{1kpc}}$, is about to reach the
maximum value (i.e. the knee of the pistol pattern seen in Figure 7), and its
growth rate slows down. Meanwhile, after star formation quenches in the
center, galaxies can still keep growing their masses and sizes in part through
circum-nuclear, in-situ star formation, and in part through galaxy mergers
(e.g. Bezanson et al., 2009; van Dokkum et al., 2010; Oser et al., 2012; Ji &
Giavalisco, 2022). However, unlike the growth of the central regions which is
likely fed by dissipative accretion of gas (see next section), this after-
quenching growth also includes, and very likely is dominated by less
dissipative processes such as minor mergers that are more likely to affect the
outer regions. These two effects combined can lead to the observed $\rm{\Delta
log\Sigma_{1kpc}\lesssim\Delta log\Sigma_{e}}$ that we find for the galaxies
with small $\rm{\Delta MS}$.

Figure 16: The difference between the inferred changes of $\Sigma_{\rm{1kpc}}$
and of $\Sigma_{e}$, i.e. $\rm{\Delta log\Sigma_{1kpc}-\Delta log\Sigma_{e}}$.
If it equals to zero (the black horizontal dashed line), this means the
changes of $\Sigma_{\rm{1kpc}}$ and of $\Sigma_{e}$ are the same. The more
negative the value is, the less growth took place within the central radius of
1 kpc compared to that within the $R_{\rm{e}}$ since $z_{\rm{form}}$. The
dark-colored circles show the result of using the sample of our own selection
for the reconstruction of the structural evolution, while the light-colored
circles show the results of using the enlarged sample. The plot shows that,
relative to the regions within $R_{\rm{e}}$, galaxies preferentially grow in
mass density in the central regions of 1 kpc when they are still in star-
forming phase, while the growth slows down as they become quiescent.

###  5.3 On the Co-happening of Structural Transformations and Quenching

The physics behind the empirically well-established correlations between
galaxy structural and star-formation properties has long been eluded us.
Critical, but yet open questions are whether a causal link exists between
structural transformations and quenching, or whether they both are the results
of a third, unidentified process, and what the relative timing of the two
phenomena is (e.g. Khochfar & Silk, 2006; Wellons et al., 2015; Zolotov et
al., 2015; Tacchella et al., 2016; Lilly & Carollo, 2016). To empirically
disentangle the timing sequence and also to investigate the possibility of a
direct causal link between quenching and structural transformations from other
possibilities, the first key step is to eliminate any contribution from the
progenitor effect (Lilly & Carollo, 2016; Ji et al., 2022), as we already
explained in section 1, namely to select galaxies with similar $z_{\rm{form}}$
and assembly histories.

So far, we have used the SFHs to separate galaxies formed at different epochs
and having different assembly histories. We have shown that, while the
progenitor effect clearly exists and is observed (section 4.1), it alone is
not enough to explain the observed correlations among $\rm{\Delta MS}$,
$\Sigma_{\rm{1kpc}}$, $\Sigma_{e}$, $\rm{M_{1kpc}/M_{*}}$ and $R_{\rm{e}}$
(section 4.2 and 5.2), pointing to the possibility of a physical link between
quenching and structural transformations. Our findings are in broad agreement
with the theoretical study of Wellons et al. (2015), where they found that
both the physical compaction of galaxies and the progenitor effect are
responsible for the formation of compact QGs in the Illustris cosmological
simulations.

Generally speaking, galaxies at cosmic noon have much larger gas fractions
than local ones, around 50% and reaching up to $\approx 80$% (Tacconi et al.,
2020), and the rate of gas inflow is also higher (e.g. Dekel et al., 2013;
Scoville et al., 2017). As the result of direct accretion, dynamical
instabilities or wet mergers, inflowing gas would sink to the center of
gravitational potential, trigger central starbursts and build up dense central
regions of galaxies, a process often referred to as wet compaction. Recent
ALMA observations of galaxies at this epoch indeed found that the distribution
of cold gas is more compact than that of stars (e.g. Barro et al., 2016;
Spilker et al., 2016; Tadaki et al., 2017; Kaasinen et al., 2020; Gómez-
Guijarro et al., 2022), supporting the notion that high-redshift galaxies
build up their centers through dissipative processes associated with cold gas,
and hence adding empirical support to the scenario of wet compaction.

Figure 17: The inferred evolutionary vectors of the structure of individual
galaxies from $z_{\rm{form}}$ to $z_{\rm{obs}}$. We use the same color scheme
as Figure 15 to divide the galaxies into four groups according to their
$\rm{\Delta MS}$. The evolutionary path of each galaxy is plotted as a light-
colored arrow that starts from its inferred structural properties at
$z_{\rm{form}}$ and ends at its observed properties at $z_{\rm{obs}}$. The
large, thick arrows show the median structural evolution of the groups of
galaxies with different $\rm{\Delta MS}$. In the panels of $\Sigma_{e}$ and
$\Sigma_{\rm{1kpc}}$, we also use the black arrow to show the inferred
structural evolution due to the highly dissipative process, i.e. wet
compaction, from Zolotov et al. (2015) and Barro et al. (2017). We see great
agreement between our purely empirical constraints of the structural
transformation of galaxies and the predictions of wet compaction made by
hydrodynamical simulations.

Thanks to the reconstruction of the structural evolution of individual
galaxies that we have described in section 5.1 and 5.2, we are able to
empirically derive their trajectories, from $z_{\rm{form}}$ to $z_{\rm{obs}}$,
in the planes of $M_{*}$ vs. structural property proxies, which are shown in
Figure 17. While there is a relatively large dispersion among the individual
evolutionary vectors, the median trajectories gradually change as a function
of $\rm{\Delta MS}$ toward the direction expected from wet compaction
processes and they are consistent with what we found in section 5.2 and Figure
15. The slope of the evolutionary vectors, which shows the relationship
between the growth in stellar mass since $z_{\rm{form}}$, i.e. $\rm{\Delta
logM_{*}}$, and the change of structural properties, can be used to
quantitatively compare with the predictions of theories. Intriguingly, Figure
17 shows that for galaxies with $\rm{\Delta MS}$ $<0.1$ our reconstructed
median evolutionary trajectory of the central density, both $\Sigma_{e}$ (top-
right panel) and $\Sigma_{\rm{1kpc}}$ (bottom-left panel), closely mirrors the
one predicted by the simulations of wet compaction (Zolotov et al., 2015). We
show this in a more quantitative way in Figure 18, where we plot the
distributions of the slope of the inferred evolutionary vectors, i.e.
$\rm{\Delta log\Sigma_{e}/\Delta logM_{*}}$ and $\rm{\Delta
log\Sigma_{1kpc}/\Delta logM_{*}}$, of galaxies with $\rm{\Delta MS}$ $<0.1$.
To estimate the uncertainties of the distributions, similarly to what we did
in section 5.2, we bootstrap the sample 1000 times and during each
bootstrapping we also resample the individual values using the measurement
uncertainties. As Figure 18 shows, the median evolutionary slope measured by
our method is in quantitative agreement with that predicted by the
hydrodynamical simulations of wet compaction.

Figure 18: The distributions of $\rm{\Delta log\Sigma_{e}/\Delta logM_{*}}$
(upper) and $\rm{\Delta log\Sigma_{1kpc}/\Delta logM_{*}}$ (bottom), i.e. the
slope of the evolutionary vector of Figure 17, for the galaxies with
$\rm{\Delta MS}$ $<0.01$ (magenta) and $\rm{\Delta MS}$ $=0.01-0.1$ (red). The
dashed curves show the results of using the enlarged sample for the
reconstruction of the structural evolution, and the solid curves show the
results of using the sample of our own selection. The medians and their
1$\sigma$ uncertainties of the 1000-time realizations are marked as the
horizontal bars (see section 5.3 for details). The vertical, black lines show
the slopes predicted by the simulations of wet compaction, i.e. the same as
those in Figure 17. The plots show quantitative agreement between our
empirical measurements and the predictions of hydrodynamical simulations of
wet compaction.

Although the consistency between our empirical constraints and the simulations
shows wet compaction is a viable mechanism, however, we stress that we do not
know what exact process(es) is responsible for the compaction. Physically, wet
compaction can be triggered by several physical processes, including galaxy
mergers (Hopkins et al., 2006), counter-rotating streams (Danovich et al.,
2015), recycled gas with low angular momentum (Dekel & Burkert, 2014) and
violent disc instabilities (Dekel et al., 2009). A common feature of all these
mechanisms, however, is that once the inflow of gas ends, it is the central
regions that first become gas-depleted, halting further stellar mass growth
(e.g. as traced by $\Sigma_{\rm{1kpc}}$) leading to inside-out quenching
(Ceverino et al., 2015; Zolotov et al., 2015; Tacchella et al., 2016). This
scenario provides a broad explanation for the correlations observed among
$M_{*}$, $\Sigma$ and SFR (Barro et al., 2013, 2016, 2017). It also is in good
quantitative agreement with what we found in Figure 16.

In conclusion, with the progenitor effect accounted for, the findings
described here suggest a direct connection, at least in the form of
simultaneous happening, of the process of quenching and the structural
transformations. This temporal coincidence does not necessarily imply a causal
link between these two phenomena. We suggest that this link is provided by the
mechanisms that first triggers gas accretion into the central regions, with
subsequent star formation and then shuts it down, eventually resulting in
quenching. The accreted mass is large, and and it becomes compact (likely due
to dissipative gaseous accretion), changing the dynamical state of the central
region (Ji & Giavalisco 2022, in preparation), and it also causes adiabatic
contraction (Giavalisco & Ji 2022, in preparation). As we have seen, the
magnitude of the structural transformations that appear to take place is
quantitatively consistent with the predictions from the wet compaction models,
where this terminology is generically representative of a family of processes
of growth of the central mass of galaxies through dissipative gaseous
accretion. Regardless of the exact physical mechanism (or mechanisms) behind
compaction, one important conclusion of our study is that quenching takes
place at the same time as the the formation of a compact, dense central core,
which reaches its maximum density as star formation terminates.

##  6 Caveats

We now explicitly address a few caveats of this study. First, the current
nonparametric SFH reconstruction requires the assumption of a prior about how
the SFR changes as a function of time on small timescales to enable the SED
fitting to converge. Using different priors can lead to systematic shifts of
the measurement of physical parameters. We have run a number of tests to
constrain the magnitude of this uncertainty. As we already mentioned in
section 3.1 and in Paper I, however, unfortunately at the moment we do not
know what the optimal prior is, a situation that can hopefully be improved
with future large, spectroscopic surveys at high redshifts. Luckily, as one
can see in Appendix A, tight correlations are observed between the parameters
of interest to this study obtained from different priors. Therefore, there is
evidence that the results of this work should not be overly sensitive to the
assumed nonparametric SFH priors. Second, throughout the paper we used
$z_{\rm{form}}$ as the proxy for the formation epoch of galaxies. This is
purely from an empirical stand point, but there could be other better
characteristic timescales (redshifts) to use. For example, in Paper I we
argued that the best characteristic redshift to mitigate the progenitor effect
should be the one that minimizes the scatter of the distribution of
$\Sigma_{\rm{1kpc}}$ (see section 3.3 of Paper I). Perhaps due to the
relatively small sample size and large scatter (both intrinsic and introduced
by the current measurements), the best characteristic redshift to use remains
inconclusive, and we plan to investigate this aspect further in the future.
Third, in this work SED modeling was done with the integrated photometry, and
we ignored possible color gradients inside the galaxies. Such gradients can
make the light distribution different from the mass distribution, hence
introduce systematics in our interpretations on the evolution of morphological
properties. We plan to expand this study to include spatially resolved
measurements using upcoming data from JWST. Finally, we remind that we ignore
entirely the environmental effect on galaxy evolution in our interpretation of
the observations, because in this work we only focus on massive galaxies whose
evolution should be primarily driven by internal processes more than the
external environment based on current observations, as mentioned in section 1.
That said, however, a detailed understanding of when and how the environmental
effect on galaxy evolution becomes important remains incomplete. A few recent
studies found evidence that the dense environment might even be able to affect
the evolution of very massive galaxies ($\log_{10}(M_{*}/M_{\sun})\geq 10.5$)
at cosmic noon by altering their molecular gas content (e.g., Wang et al.,
2018; Zavala et al., 2019). However, these studies were based on a very
limited number of known galaxy (proto-)clusters at $z>2$, statistical samples
are still required to confirm these findings. Therefore, we cannot exclude
that the environmental effect might still contribute to some part of the
apparent structural evolution described here, and this will need further
investigations in the future.

##  7 Summary

In this work we studied the relationships among the structural and star-
formation properties, and mass assembly history of a carefully characterized
sample of galaxies, with stellar mass $\log_{10}(M_{*}/M_{\sun})\geq 10.3$, at
cosmic noon epoch $z_{\rm{obs}}$ $\sim 2$. We have reconstructed high-
fidelity, nonparametric SFHs of these galaxies using the fully Bayesian SED
fitting code Prospector. We found strong correlations between the structural
properties of galaxies and their assembly history. Specifically,

  * •

for QGs (section 4.4.1), we found that the SFH of compact QGs, i.e. those
having larger $\Sigma_{\rm{1kpc}}$, $\Sigma_{e}$ and $\rm{M_{1kpc}/M_{*}}$, is
similar to that of the average QG who has assembled most of its mass through
an old burst of star formation and then its SFR gradually declined toward the
quenching phase. The SFH of extended QGs, however, is very similar to the SFH
of post-starburst galaxies, namely galaxies that have just experienced a
recent major, massive burst of star formation $<1$ Gyr ago. Their morphology
is also more disturbed compared to the compact ones.

  * •

for SFGs (section 4.4.2), we found that as they become more compact, their
SFHs are more likely to show multiple, prominent star-forming episodes. In
addition, among those with multiple star-formation episodes, we found the
fractional mass formed during the older episode, typically with an age $>1$
Gyr, also increases as the SFGs become more compact.

With the availability of SFHs, we were able to separate galaxies formed at
different epochs and with different assembly histories, hence to mitigate any
form of the progenitor effect (e.g. the dependence of the size and central
mass density on the epoch of formation) which otherwise biases the observed
correlations between galaxies’ structures and star-formation activities. We
showed that

  * •

the progenitor effect is clearly observed in the sample galaxies discussed
here (section 4.1). We showed that galaxies that formed earlier, i.e. with
larger $z_{\rm{form}}$, tend to have smaller sizes and larger densities.

  * •

once the formation epoch ($z_{\rm{form}}$) of the galaxies is controlled and
accounted for, the distributions of galaxies in the diagrams of $\rm{\Delta
MS}$, i.e. the distance from the star-forming main sequence, vs. various
compactness metrics, including $\Sigma_{\rm{1kpc}}$, $\Sigma_{e}$ and
$\rm{M_{1kpc}/M_{*}}$, still exhibit a well-known pistol pattern (section
4.2). Namely, as galaxies become quiescent they also become more compact, and
the distributions of the compactness metrics also become narrower. These
suggest the progenitor effect alone is not enough to explain the apparent
correlations between the structural and star-formation properties of galaxies
at cosmic noon.

Finally, we introduce a novel, purely empirical approach, which exploits the
SFHs to reconstruct the evolution of structural properties of individual
galaxies (section 5.1). Differently from earlier studies, our method naturally
takes the assembly history of galaxies into account, and the population
averages, which define the general trends, are obtained from the individual
galaxies as they evolve, and are not predictions from models. We showed that
since $z_{\rm{form}}$

  * •

the change of structural properties are strongly correlated with $\rm{\Delta
MS}$ (section 5.2). In particular, for SFGs, i.e. galaxies with large
$\rm{\Delta MS}$, we found that they grew in size, and their compactness
($\Sigma_{\rm{1kpc}}$, $\Sigma_{e}$ and $\rm{M_{1kpc}/M_{*}}$) is consistent
either with remaining constant or with a slight increase. QGs, i.e. those with
small $\rm{\Delta MS}$, exhibit a significant decrease in their $R_{\rm{e}}$
and increase in their compactness. Importantly, on the one hand, we found the
change of $\Sigma_{e}$ is smaller than that of $\Sigma_{\rm{1kpc}}$ for SFGs,
suggesting galaxies preferentially grow the central regions as they are still
in the star-forming phase. On the other hand, an opposite trend, namely the
change of $\Sigma_{e}$ is larger than that of $\Sigma_{\rm{1kpc}}$, was found
for QGs, suggesting that galaxies, after they go through quenching, grow more
slowly in their central radius of 1 kpc than they do within $R_{\rm{e}}$.

  * •

the evolutionary vectors in the diagrams of stellar mass vs. structural
properties enable us to compare with the predictions from cosmological
simulations (section 5.3). Our reconstructed evolutions of $\Sigma_{e}$ and of
$\Sigma_{\rm{1kpc}}$ for QGs are in quantitative agreement with the
predictions of wet compaction made by simulations.

Combining all these results together, we converge to a consistent picture
where the quenching of star formation in massive galaxies and their structural
transformations in the form of decreasing $R_{\rm{e}}$ and increasing central
stellar-mass density take place roughly at the same time. To see this, first
the progenitor effect must be accounted for when interpreting any correlation
between the structural and star-formation properties. Second, the progenitor
effect can only be partly responsible for the observed evolution, implying the
existence of some physical link between galaxies’ structural transformations
and quenching. Our reconstructed structural evolution suggests that the wet
compaction is one viable such link that helps build up the central dense
regions of galaxies via highly dissipative gaseous accretion at cosmic noon
epoch. After galaxies go through their final major episode of star formation
and eventually become quiescent, the growth of their central regions (i.e.
$<1$ kpc) slows down, while galaxy mergers can continue growing their
outskirts.

##  8 ACKNOWLEDGMENT

We thank the anonymous referee for their useful comments. This work was
completed in part with resources provided by the University of Massachusetts’
Green High Performance Computing Cluster (GHPCC).

##  Appendix A Comparison of the physical parameters derived using the
Dirichlet prior and the Continuity prior

In Figure 19 we compare the physical properties derived using the Dirichlet
prior (the fiducial one in the main text) with the ones derived using the
Continuity prior which unlike the former is strongly against the sharp change
of SFR in adjacent lookback time bins. The comparison is not only for the
basic properties such as $M_{*}$ and SFR, but also the parameters that we
introduced in Paper I to quantify the shape of reconstructed SFHs. Systematic
offsets and scatter notwithstanding, for all parameters we see strong
correlations between the two measurements, suggesting that our results of this
work should qualitatively not be sensitive to the prior assumed.

Figure 19: The comparisons of parameters derived using the Dirichlet
(horizontal axis) and the Continuity (vertical axis) priors. Black dots with
error bars show the individual measurements. In each panel, the cyan solid
line with shaded region mark the median relationship and 1$\sigma$ range. The
red dashed line shows the one-to-one relationship. The subpanel at the bottom
shows the relative differences between the two measurements, i.e. (Continuity
- Dirichlet) / Dirichlet.

##  Appendix B The stacked SFH of individual galaxy groups divided using
$\Sigma_{e}$ and $\rm{M_{1kpc}/M_{*}}$

In section 4.4 of the main text, we use $\Sigma_{\rm{1kpc}}$ to separate
compact and extended galaxies. Here we use $\Sigma_{e}$ and
$\rm{M_{1kpc}/M_{*}}$ to do the separation, and then stack the SFHs of the
individual groups of galaxies. The results are shown in Figure 20 where no
substantial change is found to our results.

Figure 20: Similar to Figure 9, but the extended and compact galaxies are
classified according to $\rm{M_{1kpc}/M_{*}}$ (the top row) and $\Sigma_{e}$
(the bottom row), respectively.

## References

  * Baldry et al. (2004) Baldry, I. K., Glazebrook, K., Brinkmann, J., et al. 2004, ApJ, 600, 681, doi: 10.1086/380092
  * Barro et al. (2013) Barro, G., Faber, S. M., Pérez-González, P. G., et al. 2013, ApJ, 765, 104, doi: 10.1088/0004-637X/765/2/104
  * Barro et al. (2016) Barro, G., Kriek, M., Pérez-González, P. G., et al. 2016, ApJ, 827, L32, doi: 10.3847/2041-8205/827/2/L32
  * Barro et al. (2017) Barro, G., Faber, S. M., Koo, D. C., et al. 2017, ApJ, 840, 47, doi: 10.3847/1538-4357/aa6b05
  * Barro et al. (2019) Barro, G., Pérez-González, P. G., Cava, A., et al. 2019, ApJS, 243, 22, doi: 10.3847/1538-4365/ab23f2
  * Bell et al. (2004) Bell, E. F., Wolf, C., Meisenheimer, K., et al. 2004, ApJ, 608, 752, doi: 10.1086/420778
  * Belli et al. (2019) Belli, S., Newman, A. B., & Ellis, R. S. 2019, ApJ, 874, 17, doi: 10.3847/1538-4357/ab07af
  * Belli et al. (2017) Belli, S., Genzel, R., Förster Schreiber, N. M., et al. 2017, ApJ, 841, L6, doi: 10.3847/2041-8213/aa70e5
  * Bertin & Arnouts (1996) Bertin, E., & Arnouts, S. 1996, A&AS, 117, 393, doi: 10.1051/aas:1996164
  * Bezanson et al. (2009) Bezanson, R., van Dokkum, P. G., Tal, T., et al. 2009, ApJ, 697, 1290, doi: 10.1088/0004-637X/697/2/1290
  * Brammer et al. (2009) Brammer, G. B., Whitaker, K. E., van Dokkum, P. G., et al. 2009, ApJ, 706, L173, doi: 10.1088/0004-637X/706/1/L173
  * Bruce et al. (2012) Bruce, V. A., Dunlop, J. S., Cirasuolo, M., et al. 2012, MNRAS, 427, 1666, doi: 10.1111/j.1365-2966.2012.22087.x
  * Bundy et al. (2010) Bundy, K., Scarlata, C., Carollo, C. M., et al. 2010, ApJ, 719, 1969, doi: 10.1088/0004-637X/719/2/1969
  * Byler et al. (2017) Byler, N., Dalcanton, J. J., Conroy, C., & Johnson, B. D. 2017, ApJ, 840, 44, doi: 10.3847/1538-4357/aa6c66
  * Calzetti et al. (2000) Calzetti, D., Armus, L., Bohlin, R. C., et al. 2000, ApJ, 533, 682, doi: 10.1086/308692
  * Cappellari (2013) Cappellari, M. 2013, ApJ, 778, L2, doi: 10.1088/2041-8205/778/1/L2
  * Cardamone et al. (2010) Cardamone, C. N., van Dokkum, P. G., Urry, C. M., et al. 2010, ApJS, 189, 270, doi: 10.1088/0067-0049/189/2/270
  * Carnall et al. (2019) Carnall, A. C., Leja, J., Johnson, B. D., et al. 2019, ApJ, 873, 44, doi: 10.3847/1538-4357/ab04a2
  * Carnall et al. (2018) Carnall, A. C., McLure, R. J., Dunlop, J. S., & Davé, R. 2018, MNRAS, 480, 4379, doi: 10.1093/mnras/sty2169
  * Ceverino et al. (2010) Ceverino, D., Dekel, A., & Bournaud, F. 2010, MNRAS, 404, 2151, doi: 10.1111/j.1365-2966.2010.16433.x
  * Ceverino et al. (2015) Ceverino, D., Dekel, A., Tweed, D., & Primack, J. 2015, MNRAS, 447, 3291, doi: 10.1093/mnras/stu2694
  * Chang et al. (2013) Chang, Y.-Y., van der Wel, A., Rix, H.-W., et al. 2013, ApJ, 773, 149, doi: 10.1088/0004-637X/773/2/149
  * Chauke et al. (2019) Chauke, P., van der Wel, A., Pacifici, C., et al. 2019, ApJ, 877, 48, doi: 10.3847/1538-4357/ab164d
  * Cheung et al. (2012) Cheung, E., Faber, S. M., Koo, D. C., et al. 2012, ApJ, 760, 131, doi: 10.1088/0004-637X/760/2/131
  * Choi et al. (2016) Choi, J., Dotter, A., Conroy, C., et al. 2016, ApJ, 823, 102, doi: 10.3847/0004-637X/823/2/102
  * Ciesla et al. (2016) Ciesla, L., Boselli, A., Elbaz, D., et al. 2016, A&A, 585, A43, doi: 10.1051/0004-6361/201527107
  * Conroy (2013) Conroy, C. 2013, ARA&A, 51, 393, doi: 10.1146/annurev-astro-082812-141017
  * Conroy & Gunn (2010) Conroy, C., & Gunn, J. E. 2010, ApJ, 712, 833, doi: 10.1088/0004-637X/712/2/833
  * Conroy et al. (2009) Conroy, C., Gunn, J. E., & White, M. 2009, ApJ, 699, 486, doi: 10.1088/0004-637X/699/1/486
  * Danovich et al. (2015) Danovich, M., Dekel, A., Hahn, O., Ceverino, D., & Primack, J. 2015, MNRAS, 449, 2087, doi: 10.1093/mnras/stv270
  * Dekel & Burkert (2014) Dekel, A., & Burkert, A. 2014, MNRAS, 438, 1870, doi: 10.1093/mnras/stt2331
  * Dekel et al. (2009) Dekel, A., Sari, R., & Ceverino, D. 2009, ApJ, 703, 785, doi: 10.1088/0004-637X/703/1/785
  * Dekel et al. (2013) Dekel, A., Zolotov, A., Tweed, D., et al. 2013, MNRAS, 435, 999, doi: 10.1093/mnras/stt1338
  * Dotter (2016) Dotter, A. 2016, ApJS, 222, 8, doi: 10.3847/0067-0049/222/1/8
  * Elbaz et al. (2007) Elbaz, D., Daddi, E., Le Borgne, D., et al. 2007, A&A, 468, 33, doi: 10.1051/0004-6361:20077525
  * Elbaz et al. (2011) Elbaz, D., Dickinson, M., Hwang, H. S., et al. 2011, A&A, 533, A119, doi: 10.1051/0004-6361/201117239
  * Elmegreen et al. (2007) Elmegreen, D. M., Elmegreen, B. G., Ravindranath, S., & Coe, D. A. 2007, ApJ, 658, 763, doi: 10.1086/511667
  * Estrada-Carpenter et al. (2020) Estrada-Carpenter, V., Papovich, C., Momcheva, I., et al. 2020, ApJ, 898, 171, doi: 10.3847/1538-4357/aba004
  * Falcón-Barroso et al. (2011) Falcón-Barroso, J., Sánchez-Blázquez, P., Vazdekis, A., et al. 2011, A&A, 532, A95, doi: 10.1051/0004-6361/201116842
  * Fang et al. (2013) Fang, J. J., Faber, S. M., Koo, D. C., & Dekel, A. 2013, ApJ, 776, 63, doi: 10.1088/0004-637X/776/1/63
  * Förster Schreiber et al. (2011) Förster Schreiber, N. M., Shapley, A. E., Genzel, R., et al. 2011, ApJ, 739, 45, doi: 10.1088/0004-637X/739/1/45
  * Franx et al. (2008) Franx, M., van Dokkum, P. G., Förster Schreiber, N. M., et al. 2008, ApJ, 688, 770, doi: 10.1086/592431
  * Genzel et al. (2008) Genzel, R., Burkert, A., Bouché, N., et al. 2008, ApJ, 687, 59, doi: 10.1086/591840
  * Gómez-Guijarro et al. (2022) Gómez-Guijarro, C., Elbaz, D., Xiao, M., et al. 2022, A&A, 658, A43, doi: 10.1051/0004-6361/202141615
  * Grogin et al. (2011) Grogin, N. A., Kocevski, D. D., Faber, S. M., et al. 2011, ApJS, 197, 35, doi: 10.1088/0067-0049/197/2/35
  * Guo et al. (2012) Guo, Y., Giavalisco, M., Ferguson, H. C., Cassata, P., & Koekemoer, A. M. 2012, ApJ, 757, 120, doi: 10.1088/0004-637X/757/2/120
  * Guo et al. (2013) Guo, Y., Ferguson, H. C., Giavalisco, M., et al. 2013, ApJS, 207, 24, doi: 10.1088/0067-0049/207/2/24
  * Guo et al. (2015) Guo, Y., Ferguson, H. C., Bell, E. F., et al. 2015, ApJ, 800, 39, doi: 10.1088/0004-637X/800/1/39
  * Guo et al. (2017) Guo, Y., Bell, E. F., Lu, Y., et al. 2017, ApJ, 841, L22, doi: 10.3847/2041-8213/aa70e9
  * Hopkins et al. (2008) Hopkins, P. F., Cox, T. J., Kereš, D., & Hernquist, L. 2008, ApJS, 175, 390, doi: 10.1086/524363
  * Hopkins et al. (2006) Hopkins, P. F., Hernquist, L., Cox, T. J., et al. 2006, ApJS, 163, 1, doi: 10.1086/499298
  * Hopkins et al. (2010) Hopkins, P. F., Bundy, K., Croton, D., et al. 2010, ApJ, 715, 202, doi: 10.1088/0004-637X/715/1/202
  * Huang et al. (2013) Huang, S., Ho, L. C., Peng, C. Y., Li, Z.-Y., & Barth, A. J. 2013, ApJ, 768, L28, doi: 10.1088/2041-8205/768/2/L28
  * Huertas-Company et al. (2013) Huertas-Company, M., Mei, S., Shankar, F., et al. 2013, MNRAS, 428, 1715, doi: 10.1093/mnras/sts150
  * Huertas-Company et al. (2015) Huertas-Company, M., Gravet, R., Cabrera-Vives, G., et al. 2015, ApJS, 221, 8, doi: 10.1088/0067-0049/221/1/8
  * Inoue et al. (2016) Inoue, S., Dekel, A., Mandelker, N., et al. 2016, MNRAS, 456, 2052, doi: 10.1093/mnras/stv2793
  * Iyer et al. (2019) Iyer, K. G., Gawiser, E., Faber, S. M., et al. 2019, ApJ, 879, 116, doi: 10.3847/1538-4357/ab2052
  * Ji & Giavalisco (2022) Ji, Z., & Giavalisco, M. 2022, arXiv e-prints, arXiv:2204.02414.  https://arxiv.org/abs/2204.02414
  * Ji et al. (2022) Ji, Z., Giavalisco, M., Kirkpatrick, A., et al. 2022, ApJ, 925, 74, doi: 10.3847/1538-4357/ac3837
  * Ji et al. (2018) Ji, Z., Giavalisco, M., Williams, C. C., et al. 2018, ApJ, 862, 135, doi: 10.3847/1538-4357/aacc2c
  * Johnson et al. (2021) Johnson, B. D., Leja, J., Conroy, C., & Speagle, J. S. 2021, ApJS, 254, 22, doi: 10.3847/1538-4365/abef67
  * Kaasinen et al. (2020) Kaasinen, M., Walter, F., Novak, M., et al. 2020, ApJ, 899, 37, doi: 10.3847/1538-4357/aba438
  * Kauffmann et al. (2003) Kauffmann, G., Heckman, T. M., White, S. D. M., et al. 2003, MNRAS, 341, 54, doi: 10.1046/j.1365-8711.2003.06292.x
  * Kawinwanichakij et al. (2017) Kawinwanichakij, L., Papovich, C., Quadri, R. F., et al. 2017, ApJ, 847, 134, doi: 10.3847/1538-4357/aa8b75
  * Kennicutt (1998) Kennicutt, Robert C., J. 1998, ARA&A, 36, 189, doi: 10.1146/annurev.astro.36.1.189
  * Khochfar & Silk (2006) Khochfar, S., & Silk, J. 2006, ApJ, 648, L21, doi: 10.1086/507768
  * Koekemoer et al. (2011) Koekemoer, A. M., Faber, S. M., Ferguson, H. C., et al. 2011, ApJS, 197, 36, doi: 10.1088/0067-0049/197/2/36
  * Kriek et al. (2009) Kriek, M., van Dokkum, P. G., Franx, M., Illingworth, G. D., & Magee, D. K. 2009, ApJ, 705, L71, doi: 10.1088/0004-637X/705/1/L71
  * Kroupa (2001) Kroupa, P. 2001, MNRAS, 322, 231, doi: 10.1046/j.1365-8711.2001.04022.x
  * Lang et al. (2014) Lang, P., Wuyts, S., Somerville, R. S., et al. 2014, ApJ, 788, 11, doi: 10.1088/0004-637X/788/1/11
  * Lee et al. (2018) Lee, B., Giavalisco, M., Whitaker, K., et al. 2018, ApJ, 853, 131, doi: 10.3847/1538-4357/aaa40f
  * Leja et al. (2019) Leja, J., Carnall, A. C., Johnson, B. D., Conroy, C., & Speagle, J. S. 2019, ApJ, 876, 3, doi: 10.3847/1538-4357/ab133c
  * Leja et al. (2018) Leja, J., Johnson, B. D., Conroy, C., & van Dokkum, P. 2018, ApJ, 854, 62, doi: 10.3847/1538-4357/aaa8db
  * Leja et al. (2017) Leja, J., Johnson, B. D., Conroy, C., van Dokkum, P. G., & Byler, N. 2017, ApJ, 837, 170, doi: 10.3847/1538-4357/aa5ffe
  * Leja et al. (2020) Leja, J., Speagle, J. S., Johnson, B. D., et al. 2020, ApJ, 893, 111, doi: 10.3847/1538-4357/ab7e27
  * Leja et al. (2022) Leja, J., Speagle, J. S., Ting, Y.-S., et al. 2022, ApJ, 936, 165, doi: 10.3847/1538-4357/ac887d
  * Lilly & Carollo (2016) Lilly, S. J., & Carollo, C. M. 2016, ApJ, 833, 1, doi: 10.3847/0004-637X/833/1/1
  * Lintott et al. (2008) Lintott, C. J., Schawinski, K., Slosar, A., et al. 2008, MNRAS, 389, 1179, doi: 10.1111/j.1365-2966.2008.13689.x
  * Lower et al. (2020) Lower, S., Narayanan, D., Leja, J., et al. 2020, ApJ, 904, 33, doi: 10.3847/1538-4357/abbfa7
  * Madau & Dickinson (2014) Madau, P., & Dickinson, M. 2014, ARA&A, 52, 415, doi: 10.1146/annurev-astro-081811-125615
  * Man & Belli (2018) Man, A., & Belli, S. 2018, Nature Astronomy, 2, 695, doi: 10.1038/s41550-018-0558-1
  * Maraston et al. (2010) Maraston, C., Pforr, J., Renzini, A., et al. 2010, MNRAS, 407, 830, doi: 10.1111/j.1365-2966.2010.16973.x
  * Matharu et al. (2019) Matharu, J., Muzzin, A., Brammer, G. B., et al. 2019, MNRAS, 484, 595, doi: 10.1093/mnras/sty3465
  * McGrath et al. (2008) McGrath, E. J., Stockton, A., Canalizo, G., Iye, M., & Maihara, T. 2008, ApJ, 682, 303, doi: 10.1086/589631
  * Merlin et al. (2021) Merlin, E., Castellano, M., Santini, P., et al. 2021, A&A, 649, A22, doi: 10.1051/0004-6361/202140310
  * Mo et al. (2010) Mo, H., van den Bosch, F. C., & White, S. 2010, Galaxy Formation and Evolution
  * Mowla et al. (2019) Mowla, L. A., van Dokkum, P., Brammer, G. B., et al. 2019, ApJ, 880, 57, doi: 10.3847/1538-4357/ab290a
  * Muzzin et al. (2013) Muzzin, A., Marchesini, D., Stefanon, M., et al. 2013, ApJ, 777, 18, doi: 10.1088/0004-637X/777/1/18
  * Nayyeri et al. (2017) Nayyeri, H., Hemmati, S., Mobasher, B., et al. 2017, ApJS, 228, 7, doi: 10.3847/1538-4365/228/1/7
  * Nelson et al. (2012) Nelson, E. J., van Dokkum, P. G., Brammer, G., et al. 2012, ApJ, 747, L28, doi: 10.1088/2041-8205/747/2/L28
  * Nelson et al. (2016) Nelson, E. J., van Dokkum, P. G., Förster Schreiber, N. M., et al. 2016, ApJ, 828, 27, doi: 10.3847/0004-637X/828/1/27
  * Newman et al. (2015) Newman, A. B., Belli, S., & Ellis, R. S. 2015, ApJ, 813, L7, doi: 10.1088/2041-8205/813/1/L7
  * Newman et al. (2018) Newman, A. B., Belli, S., Ellis, R. S., & Patel, S. G. 2018, ApJ, 862, 126, doi: 10.3847/1538-4357/aacd4f
  * Newman et al. (2014) Newman, A. B., Ellis, R. S., Andreon, S., et al. 2014, ApJ, 788, 51, doi: 10.1088/0004-637X/788/1/51
  * Newman et al. (2012) Newman, A. B., Ellis, R. S., Bundy, K., & Treu, T. 2012, ApJ, 746, 162, doi: 10.1088/0004-637X/746/2/162
  * Ocvirk et al. (2006) Ocvirk, P., Pichon, C., Lançon, A., & Thiébaut, E. 2006, MNRAS, 365, 46, doi: 10.1111/j.1365-2966.2005.09182.x
  * Oser et al. (2012) Oser, L., Naab, T., Ostriker, J. P., & Johansson, P. H. 2012, ApJ, 744, 63, doi: 10.1088/0004-637X/744/1/63
  * Pacifici et al. (2012) Pacifici, C., Charlot, S., Blaizot, J., & Brinchmann, J. 2012, MNRAS, 421, 2002, doi: 10.1111/j.1365-2966.2012.20431.x
  * Papovich et al. (2001) Papovich, C., Dickinson, M., & Ferguson, H. C. 2001, ApJ, 559, 620, doi: 10.1086/322412
  * Papovich et al. (2011) Papovich, C., Finkelstein, S. L., Ferguson, H. C., Lotz, J. M., & Giavalisco, M. 2011, MNRAS, 412, 1123, doi: 10.1111/j.1365-2966.2010.17965.x
  * Papovich et al. (2012) Papovich, C., Bassett, R., Lotz, J. M., et al. 2012, ApJ, 750, 93, doi: 10.1088/0004-637X/750/2/93
  * Patel et al. (2013) Patel, S. G., van Dokkum, P. G., Franx, M., et al. 2013, ApJ, 766, 15, doi: 10.1088/0004-637X/766/1/15
  * Peng et al. (2002) Peng, C. Y., Ho, L. C., Impey, C. D., & Rix, H.-W. 2002, AJ, 124, 266, doi: 10.1086/340952
  * Peng et al. (2010a) —. 2010a, AJ, 139, 2097, doi: 10.1088/0004-6256/139/6/2097
  * Peng et al. (2010b) Peng, Y.-j., Lilly, S. J., Kovač, K., et al. 2010b, ApJ, 721, 193, doi: 10.1088/0004-637X/721/1/193
  * Perez-Gonzalez et al. (2013) Perez-Gonzalez, P. G., Cava, A., Barro, G., et al. 2013, ApJ, 762, 46, doi: 10.1088/0004-637X/762/1/46
  * Salim et al. (2007) Salim, S., Rich, R. M., Charlot, S., et al. 2007, ApJS, 173, 267, doi: 10.1086/519218
  * Sanders et al. (1988) Sanders, D. B., Soifer, B. T., Elias, J. H., et al. 1988, ApJ, 325, 74, doi: 10.1086/165983
  * Santini et al. (2015) Santini, P., Ferguson, H. C., Fontana, A., et al. 2015, ApJ, 801, 97, doi: 10.1088/0004-637X/801/2/97
  * Schreiber et al. (2015) Schreiber, C., Pannella, M., Elbaz, D., et al. 2015, A&A, 575, A74, doi: 10.1051/0004-6361/201425017
  * Scoville et al. (2017) Scoville, N., Lee, N., Vanden Bout, P., et al. 2017, ApJ, 837, 150, doi: 10.3847/1538-4357/aa61a0
  * Shen et al. (2003) Shen, S., Mo, H. J., White, S. D. M., et al. 2003, MNRAS, 343, 978, doi: 10.1046/j.1365-8711.2003.06740.x
  * Shibuya et al. (2015) Shibuya, T., Ouchi, M., & Harikane, Y. 2015, ApJS, 219, 15, doi: 10.1088/0067-0049/219/2/15
  * Speagle et al. (2014) Speagle, J. S., Steinhardt, C. L., Capak, P. L., & Silverman, J. D. 2014, ApJS, 214, 15, doi: 10.1088/0067-0049/214/2/15
  * Spilker et al. (2016) Spilker, J. S., Bezanson, R., Marrone, D. P., et al. 2016, ApJ, 832, 19, doi: 10.3847/0004-637X/832/1/19
  * Springel et al. (2005) Springel, V., Di Matteo, T., & Hernquist, L. 2005, ApJ, 620, L79, doi: 10.1086/428772
  * Straatman et al. (2016) Straatman, C. M. S., Spitler, L. R., Quadri, R. F., et al. 2016, ApJ, 830, 51, doi: 10.3847/0004-637X/830/1/51
  * Strazzullo et al. (2013) Strazzullo, V., Gobat, R., Daddi, E., et al. 2013, ApJ, 772, 118, doi: 10.1088/0004-637X/772/2/118
  * Suess et al. (2022) Suess, K. A., Kriek, M., Bezanson, R., et al. 2022, ApJ, 926, 89, doi: 10.3847/1538-4357/ac404a
  * Tacchella et al. (2016) Tacchella, S., Dekel, A., Carollo, C. M., et al. 2016, MNRAS, 458, 242, doi: 10.1093/mnras/stw303
  * Tacchella et al. (2022) Tacchella, S., Conroy, C., Faber, S. M., et al. 2022, ApJ, 926, 134, doi: 10.3847/1538-4357/ac449b
  * Tacconi et al. (2020) Tacconi, L. J., Genzel, R., & Sternberg, A. 2020, ARA&A, 58, 157, doi: 10.1146/annurev-astro-082812-141034
  * Tadaki et al. (2017) Tadaki, K.-i., Genzel, R., Kodama, T., et al. 2017, ApJ, 834, 135, doi: 10.3847/1538-4357/834/2/135
  * Taniguchi et al. (2015) Taniguchi, Y., Kajisawa, M., Kobayashi, M. A. R., et al. 2015, PASJ, 67, 104, doi: 10.1093/pasj/psv106
  * Toft et al. (2017) Toft, S., Zabl, J., Richard, J., et al. 2017, Nature, 546, 510, doi: 10.1038/nature22388
  * Tojeiro et al. (2007) Tojeiro, R., Heavens, A. F., Jimenez, R., & Panter, B. 2007, MNRAS, 381, 1252, doi: 10.1111/j.1365-2966.2007.12323.x
  * Toomre (1964) Toomre, A. 1964, ApJ, 139, 1217, doi: 10.1086/147861
  * Valentinuzzi et al. (2010) Valentinuzzi, T., Fritz, J., Poggianti, B. M., et al. 2010, ApJ, 712, 226, doi: 10.1088/0004-637X/712/1/226
  * van der Wel et al. (2011) van der Wel, A., Rix, H.-W., Wuyts, S., et al. 2011, ApJ, 730, 38, doi: 10.1088/0004-637X/730/1/38
  * van der Wel et al. (2012) van der Wel, A., Bell, E. F., Häussler, B., et al. 2012, ApJS, 203, 24, doi: 10.1088/0067-0049/203/2/24