# Size and Spectroscopic Evolution of HectoMAP Quiescent Galaxies

Ivana Damjanov  Department of Astronomy and Physics, Saint Mary’s University,
923 Robie Street, Halifax, NS B3H 3C3, Canada<EMAIL_ADDRESS>Canada
Research Chair in Astronomy and Astrophysics, Tier II  Jubee Sohn  Center for
Astrophysics — Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138,
USA  Astronomy Program, Department of Physics and Astronomy, Seoul National
University, Gwanak-gu, Seoul 151-742, Republic of Korea  Margaret J. Geller
Center for Astrophysics — Harvard & Smithsonian, 60 Garden Street, Cambridge,
MA 02138, USA  Yousuke Utsumi  SLAC National Accelerator Laboratory, Menlo
Park, CA 94025, USA  Kavli Institute for Particle Astrophysics and Cosmology,
Stanford University, Stanford, CA 94305, USA  Ian Dell’Antonio  Department of
Physics, Brown University, Box 1843, Providence, RI 02912, USA

###### Abstract

The HectoMAP survey provides a complete, mass-limited sample of 30,231
quiescent galaxies with $i-$band Hyper Suprime-Cam Subaru Strategic Program
(HSC SSP) imaging that spans the redshift range $0.2<z<0.6$. We combine half-
light radii based on HSC SSP imaging with redshifts and $D_{n}4000$ to explore
the size – mass relation, $R_{e}=A\times M_{*}^{\alpha}$, and its evolution
for the entire HectoMAP quiescent population and for two subsets of the data.
Newcomers with $1.5<D_{n}4000<1.6$ at each redshift show a steeper increase in
$A$ as the universe ages than the population that descends from galaxies that
are already quiescent at the survey limit, $z\sim 0.6$ (the resident
population). In broad agreement with previous studies, evolution in the size –
mass relation both for the entire HectoMAP sample and for the resident
population (but not for the newcomers alone) is consistent with minor merger
driven growth. For the resident population, the evolution in the size – mass
relation is independent of the population age at $z\sim 0.6$. The contrast
between the sample of newcomers and the resident population provides insight
into the role of commonly termed “progenitor bias” on the evolution of the
size – mass relation.

galaxies:evolution, galaxies: fundamental parameters, galaxies: structure,
galaxies: stellar content, galaxies: statistics

††facilities: MMT Hectospec, Subaru Hyper Suprime-Cam††software: SciPy
(Virtanen et al., 2020), NumPy (Harris et al., 2020), astropy (Astropy
Collaboration et al., 2013, 2018; The Astropy Collaboration et al., 2022),
Scikit-learn (Pedregosa et al., 2011), PWLF: piecewise linear fitting (Jekel &
Venter, 2019), Source Extractor (Bertin & Arnouts, 1996), PSFEx (Bertin, 2011)

##  1 Introduction

During the last fifteen years, observations have provided an increasingly
detailed picture of the evolution of the quiescent galaxy population. Perhaps
the most remarkable aspect of the evolution is the size growth at fixed
stellar mass for galaxies with masses $\gtrsim 10^{10}{\rm M}_{\odot}$. From a
redshift of 1.5 to the present, the growth is a factor of 2.5-4 (Trujillo et
al., 2006, 2007; Williams et al., 2010; Damjanov et al., 2011; Newman et al.,
2012; Cassata et al., 2013; Huertas-Company et al., 2013; van der Wel et al.,
2014; Faisst et al., 2017; Damjanov et al., 2019; Mowla et al., 2019; Mosleh
et al., 2020; Kawinwanichakij et al., 2021; Yang et al., 2021; Barone et al.,
2022; Hamadouche et al., 2022 and others).

Evaluation of the size growth of the quiescent population is based on the size
– mass relation, a projection of the stellar mass fundamental plane (Hyde &
Bernardi, 2009; Bezanson et al., 2013; Zahid et al., 2016). The size – mass
relation for quiescent galaxies with stellar masses $\gtrsim 10^{10}{\rm
M}_{\odot}$ is $R_{e}=A\times M_{*}^{\alpha}$ where $R_{e}$ is the half-light
radius. Remarkably, the slope, $\alpha$ varies very little with redshift even
though $A$ increases substantially (e.g. Faisst et al., 2017; Hamadouche et
al., 2022). Many evolutionary studies have focused on $z\gtrsim 1$ where the
size evolution appears to be more rapid than it is at more recent epochs
(e.g., Cassata et al., 2011; Newman et al., 2012; Damjanov et al., 2019). One
limitation of these investigations is lack of sufficiently large quiescent
samples with complete spectroscopy in the redshift range $0.2<z<0.6$.

The consensus view is that minor mergers are the main driver of quiescent
galaxy size evolution at $0.2<z<0.6$ and possibly over a larger redshift range
(e.g., Nipoti et al., 2009; Naab et al., 2009; Nipoti et al., 2012; Newman et
al., 2012; Faisst et al., 2017; Hamadouche et al., 2022). One issue that
clouds the interpretation is the necessary distinction between objects that
are just becoming quiescent at any particular epoch and those that are
residents evolving from earlier epochs. This issue is generally called
progenitor bias (van Dokkum & Franx, 2001; Carollo et al., 2013; Belli et al.,
2015; Fagioli et al., 2016). For greater specificity in our discussion we
distinguish between newcomers (the sources of progenitor bias) and the
resident population.

We explore insights into the size evolution based on the quiescent population
in the HectoMAP survey. HectoMAP is a dense red-selected redshift survey
covering 55 square degrees of the northern sky (Geller & Hwang, 2015; Sohn et
al., 2021, 2022, in prep.). The redshift survey includes a complete mass-
limited sample of $\sim 42,500$ quiescent objects at $0.2<z<0.6$ selected on
the basis of the spectroscopic indicator $D_{n}4000$ (Section 2.1). We derive
half-light radii for 30,231 of these objects from Subaru Hyper Suprime-Cam
(HSC) Subaru Strategic Program (SSP) $i-$band imaging (Aihara et al., 2022)
that covers 80% of the HectoMAP region (Section 2.3). The MMT Hectospec
spectroscopy enables separation of likely newcomers from the resident
population already in place at the limiting redshift, $z=0.6$. We can thus
evaluate the impact of newcomers on the size evolution of the population.

Section 2 describes the data that underlie our analysis. We review the red-
selected HectoMAP survey (Section 2.1) and we use the magnitude limited SHELS
survey to comparable depth to evaluate the completeness of the HectoMAP
quiescent sample (Section 2.2). We review the HSC/SSP photometry and describe
our size measurements (Section 2.3). We use $r-$ and $i-$band Subaru
photometry to demonstrate the insensitivity of the size evolution results to
the use of $i-$band photometry throughout the redshift range.

Section 3 highlights the HectoMAP quiescent galaxy size – mass relation for
the entire complete stellar mass limited sample of $30,231$ quiescent galaxies
with publicly available $i-$band HSC/SSP imaging. Because of the large size of
the total quiescent sample we can follow the evolution of newcomers (objects
with $1.5<D_{n}4000<1.6$) as the lookback time increases by only $\sim 1.5$
Gyr (Section 4.1). We follow the quiescent population that ages from a
redshift of $z=0.6$, the resident population (Section 4.2). We compare the
size – mass relations for this population with the relation for the newcomers
and for the full quiescent sample. Finally we assess the agreement between the
evolution of these size – mass relations and broadly favored minor merger
models (Section 5.1). We discuss the results in a broader forward looking
context in Section 5.2. We conclude in Section 6.

We adopt the Planck cosmological parameters (Planck Collaboration et al.,
2016) with $H_{0}=67.74~{}\rm km~{}s^{-1}~{}Mpc^{-1}$, $\Omega_{m}=0.3089$,
$\Omega_{\Lambda}=0.6911$ and the AB magnitude system.

##  2 The Data

We use the red-selected HectoMAP redshift survey (Geller & Hwang, 2015; Sohn
et al., 2021, 2022, in prep.) along with HSC/SSP imaging (Miyazaki et al.,
2018; Aihara et al., 2022) of the HectoMAP region to trace the size evolution
of the quiescent population from $z\sim 0.2$ to $z\sim 0.6$. The redshift
survey provides the spectral indicator $D_{n}4000$ and the stellar masses that
we use to define a mass limited sample of $30,231$ quiescent objects covering
this redshift range. Because HectoMAP is color-selected, we calibrate the mass
selected sample with the complete purely magnitude limited SHELS F2 survey
(Geller et al., 2014, 2016). At $z\lesssim 0.2$ the color selection of
HectoMAP removes a significant portion of the quiescent population. We thus
restrict our analysis to the redshift range $0.2<z<0.6$.

###  2.1 HectoMAP Spectroscopy

HectoMAP is a dense, red-selected redshift survey carried out with the
Hectospec 300-fiber wide-field instrument on the 6.5-meter MMT (Fabricant et
al., 2005). The survey covers a total area of 55 deg2 in a 1.5 deg wide strip
at high declination. The right ascension and declination limits of the survey
are: $200<$ R.A. (deg) $<250$ and $42.5<$ Decl. (deg) $<44.0$ (Sohn et al.,
2021, 2022, in prep.).

Observations for HectoMAP span more than a decade. We based photometric
selection on successive SDSS data releases and we now calibrate the
completeness to SDSS DR16 (Ahumada et al., 2020). HectoMAP includes a bright
survey with $r_{petro,0}<20.5$ and a red color selection $(g-r)_{model,0}>1$
and a faint survey with $20.5<r_{petro,0}\leq 21.3$ and red selection
$(g-r)_{model,0}>1$ and $(r-i)_{model,0}>0.5$. Here $r_{petro,0}$ refers to
the Sloan Digital Sky Survey (SDSS) Petrosian magnitude corrected for Galactic
extinction and the color selection is based on the extinction-corrected SDSS
model magnitudes. For the faint portion, the $(r-i)_{model,0}$ cut removes low
redshift objects. The cut was necessary because the telescope time required
without the cut was prohibitive. We impose an additional surface brightness
limit by requiring $r_{fiber,0}<22.0$ throughout the HectoMAP survey because
red objects below this limit yield very low signal-to-noise spectra. The
limiting $r_{fiber,0}$ magnitude corresponds to the extinction-corrected flux
within the Hectospec fiber aperture (with $0\farcs 75$ radius).

Sohn et al. (2021) and Sohn et al. (2022, in prep.) describe the two data
releases that cover the entire HectoMAP redshift survey. These two papers
included detailed descriptions and analyses of the survey completeness and
uniformity. They also include detailed discussion of the redshifts, the
spectral indicator $D_{n}4000$, and the stellar mass. Here we very briefly
summarize these data along with any issues that might affect the investigation
we carry out here.

HectoMAP is a dense redshift survey. There are $\sim 2000$ redshifts per
square degree. The median depth of the survey is $z=0.31$. Within the color
selection the survey is 80% uniformly complete to the limit
$r_{petro,0}=20.5$. In the faint sample with $20.5<r_{petro,0}\leq 21.3$, the
sample is 65% complete and the completeness is somewhat lower near the edges
of the survey. The total number of redshifts is 100,097. Table 1 summarizes
these properties of the survey111Note that the numbers in this table are
related to the portion of the survey that covers redshift interval
$0.2<z<0.6$. The total number of spectra in the survey is thus larger..

The typical error in a HectoMAP redshift is $38~{}\rm km~{}s^{-1}$. There is a
small offset of $39~{}\rm km~{}s^{-1}$ between Hectospec redshifts and the
SDSS (Sohn et al., 2021). This offset is irrelevant for the analysis we pursue
here.

We compute stellar masses of HectoMAP galaxies based on SDSS DR16 $ugriz$
photometry. We use the Le Phare software package (Arnouts et al., 1999; Ilbert
et al., 2006) that incorporates stellar population synthesis models of Bruzual
& Charlot (2003), a Chabrier (2003) IMF and a Calzetti et al. (2000)
extinction law. We explore two metallicities (0.4 and 1) and a set of
exponentially decreasing star formation rates. We derive the mass-to-light
ratio from the best model and we then use this ratio to convert the observed
luminosity into a stellar mass.

###  2.2 Quiescent Population Definition and Selection

As in our previous investigations of the quiescent population, we use the
spectral indicator $D_{n}4000$ to segregate the quiescent population. The
quantity $D_{n}4000$ is a flux ratio (in $f_{\nu}$ units) in two bands:
$4000–4100$ Å and $3850–3950$ Å (Balogh et al., 1999). This ratio is a measure
of the age of a stellar population. The index is smaller for objects with a
younger population (Kauffmann et al., 2003).

For the HectoMAP spectra, the median signal-to-noise around 4000 Å is 4.5.
More than 95% of the HectoMAP spectra have adequate signal-to-noise for robust
determination of the index. Based on repeat measurements, Geller et al. (2014)
report a typical error of 4.5% in $D_{n}4000$ for the comparable SHELS survey
data. Damjanov et al. (2022) show (their Figure 3) that in the SHELS survey
the typical error in $D_{n}4000$ increases with redshift. In the Appendix A we
demonstrate that the error in $D_{n}4000$ has a negligible effect on analysis
of the size – mass relation and its evolution.

Following previous studies (e.g., Vergani et al., 2008; Damjanov et al., 2018,
2019; Hamadouche et al., 2022) we identify the quiescent population as objects
with $D_{n}4000>1.5$. Damjanov et al. (2018) use the hCOSMOS redshift survey
to test this spectroscopic selection against rest-frame UVJ color selection
based on SED fitting to 30 photometric bands covering the range 0.15 - 254
$\mu$m. They find that the spectroscopic and photometric approaches agree
well. Furthermore, in the COSMOS field, Damjanov et al. (2018) examine the
relationship between Zurich Estimator of Structural Types (ZEST; Scarlata et
al., 2007) morphologies and the spectroscopic indicator of galaxy quiescence.
About ninety percent of galaxies with $D_{n}4000>1.6$ have elliptical or
bulge-dominated morphologies as do $\sim 70\%$ of the objects with
$1.5<D_{n}4000<1.6$.

Construction of a mass complete sample of quiescent galaxies is a crucial
foundation for exploring the evolution of the population. We use the SHELS F2
sample (Geller et al., 2014, 2016) to calibrate HectoMAP. SHELS F2 is somewhat
shallower than HectoMAP, but it is not color-selected. In magnitude limited
samples like SHELS and HectoMAP the apparent magnitude limit imprints a
brighter limiting absolute magnitude along with an increasing limiting stellar
mass as the redshift increases. The upper panel of Figure 1 shows the
K-corrected $r-$band absolute magnitudes along with magnitude limits for both
SHELS (black points and black dashed line) and HectoMAP (blue points and blue
dashed curve) quiescent galaxies with $D_{n}4000>1.5$ as a function of
redshift. The apparent magnitude limit for SHELS is $r=20.75$; for HectoMAP
the limit is $r=21.3$. For HectoMAP, the color cuts are responsible for the
departures from the blue dashed curve that indicates the limiting absolute
K-corrected $r-$band magnitude. The middle panel shows the redshift
distributions for the two surveys.

We translate the samples limited in absolute magnitude into samples limited in
stellar mass. We make the conversion by computing the average mass-to-light
ratio in SHELS and HectoMAP at each redshift. The dashed black (SHELS) and
blue (HectoMAP) curves show the resulting limits. Here again the black and
blue points show quiescent galaxies in SHELS and HectoMAP, respectively. Some
objects lie below the limits as a result of photometric errors. Table 1 shows
that the fraction of galaxies below the limit is $\lesssim 10\%$ for both
SHELS F2 and HectoMAP samples.

The side panel in Figure 1 shows the aggregate distributions of stellar masses
for the SHELS and HectoMAP samples normalized to unit area for each histogram.
The mass limited samples of SHELS and HectoMAP quiescents contain,
respectively, 4217 and 52593 objects. The distributions are indistinguishable.
We examine these distributions further in Figure 2.

Table 1: Number of galaxies in the SHELS F2 and HectoMAP samples Sample | Number of Galaxies
---|---
| F2 | HectoMAP
SDSS $r-$band limit [mag] | 20.75 | 21.3
Color selection | None |  $(g-r)>1$ for $r<20.5$
|  |  $(g-r)>1\&(r-i)>0.5$ for $r>20.5$
$z$ | (0.1, 0.6) | (0.2, 0.6)
$N_{\mathrm{spec}}$ | 10848 | 64240
$N_{\mathrm{spec},\,\mathrm{D}_{n}4000}$ | 10794aaCriteria for reliable $D_{n}4000$ measurements are: 1) $D_{n}4000>0$, 2) $D_{n}4000<3$, 3) $\Delta(\mathrm{D}_{n}4000)<0.2\times\mathrm{D}_{n}4000$ | 63317aaCriteria for reliable $D_{n}4000$ measurements are: 1) $D_{n}4000>0$, 2) $D_{n}4000<3$, 3) $\Delta(\mathrm{D}_{n}4000)<0.2\times\mathrm{D}_{n}4000$
$N_{\mathrm{D}_{n}4000>1.5}$ | 4622 | 46367
$N_{\mathrm{D}_{n}4000>1.5,\,M_{\ast}}$ | 4602 | 46152
$N_{\mathrm{mass\,limited\,sample}}$ | 4088 | 42496
$N_{\mathrm{mass\,limited\,sample},R_{e}}$ | 4595bbThe $4$ deg2 area of the SHELS F2 survey is fully covered by the HSC $i-$band imaging (Damjanov et al., 2019). | 30231ccThe HSC/SSP imaging of the HectoMAP in $i-$band covers $\lesssim 80\%$ of the spectroscopic survey area.
Figure 1: Absolute K-corrected $r-$band magnitude (top panel) and stellar mass
(bottom panel) versus redshift for the quiescent galaxy population in SHELS F2
($r<20.75$; black points and black dashed lines) and HectoMAP ($r<21.3$; blue
points and blue dashed lines). The absolute magnitude limits (black dashed and
blue solid lines in the top panel), corresponding to the observed K-corrected
$r-$band absolute magnitude limits. The lower panel shows the SHELS F2 and
HectoMAP stellar mass limits (black dashed and blue solid lines in the lower
panel) derived by incorporating the average mass-to-light ratio for SHELS F2
and HectoMAP galaxies at each redshift. The horizontal and vertical histograms
show probability density distributions for the stellar mass and redshift of
SHELS F2 (black) and HectoMAP (blue) galaxies.

In contrast with the Damjanov et al. (2019) analysis of the complete SHELS
survey, the red selection of HectoMAP may affect the completeness of mass
limited samples of quiescent HectoMAP galaxies.

Figure 2 explores the completeness of the HectoMAP quiescent population. We
use the SHELS survey as a basis for assessing the completeness of HectoMAP.
For SHELS galaxies with $r<20.75$ we first show the quiescent subsample
without color selection (black histogram). We then apply the HectoMAP color
selection (orange histogram). Figure 2 shows the number of quiescent galaxies
as a function of stellar mass in successive redshift intervals. Within the
lowest redshift interval (upper left panel), the total sample (black
histogram) includes a tail toward lower stellar mass that is absent in the
red-selected sample. These objects are younger (and bluer) than those included
in the red-selected sample. This effect is also visible in Figure 1.

Figure 2: Mass distribution of SHELS F2 quiescent galaxies (black histogram),
color-selected SHELS F2 quiescent galaxies (orange histogram, $(g-r)>1$ for
$r<20.5$ and $(g-r)>1,(r-i)>0.5)$ for $20.5<r<20.75$), and HectoMAP quiescent
galaxies (cyan histogram) for four redshift bins covering the $0.1<z<0.6$.
Galaxy samples are mass complete above the completeness limit of the SHELS F2
survey (dashed black line in the bottom panel of Figure 1). For $0.2<z<0.6$,
the red-selection of the HectoMAP survey includes $>98\%$ of the quiescent
galaxies in the purely magnitude limited SHELS F2 survey.

In the three redshift intervals spanning the redshift range $0.2<z<0.6$, the
red-selected SHELS samples (orange histograms) are essentially
indistinguishable from the total (black histograms); they contain 98-100% of
the galaxies in the total sample. In this higher redshift range, the objects
with low stellar mass that populate the tail of the black histograms in the
upper left panel are fainter than the limiting magnitude of the SHELS survey.
The cyan histograms in Figure 2 show distributions for HectoMAP quiescents in
the same redshift intervals we explore for SHELS. The agreement is excellent
for $z\gtrsim 0.2$. Thus HectoMAP provides an essentially complete mass
limited quiescent sample over this range. This large sample of $42,496$
quiescent systems provides a powerful platform for examining the evolution of
this population.

The $41$ deg2 region of HectoMAP we explore here is large enough that the
resulting number densities of quiescent galaxies should be insensitive to the
impact of cosmic variance. The expected impact of cosmic variance is only 3.5%
(Driver & Robotham, 2010; Geller et al., 2016).

Figure 3 compares the number density of HectoMAP quiescents (red symbols) with
the number densities from the 5.5 deg2 PRIMUS grism survey (black points,
Moustakas et al., 2013). PRIMUS quiescents are derived from the star formation
rate versus stellar mass diagram, but Damjanov et al. (2019) show that this
difference in selection does not affect the relative number densities for
PRIMUS relative to the complete SHELS F2 redshift survey $D_{n}4000$
selection. For the PRIMUS survey the expected impact of cosmic variance is
5.5%.

Figure 3: Number density of quiescent galaxies in HectoMAP (red symbols) as a
function of redshift in bins of stellar mass compared with the number density
evolution of galaxies in the PRIMUS survey (Moustakas et al., 2013, black
simbols). In the white area the quiescent HectoMAP sample is $>99\%$ complete
at the lower stellar mass limit of the bin.

For the redshift range $0.2<z<0.6$ we compare the number densities for three
mass ranges. At $z<0.2$ the red color selection of HectoMAP biases the density
estimate because bluer quiescent objects are missing from the sample as noted
earlier. In the unshaded regions of Figure 3 where the HectoMAP samples are
mass complete, the number densities of the two surveys agree within a factor
of $\lesssim 1.5$. In the highest mass bin (right panel), the agreement is
excellent over the entire redshift range we examine in the following analysis.

###  2.3 Sizes from HSC SSP Imaging

We measure sizes for HectoMAP galaxies with redshifts following the procedure
described in detail in Damjanov et al. (2019) for the SHELS F2 field of the
SHELS survey (Geller et al., 2014, 2016). The $r-$ and $i-$band images for the
HectoMAP region are part of the HSC/SSP survey (Miyazaki et al., 2018). The
HSC $i-$band images of the Third Public Data Release (PDR3, Aihara et al.,
2022), with median seeing of $\sim 0.55$ arcsec cover 41 deg2 ($\lesssim 80\%$
of the survey area) of the HectoMAP region.

The $r-$band images cover the full HectoMAP region, but the typical seeing is
$\sim 0.2^{\prime\prime}$ worse than the $i-$ band seeing. The $r-$band seeing
is also less uniform222https://hsc-
release.mtk.nao.ac.jp/doc/index.php/quality-assurance-plots. We base our
analysis on the better and more uniform $i-$band seeing. The results are then
directly comparable with the analysis of $i-$band sizes for the SHELS F2 field
(Damjanov et al., 2019).

The image processing is based on the hscPipe system (Bosch et al., 2018), the
standard pipeline for the HSC/SSP. The pipeline facilitates reduction of
individual chips, mosaicking, and image stacking. The HSC/SSP DR3 offers two
sets of co-added images that are based on different approaches to sky
subtraction (local and global, see Figure 8 of Aihara et al., 2022). Here we
select to use $i-$band coadds with global background subtraction where
extended wings of bright objects are better preserved.

As in our earlier exploration of quiescent galaxies size and spectroscopic
evolution Damjanov et al. (2019), we estimate the sizes of HectoMAP galaxies
based on single Sérsic profile models (Sersic, 1968). Damjanov et al. (2019)
describe the approach and compare the results with previous size measurements
in detail; we briefly summarize the procedure here.

Figure 4: The ratio of $i-$band to $r$-band sizes along the galaxy major axis
as a function of redshift. The subsample show here includes 4699 HectoMAP
galaxies ($\sim 10\%$ in the quiescent sample from Table 1). The symbol for
each galaxy is color-coded by the $D_{n}4000$ index (proxy for the average
stellar population age). The average ratio of two sizes is $0.97$ (dark grey
line) and the scatter (the standard deviation of the ratio distribution) is
$\pm 0.13$ (grey band). This scatter is essentially the sum in quadrature of
the errors in the measurements in each band ($0.07$ for $i$ and $0.11$ for
$r$). The (small) difference in sizes is not a function of either redshift or
galaxy average stellar population age.

We use the SExtractor software (Bertin & Arnouts, 1996) to measure the
photometric parameters of galaxies with redshifts in HectoMAP including the
galaxy half-light radius, ellipticity, and Sérsic index. These parameters are
based on two-dimensional (2D) modeling of the galaxy surface brightness
profile following a three-step process: (a) in its first run, SExtractor
provides a catalog of sources that includes star–galaxy separation; (b) the
PSFex software (Bertin, 2011) combines clean sets of point sources from the
initial SExtractor catalog to construct a set of spatially varying point-
spread functions (PSFs) that are the input parameters for (c) the second
SExtractor run that provides a catalog with morphological parameters for all
detected sources.

Within the SPHEROID model, the SExtractor fitting procedure provides Sérsic
profile parameters and the best-fit model axial ratio b/a (SExtractor model
parameter SPHEROID_ASPECT_WORLD). The angular diameter distance implied by the
spectroscopic redshift of each galaxy translates the galaxy angular size
$\Theta_{e}$ (SExtractor model parameter SPHEROID_REFF_WORLD) into the half-
light radius along the galaxy major axis $R_{e}$ in kpc (e.g., Hogg (1999),
and references therein).

We apply our size measurement procedure to the $i-$band imaging. We also
analyze $\sim 9$ deg2 of the HSC SSP $r-$band imaging to test for sensitivity
to the use of the $i-$band data throughout the redshift range $0.2<z<0.6$.

Figure 4 shows the ratio $R_{e}(i)/R_{e}(r)$ for 4699 HectoMAP quiescent
galaxies covering the entire redshift range we analyze. The $i-$band sizes are
slightly smaller than the $r-$band sizes as expected. The solid line shows
that the mean ratio, 0.97, does not vary measurably with redshift.

The gray band shows the 1$\sigma$ error in the ratio, $\pm 0.13$, essentially
the sum in quadrature of the total errors in the individual $i-$ and $r-$band
size measurements. In each band the total error ($0.07$ for $i$ and $0.11$ for
$r$) is a combination of internal errors for individual measurements and the
average external error based on galaxies present in multiple HSC SSP cutouts
(see Damjanov et al., 2018).

We use the spectral indicator $D_{n}4000$ to select the quiescent population
and to trace its evolution. Figure 4 shows the $D_{n}4000$ for each galaxy in
the $i-$ to $r-$ band size comparison. There is no evident dependence of the
size ratio on $D_{n}4000$.

The small difference between the $i-$band and $r-$band sizes and the lack of
systematics as a function of either redshift or $D_{n}4000$ enable use of the
full HectoMAP region coverage in the $i-$band as a robust platform for
exploring the size evolution of the quiescent population.

##  3 The size – mass Relation

The size – mass relation is a fundamental ingredient of the evolutionary
picture for the quiescent population. Previous investigators demonstrate that
the slope of the relation is essentially redshift independent for $z\lesssim
1.5$ (e.g, Newman et al., 2012; van der Wel et al., 2014; Faisst et al., 2017;
Mowla et al., 2019; Barone et al., 2022). As the universe ages, the
normalization of the relation increases reflecting the size growth of the
population (Daddi et al., 2005; Trujillo et al., 2006, 2007; van Dokkum et
al., 2010; Damjanov et al., 2011; Newman et al., 2012; Huertas-Company et al.,
2013; van der Wel et al., 2014; Huertas-Company et al., 2015; Faisst et al.,
2017; Damjanov et al., 2019; Mowla et al., 2019; Kawinwanichakij et al., 2021;
Hamadouche et al., 2022, and others).

Zahid & Geller (2017) demonstrate that the size – mass relation is a function
of $D_{n}4000$. They use the SDSS to demonstrate that older objects with large
$D_{n}4000$ have smaller sizes. Quiescent galaxies with successively smaller
$D_{n}4000$ trace parallel size – mass relations that shift toward larger
sizes for smaller values of $D_{n}4000$. Damjanov et al. (2019) use SHELS F2
to show that at redshifts $0.2\lesssim z\lesssim 0.5$ there is also an anti-
correlation between $D_{n}4000$ and size.

Here we explore the constraints on the size – mass relation provided by the
larger HectoMAP quiescent sample. Figure 5 shows the size – mass relation for
HectoMAP quiescent objects. The panels show the size – mass relations for mass
limited samples. In each panel the black line connects the median values of
the relation.

In the two upper panels we fit the median values only for objects with stellar
mass above the pivot point indicated by the black arrow. We determine the
pivot point by using the software package pwlf (Jekel & Venter, 2019) to fit a
continuous piecewise linear function limited to two segments. The black dash-
dotted line in Figure 5 shows the fit given in the legend of each panel. The
gray region shows the small uncertainty in the fit. The pivot stellar mass
appears to evolve with redshift. In the higher redshift bins ($z>0.4$) shown
in the two lower panels, all of the objects in the samples have stellar masses
that exceed the pivot point.

To make a clean comparison between Figure 5 with van der Wel et al. (2014), we
combine all of the galaxies in the mass-complete sample covering the redshift
range $0.2<z<0.5$ and repeat fitting the size – mass relation. For this
ensemble sample the slope is $0.67\pm 0.01$, which is fully consistent the
results of van der Wel et al. (2014), $0.75\pm 0.06$. Mowla et al. (2019)
derive a shallower slope, $0.48\pm 0.03$, based on HST observations of COSMOS
galaxies in the same redshift range, but the error in both space-based studies
is larger. In contrast with previous work, our large quiescent sample provides
a tighter constraint on the slope. The zero points are also fully consistent:
we derive $\log(A)=0.57\pm 0.01$ for a quiescent galaxy at $5\times 10^{10}$
M⊙ and van der Wel et al. (2014) report $0.60\pm 0.02$.

We explore variations in the size – mass relation with redshift/lookback time
further in Section 4.2 where we compare these results with other subsamples of
the data. In Section 4.2 we also examine both the scatter around the fit and
the intrinsic scatter in the data as a function of redshift. Again we compare
these bulk results with various subsamples of the data.

HectoMAP spectroscopy enables clean segregation of the samples in Figure 5
based on the spectral indicator $D_{n}4000$. In each panel of Figure 5 the
colored dots show the median size as a function of stellar mass color coded by
$D_{n}4000$. The size of each colored circle indicates the error. Comparison
with Figure 9 of Damjanov et al. (2019) (based on the order of magnitude
smaller SHELS F2 dataset, Table 1) illustrates the power of large
spectroscopic samples for identifying trends in a multi-dimensional parameter
space like the stellar mass - size - stellar population age proxy
($D_{n}4000$) for the quiescent population we explore.

There are several interesting qualitative features of the distributions of
$D_{n}4000$-encoded points in Figure 5. As the universe ages, the spread
between relations for different $D_{n}4000$ values increases. The increase in
size with decreasing $D_{n}4000$ (Zahid & Geller, 2017) becomes more evident.
These trends result from (1) the broader stellar mass range sampled at lower
redshift and (2) the galaxy population entering the quiescent phase at lower
redshift and then evolving. Although the size – mass relation depends on
$D_{n}4000$, the size – mass relations for all of the $D_{n}4000$ subsamples
lie within the median absolute deviation (MAD) shown by the dashed lines.

The size of the quiescent HectoMAP sample enables extraction of other
subsamples that inform the physical interpretation of the size growth of the
population. In the following sections we describe the derivation and analysis
of subsamples that follow the evolution of the population from the highest
redshift we access, $z=0.6$.

Figure 5: Median circularized half-light radius as a function of stellar mass
and $D_{n}4000$ for quiescent HectoMAP galaxies in four redshift/lookback time
intervals. The centers of the large circles indicate the median mass and size
of galaxies in 2D bins of stellar mass and $D_{n}4000$. The circle color
indicates the median $D_{n}4000$ in the bin. The size of each circle is
proportional to the typical bootstrapped error in the median size. In each
panel the solid black line connects the median sizes in 10 equally populated
mass bins. The dashed lines show the median absolute deviation. The central
black dash-dotted line and gray region traces the best-fit linear relation
between stellar mass and size and the associated uncertainty. The background
colored points represent individual quiescent HectoMAP galaxies in the mass
limited sample appropriate for each panel. The two black arrows indicate the
stellar mass where the slope of size – mass relation changes. We fit the size-
stellar mass relation for objects above this pivot point in each redshift bin
with an arrow. The pivot mass appears to evolve with redshift. In the higher
redshift bins ($z>0.4$) our sample is above the stellar mass region of the
pivot point.

##  4 Drivers of the Size Growth of the Quiescent Population

Processes that drive the size growth of the quiescent population fall into two
broad categories: (1) progenitor bias resulting from larger, younger objects
that are newcomers to the quiescent population (e.g., Newman et al., 2012;
Carollo et al., 2013; Beifiori et al., 2014; Belli et al., 2015; Fagioli et
al., 2016; Charbonnier et al., 2017; Tacchella et al., 2017; Suess et al.,
2019; Damjanov et al., 2019; Díaz-García et al., 2019; Hamadouche et al.,
2022) and (2) processes that produce growth of the resident population that is
already quiescent (e.g., Newman et al., 2012; López-Sanjuan et al., 2012;
Belli et al., 2015; Faisst et al., 2017; Zahid et al., 2019; Suess et al.,
2019; Díaz-García et al., 2019; Damjanov et al., 2022; Hamadouche et al.,
2022).

In the following discussion we refer to newcomers (rather than continuing to
use the term progenitor bias) and resident quiescent populations to clarify
the discussion. We minimize the use of progenitor bias terminology to avoid
confusion between newcomers at each redshift and higher redshift precursors.

We demonstrate that the large sample, extended redshift range, and
availability of $D_{n}4000$ provides limits on the relative contributions of
newcomers and processes that produce growth of the aging residents in the
redshift range $0.2<z<0.6$. From here we refer to objects with
$1.5<D_{n}4000<1.6$ as newcomers. We track the size – mass relation for the
resident quiescent population that descends from $z=0.6$ progenitors. We
compare the results with the size – mass relation for the entire quiescent
population to measure the impact of objects that join the quiescent population
at $z\lesssim 0.5$. The objects that join include newcomers at $z\lesssim 0.5$
and their descendants.

###  4.1 Newcomers (Progenitor Bias)

As the universe ages, galaxies that transition from their star-forming phase
to quiescence are larger (e.g., Paulino-Afonso et al., 2017; Genel et al.,
2018, and many others). To demonstrate the impact of newcomers for $z<0.6$ we
compare the size – mass relations for newcomers onto the red sequence
($1.5<D_{n}4000<1.6$) in two narrow, but widely separated redshift bins
(Figure 6). We select bins within the range covered by HectoMAP because all of
the size measurements come from the exquisite HSC imaging. The redshift bins
that form the basis for our test are $0.23<z<0.28$ and $0.42<z<0.47$. These
bins bracket an $\sim 1.5$ Gyr change in the age of the universe.

In the range $0.42<z<0.47$ the survey is complete for stellar masses at
$\log(M_{*}/M_{\odot})\gtrsim 10.6$. The black solid line in each scatter plot
connects the median size as a function of stellar mass for this size range in
both redshift intervals. The bins are equally populated and the errors are
bootstrapped. The legend in Figure 6 gives the two fits. The slopes are the
same within the error, but the normalization differs by
$\Delta\log(R_{e,1.5-1.6})=0.11\pm 0.025$. The lower redshift population
entering quiescence is larger as expected.

The normalized histograms (probability density functions or pdf) above and to
the right of the scatter plots show the stellar mass and $R_{e}$
distributions, respectively. The histograms that accompany the low redshift
panel superimpose the low (blue) and high (orange) redshift pdfs. At the lower
redshift there are relatively more objects with lower stellar mass. The
$R_{e}$ pdfs show a striking tail toward small $R_{e}$ that distinguishes the
higher redshift population. This comparison identifies larger recently
quenched objects that are joining the quiescent population at later cosmic
times as the drivers of the size evolution of the newcomer or progenitor
population.

Figure 6: Half-light radius along the major axis as a function of stellar mass
for newcomers ($1.5<D_{n}4000<1.6$) HectoMAP galaxies in two narrow redshift
bins separated by $\sim 1.5$ Gyr. In each panel the solid black line connects
the median sizes in equally populated mass bins covering the mass range where
the survey is complete at the higher redshift. The errors in the median values
are bootstrapped. The central black dash-dotted line and gray region trace the
best-fit linear relation between stellar mass and size and the associated
uncertainties. The top histograms show normalized stellar mass distributions
and the side histograms show normalized size distributions for young
quiescents galaxies in the stellar mass range covered by the higher redshift
bin. The top and side histograms associated with the low redshift sample
(left; blue histograms) superimpose the mass and size distributions for the
higher-redshift sample (orange) in the same mass range. Note the small
$R_{e}<2.5$ kpc in the higher redshift sample.

###  4.2 Size Growth of an Aging Resident Population

A range of processes including major mergers, minor mergers, and adiabatic
expansion (e.g., Fan et al., 2008, 2010; van Dokkum et al., 2010; Trujillo et
al., 2011; López-Sanjuan et al., 2012; Cimatti et al., 2012; Huertas-Company
et al., 2013) may contribute to the size growth of the quiescent population.
For $z\lesssim 1$, several lines of investigation suggest that minor mergers
are the primary driver of growth once newcomers are taken into account (Newman
et al., 2012; Beifiori et al., 2014; Ownsworth et al., 2014; Belli et al.,
2015; Buitrago et al., 2017; Zahid et al., 2019; Hamadouche et al., 2022).

We explore the evolution of the resident population by using $D_{n}4000$ to
connect these resident quiescent galaxies (descendants) at $z<0.5$ with their
progenitors (or ancestors) at $0.5<z<0.6$. Once star formation ceases,
$D_{n}4000$ increases monotonically. We identify $1.5<D_{n}4000<1.6$ with the
cessation of star formation and entry into the quiescent phase.

Figure 7: $D_{n}4000$ as a function of lookback time for a model galaxy that
has been quiescent for 0 (purple) to 4 (red) Gyrs at redshift $z\sim 0.6$. We
use four redshift (lookback time) bins (dashed gray lines) to explore the
evolution of the size – mass relation of aging quiescent galaxy subpopulations
in Figure 9. The hashed bins delineate the subsamples we use to trace the size
– mass relation for the aging quiescent population.

Figure 7 shows the evolution of $D_{n}4000$ for objects with 5 different
values of $D_{n}4000$ at $z\sim 0.6$, the limit of the redshift survey. These
$D_{n}4000$ values correspond to periods of 0 (purple curve) to 4 Gyr (red
curve) since cessation of star formation.

We compute $D_{n}4000$ from the synthetic spectrum of a quiescent galaxy
derived from the Flexible Stellar Population Synthesis code (FSPS; Conroy et
al., 2009; Conroy & Gunn, 2010). The model galaxy has solar metallicity and
constant star formation rate for 1 Gyr.

The value of $D_{n}4000$ has some dependence on metallicity (Poggianti &
Barbaro, 1997; Balogh et al., 1999; Bruzual & Charlot, 2003). Zahid et al.
(2019) investigate the sensitivity to metallicity of evolutionary tracks like
the ones in Figure 7. They run an FSPS model with constant star formation rate
for 1 Gyr and twice solar metallicity. Their results are insensitive to this
choice relative to the assumption of solar metallicity. Their Appendix
provides details and demonstrates that statistical errors rather than the
potential impact of metallicity are the main contributor to uncertainty in the
results of this approach to analyzing the data.

Figure 8: Size-stellar mass relation for the descendants of quiescent galaxies
with $D_{n}4000$ ($z\sim 0.6$) (left-hand scale) implying a quiescent age at
$z=0.6$ ranging from 0 to 4 Gyrs in 1 Gyr steps (top to bottom). The four
columns show resident subpopulations aging according to the model in Figure 7
as the redshift (lower scale) changes. In each panel the solid black line
connects the median sizes in 10 equally populated mass bins. The errors are
bootstrapped. The region between the broken dashed lines shows the intrinsic
scatter and includes 68% of the sample in each mass bin. The central black
dash-dotted line and gray region trace the best-fit linear relation between
stellar mass and size along with the associated uncertainties. The equation
for the best-fit size-stellar mass relation in the legend of each cell
includes the effective (half-light) radius $R_{e}$ measured in kpc and stellar
mass $M_{\ast}$ in units of $10^{11}~{}M_{\sun}$. Vertical black arrows
indicate the pivot stellar mass from Figure 5.

In five rows of four panels each, Figure 8 delineates the evolution of
quiescent objects with ages $\tau=[0,1,2,3,4]$ Gyr at $z\sim 0.6$
(corresponding to the curves and bins in Figure 7) by using $D_{n}4000$ to
identify the descendants of these objects at successive redshift (or,
equivalently, look back time) intervals covering the range $0.2<z<0.5$. We use
the model in Figure 7 to identify the descendants based on their measured
$D_{n}4000$. This procedure removes all of the quiescent objects that only
enter the quiescent population at redshift $z<0.5$. The hashed boxes in Figure
7 delineate the samples of quiescents that we use to trace the evolution of
the population in each redshift bin. The $D_{n}4000$ limits of the bins are
defined by the values where of the model evolutionary curves at the redshift
limits of successive bins. In other words, the model defines the binning in
$D_{n}4000$. We can then explore the dependence of the size – mass relation on
redshift for quiescent objects in these bins.

In each row of Figure 8, the slopes (each panel shows the size – mass
relation) of the relations remain roughly constant. The slope is poorly
constrained at the highest redshift where the sample is the smallest. The
steady increase in the normalization of the relation with decreasing redshift
(left to right) along each row demonstrates the size growth of the resident
population.

Figure 9: Median half-light radius along the major axis as a function of
median stellar mass in 10 equally populated mass bins for quiescent HectoMAP
galaxies in four redshift/lookback time intervals. The colors are as in Figure
7 and denote galaxies that have been quiescent for times ranging from 0 to 4
Gyrs at $z\sim 0.6$. We track quiescent aging over the $0.2<z<0.6$ redshift
interval (Figures 7 and 8). Errors in the median values are bootstrapped. The
trend in galaxy size with stellar mass is essentially identical for all
quiescent resident populations regardless of the $D_{n}4000$ (or age since
quiescence) at $z\sim 0.6$. Vertical arrows indicate pivot stellar masses from
Figure 5.

To highlight these results, Figure 9 shows a superposition of panels in
intervals of fixed lookback time from Figure 8. The result is striking: for
the entire range of ages for the parent $z\sim 0.6$ population, the relations
for their descendants are essentially coincident at every lookback time probed
by HectoMAP. The size growth of the resident population is independent of the
galaxy age at $0.5<z<0.6$ once the impact of newcomers to quiescence is
removed at lower redshifts ($z<0.5$). In other words, the growth of the
resident population depends only on the lookback time (or, equivalently,
change in redshift) since the objects joined the quiescent population.

Figure 10: Top panels: Evolution in the slope $\alpha$ (left panel) and zero
point $A$ (right panel) for the size-stellar mass relations
$\log(R_{e}/1\,\mathrm{kpc})=A+\alpha\times\log(M_{\ast}/10^{11}\,M_{\sun})$
(same colors as Figures 8 and 9. The change in zero point demonstrates the
significant size growth. Bottom panels: Evolution in the parameters of the
size-stellar mass relation for newcomers (blue), aging quiescents from Figure
8 (red) and the entire quiescent sample from Figure 5 (black). The colored
lines show the error-weighted best fits for the evolution for the slope and
zero point of the size – mass relation as a function of redshift.

The upper panels of Figure 10 enhance the comparison in Figure 9 by displaying
the dependence of the parameters of the size – mass relation on redshift for
the $D_{n}4000$ subsamples in Figure 8. The slope (left panel) is insensitive
to redshift within the error but the size (right panel) grows significantly as
the universe ages. As suggested in Figure 8, the behavior of the size – mass
relation is remarkably similar for all of the descendant subsamples. In other
words, for galaxies that were quiescent at $z\sim 0.6$, size growth depends
only on the redshift and is independent of the time when the object became
quiescent.

The lower panels of Figure 10 follow the evolution of the size – mass relation
parameters for several subsamples of the total sample in Figure 5: the
newcomers (blue), the aging quiescents of Figure 8 (red), and the entire
quiescent sample (Figure 5; black). Table 2 lists the parameters of the size –
mass relations in these two panels. The entire sample of HectoMAP quiescent
objects (black squares) includes newcomers, the population of residents aging
from $z=0.6$ (Figure 8), and the descendants of newcomers that join the
quiescent population at redshifts $\lesssim 0.5$. The newcomers are $\sim
15\%$ of the total population and the aging quiescent population from Figure 8
constitute $\sim 40\%$ of the quiescent sample.

For all of the samples highlighted in the lower panels of Figure 10, the slope
of the size – mass relation decreases as the universe ages (lower left) and
the zero point of the relation increases (lower right). The fits listed in the
legend take the errors in the individual points into account.

The trend in the zero point $A$ of the size – mass relation with redshift
represents the characteristic size growth of a $M_{\ast}=10^{11}\,M_{\sun}$
galaxy. For the entire HectoMAP quiescent sample, the zero point trend echoes
previous results. The change in slope of the relation ($\alpha$) differs from
the constant value (within the uncertainties) reported in some earlier studies
(Section 3). More recent studies, however, reveal a increase in $\alpha$ with
redshift between $z\sim 0.3$ and $z\sim 1.1$ (Kawinwanichakij et al., 2021;
Hamadouche et al., 2022). We discuss the trend in $\alpha$ for the HectoMAP
quiescent sample further in Section 5.2.

Both the slope and zero point of the size – mass relation show the steepest
change for the newcomers. The slope and zero point change less steeply for the
aging quiescents of Figure 8 than for the entire quiescent sample of Figure 5.
The combination of newcomers and their descendants and related ‘downsizing in
time’ (Cowie et al., 1996; Neistein et al., 2006) account for this difference
(e.g., Figures 8 and 9 of Haines et al. (2017) illustrate the change with
redshift in the distribution of star forming and quiescent galaxies within the
size – mass plane between $0.5<z<1$ and $z\sim 0$). At fixed stellar mass
newcomers and their descendants are larger than the typical resident objects
that evolve as quiescent galaxies from $z=0.6$. These younger objects produce
a shallower slope and a steeper increase in the zero point. Quantitatively,
the characteristic size of an $M_{\ast}=10^{11}\,M_{\sun}$ quiescent galaxy in
the full mass limited sample increases 48% between $z=0.6$ and $z=0.2$. For
the aging resident population that is already quiescent at $z=0.6$ the
fractional increase in size is 37% over the same redshift range. The
contribution of newcomers and their descendants causes characteristic size
growth of 48% rather than by 37% from $z=0.2$ to $z=0.6$.

Table 2: Parameters of the size - mass relation $\log(R_{e}/1\,\mathrm{kpc})=A+\alpha\times\log(M_{\ast}/10^{11}\,M_{\sun})$ for the subsamples of quiescent HectoMAP population Sample | Redshift range | Slope $\alpha$ | Zero point $A$
---|---|---|---
| $0.2<z<0.3$ |  |
Quiescent sampleaaGalaxies with D${}_{n}4000>1.5$ from Figure 5. |  | $0.661\pm 0.009$ | $0.799\pm 0.003$
NewcomersbbGalaxies entering quiescent population ($1.5<\mathrm{D}_{n}4000<1.6$). |  | $0.500\pm 0.035$ | $0.887\pm 0.011$
Aging populationccSubpopulation of galaxies that have been quiescent for $0-4$ Gyr at $z\sim 0.6$ and their descendants (Figures 7 and 8). |  | $0.740\pm 0.011$ | $0.777\pm 0.003$
| $0.3<z<0.4$ |  |
Quiescent sampleaaGalaxies with D${}_{n}4000>1.5$ from Figure 5. |  | $0.749\pm 0.011$ | $0.762\pm 0.020$
NewcomersbbGalaxies entering quiescent population ($1.5<\mathrm{D}_{n}4000<1.6$). |  | $0.722\pm 0.047$ | $0.842\pm 0.008$
Aging populationccSubpopulation of galaxies that have been quiescent for $0-4$ Gyr at $z\sim 0.6$ and their descendants (Figures 7 and 8). |  | $0.782\pm 0.014$ | $0.740\pm 0.003$
| $0.4<z<0.5$ |  |
Quiescent sampleaaGalaxies with D${}_{n}4000>1.5$ from Figure 5. |  | $0.788\pm 0.014$ | $0.720\pm 0.003$
NewcomersbbGalaxies entering quiescent population ($1.5<\mathrm{D}_{n}4000<1.6$). |  | $0.656\pm 0.052$ | $0.788\pm 0.007$
Aging populationccSubpopulation of galaxies that have been quiescent for $0-4$ Gyr at $z\sim 0.6$ and their descendants (Figures 7 and 8). |  | $0.825\pm 0.020$ | $0.714\pm 0.004$
| $0.5<z<0.6$ |  |
Quiescent sampleaaGalaxies with D${}_{n}4000>1.5$ from Figure 5. |  | $0.882\pm 0.029$ | $0.614\pm 0.010$
NewcomersbbGalaxies entering quiescent population ($1.5<\mathrm{D}_{n}4000<1.6$). |  | $1.089\pm 0.133$ | $0.616\pm 0.030$
Aging populationccSubpopulation of galaxies that have been quiescent for $0-4$ Gyr at $z\sim 0.6$ and their descendants (Figures 7 and 8). |  | $0.830\pm 0.039$ | $0.631\pm 0.012$
Figure 11: Distributions of stellar mass (left panels), size (central panels),
and $D_{n}4000$ index (right panels) for two subpopulations of intermediate-
redshift quiescent galaxies: the aging population (with $D_{n}4000$ values
defined in Figure 7, top panels) and newcomers (with $1.5<D_{n}4000<1.6$,
bottom panels). Different colors correspond to the four redshift bins within
the $0.2<z<0.6$ range of the HectoMAP survey. The histograms are stacked. Each
histogram is normalized to the most populated bin. Vertical arrows show the
pivot stellar masses from Figure 5. Dashed lines show the median stellar mass
(above the pivot point at lower redshift) in each redshift bin. Comparison of
median masses in each redshift bin of the aging and newcomer populations
reveals the more prominent downsizing of the newcomers.

The histograms in Figure 11 underscore the results in the lower panels of
Figure 10. They show the distributions of stellar mass, size, and $D_{n}4000$
for the aging population (top row) and for the newcomers (bottom row). The
colored bars in the stacked histograms denote the discrete redshift slices.
The shifts of the newcomers toward lower stellar mass (colored dashed vertical
lines indicate the median mass for each redshift bin) and larger sizes are
evident in each redshift bin. The narrow $D_{n}4000$ distribution in the lower
right panel defines the sample of newcomers at each redshift.

Downsizing is a larger effect in the newcomer population (lower left) relative
to the aging residents (upper left). This effect is expected, but its
amplitude has not been measured previously in this redshift range. From $z\sim
0.55$ to $z\sim 0.25$ the change in the median stellar mass for aging
population is 0.1 dex smaller than for the population of newcomers.

Figure 12: Top panel: Intrinsic scatter (i.e., $\sigma$ of the fitted size –
mass relation $\log(R_{e})=A+\alpha\times\log(M_{\ast})+\sigma$; filled
circles) and the standard deviation in measured sizes after removal of the
best-fit relation (observed scatter, open circles) for the aging sub-
populations from Figure 8. Bottom panel: The intrinsic and observed scatter
for the three quiescent samples from the bottom panels of Figure 10.

A number of studies investigate the scatter around the size – mass relation as
a measure of the characteristic evolution of the quiescent population (e.g.,
Shen et al., 2003; van der Wel et al., 2014; Mowla et al., 2019; Yang et al.,
2021). Figure 12 examines the behavior of both the intrinsic and observed
scatter for the size – mass relations of the populations aging from $z=0.6$
(upper panels) and the total aging sample, newcomers, and the total quiescent
sample (lower panels). The intrinsic scatter (the $\sigma$ of the fitted size
– mass relations) is insensitive to redshift, in agreement with Shen et al.
(2003) and van der Wel et al. (2014). The observed scatter (the standard
deviation in observed sizes after removal of the fitted size – mass relation)
always exceeds the intrinsic scatter as expected. The observed scatter
increases mildly with redshift due to the subtle increase $D_{n}4000$
measurement uncertainty for $z>0.3$ (Damjanov et al., 2022).

The large HectoMAP quiescent sample enables the derivation of robust
parameters that characterize the size – mass relation for a range of
subsamples of the data. These subsamples encompass the newcomers and the full
mass-limited quiescent sample. We next explore the size – mass relations for
the HectoMAP quiescent subsamples relative to the minor merger model (Naab et
al., 2009; Bezanson et al., 2009; Hopkins et al., 2010) that could drive the
growth of quiescent objects in the redshift range $0.2<z<0.6$.

##  5 Discussion

Figure 13: Evolution in the characteristic size (the zero point, A) of an
$M_{\ast}=10^{11}\,M_{\sun}$ for aging quiescent galaxy samples (colored dots;
upper right panel of Figure 10). The gray shaded area highlights the range in
size growth predicted by Naab et al. (2009) minor merger models. The blue
squares show the more dramatic change in the characteristic size of a newcomer
$M_{\ast}=10^{11}\,M_{\sun}$ galaxy with $1.5<\mathrm{D}_{n}4000<1.6$ (blue
squares in the lower right panel of Figure 10). The black squares show the
behavior of the characteristic $R_{e}$ for the entire mass-complete quiescent
sample as do black squares in Figure 10. The aging quiescent population (not
the less abundant newcomers) dominates the overall quiescent size growth.

###  5.1 Consistency with Minor Merger Driven Growth at $0.2<z<0.6$

HectoMAP provides a dense sampling of the size evolution of the quiescent
population for $0.2<z<0.6$. Spectroscopy provides the redshift and the age
indicator $D_{n}4000$; HSC images provide the sizes. The density of the sample
and the inclusion of a population age indicator enable study of size growth
for populations that have different ages at the initial redshift $z\sim 0.6$.
The data also enable removal of newcomers from the aging quiescent sample.
HectoMAP shows that the growth of the aging quiescent population is
independent of the population age.

In the emerging consensus view, major mergers probably play a significant role
in the size evolution of the quiescent population at $z\gtrsim 1$, but at
lower redshift, minor mergers dominate (e.g., Trujillo et al., 2011; Newman et
al., 2012; López-Sanjuan et al., 2012; McLure et al., 2013; Conselice, 2014;
Greene et al., 2015; Oh et al., 2017; Hill et al., 2017; Matharu et al., 2019;
Zahid et al., 2019; Mosleh et al., 2020). In the redshift range $0.2<z<0.6$
covered by HectoMAP several lines of evidence already support the dominance of
minor mergers. Lotz et al. (2011, Table 4) provide a rough estimate of the
relative numbers of major and minor mergers per $M_{\ast}>10^{10}\,M_{\sun}$
galaxy at $z<1.5$. Over the redshift range of our survey each galaxy
experiences about 1 minor merger; major mergers are a factor of 4-5 less
common. The HectoMAP data enable further examination of the consistency
between the minor merger driven growth paradigm and the data.

To compare the data with a model for minor merger driven size evolution, the
gray shades area in Figure 13 highlights the range of theoretical predictions
based on the models of Naab et al. (2009). Section 2 of their paper describes
predictions based on the virial theorem and energy conservation. To compare
the model with the data we need an estimate of the minor merger rate per Gyr.
Lotz et al. (2011) summarize a range of observational studies in their Table
4. For the HectoMAP redshift interval, the Lotz et al. (2011) typical minor
merger rate estimate per galaxy is $\sim 0.3$ Gyr-1 or, equivalently, $\sim 1$
merger in the redshift range. For these minor mergers a reasonable mass ratio
for the merging systems is $0.1-0.25$.

The gray shaded region in Figure 13 covers the range of Naab et al. (2009)
model predictions (their equation 4). The underlying model assumes that the
initial and final systems are in virial equilibrium and that energy is
conserved during the minor merger.

The lower limit of the gray region in Figure 13 is for an accreted/central
mass ratio (Eq. 4 of Naab et al., 2009), $\eta=0.1$, and the upper limit is
for $\eta=0.25$. We assume that $\epsilon$, the ratio between the internal
velocity dispersion of the accreted material and the internal velocity
dispersion of the original, central galaxy is $\ll 1$; we thus neglect
$\epsilon$.

As an interesting counterpoint, Figure 13 shows the remarkably steep
dependence of the size – mass relation zero point on redshift for newcomers
(blue squares). The apparent size evolution of this population is obviously
steeper than the minor merger model prediction. Of course, the properties of
these populations should not be minor merger driven. The change in size
results from the lower stellar mass, larger objects that are just entering
quiescence as the universe ages (Figure 11).

For the resident population aging from $z\simeq 0.6$ (colored circles), the
complete overlap between the Naab et al. (2009) model predictions and the data
supports the conjecture that minor mergers are the main driver of the size
growth of the aging quiescent population. The black squares show the evolution
of the total quiescent population which includes this aging population along
with objects that first become quiescent at $z\lesssim 0.5$. As demonstrated
by the analysis accompanying Figure 10 (Section 4.2), inclusion of newcomers
in the population does not change the size evolution enough to depart from the
minor merger model predictions. As we note in Section 4.2, only 15% of the
total population are newcomers and they change the size growth from $z\sim
0.6$ to $z\sim 0.2$ by $\sim 11\%$. Recently, Hamadouche et al. (2022) report
a similar percentage of newcomers and demonstrate the dominant effect of minor
mergers based on the observed size, stellar mass, and D${}_{n}4000$ of
quiescent galaxies in the redshift range $0.7\lesssim z\lesssim 1.1$.

The lack of dependence of size evolution of resident descendants on the
quiescent age at $z=0.6$ (colored circles) may also be a signature of minor
merger driven growth. These mergers deposit material primarily in the outer
regions of the galaxy (Bezanson et al., 2009; Naab et al., 2009; Hopkins et
al., 2009a; Oser et al., 2010) and they are thus unlikely to affect the
central stellar population age or, equivalently, the $D_{n}4000$ measured in
the Hectospec 1.5′′ fiber aperture. The central population age thus can be
consistent with passive evolution models even though there is size growth
(Whitney et al., 2021). Spatially resolved spectroscopy in larger apertures
might test this conjecture by revealing the younger stellar populations
associated with merger events.

###  5.2 Limitations

The large complete HectoMAP sample of quiescent galaxies traces the structural
evolution of these objects at $z<0.6$ segregated by the average age of their
stellar population. The uncertainties in the $D_{n}4000$ index, a proxy for
the average stellar population age, have no significant effect on the analysis
(Appendix A).

As in any magnitude-limited survey, the $r=21.3$ limit translates to a stellar
mass limit that is a function of redshift (Section 2.2, Figure 1). In the
highest redshift bin ($0.5<z<0.6$), the mass-complete sample covers stellar
masses $M_{\ast}>10^{11}\,M_{\sun}$. In the lowest redshift bin ($0.2<z<0.3$)
the limiting mass of the complete sample reaches a limiting stellar mass of
$\lesssim 10^{10}\,M_{\sun}$.

By probing lower-mass quiescent systems (and thus a larger dynamic range of
stellar masses) with decreasing redshift we demonstrate the dependence of the
size – mass relation on $D_{n}4000$ (Figure 5). Including of lower stellar
mass, generally younger galaxies, changes the stellar mass range accessed as a
function of redshift and makes the slope of the size – mass relation appear
shallower at lower redshift (Figure 10).

The change in the slope of the size – mass relation is most dramatic for the
newcomers (from $\alpha(z\sim 0.55)\sim 1$ to $\alpha(z\sim 0.25)\sim 0.5$,
Table 2) because lower mass galaxies at lower redshift are also younger. The
best-fit model for the relation between $\alpha$ and redshift for newcomers
from the lower left panel of Figure 10 suggests a factor of $\gtrsim 2.7$
decrease over the redshift interval of our survey (or $\sim 3.3$ Gyr of cosmic
time). The slope of the relation decreases at a significantly slower rate for
quiescent galaxies aging from $z\sim 0.6$ (from $\alpha(z\sim 0.55)\sim 0.85$
to $\alpha(z\sim 0.25)\sim 0.75$). The best-fit evolutionary model for this
population (Figure 10) suggests a decrease in $\alpha$ of only 30% over the
same redshift interval. The descendants of galaxies with $D_{n}4000>1.5$ at
$z\sim 0.6$ are among the most massive galaxies in each redshift bin and thus
they are less susceptible to the effects of the broader stellar mass range
sampled with decreasing redshift.

The evolution of the zero point (characteristic size) for an
$M_{\ast}=10^{11}\,M_{\sun}$ quiescent galaxy, is robust to changes in the
range stellar mass range sampled as a function of redshift; the highest-
redshift points in the lower right panel of Figure 10 only increase the
uncertainty in the increase of characteristic size with redshift. Deeper
spectroscopic surveys that explore a broad and fixed stellar mass range as a
function of redshift will reduce these systematic issues and will enable a
more robust probe of the evolution in the slope of the quiescent size – mass
relation.

The evolution in the characteristic size of quiescent objects is based on the
change in the observed light profiles of the objects. In studies that use
spatially resolved spectral energy distribution (SED) fitting and/or empirical
relationship between the mass-to-light ratio $M/L$ and observed color (e.g.,
Szomoru et al., 2013; Chan et al., 2016; Suess et al., 2019; Mosleh et al.,
2020; Miller et al., 2022) the inferred mass-weighted size (or half-mass size
$R_{mass}$) of galaxies is consistently smaller than the size based on the
observed light profiles (i.e., the half-light radius $R_{e}$ we measure).
These studies rely on space-based observations (Hubble Space Telescope or,
very recently, James Webb Space Telescope, Suess et al. 2022) and almost all
of them target high-redshift ($z>1$). The results range from constant
$R_{mass}/R_{e}$ ratios, that do not alter the evolutionary trend in
characteristic size, to a reduced growth of half-mass radius with respect to
the half-light size due to the trend in $R_{mass}/R_{e}$ with redshift (for
more details, see Miller et al., 2022). On the other hand, a recent comparison
among different galaxy size estimates based on mock images of $z=0.1$ galaxies
in the EAGLE simulations demonstrates that the light-weighted structural
properties of quiescent galaxies we use here are a good proxy for their mass-
weighted properties (de Graaff et al., 2022).

With its large-area coverage and dense spectroscopy of quiescent systems at
$z<0.6$, HectoMAP has the potential to resolve the tension between results
based on various half-mass size estimates. However, the ground-based HSC SSP
imaging of the region is not adequate to this task. The resolving power of the
HSC SSP images in $grizy$ (quantified by the size of the PSF) varies
significantly across the field and among filters333https://hsc-
release.mtk.nao.ac.jp/doc/index.php/quality-assurance-
plots__pdr3/#wide_hectomap1_QA_calPsfDiff.png-Shape,444https://hsc-
release.mtk.nao.ac.jp/doc/index.php/quality-assurance-
plots__pdr3/#wide_hectomap2_QA_calPsfDiff.png-Shape. Robust estimates of
HectoMAP galaxy sizes from spatially resolved stellar mass maps require multi-
wavelength imaging with uniformly high (diffraction-limited) resolution over
the full 55 deg2 area of the survey. A photometric dataset of this quality may
only become available with future space missions (e.g., Euclid; Amiaux et al.,
2012).

Future deep and dense spectroscopic surveys with the next-generation
instruments (DESI, Zhou et al. 2022; MOONS, Taylor et al. 2018; PFS, Greene et
al. 2022) will reach beyond $z\sim 1$ enabling an extensive exploration of the
relative roles of major and minor mergers in the evolution of the quiescent
population. We show that, in agreement with the broad consensus view, minor
mergers are the dominant drivers of the size growth in quiescent population at
$z<0.6$ (e.g., Ownsworth et al., 2014). In this redshift interval, minor
mergers are several times more frequent than major mergers (Lotz et al.,
2011). Future spectroscopic surveys will probe the higher redshift, earlier
epoch where major mergers may become more dominant drivers of galaxy growth
through mass accretion (Mantha et al., 2018, and references therein). Major
mergers have a stronger effect on the central morphology, dynamics, and
stellar population of quiescent galaxies (e.g., Toomre, 1977; White, 1978;
Hopkins et al., 2009b; Bois et al., 2011) that should be directly observable
at $z\gtrsim 1$.

##  6 Conclusion

Large, complete, mass limited samples of quiescent galaxies provide an
increasingly detailed picture of their evolution. The HectoMAP survey combines
MMT spectroscopy and Subaru HSC imaging to yield a mass limited sample of
30,231 quiescent objects spanning the redshift range $0.2<z<0.6$. The
spectroscopy provides a redshift along with the age indicator, $D_{n}4000$.
HSC SSP $i-$band imaging currently provides half-light radii for quiescent
objects in 80% of the total area covered by HectoMAP. We derive stellar masses
from SDSS photometry. The HectoMAP quiescent sample is 6.6 times the size of
the largest comparable sample available previously. We explore the size – mass
relation for the entire dataset and for well-defined quiescent subpopulations.

The global size – mass relations for the HectoMAP survey underscore the known
increase in half-light radius with decreasing redshift. The large size of the
data set along with the availability of the age indicator, $D_{n}4000$,
reveals a dependence of the size – mass relation on $D_{n}4000$ that is
increasingly striking as the universe ages. As at $z\sim 0$ (based on SDSS,
Zahid & Geller, 2017) and $z\sim 0.7$ (based on LEGA-C , Wu et al., 2018),
$0.2<z<0.6$ quiescent galaxies with smaller $D_{n}4000$ are larger at fixed
stellar mass.

The HectoMAP quiescent sample is large enough to extract several subsamples
based on $D_{n}4000$ that elucidate aspects of the size evolution of the total
quiescent population. We define a sample of newcomers with $1.5<D_{n}4000<1.6$
at each redshift. These objects have just joined the quiescent population.
They constitute $\sim 15$% of the total HectoMAP quiescent sample. We examine
the evolution of this population alone by comparing samples in two redshift
bins centered at $z\sim 0.25$ and $z=0.45$. Over this $\sim 1.5$ Gyr interval,
the size growth of the newcomers is $\Delta\log{R_{e}}=0.11\pm 0.025$. In the
higher redshift bin there is a tail toward small $R_{e}$ that is absent at
lower redshift. This comparison is a unique and direct measure of the impact
of down-sizing (e.g., Figure 9 of Haines et al., 2017) in the population
entering quiescence.

At the limiting redshift of HectoMAP ($z\sim 0.6$), we segregate the quiescent
population in 5 bins based on the correspondence between age since the onset
of quiescence and $D_{n}4000$. We use the FSPS model (Conroy & Gunn, 2010) to
connect the descendants of these objects in three lower redshift intervals
with their progenitors. The objects in the five $z\sim 0.6$ age bins and their
descendants constitute 40% of the total HectoMAP quiescent sample and we
identify them as the aging quiescent subsample. Comparison of the size – mass
relations for the five $z=0.6$ age intervals yields the novel and intriguing
result that the evolution of the relation is independent of the quiescent
“age” at $z\sim 0.6$. In other words, the size growth depends only on the time
interval where we track the evolution.

We compare the parameters of the size – mass relations for the newcomers and
aging quiescents with the total HectoMAP quiescent population. The size growth
is significantly steeper for the newcomers. In the full quiescent sample, the
size of a fiducial $10^{11}\,M_{\odot}$ galaxy increases by 48% and the size
of the aging quiescents increase by 37%. Thus the newcomers contribute 11% to
the growth. This difference is often called progenitor bias (e.g., Saglia et
al., 2010); the large, complete HectoMAP sample provides a direct measure of
its impact.

The slope of the size – mass relation appears to become shallower as the
universe ages. The effect is largest for the newcomers. This result may depend
at least in part on the change in the range of stellar masses we sample as a
function of redshift. At lower redshift the sample includes objects with lower
stellar mass where the size – mass relation is known to flatten. Further
exploration of this issue obviously requires a deeper sample that encompasses
a broad and fixed stellar mass range throughout the redshift range.

We compare the suite of HectoMAP size – mass relations with the widely
supported minor merger model for its evolution. The evolution of the newcomers
alone is too steep to lie within the model boundaries. However, the
contribution of these newcomers to the growth of the entire population has a
negligible effect on the overall agreement between the evolution of the
quiescent population and the model.

The HectoMAP quiescent sample demonstrates the power of a large, complete
mass-limited quiescent sample for disentangling the impact of different
subpopulations on the evolution of the size – mass relation. Deeper redshift
surveys will further constrain the evolution by enabling exploration of large,
fixed stellar mass ranges over a sizeable redshift range. At larger redshift,
the larger impact of major mergers on growth can also be explored. Ultimately
both half-light and half-mass radii for the quiescent population will be
derived robustly from space-based photometry. Taken together these future
observations will fill out a complete picture of the evolution of the
quiescent population.

I.D. acknowledges the support of the Canada Research Chair Program and the
Natural Sciences and Engineering Research Council of Canada (NSERC, funding
reference number RGPIN-2018-05425). J.S. is supported by the CfA Fellowship.
Y.U. is supported by the U.S. Department of Energy under contract number DE-
AC02-76-SF00515. M.J.G. acknowledges the Smithsonian Institution for support.
I.D.A. gratefully acknowledges support from DOE grant DE-SC0010010 and NSF
grant AST-2108287. This research has made use of NASA’s Astrophysics Data
System Bibliographic Services.

The Hyper Suprime-Cam (HSC) collaboration includes the astronomical
communities of Japan and Taiwan as well as Princeton University. The HSC
instrumentation and software were developed by the National Astronomical
Observatory of Japan (NAOJ), the Kavli Institute for the Physics and
Mathematics of the Universe (Kavli IPMU), the University of Tokyo, the High
Energy Accelerator Research Organization (KEK), the Academia Sinica Institute
for Astronomy and Astrophysics in Taiwan (ASIAA), and Princeton University.
Funding was contributed by the FIRST program from the Japanese Cabinet Office,
the Ministry of Education, Culture, Sports, Science and Technology (MEXT), the
Japan Society for the Promotion of Science (JSPS), Japan Science and Technol-
ogy Agency (JST), the Toray Science Foundation, NAOJ, Kavli IPMU, KEK, ASIAA,
and Princeton University. This paper makes use of software developed for the
Large Synoptic Survey Telescope (LSST). We thank the LSST Project for making
their code available as free software at http://dm.lsst. org. This paper is
based [in part] on data collected at the Subaru Telescope and retrieved from
the HSC data archive system, which is operated by Subaru Telescope and
Astronomy Data Center (ADC) at National Astronomical Observatory of Japan.
Data analysis was in part carried out with the cooperation of Center for
Computational Astrophysics (CfCA), National Astronomical Observatory of Japan.

##  Appendix A The impact of the error in D${}_{n}4000$

To evaluate the effect of spectroscopic measurement uncertainties on the
quiescent evolutionary trends, we create a set of ten simulated parent
quiecent galaxy samples. We simulate ten full simulated datasets of the 79175
individual D${}_{n}4000$ measurements for HectoMAP galaxies within the
$i-$band HSC coverage. We draw ten values from a Gaussian probability
distribution function (pdf) for each $D_{n}4000$ measurement. Each Gaussian is
centered on the measured value and has a scale scale equal to the measurement
error. Thus the simulation captures the increase in the $D_{n}4000$ error for
fainter, higher redshift objects in the original HectoMAP dataset.

We then follow the analysis steps throughout this paper based on these
simulated parent quiescent samples. We evaluate the distribution of the
resulting size - mass relation parameters for all of the quiescent galaxy
(sub)sets (newcomers, aging population, and the entire quiescent population)
in the four redshift bins from Figures 5, 8, and 10.

Figure 14 shows the bottom two panels of Figure 10 with additional points
(open squares with errorbars) corresponding to the simulation results. Each
open square show the median value of the derived size – mass relation
parameter (slope $\alpha$ or zero point $A$). The errorbar shows the
$[16,84]\%$ spread around the median.

In all redshift bins and in both panels the difference between the $y-$ axis
positions of the filled squares (original galaxy samples) and open squares
(simulated samples) is minimal and firmly within the uncertainties in
parameters from the original galaxy sample. The simulations demonstrate that
the observed evolutionary trends are not affected by the spectroscopic
measurement uncertainties.

Figure 14: Evolution in the size-stellar mass relation for newcomers (blue),
aging quiescents (red) and the entire quiescent sample (black) as in Figure
10. The open squares with errorbars show the median values for fitted
parameters and their $[16,84]\%$ range based on ten simulated samples with
perturbed D${}_{n}4000$ indices. The colored lines show the error-weighted
best fits for the evolution for the slope and zero point of the size – mass
relation as a function of redshift from Figure 10. The points based on the
simulated samples follow the evolutionary trends in the slope and zeropoint we
derive with the original samples.

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