# The Nature of Low Surface Brightness Galaxies in the Hyper Suprime-Cam
Survey

Jenny E. Greene Department of Astrophysical Sciences, Princeton
University,Princeton, NJ 08544, USA  Johnny P. Greco  Center for Cosmology and
AstroParticle Physics (CCAPP), The Ohio State University, Columbus, OH 43210,
USA  Andy D. Goulding Department of Astrophysical Sciences, Princeton
University,Princeton, NJ 08544, USA  Song Huang Department of Astrophysical
Sciences, Princeton University,Princeton, NJ 08544, USA  Department of
Astronomy, Tsinghua University, Beijing 100084, China  Erin Kado-Fong
Department of Astrophysical Sciences, Princeton University,Princeton, NJ
08544, USA  Shany Danieli NASA Hubble Fellow  Department of Astrophysical
Sciences, Princeton University,Princeton, NJ 08544, USA  Jiaxuan Li Department
of Astrophysical Sciences, Princeton University,Princeton, NJ 08544, USA  Ji
Hoon Kim Subaru Telescope, National Astronomical Observatory of Japan,
National Institutes of Natural Sciences (NINS), 650 North Aohoku Place, Hilo,
HI 96720, USA  SNU Astronomy Research Center, Seoul National University, 1
Gwanak-ro, Gwanak-gu, Seoul 08826, Republic of Korea  Yutaka Komiyama Dept. of
Advanced Sciences, Faculty of Science and Engineering, Hosei University, 3-7-2
Kajino-cho, Koganei-shi, Tokyo 184-8584, Japan  Alexie Leauthaud Department of
Astronomy and Astrophysics, University of California Santa Cruz, 1156 High
St., Santa Cruz, CA 95064, USA  Lauren A. MacArthur  Department of
Astrophysical Sciences, Princeton University,Princeton, NJ 08544, USA
Cristóbal Sifón Instituto de Física, Pontificia Universidad Católica de
Valparaíso, Casilla 4059, Valparaíso, Chile

(December 2021)

###### Abstract

We present the statistical redshift distribution of a large sample of low
surface brightness (LSB) galaxies identified in the first 200 deg2 of the
Hyper Suprime-Cam Strategic Survey Program. Through cross-correlation with the
NASA-SDSS Atlas, we find that the majority of objects lie within $z<0.15$ or
$\sim 500$ Mpc, yielding a mass range of $M_{*}\approx 10^{7}-10^{9}$
$M_{\odot}$ and size range of $r_{\rm eff,g}\approx 1-8$ kpc. We find a peak
in the distance distribution within 100 Mpc, corresponding mostly to $\sim
10^{7}$ $M_{\odot}$ galaxies that fall on the known mass-size relation. There
is also a tail in the redshift distribution out to $z\approx 0.15$, comprising
more massive ($M_{*}=10^{8}-10^{9}$ $M_{\odot}$) galaxies at the larger end of
our size range. We see tentative evidence that at the higher-mass end
($M_{*}>10^{8}$ $M_{\odot}$) the LSB galaxies do not form a smooth extension
of the mass-size relation of higher surface brightness galaxies, perhaps
suggesting that the LSB galaxy population is distinct in its formation path.

##  1 Introduction

Low surface brightness (LSB) galaxies have long tantalized astronomers (LSB
galaxies; e.g., Sandage, 1976; Caldwell & Bothun, 1987; Impey et al., 1988;
Bothun et al., 1991; McGaugh et al., 1995; Dalcanton et al., 1997a). A
significant fraction of galaxies could be LSB (e.g., Dalcanton et al., 1997b).
If galaxies are selected by surface brightness alone, there is a wide range of
mass and morphology that can result. Specifically, in this work we focus on
galaxies with $M_{*}<10^{9.5}$ $M_{\odot}$ that have no bulges, as opposed to
so-called “giant” LSB disk galaxies at higher mass (such as the prototype
Malin 1 Bothun et al., 1987).

These low-mass LSB galaxies have been of interest because they may result from
the high angular momentum tail of the halo distribution (e.g., Dalcanton et
al., 1997a; Amorisco & Loeb, 2016), or extreme feedback events (e.g., El-Badry
et al., 2016; Di Cintio et al., 2017; Chan et al., 2018). Others have
suggested that it is interactions that make galaxies larger, either galaxy
merging at early times (e.g., Wright et al., 2021), falling into the cluster
environment (van Dokkum et al., 2015a, b; Safarzadeh & Scannapieco, 2017),
tidal effects (Iodice et al., 2021; Benavides et al., 2021), or (very likely)
multiple paths (e.g., Papastergis et al., 2017). Interestingly, the intrinsic
shapes of LSB galaxies appear to be thicker than “normal” dwarfs at matched
stellar mass, and this shape difference appears similar across environments
(Kado-Fong et al., 2021).

Much of the recent excitement to revisit this population has been driven by
improvements in available imaging, with wide-format CCDs on small (e.g.,
Abraham & van Dokkum, 2014; Martínez-Delgado et al., 2016; Carlsten et al.,
2020) and large (e.g., Aihara et al., 2018a; Dey et al., 2019) aperture
telescopes generating wide, deep, and high-resolution imaging data sets. Very
LSB galaxies reaching central surface brightness limits of 26-29 mag arcsec-2
are being uncovered in unprecedented numbers by modern surveys. Two types of
searches are worth highlighting here. A recent paper that reignited excitement
around LSB galaxy searches identified a very large number of such galaxies
around the Coma cluster (van Dokkum et al., 2015a). While prior work had
identified isolated examples of such galaxies (Sandage, 1976), the exquisite
surface brightness sensitivity of the Dragonfly telescope array revealed a
large population of LSB galaxies in cluster environments, which have since
been studied further (e.g., Koda et al., 2015; Mihos et al., 2015; Muñoz et
al., 2015; Yagi et al., 2016; van der Burg et al., 2016; Martínez-Delgado et
al., 2016; Lee et al., 2017) including down to groups (van der Burg et al.,
2017) and the field (e.g., Román & Trujillo, 2017; Leisman et al., 2017; Greco
et al., 2018a; Román et al., 2019).

The benefit of focusing on rich environments is the ability to assume a
distance, and thus measure physical properties. However, we are then left
wondering whether the cluster environment is a requirement to make such
extended systems. Blind searches across wide areas are complementary, and in
this paper we focus on deriving the distance distribution for one of the first
blind modern searches for LSB galaxies selected in the stellar continuum,
Greco et al. (2018b, G18 hereafter). There are other larger-area searches
(Zaritsky et al., 2019; Tanoglidis et al., 2021; Zaritsky et al., 2021) but
they do not reach the surface brightness sensitivity of G18 (§2.3). Radial
velocities for individual targets also provide robust physical properties
(e.g., van Dokkum et al., 2016; Greco et al., 2018a; Kadowaki et al., 2021),
but are quite expensive and grow prohibitive at the lower surface brightness
end of the samples. A statistical approach to estimate the distance
distribution can complement follow-up of individual objects.

Our goal here is to characterize the distance, mass, and size distribution of
the galaxies presented in G18. We follow Ménard et al. (2013) and use spatial
cross-correlation to constrain the redshift distribution statistically.
Throughout the paper, we assume a hubble constant $H_{0}=70$ km s-1 Mpc-1,
$\Omega_{M}=0.3$, $\Omega_{\Lambda}=0.7$.

Figure 1: Completeness of the G18 sample as a function of size and surface
brightness. Detailed image simulations were run to define these completeness
levels. The completeness is a weak function of size (top histogram) and a
stronger function of surface brightness (side histogram).

##  2 Samples

###  2.1 A Search with Hyper Suprime-Camera

The Hyper Suprime-Camera (HSC; Miyazaki et al., 2018; Komiyama et al., 2018)
on the Subaru Telescope provides one of the largest fields of view on an
8m-class telescope. The HSC Strategic Survey Program (HSC-SSP) is a $\sim 5$
year survey using HSC in five bands ($grizY$) (see Aihara et al., 2018a, for
an overview). The Wide layer ($\sim 1000$ deg2 to $i=25.2$ AB mag $5\,\sigma$
point source detection limit) has weak lensing cosmology as its primary focus
(e.g., Hamana et al., 2020; Miyatake et al., 2021), and is also the basis for
this paper. The 200 deg2 analyzed by G18 fall within the Public Data Release 2
(Aihara et al., 2018b) area. The data are processed using _hscPipe_ (Bosch et
al., 2018).

Figure 2: Distribution in size and surface brightness of the G18 sample. The
surface brightness histogram reveals that the blue sample ($g-i<0.5$) spans a
much narrower range in $\mu_{\rm eff,g}$ than the red ($g-i>0.8$) or green
($0.5<g-i<0.8$) samples. The galaxies of intermediate color fall between the
red and blue samples in their surface brightness distributions. The 50% and
70% completeness contours are shown.

G18 performed an LSB galaxy search with the first $\sim 200$ deg2 of HSC data.
The details of their search are described in §2 of G18, but for completeness
we summarize the main elements of their search here. First, each image is
smoothed with a Gaussian kernel that matches the point spread function (PSF)
with $i\approx 0.6$″. Bright objects are selected as groups of pixels that are
$28\sigma$ above the background level. LSB structures are flagged as groups of
at least 20 pixels that are $3\sigma$ above the background, and these are
linked to bright objects if at least $15\%$ of the pixels in an LSB structure
are above the threshold for the bright object, with these values chosen by
trial and error. With images clean of bright sources and their LSB outskirts,
G18 then identify LSB candidates in the $i-$band images using SExtractor
(Bertin & Arnouts, 1996). The images are convolved with a $1\arcsec$ kernel,
roughly twice the PSF. Larger kernels boost sensitivity but introduce blends
of faint background sources as a major contaminant. The local background is
remeasured on 22″ scales, and groups of 100 total pixels (corresponding to
radii of $\sim$roughly 1-2 times the PSF) with at least $0.7\sigma$ above the
background are chosen as potential sources. Aperture photometry for selection
only is performed with an aperture that is 2.5 times the Kron radius, with
forced photometry performed in the $g,\,r$ bands for color information. The
search is performed in independent $12\arcmin\times 12$′ patches.

Initial candidates are then selected using size and color cuts. Specifically,
the FLUX_RADIUS parameter from SExtractor is restricted to $>2.5$″, since G18
are looking for extended LSB galaxies. This cut alone reduces the candidate
list by two orders of magnitude. At the large end, G18 are limited by the
patch overlap regions (17″) and larger sizes are unreliable, so G18 impose an
upper size cut of 20″. At this stage, G18 also use the forced photometry to
remove very red galaxies. Tens of thousands of candidates remain after this
step. Finally, two-dimensional Sérsic modeling is done using imfit (Erwin,
2015). The final colors are based on a $g-$band model fixed to the structural
parameters from the $i-$band fit. After this step, the $\mu_{\rm g,eff}>24.3$
mag arcsec-2 and $r_{\rm eff}>2.5$″ cuts are applied, leaving 1521 candidates.
Two coauthors visually inspect the candidates to remove tidal features,
cirrus, and clusters of point sources that got through the above cuts.
Undoubtably, some incompleteness is introduced with this step that is not
quantified below. The final candidate list contains 781 galaxies.

###  2.2 Sample Properties

The G18 completeness (§2.3) is shown in Figure 1, which leads to a
distribution in size and surface brightness as shown in Figure 2. The sample
has a wide range in optical color, with a median of $\langle g-i\rangle=0.64$.
The red galaxies (taking the median of all galaxies redder than $g-i=0.64$
yields $\langle g-i\rangle=0.8$) show smooth and round morphologies and span
the full range of surface brightness detected in the sample. The blue galaxies
(median of the blue galaxies is $\langle g-i\rangle=0.5$) are more irregular
in shape but span a narrower range of surface brightness ($\mu_{\rm
eff,g}\lesssim 25.5$ AB mag arcsec-2). The blue LSB galaxies are actively
forming stars, as evidenced by their near-ubiquitous detection with _GALEX_ ,
as well as detected H$\alpha$ in a handful of galaxies for which we have
spectroscopic follow-up (Greco et al., 2018a). Kado-Fong et al. (2021)
investigate the intrinsic shapes of the LSB galaxies in this sample, along
with an H I-selected sample (Leisman et al., 2017). They match the mass ranges
of the H I-selected and “normal” galaxies, finding that the LSB galaxies have
intrinsic axis ratios $C/A$ that are $\sim 1.5-2$ times larger than the normal
dwarfs at matched stellar mass. With a mass distribution from this analysis,
we can check whether this shape holds robustly for optically selected samples
like this one.

Thus far, we have distances for only a handful of galaxies in the G18 sample,
from radial velocities (Greco et al., 2018a) or indirectly from globular
cluster luminosity functions (Somalwar et al., 2020). Deeper investigations
into the nature of these galaxies requires knowing their distances (at least
statistically), which is the goal of the present work. Before we proceed with
the cross correlation analysis, we first need to know the sample completeness,
presented in the next subsection.

###  2.3 Completeness

We have performed image simulations to understand our completeness as a
function of galaxy size and surface brightness. We inject $\sim 350,000$
artificial galaxies with a range of true average surface brightness of
$23\leqslant\mu_{\rm eff,g}\leqslant 28.5$ mag arcsec-2. The simulated
galaxies are modeled as single Sérsic functions, with $2\arcsec\leqslant
r_{e}\leqslant 21\arcsec$, $0.8<n<1.2$, and $0<\epsilon<0.6$ to match the G18
distribution. They are injected into the coadded final frames and recovered
using the software presented in G18.

The completeness as a function of size and surface brightness based on these
tests is shown in Figure 1. The completeness falls off towards higher surface
brightness because the sample was selected to be fainter than $\mu_{\rm
eff,g}>24.3$ mag arcsec-2. While nominally we selected galaxies larger than
$r_{\rm eff}>2.5$″, we are most complete for somewhat larger sizes, $r_{\rm
eff}>4$″, likely due to blending of smaller sources with the background that
makes it hard for the search algorithm to identify them as unique sources. At
larger size, however, the completeness is not very sensitive to size until the
very largest sizes ($r_{\rm eff}>15$ ″) where a mild decrease is seen. We are
$\sim 80\%$ complete to $\mu_{\rm eff,g}<26.5$ mag for $r_{\rm eff}<10$″. In
this size range, we are still $\sim 70\%$ complete at $\mu_{\rm eff,g}=28.5$
mag arcsec-2. For larger sizes, we are slightly less complete with surface
brightness, reaching a maximum completeness of $\sim 70\%$ for $\mu_{\rm
eff,g}<25.5$ mag, and dropping to $\sim 50\%$ at $\mu_{\rm eff,g}=28.5$ mag
arcsec-2. This size dependence is due in part to the search being performed in
individual HSC patches as discussed in §2.1.

We also perform a smaller suite of tests in which the galaxies are injected
prior to sky subtraction. In these simulations, we track the recovery of key
parameters such as Sérsic index, ellipticity, and color. We also test a wider
range of Sérsic indices $0.3<n<2.5$. These tests are detailed extensively in
Greco (2018). Here, we summarize that while we see no dependence of
completeness on Sérsic index, there is a bias in the fits that prefers lower
values of $n<1.5$, even when the injected value is higher (see Figure 5.5 in
Greco, 2018). We also find a rapidly declining completeness as a function of
ellipticity above $\epsilon\approx 0.6$ due to surface brightness projection
effects, suggesting that some edge-on disk galaxies may be missing from the
sample (Figure 5.8; see also discussion in Kado-Fong et al., 2021). We find no
color dependence in the completeness over the range of $g-i$ color observed in
the sample, nor any bias in the color recovery. Finally, Greco (2018) also
test the impact of adding additional structure (e.g., star-forming clumps) to
the blue model galaxies and also find negligible impact to the recovery
fractions.

These completeness levels highlight the power of the HSC survey (and soon the
Vera Rubin Observatory, Ivezić et al. 2019) to perform sensitive searches in
the low surface brightness regime over wide areas and thus a wide range of
environments. For context, earlier searches that quantified the number density
of LSBs (e.g., Dalcanton et al., 1997a) had a maximum completeness of $\sim
40\%$ at $23<\mu_{0}<25.5$ mag arcsec-2. Like our search, their completeness
was very poor for the smallest galaxy sizes $<2$″. van der Burg et al. (2017)
search for LSB galaxies in the ESO Kilo-Degree Survey, and achieve $\sim 80\%$
completeness at $\mu_{\rm eff,r}\approx 25.5$ mag arcsec-2, or roughly
$\mu_{\rm eff,g}\approx 26.1$ mag. Carlsten et al. (2020) used a very similar
algorithm to ours, and with the Canada France Hawai’i Telescope Legacy Survey
data were able to reach $\mu_{0}\sim 26.5$ mag arcsec-2 at $\sim 50\%$
completeness, which is roughly similar to $\mu_{\rm eff}\sim 25.8$ mag
arcsec-2 for a Sérsic index of $n=1$ (e.g., Greco et al., 2018b). Finally,
Tanoglidis et al. (2021) performed a wide-area search using the Dark Energy
Survey (DES) images (Abbott et al., 2018). Kado-Fong et al. (2021) compare the
surface brightness distributions of the G18 and Tanoglidis et al. samples, to
show that they are complete at a similar $\sim 70-80\%$ level to roughly
$\mu_{\rm eff,g}=25.75$ mag arcsec-2, and this value is comparable to
estimates from Tanoglidis et al. Dragonfly routinely reaches $\mu_{r}\sim
29.5$ mag arcsec-2 at $1\,\sigma$ on 5″ scales, so such limits are in reach
for large volume searches albeit at lower spatial resolution (Danieli et al.,
2020). Thus, the combination of depth, area, and resolution makes the HSC SSP
survey data an excellent tool for LSB galaxy searches (e.g., Huang et al.,
2018; Kado-Fong et al., 2020).

###  2.4 Volume corrections

Because of the size and surface brightness cuts we impose, we will not be
equally sensitive to all galaxies in the sample throughout the full volume. In
the standard way, we correct for this incompleteness by calculating the
maximum volume out to which each source is detectable, and then weight each
source by both the detection completeness at that size and surface brightness,
but also its volume completeness.

In practice, it is not the magnitude that determines when a galaxy falls out
of the sample in distance, but rather the size and surface brightness. To
calculate the maximum volume, we include two terms. The first is the maximum
distance at which the galaxy is large enough to remain in our sample
($r_{e}>2.5$″). The second is the maximum distance at which the galaxy has
$\mu_{\rm eff}<27$ mag arcsec-2 (our approximate $50\%$ limit). Whichever
volume is smaller sets the $V_{\rm max}$ weighting for that galaxy.

###  2.5 Redshift reference samples

In addition to our primary sample of LSB galaxies with an unknown distance
distribution, we will rely on a reference sample with known redshifts to
cross-correlate with our sample. The ideal reference sample includes very
nearby galaxies ($\sim 20$ Mpc) and extends to at least redshift $z=0.15$ (700
Mpc). In addition, the spatial sampling of the reference sample should be
well-understood, such that we can identify regions of the sky where no spectra
could have been taken due to issues like bright stars, incomplete imaging
coverage, or other non-galaxy–related issues.

We considered using the Extragalactic Distance Database (Tully et al., 2009),
but that sample starts to become incomplete above $\sim 200$ Mpc, and thus
does not cover the full distance distribution of our sample (as we justify in
§3). Similar concerns hold for the Two Mass Redshift Survey (Kochanek et al.,
2001). The Galaxy and Mass Assembly (GAMA) survey (Driver et al., 2011) only
covers a small fraction of our areal coverage. Therefore, we have relied on
two complementary samples that are both built from the Sloan Digital Sky
Survey (SDSS; York et al., 2000). As our primary reference sample, we use the
NASA-Sloan Atlas111In this work, we use Version 1_0_1:
https://www.sdss.org/dr13/manga/manga-target-selection/nsa/ (NSA; Blanton et
al., 2005, 2011). This is a catalog that painstakingly combines the redshift
survey of the SDSS Legacy galaxy sample with a magnitude limit of $r<17.77$ AB
mag (Strauss et al., 2002) and literature redshifts to fill in the range below
$z\approx 0.02$. Those redshifts and distances may come from gas, direct
distance measurements, or anything else available in the literature, and
therefore the spatial distribution of this sample is not well known, so we
cannot construct a random catalog for the NSA sample.

To complement the NSA tracer sample, therefore, we also use the SDSS Large
Scale Structure catalog from Data Release 12 (Ross et al., 2011; Ho et al.,
2012; Reid et al., 2016). This sample is a combination of the Legacy sample
referenced above and the BOSS spectroscopic survey (Dawson et al., 2013). The
spatial sampling of these redshifts is very well-understood and a catalog of
random positions is provided (Alam et al., 2017). There are two reasons that
we do not use the SDSS DR12 sample as our reference. First, the catalog only
contains galaxies with $z>0.02$, and many of our targets are closer than that
(Greco et al., 2018a). Second, because of the coverage of the catalog, we lose
roughly a quarter of our areal coverage where there are no objects in the SDSS
DR12 catalog. Thus, this sample serves mostly as a check that we recover the
same redshift distribution when we are controlling the spatial distribution of
the tracer sample.

##  3 The LSB redshift distribution

Figure 3: Cross-correlation between LSB galaxies and the NSA catalog taken in
five redshift bins (as indicated), measured separately for red ($g-i\geq
0.8$), blue ($g-i<0.5$), and green ($0.5\leq g-i<0.8$) galaxies. We show
$1\,\sigma$ errors in bold, and $2\,\sigma$ errors in semi-transparent color.
We see a clear clustering signal at $z<0.03$ in all galaxy color ranges. The
clustering strength rapidly drops thereafter for the green and then the blue
sample. There is an additional peak at $z\sim 0.1$, particularly for the red
sample. We also show the cross-correlation for red galaxies after removing LSB
galaxies that are close to clusters in projection (§4.3).

###  3.1 Cross-correlation redshifts

If an unknown sample is physically co-located with a tracer sample of known
redshift, then we detect an angular correlation signal between them, while if
they are at disjoint distances, they will show no excess clustering over a
random sample of positions. This approach has been in use for a long time,
including to understand the nature of rare objects like quasars or sub-
millimeter galaxies (e.g., Seldner & Peebles, 1979; Ho et al., 2008; Mitchell-
Wynne et al., 2012), to calibrate photometric redshifts (Schneider et al.,
2006; Quadri & Williams, 2010; Matthews & Newman, 2010), to determine redshift
distributions for classes of galaxies (e.g., Newman, 2008) or to measure
galaxy mass functions (Bates et al., 2019).

We will follow the treatment outlined in Ménard et al. (2013) and Schmidt et
al. (2013). They focus on the scenario that the target population (in our case
LSB galaxies) span a relatively narrow redshift range, such that the evolution
in galaxy bias of both the tracer and unknown populations can be ignored. They
show that the clustering in the non-linear regime can also be useful for this
application, and advocate examining the cross-correlation on scales larger
than the survey point-spread function and smaller than the typical distance
between reference objects, such that issues like Galactic reddening, cirrus or
photometric zeropoints do not become serious systematics as they would on
scales of $>1$ degree.

Taking our LSB galaxy sample as the “target” sample, with number density
$n_{t}$, and the NSA sample as the “reference” sample with number density
$n_{r}$, the correlation function of angle $\theta$ and redshift $z_{i}$ is:

$w_{tr}(\theta,z_{i})=\frac{\langle n_{t}(\theta,z_{i})\rangle}{n_{r}}-1$ (1)

where the $\langle n_{t}(\theta,z_{i})\rangle$ represents the average density
of target galaxies around reference galaxies at redshift $z_{i}$. The
$w_{tr}(\theta,z_{i})$ are estimated from pair counts between the target and
reference samples, and the random samples that define the geometry of each
survey (see §A). Once we determine $w_{tr}(\theta,z_{i})$, we will approximate
the average cluster-redshift signal in a given redshift bin as:

$\bar{w}_{tr}(z_{i})=\frac{\int_{\theta_{\rm min}}^{\theta_{\rm
max}}W(\theta)w_{tr}(\theta,z_{i})\mathrm{d}\theta}{\int_{\theta_{\rm
min}}^{\theta_{\rm max}}W(\theta)\mathrm{d}\theta}$ (2)

with $W(\theta)$ a filter that we apply to maximize the signal to noise, and
we adopt a power-law to match the common form for galaxies. In reality, the
cluster-redshift signal is an integral over the product of the bias in each
population with the dark matter autocorrelation function. However, Ménard et
al. (2013) argue that so long as the number density of the source varies more
rapidly than evolution in the bias, then we can ignore bias terms and
approximate the redshift distribution directly from the clustering amplitudes
as a function of redshift.

To measure the clustering amplitudes, we turn to our primary tracer sample,
the NSA. We measure the clustering signal using the cross-correlation
estimator introduced by Davis & Peebles (1983):

$w_{tr}(\theta_{i},z_{i})\approx\frac{D_{r}D_{t}}{D_{r}R_{t}}\frac{N_{R_{t}}}{N_{t}}-1.$
(3)

Here $D_{r}D_{t}$ represents the number of reference-target pairs separated by
angular distance $\theta$, normalized by the total number of galaxies in each
sample and $D_{r}R_{t}$ represents the number of pairs between the tracer and
random positions in the target population search area, normalized by the
number of each. This estimator does not account for the spatial selection
function of the reference sample, since we do not know the NSA selection
function.

We also use the SDSS DR12 large-scale structure catalog. Here, we do
understand the spatial distribution of the tracer SDSS sample, and so we use
the Landy & Szalay (1993) estimator:

$w_{tr}(\theta,z_{i})\approx\frac{1}{R_{r}R_{t}}\\\
\left[D_{r}D_{t}\left(\frac{{N_{R}}_{r}{N_{R}}_{t}}{N_{r}N_{t}}\right)-D_{r}R_{t}\left(\frac{{N_{r}}_{r}}{N_{r}}\right)-D_{t}R_{r}\left(\frac{{N_{R}}_{t}}{N_{t}}\right)R_{r}R_{t}\right].$
(4)

The $DD$ and $DR$ terms again represent the number of target-reference and
target-random pairs that are separated by a distance $\theta$ within a
redshift slice $z_{i}$. In this case, we have random fields for both the
reference and target samples. The cross-correlations ensuing from the SDSS
DR12 cross-correlation are shown in Appendix A.

We integrate over a uniform projected physical range in each bin. As an inner
angular scale, we take $\theta_{\rm min}=0.10$ Mpc (or 1 arcmin at $z=0.15$).
Because of how we mask luminous galaxies prior to detection, this is the
smallest angular scale over which we can hope to find a nearby LSB. As an
outer scale, we integrate out to 3 Mpc. In part, this radius is chosen because
we do not in general detect signal at larger scales, but is also motivated by
the findings of Schmidt et al. (2013). On larger scales, issues like cirrus
and photometric calibration can become an issue, so even though more real
pairs are added, the chance of increasing systematic bias and degrading the
overall signal-to-noise goes up.

We calculate the redshift distribution as
$\mathrm{d}N/\mathrm{d}z\propto\bar{w}_{tr}(z_{i})$, where $\bar{w}_{tr}$ is
the average clustering signal in a given redshift bin over the range of scales
that we consider. We normalize the redshift distribution such that $\int
dz\,dN/dz=1$. We also note that the bias in both the reference and target
samples are assumed to be constant over the full redshift range considered,
which is probably a reasonable assumption out to $z\sim 0.15$. We verify that
we can recover reasonable redshift distributions by building test samples with
GAMA, where the distances are known, and then recovering them through cross-
correlation (see Appendix A). These tests give us some confidence that even
with small and very low redshift samples, it is possible to recover redshift
distributions for faint galaxies.

Figure 4: Our inferred redshift distribution for the full sample including all
(left), red (middle), and blue (right) galaxies. We compare the results for
three different binning prescriptions (in both velocity and spatial binning),
using $\delta z=0.02,0.03$ and inner bin $250,350$ kpc. Error bars represent
the 68% range incorporating uncertainty in the $w_{tr}$ values. Following
Equation 2, we calculate the average clustering signal in each redshift bin.
In the case of the red galaxies, the overall redshift distribution is stable
to different binning decisions, but this is not the case for the blue
galaxies, where the peak distance moves a bit under different binning
prescriptions. In practice, our primary conclusions do not change if we carry
through the different redshift distributions.

###  3.2 LSB Cross-correlation with NSA

As described in §2, we do not have a detailed mask for the NSA reference
sample, and so use the Davis & Peebles estimator (Eq. 3). We do not include
all 781 LSBGs in this cross-correlation, but rather use the completeness map
shown in §2.3 to select a sample with completeness $>70\%$, which contains 339
galaxies. To calculate the cross-correlation uncertainties, we utilize the
jacknife resampling procedure in the dsigma galaxy-galaxy lensing
tool222https://github.com/johannesulf/dsigma, which computes contiguous
regions, and then allows us to divide the total area into 100 sub-regions. We
then recalculate the clustering 1000 times, removing 10 of the fields each
time, to derive the errors.

The clustering amplitude differs between red and blue LSB galaxies as seen in
normal galaxies (e.g., Zehavi et al., 2005) as well as complementary
observations of LSB galaxies (Zaritsky et al., 2019; Tanoglidis et al., 2021).
Therefore, we calculate the clustering as a function of galaxy color. Galaxy
color is measured from the matched Sérsic fits. As described in G18, there are
a number of clear differences between the red and blue samples. The blue
galaxies are clearly forming stars, as indicated by detected H$\alpha$ in a
handful of cases (Greco et al., 2018a) or by the high _GALEX_ detection
fraction among the blue galaxies. The red galaxies also extend to far lower
surface brightness ($\mu_{\rm eff,g}<28,25.5$ mag arcsec-2 for red, blue
respectively). While we see a clear bimodality in the color distributions in
G18, there are significant errors on the color measurements and some
inevitable mixing in color. We therefore focus on the extreme color ranges
where even photometric errors cannot change the galaxy color. We divide the
sample into three color ranges: 59 red ($g-i\geq 0.8$), 145 blue ($g-i<0.5$),
and 135 green ($0.5\leq g-i<0.8$) galaxies. We calculate the cross-correlation
in each redshift bin for each galaxy sample separately (Fig. 3). The
clustering strength for the red galaxies is $\sim 6-7$ times higher than for
the blue galaxies at $z<0.03$.

It is not surprising to find higher clustering among the red galaxies. As
shown by Geha et al. (2012), low-mass central galaxies are nearly always star-
forming, while low-mass satellites in groups and clusters tend to be quenched
and therefore red. The blue galaxies live in lower-density environments and
have a lower galaxy bias, and thus the clustering signal is weaker and more
challenging to detect.

We only detect a significant clustering signal for the green galaxies for
$z<0.03$ ($w_{tr}=1.0\pm 0.1$). These galaxies, which live between the red and
blue galaxies in color space, show less clustering signal than either the red
or the blue galaxies. We do not have a full explanation for their lack of
clustering in this redshift range, but discuss some possible factors here.

If the green galaxies were populated fully by scatter from red and blue
galaxies, we may expect that the differing bias and redshift distribution
between the two samples might make the signal very challenging to detect.
Another factor at play is that background spiral contamination may be most
prevalent in the green galaxies, which could be intrinsically blue galaxies at
slightly higher redshift. If we take the surface brightness distribution of
spiral galaxies as measured by MacArthur et al. (2003), then the lower end of
the distribution is $\mu_{\rm eff,B}\approx 23$ mag arcsec-2. Such a
relatively low surface brightness spiral would be scattered into our sample
through surface brightness dimming at $z>0.35$. At the same time, as the 4000Å
break moves into the $r-$band, the color will redden. This possibility is
somewhat supported by the cross-correlation signal between the green galaxies
and the SDSS DR12 LSS catalog, where we do see hints of a signal in the
$z=0.32-0.34$ bins, perhaps indicating that there is a population of higher-
redshift LSB spirals that has crept into our sample.

We attempt to construct a cleaner sample, in which we visually flag likely
contaminants (see Appendix B for examples). These include both galaxies that
appear bulge-dominated and a handful of objects that are likely to be tidal
debris (Greco et al., 2018a). We find no difference in the cross-correlation
signal for the sample with likely contaminants masked.

A final possibility is that the green galaxies are physically distinct from
the red and blue ones. They may be more likely to be “back-splash” galaxies
(Benavides et al., 2021), that have tidally interacted with a more massive
galaxy but are not technically within its virial radius, or even perhaps
beyond the $\sim 1$ Mpc scales that dominate our clustering signal. Larger
samples will be needed to determine the nature of these green galaxies, and we
will focus more strongly on the red and blue subsets for the remainder of the
paper.

We also confirm the clustering results using the SDSS DR12 Large Scale
Structure sample, that contains a detailed spatial mask. Since the area of
overlap is smaller by $\sim 25\%$, we only use these as a sanity check, but
they do confirm the trends presented here (see Appendix B for details.)

###  3.3 The redshift distribution

We now infer the redshift distribution directly from the average clustering
strength in each bin as $\mathrm{d}N/\mathrm{d}z\propto\bar{w_{tr}}$. To
calculate $\bar{w_{tr}}$, we use Eq. 2, and as our matched filter we take
$W(\theta)\propto\theta^{-0.8}$, since this is often a good fit to galaxy
correlation functions (e.g., Peebles, 1974; Zehavi et al., 2005). In Figure 4,
we plot the average cross-correlation signal for the full, red, and blue
samples, normalized such that $\int dz(dN/dz)=1$. The errors represent the 68%
distribution incorporating uncertainties in the $w_{tr}$ measurements.

The redshift distribution peaks at $z<0.02$ ($D=100$ Mpc). For the red
galaxies, we find that $\sim 30\%$ of the galaxies have $z<0.02$, $\sim 20\%$
have $0.02<z<0.05$, and $\sim 40\%$ have $z>0.05$, with nominal uncertainties
of $\sim 15\%$. The redshift distribution appears flatter for the blue
galaxies, with 20-40% of the sample falling in each of these three redshift
bins. Looking at the $\delta z=0.02,350~{}\rm{kpc}$ bins, the inferred
redshift distribution is marginally different, 10-25% of the sources falling
in the first redshift bin, 30-60% in the second redshift bin, and 20-50% in
the third redshift bin. This distribution is still consistent at $1\,\sigma$
with the default distributions. Within errors, the distributions are
consistent with each other regardless of binning. The NSA does not extend
beyond $z=0.15$, but in Appendix C we show that there is minimal correlation
signal beyond $z\sim 0.15$, and thus by happenstance the NSA does provide
optimal distance coverage for our purposes.

Galaxies within $D<100$ Mpc are typically “normal” $\sim 10^{7}$ $M_{\odot}$
galaxies, in the sense that their size is similar to known galaxies in this
mass range (e.g., Carlsten et al., 2021, C21 hereafter), whose intrinsically
low surface brightnesses place them within our selection. Galaxies at larger
distance, in contrast, are typically more massive and are outliers from the
median mass-size relation. Ultra-diffuse galaxies, as defined by van Dokkum et
al. (2015a) populate the higher-redshift tail of our distribution.

The sample of independently measured redshifts remains quite small at present,
particularly at the lower surface brightness end of our sample. However, we
have performed _Hubble Space Telescope_ follow-up of LSB dwarfs from Greco et
al. (2018b) that were close in projected distance to two $z\sim 0.08$ groups.
Somalwar et al. (2020) found globular cluster populations consistent with
these LSB galaxies likely being at the distance of their groups, providing
some evidence for a higher redshift tail that extends at least to $z\sim 0.1$.
In §4.3 we will investigate the LSB galaxies that are likely to be associated
with rich groups and clusters, to again see evidence for a tail of galaxies
extending to $z\approx 0.15$.

Figure 5: Inferred distribution in stellar mass and size given the redshift
distributions from the cross-correlation, based on $dz=0.03$ and three spatial
bins. To show that the precise redshift binning does not strongly impact our
results, in dot-dash, we show the difference in the inferred distributions for
the blue galaxies if we use instead the redshift distributions based on
$dz=0.02$ bins. On the top row we show the inferred observed distributions. On
the bottom row, we show the volume and completeness-weighted distributions. On
the bottom left, we compare with the stellar mass function inferred for an H I
selected sample of UDGs (Leisman et al., 2017; Jones et al., 2018). On the
bottom right, we compare with the slope fitted to a complementary LSB sample
selected by Prole et al. (2021) from HSC.

##  4 Physical Properties of the LSB Galaxy Sample

We have used cross-correlation with a known tracer population to calculate the
redshift distribution for the ensemble of LSB galaxies discovered in Greco et
al. (2018b). In this section, we combine the redshift distribution with the
observed angular sizes, magnitudes, and colors of each galaxy to infer their
mass and size distributions. As argued above, it is more meaningful to treat
the red and blue galaxies separately, given their differing bias and possibly
different redshift distribution. In this analysis, we utilize 762 of the 781
galaxies, excluding a small number of the lowest surface brightness sources
with problematic photometry and inferred completeness levels $c<0.01$.

###  4.1 Mass and Size

Figure 6: Bottom: Distribution in the mass-size plane for the full sample of
LSB galaxies. The color scale represents the volume weight for each mass-size
bin. Points are red ($g-i>0.8$) galaxies that are within a projected 0.7 Mpc
from groups within $z<0.15$, presuming each is at the distance of their
cluster. We show the C21 (purple) and Lange et al. (2015) (red/blue) mass-size
relations for comparison, and indicate tracks of constant surface brightness
corresponding roughly to our selection region at a typical distance of 200 Mpc
(white lines). Light green regions are excluded from our selection for either
being below our surface brightness sensitivity (upper left) or too high in
surface brightness (bottom right). Grey-scale represents the number density in
that cell. Top: We plot slices (in dashed lines) through the mass-size
relation at $10^{7},\,10^{8},\,10^{9}$ $M_{\odot}$ using the 0.2 dex log-
normal scatter from C21, and normalizing the overall number density to Baldry
et al. (2012). Above $M_{*}=10^{9}$ $M_{\odot}$ we assume that red galaxies
account for 10% of the population. Shaded regions indicate the range of number
densities possible for LSB galaxies in the same mass range. We see an excess
of red LSB galaxies over expectations based on mass-size fits to known galaxy
populations.

We first present the distributions in mass and size separately, and then
examine their joint distribution. To construct the mass and size histograms,
we follow C21 and use the color to assign a stellar mass following Into &
Portinari (2013). It is hard to get a robust estimate of the scatter
introduced by this relation, since color-mass relations are calibrated for
more massive systems. In a forthcoming work, we directly compute the color-
mass relation of a set of H I-selected UDGs and NSA dwarfs (Kado-Fong et al.,
2022; in preparation). We compute the stellar mass of each galaxy via UV-
optical SED fitting assuming a Kroupa (2001) IMF using Galex and HSC-SSP
imaging in conjunction with the H I-derived distances. We find no significant
difference between the color-mass relation of the UDG and spectroscopic dwarf
samples. Our measured color-mass relation lies between the Into & Portinari
(2013) and Bell et al. (2003) color-mass relations (the former predicts lower
mass-to-light ratios at fixed $g-r$ color), and falls within 2$\sigma$ of both
relations. In particular, the deviation from the Into & Portinari (2013)
relation for $\Delta\Psi=\Delta\log_{10}(M_{\star}/L_{g})$ has a mean value of
$\langle\Delta\Psi\rangle=0.10(0.08)$ dex for the UDG (spectroscopic) sample
with a standard deviation of $\sigma_{\Delta\Psi}=0.08(0.12)$ dex. Given that
the 4000Å break is within the $g-$band over our full distance range, and given
our ignorance of the galaxy spectra, we ignore $k$-correction and treat the
observed colors as the rest-frame values.

To construct the mass and size distributions, we first create an interpolated
redshift distribution. We then randomly assign a distance to each galaxy drawn
from that distribution over a thousand draws. We incorporate scatter in color
and size from G18. We generate histograms (unweighted), and then we correct
each galaxy for surface brightness incompleteness (Figure 1) and weight by the
maximum observable volume as described in §2.3 (Figure 5). For completeness,
we also carry through these calculations using the $\rm{d}z=0.02$ binning for
the blue galaxies, since the redshift distributions are marginaly different,
to show that the final results are insensitive to our binning choices.

We had some concern that we would end up with unphysically large galaxies,
given the long tail in our redshift distribution. Thus, we also experimented
with ranking the galaxies by apparent size, and enforcing an upper size limit
(and thus upper distance limit) for each galaxy. Almost none of the
conclusions presented here change with that constraint, but the slope in the
size distribution does become steeper than that observed by other groups. If
there is an upper size limit, it is not known, and since $80\pm 4\%$ ($90\pm
3\%$) of our galaxies are smaller than 8 (10) kpc, we do not impose this
additional constraint.

To examine the maximum galaxy size with a bit more care, we assume that our
inferred redshift distribution is correct, and that there are G18 galaxies
extending to $z\approx 0.15$. We look at the inferred number of galaxies with
$7<r_{\rm e}/\rm{kpc}<8$ and $8<r_{\rm e}/\rm{kpc}<9$ accounting for both the
uncertainty in the redshift distribution and in the size measurements, which
are typically $\sim 1\arcsec$ (Greco et al., 2018b). We focus here only on the
red galaxies because their clustering signal is much more robustly measured.
Also, it is very hard to imagine a background contaminant to these galaxies,
since such a population would constitute physically huge and red galaxies. For
the red subsample, we find 10$\pm 3$ galaxies with $7<r_{\rm e}/\rm{kpc}<8$, a
$3\,\sigma$ detection. We conservatively do not apply a completeness
correction to derive this number. We find $8\pm 3$ galaxies in the next higher
bin, so we do not have a significant detection for $r_{\rm e}>8$ kpc.
Therefore, the maximum galaxy size in this sample is $\approx 8$ kpc. This is
an extreme galaxy size, even larger than most UDGs in Coma. On the other hand,
these galaxies are rare, with number densities $n<10^{-5}$ Mpc-3.

We also briefly investigate whether a systematic error in our sizes would
shift the size distribution dramatically, by shifting all the sizes downward
by $\sim 0.5$″, or half the size uncertainty. In this case, we find $9\pm 3$
galaxies with $7<R_{e}/\rm{kpc}<8$, maintaining the $3\,\sigma$ detection.

We thus infer a stellar mass range of $10^{7}-10^{9}$ $M_{\odot}$ and sizes of
$r_{e}\approx 1-8$ kpc. This mass range is comparable to other LSB dwarf
galaxy samples where distances, masses, and sizes are inferred from clusters
(e.g., van Dokkum et al., 2015a; van der Burg et al., 2016; Koda et al., 2015)
or distances are measured from H I (e.g., Leisman et al., 2017; Karunakaran et
al., 2020; Kado-Fong et al., 2021) or optical spectroscopic follow-up
(Kadowaki et al., 2017, 2021). A substantial fraction of our sources do
technically qualify as ultra-diffuse galaxies based on our surface brightness
limit of $\mu_{\rm g,eff}>24.3$ mag arcsec-2 and inferred size $r_{e}>1.5$
kpc. Specifically, $65-80\%$ of the red galaxies, and $60-77\%$ of the blue
galaxies fall above the $1.5$ kpc size boundary by number.

The differing distributions in redshift, angular size, magnitude, and color
between the red and blue samples translate into small differences in the mass
distributions for the two samples. Specifically, the peak mass of the red
galaxies is shifted to slightly higher mass due to the second higher-redshift
peak in the inferred redshift distribution. There is a corresponding very
small shift to larger size for the red galaxies as well.

We compare our inferred stellar mass function with two literature mass
functions (Figure 5, bottom left). First, we compare to the galaxy mass
function from Baldry et al. (2012) (see also Wright et al., 2017). We see that
at $M_{*}\sim 10^{7}$ $M_{\odot}$, our selection finds the same number density
of galaxies as are inferred to be in the field. However, as we go to higher
mass, our surface brightness selection increasingly selects rarer subsets of
galaxies. We also compare to the mass function from Jones et al. (2018) for H
I selected UDGs from Leisman et al. (2017). We find reasonable agreement with
their mass function above $M_{*}\approx 10^{8}$ $M_{\odot}$, which is rather
remarkable given the differences in how the stellar masses are calculated. In
any case, we come to a similar conclusion to Jones et al. in that the LSB
galaxies make up a small fraction of the galaxy population at this mass.

The LSB size distribution is well-described by a power-law. Fitting only for
galaxies with $r_{\rm eff}>1$ kpc, we fit a slope of $\alpha=-3.3\pm 0.6$ for
$N\propto r_{\rm e}^{\alpha}$, consistent with the slope measured by Prole
(2021, $-3.5\pm 0.3$) to their own HSC-selected LSB sample (Prole et al.,
2021). A similar slope was reported from the cluster sample of van der Burg et
al. (2017) and Mancera Piña et al. (2019), while Amorisco & Loeb (2016)
predict a similar slope if UDGs form in halos with the highest spin. It is
worth noting that our slope is sensitive to the exact size range that we fit
over, and in particular the slope is shallower at smaller size.

We then combine the mass and size constraints in Figure 6. Completeness
corrections and volume weights are applied. First of all, we see that the
slope of the mass-size distribution for this sample of LSB galaxies appears to
be steeper than the measured slope from C21 or Lange et al. (2015). This slope
is not an attribute of the sample, but rather a direct artifact of our surface
brightness selection. To make this point, we indicate excluded regions in
surface brightness for galaxies at 200 Mpc. Technically, all cells above the
$\mu_{e}=24$ and $r_{e}>1.5$ kpc lines qualify as UDGs as defined by van
Dokkum et al. (2015a). Nearly all the galaxies with $M_{*}>10^{8}$ $M_{\odot}$
are in this category. This tail of more massive and more extended galaxies is
a direct consequence of the cross-correlation signal that we detect at
$z>0.05$. Most known galaxies with $M_{*}<10^{7.5}$ $M_{\odot}$ fall within
our selection, so we are not selecting outliers in that mass range, at least
according to the mass-size relations of C21. At higher mass, in contrast, we
are mostly sensitive to galaxies that are considerably larger than the ridge-
line of the mass-size relation.

Direct comparisons with typical galaxies are made more difficult by the lack
of a robust measurement of the mass-size relation across all environments and
the full mass range probed by our sample (for instance, Shen et al., 2003;
Lange et al., 2015, are both incomplete below $\sim 10^{9}$ $M_{\odot}$).
While C21 showed that there is little difference in the mass-size relation
between early and late-type galaxies for masses $10^{5.5}-10^{8.5}$
$M_{\odot}$ (see also Kormendy & Freeman, 2016), the situation changes
dramatically at higher mass. Early-type galaxies flatten to a constant size of
$\sim 1$ kpc between $\sim 10^{8.5}-10^{10}$ $M_{\odot}$. This flattening has
been seen in large environment-blind volumes (e.g., Shen et al., 2003), in
individual cluster galaxies (e.g., Smith Castelli et al., 2008; Misgeld et
al., 2008; Eigenthaler et al., 2018), and even out to intermediate redshift
(Nedkova et al., 2021). On the other hand, late-type galaxies continue to grow
in size over this $10^{8.5}-10^{10}$ $M_{\odot}$ mass range. The fits from
Lange et al. (2015, red and blue dashed lines in Figure 6) show that between
the high-mass end of C21 and $\sim 5\times 10^{9}$ $M_{\odot}$, galaxies are
$\sim 1$ kpc in size. We now turn to examine the size distributions at fixed
mass.

Table 7: LSB Galaxy Mass Function

log Mass | # (low) | # (med) | # (high)
---|---|---|---
($M_{\odot}$) | ($10^{-3}$ Mpc-2 dex-1) | ($10^{-3}$ Mpc-2 dex-1) | ($10^{-3}$ Mpc-2 dex-1)
7.00 | 5.032 | 15.203 | 38.669
7.20 | 3.171 | 9.573 | 22.373
7.40 | 2.409 | 5.662 | 13.401
7.60 | 1.387 | 3.081 | 6.727
7.80 | 0.813 | 1.931 | 4.037
8.00 | 0.551 | 1.154 | 2.264
8.20 | 0.389 | 0.721 | 1.288
8.40 | 0.251 | 0.442 | 0.781
8.60 | 0.174 | 0.292 | 0.454
8.80 | 0.098 | 0.165 | 0.261
9.00 | 0.053 | 0.090 | 0.145
9.20 | 0.024 | 0.044 | 0.072
9.40 | 0.011 | 0.019 | 0.033
7.00 | 1.516 | 3.464 | 8.090
7.20 | 1.080 | 2.290 | 5.000
7.40 | 0.715 | 1.427 | 2.867
7.60 | 0.459 | 0.867 | 1.776
7.80 | 0.293 | 0.563 | 1.089
8.00 | 0.181 | 0.343 | 0.643
8.20 | 0.107 | 0.189 | 0.374
8.40 | 0.065 | 0.117 | 0.202
8.60 | 0.040 | 0.066 | 0.112
8.80 | 0.020 | 0.035 | 0.060
9.00 | 0.009 | 0.018 | 0.032
9.20 | 0.004 | 0.008 | 0.015
9.40 | 0.001 | 0.004 | 0.008

Note. — Top is red mass function, bottom is blue. (1) Log of the stellar mass.
(2) $1\sigma$ low value of mass function. (3) Median mass function. (4)
$1\sigma$ high value of mass function.

###  4.2 Size Distribution at Fixed Mass

We do not yet know whether LSB galaxies are outliers in the mass-size relation
(e.g., van der Burg et al., 2016; Danieli & van Dokkum, 2019). In general,
both in dwarfs and more massive galaxies, the size distribution at fixed mass
appears to be well-fit by a log-normal (e.g., Shen et al., 2003, C21). We
adopt the 0.2 dex scatter that was measured by C21, understanding that this
value may not apply above $M_{*}\approx 10^{8.5}$ $M_{\odot}$ (although it is
similar to the dispersion measured for more massive galaxies Kawinwanichakij
et al., 2021). We can then ask whether the LSB galaxy distribution is well-
modeled as the tail of the log-normal that describes known galaxy samples.

We therefore examine the size distribution of our sample galaxies in fixed
mass slices, and compare with “normal” galaxies (Figure 6). We plot the size
distribution at fixed masses of $M_{*}=10^{7},10^{8},10^{9}$ $M_{\odot}$ for
normal (dashed) and LSB (filled region) galaxies. The normal galaxies are
assumed to have a log-normal size distribution, with the mean size and width
given by C21. The normalization of the distribution is set such that the
integral matches the number density at that mass from Baldry et al. (2012).
The normalization of the LSB region is set by the mass functions presented in
Figure 5. Note that for $M_{*}<10^{8.5}$ $M_{\odot}$, C21 show that
quenched/red and star-forming/blue galaxies have the same size distribution at
fixed mass. We combine the red and blue samples for those mass bins. At the
highest mass bin, red galaxies are $\sim 1$ kpc in size, while blue galaxies
continue to grow larger with mass. Therefore, we compare those populations
separately, assuming that red galaxies comprise $\sim 10\%$ of the galaxy
population at $M_{*}\sim 10^{9}$ $M_{\odot}$ (e.g., Blanton & Moustakas,
2009).

Starting with the left-most slice through the mass-size plane, we see that the
$M_{*}=10^{7}$ $M_{\odot}$ LSB galaxies are comparable to the full galaxy
sample. Both the number density and the size distributions match. We do not
detect galaxies with low enough surface brightness to find any excess relative
to the measured log-normal size distribution at this mass, should it exist. At
$M_{*}=10^{8}$ $M_{\odot}$ there is a hint that the LSB galaxies may extend a
bit beyond the normal galaxies, but better surface brightness sensitivity
would still be needed to say for sure. Only at $M_{*}=10^{9}$ $M_{\odot}$,
where our sensitivity allows us to probe well beyond the peak in the mass-size
relation, do we see clear evidence that the red galaxies are considerably
larger than expected from the log normal size distribution. Normal red
galaxies at $M_{*}\sim 10^{8.5}-10^{9.5}$ $M_{\odot}$ have $r_{\rm e}\approx
1$ kpc in size, while our LSB galaxies range from $r_{\rm e}=3-8$ kpc in this
mass range (as argued above, the detection of galaxies with $r_{\rm e}>8$ kpc
is marginal). Given the observed number densities, this size range is totally
inconsistent with a 0.2 dex log-normal size distribution centered at 1 kpc.

It is hard to say whether something special is happening at $\sim 10^{9}$
$M_{\odot}$. On the one hand, transition happens at this mass. Below $\sim
10^{9}$ $M_{\odot}$, all galaxies have very similar mass-size relations. Above
this mass, star-forming galaxies are becoming thin disks (Kado-Fong et al.,
2020) while quenched galaxies have a flat mass-size relation. So perhaps, the
excess of very extended LSB galaxies is also associated with this special mass
range. On the other hand, there may be an extended tail of puffy galaxies at
higher and lower mass that fall outside of the surface brightness sensitivity
of our survey (see, e.g., Danieli & van Dokkum, 2019).

###  4.3 A Likely Group and Cluster Sample

The G18 sample is “blind” to environmental density, which is one of its
strengths. However, as discussed by Tanoglidis et al. (2021), cross-matching
the full sample with known overdensities can also prove interesting. First, it
provides a sanity check on derived physical properties, since we will show
that the cluster sample is too numerous to have happened by chance. Second, we
can ask what fraction of the red LSB galaxies are likely associated with
groups and clusters. Third, we double check that the cross-correlation signal
persists even with the cluster galaxies removed.

To identify groups and clusters, we use the Yang et al. (2007) group and
cluster catalog built from the seventh data release of the SDSS (Abazajian et
al., 2009). Halo masses in the catalog are calculated based on group richness
and abundance matching. We select all halos with $M_{h}>10^{13}$ $M_{\odot}$,
and then identify all the LSB galaxies that lie within a projected 0.7 Mpc
distance of these groups and clusters. We have chosen a fixed physical
aperture for simplicity, and a relatively small size to limit contamination
from chance alignments. To boost the likelihood that the galaxies are
associated with clusters, we also restrict to the red LSB galaxies
($g-i>0.8$). There are 36 red galaxies matching these criteria. We find
galaxies projected within clusters across the full redshift range to $z=0.15$.
These galaxies encompass $\sim 16\%$ of the LSB galaxies in this color range,
which is a lower limit to the fraction of red galaxies residing in groups and
clusters, and similar to the 30% within 1 Mpc of clusters reported by
Tanoglidis et al. (2021). The projected cluster sample are plotted as points
on the mass-size plane in Figure 6 under the assumption that all of the
satellites fall at the distance of their group. Their mass distribution skews
a bit more massive than the overall sample, while the size distribution ranges
from $1-10$ kpc (Figure 6). Most of these galaxies would qualify as UDGs,
should they indeed be at the distance of their associated group.

We evaluate the probability to find this number of projections purely by
chance. We perform the following Monte Carlo test 100 times. We select 36
random positions within our footprint, and match those positions with the same
Yang catalog as the real galaxy positions. We find a mean of $17\pm 4$
matches, making our detection of 36 matched galaxies a $4\sigma$ outlier. We
thus conclude that we are detecting a real sub-population of red UDGs in
clusters. The fact that we find matches with clusters out to $z\sim 0.15$ may
provide some additional confirmation of our inferred redshift range, although
follow-up of those most distant cluster galaxy candidates would provide nice
confirmation.

We also recalculate the cross-correlation for the LSB sample excluding those
close in projection to clusters (Figure 3). We still see a significant signal
in the red galaxies ($w_{tr}=1.1\pm 0.1$ within 100 kpc and $z<0.015$). The
clustering amplitude for the cluster-free sample is significant at $z<0.045$
and $z=0.12-0.15$. Thus, even excluding galaxies belonging to groups and
clusters in projection, we recover a clustering signal for the red galaxies.

##  5 Discussion and Summary

This paper aims to determine the distance distribution, via cross-correlation,
of the LSB galaxies selected by Greco et al. (2018b) from all environments
within the HSC-SSP Survey. We find a relatively broad distance distribution
extending to $z\sim 0.15$ but peaked at $D<100$ Mpc. Spectroscopic follow-up
for two G18 galaxies find $D<40$ Mpc, corresponding to the first prominent
distance peak in our distribution. This distance distribution is consistent
with inferences by Prole et al. (2021) and Tanoglidis et al. (2021), based on
similarly selected samples from HSC and DES respectively. It is broader than
the distance distribution inferred for the Systematically Measuring Ultra-
Diffuse Galaxies (SMUDGes, Zaritsky et al., 2019, 2021) program, which does
not seem to find very many galaxies more distant than 100 Mpc (Barbosa et al.,
2020; Kadowaki et al., 2021). It is also broader than the distance
distribution inferred for a 36 deg2 area around the NGC 1052 group inferred by
Román et al. (2021) from cross-correlation, but no spectroscopic tracers with
$z>0.04$ were included in that work. From this distance range, we are also
able to infer physical parameter distributions for the sample, including
number densities, masses, and sizes.

We find typical stellar masses of $\sim 10^{7}-10^{9}$ $M_{\odot}$ and a range
of sizes from $1-8$ kpc. Our mass ranges are consistent with prior work from
cluster-based samples (e.g., Koda et al., 2015; van der Burg et al., 2016;
Danieli & van Dokkum, 2019), as well as H I-selected field samples Leisman et
al. (2017); Kado-Fong et al. (2021). We infer that a large fraction ($\sim
70\%$) of the galaxies in this sample qualify as an “ultra-diffuse” galaxy by
the van Dokkum et al. (2015a) definition, but we also find a significant
number of low-mass galaxies of $\sim 10^{7}$ $M_{\odot}$ that fall on the
“normal” mass-size relation. We find that the number of galaxies at a given
size falls as a power-law with $n\propto r^{-3.3\pm 0.6}$, with a slope that
is similar to that reported by Prole (2021).

Our inferred number densities are consistent with known galaxy populations at
$M_{*}\approx 10^{7}$ $M_{\odot}$, but then at higher masses become a rarer
sub-constituent as they grow further and further from the mass-size ridge-
line. They constitute only $10^{-3}$ of the galaxy population at $M_{*}\approx
10^{9}$ $M_{\odot}$ (Figure 5). Our mass function is surprisingly consistent
with that of H I-selected UDG samples for $M_{*}>10^{8}$ $M_{\odot}$,
suggesting that we are probing similar populations, and that star-forming and
quiescent LSB galaxies have similar space densities.

A driving question as we continue to discover larger samples of LSB galaxies
is whether they simply represent the smooth tail of the galaxy size
distribution, or whether a distinct formation path is needed to describe them
(see also Van Nest et al., 2021). A related question is whether LSB galaxies
are over-represented in rich environments as suggested by Koda et al. (2015)
and van der Burg et al. (2016), although there is some evidence that the
number density of LSB galaxies may rise in lower-mass groups (Mancera Piña et
al., 2018).

At the highest masses in our sample ($M_{*}>10^{8}$ $M_{\odot}$), we do find
tentative evidence that the LSB galaxies constitute an excess at large size
compared to the default assumption of a log-normal distribution (see also
Danieli & van Dokkum, 2019). At $M_{*}\approx 10^{9}$ $M_{\odot}$, the galaxy
population is just beginning to bifurcate into blue disks and red compact
systems (e.g., Lange et al., 2015). The observed excess of red LSB galaxies at
this mass scale may be somehow related to this transition. Alternatively,
there may be such an excess of large outliers from the mass-size ridge-line at
all masses that our survey is simply not sensitive enough to detect. In this
case, the question still remains whether the extreme sizes are providing hints
about the dark matter halo masses (e.g., Kravtsov, 2013) and/or spin
distributions (e.g., Dalcanton et al., 1997a; Amorisco & Loeb, 2016). Detailed
modeling of kinematics for more LSB galaxies may help make progress on this
point (e.g., Greco et al., 2018c; Wasserman et al., 2019; Mancera Piña et al.,
2020).

By cross-matching our redder galaxies with groups and clusters, we also find
that the galaxies projected near groups do tend to fall towards the more
extended and more massive end of our sample. However, only 16% of the red
galaxies are associated with $M_{h}>10^{13}$ $M_{\odot}$ groups. Furthermore,
the cross-correlation of red galaxies excluding the cluster-associated LSB
galaxies still shows signal at $z>0.1$, suggesting that not all of the UDG-
like galaxies in our sample can be associated with rich clusters. They may,
however, all be satellites. Only a small handful of “isolated” and “red” LSB
galaxies are currently known (e.g., Martínez-Delgado et al., 2016; Polzin et
al., 2021), most of which are very near to the Local Group and may have
tidally interacted with it (e.g., Teyssier et al., 2012). Prole et al. (2021)
estimate that 25% of the isolated LSB population are red.

We are currently in the process of repeating the Greco et al. (2018b) search
on the full HSC data set. This will represent a factor of four increase in
area, and thus a substantive improvement in statistical power. At that time,
we will more fully model the expected mass and redshift distribution to
address the question of whether these LSBs are true mass-size outliers. We
will also be in a position to divide the sample into finer color and size
bins. The upcoming Vera Rubin Observatory Legacy Survey of Space and Time
(LSST Science Collaboration et al., 2009) will provide data of comparable
depth over much wider areas, with interesting new prospects for cross-
correlation. At the same time, we anticipate deriving direct distances via
surface brightness fluctuations (e.g., Greco et al., 2021) to a number of the
nearest galaxies (within 15 Mpc) of the new sample, to complement the radial
velocities currently available from gas (Karunakaran et al., 2020). These
larger samples, with multi-wavelength data, will address the key factors that
lead galaxies to be LSB, whether they be internal or environmental. These
surveys will enable a full census of dwarf galaxies in this mass and surface
brightness range.

##  6 acknowledgements

JPG thanks Yi-Kuan Chiang for valuable discussions about cross correlation
redshifts. We thank the anonymous referee for excellent and timely comments.
JEG is supported in part by NSF grant AAG-1007052. SD is supported by NASA
through Hubble Fellowship grant HST-HF2-51454.001-A awarded by the Space
Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Incorporated, under NASA contract
NAS5-26555.

## References

  * Abazajian et al. (2009) Abazajian, K. N., Adelman-McCarthy, J. K., Agüeros, M. A., et al. 2009, ApJS, 182, 543, doi: 10.1088/0067-0049/182/2/543
  * Abbott et al. (2018) Abbott, T. M. C., Abdalla, F. B., Allam, S., et al. 2018, ApJS, 239, 18, doi: 10.3847/1538-4365/aae9f0
  * Abraham & van Dokkum (2014) Abraham, R. G., & van Dokkum, P. G. 2014, PASP, 126, 55, doi: 10.1086/674875
  * Aihara et al. (2018a) Aihara, H., Arimoto, N., Armstrong, R., et al. 2018a, PASJ, 70, S4, doi: 10.1093/pasj/psx066
  * Aihara et al. (2018b) Aihara, H., Armstrong, R., Bickerton, S., et al. 2018b, PASJ, 70, S8, doi: 10.1093/pasj/psx081
  * Alam et al. (2017) Alam, S., Ata, M., Bailey, S., et al. 2017, MNRAS, 470, 2617, doi: 10.1093/mnras/stx721
  * Amorisco & Loeb (2016) Amorisco, N. C., & Loeb, A. 2016, MNRAS, 459, L51, doi: 10.1093/mnrasl/slw055
  * Baldry et al. (2012) Baldry, I. K., Driver, S. P., Loveday, J., et al. 2012, MNRAS, 421, 621, doi: 10.1111/j.1365-2966.2012.20340.x
  * Barbosa et al. (2020) Barbosa, C. E., Zaritsky, D., Donnerstein, R., et al. 2020, ApJS, 247, 46, doi: 10.3847/1538-4365/ab7660
  * Bates et al. (2019) Bates, D. J., Tojeiro, R., Newman, J. A., et al. 2019, MNRAS, 486, 3059, doi: 10.1093/mnras/stz997
  * Bell et al. (2003) Bell, E. F., McIntosh, D. H., Katz, N., & Weinberg, M. D. 2003, ApJS, 149, 289, doi: 10.1086/378847
  * Benavides et al. (2021) Benavides, J. A., Sales, L. V., Abadi, M. G., et al. 2021, Nature Astronomy, 5, 1255, doi: 10.1038/s41550-021-01458-1
  * Bertin & Arnouts (1996) Bertin, E., & Arnouts, S. 1996, A&AS, 117, 393, doi: 10.1051/aas:1996164
  * Blanton et al. (2011) Blanton, M. R., Kazin, E., Muna, D., Weaver, B. A., & Price-Whelan, A. 2011, AJ, 142, 31, doi: 10.1088/0004-6256/142/1/31
  * Blanton et al. (2005) Blanton, M. R., Lupton, R. H., Schlegel, D. J., et al. 2005, ApJ, 631, 208, doi: 10.1086/431416
  * Blanton & Moustakas (2009) Blanton, M. R., & Moustakas, J. 2009, ARA&A, 47, 159, doi: 10.1146/annurev-astro-082708-101734
  * Bosch et al. (2018) Bosch, J., Armstrong, R., Bickerton, S., et al. 2018, PASJ, 70, S5, doi: 10.1093/pasj/psx080
  * Bothun et al. (1991) Bothun, G. D., Impey, C. D., & Malin, D. F. 1991, ApJ, 376, 404, doi: 10.1086/170290
  * Bothun et al. (1987) Bothun, G. D., Impey, C. D., Malin, D. F., & Mould, J. R. 1987, AJ, 94, 23, doi: 10.1086/114443
  * Caldwell & Bothun (1987) Caldwell, N., & Bothun, G. D. 1987, AJ, 94, 1126, doi: 10.1086/114550
  * Carlsten et al. (2020) Carlsten, S. G., Greco, J. P., Beaton, R. L., & Greene, J. E. 2020, ApJ, 891, 144, doi: 10.3847/1538-4357/ab7758
  * Carlsten et al. (2021) Carlsten, S. G., Greene, J. E., Greco, J. P., Beaton, R. L., & Kado-Fong, E. 2021, arXiv e-prints, arXiv:2105.03435.  https://arxiv.org/abs/2105.03435
  * Chan et al. (2018) Chan, T. K., Kereš, D., Wetzel, A., et al. 2018, MNRAS, 478, 906, doi: 10.1093/mnras/sty1153
  * Dalcanton et al. (1997a) Dalcanton, J. J., Spergel, D. N., Gunn, J. E., Schmidt, M., & Schneider, D. P. 1997a, AJ, 114, 635, doi: 10.1086/118499
  * Dalcanton et al. (1997b) Dalcanton, J. J., Spergel, D. N., & Summers, F. J. 1997b, ApJ, 482, 659, doi: 10.1086/304182
  * Danieli & van Dokkum (2019) Danieli, S., & van Dokkum, P. 2019, ApJ, 875, 155, doi: 10.3847/1538-4357/ab14f3
  * Danieli et al. (2020) Danieli, S., Lokhorst, D., Zhang, J., et al. 2020, ApJ, 894, 119, doi: 10.3847/1538-4357/ab88a8
  * Davis & Peebles (1983) Davis, M., & Peebles, P. J. E. 1983, ApJ, 267, 465, doi: 10.1086/160884
  * Dawson et al. (2013) Dawson, K. S., Schlegel, D. J., Ahn, C. P., et al. 2013, AJ, 145, 10, doi: 10.1088/0004-6256/145/1/10
  * Dey et al. (2019) Dey, A., Schlegel, D. J., Lang, D., et al. 2019, AJ, 157, 168, doi: 10.3847/1538-3881/ab089d
  * Di Cintio et al. (2017) Di Cintio, A., Brook, C. B., Dutton, A. A., et al. 2017, MNRAS, 466, L1, doi: 10.1093/mnrasl/slw210
  * Driver et al. (2009) Driver, S. P., Norberg, P., Baldry, I. K., et al. 2009, Astronomy and Geophysics, 50, 5.12, doi: 10.1111/j.1468-4004.2009.50512.x
  * Driver et al. (2011) Driver, S. P., Hill, D. T., Kelvin, L. S., et al. 2011, MNRAS, 413, 971, doi: 10.1111/j.1365-2966.2010.18188.x
  * Eigenthaler et al. (2018) Eigenthaler, P., Puzia, T. H., Taylor, M. A., et al. 2018, ApJ, 855, 142, doi: 10.3847/1538-4357/aaab60
  * El-Badry et al. (2016) El-Badry, K., Wetzel, A., Geha, M., et al. 2016, ApJ, 820, 131, doi: 10.3847/0004-637X/820/2/131
  * Erwin (2015) Erwin, P. 2015, ApJ, 799, 226, doi: 10.1088/0004-637X/799/2/226
  * Farrow et al. (2015) Farrow, D. J., Cole, S., Norberg, P., et al. 2015, MNRAS, 454, 2120, doi: 10.1093/mnras/stv2075
  * Geha et al. (2012) Geha, M., Blanton, M. R., Yan, R., & Tinker, J. L. 2012, ApJ, 757, 85, doi: 10.1088/0004-637X/757/1/85
  * Greco (2018) Greco, J. P. 2018, PhD thesis, Princeton University, United States
  * Greco et al. (2018a) Greco, J. P., Goulding, A. D., Greene, J. E., et al. 2018a, ApJ, 866, 112, doi: 10.3847/1538-4357/aae0f4
  * Greco et al. (2021) Greco, J. P., van Dokkum, P., Danieli, S., Carlsten, S. G., & Conroy, C. 2021, ApJ, 908, 24, doi: 10.3847/1538-4357/abd030
  * Greco et al. (2018b) Greco, J. P., Greene, J. E., Strauss, M. A., et al. 2018b, ApJ, 857, 104, doi: 10.3847/1538-4357/aab842
  * Greco et al. (2018c) Greco, J. P., Greene, J. E., Price-Whelan, A. M., et al. 2018c, PASJ, 70, S19, doi: 10.1093/pasj/psx051
  * Hamana et al. (2020) Hamana, T., Shirasaki, M., Miyazaki, S., et al. 2020, PASJ, 72, 16, doi: 10.1093/pasj/psz138
  * Ho et al. (2008) Ho, S., Hirata, C., Padmanabhan, N., Seljak, U., & Bahcall, N. 2008, Phys. Rev. D, 78, 043519, doi: 10.1103/PhysRevD.78.043519
  * Ho et al. (2012) Ho, S., Cuesta, A., Seo, H.-J., et al. 2012, ApJ, 761, 14, doi: 10.1088/0004-637X/761/1/14
  * Huang et al. (2018) Huang, S., Leauthaud, A., Greene, J. E., et al. 2018, MNRAS, 475, 3348, doi: 10.1093/mnras/stx3200
  * Impey et al. (1988) Impey, C., Bothun, G., & Malin, D. 1988, ApJ, 330, 634, doi: 10.1086/166500
  * Into & Portinari (2013) Into, T., & Portinari, L. 2013, MNRAS, 430, 2715, doi: 10.1093/mnras/stt071
  * Iodice et al. (2021) Iodice, E., La Marca, A., Hilker, M., et al. 2021, A&A, 652, L11, doi: 10.1051/0004-6361/202141086
  * Ivezić et al. (2019) Ivezić, Ž., Kahn, S. M., Tyson, J. A., et al. 2019, ApJ, 873, 111, doi: 10.3847/1538-4357/ab042c
  * Jones et al. (2018) Jones, M. G., Papastergis, E., Pandya, V., et al. 2018, A&A, 614, A21, doi: 10.1051/0004-6361/201732409
  * Kado-Fong et al. (2020) Kado-Fong, E., Greene, J. E., Huang, S., et al. 2020, ApJ, 900, 163, doi: 10.3847/1538-4357/abacc2
  * Kado-Fong et al. (2021) Kado-Fong, E., Petrescu, M., Mohammad, M., et al. 2021, arXiv e-prints, arXiv:2106.05288.  https://arxiv.org/abs/2106.05288
  * Kadowaki et al. (2017) Kadowaki, J., Zaritsky, D., & Donnerstein, R. L. 2017, ApJ, 838, L21, doi: 10.3847/2041-8213/aa653d
  * Kadowaki et al. (2021) Kadowaki, J., Zaritsky, D., Donnerstein, R. L., et al. 2021, arXiv e-prints, arXiv:2110.00015.  https://arxiv.org/abs/2110.00015
  * Karunakaran et al. (2020) Karunakaran, A., Spekkens, K., Zaritsky, D., et al. 2020, ApJ, 902, 39, doi: 10.3847/1538-4357/abb464
  * Kawinwanichakij et al. (2021) Kawinwanichakij, L., Silverman, J. D., Ding, X., et al. 2021, ApJ, 921, 38, doi: 10.3847/1538-4357/ac1f21
  * Kochanek et al. (2001) Kochanek, C. S., Pahre, M. A., Falco, E. E., et al. 2001, ApJ, 560, 566, doi: 10.1086/322488
  * Koda et al. (2015) Koda, J., Yagi, M., Yamanoi, H., & Komiyama, Y. 2015, ApJ, 807, L2, doi: 10.1088/2041-8205/807/1/L2
  * Komiyama et al. (2018) Komiyama, Y., Obuchi, Y., Nakaya, H., et al. 2018, PASJ, 70, S2, doi: 10.1093/pasj/psx069
  * Kormendy & Freeman (2016) Kormendy, J., & Freeman, K. C. 2016, ApJ, 817, 84, doi: 10.3847/0004-637X/817/2/84
  * Kravtsov (2013) Kravtsov, A. V. 2013, ApJ, 764, L31, doi: 10.1088/2041-8205/764/2/L31
  * Kroupa (2001) Kroupa, P. 2001, MNRAS, 322, 231, doi: 10.1046/j.1365-8711.2001.04022.x
  * Landy & Szalay (1993) Landy, S. D., & Szalay, A. S. 1993, ApJ, 412, 64, doi: 10.1086/172900
  * Lange et al. (2015) Lange, R., Driver, S. P., Robotham, A. S. G., et al. 2015, MNRAS, 447, 2603, doi: 10.1093/mnras/stu2467
  * Lee et al. (2017) Lee, M. G., Kang, J., Lee, J. H., & Jang, I. S. 2017, ApJ, 844, 157, doi: 10.3847/1538-4357/aa78fb
  * Leisman et al. (2017) Leisman, L., Haynes, M. P., Janowiecki, S., et al. 2017, ApJ, 842, 133, doi: 10.3847/1538-4357/aa7575
  * LSST Science Collaboration et al. (2009) LSST Science Collaboration, Abell, P. A., Allison, J., et al. 2009, arXiv e-prints, arXiv:0912.0201.  https://arxiv.org/abs/0912.0201
  * MacArthur et al. (2003) MacArthur, L. A., Courteau, S., & Holtzman, J. A. 2003, ApJ, 582, 689, doi: 10.1086/344506
  * Mancera Piña et al. (2019) Mancera Piña, P. E., Aguerri, J. A. L., Peletier, R. F., et al. 2019, MNRAS, 485, 1036, doi: 10.1093/mnras/stz238
  * Mancera Piña et al. (2018) Mancera Piña, P. E., Peletier, R. F., Aguerri, J. A. L., et al. 2018, MNRAS, 481, 4381, doi: 10.1093/mnras/sty2574
  * Mancera Piña et al. (2020) Mancera Piña, P. E., Fraternali, F., Oman, K. A., et al. 2020, MNRAS, 495, 3636, doi: 10.1093/mnras/staa1256
  * Martínez-Delgado et al. (2016) Martínez-Delgado, D., Läsker, R., Sharina, M., et al. 2016, AJ, 151, 96, doi: 10.3847/0004-6256/151/4/96
  * Matthews & Newman (2010) Matthews, D. J., & Newman, J. A. 2010, ApJ, 721, 456, doi: 10.1088/0004-637X/721/1/456
  * McGaugh et al. (1995) McGaugh, S. S., Bothun, G. D., & Schombert, J. M. 1995, AJ, 110, 573, doi: 10.1086/117543
  * Ménard et al. (2013) Ménard, B., Scranton, R., Schmidt, S., et al. 2013, ArXiv e-prints
  * Mihos et al. (2015) Mihos, J. C., Durrell, P. R., Ferrarese, L., et al. 2015, ApJ, 809, L21, doi: 10.1088/2041-8205/809/2/L21
  * Misgeld et al. (2008) Misgeld, I., Mieske, S., & Hilker, M. 2008, A&A, 486, 697, doi: 10.1051/0004-6361:200810014
  * Mitchell-Wynne et al. (2012) Mitchell-Wynne, K., Cooray, A., Gong, Y., et al. 2012, ApJ, 753, 23, doi: 10.1088/0004-637X/753/1/23
  * Miyatake et al. (2021) Miyatake, H., Sugiyama, S., Takada, M., et al. 2021, arXiv e-prints, arXiv:2111.02419.  https://arxiv.org/abs/2111.02419
  * Miyazaki et al. (2018) Miyazaki, S., Komiyama, Y., Kawanomoto, S., et al. 2018, PASJ, 70, S1, doi: 10.1093/pasj/psx063
  * Muñoz et al. (2015) Muñoz, R. P., Eigenthaler, P., Puzia, T. H., et al. 2015, ApJ, 813, L15, doi: 10.1088/2041-8205/813/1/L15
  * Nedkova et al. (2021) Nedkova, K. V., Häußler, B., Marchesini, D., et al. 2021, MNRAS, 506, 928, doi: 10.1093/mnras/stab1744
  * Newman (2008) Newman, J. A. 2008, ApJ, 684, 88, doi: 10.1086/589982
  * Papastergis et al. (2017) Papastergis, E., Adams, E. A. K., & Romanowsky, A. J. 2017, A&A, 601, L10, doi: 10.1051/0004-6361/201730795
  * Peebles (1974) Peebles, P. J. E. 1974, A&A, 32, 197
  * Polzin et al. (2021) Polzin, A., van Dokkum, P., Danieli, S., Greco, J. P., & Romanowsky, A. J. 2021, ApJ, 914, L23, doi: 10.3847/2041-8213/ac024f
  * Prole (2021) Prole, D. J. 2021, MNRAS, 506, L59, doi: 10.1093/mnrasl/slab073
  * Prole et al. (2021) Prole, D. J., van der Burg, R. F. J., Hilker, M., & Spitler, L. R. 2021, MNRAS, 500, 2049, doi: 10.1093/mnras/staa3296
  * Quadri & Williams (2010) Quadri, R. F., & Williams, R. J. 2010, ApJ, 725, 794, doi: 10.1088/0004-637X/725/1/794
  * Reid et al. (2016) Reid, B., Ho, S., Padmanabhan, N., et al. 2016, MNRAS, 455, 1553, doi: 10.1093/mnras/stv2382
  * Román et al. (2019) Román, J., Beasley, M. A., Ruiz-Lara, T., & Valls-Gabaud, D. 2019, MNRAS, 486, 823, doi: 10.1093/mnras/stz835
  * Román et al. (2021) Román, J., Castilla, A., & Pascual-Granado, J. 2021, A&A, 656, A44, doi: 10.1051/0004-6361/202142161
  * Román & Trujillo (2017) Román, J., & Trujillo, I. 2017, MNRAS, 468, 703, doi: 10.1093/mnras/stx438
  * Ross et al. (2011) Ross, A. J., Ho, S., Cuesta, A. J., et al. 2011, MNRAS, 417, 1350, doi: 10.1111/j.1365-2966.2011.19351.x
  * Safarzadeh & Scannapieco (2017) Safarzadeh, M., & Scannapieco, E. 2017, ApJ, 850, 99, doi: 10.3847/1538-4357/aa94c8
  * Sandage (1976) Sandage, A. 1976, AJ, 81, 954, doi: 10.1086/111975
  * Schmidt et al. (2013) Schmidt, S. J., Ménard, B., Scranton, R., Morrison, C., & McBride, C. K. 2013, MNRAS, 431, 3307, doi: 10.1093/mnras/stt410
  * Schneider et al. (2006) Schneider, M., Knox, L., Zhan, H., & Connolly, A. 2006, ApJ, 651, 14, doi: 10.1086/507675
  * Seldner & Peebles (1979) Seldner, M., & Peebles, P. J. E. 1979, ApJ, 227, 30, doi: 10.1086/156699
  * 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
  * Smith Castelli et al. (2008) Smith Castelli, A. V., Bassino, L. P., Richtler, T., et al. 2008, MNRAS, 386, 2311, doi: 10.1111/j.1365-2966.2008.13211.x
  * Somalwar et al. (2020) Somalwar, J. J., Greene, J. E., Greco, J. P., et al. 2020, ApJ, 902, 45, doi: 10.3847/1538-4357/abb1b2
  * Strauss et al. (2002) Strauss, M. A., Weinberg, D. H., Lupton, R. H., et al. 2002, AJ, 124, 1810, doi: 10.1086/342343
  * Tanoglidis et al. (2021) Tanoglidis, D., Drlica-Wagner, A., Wei, K., et al. 2021, ApJS, 252, 18, doi: 10.3847/1538-4365/abca89
  * Teyssier et al. (2012) Teyssier, M., Johnston, K. V., & Kuhlen, M. 2012, MNRAS, 426, 1808, doi: 10.1111/j.1365-2966.2012.21793.x
  * Tully et al. (2009) Tully, R. B., Rizzi, L., Shaya, E. J., et al. 2009, AJ, 138, 323, doi: 10.1088/0004-6256/138/2/323
  * van der Burg et al. (2016) van der Burg, R. F. J., Muzzin, A., & Hoekstra, H. 2016, A&A, 590, A20, doi: 10.1051/0004-6361/201628222
  * van der Burg et al. (2017) van der Burg, R. F. J., Hoekstra, H., Muzzin, A., et al. 2017, A&A, 607, A79, doi: 10.1051/0004-6361/201731335
  * van Dokkum et al. (2016) van Dokkum, P., Abraham, R., Brodie, J., et al. 2016, ApJ, 828, L6, doi: 10.3847/2041-8205/828/1/L6
  * van Dokkum et al. (2015a) van Dokkum, P. G., Abraham, R., Merritt, A., et al. 2015a, ApJ, 798, L45, doi: 10.1088/2041-8205/798/2/L45
  * van Dokkum et al. (2015b) van Dokkum, P. G., Romanowsky, A. J., Abraham, R., et al. 2015b, ApJ, 804, L26, doi: 10.1088/2041-8205/804/1/L26
  * Van Nest et al. (2021) Van Nest, J. D., Munshi, F., Wright, A. C., et al. 2021, arXiv e-prints, arXiv:2108.12985.  https://arxiv.org/abs/2108.12985
  * Wasserman et al. (2019) Wasserman, A., van Dokkum, P., Romanowsky, A. J., et al. 2019, ApJ, 885, 155, doi: 10.3847/1538-4357/ab3eb9
  * Wright et al. (2021) Wright, A. C., Tremmel, M., Brooks, A. M., et al. 2021, MNRAS, 502, 5370, doi: 10.1093/mnras/stab081
  * Wright et al. (2017) Wright, A. H., Robotham, A. S. G., Driver, S. P., et al. 2017, MNRAS, 470, 283, doi: 10.1093/mnras/stx1149
  * Yagi et al. (2016) Yagi, M., Koda, J., Komiyama, Y., & Yamanoi, H. 2016, ApJS, 225, 11, doi: 10.3847/0067-0049/225/1/11
  * Yang et al. (2007) Yang, X., Mo, H. J., van den Bosch, F. C., et al. 2007, ApJ, 671, 153, doi: 10.1086/522027
  * York et al. (2000) York, D. G., Adelman, J., Anderson, Jr., J. E., et al. 2000, AJ, 120, 1579, doi: 10.1086/301513
  * Zaritsky et al. (2021) Zaritsky, D., Donnerstein, R., Karunakaran, A., et al. 2021, arXiv e-prints, arXiv:2109.03345.  https://arxiv.org/abs/2109.03345
  * Zaritsky et al. (2019) Zaritsky, D., Donnerstein, R., Dey, A., et al. 2019, ApJS, 240, 1, doi: 10.3847/1538-4365/aaefe9
  * Zehavi et al. (2005) Zehavi, I., Zheng, Z., Weinberg, D. H., et al. 2005, ApJ, 630, 1, doi: 10.1086/431891

##  Appendix A Validation with GAMA

Figure 7: Top: Cross-correlation for red (left) and blue (right) GAMA test
galaxies, with $18<M_{g}<19$ mag. Bottom: Resulting inferred redshift
distributions for red (left) and blue (right) populations, with the input
distribution shown in histogram. Regardless of binning, we see a clear signal
in the red galaxies at $z\sim 0.05$, in accord with the input redshift
distribution. The situation is more challenging for the blue subset, but the
mean redshift in the distribution still does match the input distribution.

We present a test to gain some intuition about how the technique will work for
our low redshift sample over a limited volume.

We construct a test sample using the GAMA survey (Driver et al., 2009),
selecting galaxies with known redshifts to test our ability to recover the
input redshift distribution (see a more sophisticated analysis in Schmidt et
al., 2013). We use the NASA-SDSS Atlas (NSA) as the tracer sample, and to
maximize spatial overlap we focus only on the GAMA equatorial fields G09, G12,
and G15, for roughly 180 deg2, which is comparable to our HSC area of 200
deg2. Recall that we do not have masks or random catalogs for the NSA, but the
GAMA team do provide random catalogs (Farrow et al., 2015).

From the GAMA survey, we select a sample of low-mass galaxies with $18<r<19$
mag and $0.04<z<0.065$ and we start with red ($g-r>0.8$) galaxies for a total
of 255 galaxies). We restrict our attention to a narrow range in magnitude to
mimic our galaxies of interest. The input redshift distribution is basically a
delta function (Fig. 7). We find that we can recover the mean redshift of
$\langle z\rangle=0.054$ with $\langle z_{\rm out}\rangle=0.052\pm 0.015$. We
cannot recover the comparable redshift distribution for the blue galaxies, so
we widen the redshift distribution to $0.04<z<0.085$ and for 730 blue
($g-r<0.5$) galaxies we find $\langle z_{\rm out}\rangle=0.064\pm 0.014$ when
the truth is $\langle z\rangle=0.066$.

These tests demonstrate that when we input narrow redshift signals, even at
very low redshift, we are able to recover them. We do not generate spurious
signals at higher redshift. These tests give us some confidence that our
result, namely the apparently high-redshift tail to the LSB galaxy
distribution, is unlikely to be an artifact of our methodology.

##  Appendix B Green Galaxies

Here we present examples of green galaxies deemed to be in the sample (Figure
8) and green galaxies deemed to be in the background (Figure 9).

Figure 8: Examples of “green” galaxies that are considered to be in the main
sample, to be compared with the rejected examples in the next figure. Images
are 24″ on a side.

Figure 9: Examples of “green” galaxies that we flag as likely background. We
remove them on the basis of an apparent redder bulge-like component along with
clear spiral structure. Images are 24″ on a side.

##  Appendix C SDSS DR12 as the reference population

We look at the cross-correlation signal with the LSB galaxies and galaxies in
the SDSS DR12 large-scale structure catalog (Reid et al., 2016). In this case,
we do know the spatial mask, so we can construct random catalogs for the SDSS
DR12 sample. We see no evidence of correlation beyond $z\sim 0.15$.

Figure 10: The cross-correlation signal between the LSB galaxies and the SDSS
DR12 large-scale structure catalog. We see a clear correlation with the red
galaxies but not the green or blue at $z>0.04$.