# Brighter and More Massive Galaxies in the Vicinity of Ly$\alpha$ Nebulae

Natalie K. Wells Department of Astronomy, New Mexico State University, P.O.
Box 30001, MSC 4500, Las Cruces, NM, 88003, USA  Moire K. M. Prescott
Department of Astronomy, New Mexico State University, P.O. Box 30001, MSC
4500, Las Cruces, NM, 88003, USA  Kristian M. Finlator Department of
Astronomy, New Mexico State University, P.O. Box 30001, MSC 4500, Las Cruces,
NM, 88003, USA

(Received June 7, 2022; Accepted November 15, 2022)

###### Abstract

It has been well established in the local universe that galaxy properties
differ based on the large-scale environment in which they reside. As luminous
Lyman-alpha (Ly$\alpha$) nebulae have been shown to trace overdense
environments at z$\sim$2-3, comparing the properties of galaxies within
Ly$\alpha$ nebulae systems to those in the field can provide insight into how
and when locally-observed trends between galaxy properties and environment
emerged. Six Ly$\alpha$ nebulae were discovered at z$\sim$2.3 in a blind
search of the GOODS-S extragalactic field, a region also covered by the 3D-HST
spectroscopic survey. Utilizing 3D-HST data, we identified 86 galaxies in the
vicinity of these nebulae and used statistical tests to compare their physical
properties to galaxies elsewhere in the field. Galaxies lying within 320
proper kpc of a Ly$\alpha$ nebula are roughly half a magnitude brighter than
those in the field, with higher stellar masses, higher star formation rates,
and larger effective radii. Even when considering the effects of sample
incompleteness, our study suggests that galaxies in overdensities at
z$\sim$2.3 traced by Ly$\alpha$ nebulae are being influenced by their
environment. Furthermore, Ly$\alpha$ nebula-associated galaxies lie on the
same main sequence of star formation as field galaxies, but have a larger
proportion of high-mass galaxies, consistent with the idea that galaxy
evolution is accelerated in rich environments. Expanded surveys for Ly$\alpha$
nebulae in other deep extragalactic fields and galaxy spectroscopic follow-up
with JWST will better constrain the demographics of Ly$\alpha$ nebula-
associated galaxies.

high redshift galaxies, galaxy environments, galaxy evolution, protoclusters

††journal: The Astrophysical Journal††software: Astropy (Collaboration et al.,
2013, 2018), SciPy (Virtanen et al., 2020), NumPy (Harris et al., 2020),
pandas (McKinney, 2010), Matplotlib (Hunter, 2007)

##  1 Introduction

In the local universe (z$<$1), a galaxy’s structure is strongly dependent on
its surroundings: elliptical galaxies are found predominantly in clusters,
while disk galaxies are more common in isolation. This “morphology-density
relation” was first established by Oemler (1974) and Dressler (1980), and its
physical origins are still the subject of investigation. The star formation
rate (SFR) is also linked to environment and morphology at low redshifts, with
cluster ellipticals tending to be quiescent and red, and field spirals showing
blue colors and substantial SFRs (e.g. Lewis et al., 2002; Gomez et al., 2003;
Hogg et al., 2004; Kauffmann et al., 2004; Peng et al., 2010).

Study of the more distant universe gives us a window back in time to when
these environmental differences in galaxy properties first emerged. In high
redshift studies (z=2-4), galaxies in dense protocluster environments have
widely been found to have higher stellar masses when compared to a control
group (Koyama et al., 2013; Hatch et al., 2011; Steidel et al., 2005; Cooke et
al., 2014; Shimakawa et al., 2018; Ito et al., 2020). Somewhat less agreement
is found, however, when examining growth histories. For example, Koyama et al.
(2013) and Cooke et al. (2014) found that protocluster and field galaxies
follow the same SFR-stellar mass relation at z$\sim$2.2 and z$\sim$2.5,
respectively, as did Shimakawa et al. (2018) when comparing lower and higher
density regions of a protocluster at z$\sim$2.5. However, Shimakawa et al.
(2018) found boosted SFRs for a given stellar mass among galaxies in the
densest subset of their sample. By contrast, Hatch et al. (2011) found lower
specific SFRs among members of two protoclusters at z$\sim$2.2 compared to the
field. Finally, Koyama et al. (2013) saw redder colors among protocluster
galaxies, whereas Hatch et al. (2011) found no such distinction between their
protocluster and field groups.

The tendency for high redshift protocluster environments to host a larger
fraction of high mass, actively star-forming galaxies differs from the trend
seen in the local universe, and may indicate that star formation begins
earlier in galaxies in rich environments (e.g., Hatch et al., 2011; Ito et
al., 2020), consistent with theoretical predictions (e.g. Chiang et al.,
2017). Additional high redshift studies with complementary approaches to
galaxy sample selection are needed to disentangle the influences that
intrinsic and environmental differences have on the assembly histories of
galaxies and clusters. Unique among the studies already mentioned, which
select protocluster galaxies based on strong line emission or UV continuum
emission, this work uses proximity to Ly$\alpha$ nebulae. Ly$\alpha$ nebulae
(also known as Lyman-alpha blobs, LABs, or simply “blobs”) are extended
sources of Ly$\alpha$ emission found at z$\sim$2-6. Typically more than 100
kiloparsecs across, these glowing clouds of gas are some of the largest
objects known and have been associated with dense protocluster environments
(e.g. Steidel et al., 2000; Matsuda et al., 2004; Saito et al., 2006; Prescott
et al., 2008; Yang et al., 2009, 2010; Hennawi et al., 2015; Cai et al.,
2017a, b; Battaia et al., 2018). Due to the expansion of the Universe, the
ultraviolet light from Ly$\alpha$ nebulae is redshifted to optical
wavelengths, making it possible to observe these giant high-redshift
structures with sensitive optical ground- and space-based telescopes.

In order to probe the origin of environment-dependent trends in galaxy
properties, we use these Ly$\alpha$ nebulae as signposts of overdense regions
and leverage high resolution Hubble Space Telescope (HST) observations of the
GOODS-S extragalactic field to compare a composite group of galaxies in the
vicinity of 6 known Ly$\alpha$ nebulae to a control group in the field at
z$\sim$2.3. We discuss data sources and quality requirements in Section 2,
sample selection in Section 3, and analysis methods in Section 4. We report
our results and describe the various tests of the robustness of these findings
in Section 5. We discuss our results and conclude in Sections 6 and 7.
Throughout, we assume the standard $\Lambda$CDM cosmology, i.e.,
$\Omega_{m}=0.27$, $\Omega_{\Lambda}=0.73$, $h=0.7$. The angular scale at
$z=2.3$ is 8.4 kiloparsecs/arcsec (kpc/′′). All magnitudes are in the AB
system (Oke, 1974).

##  2 Data

3D-HST is a Hubble Space Telescope (HST) spectroscopic survey covering four
well-studied extragalactic fields: AEGIS, COSMOS, GOODS-S, and UDS (Skelton et
al., 2014). It builds upon the Cosmic Assembly Near-infrared Deep
Extragalactic Legacy Survey (CANDELS), which obtained Hubble Wide Field Camera
3 (WFC3) imaging of the same regions, as well as the GOODS-N field (Grogin et
al., 2011). The online 3D-HST catalog1113D-HST website and online catalog:
https://3dhst.research.yale.edu/Data.php provides imaging, photometry,
spectra, emission line measurements, redshifts, and other derived properties
for $\sim$100,000 galaxies in all five CANDELS fields using data from 3D-HST,
CANDELS, and other space- and ground-based surveys (Momcheva et al., 2016;
Skelton et al., 2014; Brammer et al., 2012).

Building on CANDELS data, van der Wel et al. (2012) derived structural
parameters for the galaxies in 3D-HST using the best Sérsic model fits to
CANDELS imaging. Their catalogs, available online222CANDELS structural
catalogs: https://www2.mpia-hd.mpg.de/homes/vdwel/3dhstcandels.html,
complement those of 3D-HST and are line-matched using the same photometric
IDs. Guo et al. (2013) provide additional data on objects in GOODS-S in their
online catalog333CANDELS GOODS-S multi-wavelength catalog:
https://cdsarc.cds.unistra.fr/viz-bin/cat/J/ApJS/207/24, including limiting
F160W magnitudes.

###  2.1 Galaxy Properties and Quality Cuts

For all galaxies in our sample, we gathered WFC3 F160W $(H)$ and F814W $(I)$
fluxes from the 3D-HST GOODS-S photometric catalog (release V4.1), converting
to AB magnitudes. We consider only those galaxies with good photometry (i.e.
only those whose $use\\_phot$ flag was set to 1; Skelton et al., 2014). This
is especially important given that over 90% of sources near z=2.3 have only a
photometric redshift, upon which we rely when selecting nebula-associated
galaxies and their counterparts in the field. We also obtained the following
stellar population parameters, which were estimated from the best fits to
multiband photometry using FAST (Kriek et al., 2009): stellar mass, SFR,
specific SFR, stellar age (since the onset of star formation), star formation
decay timescale $\tau$, and dust attenuation AV. To ensure we were only using
galaxies with high quality SED fits, we required the reported $\chi^{2}$ of
the FAST fit to be less than 2. This cut retained 79% of the GOODS-S
photometric galaxy sample.

From the 3D-HST GOODS-S grism catalog (release V4.1.5), we collected fluxes
and equivalent widths for the OII, OIII, and H$\beta$ emission lines,
requiring S/N$>$2\. The “z_best” grism catalog provides redshifts as well as
combined UV+IR star formation rates. The “best” redshift available may be a
spectroscopic redshift or, in its absence, a grism redshift. In the absence of
both, a photometric redshift is provided (Momcheva et al., 2016). UV+IR star
formation rates based on fluxes with low S/N were rejected by requiring the
SFR flag to be set to zero.

Finally, we obtained Sérsic indices $n$ and effective radii re from van der
Wel et al. (2012). We required the provided flag to be set to zero, indicating
a good fit. We also cross-matched objects in the 3D-HST GOODS-S catalogs with
that of Guo et al. (2013) to associate the F160W limiting magnitude of the
surrounding region to each galaxy in our sample.

Given data on over 50,000 sources within the GOODS-S field, we proceeded to
compare a range of properties between galaxies in the vicinity of Ly$\alpha$
nebulae and similarly selected galaxies in the field.

##  3 Sample Selection

###  3.1 Ly$\alpha$ Nebula Sample

Our Ly$\alpha$ nebula sample is taken from a blind survey of four
extragalactic fields (Yang et al., 2010). The nebulae were selected using a
narrowband filter corresponding to z=2.3${\pm}$0.037. Based on the derived
number density from this blind search, the authors estimated a halo mass
hosting a Ly$\alpha$ nebula to be $\geq$1013 M$\odot$. Of the 25 Ly$\alpha$
nebulae detected, sixteen lie within the Chandra Deep Field South (CDFS), and
six of these overlap the 3D-HST GOODS-S catalog.

(a)  (b)

Figure 1: _Left:_ The GOODS-S extragalactic field ($\sim$170 arcmin2), showing
F160W limiting magnitude of galaxies in shades of gray (Guo et al., 2013), and
locations of Ly$\alpha$ nebulae as red crosses (Yang et al., 2010). _Right:_
Selected blob ($yellow$) and field ($black$) galaxies at z$\sim$2.3. Galaxies
in the shallower lower third of the field and in the deeper Hubble Ultra Deep
Field (Koekemoer et al., 2013; Beckwith et al., 2006) are eliminated after
imposing a limiting magnitude constraint (Section 3.2). Figure 2: Close-up
view of the blob galaxy sample. The blob locations are labeled with red
crosses, and associated galaxies (as defined in Section 3.2) are indicated as
black circles.

###  3.2 Galaxy Sample

We defined Ly$\alpha$ nebula-associated galaxies to be those within 40
arcseconds of any of the six Ly$\alpha$ nebulae locations and with
z=2.3$\pm$0.15, forming a composite “blob galaxy” sample. We selected these
radius and redshift ranges based on a number of considerations. The redshift
range within which we can expect to find galaxies associated with a nebula
must be large enough to capture true members, given the redshift
uncertainties, but not so large as to include a great number of interlopers.
The best available redshift measurements for the vast majority of galaxies
near z=2.3 in the 3D-HST catalog are photometric, with typical uncertainties
of $\Delta$z=0.15, corresponding to $\Delta$z/(1+z)=0.05 at this redshift,
roughly an order of magnitude larger than any $\Delta$z expected from peculiar
motion (even considering cluster members).

A similar balance must be struck when defining a radius around each Ly$\alpha$
nebula within which we select blob galaxies. Thanks to space-based imaging,
uncertainty in on-sky position is negligible. From our analysis, a 40” radius
($\sim$320 proper kpc at z$\sim$2.3) emerged as the distance beyond which
statistically significant differences between blob and field galaxies
disappear (see Section 5.2.3). Finally, survey depth varies in GOODS-S (Figure
1, _left_), so in order to make a fair comparison of the galaxy population
across the field, we rejected both shallower and deeper regions by only
selecting galaxies with an associated F160W limiting magnitude between 28.25
and 28.75 (Figure 1, _right_). However, we note that when we do not correct
for field depth in this way, we see qualitatively the same results.

Eighty-six galaxies make up the composite blob sample (Figure 2). We defined
our control sample as all other GOODS-S sources lying outside of the defined
blob regions whose redshifts and limiting magnitudes fell within the same
specified ranges, yielding a population of 1,099 galaxies. In total, 1% of the
galaxy redshifts used in this work are are spectroscopic, 7% are grism, and
91% are photometric. Only a fraction of the blob and field galaxies have data
for each property of interest that meet the quality requirements outlined in
Section 2.1. Thus, the number of galaxies $n$ included in a given analysis is
specified where applicable.

##  4 Analysis

We utilized two non-parametric statistical tests in order to compare
properties between blob and field galaxies: the 2-sample Kolmogorov–Smirnov
(KS) test and the k-sample Anderson-Darling (AD) test, implemented using the
$ks\\_2samp$ and $anderson\\_ksamp$ functions in SciPy’s $stats$ module
(Virtanen et al., 2020). Given two sets of observations of a single,
continuous variable, these tests compare the distributions of the two samples
without assuming anything about the nature of the parent distribution.
Graphically, both methods arrange the data from each sample into empirical
cumulative probability distributions (CPDs). The KS test reports the largest
difference between the two CPDs at any one interval, and from this statistic a
$p$-value between 0 and 1 is calculated given the size $n$ of each sample
(Hodges, 1958). Meanwhile, the AD test calculates the sum of the squared
differences between the CPDs at _every_ interval, and gives more weight to
differences in the tails of the distributions. As with the KS test, a
$p$-value is produced based on this statistic and the sample size (Scholz &
Stephens, 1987). In both cases, the reported $p$-value indicates the
probability that such a difference would be seen by chance assuming that the
two samples are drawn randomly from the same parent distribution (the null
hypothesis). A low $p$-value, resulting from a large difference between the
CPDs, indicates that we have reason to reject the null hypothesis. For
example, a $p$-value of $<$0.05 indicates that there is less than a 5% chance
that two given samples come from the same population. This is a commonly
accepted threshold for statistical significance that is used in works
comparable to this one (e.g. Hatch et al., 2011; Cooke et al., 2014; Shimakawa
et al., 2018), and is what we adopt here.

##  5 Results

###  5.1 Fiducial Results

Resulting $p$-values from both statistical tests are displayed for every
property we examined in Figure 3, with the differential and cumulative
distributions of selected properties shown in Figures 4, 5, and 6. We see
statistically significant differences between the blob and field galaxies in
five properties: F160W and F814W magnitudes, stellar masses, star formation
rates (SFR), and effective radii (re). In these five cases, the size of the
blob galaxy sample is around 75, and the size of the field galaxy sample is
around 1000, except for re, which has sample sizes $\sim$30% smaller. The
differences seen in this group of properties are well distinguished from the
other galaxy trait comparisons we performed: in all five cases where the
$p$-value from the KS test falls below the $p$=0.05 threshold, we see that the
$p$-value from the AD test does as well (we include re in this group, with KS
test $p$=0.06, AD test $p$=0.03). In all other cases, the $p$-values from both
tests fall well above this threshold. It is important to note sample sizes in
each case, as these vary widely based on the availability and quality of data.
Properties which have a blob galaxy sample smaller than $\sim$30, such as the
emission line measurements, have too few blob galaxies for a reliable KS test
result (Razali & Wah, 2011), meaning that if a true difference were present,
we would be unable to detect it with so few galaxies. As a concrete example,
unlike the FAST-derived SFR measurement where the blob galaxy sample size is
$n$=72 (KS test $p$=0.02), we have only 28 quality blob galaxy measurements
for the combined UV+IR SFR (SFRUV+IR), with KS test $p$=0.13. When we compared
the FAST SFR of these same blob galaxies ($n$=28) to the field, the resulting
KS test shows no significant difference ($p$=0.19), even though we know that
with a larger blob galaxy sample size ($n$=72), a significant difference is
indeed detected ($p$=0.02) (Figure 3).

Figure 3: KS (_dark purple_) and AD (_light purple_) test $p$-values for all
properties compared between galaxies near Ly$\alpha$ nebulae and those
elsewhere in the GOODS-S field at z$\sim$2.3 (see Section 2.1). Both tests
indicate statistically significant differences between the two groups in
magnitude (F160W and F814W), stellar mass, star formation rate (SFR), and
effective radius (re). Among the other traits, $p$-values from both tests are
well above the $p$=0.05 threshold (_dashed line_).

Our most robust result is that blob galaxies are brighter in both filters than
those in the field. Normalized histograms comparing the filter magnitudes are
shown in Figure 4, with blob galaxy distributions in $red$, field galaxy
distributions in $blue$, and sample sizes and KS/AD test $p$-values shown in
the inset boxes. In both filters, we see a proportionally larger number of
bright blob galaxies and a systematic offset towards brighter magnitudes in
the blob galaxy group with respect to the field. It must be noted that the
distributions are incomplete at the faint end; the 50% and 75% F160W
completeness thresholds for GOODS-S are indicated with light and dark gray
dashed lines respectively. However, given that both samples are depth-
corrected, the two groups should be incomplete in the same way, so, all else
being equal, we would expect the distributions to fall off at the faint end in
the same manner. Instead, they do not: in F160W the offset to brighter
magnitudes becomes apparent before 26.5 mag, the 50% completeness threshold
for this filter in this region. The same type of offset is seen also in the
F814W data. It does appear, then, that blob distributions have proportionally
more bright galaxies, as well as fewer faint ones. At the same time, while
blob galaxies appear to be systematically brighter than those in the field,
they do not exhibit a statistically significant difference in F814W-F160W
color (Figure 4).

Figure 4: Comparisons of the distributions of galaxies near Ly$\alpha$ nebulae
(_red_) and those in the field (_blue_) for three properties: F160W mag, F814W
mag, and F814W-F160W color index. The 75% and 50% F160W completeness
thresholds are shown as dark and light gray dashed lines, respectively. The
median value of each group is indicated with a triangle. Sample sizes and
$p$-values from the KS and AD statistical tests are provided in the inset
boxes. In both filters, a systematic offset towards brighter magnitudes is
seen among the blob galaxies with respect to those in the field, with
correspondingly low $p$-values. No significant difference is seen in
F814W-F160W color.

Unsurprisingly, given their systematically brighter magnitudes, blob galaxies
appear to be consistently shifted towards higher stellar masses and SFRs
relative to galaxies in the field when we examine these FAST-derived
properties (Figure 5). No such offset is seen when comparing SFR per unit
stellar mass (specific SFR) between the two groups, suggesting that while the
blob group is skewed towards higher masses, the two groups may follow the same
SFR-mass relation; we return to this point in Section 6.

Figure 5: Comparisons of the distributions of galaxies near Ly$\alpha$ nebulae
(_red_) and those in the field (_blue_) for three properties: stellar mass,
star formation rate (SFR), and SFR per unit stellar mass (or specific star
formation rate, sSFR). The median value of each group is indicated with a
triangle. Sample sizes and $p$-values from the KS and AD statistical tests are
provided in the inset boxes. A systematic offset towards higher mass and SFR
is seen among the blob galaxies with respect to those in the field, with
correspondingly low $p$-values. No statistically significant difference is
seen in sSFR between the groups.

Finally, the effective radii of blob galaxies appear to be systematically
larger than those in the field. The corresponding histograms are shown in
Figure 6.

Figure 6: Comparison of blob (_red_) and field (_blue_) galaxy effective
radius distributions. The median value of each group is indicated with a
triangle. Sample sizes and $p$-values from the KS and AD statistical tests are
provided in the inset boxes. A systematic offset towards larger effective
radii is seen among the blob galaxies with respect to those in the field,
accompanied by correspondingly low $p$-values.

Thus far, we have demonstrated that there is a statistical difference between
blob and field galaxies in these five key properties. Next, we aim to quantify
the nature and magnitude of these differences. The non-parametric KS and AD
tests are sensitive to differences in shape, center and spread between two
distributions, but the resulting statistic and $p$-value do not indicate which
one (or more) of these differences is at play, or how large it is. If our
visual impression that the blob distributions are systematically offset from
the field is correct, with the primary difference being their centers and not
their shape or spread, then we can directly compare their medians. We compare
medians instead of means to avoid being sensitive to a few outliers in our SFR
and re data. For a given property we added the difference between the blob and
field galaxy medians to every individual blob galaxy measurement, thereby
preserving the blob distribution’s shape and spread but shifting its median to
coincide with that of the field. When we redo the KS and AD tests, we find
that for all five properties (F160W, F814W, stellar mass, SFR, and effective
radius), the KS test $p$-values rise to 0.96, 0.94, 0.70, 0.67, and 0.72
respectively, and the AD test $p$-values exceed 0.25. This indicates that the
shape and spread of the distributions do not differ significantly, and that
the difference in median is driving the statistical difference.

Thus, this simple exercise indicates that galaxies near blobs are roughly 0.49
magnitudes or 1.6 times brighter in F160W and 0.59 magnitudes or 1.7 times
brighter in F814W, have 1.9 times more stellar mass, have star formation rates
that are 1.4 times higher, and have effective radii that are 1.3 times larger
than galaxies in the field. In what follows, we explore whether these results
could be influenced by issues of incompleteness or sample selection.

###  5.2 Verifying the Results

In this section we describe the various measures we took to test the
reliability of our results. In particular, we investigated the consequences of
imposing completeness limits, how the size of the region within which we
select blob galaxies affects our results, and whether a subset of the six blob
regions is primarily responsible for the outcome. We also estimated the
expected false positive rate by selecting six random blob-free locations
within GOODS-S and analyzing the resulting “mock” blob and field galaxies in
identical fashion.

####  5.2.1 Sample Completeness

The faint end of our galaxy sample will necessarily be affected by
incompleteness. However, given that our sample is drawn from regions of
consistent survey depth (F160W limiting magnitude mlim = 28.50${\pm}$0.25
mag), we expect our blob and field galaxy populations to be incomplete to the
same degree, allowing them to be fairly compared with one another while
keeping in mind this limitation. Nevertheless, we confirmed that our
qualitative results are robust to completeness issues via two distinct
experiments: (1) removing the faintest galaxies and (2) correcting the sample
for incompleteness by adding simulated galaxies to the faintest bins. In what
follows, we use the completeness fraction as a function of F160W magnitude for
the GOODS-S field (Skelton et al., 2014).

Figure 7: Comparison of blob and field galaxy F160W distributions when
cropping the sample at the 50% (_top_) and 75% (_middle_) completeness limits.
Alternatively, we simulated a sample corrected for incompleteness by adding
the appropriate number of mock galaxies to the faintest bins (_bottom_).

First, when only considering blob and field galaxies with F160W magnitudes
brighter than the 50% completeness limit, the AD and KS $p$-values remain
significant at 0.02 and 0.06 respectively. When we crop the sample at the 75%
completeness limit, both $p$-values rise above 0.25 (Figure 7, _top_ and
_middle_ panels). As discussed in Section 5, though our sample is corrected
for depth, the blob and field galaxy F160W and F814W distributions do not fall
off in the same manner at the faint end as would be expected, but rather there
appear to be proportionally fewer faint galaxies in the blob regions (Figure
4). When we remove the faintest galaxies to increase the completeness of our
sample, the signal weakens, emphasizing that the faint end is contributing to
the differences we detect.

In the second approach, we simulated a complete F160W sample. Using Skelton et
al.’s (2014) completeness fraction for each half-magnitude bin, we calculated
the number of missing galaxies in a given bin, and drew a corresponding number
of galaxy magnitudes randomly, assuming a uniform distribution within the bin.
After artificially recovering the faintest galaxies in our sample in this way,
differences between the blob and field F160W distributions become even more
pronounced, with both AD and KS $p$-values falling below 0.01 (Figure 7,
_bottom_ panel).

These incompleteness tests suggest that the existing data are already
revealing real differences between blob and field galaxies. In the remainder
of the paper, we therefore choose to stay close to the data, focusing on the
fiducial depth-corrected sample. We leave a more in-depth investigation of the
impact of incompleteness on all other measured quantities to future work.

####  5.2.2 The Blob Region Radius

In using Ly$\alpha$ nebulae as markers of overdensities, we must define the
angular extent of such regions. A small “blob region” radius, centered on a
Ly$\alpha$ location, is more likely to capture a high fraction of true
inhabitants of the dense environment, but restricts our sample size. As the
radius is extended, we benefit from a larger blob galaxy sample, but that
sample is more likely to contain a greater fraction of interlopers - galaxies
that are not subject to the same potential influences as those embedded in an
overdense environment. Making no _a priori_ assumptions about what the
plausible physical size of such a dense region might be, we tested blob region
radii from 10”-100” ($\sim$84-840 kpc). This range straddles the expected
virial radius of $R_{vir}$=219 kpc for a galaxy group mass halo ($\sim$1013
M$\odot$) at z$\sim 2.3$, based on a standard spherical collapse calculation
(Bryan & Norman, 1998). In steps of 5”, we compared the properties of each new
set of trial blob and field galaxies using the KS and AD tests as before. This
allowed us to see how the $p$-values changed as the defined blob region varied
in size. For simplicity and as the more conservative option we plot only the
KS test $p$-values in this subsection (Figures 8-11). However, in general we
found the AD test to produce lower $p$-values given the same samples.

In Figure 8 we see that comparisons in the five selected properties produce
$p$-values that dip below $p$=0.05 when the blob region reaches $\sim$40” in
radius. For r=10-35”, the blob galaxy sample size grows from $\sim$12 to
$\sim$65 when considering the filter magnitudes; from 11 to 61 when
considering the mass and SFR; and from 8 to 35 when considering the effective
radius. The larger $p$-values seen in this range show the limit of the KS test
for small sample sizes - we have demonstrated that the two populations differ
in these properties when r=40” (e.g. Figure 3), but these differences cannot
be detected at smaller radii given the limited blob galaxy sample size.

Figure 8: KS test results comparing selected properties of sets of blob and
field galaxies for increasing blob region radius. It is not until the defined
blob region reaches $\sim$40” in radius that the blob-associated galaxy sample
size is large enough for the KS test to detect the difference between galaxies
in the blob region and those in the field.

A potential problem with varying the size of the blob region radius in this
way is that as the blob galaxy sample grows the field galaxy sample shrinks,
with members of the latter group switching to the former as the blob region
radius increases. Consequentially, the blob sample is not being compared to a
static group, but rather one whose distribution changes slightly at every
step. In order to control for this, we performed the same test but this time
defined field galaxies to be those that lie greater than 60” from any
Ly$\alpha$ nebula location, giving us a consistent field group to which to
compare blob galaxies. As we scanned through blob region radii of 10”, 20”,
30”, 40”, and 50”, those galaxies that fell in the gap between the blob region
radius and our field galaxy radius were simply ignored. This method produced
results that were virtually identical to Figure 8, demonstrating that the
impact of slight changes in the field galaxy distribution is negligible when
using the previous approach.

####  5.2.3 The Core Blob Galaxies

Given that the $p$-values for the five properties in Figure 8 dip below and
generally stay under $p$=0.05 from r=30” out to r=100”, we then investigated
whether 1) the galaxies closest to the blobs were responsible for this
outcome, differing from the field enough to depress the $p$-values at large
blob region radii despite the likely possibility that larger radii will
incorporate more and more interlopers into the blob galaxy sample, or whether
2) galaxies at relatively large distances from the blobs were contributing to
this effect. To this end, we progressively removed the most central blob
galaxies from the analysis entirely, and, allowing the blob region radius to
vary as before, compared the properties of the blob and field galaxy subsets.
The results of this test are shown in Figure 9. With more and more central
blob galaxies missing, the $p$-value curves steadily rise, demonstrating that
the core blob galaxies are indeed responsible for the low $p$-values seen in
Figure 8. When central galaxies within 40” of any blob are absent,
statistically significant differences in the five selected properties
disappear. This is not due to prohibitively small sample sizes, as the blob
sample size at this stage (those galaxies with 40”$<$r$<$50”) is large enough
(n=43) for a reliable result. This demonstrates that it is only galaxies
within a 40” radius of any blob that appear to be distinct from the field.

Thus, when defining the blob region radius for our fiducial analysis (Section
3.2), we converged on r=40”, this being the smallest radius that allows a blob
galaxy sample of sufficient size (Section 5.2.2), as well as being the maximum
size of the Ly$\alpha$ nebulae’s apparent region of influence.

Figure 9: Resulting $p$-value curves when galaxies within 10” (_first panel_),
20” (_second_), 30” (_third_), and 40” (_fourth_) of any blob are absent from
the analysis. When galaxies within a 40” blob region radius are missing
(_fourth panel_), statistical differences in the five selected properties
disappear, indicating that galaxies within this radius are driving the
differences we detect, and not those at larger distances from the Ly$\alpha$
nebulae.

####  5.2.4 All Blobs are to Blame

We considered the possibility that just one or two of the six blob regions
contained galaxies sufficiently distinct from the field so as to sway the
results, given that we are using a composite blob galaxy sample. Again we
allowed the blob region radius around each blob position to grow from 10” to
100”, comparing the resulting blob and field galaxies after having removed
each blob and blob pair (since two pairs of blobs lie within 55” of each
other, see Figure 2) from the analysis in turn. Without the contribution of
the galaxies associated with one particular blob or another, we still see
significant differences between the blob and field galaxies at r$\approx$40”,
as shown in Figure 10, although LAB-06 and LAB-14 may be contributing more to
the signal. All of the blob regions, then, appear to contain galaxies that are
distinct from the field.

Figure 10: Removing galaxies in each blob region from the analysis in turn
does not eliminate all statistically significant differences between the blob
galaxy and field galaxy groups (although LAB-06 and LAB-14 may be contributing
more to the signal), indicating that all six blob regions, and not a single
special location, are responsible for the observed differences.

####  5.2.5 Random Blob Locations

As an independent check of our KS test results, we examined the same $p$-value
curves over increasing blob region radii when using random Ly$\alpha$ nebula
locations, comparing the resulting groups of mock blob and field galaxy
properties in order to see how likely it is that we would see results similar
to Figure 8 by chance. Yang et al. (2010) surveyed the entire footprint of
GOODS-S with 3D-HST coverage, so if random blob-free positions at z=2.3 show
galaxy properties that are similarly offset from those in the field, this
would cast doubt on our ability to use Ly$\alpha$ nebulae to identify special
areas of influence at this epoch.

To carry out this test, we generated 100 trial sets of 6 random blob positions
in GOODS-S, computing the KS test results and p-value curves in each case. The
top panel of Figure 11 shows a typical result, with high and meandering
$p$-values as a function of blob region radius. By contrast, the real blob
locations produced $p$-values that dipped below $p$=0.05 at or before r=40”
for 4 out of 5 properties of interest (Figure 8). Out of those 100 trials,
only 3 produced $p$-value that meet those same criteria (Figure 11, _lower
three panels_).

The same test was repeated after removing what we may call the “true” blob
galaxies, those we defined as lying within 40” of any of the 6 actual
Ly$\alpha$ nebula locations with 2.15$\leq$z$\leq$2.45, to ensure that those
galaxies were not contaminants in the random location trials. No significant
difference was seen in the resulting $p$-value curve plots.

These results correspond to a 3% false positive rate, consistent with or
better than the expected 5% threshold employed in the KS test. In addition, if
we use a stricter definition of significance satisfied by our real data,
requiring $p<$0.05 across the range of r=40-50” for 4 out of 5 properties, we
find zero false positives out of 100 trials. Thus, the environments around the
six true Ly$\alpha$ nebulae do indeed appear to be “special” places in the
GOODS-S field at z$\sim$2.3.

Figure 11: Examples of KS $p$-value curves from 100 trials using random
Ly$\alpha$ blob locations. The first panel is an example of the typical
result. The remaining panels are the three cases which met our “false
positive” criteria, as discussed in the text, i.e., appearing most similar to
our results when using the true Ly$\alpha$ blob locations (Figure 8).

##  6 Discussion

Our investigation has found significant differences between galaxies
associated with Ly$\alpha$ nebulae at z$\sim$2.3 versus those elsewhere in the
170 arcmin2 GOODS-S field. Under the assumption that Lya nebulae trace
overdensities, our results imply that at this redshift, galaxies in denser
environments are brighter in F160W and F814W magnitude (n=86, n=74), with
higher SFR (n=72) and larger effective radii (n=44). We found that galaxies
near Ly$\alpha$ nebulae populate the same main sequence of star formation
(n=72; Figure 5, _bottom_ ; Figure 12), but that nebula regions have
proportionally more high-mass galaxies as well as fewer low-mass ones (n=72;
Figure 5, _top_). We note that our results were based on an empirically-
derived radius criterion of 40” ($\sim$320 proper kpc at z$\sim$2.3). Assuming
these z$\sim$2.3 Ly$\alpha$ nebulae inhabit galaxy group scale 1013 M$\odot$
mass halos (Yang et al., 2010), a standard spherical collapse calculation
(Bryan & Norman, 1998) suggests virial radii of 219 proper kpc. Interestingly,
this value is within 50% of the radius where our analysis robustly detects
differences in the properties of blob-associated galaxies versus those in the
field.

Our findings are in broad agreement with other studies of the dependence of
protocluster galaxy properties on environment at high redshift (z=2-4). Many
studies have reported that protocluster members skew towards higher stellar
masses (Koyama et al., 2013; Hatch et al., 2011; Steidel et al., 2005; Cooke
et al., 2014; Shimakawa et al., 2018; Ito et al., 2020). Protocluster and
field galaxies also generally follow the same SFR-stellar mass relation (the
so-called star forming main sequence), however here there is somewhat less
agreement in the literature. For example, Steidel et al. (2005) examined 72
UV-selected galaxies in a z$\sim$2.3 overdensity and found the mean stellar
mass was 2x higher than a comparison group. Koyama et al. (2013) observed 83
H-alpha emitting galaxies (HAEs) in and around a protocluster at z$\sim$2.2,
finding that galaxies residing in the densest regions of the cluster appeared
to follow the same SFR-stellar mass relation as those in the outskirts, and
that the high-density population contained a greater proportion of high-mass
galaxies. Cooke et al. (2014) compared HAEs within a 7 arcmin2 overdensity at
z$\sim$2.5 to a control group in the field and reported that protocluster and
field galaxies showed the same star-forming main sequence with protocluster
members skewing towards higher stellar masses, again in good accord with our
own results. When considering $>$100 HAEs in another z$\sim$2.5 protocluster
and dividing the sample into regions of higher and lower density, Shimakawa et
al. (2018) observed the same star-forming main sequence in each but saw a
larger proportion of high-mass, high-SFR galaxies in the higher density
subset. However, when comparing the lower density group to a superdense subset
of the higher density group, Shimakawa et al. (2018) found that the SFRs of
the superdense group were boosted for their given stellar mass. Finally, when
comparing HAEs within 2 protoclusters at z$\sim$2.2 to a control field sample,
Hatch et al. (2011) reported that the protocluster group was 0.8 mag brighter
than the control, corresponding to a $\sim$2x greater stellar mass. However,
the Hatch et al. (2011) samples did not differ in SFR and the specific SFR of
the protocluster population was determined to be lower than that of the field,
in contrast to our results.

At the same time, results on other physical properties are more mixed. For
example, our study did not detect a difference in F814W-F160W color, dust
extinction (AV), or stellar age, but did find larger effective radii among
galaxies in overdensities. Hatch et al. (2011) also observed no significant
difference in color when comparing HAEs within 2 protoclusters at z$\sim$2.2
to a control field sample, but by contrast, Koyama et al. (2013) found the
HAEs in their high-density group to be redder. Cooke et al. (2014) found a 2x
larger median AV among protocluster members, while Steidel et al. (2005)
reported a 2x greater inferred age for the overdensity group. Building on
Steidel et al. (2005), Peter et al. (2007) compared the size distributions of
85 UV-selected star-forming galaxies near the same z$\sim$2.3 protocluster to
63 control galaxies and found no significant difference, yet Shimakawa et al.
(2018) found larger sizes for HAEs in the denser regions of the z$\sim$2.5
protocluster. Thus, additional study and larger galaxy samples will be needed
to to better understand how galaxy properties correlate with environment at
these redshifts.

Taken together, our results add to a suite of high redshift studies
consistently finding that galaxies in protoclusters have greater stellar
masses than their counterparts in the field. Although varied in their approach
to selecting and studying overdense regions, all of these snapshots are
consistent with the idea of galaxies in rich environments having matured
earlier than their peers in the field, something that is supported by several
theoretical studies (Muldrew et al., 2015; Chiang et al., 2017; Lovell et al.,
2018). In this picture, protocluster galaxies virialize and commence star
formation sooner, leading them to exhaust their gas reservoirs earlier and
evolve into the massive, inactive elliptical galaxies that we observe haunting
cluster environments in the local universe. The broad consistency of our
results with other complementary protocluster studies lends additional support
to the use of Ly$\alpha$ blobs as markers of dense environments.

Figure 12: SFR versus stellar mass of galaxies associated with blobs ($red$)
and those in the field ($blue$). Triangle symbols indicate a non-zero SFR
below $10^{-4.5}$ solar masses/year. Both populations appear to follow the
same star-forming main sequence, corroborating the result that there is no
statistical difference in their specific SFR distributions (Figure 5, _bottom
panel_).

##  7 Conclusions

Ly$\alpha$ nebulae reside in overdense regions and therefore provide an
opportunity to study the impact of such dense environments on the local galaxy
population. Using 3D-HST GOODS-S catalog data, we compared traits between
galaxies in the vicinity of six Ly$\alpha$ nebulae and those in the field at
z$\sim$2.3. We found statistically significant differences between the two
groups in five properties: nebula-associated galaxies are systematically
brighter (F160W and F814W) than the field, with correspondingly higher stellar
masses, SFRs, and effective radii, but both populations exhibit the same SFR-
stellar mass relation. These results are broadly consistent with other studies
examining galaxy traits in dense protocluster environments from z=2-4, which
have found that the specific star formation rate does not appear to be
affected in such regions, but that the fraction of high-mass, high-SFR
galaxies is greater in protocluster populations. This suggests that, while
star formation proceeds in the same fashion regardless of environment at this
epoch, galaxies in dense environments have formed earlier and so are “ahead”
in their development compared to their peers, a scenario predicted by several
theoretical studies. The larger and higher quality galaxy samples expected
from the James Webb Space Telescope (JWST) will add clarity to our
understanding of high-redshift galaxies and the effects of cluster and group
environments on their evolution.

The authors would like to thank Kelly Sanderson, Audrey Dijeau, and Daniel
Godines Alcantara for helpful discussions. N.K.W. and M.K.M.P. acknowledge
support from NSF grant AAG-1813016. This work is based on data obtained for
the 3D-HST Treasury Program (GO 12177 and 12328) and the CANDELS Multi-Cycle
Treasury Program by the NASA/ESA Hubble Space Telescope, which is operated by
the Association of Universities for Research in Astronomy, Incorporated, under
NASA contract NAS5-26555.

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