# Resolved UV and optical color gradients reveal environmental influence on
galaxy evolution at redshift z$\sim$1.6

Cramer, W. J  Department of Physics and Astronomy, Notre Dame University,
South Bend, IN 46617, USA  School of Earth and Space Exploration, Arizona
State University, Tempe, AZ 85287, USA  Noble, A. G  School of Earth and Space
Exploration, Arizona State University, Tempe, AZ 85287, USA  Beus Center for
Cosmic Foundations, Arizona State University, Tempe, AZ 85287-1404, USA
Rudnick, G  Department of Physics & Astronomy, University of Kansas, 1251
Wescoe Hall Drive, Malott room 1082, Lawrence, KS 66045  Pigarelli, A  School
of Earth and Space Exploration, Arizona State University, Tempe, AZ 85287, USA
Wilson, G  Department of Physics and Astronomy, University of California,
Riverside, 900 University Avenue, Riverside, CA 92521, USA  Bahé, Y. M
Laboratory of Astrophysics, Ecole Polytechnique Fédérale de Lausanne (EPFL),
Observatoire de Sauverny, 1290 Versoix, Switzerland  Leiden Observatory,
Leiden University, PO Box 9513, NL-2300 RA Leiden, the Netherlands  Cooper, M.
C  Department of Physics & Astronomy, University of California, Irvine, 4129
Reines Hall, Irvine, CA 92697, USA  Demarco, R  Institute of Astrophysics,
Facultad de Ciencias Exactas, Universidad Andrés Bello, Sede Concepción,
Talcahuano, Chile  Matharu, J  Cosmic Dawn Center (DAWN), Denmark  Niels Bohr
Institute, University of Copenhagen, Jagtvej 128, DK-2200 Copenhagen N,
Denmark  Miller, T. B  Center for Interdisciplinary Exploration and Research
in Astrophysics (CIERA), Northwestern University, 1800 Sherman Avenue,
Evanston, IL 60201, USA  Muzzin, A  Department of Physics and Astronomy, York
University, 4700 Keele St., Toronto, Ontario, M3J 1P3, Canada  Nantais, J
Facultad de Ciencias Exactas, Departamento de Ciencias Físicas, Instituto de
Astrofísica, Universidad Andrés Bello, Fernández Concha 700, Las Condes, RM
7591538, Chile  Sportsman, W  Department of Physics & Astronomy, University of
Kansas, 1251 Wescoe Hall Drive, Malott room 1082, Lawrence, KS 66045  van
Kampen, E  European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748,
Garching bei München, Germany  Webb, T. M. A  Department of Physics, McGill
Space Institute, McGill University, 3600 rue University, Montréal, Québec,
Canada, H3A 2T8  Yee, H. K. C  The David A. Dunlap Department of Astronomy and
Astrophysics, University of Toronto, 50 St George St., Toronto, ON M5S 3H4,
Canada

###### Abstract

The changes in colors across a galaxy are intimately connected to the galaxy’s
formation, growth, quenching history, and dust content. A particularly
important epoch in the growth of galaxies is near $z\sim 2$ often referred to
as ‘cosmic noon’, where galaxies on average reach the peak of their star
formation. We study a population of 125 cluster galaxies at $z\sim 1.6$ in
three Hubble Space Telescope (HST) filters, F475W, F625W, and F160W, roughly
corresponding to the rest-frame FUV, NUV, and r band, respectively. By
comparing to a control sample of 200 field galaxies at similar redshift, we
reveal clear, statistically significant differences in the overall spatially
resolved colors and color gradients in galaxies across these two different
environments. On average, cluster galaxies have redder UV colors in both the
inner and outer regions bounded by $r_{\mathrm{50}}$, as well as an overall
wider dispersion of outside-in color gradients. The presence of these observed
differences, along with evidence from ancillary data from previous studies,
strongly suggests that the environment drives these population-level color
differences, by affecting the stellar populations and/or dust content.

††software: SwARP (version: 2.19.1; Bertin et al. (2002)), Source Extractor
(version 2.24.2; (Bertin & Arnouts, 1996), EzGal (Mancone & Gonzalez, 2012),
stsynphot (STScI Development Team, 2020), photutils (Bradley et al., 2023),
FAST (Kriek et al., 2018)

##  1 Introduction

A major question in astronomy is the role that the environment a galaxy lives
in plays on its evolution. At low redshift, the interactions between galaxies
in dense environments, as well as the interactions of galaxies with the gas
between them, significantly change the population of observed galaxies.
Overall, galaxies in clusters are more likely, when compared to groups and the
field, to be red and quenched than blue and star forming (Gunn & Gott, 1972;
Dressler, 1980; Poggianti et al., 1999).

One commonly invoked driver of this population difference in clusters at low
redshift is ram pressure stripping (Abadi et al., 1999; Boselli & Gavazzi,
2006; Tonnesen et al., 2007). As ram pressure strips the material for
replenishing star formation —gas— from the outside-in, quenching driven by ram
pressure would also be expected to follow this path. Indeed, several high-
resolution studies of the stellar populations in individual galaxies at low
redshift have found positive color gradients (redder outskirts than inner
region) consistent with this radial trend in quenching (Pappalardo et al.,
2010; Abramson et al., 2011; Merluzzi et al., 2016; Fossati et al., 2018;
Cramer et al., 2019). However, during active stripping, observations of large
samples of jellyfish galaxies in clusters have found overall star formation
rates are enhanced when compared to the main sequence at low redshift (Vulcani
et al., 2018). A similar enhancement was also recently found in jellyfish
galaxies in clusters out to $z=0.55$ compared to field galaxies (Vulcani et
al., 2024). Depending on the strength and stage of stripping, we thus expect a
study of cluster galaxies of sufficient sample size may find observable color
differences compared to similar field galaxies, which are generally free of
the influence of ram pressure.

Field galaxy color gradients are much more extensively studied than those of
cluster galaxies. At low redshift, both quiescent and star forming galaxies
have been found to be much more likely to have bluer outskirts than the inner
region (i.e. a negative color gradient) rather than redder outskirts than the
inner region (a positive color gradient) (Kormendy & Djorgovski, 1989; Wu et
al., 2005; Tortora et al., 2011). Recent studies with HST have also found
similar negative color gradients for all types of field galaxies beyond $z>1$
(Szomoru et al., 2013; Liu et al., 2016, 2017). While the conclusions of these
studies are limited by the difficulty of disentangling the degenerate effects
of dust and quenching on optical and UV colors, galaxies at this redshift are
primarily thought to grow from the inside-out (Nelson et al., 2016; Hagen et
al., 2016; Spilker et al., 2019), supporting the hypothesis that negative
color gradients may be connected to star formation. Certain combinations of
colors can be used to disentangle the dust-quenching degeneracy directly, most
popularly using the UVJ diagram, which has been used to distinguish between
star forming and quiescent galaxies at high-redshift (Wuyts et al., 2007;
Williams et al., 2009; Brammer et al., 2009; Whitaker et al., 2011; Muzzin et
al., 2013b).

Arguably, the most impactful epoch in the evolutionary history of galaxies on
a cosmic scale is $z\sim 2$, sometimes referred to as “cosmic noon”, where
galaxies reach the peak of their star formation (Shapley, 2011; Madau &
Dickinson, 2014). However, to study color gradients and distinguish between
outside-in and inside-out growth via star formation at high redshift requires
observations that include the rest-frame near infrared. Until recently
becoming available with JWST, these observations have been limited by
resolution and sensitivity. The only previously available instrument able to
even access the rest-frame $J$-band at high-redshift was Spitzer/IRAC, and
that instrument had very poor resolution for use on high redshift galaxies. A
recent study with JWST found that, of a sample of 54 star-forming field
galaxies from $1.7<z<2.3$, 85% of these galaxies had color gradients
consistent with bluer outskirts than the inner galaxy and 70% of the sample
had this negative color gradient driven primarily by dust (Miller et al.,
2022). This was the first data presented on resolved UVJ color gradients of
field galaxies at or near cosmic noon.

There are few to no studies of sufficient sample size to study the color
gradients of cluster galaxies, aside from the most extreme examples of
stripping manifested in jellyfish galaxies. Few to no studies of cluster
galaxies have sufficient sample size at any redshift to make a statistically
robust comparison to field galaxies. It is unclear at what epoch in the
buildup of galaxy clusters ram pressure becomes a significant influence on
galaxy evolution. Simulations have predicted that the ICM becomes built up
enough for ram pressure to become significant for galaxy evolution in clusters
around $z\sim 2$ (see the recent review by Boselli et al. 2022). At high
redshift, direct detection of ‘jellyfish’ like ram pressure stripped tails
seen at low redshift (e.g. Yagi et al. 2010; Sun et al. 2010; Kenney et al.
2014; Cramer et al. 2019; Jáchym et al. 2019) is difficult. Therefore, a
population wide difference in color gradients between cluster and field could
be a powerful tool for constraining the role of the cluster environment in
driving galaxy properties and evolution near cosmic noon. In order to conduct
such an investigation, observations of high enough resolution to study
gradients at this redshift are needed, especially with filters that can trace
young star formation directly, such as in the UV.

As such, in this paper we present new UV observations of galaxies within three
$z\sim 1.6$ clusters. The environmental quenching efficiency in these
clusters, based on the number of non-star forming galaxies and defined as
$f_{\text{eqe }}=\left(f_{\text{passive,cluster }}-f_{\text{passive, field
}}\right)/\left(f_{\text{star-forming, field }}\right)$, is estimated to be
$16\pm 16\%$. Although there is significant uncertainty, this suggests that
environmental quenching could be present, but may be low in these clusters at
this redshift (Nantais et al., 2017).

Later observations of these clusters have confirmed the presence of galaxies
actively experiencing environmental influence. Noble et al. (2017, 2019)
observed $\sim$25 galaxies across these three clusters with ALMA, and found
asymmetric gas morphology and overall elevated gas fractions, suggestive of
the influence of ram pressure. Indeed, studies that identified a number of
galaxies affected by ram pressure also observed enhanced molecular gas
fractions (M${}_{\mathrm{H_{2}}}$ / M⋆) in the disk when compared to a control
sample of field galaxies, often preferentially on the leading side of the
galaxy, i.e. the side experiencing maximum ram pressure (Jáchym et al., 2014,
2017; Verdugo et al., 2015; Lee et al., 2017; Moretti et al., 2018; Cramer et
al., 2020, 2021; Roberts et al., 2022; Vulcani et al., 2024).

In Cramer et al. (2023), we conducted a follow-up kinematic study of this
sample of ALMA detected $z\sim 1.6$ cluster galaxies and found that the
overall rates and degrees of kinematic asymmetry in the molecular gas
component were dramatically elevated when compared with field galaxies at
similar redshift (Cramer et al., 2023). While these asymmetries cannot be
incontrovertibly attributed to ram pressure stripping because of the limited
sensitivity and resolution resulting in only a few tenuous detections of
stripped tails, the observed kinematic asymmetry patterns in the disk are
extremely similar to those observed in low-redshift jellyfish galaxies
(Bacchini et al., 2023). Combining the evidence of a moderate elevation in
quenching with the direct evidence of ram pressure stripping, suggests that
these clusters supports that these clusters are in early stages of
environmentally driven quenching. They are thus ideal candidates for a follow-
up HST study of resolved color gradients which is the subject of this work.

##  2 Observations

The three clusters of galaxies studied in this paper, J022426–032330 (J0224),
J033057–284300 (J0330), and J022546–035517 (J0225), were discovered within the
42 deg2 Spitzer Adaptation of the Red-sequence Cluster Survey (SpARCS) fields
(see additional information in Table 1 of Nantais et al. 2016). All three
clusters are spectroscopically confirmed with redshifts of $z=1.63$, $z=1.63$,
and $z=1.59$, respectively (Lidman et al., 2012; Muzzin et al., 2013a; Nantais
et al., 2016), and contain, in total, 91 spectroscopically confirmed members
within $\delta_{\mathrm{zspec}}=0.015$ of the cluster BCG (Nantais et al.,
2016, 2017). In total, there are observations over 16 bands spanning optical
to NIR ($ugrizYK_{s}$ and F160W) as well as IR and FIR
(3.6/4.5/5.8/8.0/24/250/350/500$\mu$m), that are used to estimate photometric
redshifts with EAZY for the entire cluster field (Wilson et al., 2009; Muzzin
et al., 2009, 2013a; Nantais et al., 2016). The photometric cluster membership
criterion is $(z_{\text{phot}}-z_{\text{cluster}})/(1+z_{\text{cluster}})\leq
0.05$, which for these clusters is approximately $1.5<z_{\text{phot}}<1.76$.

###  2.1 HST Observations

The central cluster regions have deep HST imaging with the F160W filter from
the WFC3-IR camera from the “See Change” program (GO-13677 and GO-14327;
Hayden et al. 2021). Furthermore, the clusters were also imaged with the F475W
(rest-frame FUV) and F625W (rest-frame NUV) filters with ACS (GO-16300, PI:
Noble). Observations in each of these two filters consisted of data from 9
total executions across 3 orbits, for a total of 6882 seconds of exposure time
in J0225 and J0330. Due to lost guiding during individual exposures in some
orbits of the observations of J0224, we applied for and received additional
orbits to make up for the lost exposure time. This resulted in the F475W and
F625W images of J0224 having a total exposure time of 8319s and 6579s
respectively. The data were reduced using the Drizzlepac software by aligning
the individual frames using tweakreg, and drizzling the aligned frames using
astrodrizzle with a pixel size of 0.035″ for F475W and F625W, and 0.05″ for
F160W. The images were then matched to a common grid using the Astromatic
software SWarp (Bertin et al., 2002), to allow for pixel-by-pixel analysis and
comparison. In Figure 1 we show a color composite image with the reduced
observations from HST F475W, F625W, and F160W filters of the J0224 cluster.

The images were then PSF-matched based on empirically created effective point
spread functions (ePSFs) based on the selected point-like stars in the image
with the photutils (Bradley et al., 2023) Python package. Stars in each
cluster image were visually inspected in each filter to ensure a clean final
ePSF. The J0224 ePSF was created from 9 stars, J0225 from 8 stars, and the 3
stars in the J0330 image were not sufficient to create an ePSF. We chose to
use the J0224 ePSF for the J0330 image matching since it was derived from the
most stars. The impact of this choice should be minimal as the J0224 PSF stars
in the images failed to strongly capture the diffraction spikes of the
secondary mirror, so the derived PSF is close to a Gaussian. Furthermore, the
position angle of the J0224 and J0330 observations is similar, so any
diffraction spikes would have been in similar locations. We employed the
pypher (Boucaud et al., 2016) Python package to create matching kernels, to
redistribute the flux from one band of imaging to match the flux distribution
of the target band of imaging, with a regularization parameter of 10-4. The
images were then convolved using the convolve method from astropy.

###  2.2 Hubble Frontier Fields Catalogues

In order to have as close a comparison sample of field galaxies as possible at
a similar redshift as the $z\sim 1.6$ clusters, we required a dataset with the
same, or at least similar filters. Furthermore, we also needed high-resolution
observations for a resolved study of radial color gradients, and relatively
deep observations to allow for a large comparison sample. The HFF-DeepSpace
(HFF-DS) program, data release v3.9 (Shipley et al., 2018), provides the best
sample for satisfying these three criteria. The program includes 6 parallel
pointings of the outskirts of galaxy clusters near $z\sim 0.5$. The high
redshift galaxies that fall in these fields represent an unbiased field galaxy
sample. The imaging with ACS includes observations with the F435W, F606W, and
F160W filters. In total, the program includes 840 orbits of Director’s
Discretionary time, resulting in extremely deep imaging which allows us to
build a large sample of 200 $z\sim 1-2$ field galaxies in the parallel fields.
111The reduced, PSF-matched images that we use are publicly available upon
request at http://cosmos.phy.tufts.edu/~danilo/HFF/Download.html.

##  3 Methodology

Our goal is to study the radial colors and color gradients of high-z cluster
galaxies and compare them to the field, in order to search for population
level differences in the two samples. In the following sections, all
magnitudes are given in units of AB magnitudes.

###  3.1 Image Processing and Source Identification

While the HST reduction process includes sky subtraction, some residual sky
was still evident in the F475W and F625W images from the high-$z$ cluster
data. We applied a Gaussian smoothing kernel with a size of 3 pixels, and a
threshold of 3$\sigma$ to a source subtracted image (based on a segmentation
map from Source Extractor (Bertin & Arnouts, 1996)), and subtracted this
smoothed image from the image data, in order to account for this background.
The field data available from the HFF-DS sample had already been background
subtracted in the same manner. We then identify all sources above a threshold
of 5$\sigma$, and deblend them using photutils.segmentation (Bradley et al.,
2023). We calculate the Kron flux and the major and minor axis of the ellipse
encompassing the Kron flux.

We then cross match the cluster sample we detect within the F160W field of
view with spectroscopic and photometric catalogues from Nantais et al. (2016)
to isolate galaxies within the redshift range of the high-$z$ clusters
($1.5<z<1.76$). We find a total of 65 spectroscopically confirmed galaxies,
and an additional 60 galaxies with photometric redshifts within this range.
While spectroscopic redshifts are more reliably constrained than photometric
redshifts, we find no statistically significant effect on any of our analysis
from comparing the full sample we use with one with only spectroscopically
confirmed cluster members (see Appendix and Figures 8 & 9 for details). For
the field sample, we select all galaxies between $1.2<z_{\text{phot}}<2.0$, a
total of 200 galaxies; having a large sample of galaxies with which to compare
is important for this analysis.

###  3.2 Aperture flux extraction

In order to study color gradients, we calculate the flux within an outer
annulus and an inner aperture for each galaxy. To constrain radial evolution
within a galaxy, it is especially important to effectively compare the flux
within annuli with geometry consistent with the overall ellipticity of each
galaxy. Commonly used source finding software packages like Source Extractor
(Bertin & Arnouts, 1996) and photutils can both be configured to output the
parameters of a circular aperture containing a user specified fraction of the
total flux, but not an elliptical aperture. We modified the photutils code to
use an elliptical optimizing function, instead of a circular optimizing
function, to fit proper elliptical apertures to each galaxy. We find the
ellipses encompassing 50%, and between 50% and 90%, of the total Kron flux in
the F160W filter for each galaxy, then apply these same outer and inner
ellipses to each accompanying filter pair. Thus, the inner and outer region of
a galaxy in all three filters is measured with the same apertures, as set by
the F160W surface brightness profile of each galaxy. An example of these
apertures overlayed on a cluster galaxy from our sample is shown in Figure 1.

Figure 1: A composite color image of one of the three SpARCS clusters we
study, J0224, with HST imaging in filters F475W (blue), F625W (green), and
F160W (red). Members of this cluster considered for this paper are indicated
with white ellipse. On the right is an example of the inner aperture
($r<r_{50}$, the orange shaded region) and outer annulus ($r_{50}<r<r_{90}$,
the pink shaded region) determined for this galaxy, used for the analysis
presented in this paper. The size and ellipticity of the two apertures is
determined based on the total flux in the F160W filter, and then applied to
both the F625W and F475W images.

###  3.3 Flux error estimation

Determining the uncertainties for our measured fluxes is complicated by the
correlated nature of the noise in our images that results from the sub-pixel
drizzling process. The measured pixel-to-pixel variation therefore
underestimates the true uncertainty. To address this, we make an empirical
estimate of the uncertainties using the technique from Labbé et al. (2003) and
Shipley et al. (2018), which works under the assumption that the uncertainty
in the measured fluxes is dominated by the background uncertainty, which is
true for our images. We refer the reader to those papers for a detailed
description of the technique. In summary, we placed empty apertures throughout
each image, ensuring that the apertures do not overlap with detected objects.
For a given sized aperture, the distribution of the measured flux in these
empty apertures has a Gaussian distribution. The standard deviation of this
distribution gives the flux uncertainty for apertures of identical area
($\sigma_{f}$). This procedure was repeated for many different aperture sizes
and a 2nd order polynomial was fit to the trend of $\sigma_{f}$ vs. aperture
size. As shown in Labbé et al. (2003), this is a good fit to the trend, which
we verified with our images. For every aperture in which we measured a flux in
our image we use these fits to find the uncertainty in apertures of identical
area in pixels. This becomes our quoted uncertainty.

###  3.4 Sample matching

The HFF-DS catalogue is the best sample for a field comparison in terms of
depth of observations and thus the abundance of galaxies in our desired
redshift range at HST resolution. However, there are several key differences
between our sample and that from HFF-DS, that need to be addressed for a
proper comparison. The field sample of the HFF-DS survey and our data have
slightly different filters (F435W & F606W versus F475W & F625W). Furthermore,
to maximize the available sample of control galaxies, we include HFF-DS
galaxies from $1.2<z<2.0$ while cluster galaxies are between $1.53<z<1.69$. In
order to make the closest possible comparison between the two samples, we
correct measured HFF-DS galaxy fluxes with both a filter and redshift
correction.

| Detected in | Non-detected | Non-detected | Non-detected
---|---|---|---|---
| all three filters | only F475W | only F625W | F475W & F625W
| Cluster galaxy
---
(inner region)
94 | 22 | 1 | 8
| Cluster galaxy
---
(outer region)
90 | 20 | 4 | 11
| Field galaxy
---
(inner region)
188 | 11 | 0 | 1
| Cluster galaxy
---
(outer region)
184 | 13 | 0 | 3
Table 1: Number of detections in the inner and outer galaxy regions above
$3\sigma$ in both the cluster and fields samples. The colors of these regions
are plotted in Figure 3.

The filter correction was done by analyzing a series of both SSP and
$\tau=0.6$ Gyr models from EzGal (Mancone & Gonzalez, 2012) at a range of ages
(see Figure 2). We estimate the predicted flux within the F435W, F475W, F606W,
and F625W filters using stsynphot (STScI Development Team, 2020). We then fit
the scatter plot of points consisting of the 22 different models with a
quadratic function, which allows us to determine the conversion between
filters (see Figure 2). We find the following relations:

$\displaystyle(F435W-F475W)=0.052(F435W-F606W)^{2}$
$\displaystyle+0.19(F435W-F606W)+0.010$

$\displaystyle(F606W-F625W)=-0.043(F435W-F606W)^{2}$
$\displaystyle+0.21(F435W-F606W)-0.001$

The 1$\sigma$ errors on these fits are added in quadrature with the flux
errors for each galaxy from the empty aperture method for the rest of this
analysis. The filter correction is then adjusted for the redshift of the field
galaxies, ranging from $z=1.2-2.0$ rounded to the nearest 0.1, once more using
stsynphot to shift the template spectra to each redshift step before
calculating the estimated filter flux conversions.

After equalizing these filters, for simplicity we will hereafter refer to the
these filters by their nearest overlapping rest-frame color band, the F475W
filter as FUV, the F625W as NUV, and the F160W as r.

We also impose a restriction on galaxy stellar mass in each sample to be
between $M_{\star}=10^{9}-10^{11}~{}{\rm M}_{\odot}$, based on estimates from
FAST modelling (Kriek et al., 2018) presented in Nantais et al. (2016) for the
cluster, and Shipley et al. (2018) for the field. As the HFF-DS observations
are deeper than the cluster observations, which are mass limited to
$M_{\star}\sim 10^{9}~{}{\rm M}_{\odot}$, this cut is necessary to make sure
we are comparing similar types of galaxies. The stellar mass distributions of
our samples are similar, but there are about two times as many galaxies of
$\sim 10^{9}~{}{\rm M}_{\odot}$ in the field sample as the cluster, possibly
due to the difference in the depth of the observations, or the larger total
area covered by the HFF-DS survey. We re-ran our analysis, randomly removing
low mass field galaxies until the total number in the low mass bins matched
the number of galaxies in the cluster sample, and found no statistically
significant difference in any of our results and conclusions.

The full details of the number of galaxies with fluxes detected in one, two,
or all three filters are shown in Table 1.

Figure 2: Quadratic fits to the predicted color from SSP and $\tau=0.6$ Gyr
models produced at ages of 0.01, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.8, 1.0, 1.5,
2.0, 3.0, 4.0, 5.0, and 6.0 Gyr by EzGal using a BC03 IMF (Chabrier, 2003)
(see text for additional details). Axes are in units of AB magnitudes. With
these fits, we establish a formula for converting the F435W to the F475W
filter, and the F606W to the F625W filter, based on the observed F435W - F606W
color, in order to directly compare fluxes between the cluster and field.

##  4 Discussion

The result of the color analysis of both cluster and field galaxies is shown
in Figure 3. Simply by eye, it is apparent that field galaxies are overall
more tightly distributed than cluster galaxies, with both inner and outer
region colors centered between approximately 0 and 0.5 (FUV-NUV), and between
0.5 and 2.5 (NUV-r). The distribution of cluster galaxy colors, both inner and
outer, is much more dispersed than field galaxies, although the largest
concentration of points is also centered near FUV-NUV of 0.5 to 0, and NUV-r
between 2.5 to 0.5. Both cluster and field galaxies have a similar total range
of NUV-r values, ranging from approximately 6 to 0. It is rare to find galaxy
regions with FUV-NUV $<$ 0 in either sample, although it is slightly more
common in cluster galaxies. Cluster galaxies also cover a wider total range of
FUV-NUV values for a given NUV-r color, but we note that the cluster sample
errors are larger.

While these insights from visual inspection can be useful for assessing the
potential differences between the distributions within the cluster and field
samples, proper statistical analysis of these distribution is necessary in
order to draw strong conclusions.

Figure 3: A comparison of the inner (orange) and outer (pink) NUV-r vs. FUV-
NUV colors for the field sample and the cluster sample. Left facing triangles
show aperture fluxes at $3\sigma$ limits in either the FUV flux, or the NUV
flux. Downward facing arrows show limits in just the NUV flux. Open circles
show aperture fluxes with limit fluxes in both FUV and NUV. Contours based on
a kernel density estimate (KDE) are also shown to better illustrate the
comparison. Overall, the field sample shows a tighter overall distribution,
while the cluster sample shows a wider range of values in the FUV-NUV color.
Overall error levels on colors are quite low, averaging $\sim$0.05 mags for
the field sample, and $\sim$0.1 mags for the cluster sample.

###  4.1 Statistical Survival Analysis

It is important to compare the overall properties of galaxies in the clusters
we study with the matched field sample in a statistically sound manner. In
order to do this, and understand the impact of limit cases in our statistical
analysis, we use a method called survival analysis. Survival analysis
originates from public health, and is commonly used to measure mortality over
the course of a drug trial, where “censored” data need to be considered to
account for patients that leave the trial early. It can be used in our case to
consider the color distributions of the cluster and field sources, while
incorporating the galaxies and galaxy regions with only upper limits on the
flux as censored data. While more well-known in the public health field, this
analysis technique has been used for similar purposes to that we described
here in a number of other astronomy papers (e.g. Mok et al., 2016; Odekon et
al., 2016). All galaxies in our samples are detected in r, but $\sim 20\%$ of
the sample have an upper limit flux in FUV or NUV, that are important to
consider, as limits still provide useful color constraints.

We fit a survival statistics Kaplan-Meier function, a non-parametric, maximum
likelihood statistical estimator, to both the field and cluster samples for
each color. We then use a log-rank test to assess the similarity in the
distribution of both functions, and calculate a p-value to estimate whether
the two populations are significantly different. The survival statistics
method and the logrank test do not incorporate measurement errors directly. We
are able to calculate the 95% confidence interval (shown in Figure 5), based
on the sample size, sampling of the data, and the number of limit cases, but
not the measurement errors. In order to assess the impact of these errors on
the p-value we calculate, we implement a Monte-Carlo approach, using the
1$\sigma$ errors from our empty aperture analysis. We vary each filter
component of each color within its 1$\sigma$ error, then proceed with a
Kaplan-Meier fit, and calculate a p-value with a log-rank method, over 100
iterations. This allows us to estimate the variance of the p-value within the
1$\sigma$ flux errors. We find that the individual flux errors do not
contribute to a significant change in the overall confidence level of the
p-values we calculate. Plots showing the Monte-Carlo generated Kaplan-Meier
fits can be found in the Appendix (Figures 10 & 11).

| Cluster Median | Field Median | Logrank | Logrank | KS Test
---|---|---|---|---|---
| Color | Color | P-value | P-value STD | P-value
FUV - NUV (inner) | 0.6 | 0.3 |  $<$0.001 |  $<$0.001 |  $<$0.001
FUV - NUV (outer) | 0.4 | 0.2 |  $<$0.001 |  $<$0.001 |  $<$0.001
$\Delta$ FUV - NUV | 0.02 | -0.04 |  $<$0.001 |  $<$0.001 | 0.2
NUV - r (inner) | 3.1 | 2.4 | 0.06 | 0.02 | 0.11
NUV - r (outer) | 2.1 | 2.1 | 0.8 | 0.1 | 0.5
$\Delta$ NUV - r | -0.04 | -0.2 | 0.008 | 0.02 | 0.003
Table 2: Results of survival analysis of field and cluster galaxy samples’
colors. From left to right, in column (1) the median value of the color for
cluster galaxies, in column (2) the median value of the color for field
galaxies. In column (3) the p-value from a logrank comparison, and in column
(4), the standard deviation in the p-value based on a Monte-Carlo approach for
estimating the effect of the flux errors on the CMDs (described further in the
Appendix). In column (5) are the p-values resulting from a KS test between the
curves, considering only the uncensored data points.

We compare the cumulative distribution functions (CDFs), derived from the
survival statistics methods, of the cluster and field galaxies in two galaxy
regions, from $r<r_{50}$, and from $r_{50}$ to $r_{90}$. These are the inner
and outer FUV-NUV color, the inner and outer NUV-r color, and the color radial
gradient from subtracting FUVouter-NUVouter from FUVinner-NUVinner, as well as
NUVouter-router from NUVinner-rinner. The results are shown in Figures 5 & 5.

It has been found that the results of the logrank test can be less certain
when comparing two survival curves that cross each other (Lin & Xu, 2009;
Dormuth et al., 2022). While an analysis of a large number of published papers
that have data with crossing survival curves found a significant fraction
still use the logrank test in this situation (Li et al., 2015), we wanted to
be careful that our results were not significantly impacted by this
possibility. For this reason, we also used a simple Kolmogorov-Smirnov (K-S)
test to compare the cluster and field color distributions (the limitation of
this approach is that it does not incorporate the flux limit cases). The
results of the logrank test and the K-S tests are shown in Table 2.

For the logrank and K-S test of the FUV-NUV color distributions we find
p-values of less than $\sim 0.001$ for both inner and outer regions,
indicating a $>3\sigma$ level difference between the distributions of cluster
and field galaxy UV colors. Furthermore, cluster galaxies are redder than
field galaxies in both outer and inner regions, based on the median UV color
of each population. In contrast, in both outer and inner regions, there is no
significant difference in the NUV-r color between field and cluster galaxies
as seen in Figure 5 (right).

The CDF comparison of the color gradient in (NUVouter-router) - (NUVinner-
rinner) indicates a statistically significant difference in color gradient
distribution when comparing cluster to field galaxies. The color gradient for
(FUVouter-NUVouter) - (FUVinner-NUVinner) appears to show statistically
significant differences between the two samples according to the logrank test.
However, the K-S test produces a p-value in which the null hypothesis cannot
be rejected. As we are wary of the possible effect of the crossing survival
curves, we cannot strongly conclude that the cluster and field sample have
different overall UV gradients. However, we do note that regardless of the
overall distribution, the range of the FUV-NUV gradients in clusters indicates
towards the cluster sample containing galaxies with steeper outside-in
reddening gradients than the field.

Understood together, these results indicate that cluster galaxies are more
likely to have redder inner and outer regions in the UV than field galaxies.
Only one of the two statistical tests we apply supports there being a redder
inside-out color gradient across cluster galaxies. In contrast, cluster and
field galaxies have statistically similar overall NUV-r colors in the inner
and outer regions.

Figure 4: A comparison of the cumulative distribution function best fits to
the cluster and field samples. The shaded regions show the 95% confidence
interval, which is calculated based on the sample size, sampling of the data,
and the number of limit cases. The x-axis is in units of AB magnitudes. We
show a comparison of the cumulative distributions of the inner regions (top)
and the outer regions (bottom) for cluster (in green) and field (in orange)
galaxies. For the FUV-NUV color, both the inner and outer comparisons show
statistically significant differences based on the p-value of a log-rank
comparison. For the NUV-r color, the inner and outer comparisons show no
statistically significant differences based on the p-value.

Figure 5: Top: A comparison of the cumulative distribution function fits to
the cluster and field samples. The shaded regions show the 95% confidence
interval, which is calculated based on the sample size, sampling of the data,
and the number of limit cases in the sample. The x axis is in units of AB
magnitudes. We show a comparison of the cumulative distributions of the outer-
inner color gradient distributions for cluster and field galaxies. For the
FUV-NUV color (left), only the logrank test supports a different cluster vs.
field gradient distribution, but nonetheless, a number of the cluster galaxies
do have steeper redder outside to bluer inside trends. For the NUV-r color
(right), the overall outer-inner distribution is statistically significantly
different, but without a clear bias towards more negative or positive
gradients.

###  4.2 Quadrant Analysis

Along with investigating the individual color gradients, studying the combined
information for each galaxy of both (FUVouter-NUVouter) - (FUVinner-NUVinner)
and (NUVouter-router) - (NUVinner-rinner), hereafter referred to as
$\Delta$FUV-NUV and $\Delta$NUV-r, has the potential to provide additional
insight into galaxy properties. Based on the four possible combinations of
gradients for $\Delta$FUV-NUV and $\Delta$NUV-r, which are positive-positive
(quadrant I), positive-negative (quadrant II), negative-positive (quadrant
III), and negative-negative (quadrant IV) gradients, each falls in a different
region of the color gradient quadrant plot, illustrated in Figure 7 & 7.
Comparing the overall distributions of cluster and field galaxies across these
quadrants highlights the differences in galaxy colors driven by the
environment.

We find that overall the distribution of cluster galaxies has a larger UV
color range, and is more dispersed than field galaxies. The total fraction of
galaxies in quadrants I & III when comparing the cluster and field are
similar. The most prominent difference between cluster and field is a
$\sim$20% difference in the fraction of galaxies falling in quadrant IV in the
field, versus quadrant II in the cluster.

A galaxy falling in quadrant I, with positive-positive gradients, could be the
result of ram pressure stripping of sufficient strength and on a timescale
long enough to result in a color difference in the outer aperture, driven by
quenching. At low-redshift, Abramson et al. (2011) found that in a known ram
pressure stripped galaxy, $\Delta$FUV-NUV and $\Delta$NUV-$r$ were both
positive, meaning the galaxy would have redder outskirts than the inner region
for both gradients. However, unlike at low-redshift where it is possible to
isolate what appear to be dust-free regions, where ram pressure has stripped
away all or most of the dust, for our sample at high-redshift the resolution
is too poor to make a solid determination of the dust content in the inner or
outer apertures with these filters. Thus, an observed positive-positive
gradient could also be the result of stronger dust extinction in the outer
parts of the galaxy, as well as from stronger inner vs. outer star formation
from something like a central starburst. However, Miller et al. (2022) found
that only 15$\pm$6% of star-forming $1.3<z<2.0$ galaxies had color gradients
consistent with stronger inner than outer star formation, and another 8$\pm$4%
were consistent with stronger outer dust extinction. A significant
overabundance of cluster galaxies in this quadrant when compared with the
field could thus be evidence for significant enough ram pressure over time to
result in quenching. We do not find any significant difference between cluster
and field population within quadrant I, indicating this quenching pathway is
currently not significant in these clusters.

Quadrant IV, where both $\Delta$FUV-NUV and $\Delta$NUV-r are negative (redder
inner colors), is the most commonly observed combination of color gradients in
the field comparison sample. Miller et al. (2022) also predicted that the
majority of field galaxies would fall in this quadrant as the result of either
stronger dust extinction in the center versus the outskirts (69% of galaxies),
or stronger outer star formation (8% of galaxies).

Quadrants II (where $\Delta$FUV-NUV is positive and $\Delta$NUV-r is negative)
and III (where $\Delta$FUV-NUV is negative and $\Delta$NUV-r is positive) are
difficult to interpret given the limitations in terms of the color bands we
have access to, and the complications of dust extinction. We thus hesitate to
speculate on the physical drivers of galaxies being found in these quadrants,
and note instead simply that the environment, through any combination of a
number of effects like (but not limited to) stripping, strangulation, pre-
processing, and tidal interactions, clearly seems to drive an observable
difference in the number of galaxies that fall here.

We discuss the implications of the quadrant analysis further in Section 5.1.

Figure 6: Top: When comparing the color gradient in $\Delta$FUV-NUV (along the
y-axis) and $\Delta$NUV-r (along the x-axis), there are four possible
combinations, depending on whether each gradient is positive or negative. We
indicate these four quadrants here, labeled I-IV, along with an illustration
of the relative color gradients. According to an analysis of very similar
colors at $z\sim 0$ from Abramson et al. (2011), actively or recently ram
pressure stripped galaxies should fall in quadrant I. From surveys of field
galaxies at $z\sim 2$, showing galaxies having bluer outskirts than insides,
the majority of galaxies would lie in quadrant IV, absent significant
evolutionary pressure from the environment.

Figure 7: A comparison of the $\Delta$FUV-NUV and $\Delta$NUV-r color
gradients of field galaxies (left) and cluster galaxies (right). Axes are in
units of AB magnitudes. We also show density contours based on a kernel
density estimate (KDE) of the plotted points for another visual representation
of the distributions. Overall, both cluster and field have similar amounts of
galaxies falling in quadrants I and III. The major difference between cluster
and field appears to be the number of galaxies in quadrant IV and quadrant II
in the cluster. There is also an overall much wider range in color gradient
values in the cluster.

##  5 Interpretation

We find statistically significant differences between the cluster and field
population based on the resolved UV/optical color properties. This in and of
itself is a significant advancement in our understanding of the influence of
the environment at high redshift based on resolved observations.

The interpretation of the physical drivers behind these differences is
complicated. This is driven primarily by the difficulty in distinguishing
between reddening driven by dust, and/or stellar population age, both of which
affect the FUV-NUV and the NUV-r colors in similar ways. There are
statistically significant differences in both the inner and outer FUV-NUV
colors for cluster and field galaxies, which could be interpreted as either a
more quenched population, or increased dust extinction in cluster galaxies.
However, there is no overall difference in NUV-r population level colors. FUV-
NUV is more sensitive to color changes from recent star formation and dust
extinction, indicating that perhaps this wavelength range is necessary to
detect differences on the population level between cluster and field at this
redshift.

The color gradient distributions are another complicating piece of the overall
picture to consider. While the overall population level FUV-NUV color shows a
difference between the cluster and field, there is no detectable difference in
the color gradient that can be confirmed by both the logrank and K-S test. We
may have expected environmental influences that act radially from outside-in
like ram pressure, which are more significant in clusters, to result in an
observable difference in the cluster galaxy population, but none is detected
at this level. We do find a difference in the NUV-r color distribution, but
not clearly favoring a particular direction for positive vs. negative color
gradients. Clearly, more filters, and rigorous stellar population modelling
based on more filters, is needed to further identify the forces within the
environment driving these color differences at this redshift.

###  5.1 Quadrant Analysis Interpretation

The quadrant plot clearly shows a difference in the distribution and overall
range of color gradients observed between the cluster and field populations.
Ram pressure stripped and quenched cluster galaxies are likely to be observed
in quadrant I based on low-redshift galaxy studies (Abramson et al., 2011;
Merluzzi et al., 2016; Cramer et al., 2019). Although no large scale studies
of color gradients at a population level have been conducted at low or high
redshift in clusters, studies of other signatures of ram pressure stripping
and quenching have found it affects a very significant number of star forming
galaxies in low redshift clusters like Virgo (e.g. Chung et al., 2009; Zabel
et al., 2022; Villanueva et al., 2022; Jiménez-Donaire et al., 2023; Zinger et
al., 2024). Furthermore, previous studies of galaxies in these three $z\sim
1.6$ clusters found evidence supporting the effects of ram pressure acting on
the molecular gas content (Noble et al., 2019) and the molecular gas
kinematics (Cramer et al., 2023). We expected, given this supporting evidence,
that we might find an overabundance of galaxies in quadrant I in the cluster
sample. Our analysis, however, finds no significant overabundance of galaxies
in this quadrant when compared with the field.

We do, however, find that an additional $\sim$20% of galaxies in the cluster
are found in quadrant II (bluer outside-in NUV-r color gradients, and redder
outside-in FUV-NUV color gradients) compared to the field. This could be
related to cases observed at low redshift of the more complex cases of ram
pressure interaction, such as star formation triggering (Moretti et al., 2020;
Cramer et al., 2020, 2021). It is also consistent with results from
simulations that show that around $z\sim 2$ (Boselli et al., 2022), ram
pressure is only beginning to actively morph galaxies in these clusters. In
the case of these clusters, this supports that perhaps this is not yet
resulting in significant outside-in quenching that would be seen in quadrant
I, but is still influencing the stellar populations in ways like star
formation triggering. This could be related to early stages of the development
of the ICM near this redshift, that would need to be confirmed with deep X-ray
observations of these clusters.

This interpretation is highly speculative. However, the fact remains that,
regardless of the physical or observational driver, there are clear
differences in the colors and color gradients of cluster galaxies when
compared with the field. Similar large sample-size studies of cluster galaxy
color gradients at low redshift, especially in clusters where ram pressure
stripped galaxies are easily identified based on previous studies of gas
morphology (e.g. the Virgo cluster Chung et al. 2009; Brown et al. 2021),
could be extremely helpful in determining where different physical drivers
move galaxies within the quadrant plot. Furthermore, it would also be useful
to study whether high-redshift clusters have a different distribution of
galaxies within the quadrant plot when compared with low-redshift clusters.
This could have direct implications for better understanding the evolution of
galaxies within clusters over cosmic time. We hope to implement this sort of
color gradient analysis across a range of redshifts in future work, and that
this study provides additional motivation for others to pursue these kinds of
analyses, to more fully probe the parameter space.

###  5.2 Future Work with JWST

While the color gradient quadrant analysis of $\Delta$FUV-NUV and
$\Delta$NUV-$r$ can be informative for determining physical drivers behind
galaxy locations in the diagram, they are also affected by the degenerate
effects of dust extinction and quenching. A solution to determining the
relative degree to which each is contributing to the observed color
differences will need to come from observations that can distinguish between
the two scenarios. The UVJ diagram is one method for disentangling the effects
of dust extinction from differences in stellar population. Galaxies with high
dust extinction are reddened in both the U - V and V - J colors (Forrest et
al., 2016; Martis et al., 2016). Galaxies with older stellar populations are,
when compared with the U - V, bluer in the V - J color (Carnall et al., 2019;
Belli et al., 2019). Moreover, UVJ colors have also been shown to correlate
with specific star formation rate, allowing further constraints on quenching
in cluster vs. field (Leja et al., 2019; Akins et al., 2022). At $z>1.6$, the
rest-frame $J$ band is redshifted to $>$3 $\mu$m, and therefore JWST NIRCam
imaging is the only high-resolution instrument that can provide resolved
observations in this regime.

Recent work by Miller et al. (2022), using a subset of star-forming galaxies
ranging from $1.3<z<2.0$ from the JWST CEERS survey (Finkelstein et al.,
2023), used a combination of JWST filters nearest to the redshifted UVJ color
bands, and flexible modelling to interpolate to the rest-frame UVJ colors, to
measure the resolved color profiles for $45$ star-forming field galaxies. By
using a similar observing setup with JWST NIRCam as Miller et al. (2022), a
similar color analysis of these cluster galaxies could be easily compared with
the field sample.

##  6 Summary

We have conducted a rigorous statistical analysis of the colors and color
gradients, derived from HST observations, of high-redshift galaxies in
clusters and in the field. By carefully controlling for confounding variables
in these two samples, like filter differences and redshift, differences in
these colors and color gradients are informing us about differences in the
evolutionary pathway of galaxies in low and high density environments.

$\bullet$ Environment drives color differences

We find that cluster galaxies at $z\sim 1.6$ have statistically significantly
redder UV colors in inner ($<r_{50}$) and outer ($r_{50}<r<r_{90}$) regions
when compared to field galaxies at similar redshift and stellar mass. In
contrast, there is no statistically significant difference detected in the
NUV-r color in the inner or outer regions of galaxies. This large scale study
of the color gradients of cluster galaxies and field galaxies at this redshift
is the first of its kind, and reveals clear differences between the two
populations.

$\bullet$ Constraints on outside-in quenching

Previous study of these clusters found evidence consistent with ram pressure
stripping of gas (Noble et al., 2017, 2019; Cramer et al., 2023). However, we
find no overabundance of cluster galaxies with outside-in color gradients
reflective of quenching driven by prolonged strong ram pressure stripping
(like is seen in low redshift galaxies) at this redshift. We do, however, find
that an additional $\sim$20% of galaxies in the cluster are found in quadrant
II (bluer outside-in NUV-r color gradients, and redder outside-in FUV-NUV
color gradients) compared to the field. This could be consistent with ram
pressure influencing the stellar population of galaxies in these clusters
through star formation triggering, but merits further investigation.

This work was supported from HST program number GO-16300, and is based on data
and catalog products from HFF-DeepSpace, funded by the National Science
Foundation and Space Telescope Science Institute (operated by the Association
of Universities for Research in Astronomy, Inc., under NASA contract
NAS5-26555). A.N. additionally acknowledges support from the Beus Center for
Cosmic Foundations at Arizona State University, from the National Science
Foundation through grant AST-2307877 and through award SOSPA7-025 from the
NRAO. GHR acknowledges support from NASA through HST-GO-16300.004, NASA Keck
RSA-1699004, from the NSF through AST-2206473, and through NASA via ADAP grant
80NSSC19K0592. He also acknowledges the generous support from the
International Space Sciences Institute (ISSI) via the “Cosweb” team. YMB
gratefully acknowledges funding from the Netherlands Organization for
Scientific Research (NWO) under Veni grant number 639.041.751 and financial
support from the Swiss National Science Foundation (SNSF) under funding
reference 200021_213076. JN acknowledges support from the Universidad Andres
Bello internal grant DI-07-22/R and Fondecyt Regular #1230591 (PI Lucia
Guaita). GW gratefully acknowledges support from the National Science
Foundation through grant AST-2205189 and from HST program number GO-16300.
R.D. gratefully acknowledges support by the ANID BASAL project FB210003. Based
on observations made with the NASA/ESA Hubble Space Telescope, and obtained
from the Hubble Legacy Archive, which is a collaboration between the Space
Telescope Science Institute (STScI/NASA), the Space Telescope European
Coordinating Facility (ST-ECF/ESA) and the Canadian Astronomy Data Centre
(CADC/NRC/CSA). This research made use of photutils, an Astropy package for
detection and photometry of astronomical sources (Bradley et al., 2023). Many
thanks to the lead developer of photutils, Larry Bradley, for his help in
developing additional code for our particular need.

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##  Appendix A Full photo-z and spec-z cluster galaxy sample vs. just spec-z

To assess whether there is any significant risk to our analysis from
contamination of non-cluster members based on poorly constrained redshift
estimates from photo-$z$ methods, we compare with a rerun analysis using a
sample containing only galaxies with a spectroscopic redshift between
$1.53<z<1.69$. This restricted sample contains $\sim$50% of the larger sample
used in the main text. As shown in Figures 8 & 9, there is no significant
difference in the distribution of aperture fluxes in the color diagram, or in
the distribution of color gradients in the quadrant plot. Furthermore, a
reproduced survival analysis found no difference in the p-value significance
of any of the 6 tested parameters. Thus, we are confident that our conclusions
are not impacted by contamination from uncertain photometric redshift
estimates.

Figure 8: A comparison of the inner (orange) and outer (pink) colors in NUV-r
vs. FUV-NUV for the full (photo and spec-z) cluster sample used in the
analysis from the main text (left), as well as a spec-z only sample (right).
Axes are in units of AB magnitudes. Right facing triangles show aperture
fluxes with limits in either the FUV flux, or the NUV flux. Downward facing
arrows show limits in just the NUV flux. Open circles show aperture fluxes
with limit fluxes in both FUV and NUV. Contours based on a kernel density
estimate (KDE) are also shown to better illustrate the comparison. Figure 9: A
comparison of the $\Delta$FUV-NUV and $\Delta$NUV-r color gradients for the
full (photo and spec-z) cluster sample used in the analysis from the main text
(right), as well as a spec-z only sample (left). Axes are in units of AB
magnitudes. We also show contours based on a kernel density estimate (KDE) of
the plotted points for another visual representation of the distributions.
Overall, both samples have similar distributions across all four quadrants, as
well as equal representation in each quadrant within the estimated error from
a Monte Carlo Markov Chain.

##  Appendix B Survival Statistics Error Estimation

Survival analysis does not directly incorporate errors in its Kaplan-Meier
fit, or a log-rank comparison of those fits. In order to assess the impact of
these errors on the p-value we calculate, we implement a Monte-Carlo approach,
using the 1$\sigma$ errors from our empty aperture analysis. We vary each
filter component of each color within the 1$\sigma$ error, then proceed with a
Kaplan Meier fit, and calculate a p-value with a log-rank method, over 100
iterations. This allows us to estimate the variance of the p-value within the
1$\sigma$ flux errors. These Kaplan Meier fits are shown in Figures 10 & 11.
We find that the individual flux errors do not contribute to a significant
change in the overall confidence level of the p-values we calculate.

Figure 10: Here we show the cumulative distribution functions of the inner
region of cluster galaxies (blue) and field galaxies (orange) in the FUV-NUV,
and NUV-r, inner and outer color, generated with survival statistics. Survival
statistics does not incorporate errors on the input values directly. Thus, to
test the robustness of our results, we run a Monte Carlo, with 100 iterations
within the 1$\sigma$ flux errors, of the survival statistics cumulative
distribution function for both cluster and field. All iterations are plotted
here. We then calculate the rms of the p-value for each pair of Monte Carlo
generated distributions, in order to calculate an error estimate for the
p-value. Figure 11: Same caption as Figure 10, but for the $\Delta$FUV-NUV and
$\Delta$NUV-r color gradients.


# Automorphic Forms and Holomorphic Functions on the Upper Half-plane

Md. Shafiul Alam  Department of Mathematics, University of Barishal,
Barishal-8254, Bangladesh<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We define a set of holomorphic functions in terms of the Hauptmodul of a
quotient Riemann surface and prove that these functions are holomorphic on the
upper half-plane. It is also shown that these functions are automorphic forms
of weight $k$ with respect to a Fuchsian group.

###### Key words and phrases:

Automorphic form, Holomorphic function, Fuchsian group, Hauptmodul

###### 2020 Mathematics Subject Classification:

30F35; 11F12; 30C15.

##  1\. Introduction

The group $\operatorname{SL}(2,\mathbb{R})$ is defined by

$\operatorname{SL}(2,\mathbb{R})=\Bigg{\\{}\displaystyle\begin{pmatrix}a&b\\\
c&d\end{pmatrix}:a,b,c,d\in\mathbb{R},\leavevmode\nobreak\ ad-bc=1\Bigg{\\}}$

and the group
$\operatorname{PSL}(2,\mathbb{R})=\operatorname{SL}(2,\mathbb{R})/\\{\pm
I_{2}\\}$, where $I_{2}$ is the $2\times 2$ identity matrix (see [12, Chapter
VII]). Let $\mathbb{H}$ denote the upper half-plane
$\\{\tau\in\mathbb{C}:{\operatorname{Im}\,}\tau>0\\}$. The boundary of
$\mathbb{H}$ is $\mathbb{R}\cup\infty$. The group
$\operatorname{PSL}(2,\mathbb{R})$ acts on $\mathbb{H}$ as follows:

$\tau\mapsto\gamma\cdot\tau=\frac{a\tau+b}{c\tau+d},\,\text{for}\,\gamma=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\operatorname{PSL}(2,\mathbb{R}),\,\tau\in\mathbb{H}.$

All transformations of $\operatorname{PSL}(2,\mathbb{R})$ are conformal.

A Fuchsian group is a discrete subgroup of $\operatorname{PSL}(2,\mathbb{R})$,
i.e., it is a group of orientation-preserving isometries of $\mathbb{H}$.
Study of Fuchsian group is a very interesting topic in many fields of
Mathematics. Many mathematicians studied Fuchsian group and various subgroups
of Fuchsian group, for example, see [7], [11], [15] and [14]. The Hecke group
which is a subgroup of Fuchsian group is studied in [1] and [2] to investigate
Ramanujan’s modular equations.

Let $\gamma=\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\operatorname{PSL}(2,\mathbb{R})$ and let
$\operatorname{tr}(\gamma)$ denote the trace of $\gamma$, then the element
$\gamma$ is said to be elliptic, parabolic and hyperbolic when
$|\operatorname{tr}(\gamma)|<2,\,|\operatorname{tr}(\gamma)|=2$ and
$|\operatorname{tr}(\gamma)|>2$, respectively. If
$\Gamma\subset\operatorname{PSL}(2,\mathbb{R})$ is a Fuchsian group and
$\gamma\in\Gamma$ is an elliptic element, then a point $\tau\in\mathbb{H}$ is
called an elliptic point of $\Gamma$ if $\gamma(\tau)=\tau$. Also, for a
parabolic element $\sigma\in\Gamma$, a point $x\in\mathbb{R}\cup\\{\infty\\}$
is called a cusp of $\Gamma$ if $\sigma(x)=x$. If a Fuchsian group $\Gamma$
acts on $\mathbb{H}$ properly discontinuously, then we have the quotient
Riemann surface $\Gamma\backslash\mathbb{H}$. For a detailed discussion, see
[3] and [9].

Let $\mathbb{H}^{*}$ denote the union of the upper half-plane $\mathbb{H}$ and
the set of cusps of a Fuchsian group $\Gamma$. Suppose $\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\Gamma$, $\tau\in\mathbb{H}$ and
$f:\mathbb{H}\rightarrow{\mathbb{C}}$ is a holomorphic function. Then the
function $f$ is called an automorphic form of weight $k$ with respect to
$\Gamma$ if

$f\bigg{(}\frac{a\tau+b}{c\tau+d}\bigg{)}=(c\tau+d)^{k}f(\tau).$

If $k=0$, then

$f\bigg{(}\frac{a\tau+b}{c\tau+d}\bigg{)}=f(\tau)$

and $f$ is called an automorphic function. When the genus of the quotient
Riemann surface $\Gamma\backslash\mathbb{H}^{*}$ is zero, an automorphic
function is called a Hauptmodul. If an automorphic function has no poles, then
it is constant according to the consequence of maximum modulus principle. For
details, we refer the reader to [5], [6], [8], and [10].

Let $F$ be the fundamental domain for the Fuchsian group $\Gamma$. Let $X$ and
$\hat{X}$ denote the quotient Riemann surfaces $\Gamma\backslash\mathbb{H}$
and $\Gamma\backslash\mathbb{H}^{*}$, respectively. If $F$ is compact, then it
has finitely many vertices which are elliptic points and cusps of
$\hat{X}=\Gamma\backslash\mathbb{H}^{*}$. Let $P_{1},\ldots,P_{r}$ be the
vertices whose orders are $n_{1},n_{2},\ldots,n_{r}$, respectively. If the
number of elliptic elements and cusps of $\Gamma$ are $m$ and $l$,
respectively, then $m+l=r$. If $g$ is the genus of $\hat{X}$, then we say that
$\Gamma$ has signature $(g;n_{1},\ldots,n_{n})$. For more detailed discussion,
reader may consult Section 2.1 of [4], Chapter 4 of [9], and Section 2 of
[15]. Let us denote by $A_{k}$ the space of automorphic forms of weight $k$
with respect to $\Gamma$. The basis for $A_{k}$ on a Shimura curve $X$ with
genus $0$ is determined in Theorem 4 of [16]. The following theorem is written
according to Theorem 2.23 of [13] to determine the dimension of $A_{k}$.

######  Theorem 1 ([13, Theorem 2.23]).

For a Fuchsian group $\Gamma$ with signature $(g;n_{1},\ldots,n_{r})$, let $g$
be the genus of the compact quotient Riemann surface
$\hat{X}=\Gamma\backslash\mathbb{H}^{*}$. Then, the dimension,
$\operatorname{dim}A_{k}$, of $A_{k}$ for an even integer $k$ is given by

$\displaystyle\operatorname{dim}A_{k}=\begin{cases}0&\text{if $k<0$,}\\\
1&\text{if $k=0$,}\\\ g&\text{if $k=2$,}\\\
(g-1)(k-1)+\displaystyle\sum_{i=1}^{r}\bigg{\lfloor}\frac{k}{2}\big{(}1-\frac{1}{n_{i}}\big{)}\bigg{\rfloor}&\text{if
$k\geq 4$}.\end{cases}$

In the following section, we present our main results and their proofs.

##  2\. Main Results

Let $(0;n_{1},\ldots,n_{r})$ be the signature of a Fuchsian group $\Gamma$,
i.e., the genus of the quotient Riemann surface
$\hat{X}=\Gamma\backslash\mathbb{H}^{*}$ is $0$ and let
$d=\operatorname{dim}A_{k}$. Then, for an even integer $k\geq 4$, we have from
Theorem 1

$d=1-k+\displaystyle\sum_{i=1}^{r}\bigg{\lfloor}\frac{k}{2}\big{(}1-\frac{1}{n_{i}}\big{)}\bigg{\rfloor}.$

In the following theorem, we define the functions $h_{j}$ for $j=0,\ldots,d-1$
so that the functions are holomorphic on $\mathbb{H}$. Also, these functions
are automorphic forms of weight $k$ with respect to $\Gamma$.

######  Theorem 2.

Consider the Fuchsian group $\Gamma$ with signature $(0;n_{1},\ldots,n_{r})$
and the compact quotient Riemann surface
$\hat{X}=\Gamma\backslash\mathbb{H}^{*}$. Let $\tau_{1},\ldots,\tau_{r}$ be
the inequivalent vertices (elliptic points or cusps of $\hat{X}$) of the
fundamental domain of $\Gamma$ of orders $n_{1},\ldots,n_{r}$, respectively,
and let $w(\tau)$ be a Hauptmodul of $\hat{X}$. For an even integer $k\geq 4$,
let

$\displaystyle
a_{i}=\Big{\lfloor}\frac{k}{2}\Big{(}1-\frac{1}{n_{i}}\Big{)}\Big{\rfloor}\quad\text{and}\quad
d=\operatorname{dim}A_{k}=1-k+\sum_{i=1}^{r}a_{i}.$

If $w(\tau_{i})=w_{i}$ for $i=1,\ldots,r$ and the functions

(2.1)
$h_{j}(\tau)=\frac{\big{(}w^{\prime}(\tau)\big{)}^{k/2}\big{(}w(\tau)\big{)}^{j}}{\displaystyle\prod_{i=1,w_{i}\neq\infty}^{r}\big{(}w(\tau)-w_{i}\big{)}^{a_{i}}}$

for $j=0,\ldots,d-1$ and $\tau\in\mathbb{H}$, then $h_{j}(\tau)$ is
holomorphic on $\mathbb{H}$.

###### Proof.

We need to consider the following three cases:

  1. (i)

the Hauptmodul $w(\tau)$ does not have any pole at the points $\tau_{i}$ for
$i=1,\ldots,r$;

  2. (ii)

the Hauptmodul $w(\tau)$ has a pole at one of the points $\tau_{i}$ for
$i=1,\ldots,r$;

  3. (iii)

the Hauptmodul $w(\tau)$ has a pole at another point, say $\tau=\tau_{0}$,
except the points $\tau_{1},\ldots,\tau_{r}$.

If a function has a zero of order $\geq 0$ and has a pole of order $\leq 0$ at
a point, then there is no principal part in the expansion of the function at
that point, i.e., the function is holomorphic. Thus, we have to show that
$h_{j}$ has a zero of order $\geq 0$ at $\tau=\tau_{i}$ for Case (i), $h_{j}$
has a pole of order $\leq 0$ at $\tau=\tau_{i}$ for Case (ii) and $h_{j}$ has
a pole of order $\leq 0$ at $\tau=\tau_{0}$ for Case (iii).

Case (i): If $w(\tau)$ does not have any pole at $\tau_{i}$, then
$w(\tau_{i})=w_{i}\neq\infty$ for $i=1,\ldots,r$. Since $\tau_{i}$ is a vertex
of order $n_{i}$, in a neighbourhood of $\tau=\tau_{i}$, we have

(2.2) $\displaystyle
w(\tau)-w(\tau_{i})=b_{i}(\tau-\tau_{i})^{n_{i}}+O\big{(}(\tau-\tau_{i})^{n_{i}+1}\big{)}$

or,

(2.3) $\displaystyle w(\tau)-w_{i}=(\tau-\tau_{i})^{n_{i}}w^{*}(\tau),$

where $b_{i}\in\mathbb{C}\setminus\\{0\\}$, $w^{*}(\tau)$ is analytic in a
neighbourhood of $\tau=\tau_{i}$ and $w^{*}(\tau_{i})\neq 0$ for
$i=1,\ldots,r$. Therefore, in a neighbourhood of $\tau=\tau_{i}$, one can
define a single-valued analytic $n_{i}$-th root of $(w-w_{i})$ and this can be
done at all points which are equivalent to $\tau_{i}$ under the action of the
Fuchsian group $\Gamma$. Since $w(\tau)-w_{i}\neq 0$ for $\tau\neq\tau_{i}$
and $(w-w_{i})$ is analytic on the other part of $\mathbb{H}$, its $n_{i}$-th
root is analytic at each point of the remainder of $\mathbb{H}$. As
$(w(\tau)-w_{i})^{n_{i}}$ is locally analytic and single-valued at each
$\tau\in\mathbb{H}$, so it follows from monodromy theorem that a single-valued
and analytic $n_{i}$-th root of $(w-w_{i})$ can be defined on the whole
$\mathbb{H}$.

From (2.3), we observe that $\big{(}w(\tau)-w_{i}\big{)}$ has a zero of order
$n_{i}$ at $\tau=\tau_{i}$ and
$\displaystyle\prod_{i=1,w_{i}\neq\infty}^{r}\big{(}w(\tau)-w_{i}\big{)}^{a_{i}}$
has a zero of order $n_{i}a_{i}=\displaystyle
n_{i}\Big{\lfloor}\frac{k}{2}\Big{(}1-\frac{1}{n_{i}}\Big{)}\Big{\rfloor}$ at
$\tau=\tau_{i}$. Also, we have from (2.2)

(2.4) $\displaystyle
w^{\prime}(\tau)=b_{i}n_{i}(\tau-\tau_{i})^{n_{i}-1}+O\big{(}(\tau-\tau_{i})^{n_{i}}\big{)}.$

Consequently, at $\tau=\tau_{i}$, $\big{(}w^{\prime}(\tau)\big{)}^{k/2}$ has a
zero of order $\displaystyle\frac{k}{2}(n_{i}-1)$. Since

$\displaystyle\frac{k}{2}(n_{i}-1)-\displaystyle
n_{i}\Big{\lfloor}\frac{k}{2}\Big{(}1-\frac{1}{n_{i}}\Big{)}\Big{\rfloor}\geq
0,$

we conclude from (2.1) that $h_{j}$ has a zero of order $\geq 0$ at
$\tau=\tau_{i}$. Hence $h_{j}$ is holomorphic on $\mathbb{H}$.

Case (ii): Assume that $w(\tau)$ has a pole at one of the points $\tau_{i}$
for $i=1,\ldots,r$. Without loss of generality, suppose $w(\tau)$ has a pole
at $\tau_{1}$, i.e., $w(\tau_{1})=w_{1}=\infty$. Since $\tau_{1}$ is a vertex
of order $n_{1}$, it follows that

$\displaystyle
w(\tau)=\frac{b_{1}}{(\tau-\tau_{1})^{n_{1}}}+O\big{(}(\tau-\tau_{1})^{1-n_{1}}\big{)},\quad
b_{1}\in\mathbb{C}\setminus\\{0\\}$

and

$\displaystyle
w^{\prime}(\tau)=-\frac{b_{1}n_{1}}{(\tau-\tau_{1})^{n_{1}+1}}+O\big{(}(\tau-\tau_{1})^{-n_{1}}\big{)}.$

In this case, from (2.1) we have

(2.5)
$h_{j}(\tau)=\frac{\big{(}w^{\prime}(\tau)\big{)}^{k/2}\big{(}w(\tau)\big{)}^{j}}{\displaystyle\prod_{i=2,w_{i}\neq\infty}^{r}\big{(}w(\tau)-w_{i}\big{)}^{a_{i}}}.$

Now, suppose that $h_{j}(\tau)$ defined in (2.5) has a pole of order $N$ at
$\tau=\tau_{1}$. Since $w(\tau)$ has a pole of order $n_{1}$ at
$\tau=\tau_{1}$, $\big{(}w^{\prime}(\tau)\big{)}^{k/2}$ has a pole of order
$\displaystyle\frac{k}{2}(n_{1}+1)$ and
$\displaystyle\prod_{i=2,w_{i}\neq\infty}^{r}\big{(}w(\tau)-w_{i}\big{)}^{a_{i}}$
has a pole of order $\displaystyle n_{1}\sum_{i=2}^{r}a_{i}$ at
$\tau=\tau_{1}$. As $j$ varies from $0$ to $d-1$, so the maximum value of $j$
is $d-1=\displaystyle\sum_{i=1}^{r}a_{i}-k$. Hence $\big{(}w(\tau)\big{)}^{j}$
has a pole of order at most
$n_{1}\Big{(}\displaystyle\sum_{i=1}^{r}a_{i}-k\Big{)}$ at $\tau=\tau_{1}$.
Therefore, we have

$\displaystyle N$
$\displaystyle\leq\frac{k}{2}(n_{1}+1)+n_{1}\Big{(}\displaystyle\sum_{i=1}^{r}a_{i}-k\Big{)}-n_{1}\sum_{i=2}^{r}a_{i}$
$\displaystyle=\frac{k}{2}(n_{1}+1)+n_{1}\Big{(}\sum_{i=1}^{r}\Big{\lfloor}\frac{k}{2}\Big{(}1-\frac{1}{n_{i}}\Big{)}\Big{\rfloor}-k\Big{)}-n_{1}\sum_{i=2}^{r}{\Big{\lfloor}\frac{k}{2}\Big{(}1-\frac{1}{n_{i}}\Big{)}\Big{\rfloor}}$
$\displaystyle=-\frac{k}{2}\Big{(}n_{1}-1\Big{)}+n_{1}\Big{\lfloor}\frac{k}{2}\Big{(}1-\frac{1}{n_{1}}\Big{)}\Big{\rfloor}\leq
0.$

Since $N\leq 0$, it follows that there is no principal part in the expansion
of $h_{j}$, i.e., $h_{j}$ is holomorphic on $\mathbb{H}$.

Case (iii): Suppose that $w(\tau)$ has the value $\infty$ at the point
$\tau=\tau_{0}$ and $w(\tau_{i})\neq\infty$ for $i=1,\ldots,r$. Therefore,
$w(\tau)$ has a simple pole at $\tau_{0}$ and we have

(2.6) $\displaystyle w(\tau)=\frac{b_{0}}{(\tau-\tau_{0})}+O(1),\quad
b_{0}\in\mathbb{C}\setminus\\{0\\}$

and

(2.7) $\displaystyle
w^{\prime}(\tau)=-\frac{b_{0}}{(\tau-\tau_{0})^{2}}+O(1).$

Let $N_{0}$ be the order of the pole of $h_{j}$ defined in (2.1) at
$\tau=\tau_{0}$. From (2.6) and (2.7), we observe that
$\big{(}w^{\prime}(\tau)\big{)}^{k/2}$ has a pole of order $k$,
$\displaystyle\prod_{i=1,w_{i}\neq\infty}^{r}\big{(}w(\tau)-w_{i}\big{)}^{a_{i}}$
has a pole of order $\displaystyle\sum_{i=1}^{r}a_{i}$ and $w^{j}$ has a pole
of order at most $d-1=\displaystyle\sum_{i=1}^{r}a_{i}-k$. Therefore, from
(2.1), it follows that

$\displaystyle N_{0}\leq k+\sum_{i=1}^{r}a_{i}-k-\sum_{i=1}^{r}a_{i}=0,$

which implies that $h_{j}$ is holomorphic on $\mathbb{H}$ in this case also. ∎

######  Lemma 1.

The functions $h_{j}$ for $j=0,\ldots,d-1$ defined in (2.1) is an automorphic
form of weight $k$ with respect to the Fuchsian group $\Gamma$.

###### Proof.

For $j=0,\ldots,d-1$ and
$a_{i}=\Big{\lfloor}\frac{k}{2}\Big{(}1-\frac{1}{n_{i}}\Big{)}\Big{\rfloor}$,
we have to show that

$h_{j}\Big{(}\frac{a\tau+b}{c\tau+d}\Big{)}=(c\tau+d)^{k}h_{j}(\tau),$

where $\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\Gamma$ and
$\tau\in\mathbb{H}$. Since $w(\tau)$ is a Hauptmodul of $\hat{X}$, i.e.,
$w(\tau)$ is an automorphic function, thus we have

$w\Big{(}\frac{a\tau+b}{c\tau+d}\Big{)}=w(\tau)$

and

$w^{\prime}\Big{(}\frac{a\tau+b}{c\tau+d}\Big{)}=(c\tau+d)^{1/2}w^{\prime}(\tau).$

Now,

$\displaystyle h_{j}\Big{(}\frac{a\tau+b}{c\tau+d}\Big{)}$
$\displaystyle=\frac{\displaystyle\bigg{(}w^{\prime}\Big{(}\frac{a\tau+b}{c\tau+d}\Big{)}\bigg{)}^{k/2}\bigg{(}w\Big{(}\frac{a\tau+b}{c\tau+d}\Big{)}\bigg{)}^{j}}{\displaystyle\prod_{i=1,w_{i}\neq\infty}^{r}\bigg{(}w\Big{(}\frac{a\tau+b}{c\tau+d}\Big{)}-w_{i}\bigg{)}^{a_{i}}}$
$\displaystyle=\frac{(c\tau+d)^{k}\big{(}w^{\prime}(\tau)\big{)}^{k/2}\big{(}w(\tau)\big{)}^{j}}{\displaystyle\prod_{i=1,w_{i}\neq\infty}^{r}\big{(}w(\tau)-w_{i}\big{)}^{a_{i}}}$
$\displaystyle=(c\tau+d)^{k}h_{j}(\tau).$

Thus, $h_{j}$ is an automorphic form of weight $k$ with respect to $\Gamma$. ∎

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