# On the Reionization-Era Globular Cluster in Low-Mass Galaxy Eridanus II

Daniel R. Weisz Department of Astronomy
University of California
Berkeley, CA 94720, USA  Alessandro Savino Department of Astronomy
University of California
Berkeley, CA 94720, USA  Andrew E. Dolphin Raytheon Technologies, 1151 E.
Hermans Road, Tucson, AZ 85756, USA  Steward Observatory, University of
Arizona, 933 N. Cherry Avenue, Tucson, AZ 85719, USA

(Received July 6, 2022; Accepted March 10, 2023)

###### Abstract

Using color-magnitude diagrams from deep archival Hubble Space Telescope
imaging, we self-consistently measure the star formation history of Eridanus
II (Eri II), the lowest-mass galaxy ($M_{\star}(z=0)\sim 10^{5}M_{\odot}$)
known to host a globular cluster (GC), and the age, mass, and metallicity of
its GC. The GC ($\sim 13.2\pm 0.4$ Gyr, $\langle$[Fe/H]$\rangle=-2.75\pm 0.2$
dex) and field (mean age $\sim 13.5\pm 0.3$ Gyr,
$\langle$[Fe/H]$\rangle=-2.6\pm 0.15$ dex) have similar ages and
metallicities. Both are reionization-era relics that formed before the peak of
cosmic star and GC formation ($z\sim 2-4$). The ancient star formation
properties of Eri II are not extreme and appear similar to $z=0$ dwarf
galaxies. We find that the GC was $\lesssim 4$ times more massive at birth
than today and was $\sim$10% of the galaxy’s stellar mass at birth. At
formation, we estimate that the progenitor of Eri II and its GC had $M_{\rm
UV}\sim-7$ to $-12$, making it one of the most common type of galaxy in the
early Universe, though it is fainter than direct detection limits, absent
gravitational lensing. Archaeological studies of GCs in nearby low-mass
galaxies may be the only way to constrain GC formation in such low-mass
systems. We discuss the strengths and limitations in comparing archaeological
and high redshift studies of cluster formation, including challenges stemming
from the Hubble Tension, which introduces uncertainties into the mapping
between age and redshift.

Dwarf galaxies (416), Globular star clusters (656), Local Group (929),
Hertzsprung Russell diagram (725)

††facilities: HST(ACS)††software:  This research made use of routines and
modules from the following software packages: Astropy (Astropy Collaboration
et al., 2013), DOLPHOT (Dolphin, 2016), IPython (Perez & Granger, 2007),
Matplotlib (Hunter, 2007), NumPy (van der Walt et al., 2011) and SciPy
(Virtanen et al., 2020)

##  1 Introduction

Metal-poor globular clusters (GCs) are likely relics of the early Universe.
They are thought to have formed in low-mass halos at high-redshifts and may
represent an extreme mode of star formation that is non-existant or rare at
low-redshifts (e.g., Peebles & Dicke, 1968; Fall & Rees, 1985; Elmegreen &
Efremov, 1997).

One obstacle to deciphering their origins is a paucity of suitable constraints
on both metal-poor GCs and their host galaxies. The sensitivity and resolving
power of the Hubble Space Telescope (HST) and the James Webb Space Telescope
(JWST) have revealed a number of possible young clusters and/or compact star
forming clumps that may be progenitors of the GCs found in the local Universe
(e.g., Vanzella et al., 2017, 2019; Zick et al., 2020; Vanzella et al., 2022a;
Pascale et al., 2023; Vanzella et al., 2022b; Welch et al., 2023). However,
given the high stellar masses and star formation rates (SFRs) associated with
these objects, they are likely one of the most massive and extreme examples of
cluster formation in the early Universe, and are unlikely to sample formation
of typical metal-poor GCs. Moreover, their host galaxies are likely
progenitors of MW-like systems at $z=0$ rather that the very low-mass systems
thought to host metal-poor GCs (e.g., Kravtsov & Gnedin, 2005). Due to their
faintness, direct detection and characterization of the the lower mass GCs and
their host galaxies is likely to prove challenging even with the sensitivity
of JWST.

At low redshift, we have access to a larger population of metal-poor GCs,
which can be studied in great detail. However, the majority of them reside in
massive galaxies such as the Milky Way (MW) where the process of hierachical
accretion of GCs, and in some cases their disruption into streams, obfuscates
the connection between metal-poor GCs and their birth galaxies (e.g., Kravtsov
& Gnedin, 2005; Zaritsky et al., 2016; Choksi et al., 2018; El-Badry et al.,
2019; Kruijssen et al., 2019; Garcia et al., 2022; Chen & Gnedin, 2023).

Historically, only a handful of GCs are known reside in nearby dwarf galaxies
(e.g., Humason et al. 1956; Ables & Ables 1977; Hoessel & Mould 1982; Hodge et
al. 1999; Grebel et al. 2000; Cook et al. 2012; Caldwell et al. 2017; Cole et
al. 2017; Forbes et al. 2018; Beasley et al. 2019; Larsen et al. 2020; Pace et
al. 2021; Larsen et al. 2022), though new search efforts are rapidly
increasing this number (e.g., Minniti et al., 2021; Carlsten et al., 2022).
The sample of GCs in low-mass galaxies has provided rich insight into the
formation and evolution of metal-poor GCs in their birth environment. For
example, a series of spectroscopic studies have shown that GCs in moderate-
mass dwarf galaxies can only have been a factor of a few times more massive in
the past, providing stringent constraints on the mass budget problem in these
environments (e.g., Larsen et al., 2012, 2014). Similarly, the locations of
the GCs within low-mass galaxies can be used to constrain the cluster
formation mechanisms (e.g., Zaritsky, 2022), while basic properties such as
age and metallicity can be used to constrain a wide variety of astrophysics
from the contribution of GCs to reionization to the halo masses and accretion
histories of the host galaxies.

The Local Group (LG) ultra-faint dwarf galaxy (UFD) Eridanus II (Eri II;
Bechtol et al. 2015; Koposov et al. 2015) provides a unique opportunity to
simultaneously study a metal-poor GC and its birth galaxy in the lowest mass
galaxy known that hosts a GC (e.g., Forbes et al., 2018). Eri II has deep HST
imaging of its resolved stellar population enabling an archaeological
characterization of the galaxy and GC in the early Universe. Prior HST-based
studies have shown Eri II to be an ancient ($>13$ Gyr old) and metal-poor
([Fe/H] $\lesssim-2.3$) galaxy (e.g., Gallart et al., 2021; Simon et al.,
2021; Fu et al., 2022). However, there has been less focus on the GC and its
relationship to the host galaxy and cluster formation more broadly.

In this paper, we self-consistently characterize the stellar populations of
Eri II’s GC and field populations from color-magnitude diagram (CMD) analysis
and combine our measurements with simple assumptions to constrain formation
scenarios for a metal-poor GC in an extremely low-mass galaxy. We describe our
photometric reduction in §2 and summarize our analysis methodology in §3. We
present our results and place them into a broader context in §4.

##  2 Data & the Color-Magnitude Diagrams

We use archival HST/ACS F475W, F606W, and F814W imaging that was taken as part
of two separate HST programs (GO-14224 and GO-14234). The deep imaging from
both programs was designed to extend well-below the ancient main sequence
turnoff (MSTO), enabling a ‘gold standard’ reconstruction of Eri II’s
properties at all cosmic epochs.

We performed point spread function (PSF) photometry on each flc image using
DOLPHOT, a widely-used PSF fitting package that has an HST/ACS specific module
(Dolphin, 2000, 2016). We used the DOLPHOT parameters and culling criteria
specified in Williams et al. (2014), which were optimized for the disk of M31
and are well-suited for the comparatively less crowded field and GC
populations of Eri II.

We characterized uncertainties in the photometry of the GC and field
population separately. Following resolved cluster analysis in Johnson et al.
(2016), we generate $50,000$ artificial star tests (ASTs) that trace the light
profile of the GC (Simon et al., 2021). For the field population, we uniformly
distributed $500,000$ ASTs throughout the non-cluster region of the HST
images. The GC and field photometry is 100% complete $>1$ mag below the oldest
MSTO.

Figure 1: The HST/ACS F475W-F814W CMDs of Eri II and its cluster. The GC CMD
contains stars within $2r_{h}$ (half-light radius) and it has been excluded
from the field CMD on the left. The middle panels show a BaSTI isochrone at
our measured HB distance. The bottom panels are zoomed in on the MSTO regions
use in our fitting. Over-plotted are varying [Fe/H] BaSTI models at a fixed
age of 13.5 Gyr. The cluster and field population appear qualitatively similar
in age and metallicity.

Figure 1 shows the F475W-F814W CMDs of the field population and GC. The GC CMD
includes stars within $2r_{h}$ and field population CMD is all stars outside
this radius. Our field CMD is visually similar to those presented and
discussed in Gallart et al. (2021), while our cluster CMD is similar to that
presented in Martínez-Vázquez et al. (2021). The overlaid BaSTI isochrones
(Hidalgo et al., 2018) qualitatively show that Eri II is predominantly an
ancient, metal-poor population as found in other studies. The presence of a
blue and red horizontal branch may indicate a population of mixed age and/or
broad metallicities, the latter of which is also reported in the literature
(e.g., Li et al., 2017; Fu et al., 2022), or could reflect physics such as
variable mass loss or helium abundance (e.g., Milone et al., 2017; Savino et
al., 2019). There is a clear blue plume (i.e., a vertical sequence of stars
extending above the ancient MSTO), which may indicate a younger population
and/or blue stragglers. Both Gallart et al. (2021) and Simon et al. (2021)
show that the number of stars in this region of the CMD are consistent with
expectations for blue stragglers.

Eri II’s GC also appears to be ancient and metal-poor. Simon et al. (2021) and
Martínez-Vázquez et al. (2021) comment on the visual similarity between the
field and cluster populations. The former study notes there may be a small
offset in the colors of the GC and field populations near the sub-giant
branch. From the bottom panel of Figure 1, it appears that the GC may be
slightly more metal-poor than the field population; formal CMD fitting, i.e.,
a goal of this paper, is required to quantify such subtle differences.

##  3 Methodology

###  3.1 Distance

Literature distances to Eri II vary from $\mu=22.60$ to $22.90$ (e.g., Bechtol
et al., 2015; Koposov et al., 2015; Simon et al., 2021; Gallart et al., 2021;
Martínez-Vázquez et al., 2021; Nagarajan et al., 2021). Three of these
distances are based on RR Lyrae, which are considered ‘gold standards’ for Pop
II stars. However, they are in modest tension: $\mu=24.65$ (Simon et al.,
2021), $\mu=24.67$ (Nagarajan et al., 2021), and $\mu=24.84$ (Martínez-Vázquez
et al., 2021), making it challenging to know which one to use. These
discrepancies could be due to a variety of factors (e.g., light curve
sampling, analysis techniques, zero point differences). Resolving these
tensions are beyond the scope of this paper. For simplicity, we instead derive
our own distance to Eri II using the luminosity of the HB following McQuinn et
al. (2015) and Weisz et al. (2019).

Briefly, we convert F606W to a pseudo V-band using the filter transformations
provided in Saha et al. (2011). F606W is closer to V-band than F475W making it
a more suitable choice. We fit the pseudo V-band luminosity function of the
region around the HB using a mixture of a Gaussian and exponential function.
We find the most likely pseudo V-band HB magnitude to be $m_{V}=23.30$, which
yields $\mu=22.79\pm 0.1$ for $A_{\rm V}=0.038$ mag (Schlafly & Finkbeiner,
2011) and $M_{\rm V,0,HB}=+0.47$ (Weisz et al., 2019). This distance is
consistent with most values reported in the literature, including falling
within 1-1.5$\sigma$ of all of the RR Lyrae distances. Isochrones at this
distance and extinction show a good match to the observed CMDs (Figure 1). We
adopt $\mu=22.79\pm 0.1$ for the remainder of this analysis.

###  3.2 Modeling the Field and Cluster Color-Magnitude Diagrams

We measure the SFH of the field population and the age, mass, and metallicity
of the GC using MATCH (Dolphin, 2002) and the F475W-F814W CMD, for which the
broad color baseline is particularly sensitive to age and metallicity at the
main sequence turnoff (e.g., Cole et al., 2007; Monelli et al., 2010; Gallart
et al., 2015). MATCH is a widely used package for modeling CMDs of nearby
galaxies and star clusters (Weisz et al., 2014a, 2016; Johnson et al., 2016;
Skillman et al., 2017). We follow implementation details described in these
papers.

Briefly, for a given set of stellar evolution models, a specified initial mass
function (IMF), distance, extinction, and binary fraction, MATCH generates a
set of synthetic simple stellar populations (SSPs) over a range of ages and
metallicities. These SSPs are linearly combined to form a model composite CMD.
The CMD is partitioned into small color and magnitude bins (i.e,. a Hess
diagram). The composite CMD is convolved with the error distribution from the
ASTs and a contamination model CMD (e.g., foreground stars) is linearly added.
The resulting synthetic CMD is compared to the observed CMD using a Poisson
likelihood function. MATCH repeats this process either as a full grid search
or by a maximum likelihood optimization.

For this analysis, we adopt a Kroupa IMF (Kroupa, 2001) from 0.08 to 120
$M_{\odot}$, $\mu=22.79$ mag, $A_{\rm V}=0.038$ (Schlafly & Finkbeiner, 2011),
and a binary fraction of 0.35 with a uniform mass ratio distribution. We use
the BaSTI (Hidalgo et al., 2018) stellar models, as they include heavy element
diffusion and their publicly available models encompass the low stellar
metallicites ([Fe/H] $<-3$) found in Eri II (Li et al., 2017; Fu et al.,
2022). We measured the SFH in two iterations. First, we generate a grid of
SSPs with coarse time ($\log(\Delta t)=0.05$ dex) and metallicity
($\log(\Delta[M/H])=0.1$ dex) resolutions over the ranges
$10.15\leq\log(t)\leq 9.0$ and $-3.5\leq\log([M/H])\leq-1.5$. This allowed us
to get a broad sense of the SFH and evaluate the significance of the ‘blue
plume’ population. We found that the majority of field star formation occurred
in the oldest time bin, while $\sim 4$% of the stellar mass formed 2-3 Gyr
ago. This younger population appears consistent with blue straggler
contributions and is consistent with the findings of other analyses of Eri II.
Given the high SNR of the Eri II data, we opted to measure the SFH at much
higher time resolution. In this case, we generate a grid of SSPs with fine
time ($\log(\Delta t)=0.01$ dex) and metallicity ($\log(\Delta[M/H])=0.05$
dex) over the ranges $10.14\leq\log(t)\leq 9.8$ and $-3.5\leq\log(t)\leq-1.5$.
The lower limit on age was selected due to make for reasonable computational
efficiency, even though it formally excludes the blue straggler contribution.
This has no impact on any of our results. We restrict our analysis to the
region of the field and GC CMDs around the MSTO (see Figure 2).

Figure 2: Fit quality for Eri II (top row) and its GC (bottom row). Plotted
are the Hess diagrams for the data (left), the best fit models (middle), and
residual normalized to $\sigma$ (right). The best fit models for both field
and cluster population show no significant and/or systematics and overall
appear to be good fits to the data.

For the field SFH, we exclude stars within $2r_{h}$ of the GC. We include a
model for Galactic foreground contamination (de Jong et al., 2010), even
though only a few $\sim 5$ MW interlopers are expected in this field of view
and to have passed all photometric quality cuts. We place a prior on the age-
metallicity relationship such that it must increase monotonically, with an
allowed dispersion of 0.4 dex, which is based on RGB star metallicity studies
of Eri II. Given all these parameters, MATCH finds the maximum likelihood
solution. To compute random uncertainties, we use a Hamiltonian Monte Carlo
(Duane et al., 1987, HMC;) approach described in Dolphin (2013) and
implemented in Weisz et al. (2014a). We initialize the HMC chains at the
maximum likelihood point and draw 5000 samples after burn-in. Random
uncertainties are the narrowest 68% interval around the best fit SFH.

We model the GC using only stars located within $2r_{h}$. We use the field
population from $>2r_{h}$ as a background CMD and allow MATCH to linearly
scale its contribution while solving for the GC parameters. This has proven to
be an effective decontamination method in practice (e.g., for clusters in the
disk of M31; Weisz et al., 2015; Johnson et al., 2016). We assume the cluster
is an SSP and conduct a full grid search over the entire age and metallicity
grid defined above. Assuming flat priors in age and metallicty, we compute the
marginalized distributions for age and metallicty of the GC. For
uncertainties, we report the best fit age of each GC and the 68% confidence
around the best fit and add the grid resolution (0.01 dex in age, 0.05 dex in
metallicity) in quadrature to account for the finite width of the basis
functions (e.g., Weisz et al., 2016).

Figure 2 shows the observed, best fit model, and residual Hess diagrams for
the field and GC CMDs. For both the field and GC, the residuals show that age
and metallicity sensitive regions of the observed CMD, i.e., the MSTO and SGB,
are matched by the model at the $<2$-$\sigma$ level. The highest areas of
disagreement are a handful of pixels below the oldest MSTO that is not age
sensitive. Overall, the observed CMDs are well-described by the best-fit
models.

Figure 3: The cumulative (top) and differential (bottom) SFHs of Eri II. The
black lines are the best fit SFHs and the uncertainties reflect the 68%
confidence intervals. The dark blue line shows the best fit GC age for each
model and the light blue shaded region is the 68% confidence interval. The red
dashed lines indicates the timing of reionization assuming a Planck
Collaboration et al. (2020) cosmology.

##  4 Results

Figure 3 shows the SFH of Eri II’s field population for the BaSTI and MIST
models with the corresponding GC age over-plotted. Uncertainties for the field
SFH and GC are the 68% confidence intervals. We list the measured properties
of the GC in Table 1.

The field population of Eri II is ancient: $\sim 80$% of the stellar mass
formed prior to 13.5 Gyr ago ($z\sim 10$), which is nearly identical to field
SFHs for Eri II in the literature (Alzate et al., 2021; Gallart et al., 2021;
Simon et al., 2021). We find a small secondary burst of star formation $\sim
11-12$ Gyr ago, but it is of low significance. This secondary event could
contribute to Eri II’s broad HB morphology, though the metallicity spread in
Eri II may also contribute. The mass-weighted age of Eri II’s field population
is 13.5$\pm 0.3$ Gyr. As with many UFDs in the MW halo (e.g., Brown et al.,
2014; Weisz et al., 2014a; Sacchi et al., 2021), Eri II appears to be a
reionization-era galaxy within the Planck-based $\Lambda$CDM cosmology.

We find the mean metallicity of the field population to be
$\langle$[Fe/H]$\rangle=-2.6\pm 0.15$ dex, which is similar to RGB star
metallicities (e.g., Li et al., 2017; Fu et al., 2022). If Eri II is
$\alpha$-enhanced, the values of [Fe/H] may be a few tenths of a dex lower;
BaSTI $\alpha$-enhanced models were not available at the time of our analysis,
however.

We measure the stellar mass within the ACS field to be $M_{\star}\sim
1.9_{-0.03}^{+0.04}\times 10^{5}$ $M_{\odot}$. Following Gallart et al.
(2021), we multiply a factor of 2 to account for the fact that the ACS field
encompasses $\sim 50$% of the galaxy’s integrated optical light get a total
stellar mass of $M_{\star}\approx 3.8\times 10^{5}$ $M_{\odot}$. This is the
stellar birth mass of the galaxy as the CMD fitting code uses stellar birth
masses and does not account for mass loss due to stellar evolution. We discuss
the present day stellar mass in §5.3.

Eri II’s GC is ancient, metal-poor, and has a modest stellar mass (Table 1).
We find a GC metallicity of $\langle$[Fe/H]$\rangle=-2.75\pm 0.2$ dex, which
is marginally more metal-poor than the field population, though consistent
within uncertainties. The best fit cluster age, $13.2\pm 0.4$ Gyr, is slightly
younger than the field, though formally consistent ages within 1-$\sigma$
uncertainties.

As a check on robustness of our modeling, we also fit the field population and
GC using the MIST stellar libraries (Choi et al., 2016). The MIST-based field
SFH is nearly identical ($\sim 90$% of the stellar mass formed prior to
$\sim$13.5 Gyr ago) to our solution using the BaSTI models. Similarly, the
MIST-based age of the GC is 13.2$\pm 0.4$ Gyr, which is the same as our BaSTI
fit. The MIST results are systematically several tenths of a dex more metal-
rich. The similarity of SFHs but differences in recovered metallicities when
using different stellar libraries has previously been identified and
quantified in more detail (e.g., see the Appendix in Skillman et al. 2017). In
both cases, the formation epoch of the GC is clearly older than than the peak
GC formation epoch ($z\sim 2-4$) predicted by several recent models of GC
formation (e.g., Choksi et al., 2018; El-Badry et al., 2019; Reina-Campos et
al., 2019); though some models find the peak of metal-poor GC formation to be
at $z\gtrsim 5$ (e.g., Renzini, 2017; Boylan-Kolchin, 2018a).

We measure the stellar mass of the GC to be $M_{\star}\approx 8.6\times
10^{3}$$M_{\odot}$. This mass is based on CMD modeling, which models the
observed star counts at the present-day and sums their birth masses to get a
total stellar mass of the population. That is the CMD mass is not the present-
day mass because it uses initial masses from the stellar models and it is not
the birth mass because it does not account of dynamical mass loss effects.
However, with some modest assumptions we can translate the CMD-based mass into
the present-day and birth mass of the cluster. We calculate these quantities
in the next section.

Quantity | Field | GC
---|---|---
Age (Gyr) | $13.5\pm 0.3$ | $13.2\pm 0.4$
$\langle$[Fe/H]$\rangle$ (dex) | $-2.6\pm 0.15$ | $-2.75\pm 0.2$
$M_{\star,CMD}$ ($10^{3}M_{\odot}$) | 380 | 8.6
$M_{\star,Birth}$ ($10^{3}M_{\odot}$) | 380 | $\lesssim 34$
$M_{\star,z=0}$ ($10^{3}M_{\odot}$) | 150 | 5.2
$M_{V}/L_{V}(z=0)$ | $\approx 2.5$ | $\approx 2.5$
$\dot{M}_{\star}$ ($10^{-3}$ $M_{\odot}$ yr${}^{-1})$ | $1-7$ | $1-5$
$\Sigma_{\rm SFR}$ ($M_{\odot}$ yr-1 kpc-2) | $0.001-0.9$ | $5-500$
$M_{\rm UV,formation}$ (mag) | $\sim-9\pm 2.5$ | $\sim-12.5$
Table 1: Measured or estimated properties of the GC in Eri II. We list three
types of stellar mass for the cluster: the stellar mass measured directly from
the CMD ($M_{\star,CMD}$); the stellar mass at GC birth $M_{\star,Birth}$,
which includes a correction for mass loss due to dynamical effects; the
present-day stellar mass $M_{\star,z=0}$, which includes dynamical mass loss
and mass loss due to stellar evolution. We discuss how each quantity is
estimated in §4.

##  5 Discussion

###  5.1 Cluster Birth Mass and Constraints on Dynamical Mass Loss

The birth masses and amount of dynamical mass loss in GC are key to several
open questions including the origin of multiple populations, the contribution
of GCs to reionization, and their long-term survival (e.g., Schaerer &
Charbonnel, 2011; Conroy, 2012; Bastian & Lardo, 2015; Kruijssen, 2015;
Renzini et al., 2015; Bastian & Lardo, 2018; Boylan-Kolchin, 2018b; Krumholz
et al., 2019).

However, the mass lost over the lifetime of a GC due to dynamical effects is
challenging to measure resulting in a paucity of empirical constraints (see
discussion in Bastian & Lardo (2018) and references therein), particularly in
in low-mass galaxies. One of the best constraints in a low-mass system comes
from Larsen et al. (2012) which estimates dynamical mass loss from the GCs in
Fornax. They compare the number and mass of metal-poor stars in the field of
Fornax to its GC population to find that the GCs were no more than 3-5 times
more massive at birth compared to the present day. Using the same technique,
Larsen et al. (2014) find comparable amounts of mass loss in slightly more
massive dwarf galaxies WLM and IKN.

We use a similar approach to estimate the amount of mass lost by the GC in Eri
II. Our CMD modeling provides the total mass of field stars as a function of
metallicity, i.e., a mass-weighted metallicity distribution function (MDF).
Within the 1-$\sigma$ range of the cluster metallicity, i.e., [Fe/H]$=-2.55$
to $-2.95$ dex, we find the cluster-to-galaxy stellar mass ratio to $\sim
3.5\pm 0.5$. This estimate includes the areal correction applied to the field
population. This is an upper limit on the amount of stellar mass lost by the
GC through dynamical effects over its lifetime , i.e., if the GC was entirely
responsible for field stars in this metallicity range then it would have been
3-4 times more massive at birth.

We multiply the CMD-based mass by 4 and and find a maximum birth mass of
$M_{\star,Birth}\lesssim 3.4\pm 0.4\times 10^{4}$ $M_{\odot}$. At birth, the
GC was $\sim 10$% of the galaxy stellar mass.

Our upper limit on dynamical mass loss for Eri II’s GC is comparable to the
upper limit of 3-5 reported by Larsen et al. (2012) for Fornax, WM, and IKN.
The consistency between mass loss in Eri II and Fornax (the closest to Eri II
in stellar mass) is interesting as Fornax has 100 times more stellar mass than
Eri II at the present day, which would suggest a stronger disruption
mechanisms (e.g., Kruijssen, 2015). However, because this method only provides
for upper limits, it is not possible to identify subtle differences in mass
loss as a function of galaxy mass. Nevertheless, there are now two data points
in the low-mass galaxy regime that suggest dynamical mass loss is no more than
a factor of a few.

A moderate birth mass, and moderate mass loss, for Eri II’s GC is broadly in
line with some theoretical predictions for GC formation in low-mass halos
(e.g., Kruijssen, 2015; Choksi et al., 2018; Kruijssen, 2019). In massive
galaxies like the MW, lower mass GCs ($M_{\star,birth}<10^{5}$) would be
unlikely to survive for a Hubble time in the harsh galactic environment (e.g.,
due to tidal shocks, collisions with gas clouds, migration, the density of the
host galaxy/halo). However, these effects are thought to be much weaker in a
low-mass galaxy like Eri II, which enables the long-term survival of moderate
mass GCs. The long lifetime of Eri II’s GC and modest dynamical mass loss
support this broad picture.

###  5.2 Connections to Select Theoretical Models of Cluster Formation in Low-
Mass Halos

The GC in Eri II provides a useful data point for constraining various
theoretical models of GC formation in the low-mass galaxy regime. For example,
Kruijssen (2019) invoke the metallicity floor of GCs (i.e., the idea that
there are few known GC with [Fe/H] $<-2$ to posit the properties (e.g., mass
of galaxy and GC) and formation epoch of GCs that can survive a Hubble time.
Eri II and its GC are both lower in mass and metallicity than predicted by
this models and formally fall into the ‘zone of avoidance’ in which such a GC
is predicted to be rare. In this regime of predicted spare populations,
Kruijssen et al. (2019) stress the importance of stochasticity, which means
that although Eri II is rare, it is fully consistent with these model
predictions.

GCs have been considered as contributors to cosmic reionization (e.g.,
Renzini, 2017; Boylan-Kolchin, 2018b). For example, models presented in
Boylan-Kolchin (2018a) suggest that even with modest values of birth-to-
present day masses (i.e., $<5$) GCs in faint galaxies were likely to be
significant contributors to the faint-end of the ultra-violet (UV) luminosity
function ($z\sim 4-10$). Our characterization of Eri II’s GC is compatible
with this scenario. The formation epoch of the GC ($z\sim 8$) is within the
canonical reionization epoch, and the birth-to-present day mass ratio of 4 is
also within range of these models. The clear caveat is that Eri II is only a
single system and a key component of Boylan-Kolchin (2018a)’s models are a
solid accounting of the GC number density, which our data do not help
constrain.

However, there is no consensus that GCs do contribute to reionization. For
example, GC formation models presented in Choksi et al. (2018) and El-Badry et
al. (2019) predict that the bulk of GC formation took place after the Universe
was reionized (i.e., $z\lesssim 5$). In these models, the GC in Eri II is
simply a rare occurrence at at high-z.

It is challenging to provide useful constraints on GC formation models from a
single cluster in a single galaxy. However, the combination of this type of
archaeological analysis with the increased discovery rate of GCs in the local
and early Universe (e.g., Carlsten et al., 2022; Vanzella et al., 2022a;
Pascale et al., 2023; Vanzella et al., 2022b; Welch et al., 2023), should help
to increase empirical constraints on metal-poor GC formation across the galaxy
mass spectrum and cosmic time.

###  5.3 Present Day Mass and Mass-to-light Ratio

Several studies exploit the fragility of the low-mass GC in Eri II to
constrain the shape of the inner dark matter profile of Eri II. Conclusions
remain mixed; some studies suggest that a dark matter core is necessary for
the long-term survival of Eri II’s GC (e.g., Amorisco, 2017; Contenta et al.,
2018; Orkney et al., 2022), others favor a cuspy dark matter profile (e.g.,
Zoutendijk et al., 2021).

One key quantity in these models is the present day mass-to-light ratio of the
GC. From the CMD-based GC mass, we can estimate the present-day stellar mass
by accounting for stellar evolution effects. The synthesis models of Conroy et
al. (2009) stellar evolution mass loss to be $\sim$40%. Applying this to our
CMD-based mass yields a present-day GC stellar mass of $M_{\star,z=0}\sim
5.2\times 10^{3}$ $M_{\odot}$. At $z=0$, the GC is only $\sim 4$% of the
galaxy’s present day stellar mass, whereas at birth it was $\sim 10$%. The
latter is a more typical value for GCs in other galaxies at $z=0$ (e.g.,
Forbes et al., 2018).

Using our present day stellar mass of the GC, we find $M/L_{V}\sim 2.5$ for
$M_{V}=-3.5$ mag (Crnojević et al., 2016) and $M/L_{V}\sim 5$ for $M_{V}=-2.7$
(Simon et al., 2021). The former is within the scatter of the canonical mass-
to-light ratios expected for an old (i.e., $M/L_{V}=2$, $\sigma\sim$1; e.g.,
Conroy et al., 2009; Martin et al., 2008), low-mass stellar population,
whereas the latter is marginally high.

As a check on the GC luminosity, we sum the fluxes of all GC stars within
$2r_{h}$ in F606W down to $m_{F606W}=28$, which our ASTs indicate is 100%
complete. We then transform F606W to V-band (Saha et al., 2011) and subtract
the light contribution from 30% of the GC stars that CMD modeling indicates
are field star contaminants and the handful of MW foreground stars expected in
the small HST field of view. CMD simulations indicate that unseen lower mass
stars contribute 15% more light than we measure. In total, we find
$M_{V,GC}\sim-3.4$ within $2r_{h}$, in good agreement with Crnojević et al.
(2016), and yields a reasonable $M/L_{V}$.

For completeness, we also compute the mass-to-light ratio of the field
population. Assuming that 40% of the CMD-based stellar mass of the galaxy is
lost to stellar evolution effects, we find a present day stellar mass
$M_{\star,z=0}\approx 1.5\times 10^{5}$ $M_{\odot}$ and
$M_{\star,z=0}/L_{V}\approx 2.5$ for a galaxy luminosity of $M_{V}=-7.1$
(Crnojević et al., 2016).

###  5.4 Comparison to GCs in the Local Group

The GC in Eri II is one of the oldest, most metal-poor, and lowest-mass GCs
known in nearby dwarf galaxies (e.g., Forbes et al. 2018; Beasley et al. 2019;
Larsen et al. 2020, 2022). GCs in more luminous dwarfs such as Fornax, NGC
6822, NGC 147, NGC 185, and WLM, tend to be more massive, metal-rich, and/or
younger. Peg DIG is the lowest-mass ($M_{\star,z=0}\sim 10^{7}$ $M_{\odot}$)
gas-rich dwarf galaxy to host a GC. But the GC is younger, more metal-rich,
and 10$\times$ more massive than Eri II’s GC (Cole et al., 2017). A small
number of faint GCs have been identified in dwarf galaxies of moderate
luminosity. For example, Cusano et al. (2016) report a putative faint GC
($M_{\rm V}=-4.6$) in And XXV ($M_{\rm V}=-9.1$) and Caldwell et al. (2017)
re-discovered a faint GC ($M_{\rm V}=-3.5$; comparable to the GC in Eri II) in
And I ($M_{\rm V}=-11.7$). The M31 galaxy and GC luminosities have been
updated with the RR Lyrae distances from Savino et al. (2022).

Even within more massive galaxies, only a handful of more metal-poor GCs are
known. For example, Larsen et al. (2020) report an M31 GC with
$\langle$[Fe/H]$\rangle=-2.91\pm 0.04$ dex, with a dynamical mass of $\sim
10^{6}$ $M_{\odot}$. Though comparable in metallicity to Eri II’s GC, this M31
GC is $\sim 100$ times more massive than Eri II’s and is more luminous than
the entirety of Eri II as a galaxy, suggesting it originated in a different
environment (i.e., this single GC may have more stellar mass than all of Eri
II as a galaxy).

More recently, Martin et al. (2022) identified a disrupted GC in the MW with
$\langle$[Fe/H]$\rangle=-3.38\pm 0.2$ dex with a stellar mass of at least
$M_{\star,z=0}\sim 8\times 10^{3}$$M_{\odot}$. This is more metal-poor than
Eri II’s GC, but may have a comparable stellar mass, though Martin et al.
(2022) note severval challenges in accurately measuring the mass of a
disrupted object. Regardless, the mass and metallicity suggest this disrupted
GC could have originated in an accreted dwarf galaxy. However, backing out the
progenitor characteristics of a faint substructures in the MW halo is
extremely challenging, even with full phase space information available (e.g.,
Brauer et al., 2022), highlighting the particular importance of Eri II for
connecting metal-poor GCs to their host galaxies.

###  5.5 Eri II and its GC in the high-redshift Universe

The sensitivity and resolving power of HST and the James Webb Space Telescope
have the potential to directly capture the birth of GCs in the early Universe.
Recently, several candidate proto-GCs at high redshift have been recently
reported in the literature. For example, Vanzella et al. (2019) use
gravitational lensing to identify a putative proto-GC forming in situ in a
dwarf galaxy at $z\sim 6$. The proto-GC is reported to have $r_{h}\lesssim 13$
pc, $M_{\star}\sim 10^{6}$ $M_{\odot}$, $\Sigma_{\rm SFR}\gtrsim 10^{2.5}$
$M_{\odot}$ yr-1 kpc-2 and $M_{\rm UV}(z\sim 6)\sim-15.6$. The host is
reported as dwarf galaxy with $r_{h}\sim 450$ pc, $M_{\star}\sim 4\times
10^{8}$ $M_{\odot}$, $\Sigma_{\rm SFR}\gtrsim 10^{1.5}$ yr-1 kpc-2 and $M_{\rm
UV}(z\sim 6)\sim-17.3$. In many cases, these quantities reflects best
estimates given the coarse constraints of the data (objects are spatially
unresolved meaning clusters and galaxies can be hard to distinguish; e.g.,
Bouwens et al. 2022).

For comparison, we can make estimates for the high-z properties of Eri II and
its GC. For the field population, our CMD modeling (plus fact of $\sim 2$
correction for surveyed area) yields a galaxy-wide SFR of $\dot{M}_{\star}\sim
1.3\pm 0.5\times 10^{-3}$ $M_{\odot}$ yr-1, averaged over our time resolution
of 300 Myr. We cannot rule out a shorter duration of star formation, which
would increase $\dot{M}_{\star}$. For example, Gallart et al. (2021) posit
that the duration of star formation could have been as short as 100 Myr, while
analytic chemical evolution models suggest an Eri II-like MDF could be the
result of Type II enrichment in as a little at $\sim 50$ Myr (e.g., Weinberg
et al., 2017). These limits suggest a range of SFRs from $\dot{M}_{\star}\sim
1-7\times 10^{-3}$ $M_{\odot}$ yr-1 which are similar to what is found in
star-forming dwarf galaxies in the very local Universe at the present day
(e.g., Lee et al., 2011; Johnson et al., 2013). Cosmological simulations
suggest galaxies in the same mass range as Eri II should have very bursty SFHs
with rapid fluctuations in SF on the order of $10-20$ Myr (e.g., Stinson et
al., 2007; Governato et al., 2010; El-Badry et al., 2016; Kimm et al., 2016;
Fitts et al., 2017; Wheeler et al., 2019; Applebaum et al., 2021; Sameie et
al., 2022). Unfortunately, our data cannot resolve such fine detail in the
SFHs.

These same cosmological simulations suggest that star formation is generally
concentrated in the central regions of low-mass galaxies at early times.
Motivated by these simulations, we assume that star formation occurred over a
region 50-500 pc in radius (e.g., Fitts et al., 2017; Applebaum et al., 2021),
from which we estimate an average star formation rate surface density of
$\Sigma_{\rm SFR}\sim 0.001-0.9$ $M_{\odot}$ yr-1 kpc-2.

To facilitate further comparison to high-z observations of proto-GC
candidates, we reconstruct the rest frame UV luminosity of Eri II’s field
population using its SFH and well-established techniques from the literature
(e.g., Weisz et al., 2014b; Boylan-Kolchin et al., 2015). During its initial
epoch of star formation, we find $M_{\rm UV}\sim-9\pm 2.5$, assuming SF
durations ranging from $50-300$ Myr with stochastic bursts of $\sim 20$ Myr in
duration.

We can make similar estimate for UV luminosity of the GC following Boylan-
Kolchin (2018c). A formation duration of 1-5 Myr for the GC yields an
$\dot{M\star}\sim 1-5\times 10^{-3}$ $M_{\odot}$ yr-1. A GC birth radius of
1-10 pc gives $\Sigma_{\rm SFR}\sim 5-500$ $M_{\odot}$ yr-1 kpc-2. Finally,
assuming the GC is an single stellar population, the restframe UV luminosity
of the GC at birth was $M_{\rm UV}\sim-12.5$ mag, which faded to $M_{\rm
UV}\sim-8.5$ mag 100 Myr after it formed.

Eri II and its GC are fainter than what can be directly observed in the early
Universe. For example, they are orders of magnitude less massive and luminous
than what Vanzella et al. (2019), which is likely observing a MW progenitor
(e.g., Boylan-Kolchin et al., 2015), where Eri II is clearly a dwarf galaxy at
all redshifts. Through lensing with magnifications of $\mu\lesssim 30$,
Bouwens et al. (2022) reports the detection of objects as faint as
$M_{UV}\sim-13$ at $z\sim 6-8$, which approaches the maximum UV luminosity of
the GC in Eri II.

At birth, the GC was $\sim 9$% of the stellar mass of the galaxy. If the
cluster formed before the majority of the field population, as often appears
the case in the recent simulations of Sameie et al. (2022), then it may have a
higher fractional mass. Our data cannot rule out the possibility of non-
concurrent GC an galaxy formation. On the other hand, if the amount of
dynamical mass loss was lower than our estimate, the cluster birth mass would
be lower, and it would constitute a lower fraction of the system’s total
stellar mass. The $\sim 10$% ratio of GC-to-galaxy stellar mass at birth is
similar to what is known in the local Universe, i.e., present day mass ratios
(e.g., Forbes et al., 2018).

In a broader context, Eri II appears to be a fairly common type of high-z
galaxy based on its location on the high-z UV luminosity function (UVLF; e.g.,
Atek et al. 2018, Bouwens et al. 2022). Given its modest luminosity, it likely
resides near the peak of the UVLF, assuming the high-z galaxy UVLF turns over
as many studies argue it should (e.g., O’Shea et al., 2015; Gnedin, 2016; Yung
et al., 2019). The exact luminosity of the putative turnover is not yet known
as it appears to be fainter than the limits of the deepest HST imaging (e.g.,
Atek et al., 2018; Bouwens et al., 2022).

###  5.6 The Age of Eri II in a cosmological context

While our archaeological approach to characterizing GCs provides access to
low-mass galaxy and clusters not detectable virutally anywhere else in the
Universe, there are some important caveats. First, even with the exquisitely
deep HST imaging, our time resolving power is limited to a few hundred Myr.
Thus, we cannot discern between a short formation epoch (i.e., everything
formed within $\sim 10$ Myr) vs a more protracted episode. This results in
large uncertainties on certain quantities of interest such as $\Sigma_{\rm
SFR}$ (Table 1). Higher S/N imaging would improve the time resolution, but
ultimately, the increased time resolution is limited by the small number of
stars on the CMD. Incorporating other information (e.g., external knowledge of
the metallicity distribution function; de Boer et al. 2012, Dolphin 2016)
should also improve the time resolution of the SFH and cluster age.

A second limitation is due to uncertainties in the stellar models themselves
(e.g., Weisz et al., 2011; Dolphin, 2013). Though the multiple models we
tested in this paper yield consistent ages, even slight differences of a few
hundred Myr can translate to large uncertainties in redshift space. Though our
knowledge of stellar physics continues to improve, model systematics remain
important to precisely mapping stellar ages to redshifts.

Finally, note that interpreting the age of Eri II, along with other UFDs in
the LG is complicated in light of uncertainties in our knowledge of cosmology
due to the Hubble tension. For example, new physics solutions, such as some
variations of early dark energy (EDE) result in Universe that is substantially
younger ($\sim 12.8$ Gyr; e.g., Smith et al. 2022) compared to its age from
vanilla $\Lambda$CDM and a Planck cosmology ($\sim 13.8$ Gyr; e.g., Planck
Collaboration et al. 2020). However, Eri II’s formation epoch ($\sim 13.5$
Gyr) is older than the age of an EDE Universe. Though multiple stellar models
(and CMD fitting techniques) find consistently old age for Eri II, additional
uncertainties in stellar physics (e.g., convection) may also need to be
considered determining absolute ages (e.g., Chaboyer et al., 1996, 2017;
O’Malley et al., 2017; Joyce et al., 2022). Alternatively, since virtually all
known UFDs around the MW appear to be older than $\sim 13$ Gyr (e.g., Brown et
al., 2014; Weisz et al., 2014a; Sacchi et al., 2021), it may indicate that
physics-based solutions to the Hubble tension resulting in ages younger than
$\sim$13 Gyr may not be consistent with all available measurements of the
Universe’s age. Similarly, Boylan-Kolchin & Weisz (2021) note that
uncertainties in the cosmological model due to the Hubble tension introduces a
minimum uncertainty of 5% in the age-redshift mapping, which sets a floor on
any comparison between ages of stars and galaxies and ages inferred from
redshifts in the early Universe.

We thank the anonymous referee for a constructive and thorough review. The
authors thank Nick Choksi, Mike Boylan-Kolchin, Kareem El-Badry, and Diederik
Kruijssen for helpful comments on early drafts. Support for this work was
provided by NASA through grants HST-GO-13768, HST-GO-15746, HST-GO-15901, HST-
GO-15902, and HST-AR-16159 from the Space Telescope Science Institute, which
is operated by AURA, Inc., under NASA contract NAS5-26555. This research has
made use of NASA’s Astrophysics Data System Bibliographic Services

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