# Low-luminosity galaxies in the early universe have observed sizes similar to
star cluster complexes

R.J. Bouwens11affiliation: Leiden Observatory, Leiden University, NL-2300 RA
Leiden, Netherlands , G.D. Illingworth22affiliation: UCO/Lick Observatory,
University of California, Santa Cruz, CA 95064 , P.G. van Dokkum33affiliation:
Department of Astronomy, Yale University, New Haven, CT 06520 , B.
Ribeiro11affiliation: Leiden Observatory, Leiden University, NL-2300 RA
Leiden, Netherlands , P.A. Oesch44affiliation: Observatoire de Genève, 1290
Versoix, Switzerland , M. Stefanon11affiliation: Leiden Observatory, Leiden
University, NL-2300 RA Leiden, Netherlands

###### Abstract

We compare the sizes and luminosities of faint $z=6$-8 galaxies magnified by
the Hubble Frontier Fields (HFF) clusters with star-forming regions, as well
as more evolved objects, in the nearby universe. Our high-redshift comparison
sample includes 333 $z=6$-8 galaxies, for which size measurements were made as
part of a companion study where lensing magnifications were estimated from
various public models. Accurate size measurements for these sources are
complicated by the lens model uncertainties, but other results and arguments
suggest that faint galaxies are small, as discussed in a companion study. The
measured sizes for sources in our comparison sample range from $<$50 pc to
$\sim$500 pc. For many of the lowest luminosity sources, extremely small sizes
are inferred, reaching individual sizes as small as 10-30 pc, with several
sources in the 10-15 pc range with our conservative magnification limits. The
sizes and luminosities are similar to those of single star cluster complexes
like 30 Doradus in the lower-redshift universe and – in a few cases – super
star clusters. The identification of these compact, faint star-forming sources
in the $z\sim 6$-8 universe also allows us to set upper limits on the proto-
globular cluster LF at $z\sim 6$. By comparisons of the counts and sizes with
recent models, we rule out (with some caveats) proto-globular cluster
formation scenarios favoring substantial ($\xi=10$) post-formation mass loss
and set useful upper limits on others. Our size results suggest we may be very
close to discovering a bona-fide population of forming globular clusters at
high redshift.

##  1\. Introduction

There are a wide variety of evolved stellar systems in the nearby universe
(Norris et al. 2014), from globular clusters (Brodie & Strader 2006; Kruijssen
2014; Renzini et al. 2015) to compact elliptical galaxies (e.g., Faber 1973),
ultra-faint dwarfs (e.g., Simon & Geha 2007), and ultra-diffuse spheroids
(e.g., van Dokkum et al. 2015), each of which presumably has its own
characteristic formation pathway. The high stellar densities in many of these
systems in combination with their old ages (e.g., Forbes & Bridges 2010)
suggest that the majority of their star formation occured at $z\gtrsim 1.5$
when the gas densities in the universe were in general much higher.

One potentially promising way forward for investigating the formation of these
local systems is by obtaining a sensitive, high-resolution view of the distant
universe. Fortunately, such observations can be obtained by combining the
power of long exposures with the Hubble Space Telescope with the magnifying
effect of gravitational lensing, as recently implemented in the ambitious
Hubble Frontier Fields (HFF) program (Coe et al. 2015; Lotz et al. 2017). Such
sensitive observations allow us to probe to very low luminosities, as is
likely necessary to detect many of the progenitors of local systems. The high
lensing magnifications from massive galaxy clusters stretch many galaxies by
substantial factors, allowing them to be studied at very high spatial
resolution. As we discussed in Bouwens et al. (2021a), this stretching can
reliably be estimated up to linear magnifications of $\sim$30$\times$ (or
total magnification factors of $\sim$50$\times$: see also Bouwens et al. 2017b
where similar, though smaller limits were presented with the then-current
models, and Meneghetti et al. 2017).

The promise of the HFF program to examine faint sources in the distant
universe at extremely high resolution was bourne out by the studies of
Kawamata et al. (2015) and Bouwens et al. (2017a), who identified small $<$40
pc star-forming sources in the $z\sim 6$-9 universe and remarked that the size
of such systems approached that of giant molecular clouds or star-forming
clumps in the local universe (Kennicutt et al. 2003). Vanzella et al. (2017a)
speculated that a few of these small star-forming sources behind the HFF
clusters could correspond to proto-globular clusters. They supported their
hypothesis with MUSE spectroscopy of the sources, noting small probable
dynamical masses (due to the small measured velocity dispersions) and probable
physical associations with brighter neighbors (due to their similar
redshifts). In two follow-up studies, Vanzella et al. (2019, 2020) identified
two candidate proto-globular cluster systems with inferred sizes $<$13 pc and
5-10 pc, which is very close to the expected sizes of young massive clusters
(Kruijssen 2014).

We can now build on this recent work, using large catalogs of $z\sim 6$-9
galaxies constructed from the deep HST observations over the six HFF clusters.
Lensing models have also improved, as a result of a larger number of
approaches and improved techniques. The median of these models is expected to
be more robust than any individual model and the scatter provides an estimate
of the uncertainties. Thanks to the improvements in the lensing models and the
large number of clusters targeted, it has become possible to measure the sizes
for large samples of faint $z=6$-9 galaxies and to very small sizes. Kawamata
et al. (2018) report results for 181 $z=6$-9 galaxies, while Bouwens et al.
(2021a) report results for 333 $z=6$-8 galaxies. Both studies report results
on all six HFF clusters and find substantial numbers ($>$30) of $z=6$-9 star-
forming sources with plausible half-light radii $\leq$50 pc. This remains the
case even when we adopt the conservative approach discussed in Bouwens et al.
(2021a) to limit linear and total magnification factors to 30 and 50,
respectively.

Given the small inferred sizes of the fainter lensed sources identified by
Kawamata et al. (2018) and Bouwens et al. (2021a), it is interesting to place
these sources in the context of various stellar systems that they may evolve
into today, as well as other small star-forming systems like star clusters or
cluster complexes. An initial look at such comparisons were already executed
in an earlier unpublished study by our group (Bouwens et al. 2017) and also by
Kikuchihara et al. (2020). An important early inference from these studies was
that lensed $z=6$-8 galaxies have sizes and masses that appear to lie in the
range of $\sim$50-500 pc and 107 to 108 $M_{\odot}$, lying somewhere between
ultra compact dwarfs/globular clusters and compact elliptical galaxies in
size/mass space.

The purpose of the present work is to revisit these comparisons, utilizing the
large selection of $z=6$-8 galaxies with size measurements from Bouwens et al.
(2021a) while leveraging a variety of new results – both from simulations
(Ploeckinger et al. 2019; Pfeffer et al. 2019) and from local observations
(Zick et al. 2018) – to allow for an improved interpretation of the
observational results. Two aspects are of particular interest: (1) comparing
the sizes of sources in these selections with star cluster complexes like 30
Doradus in the LMC and (2) using the Bouwens et al. (2021a) searches to set
constraints on proto-globular cluster formation models such as described by
Boylan-Kolchin (2017), Pfeffer et al. (2019), J. Pfeffer et al. (2021, in
prep), and Pozzetti et al. (2019). Throughout the paper, we assume a standard
“concordance” cosmology with $H_{0}=70$ km s-1 Mpc-1, $\Omega_{\rm m}=0.3$ and
$\Omega_{\Lambda}=0.7$, which is in good agreement with recent cosmological
constraints (Planck Collaboration et al. 2016). Magnitudes are in the AB
system (Oke & Gunn 1983).

Table 1Catalog of Tiny Star-Forming Sourcesa,ba,bfootnotemark:  at $z\sim 6$-8

ID | R.A. | Decl |  $M_{UV}$ |  $\mu$ccMedian magnification factors (and $1\sigma$ uncertainties) derived weighting equally the latest public version 3/4 parametric models from each lensing methodology (§4.1 from Bouwens et al. 2021a). |  $\mu_{1D}$dd$\mu_{1D}$ are the median one-dimensional magnification factors (and $1\sigma$ uncertainties) along the major shear axis $\mu^{1/2}S^{1/2}$ weighting equally the parametric models from each lensing methodology. This is the same quantity as $\mu_{tang}$ reported by Vanzella et al. (2017a). |  $r_{e}$ (pc)eeInferred half-light radius in physical units. The quoted uncertainties include both uncertainties in the spatial fits and uncertainties in the lensing models. These half-light radii are derived by fitting a Sersic profile to the imaging observations while fixing the Sersic parameter $n$ to 1, as discussed in Bouwens et al. (2021a).
---|---|---|---|---|---|---
A2744Z-4222023578 | 00:14:22.20 |  $-$30:23:57.8 |  $-$15.6${}_{-1.2}^{+0.7}$ | 50.1${}_{-33.6}^{+41.4}$ | 8.5${}_{-3.8}^{+4.4}$ | 31${}_{-13}^{+31}$
A2744Z-4222023578 | 00:14:22.20 |  $-$30:23:57.8 |  $-$15.6 | 50.0ffTotal magnification factor $\mu$ set to 50.0 where the median magnification models begin losing their predictive power (Bouwens et al. 2021a). Because of this use of a limit on the magnification factor, no uncertainty is quoted on the inferred $UV$ luminosity $M_{UV}$. Nevertheless, a useful indication of the uncertainties can be seen, if we take the median $1\sigma$ uncertainties in the quoted magnification factors before imposing a limit, which would give a value of 0.59 dex. | 8.5${}_{-3.8}^{+4.4}$ | 31${}_{-13}^{+31}$
A2744Y-4204124034* | 00:14:20.41 |  $-$30:24:03.4 |  $-$14.0${}_{-1.6}^{+1.2}$ | 75.0${}_{-58.0}^{+151.2}$ | 15.1${}_{-9.5}^{+24.0}$ | 14${}_{-9}^{+32}$
A2744Y-4204124034* | 00:14:20.41 |  $-$30:24:03.4 |  $-$14.4 | 50.0ffTotal magnification factor $\mu$ set to 50.0 where the median magnification models begin losing their predictive power (Bouwens et al. 2021a). Because of this use of a limit on the magnification factor, no uncertainty is quoted on the inferred $UV$ luminosity $M_{UV}$. Nevertheless, a useful indication of the uncertainties can be seen, if we take the median $1\sigma$ uncertainties in the quoted magnification factors before imposing a limit, which would give a value of 0.59 dex. | 15.1${}_{-9.5}^{+24.0}$ | 14${}_{-9}^{+32}$
M0416I-6095704260* | 04:16:09.57 |  $-$24:04:26.0 |  $-$15.0${}_{-0.2}^{+0.3}$ | 12.5${}_{-2.5}^{+3.3}$ | 8.0${}_{-2.0}^{+2.5}$ | 28${}_{-9}^{+13}$
M0416I-6118103480$\ddagger$$\ddagger$Source also has an inferred size of $\leq$40 pc in the Kawamata et al. (2018) catalog. | 04:16:11.81 |  $-$24:03:48.1 |  $-$15.0${}_{-1.0}^{+1.3}$ | 33.6${}_{-19.9}^{+81.6}$ | 24.5${}_{-15.2}^{+85.1}$ | 16${}_{-13}^{+43}$
M0416I-6130803432*$\ddagger$$\ddagger$Source also has an inferred size of $\leq$40 pc in the Kawamata et al. (2018) catalog. | 04:16:13.08 |  $-$24:03:43.2 |  $-$17.3${}_{-0.2}^{+0.2}$ | 3.5${}_{-0.6}^{+0.7}$ | 3.4${}_{-0.6}^{+0.3}$ | 38${}_{-4}^{+8}$
M0416I-6115434445$\dagger$,$\ddagger$$\dagger$,$\ddagger$footnotemark:  | 04:16:11.54 |  $-$24:03:44.5 |  $-$14.5${}_{-1.0}^{+0.9}$ | 43.0${}_{-25.4}^{+54.2}$ | 31.0${}_{-18.4}^{+48.0}$ | 21${}_{-14}^{+44}$
M0416I-6115434445$\dagger$,$\ddagger$$\dagger$,$\ddagger$footnotemark:  | 04:16:11.54 |  $-$24:03:44.5 |  $-$14.5${}_{-1.0}^{+0.9}$ | 43.0${}_{-25.4}^{+54.2}$ | 30.0ggLinear magnification factor $\mu_{1D}$ set to 30.0 where the median magnification models begin losing their predictive power (Bouwens et al. 2021a). Because of this use of a limit on the magnification factor, no uncertainty is quoted on the inferred size $r_{e}$. Nevertheless, a useful indication of the uncertainties can be seen, if we take the median $1\sigma$ uncertainties in the quoted magnification factors before imposing a limit, which would give a value of 0.52 dex. | 22
M0416I-6106703335* | 04:16:10.67 |  $-$24:03:33.5 |  $-$16.3${}_{-0.2}^{+0.3}$ | 11.1${}_{-2.0}^{+3.7}$ | 8.8${}_{-0.7}^{+3.1}$ | 33${}_{-11}^{+10}$
M0416Z-6114803434 | 04:16:11.48 |  $-$24:03:43.5 |  $-$17.0${}_{-0.5}^{+0.3}$ | 21.3${}_{-7.6}^{+6.3}$ | 14.9${}_{-4.8}^{+7.8}$ | 38${}_{-16}^{+26}$
M0416Y-6088104378* | 04:16:08.81 |  $-$24:04:37.9 |  $-$13.4${}_{-1.1}^{+1.0}$ | 62.1${}_{-39.5}^{+90.8}$ | 38.3${}_{-27.0}^{+45.7}$ | 11${}_{-7}^{+33}$
M0416Y-6088104378* | 04:16:08.81 |  $-$24:04:37.9 |  $-$13.7 | 50.0ffTotal magnification factor $\mu$ set to 50.0 where the median magnification models begin losing their predictive power (Bouwens et al. 2021a). Because of this use of a limit on the magnification factor, no uncertainty is quoted on the inferred $UV$ luminosity $M_{UV}$. Nevertheless, a useful indication of the uncertainties can be seen, if we take the median $1\sigma$ uncertainties in the quoted magnification factors before imposing a limit, which would give a value of 0.59 dex. | 30.0ggLinear magnification factor $\mu_{1D}$ set to 30.0 where the median magnification models begin losing their predictive power (Bouwens et al. 2021a). Because of this use of a limit on the magnification factor, no uncertainty is quoted on the inferred size $r_{e}$. Nevertheless, a useful indication of the uncertainties can be seen, if we take the median $1\sigma$ uncertainties in the quoted magnification factors before imposing a limit, which would give a value of 0.52 dex. | 14
M0717Z-7354743496 | 07:17:35.47 | 37:43:49.7 |  $-$13.9${}_{-1.4}^{+1.6}$ | 57.7${}_{-42.1}^{+205.4}$ | 46.8${}_{-34.0}^{+142.4}$ | 27${}_{-21}^{+103}$
M0717Z-7354743496 | 07:17:35.47 | 37:43:49.7 |  $-$14.0 | 50.0ffTotal magnification factor $\mu$ set to 50.0 where the median magnification models begin losing their predictive power (Bouwens et al. 2021a). Because of this use of a limit on the magnification factor, no uncertainty is quoted on the inferred $UV$ luminosity $M_{UV}$. Nevertheless, a useful indication of the uncertainties can be seen, if we take the median $1\sigma$ uncertainties in the quoted magnification factors before imposing a limit, which would give a value of 0.59 dex. | 30.0ggLinear magnification factor $\mu_{1D}$ set to 30.0 where the median magnification models begin losing their predictive power (Bouwens et al. 2021a). Because of this use of a limit on the magnification factor, no uncertainty is quoted on the inferred size $r_{e}$. Nevertheless, a useful indication of the uncertainties can be seen, if we take the median $1\sigma$ uncertainties in the quoted magnification factors before imposing a limit, which would give a value of 0.52 dex. | 42
M0717I-7374244282 | 07:17:37.42 | 37:44:28.2 |  $-$15.0${}_{-0.8}^{+0.8}$ | 9.6${}_{-5.0}^{+10.8}$ | 3.6${}_{-0.9}^{+1.5}$ | 34${}_{-12}^{+18}$
M0717Z-7390844017$\ddagger$$\ddagger$Source also has an inferred size of $\leq$40 pc in the Kawamata et al. (2018) catalog. | 07:17:39.08 | 37:44:01.7 |  $-$17.1${}_{-1.8}^{+0.4}$ | 21.8${}_{-17.8}^{+10.7}$ | 17.7${}_{-13.3}^{+14.4}$ | 24${}_{-12}^{+85}$
M0717Z-7401344384 | 07:17:40.13 | 37:44:38.5 |  $-$16.1${}_{-0.7}^{+0.0}$ | 8.3${}_{-3.9}^{+0.3}$ | 6.4${}_{-2.3}^{+1.5}$ | 39${}_{-9}^{+24}$
M0717Z-7311744437* | 07:17:31.17 | 37:44:43.8 |  $-$16.0${}_{-0.4}^{+0.8}$ | 8.2${}_{-2.6}^{+8.8}$ | 4.0${}_{-0.7}^{+3.1}$ | 27${}_{-14}^{+15}$
M0717Y-7336744331 | 07:17:33.67 | 37:44:33.1 |  $-$14.4${}_{-1.3}^{+2.3}$ | 43.1${}_{-29.9}^{+312.6}$ | 7.6${}_{-3.7}^{+40.9}$ | 31${}_{-26}^{+58}$
M0717Y-7329744137 | 07:17:32.97 | 37:44:13.7 |  $-$15.3${}_{-0.6}^{+0.9}$ | 13.7${}_{-5.9}^{+16.8}$ | 7.5${}_{-3.7}^{+8.6}$ | 34${}_{-20}^{+48}$
M1149I-9384023344 | 11:49:38.40 | 22:23:34.5 |  $-$13.0${}_{-2.4}^{+2.6}$ | 124.7${}_{-111.1}^{+1226.2}$ | 90.7${}_{-81.8}^{+949.9}$ | 26${}_{-24}^{+342}$
M1149I-9384023344 | 11:49:38.40 | 22:23:34.5 |  $-$14.0 | 50.0ffTotal magnification factor $\mu$ set to 50.0 where the median magnification models begin losing their predictive power (Bouwens et al. 2021a). Because of this use of a limit on the magnification factor, no uncertainty is quoted on the inferred $UV$ luminosity $M_{UV}$. Nevertheless, a useful indication of the uncertainties can be seen, if we take the median $1\sigma$ uncertainties in the quoted magnification factors before imposing a limit, which would give a value of 0.59 dex. | 30.0ggLinear magnification factor $\mu_{1D}$ set to 30.0 where the median magnification models begin losing their predictive power (Bouwens et al. 2021a). Because of this use of a limit on the magnification factor, no uncertainty is quoted on the inferred size $r_{e}$. Nevertheless, a useful indication of the uncertainties can be seen, if we take the median $1\sigma$ uncertainties in the quoted magnification factors before imposing a limit, which would give a value of 0.52 dex. | 79
M1149Y-9377423253* | 11:49:37.74 | 22:23:25.4 |  $-$14.9${}_{-0.3}^{+0.1}$ | 16.2${}_{-4.1}^{+2.0}$ | 7.9${}_{-3.9}^{+1.3}$ | 19${}_{-4}^{+20}$
A370I-9544334397* | 02:39:54.43 |  $-$1:34:39.7 |  $-$12.7${}_{-1.2}^{+3.5}$ | 228.9${}_{-151.9}^{+5472.8}$ | 23.1${}_{-11.0}^{+484.7}$ | 10${}_{-10}^{+21}$
A370I-9544334397* | 02:39:54.43 |  $-$1:34:39.7 |  $-$14.4 | 50.0ffTotal magnification factor $\mu$ set to 50.0 where the median magnification models begin losing their predictive power (Bouwens et al. 2021a). Because of this use of a limit on the magnification factor, no uncertainty is quoted on the inferred $UV$ luminosity $M_{UV}$. Nevertheless, a useful indication of the uncertainties can be seen, if we take the median $1\sigma$ uncertainties in the quoted magnification factors before imposing a limit, which would give a value of 0.59 dex. | 23.1${}_{-11.0}^{+484.7}$ | 10${}_{-10}^{+21}$
A370Z-9535934395 | 02:39:53.59 |  $-$1:34:39.6 |  $-$15.5${}_{-0.1}^{+2.1}$ | 10.9${}_{-1.2}^{+65.5}$ | 7.7${}_{-0.8}^{+35.5}$ | 31${}_{-26}^{+28}$
AS1063I-8439532143 | 22:48:43.95 |  $-$44:32:14.3 |  $-$13.5${}_{-2.8}^{+0.6}$ | 42.3${}_{-39.0}^{+28.4}$ | 19.0${}_{-16.3}^{+12.6}$ | 33${}_{-15}^{+219}$
AS1063I-8469531055 | 22:48:46.95 |  $-$44:31:05.5 |  $-$14.9${}_{-1.8}^{+0.2}$ | 12.9${}_{-10.4}^{+2.3}$ | 10.3${}_{-7.9}^{+1.8}$ | 35${}_{-7}^{+115}$
AS1063Y-8436531406* | 22:48:43.65 |  $-$44:31:40.6 |  $-$12.5${}_{-1.6}^{+1.0}$ | 175.3${}_{-134.0}^{+257.9}$ | 77.7${}_{-57.8}^{+81.7}$ | 9${}_{-5}^{+30}$
AS1063Y-8436531406* | 22:48:43.65 |  $-$44:31:40.6 |  $-$13.9 | 50.0ffTotal magnification factor $\mu$ set to 50.0 where the median magnification models begin losing their predictive power (Bouwens et al. 2021a). Because of this use of a limit on the magnification factor, no uncertainty is quoted on the inferred $UV$ luminosity $M_{UV}$. Nevertheless, a useful indication of the uncertainties can be seen, if we take the median $1\sigma$ uncertainties in the quoted magnification factors before imposing a limit, which would give a value of 0.59 dex. | 30.0ggLinear magnification factor $\mu_{1D}$ set to 30.0 where the median magnification models begin losing their predictive power (Bouwens et al. 2021a). Because of this use of a limit on the magnification factor, no uncertainty is quoted on the inferred size $r_{e}$. Nevertheless, a useful indication of the uncertainties can be seen, if we take the median $1\sigma$ uncertainties in the quoted magnification factors before imposing a limit, which would give a value of 0.52 dex. | 23

**footnotetext: One of the sources where the upper $1\sigma$ limit on the
inferred half-light radius is less than 50 pc.

$\dagger$$\dagger$footnotetext: Tiny star-forming source also presented in
Vanzella et al. (2017a).

##  2\. High-Redshift Samples and Size Measurements

Here we provide a brief summary of the $z=6$-8 selections from Bouwens et al.
(2021a) that we utilize in interpreting their properties relative to various
star-forming or evolved sources in the local universe. For a more detailed
description of the procedure used in selecting sources or deriving
sizes/luminosities, the interested reader is referred to our companion paper
(Bouwens et al. 2021a).

###  2.1. Selection and Size Measurements for $z\sim 6$-8 Galaxies

Briefly, the $z\sim 6$-8 samples we utilize are selected from the v1.0
reductions of the HST data over all six HFF clusters. A combination of Lyman-
break and photometric-redshift selection criteria are applied in creating
source catalogs from the HST images. Bright foreground galaxies and also the
intracluster light were modeled and subtracted from the HST images, before
construction of the catalogs.

In deriving the luminosities and sizes of individual sources, source
magnifications $\mu$ and shear factors $S$ are estimated from all of the v3 or
v4 public parametric lensing models. The shear factor $S$ utilized here was
defined as

$S=\left\\{\begin{array}[]{lr}\frac{1-\kappa-\gamma}{1-\kappa+\gamma},&\text{for}\frac{1-\kappa-\gamma}{1-\kappa+\gamma}\geq
1\\\
\frac{1-\kappa+\gamma}{1-\kappa-\gamma},&\text{for}\frac{1-\kappa-\gamma}{1-\kappa+\gamma}<1\end{array}\right.$
(1)

(see Bouwens et al. 2017a and Bouwens et al. 2020). $\kappa$ is the
convergence and $\gamma$ is the shear. From the estimated magnifications $\mu$
and shear factors $S$, we can derive a median linear magnification along the
major and minor shear axes, i.e., $(\mu S)^{1/2}$ and $(\mu/S)^{1/2}$, which
we use in measuring the physical size of the source along both axes. We find
that these linear magnification factors can now be reliably estimated to
magnification factors of $\sim$30 (§4.2 from Bouwens et al. 2020) and possibly
to slightly larger factors using a median of the latest parametric lensing
models available for the HFF clusters (e.g., as in Livermore et al. 2017;
Bouwens et al. 2017b).

The fits to the sizes of individual sources are then performed using an
Markov-Chain Monte-Carlo procedure, where we take a model Sersic profile,
transform the profile according to the median magnification factors, convolve
the derived profile with the PSF, compare the result with the observations,
and compute a residual. We use our MCMC procedure to minimize the sum of the
square of the residuals.

A full compilation of the coordinates, luminosities, and sizes of the $z\sim
6$-8 galaxies Bouwens et al. (2021a) identify behind all six HFF clusters is
given in their Table 2. The uncertainties Bouwens et al. (2021a) compute on
the derived sizes include both the formal uncertainties on the size fits and
the $1\sigma$ error $\mu_{1D}$ computed based on the range in linear
magnifications predicted by the parametric lensing models. Table 1 of Bouwens
et al. (2021a) tabulates the sizes and luminosities of those sources with
linear $\mu_{1D}$ and total magnification factors $\mu$ $>$30 and $>$50,
respectively, a second time but setting the magnification factors to 30 and
50, respectively, to provide more conservative size and luminosity estimates,
given the breakdown in the predictive power of these models at very high
magnification factors. When adopting an upper limit of 30 and 50 on the linear
and total magnification factors, respectively, no uncertainties are quoted on
the sizes $r_{e}$ or luminosities $M_{UV}$, respectively, of sources.
Nevertheless, if we take the median $1\sigma$ scatter in the quoted
magnification factors before imposing these limits, the typical uncertainty
would be 0.52 and 0.59 dex, respectively. The total magnification $\mu$ and
linear magnification $\mu_{1D}$ are treated as independent for the purposes of
imposing the upper limits.

There are 23 objects from the Bouwens et al. (2021a) compilation with sizes
$<$40 pc, with the three smallest sources having sizes of 11${}_{-7}^{+33}$ pc
(M0416Y-6088104378), 11${}_{-7}^{+33}$ pc 10${}_{-10}^{+21}$
(A370I-9544334397), and 9${}_{-5}^{+30}$ pc (AS1063Y-8436531406). For
convenience, we include a list of these sources in Table 1. The uncertainties
in the measured sizes for these small sources are clearly substantial. Of the
23 sources in Table 1, only three have measured sizes less than 40 pc after
allowing for the positive $1\sigma$ fluctuations in the measured sizes. Nine
sources are within an upper bound of 50 pc (a slightly less stringent limit)
allowing for the $1\sigma$ uncertainties. These 9 sources are indicated in
Table 1 with a “*” to indicate that their small size measurements are more
robust. As in Table 2 of Bouwens et al. (2021a), we also include sources a
second time when their linear and total magnification factors exceed 30 and
50, respectively, but setting these factors to 30 and 50, respectively.

###  2.2. Comparison with Previous Compilations of Small Star-Forming Sources

Here we compare the sizes of small galaxies in our $z\sim 6$-8 sample to
similarly small ($\sim$10-100 pc) sources identified by Vanzella et al.
(2017a) and to new results recently obtained by Kawamata et al. (2018: which
are an update to the earlier Kawamata et al. 2015 results). In the companion
paper to this one (Bouwens et al. 2021a), we provided a similar comparison but
focusing on our entire lensed sample of $z\sim 6$-8 sources.

Two of the three star-forming candidates that Vanzella et al. (2017a) identify
over MACS0416 are also included in our compilation of small sources (Table 1).
M0416I-6115434445 corresponds to GC1 from Vanzella et al. (2017a), while
M0416Z-6114803434 corresponds to D1 from Vanzella et al. (2017a) and Vanzella
et al. (2019). D2 from Vanzella et al. (2017a) is also in the Bouwens et al.
(2021a) source catalogs.

For GC1, D1, and D2, we measure half-light radii of 21${}_{-14}^{+44}$ pc,
38${}_{-16}^{+26}$ pc, 72${}_{-4}^{+13}$ pc, while Vanzella et al. (2017a)
measure 16$\pm$7 pc, 140$\pm$13 pc, and $<$100 pc, respectively, for these.
Vanzella et al. (2019) later revisited their structural analysis of D1 and
derived a half-light radius of $<$13 pc for the core of D1 and 44 pc for the
source as a whole. The sizes we infer for M0416I-6115434445/GC1 and
M0416I-6103003258/D2 are in excellent agreement with those from Vanzella et
al. (2017a). For the third source M0416Z-6114803434/D1, the half-light radius
Bouwens et al. (2021a) estimate is in good agreement with the size Vanzella et
al. (2019) derive for the source as a whole.

The larger size estimate for M0416Z-6114803434/D1 from Vanzella et al. (2017a)
likely results from the $n=3.0\pm 0.3$ Sersic parameter they find from their
size fits, which might suggest a larger size for the source. Vanzella et al.
(2019), however, in a follow-up study argue that perhaps a preferable
interpretation of observations is in terms of a two component fit, with a
bright central component for the source. The presence of a compact, central
high surface-brightness region with an estimated size of $<$13 pc is
fascinating in the context of our discussion in §5.1 and its relevance to the
suspected presence of high surface brightness star-forming complexes as per
Figure 4. This conceptual model is discussed in more depth in §5.1.

We can also compare with the compilation of small $z=6$-8 sources identified
by Kawamata et al. (2018). Of the 23 $z=6$-8 sources in the Kawamata et al.
(2018) catalogs with source sizes less than 50 pc or smaller and which also
appear in our catalogs, we measure a median size of 63 pc. For the eight
$z=6$-8 galaxies in our catalogs with inferred source sizes $<$50 pc which
also appear in the Kawamata et al. (2018) catalogs, the median size Kawamata
et al. (2018) derive is 35 pc. This demonstrates that the derived sizes for
the smallest sources in each of our catalogs is reproducible (at least broadly
speaking). Overall, for the 24 sources in our collective $z=6$-8 catalogs
where Kawamata et al. (2018) or we derive sizes less than 50 pc, we compute an
RMS source-to-source scatter of 0.31 dex. This is similar to the 0.30-dex
scatter we find in size measurements for the 92 sources in common between our
samples. The size comparisons with Kawamata et al. (2018) are quite
reassuring.

We remark that the higher significance $5\sigma$ detections that Kawamata et
al. (2018) require in multiple near-IR bands for their own selections of
$z=6$-9 galaxies appear to be a major reason why their catalogs include a
smaller number of sources (181 sources) than our own (333 sources). In our own
selections, we only require sources be detected at 6$\sigma$, based on the
$\chi^{2}$ parameter results we obtain by adding in quadrature the detection
significance in the $Y_{105}J_{125}JH_{140}H_{160}$ near-IR bands. We made use
of a similar procedure in a variety of previous analyses (e.g., Bouwens et al.
2011, 2015, 2021b). This results in our selection including many more sources
at $>$28 mag than are present in the Kawamata et al. (2018) selection and thus
being $\sim$1.8$\times$ larger.

Figure 1.— Median size vs. luminosity relation of galaxies identified in blank
field studies, i.e., the XDF/HUDF and CANDELS (left panel) and those of star
cluster complexes (right panel). The canonical size-luminosity relation for
high-redshift galaxies is presented using both the Shibuya et al. (2015) fit
results (black line) and median sizes at $z\sim 6$ (green circles), $z\sim 7$
(blue circles), and $z\sim 8$ (red circles). The black dotted line shows an
extrapolation of the best-fit Shibuya et al. (2015) $r\propto L^{0.26\pm
0.03}$ trend to lower luminosities, as Bouwens et al. (2021a) find in their
§3. The shaded grey area shows the approximate region occupied by lensed
$z\sim 6$-8 sources identified by Bouwens et al. (2021a). The right panel
shows the regions in size-luminosity space occupied by star clusters (and
super star clusters) and cluster complexes. The dark gray region indicates the
size-luminosity relation for star cluster complexes in $z=0$-3 galaxies
inferred by Livermore et al. (2015) by fitting to the $z=0$ results from SINGS
(Kennicutt et al. 2003), as well as the results of Jones et al. (2010),
Swinbank et al. (2012), Livermore et al. (2012), Wisnioski et al. (2012), and
Livermore et al. (2015). The light gray region indicates the size-luminosity
relation for star cluster complexes extrapolating this relation to $z=3$-6.
The solid red circles correspond to the measured sizes and equivalent $UV$
luminosities of the star cluster complexes identified in the highly magnified
$z=4.92$ galaxy behind MS1358+62 (Franx et al. 1997) by Swinbank et al. (2009)
and Jones et al. (2010), while the solid green triangle and cyan circle
correspond to the sizes and luminosities of 30 Doradus and IIZw40,
respectively (Meylan 1993; Vanzi et al. 2008). The magenta lines enclose the
luminosities and sizes measured for star clusters and super star clusters at
$z\sim 0$ (Meurer et al. 1995). Figure 2.— Comparison of the inferred sizes
[in pc] and luminosities of lensed galaxies in the HFF clusters (red circles)
with star clusters (demarcated by the magenta lines: §3.1), super star
clusters (demarcated by the violet lines: §3.1.1), and star cluster complexes
(demarcated by the black lines: §3.1.2 and Figure 1). The right vertical axis
shows the corresponding angular size of sources in mas if they are at $z\sim
7$. $1\sigma$ errors on the inferred sizes and $UV$ luminosities are quoted
based on the 68% confidence intervals in the size fits and the range of
magnification factors in the parametric lensing models, as in Figure 4 (middle
panel) of Bouwens et al. (2021a). For those sources where the median total and
linear magnification factors exceed 30 and 50, respectively, the sizes and
luminosities are derived assuming maximum magnification factors of 30 and 50,
respectively (but keeping the same uncertainties on the size and luminosity
measurements as they were estimated prior to imposing upper limits on the
magnification factors). The sources where upper limits have been applied to
the magnification factors are shown with the red squares. The cyan, blue, and
green circles show the sizes and luminosities reported by proto-globular
clusters and star cluster candidates claimed by Vanzella et al. (2017a, 2017b,
2019, 2020), Johnson et al. (2017), and Zick et al. (2020) while the green
triangle shows the size and luminosity of the 30 Doradus star complex. The
conversion between a given $UV$ luminosity and a stellar mass is made assuming
a star formation duration of 10 Myr. While most of the lensed $z=6$-8 sources
in the HFF observations appear to have sizes and luminosities consistent with
star cluster complexes seen in $z=0$-3 galaxies (Bastian et al. 2006; Jones et
al. 2010; Wisnioski et al. 2012; Swinbank et al. 2012; Livermore et al. 2012;
Adamo et al. 2013; Livermore et al. 2015; Johnson et al. 2017; Dessauges-
Zavadsky et al. 2017), a few sources lie in the super star cluster region of
parameter space. We use the few sources found in the star cluster region to
place constraints on the volume density of proto-globular clusters at $z\sim
6$ (§5.2 and Figure 5).

##  3\. Comparison to Star Clusters, Super Star Clusters, and Star Cluster
Complexes

Given the extremely small sizes (i.e., $<$40 pc) inferred for some fraction
(here, 7% of our sample of 333 galaxies) of the lensed sources identified
behind the HFF clusters (Bouwens et al. 2017a; Vanzella et al. 2017a, 2017b,
2019), it is interesting to compare these sources with more familiar star-
forming entities in the nearby universe in terms of their sizes and
luminosities. These entities broadly fall into two classes by size, star
clusters and star cluster complexes.

###  3.1. Star Clusters and Star Cluster Complexes

####  3.1.1 Star Clusters and Super Star Clusters

The first of these classes are the smaller star-forming objects, star clusters
and super star clusters. Meurer et al. (1995) provide a convenient summary of
where star clusters lie in terms of their effective radii and $UV$
luminosities $M_{UV}$ in their Figure 14. $UV$ luminosities of the star
clusters extend from $-$9 mag to $-19$ mag, masses range from $10^{4}$ to
$10^{8}$ $M_{\odot}$ (Maraston et al. 2004; Cabrera-Ziri et al. 2014, 2016),
while the typical effective radii of star clusters range from 0.5 pc to 4 pc
(e.g., Lada & Lada 2003).

The magenta line in the right panel of Figure 1 demarcates the approximate
sizes and luminosities found for star clusters and super star clusters in the
nearby universe. For comparison, the size-luminosity relation derived in
Bouwens et al. (2021a: §3) for bright star-forming galaxies at $z=6$-8 is also
presented in the left panel of Figure 1, with an extrapolation to lower
luminosities. We also show broadly, as a grey shaded area, the region occupied
by a fainter sample of 333 galaxies, from Bouwens et al. (2021a).

The most massive ($>$105 $M_{\odot}$) star clusters are often called super
star clusters, with the effective radii extending to a maximum size of
$\sim$20 pc (e.g., Meurer et al. 1995; Rejkuba et al. 2007; Murray 2009;
Bastian et al. 2013). Meurer et al. (1995) classify any star clusters with
$UV$ luminosities greater than $-$14 mag as super star clusters. We refer the
interested readers to other excellent discussions on this topic by Rejkuba et
al. (2007), Murray (2009), and Bastian et al. (2013).

####  3.1.2 Star Cluster Complexes

The larger class of star-forming regions are star cluster complexes seen in
star-forming galaxies at $z=0$ (Kennicutt et al. 2003; Bastian et al. 2005).
These are also seen out to $z\sim 3$ in strongly lensed galaxies (Jones et al.
2010; Wisnioski et al. 2012; Livermore et al. 2012, 2015; Swinbank et al.
2012; Adamo et al. 2013; Vanzella et al. 2017b; Dessauges-Zavadsky et al.
2017).

Star cluster complexes – often referred to as cluster complexes in nearby
galaxies – are known to show a range of surface brightnesses at all redshifts
where they are observed, i.e., $z\sim 0$-3 (Bastian et al. 2005, 2006; Jones
et al. 2010; Swinbank et al. 2012; Wisnioski et al. 2012; Rodríguez-Zaurín et
al. 2011; Kennicutt et al. 2003). Star cluster complexes are also described as
star-forming clumps (or giant HII regions) when observed in distant galaxies.
A simple fit to the mean surface brightness of star cluster complexes as
function of redshift yields the following relation (Livermore et al. 2015):

$\log\left(\frac{\Sigma_{clump}}{M_{\odot}\,\textrm{yr}^{-1}\,\textrm{kpc}^{-2}}\right)=(3.5\pm
0.5)\log(1+z)-(1.7\pm 0.2)$ (2)

While many other observations of star cluster complexes at intermediate to
high redshifts are also consistent with the above trend (Franx et al. 1997;
Swinbank et al. 2009; Wuyts et al. 2014; Johnson et al. 2017), some star
cluster complexes at $z\sim 0$ have been reported to show much higher (by
factors of $\sim$100) surface densities of star formation (Fisher et al.
2017).

The implied evolution in the surface brightness of star cluster complexes is
essentially identical to what one would infer from dimensional arguments. The
sizes of collapsed sources is generally found to scale as $(1+z)^{-1}$ (e.g.,
Bouwens et al. 2004; Oesch et al. 2010; Ono et al. 2013; Holwerda et al. 2015;
Shibuya et al. 2015) and the evolution in dynamical time goes as
$(1+z)^{-1.5}$, such that $\Sigma_{SFR}\propto r^{-2}t_{dyn}^{-1}$
$\propto(1+z)^{3.5}$. Nevertheless, it should be recognized that the best-fit
evolution in $\Sigma_{clump}$ with redshift likely suffers from surface
brightness selection effects (given the bias towards selection of the highest
surface brightness star cluster complexes at a given redshift), so the
evolution suggested by Eq. 2 should only be considered indicative. The black
and magenta lines indicate the region in parameter space where we would expect
star cluster complexes and star clusters, respectively, to reside (see Figure
1: right panel).

We include a gray-shaded trapezoid in Figure 1 to show the region in size-
luminosity parameter space that star cluster complexes in $z\sim 0$-3 galaxies
have been found to inhabit. The light gray region shows an extrapolation of
this relation to $z=3$-6. The solid red circles correspond to the measured
sizes and equivalent $UV$ luminosities of the star cluster complexes
identified in the highly magnified $z=4.92$ galaxy behind MS1358+62 (Franx et
al. 1997) by Swinbank et al. (2009) and Jones et al. (2010), while the solid
cyan circle and green triangle correspond to the sizes and luminosities of
IIZw40 and 30 Doradus, respectively (English & Freeman 2003; Vanzi et al.
2008).

Figure 3.— Similar to Figure 2, but focusing on comparisons with the sizes and
masses of various evolved stellar systems in the nearby universe, including
E/S0 (black circles), ultra diffuse elliptical galaxies (UDG: cyan squares),
dwarf spheroids (dSphs: gray squares), dwarf ellipticals (dEs: gray
triangles), compact ellipticals (cEs: green star-like symbols), and globular
clusters/ultra compact dwarfs (GC/UCD: open blue circles). The conversion
between a given $UV$ luminosity and a stellar mass is made assuming a star
formation duration of 100 Myr (as found to be appropriate by Kikuchihara et
al. 2020). The left and right red arrows shows the expected change in the
inferred masses when changing the duration of star formation from 100 Myr to
10 Myr and 400 Myr, respectively. The layout of this figure is similar to
Figure 8 in Brodie et al. (2011) and Figures 11 of Norris et al. (2014). See
also Figure 10 of Bouwens et al. (2017c) and Figures 11-13 of Kikuchihara et
al. (2020). While the smallest lensed $z=6$-8 sources in the HFF observations
appear to have sizes and luminosities consistent with that of globular
clusters or ultra-compact dwarfs, most of the lensed $z=6$-8 galaxies have
sizes and luminosities that lie in the region between globular clusters and
that of elliptical galaxies and seem to best match that seen in star cluster
complexes (Figure 2). They presumably undergo further dynamical evolution
and/or accretion before becoming the evolved descendants we see today.

Wisnioski et al. (2012) find that surface brightness of star cluster complexes
depend on luminosity as $L^{0.26}$, such that the most luminous star cluster
complexes also have the highest surface brightnesses.

###  3.2. Putting Lensed Star-Forming Sources in Context

Having presented size, luminosity, and mass relations for star clusters and
cluster complexes, we now compare them with our lensed samples of $z\sim 6$-8
sources from Bouwens et al. (2021a). We present such a comparison in Figure 2,
including also the candidates identified by Vanzella et al. (2017a, 2017b,
2019, 2020), Johnson et al. (2017), and Zick et al. (2020). The sizes and
luminosities in Figure 2 (and Figure 3) use values for the linear and total
magnification factor of 30 and 50, respectively, for sources where the median
magnification factors from the lensing models exceed 30 and 50, respectively.

The black and magenta lines indicate the region in parameter space where we
would expect star cluster complexes and star clusters, respectively, to reside
(as we showed in Figure 1). Most strikingly, the size and luminosities of our
sample of 333 lensed $z\sim 6$-8 sources are very similar overall to what is
found for star cluster complexes in $z\sim 2$-3 galaxies.

Interestingly, in a few cases, the overall sizes and luminosities of these
sources even extend into the parameter space occupied by super star clusters.
It is difficult to be sure about a star cluster identification at the spatial
resolutions available with HST (or ground-based telescopes) even with lensing
magnification. Nevertheless, there are a few sources that have been
identified, i.e., the core of the Vanzella et al. (2017a) D1 source and knot A
of the Sunburst arc (Dahle et al. 2016; Vanzella et al. 2020), where there is
particularly strong evidence for the sources having very high magnification
factors and very small sizes thanks to the availability of multiple images of
each source behind a given lensing cluster (Vanzella et al. 2019, 2020). The
core of the D1 source has an inferred size $<$13 pc, while knot A of the
Sunburst arc has a likely size of 5-10 pc.

We will discuss these similarities in more detail in §5.

##  4\. Connection to Sources in the Local Universe

Beyond exploring potential similarities with star clusters and cluster
complexes, it is interesting to ask how the faint lensed sources we are
finding compare with various stellar systems found in the nearby universe.

For this exercise, we use the compilation that Norris et al. (2014) and M.
Norris (2021, private communication) provide of the sizes and masses for a
wide variety of local sources. This compilation includes elliptical and S0
galaxies (e.g., Cappellari et al. 2011; McDermid et al. 2015), ultra-diffuse
ellliptical galaxies (e.g., van Dokkum et al. 2015), dwarf ellipticals and
spheroids (e.g., Misgeld et al. 2008), compact ellipticals such as Messier 32
(e.g., Chilingarian et al. 2009), ultra-compact dwarfs (e.g., Evstigneeva et
al. 2007; Misgeld et al. 2011), and globular clusters (e.g., Hasegan et al.
2005; Firth et al. 2007; Mieske et al. 2007; Francis et al. 2012).

Figure 3 shows the inferred sizes and masses for our lensed $z\sim 6$-8 sample
relative to the Norris et al. (2014) compilation. Other noteworthy especially
small candidates from the literature (e.g., Vanzella et al. 2017a, 2017b,
2019, 2020; Johnson et al. 2017; Zick et al. 2020) are also presented. The
indicative masses that we use for our lensed sample are computed assuming a
fixed stellar population duration of 100 Myr in converting from their measured
$UV$ luminosities $M_{UV}$. Kikuchihara et al. (2020) found that this stellar
population age was a good match to the mass-to-light ratios derived through
detailed stellar population modeling of the full HST+Spitzer/IRAC photometry
from lensed $z=6$-8 sources.

Interestingly, some lensed sources in our samples have size and luminosities
in the regime of ultra-compact dwarf galaxies or globular clusters, with
measured sizes $<$40 pc. Kawamata et al. (2015) had previously reported two
sources with such small sizes. Now, largely due to the new compilations by
Bouwens et al. (2017a), Kawamata et al. (2018), and Bouwens et al. (2021a) and
candidates by Johnson et al. (2017), Vanzella et al. (2017a, 2017b, 2019,
2020), we now recognize that many, if not most, very faint galaxies at $z\sim
4$-8 may be small, but we cannot quantify the fraction yet with certainty
(e.g., Figure 5 from Bouwens et al. 2020). Bouwens et al. (2021a) provides an
extensive discussion of the evidence for why many faint $z\sim 6$-8 galaxies
seem to be very small, even if the individual measurements are not yet
definitive.

Figure 4.— Cartoon schematic showing the formation of a single star cluster
complex within a high-redshift dwarf galaxy. The star cluster complex forms
out of overdense baryonic material. As feedback from the star cluster complex
could temporarily inhibit star formation in other regions of the galaxy (e.g.,
Bastian 2008; Ploeckinger et al. 2019), galaxies may be much smaller in terms
of their readily-visible spatial extent than they actually are. The spatial
size of distant star-forming galaxies would be made to look smaller than they
are due to dominant impact of the youngest star cluster complexes on the $UV$
morphologies of dwarf galaxies (e.g., Overzier et al. 2008; Ma et al. 2018),
as occurs e.g. in nearby tadpole galaxies (e.g. Kiso 5639: Elmegreen et al.
2016) or blue compact dwarf galaxies (Elmegreen et al. 2012b; Papaderos et al.
2008). In particular, Figures 4 and 5 of Elmegreen et al. (2016) would be
strikingly similar to the compact objects seen at high redshift. The lensed
$z\sim 5$ galaxy MS1358+62 (Franx et al. 1997), with a dominant star cluster
complex $<$200 pc in size (see Swinbank et al. 2009; Zitrin et al. 2011), is
another dramatic example.

##  5\. Discussion

###  5.1. Just a Single Prominent Star Cluster or Star Cluster Complex Per
Faint Galaxy?

Interestingly most of the lensed $z\sim 6$-8 sources behind the HFF clusters
show very similar sizes and luminosities as star cluster complexes identified
in $z\sim 2$-3 galaxies (Figure 2); in particular, they are significantly
smaller in size than the extrapolated size-luminosity relation for $z=6$-8
galaxies from blank field studies where radius scales as $L^{1/4}$ or
$L^{1/3}$ (e.g., de Jong & Lacey 2000; van der Wel et al. 2014).

As has been speculated in the past (e.g., Bouwens et al. 2017a; Vanzella et
al. 2017a), one might ask if we are simply seeing individual star cluster
complexes in $z\sim 6$-8 galaxies and not the galaxies as a whole. Only a
fraction of the baryonic material associated with dwarf galaxies, could be
lighting up in the rest-$UV$ in the observations, i.e., we might be seeing a
single dominant star cluster complex (e.g., Figure 4). In the Overzier et al.
(2008) study of local Lyman-Break Analogue galaxies, it was noted how our
entire rest-$UV$ view of these galaxies could be dominated by a few cluster
complexes. Overzier et al. (2008) speculated about how fainter Lyman-break
analogues might look and whether they might be dominated by a single super
star cluster or cluster complex (see their §5.2.2).

More recent examples of galaxies where the rest-$UV$ view of galaxies are
dominated by a small region of the galaxy include Kiso 5639 (Elmegreen et al.
2016). At lower luminosities closer to the $z=6$-8 galaxies we are
considering, possible $z\sim 0$ analogues could include the recent extremely
metal poor, high sSFR galaxies shown in Figure 1 of Isobe et al. (2020). There
metal poor galaxies appear to be significantly less spatially extended than
the seemingly associated galaxies they seem to be a part. While no rest-$UV$
observations are presented by Isobe et al. (2020) to evaluate their spatial
extent in the rest-$UV$, the very strong line emission from these galaxies
suggests they may dominate the morphology of these galaxes at such
wavelengths, making such sources very compact in the rest-$UV$, with sizes
predominantly in the range 40 to 500 pc. Consistent with the suggested clumpy
star-cluster complex scenario, observations show an increasing fraction of
light in compact star-forming regions, from low redshift to high redshift
(Ribeiro et al. 2017).

How likely is it for lower-mass galaxies in the $z=6$-8 universe to host just
a single dominant star cluster complex? There are several recent observational
and theoretical results we can consider to try to address this question.

From a theoretical perspective, addressing this question properly would
require high resolution hydrodynamical simulations (e.g., Ma et al. 2018,
2020; Ploeckinger et al. 2019). Nevertheless, one plausible scenario is to
imagine the collapse of an overdensity resulting in the formation of a star
cluster complex and feedback from that star cluster complex preventing star
formation from occurring at any other position in a dwarf galaxy (Figure 4).
While SNe winds might be one possible mechanism (i.e., 4 Myr feedback time
scale for wind speeds of 50 km/s and source sizes of $\sim$200 pc: see Bastian
2008), another mechanism for inhibiting star formation in other parts of lower
mass galaxies would be through radiative feedback. Ploeckinger et al. (2019)
show that this mechanism could well be more effective at inhibiting star
formation in other regions of lower-mass galaxies.

Zick et al. (2018) looked in detail at the probable appearance of forming
globular clusters in the Fornax dwarf galaxy using information in the color
magnitude diagrams of field stars in Fornax and stars in its 5 globular
clusters. Zick et al. (2018) demonstrated that the forming globular clusters
could have been 10-100 times brighter than the Fornax dwarf galaxy as a whole
during its formation epoch and $>$104 brighter in terms of their surface
brightness, implying that the only readily detectable regions of dwarf
galaxies in deep extragalactic views are the forming globular clusters
themselves.

New simulation results from Pfeffer et al. (2019) using the E-MOSAICS
(MOdeling Star cluster population Assembly in Cosmological Simulations with
EAGLE: Pfeffer et al. 2018; Kruijssen et al. 2019) project offer a somewhat
different perspective on the detectability of forming globular clusters. Given
the sophistication of the E-MOSAICS project and the care taken to incorporate
star cluster formation, evolution, and disruption into the EAGLE galaxy
formation simulations (Schaye et al. 2015), results from the project should
give us a fairly realistic gauge of what we are observing in current data
sets. Most notably, Pfeffer et al. (2019) find that star clusters form in
those regions of galaxies where substantial clustered and unclustered star
formation is taking place. This would suggest that compact star-forming
regions in the distant universe most likely correspond to cluster complexes
and not proto-globular clusters forming in isolation. It is unclear at present
how to reconcile these results with the inferences made by Zick et al. (2018)
for globular clusters forming in the Fornax dwarf galaxy.

While hypothesizing that a single star-cluster complex may dominate the
rest-$UV$ view of many lensed $z=6$-8 galaxies, we suspect that the underlying
host galaxies are likely larger in terms of their physical extent (cf., the
$z=4.92$ Franx et al. 1997 MS1358+62 example). It would be easy for the lower
surface brightness regions to be missed due to cosmic surface brightness
dimming (e.g., see Ma et al. 2018).

###  5.2. Limits on the Volume Densities of Forming Globular Clusters in the
$z\sim 6$ Universe

We can also use the size measurements from Bouwens et al. (2021a) to set
constraints on the luminosity function of proto-globular clusters in the early
universe. Clearly, the large ages of stars in most globular clusters together
with the high gas densities appropriate for globular cluster formation
(Goddard et al. 2010; Adamo et al. 2011; Silva-Villa & Larsen 2011) – as well
as the high prevalence of globular clusters even in lower-mass galaxy halos
(Spitler & Forbes 2009; Harris et al. 2013) – strongly suggest a $z\gtrsim
1.5$ formation era. Having observational constraints on the formation of these
sources in the early universe is both valuable and interesting.

Given the proximity in time of powerful facilities like the James Webb Space
Telescope JWST, there are now numerous predictions for the number of such
clusters which might be found in a typical search field with the JWST (Renzini
2017; Boylan-Kolchin 2017, 2018; Elmegreen et al. 2012a) as well as candidate
proto-globular clusters identified in separate studies (Vanzella et al.
2017a).

Table 2Volume Density Constraints on proto-globular clusters forming at $z\sim
6$

$M_{UV}$ |  $\phi$ (Mpc-3)
---|---
This WorkaaAll sources with inferred half-light radii less than 40 pc are
included in this table. As we discuss in §3 and §4, these small star-forming
sources could be characterized as star cluster complexes, super star clusters,
proto-globular clusters, or especially compact galaxies.
$-$17.50 |  $<$0.0001bbIn cases where the median total magnification and linear magnification exceeds 50 and 30, resepectively, we quote alternative estimates for the absolute magnitude $M_{UV}$ and physical size $r_{e}$ with the total and linear magnification fixed to 50 and 30, respectively. This alternate estimates of $M_{UV}$ and $r_{e}$ are provided for sources given the challenges in relying upon magnification factors in excess of these values (§4.2 from Bouwens et al. 2021a and Bouwens et al. 2017b).
$-$16.50 |  $<$0.0007bb$1\sigma$ upper limits
$-$15.50 |  $<$0.004bb$1\sigma$ upper limits
$-$14.50 |  $<$0.047bb$1\sigma$ upper limits
$-$13.50 |  $<$0.14bb$1\sigma$ upper limits
Estimated From Vanzella et al. (2017a)
$-$15.50 |  $<$0.0106b,cb,cfootnotemark:
Volume Densities Probed with the HFF programddGray region in Figure 5
$-$17.50 | 0.00014
$-$16.50 | 0.00041
$-$15.50 | 0.0020
$-$14.50 | 0.011
$-$13.50 | 0.062
$-$12.50 | 0.9

aafootnotetext: These upper limits are derived using Bouwens et al. (2017b)
$z\sim 6$ LF results and the fraction of faint $z\sim 6$ galaxies whose
measured size is 40 pc or smaller. These upper limits are based on median
linear magnification factors from the parametric models. If we only include
those 21 sources (out of 23) which satisfy our $<$40 pc-criterion when
utilizing a maximum linear magnification factor of 30, the upper limits we
derive would be 10% more stringent than presented here.

ccfootnotetext: We use the same criteria in establishing the upper limits on
the proto-globular cluster volume densities as we use for the Bouwens et al.
(2021a) observational results

Figure 5.— (upper) Fraction of the lensed $z\sim 6$ sources in the HFF
observations which are measured by Bouwens et al. (2021a) to have half-light
radii $\leq$40 pc. (middle) Constraints on the volume density of forming
proto-globular clusters at $z\sim 6$ using searches for small sources behind
the HFF clusters. The plotted red upper limits combine the $UV$ LF constraints
we obtained from all six HFF clusters with the fraction of sources for which
the available constraints suggest sizes $\leq$40 pc. The blue upper limit
gives the volume density constraints we infer here for the proto-globular
cluster reported by Vanzella et al. (2017a). The dotted black line shows a
recent determination of the $z\sim 6$ LF for galaxies from the Hubble Frontier
Fields from Bouwens et al. (2017b). This LF was used to help derive the
observational constraints (together with the results in the top panel).
(lower) Comparison of the observational constraints on the proto globular
cluster volume densities with the predicted LF of proto-globular cluster
candidates from J. Pfeffer et al. (2021, in prep: thin black line), Pfeffer et
al. (2019: thick black line), and Boylan-Kolchin (2017, 2018) assuming the
mass ratio $\xi=1$ (green solid line: see §5.2) and $\xi=10$ (green dashed
line), where $\xi$ is the ratio of stellar mass in the initial globular
cluster at its birth and that present at $z=0$. The shaded gray region shows
the volume densities and luminosities where the HFF program does not provide
us with sufficient volume to probe; its upper envelope is equal to the
reciprocal of the total volume computed to be available over the first six HFF
clusters in the Bouwens et al. (2021, in prep) analysis.

To provide constraints on the volume density of forming globular clusters in
the early Universe, we explicitly consider the size constraints we have
available for the full sample of $z\sim 6$ sources over all six HFF clusters
from Bouwens et al. (2021a) vs. that expected for star clusters (Figure 2). If
the size measurements for a source yield a half-light radius measurement of
$<$40 pc, we consider it as a possible globular cluster candidate. In this
discussion we use the full sample of sources with sizes $<$40 pc from Bouwens
et al. (2021a), but also show the (small) impact of limiting ourselves to
linear and total magnifications of 30 and 50, respectively. This target size
range factors in the $\sim$10-20-pc maximum observed sizes of globular
clusters (§3.1.1), with a few notable exceptions (Pal 5 and Pal 14: Baumgardt
& Hilker 2018; Kikuchihara et al. 2020) and a factor-of-$\sim$2 evolution in
the size of the globular clusters themselves that can occur following their
initial formation time. Such size evolution could occur due to two-body
relaxation, mass loss by stellar evolution, and galaxy tides (Hills 1980;
Goodman 1984; Baumgardt & Makino 2003; Shin et al. 2013). A $<$40-pc size
limit also allows us to be as inclusive as possible in selecting possible
globular cluster candidates to ensure that the upper limit we set on the
volume density of forming globular cluster candidates is robust. This is a
careful, even conservative approach, that is particularly appropriate with our
sample given the significant systematic and random uncertainties in our direct
size measurements, as we discussed extensively in Bouwens et al. (2021a).
Using this definition, one of the $z\sim 6$ sources with size measurements
from Kawamata et al. (2015), i.e., HFF1C-i10, would already qualify as a
proto-globular cluster candidate.

In the upper panel of Figure 5, we present the fraction of $z\sim 6$ sources
which could correspond to forming globular clusters vs. $UV$ luminosity (i.e.,
those with $r_{e}\leq 40$ pc: see Table 1). We then show in the middle panel
of Figure 5, the implied upper limits we obtain on the volume density of
proto-globular clusters forming in the distant universe. Given the challenges
in being sure that any given system corresponds to a proto-globular cluster
(or the challenges in being sure that specific lensed sources are in fact
small: see §4.4-4.5 from Bouwens et al. 2020), we include our constraints as
upper limits. The upper limits and fractions we present in Figure 5 are based
on median linear magnification factors from the parametric models. If we only
include those 21 sources (out of 23) which satisfy our $<$40 pc-criterion when
utilizing a maximum linear magnification factor of 30, the upper limits we
derive would be 10% more stringent than what present in Table 1 and Figure 5.

In the same panel, we also include the one proto-globular cluster candidate
GC1 identified by Vanzella et al. (2017a) at $z\sim 6$ again as an upper limit
on the volume density (see also Vanzella et al. 2019). For consistency with
the candidates included from our own study, we only include this candidate
since its inferred size of 16$\pm$7 pc satisifes our criterion of a source
size $<$40 pc. In Table 2, we provide the estimated upper limits on the volume
density of sources for comparison.

An important uncertainty in attempting to quantify the volume density of
globular clusters forming in the distant universe regards the possibility that
globular clusters may not be identifiable separately from the host galaxy of
which they are a part. As we discusssed in §5.1, Zick et al. (2018) made use
of the available resolved stellar population information on the Fornax dwarf
galaxy to demonstrate that globular clusters forming in that system could be
clearly identifiable as separate systems during their formation phase. The
E-MOSAICS simulation results presented by Pfeffer et al. (2019) suggest that
such identifications could be much more challenging, however, due to star
clusters typically forming in significant associations. While such could make
the identification of single star clusters more challenging, we will need to
postpone a detailed consideration of these issues to a future analysis where
we can simulate these issues in detail in the context of a full globular
cluster+galaxy formation simulation like E-MOSAICS.

For comparison with our proto-globular cluster constraints, we also include on
the lower panel of Figure 5 several predictions for the number density of
proto-globular cluster and evolved globular cluster systems as a function of
$UV$ luminosity present in the $z\geq 6$ universe using several different
proto-globular cluster formation models. The first predictions are from the
E-MOSAICS project (Pfeffer et al. 2018; Kruijssen et al. 2019), which
explicitly adds star cluster formation to the EAGLE galaxy formation
simulations, while matching of the observed scaling relations and prevalence
of young and evolved star clusters. Using results from E-MOSAICS executed on a
(34.4 comoving Mpc)3 volume, we show predictions for the $UV$ luminosity
function of the forming globular clusters at $z\sim 6$ (thin black line: J.
Pfeffer et al. 2020). Also shown are the predicted from Pfeffer et al. (2019)
zoom-in simulation results (thick black line) on 25 Milky-Way-like galaxies
normalized to match the new J. Pfeffer (2021, in prep) results at the low
luminosity end.

Additionally, we show the predicted globular cluster $UV$ LF from the Boylan-
Kolchin (2017, 2018) model. The Boylan-Kolchin (2018) approach has particular
value due to its relative simplicity and the convenient Schechter function
approximation provided for the proto-globular cluster LF. Shown are the
predicted globular cluster LF for two different values of the ratio of stellar
mass in the initial globular cluster at its birth and that present at $z=0$,
i.e., $<m_{GC}(\textrm{birth})>/<m_{GC}(z=0)>$. Following Boylan-Kolchin
(2017, 2018), we use the symbol $\xi$ to describe this ratio and present both
the $\xi=1$ case (where $\phi^{*}=4\times 10^{-3}$ Mpc-3, $M^{*}=-16.9$,
$\alpha=-1.7$) and $\xi=10$ case (where $\phi^{*}=4\times 10^{-3}$ Mpc-3,
$M^{*}=-19.4$, $\alpha=-1.7$) in Figure 5. The $\xi=10$ case involves
substantial mass loss after the initial globular burst would favor very bright
proto-globular clusters in the early universe. Such a scenario is motivated
e.g. by Schaerer & Charbonnel (2011) based on chemical complexity of the
enrichment in globular clusters (see also D’Ercole et al. 2008; Renzini et al.
2015). E-MOSAICS features very little mass loss from forming globular clusters
and therefore the effective mass-loss factor $\xi$ is closest to 1. The
predicted $UV$ LFs of other models in the literature, e.g., Pozzetti et al.
(2019), lie between the $\xi=1$ and $\xi=10$ results from Boylan-Kolchin
(2017, 2018).

For context, we also show on the lower panel of Figure 5 the volume densities
to which we would be able to search for proto-globular clusters of specific
luminosities with the full HFF program. The volume densities are computed as
in Bouwens et al. (2017b) and hence would be for a probe of proto-globular
clusters at $z\sim 6$. They are based on the results over all six HFF clusters
(Bouwens et al. 2021, in prep). These volume densities are also compiled in
Table 2 for convenience. The search volume available for proto-globular
clusters in the $z=6$-10 universe is $\sim$3$\times$ larger than in the $z\sim
6$ universe (i.e., $z\sim 5.5$-6.3 using the Bouwens et al. 2017b criteria).

Remarkably, the predictions from E-MOSAICS (Pfeffer et al. 2019; J. Pfeffer et
al. 2021, in prep) lie quite close to the upper limits we can set on the basis
of existing HFF search results at $-$15 mag and brightward. E-MOSAICS predicts
$\sim$1.5 proto-globular cluster candidates at both $-$17.5 and $-$16.5 mag in
the total magnified volume behind the HFF clusters, consistent with the number
of candidates found here from the Bouwens et al. (2021a) compilation. The
$\xi=1$ model from Boylan-Kolchin (2017, 2018) predicts a proto-globular
cluster LF that is $\sim$0.4-0.5 dex higher than the E-MOSAICS model. This is
clearly in excess of our HFF constraints brightward of $-$16 mag, predicting
$\sim$3 and $\sim$10 sources at $-17.5$ mag and $-16.5$ mag, respectively,
within the volume of the HFF program. As mentioned above, the Pozzetti et al
(2019) model predictions lie between the $\xi=1$ and the $\xi=10$ lines. As
such the Pozzetti et al. (2019) model predicts an even larger number of proto-
globular cluster candidates ($\approx$2-3$\times$ larger) to be present in the
HFF volume (not shown in Figure 5).

At slightly fainter luminosities, i.e., $-$15.5 mag, our observational results
(red downward arrows) are also more consistent with the minimal $\xi=1$
scenario sketched out by Boylan-Kolchin (2017: green solid line). If the
Boylan-Kolchin (2018) $\xi=1$ scenario is correct, three $-15.5$-mag sources
identified with the HFF program are expected to correspond to proto-globular
clusters in formation. For the E-MOSAICS model, one $-15.5$-mag source would
correspond to a proto-globular cluster. While still small numbers, these
constitute $\sim$50%-100% of the tiny star-forming sources that we have
identified at those low luminosities. These results indicate that we observe
plausible consistency between the current globular cluster formation models
and what we derive from our size and LF results. Faintward of $-$14 mag, the
predicted proto-globular prevalence from the E-MOSAICS and Boylan-Kolchin
(2017, 2018) models appear to be too low to be well probed by the HFF program.

Interestingly, and rather definitively, the $\xi=10$ model of Boylan-Kolchin
(2017) exceeds the upper limits we can set from the HFF observations at all
luminosities. As such, we can likely already rule the $\xi=10$ model out. This
adds to other independent evidence against such models (Bastian & Lardo 2015;
Kruijssen 2015; Webb & Leigh 2015; Martocchia et al. 2017; Elmegreen 2017).

The present comparisons suggest that observers may be on the brink of
exploring the formation of globular clusters in the distant universe with
current and especially using future observations with JWST (see also
discussion in Renzini 2017, Vanzella et al. 2019, and Pozzetti et al. 2019).
With JWST, not only will we be able to search for proto-globular cluster
candidates much more efficiently, but we will be able to characterize each
candidate system in detail using high S/N spectrocopy, probing the velocity
dispersion (and thus allowing for a measurement of the dynamical mass) as well
as the chemical maturity of such systems.

###  5.3. Challenges Related to the Size Measurements We Utilize

The comparisons we present here of source sizes at $z\sim 6$-8 and those of
local star forming regions exemplifies the power of using lensing clusters.
However, as is discussed in §4.5 of Bouwens et al. (2021a), uncertainties in
the magnification factors can have a significant impact on the measured sizes,
resulting in sources whose sizes are both larger and smaller than reality. The
tests discussed in Bouwens et al. (2021a), which build on the work of Bouwens
et al. (2017a) and (2017b), suggest that recovery of the true sizes and
luminosities of the original lensed galaxies is nevertheless possible up to
linear magnification factors of $\sim$30 (for sizes), and total magnification
factors of $\sim$50 (for luminosities), but become less unreliable at higher
magnification factors. The collective impact of uncertainties in the lensing
models on the derived size distributions is illustrated well with Figure 5 of
Bouwens et al. (2021a).

Nonetheless, there is an abundance of evidence that many faint sources are
small, and that the broad distribution of sources we show here in Figures 2
and 3 are likely representative of the real distribution of sources. One
especially strong piece of evidence is from the observed surface density of
faint $z\sim 6$-8 galaxies in the highest magnification regions behind lensing
clusters and the fact that this surface density appears to show no dependence
on the shear factor (Bouwens et al. 2017a). This could only be the case if
faint galaxies had intrinsically small sizes, i.e., $<$165 pc (86% confidence:
Bouwens et al. 2017a).

Second, a number of sources with especially small size measurements benefit
from the existence of other multiple images of the same source. The existence
of such images allow us to set tighter constraints on the magnification of
sources at very high magnification factors. Examples of such sources include
both the central proto-globular candidate in D1 (Vanzella et al. 2017, 2019)
and the candidate YMC in knot A of the Sunburst arc (Vanzella et al. 2020) and
give us confidence that many more such small sources exist amongst the lensed
population of $z=6$-8 galaxies behind the HFF clusters.

Third, a high prevalence of faint $z=6$-8 galaxies is found in high
magnification regions behind the HFF clusters (Bouwens et al. 2017a,b;
Ishigaki et al. 2018; Bouwens et al. 2020). This could only be the case if
sources were significantly smaller than found from a simple extrapolation of
the Shibuya et al. (2015) relation (unless the LF showed a concave-upwards
form, which seems inconsistent with galaxy formation models). The discussion
in §5.3 from Bouwens et al. (2021a), as well as their Figure 10, illustrates
this well.

Finally, there are also the size-luminosity results at $z\sim 4$ (Bouwens et
al. 2020) which are less impacted by incompleteness systematics and provide
clearer evidence for lower luminosity sources having very small sizes. While
there are a number of uncertainties in the size-luminosity relation we derive
from direct measurements, the above results collectively provide strong
evidence that (1) galaxies fainter than about $-$17 $M_{UV}$ have very small
sizes, and (2) do so with a slope that appears to be steeper than for luminous
galaxies. An important caveat here is that our size measurements are in the
rest-$UV$ and thus may not reflect the true spatial extent of a galaxy, but
merely that of the dominant star forming region, as we discussed in §5.1 and
showed schematically in Figure 4.

##  6\. Summary

We have made use of a significant sample of 333 lensed star-forming galaxies
behind the HFF clusters and compared their inferred sizes and luminosities to
various classes of sources in the local universe to help put these high-
redshift sources in context. Both the identification of these $z\sim 6$-8
sources and their size measurements are described in a companion paper
(Bouwens et al. 2021a).

The results presented here broadly assume that that the distribution of sizes
and luminosities inferred by Bouwens et al. (2021a) are representative of
reality. Indeed, there is a wide variety of indirect evidence supporting the
general conclusion from Bouwens et al. (2021a) that most lower luminosity,
$z\sim 6$-8 sources are small, as discussed here in §5.3. Nevertheless, it is
worth being aware of the significant impact that issues like incompleteness
and uncertainties in the lensing models can have on the measured sizes and
luminosities, as e.g. discussed by Bouwens et al. (2021a). The impact of these
uncertainties is especially large for the lowest luminosity sources, so some
caution is required in interpreting the present results.

Our analysis shows that $z=6$-8 sources from the Bouwens et al. (2021a)
samples have measured sizes and luminosities very similar to that derived for
star cluster complexes identified in galaxies at $z=0$-3 (Jones et al. 2010;
Livermore et al. 2012, 2015; Wisnioski et al. 2012; Swinbank et al. 2012;
Johnson et al. 2017). In fact, the typical $-$15 mag galaxy in our samples has
a smaller half-light radius than 30 Doradus, which has a measured half-light
radius of $\sim$100 pc. This could be interpreted to suggest that lower
luminosity galaxies in the early universe may often contain a single prominent
star cluster complexes which dominates the observed UV morphology (Figure 4).

We also placed the measured size and luminosities of lensed $z=6$-8 galaxies
in our samples with the sizes and masses of stellar systems in the nearby
universe (§4). Most of the sources have inferred masses and luminosities that
place them in the region of parameter space where star cluster complexes lie
(Figure 2), which occurs midway between ultra-compact dwarfs and elliptical
galaxies (Figure 3). This suggests that many low-luminosity galaxies may be
dominated by a single star cluster complex in terms of their observed
morphologies (Figure 4).

Nevertheless, we remark that for a small minority of sources in our sample,
their properties are consistent with potentially corresponding to super star
clusters and – as such – they could correspond to proto-globular clusters
(Figure 2: §3). There are 23 sources in the Bouwens et al. (2021a) selection
with median size measurements equal to 40 pc or smaller. There are 3 sources
from the Bouwens et al. (2021a) selection that are likely to have sizes as
small as 10-20 pc.

We combine current constraints on the fraction of especially small sources
behind the HFF clusters with new state-of-the-art constraints on the $UV$ LF
of sources at $z\sim 6$ from the HFF clusters (Bouwens et al. 2017b) to derive
constraints on the proto-globular cluster LF at high redshift (§5.2).
Comparing this LF with predictions from the recent E-MOSAICS project (Pfeffer
et al. 2019; J. Pfeffer et al. 2021, in prep), models from Boylan-Kolchin
(2017, 2018), and Pozzetti et al. (2019), we find that with current
observations from the HFF clusters we are probably very close to identifying
bona-fide globular clusters in formation in the early universe (if some such
sources have not been identified already with our probe or that of Vanzella et
al. 2019, 2020).

For example, the E-MOSAICS project predicts that $\sim 1.5$ proto-globular
clusters in formation should be visible in the HFF observations at $-$17.5 mag
and $-$16.5 mag, respectively, while the $\xi=1$ model of Boylan-Kolchin
(2017, 2018) predicts $\sim$3, $\sim$10, and $\sim$3 forming proto-globular
clusters at $-$17.5 mag, $-$16.5 mag, and $-$15.5 mag. Tantalizingly enough,
our analysis of the Bouwens et al. (2021a) observational results shows some
candidates with plausible proto-globular cluster sizes at $<$$-$15 mag (see
lowest panel of Figure 5) in the same luminosity range where such sources are
predicted.

Despite plausible consistency between our search results and the predictions
of e.g. the E-MOSAICS project, the present results already place strong
constraints on more extreme globular cluster formation scenarios, e.g., those
involving substantial ($\xi=10$) mass loss after the initial formation burst.
Our results appear to rule out those scenarios entirely. While this inference
is contingent on forming proto-globular clusters having sufficiently high
surface brightness to be identified as separate from the host galaxy (see also
Bastian & Lardo 2015; Kruijssen 2015; Webb & Leigh 2015; Martocchia et al.
2017 for further evidence), such issues are unlikely to cause all such forming
globular clusters to be hidden.

As in our own previous work or other work (i.e., Bouwens et al. 2017a, 2021a;
Vanzella et al. 2017a, 2017b, 2019; Johnson et al. 2017), one should treat the
present results with some caution (§5.3) as a result of the impact of
uncertainties in the lensing models. The present results require that
parametric lensing models be predictive to magnification factors to
$\gtrsim$30\. Fortunately, this condition is likely satisfied given the end-
to-end simulation+recovery results obtained by Meneghetti et al. (2017) and
own tests comparing the HFF models.

In the future, we aim to gain more insight into the nature of small star-
forming galaxies in the distant universe through more extensive spectroscopic
observations, larger samples, and NIRSPEC/IFU observations with JWST.

We acknowledge stimulating discussions with Angela Adamo, Nate Bastian, Mike
Boylan-Kolchin, James Bullock, Rob Crain, Bruce Elmegreen, Phil Hopkins,
Xaiocheng Ma, Mike Norman, Elliot Quataert, Alvio Renzini, Britton Smith, Eros
Vanzella, Dan Weisz, Shelley Wright, and Tom Zick. Nate Bastian and Angela
Adamo provided us with extremely valuable feedback on the scientific content
and language used in an earlier version of this paper, especially with regard
to star clusters and star cluster complexes. We are grateful to Mark Norris
for sending us a compilation of the sizes, luminosities, and masses of many
evolved stellar systems in the nearby universe. We feel very appreciative to
Joel Pfeffer for sending us pre-submission results on the $UV$ LF for proto-
globular clusters from the E-MOSAICS project and providing us with pre-
submission feedback on this manuscript. We acknowledge the support of NASA
grants HST-AR-13252, HST-GO-13872, HST-GO-13792, and NWO grants
600.065.140.11N211 (vrij competitie) and TOP grant TOP1.16.057. This research
is based on observations made with the NASA/ESA Hubble Space Telescope
obtained from the Space Telescope Science Institute, which is operated by the
Association of Universities for Research in Astronomy, Inc., under NASA
contract NAS 5–26555.

## References

  * Adamo et al. (2011) Adamo, A., Östlin, G., & Zackrisson, E. 2011, MNRAS, 417, 1904
  * Adamo et al. (2013) Adamo, A., Östlin, G., Bastian, N., et al. 2013, ApJ, 766, 105
  * Bastian et al. (2005) Bastian, N., Gieles, M., Efremov, Y. N., & Lamers, H. J. G. L. M. 2005, A&A, 443, 79
  * Bastian et al. (2006) Bastian, N., Emsellem, E., Kissler-Patig, M., & Maraston, C. 2006, A&A, 445, 471
  * Bastian (2008) Bastian, N. 2008, MNRAS, 390, 759
  * Bastian et al. (2013) Bastian, N., Schweizer, F., Goudfrooij, P., Larsen, S. S., & Kissler-Patig, M. 2013, MNRAS, 431, 1252
  * Bastian & Lardo (2015) Bastian, N., & Lardo, C. 2015, MNRAS, 453, 357
  * Baumgardt & Makino (2003) Baumgardt, H., & Makino, J. 2003, MNRAS, 340, 227
  * Baumgardt & Hilker (2018) Baumgardt, H., & Hilker, M. 2018, MNRAS, 478, 1520
  * Bouwens et al. (2004) Bouwens, R. J., Illingworth, G. D., Blakeslee, J. P., Broadhurst, T. J., & Franx, M. 2004, ApJ, 611, L1
  * Bouwens et al. (2011) Bouwens, R. J., Illingworth, G. D., Oesch, P. A., et al. 2011, ApJ, 737, 90
  * Bouwens et al. (2015) Bouwens, R. J., Illingworth, G. D., Oesch, P. A., et al. 2015, ApJ, 803, 34
  * Bouwens et al. (2017) Bouwens, R. J., Illingworth, G. D., Oesch, P. A., et al. 2017a, ApJ, 843, 41
  * Bouwens et al. (2017) Bouwens, R. J., Oesch, P. A., Illingworth, G. D., Ellis, R. S., & Stefanon, M. 2017b, ApJ, 843, 129
  * Bouwens et al. (2017) Bouwens, R. J., Illingworth, G. D., Oesch, P. A., et al. 2017c, arXiv:1711.02090
  * Bouwens et al. (2021a) Bouwens, R. J., Illingworth, G.D., van Dokkum, P.G., et al. 2021a, ApJ, submitted

  * Bouwens et al. (2021b) Bouwens, R. J., Oesch, P. A., Stefanon, M., et al. 2021b, ApJ, in press, arXiv:2102.07775

  * Boylan-Kolchin (2017) Boylan-Kolchin, M. 2017, MNRAS, 472, 3120
  * Boylan-Kolchin (2018) Boylan-Kolchin, M. 2018, MNRAS, 479, 332
  * Brodie & Strader (2006) Brodie, J. P., & Strader, J. 2006, ARA&A, 44, 193
  * Brodie et al. (2011) Brodie, J. P., Romanowsky, A. J., Strader, J., & Forbes, D. A. 2011, AJ, 142, 199
  * Cabrera-Ziri et al. (2014) Cabrera-Ziri, I., Bastian, N., Davies, B., et al. 2014, MNRAS, 441, 2754
  * Cabrera-Ziri et al. (2016) Cabrera-Ziri, I., Bastian, N., Hilker, M., et al. 2016, MNRAS, 457, 809
  * Cappellari et al. (2011) Cappellari, M., Emsellem, E., Krajnović, D., et al. 2011, MNRAS, 413, 813
  * Chilingarian et al. (2009) Chilingarian, I., Cayatte, V., Revaz, Y., et al. 2009, Science, 326, 1379
  * Coe et al. (2015) Coe, D., Bradley, L., & Zitrin, A. 2015, ApJ, 800, 84
  * Dahle et al. (2016) Dahle, H., Aghanim, N., Guennou, L., et al. 2016, A&A, 590, L4
  * de Jong & Lacey (2000) de Jong, R. S., & Lacey, C. 2000, ApJ, 545, 781
  * D’Ercole et al. (2008) D’Ercole, A., Vesperini, E., D’Antona, F., McMillan, S. L. W., & Recchi, S. 2008, MNRAS, 391, 825
  * Dessauges-Zavadsky et al. (2017) Dessauges-Zavadsky, M., Schaerer, D., Cava, A., Mayer, L., & Tamburello, V. 2017, ApJ, 836, L22
  * Elmegreen et al. (2012) Elmegreen, B. G., Malhotra, S., & Rhoads, J. 2012a, ApJ, 757, 9
  * Elmegreen et al. (2012) Elmegreen, B. G., Zhang, H.-X., & Hunter, D. A. 2012b, ApJ, 747, 105
  * Elmegreen et al. (2016) Elmegreen, D. M., Elmegreen, B. G., Sánchez Almeida, J., et al. 2016, ApJ, 825, 145
  * Elmegreen (2017) Elmegreen, B. G. 2017, ApJ, 836, 80
  * English & Freeman (2003) English, J. & Freeman, K. C. 2003, AJ, 125, 1124
  * Evstigneeva et al. (2007) Evstigneeva, E. A., Drinkwater, M. J., Jurek, R., et al. 2007, MNRAS, 378, 1036
  * Faber (1973) Faber, S. M. 1973, ApJ, 179, 423
  * Firth et al. (2007) Firth, P., Drinkwater, M. J., Evstigneeva, E. A., et al. 2007, MNRAS, 382, 1342
  * Fisher et al. (2017) Fisher, D. B., Glazebrook, K., Damjanov, I., et al. 2017, MNRAS, 464, 491
  * Forbes & Bridges (2010) Forbes, D. A., & Bridges, T. 2010, MNRAS, 404, 1203
  * Franx et al. (1997) Franx, M., Illingworth, G. D., Kelson, D. D., van Dokkum, P. G., & Tran, K.-V. 1997, ApJ, 486, L75
  * Francis et al. (2012) Francis, K. J., Drinkwater, M. J., Chilingarian, I. V., Bolt, A. M., & Firth, P. 2012, MNRAS, 425, 325
  * Goddard et al. (2010) Goddard, Q. E., Bastian, N., & Kennicutt, R. C. 2010, MNRAS, 405, 857
  * Goodman (1984) Goodman, J. 1984, ApJ, 280, 298
  * Harris et al. (2013) Harris, W. E., Harris, G. L. H., & Alessi, M. 2013, ApJ, 772, 82
  * Haşegan et al. (2005) Haşegan, M., Jordán, A., Côté, P., et al. 2005, ApJ, 627, 203
  * Hills (1980) Hills, J. G. 1980, ApJ, 235, 986
  * Holwerda et al. (2015) Holwerda, B. W., Bouwens, R., Oesch, P., et al. 2015, ApJ, 808, 6
  * Ishigaki et al. (2018) Ishigaki, M., Kawamata, R., Ouchi, M., et al. 2018, ApJ, 854, 73
  * Isobe et al. (2020) Isobe, Y., Ouchi, M., Kojima, T., et al. 2020, submitted, arXiv:2004.11444
  * Johnson et al. (2017) Johnson, T. L., Rigby, J. R., Sharon, K., et al. 2017, ApJ, 843, L21
  * Jones et al. (2010) Jones, T. A., Swinbank, A. M., Ellis, R. S., Richard, J., & Stark, D. P. 2010, MNRAS, 404, 1247
  * Lada & Lada (2003) Lada, C. J., & Lada, E. A. 2003, ARA&A, 41, 57
  * Livermore et al. (2012) Livermore, R. C., Jones, T., Richard, J., et al. 2012, MNRAS, 427, 688
  * Livermore et al. (2015) Livermore, R. C., Jones, T. A., Richard, J., et al. 2015, MNRAS, 450, 1812
  * Livermore et al. (2017) Livermore, R., Finkelstein, S., Lotz, J. 2017, ApJ, 835, 113
  * Lotz et al. (2017) Lotz, J. M., Koekemoer, A., Coe, D., et al. 2017, ApJ, 837, 97
  * Kawamata et al. (2015) Kawamata, R., Ishigaki, M., Shimasaku, K., Oguri, M., & Ouchi, M. 2015, ApJ, 804, 103
  * Kawamata et al. (2018) Kawamata, R., Ishigaki, M., Shimasaku, K., et al. 2018, ApJ, 855, 4
  * Kennicutt et al. (2003) Kennicutt, R. C., Jr., Armus, L., Bendo, G., et al. 2003, PASP, 115, 928
  * Kikuchihara et al. (2020) Kikuchihara, S., Ouchi, M., Ono, Y., et al. 2020, ApJ, 893, 60
  * Kruijssen (2014) Kruijssen, J. M. D. 2014, Classical and Quantum Gravity, 31, 244006
  * Kruijssen (2015) Kruijssen, J. M. D. 2015, MNRAS, 454, 1658
  * Kruijssen et al. (2019) Kruijssen, J. M. D., Pfeffer, J. L., Crain, R. A., et al. 2019, MNRAS, 486, 3134
  * Ma et al. (2018) Ma, X., Hopkins, P. F., Boylan-Kolchin, M., et al. 2018, MNRAS, 477, 219
  * Ma et al. (2020) Ma, X., Grudić, M. Y., Quataert, E., et al. 2020, MNRAS, 493, 4315
  * Maraston et al. (2004) Maraston, C., Bastian, N., Saglia, R. P., et al. 2004, A&A, 416, 467
  * Martocchia et al. (2017) Martocchia, S., Bastian, N., Usher, C., et al. 2017, MNRAS, 468, 3150
  * McDermid et al. (2015) McDermid, R. M., Alatalo, K., Blitz, L., et al. 2015, MNRAS, 448, 3484
  * Meneghetti et al. (2017) Meneghetti, M., Natarajan, P., Coe, D., et al. 2017, MNRAS, 472, 3177
  * Meurer et al. (1995) Meurer, G. R., Heckman, T. M., Leitherer, C., et al. 1995, AJ, 110, 2665
  * Meylan (1993) Meylan, G. 1993, The Globular Cluster-Galaxy Connection, 48, 588
  * Mieske et al. (2007) Mieske, S., Hilker, M., Jordán, A., Infante, L., & Kissler-Patig, M. 2007, A&A, 472, 111
  * Misgeld et al. (2008) Misgeld, I., Mieske, S., & Hilker, M. 2008, A&A, 486, 697
  * Misgeld et al. (2011) Misgeld, I., Mieske, S., Hilker, M., et al. 2011, A&A, 531, A4
  * Murray (2009) Murray, N. 2009, ApJ, 691, 946
  * Norris et al. (2014) Norris, M., Kannappan, S., Forbes, D.A., et al. 2014, MNRAS, arXiv:1406.6065v1
  * Oesch et al. (2010) Oesch, P. A., Bouwens, R. J., Illingworth, G. D., et al. 2010a, ApJ, 709, L16
  * Oke & Gunn (1983) Oke, J. B., & Gunn, J. E. 1983, ApJ, 266, 713
  * Ono et al. (2013) Ono, Y., Ouchi, M., Curtis-Lake, E., et al. 2013, ApJ, 777, 155
  * Overzier et al. (2008) Overzier, R. A., Heckman, T. M., Kauffmann, G., et al. 2008, ApJ, 677, 37
  * Papaderos et al. (2008) Papaderos, P., Guseva, N. G., Izotov, Y. I., et al. 2008, A&A, 491, 113
  * Pfeffer et al. (2018) Pfeffer, J., Kruijssen, J. M. D., Crain, R. A., et al. 2018, MNRAS, 475, 4309
  * Pfeffer et al. (2019) Pfeffer, J., Bastian, N., Crain, R. A., et al. 2019, MNRAS, 487, 4550
  * Planck Collaboration et al. (2016) Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2016, A&A, 594, A13
  * Ploeckinger et al. (2019) Ploeckinger, S., Schaye, J., Hacar, A., et al. 2019, MNRAS, 484, 4379
  * Pozzetti et al. (2019) Pozzetti, L., Maraston, C., & Renzini, A. 2019, MNRAS, 485, 5861
  * Rejkuba et al. (2007) Rejkuba, M., Dubath, P., Minniti, D., & Meylan, G. 2007, A&A, 469, 147
  * Renzini et al. (2015) Renzini, A., D’Antona, F., Cassisi, S., et al. 2015, MNRAS, 454, 4197
  * Renzini (2017) Renzini, A. 2017, MNRAS, 469, L63
  * Rodríguez-Zaurín et al. (2011) Rodríguez-Zaurín, J., Arribas, S., Monreal-Ibero, A., et al. 2011, A&A, 527, A60
  * Ribeiro et al. (2017) Ribeiro, B., Le Fèvre, O., Cassata, P., et al. 2017, A&A, 608, A16
  * Schaerer & Charbonnel (2011) Schaerer, D., & Charbonnel, C. 2011, MNRAS, 413, 2297
  * Schaye et al. (2015) Schaye, J., Crain, R. A., Bower, R. G., et al. 2015, MNRAS, 446, 521 T., Trenti, M., et al. 2014, ApJ, 786, 57
  * Shibuya et al. (2015) Shibuya, T., Ouchi, M., & Harikane, Y. 2015, ApJS, 219, 15
  * Shin et al. (2013) Shin, J., Kim, S. S., Yoon, S.-J., & Kim, J. 2013, ApJ, 762, 135
  * Silva-Villa & Larsen (2011) Silva-Villa, E., & Larsen, S. S. 2011, A&A, 529, A25
  * Simon & Geha (2007) Simon, J. D., & Geha, M. 2007, ApJ, 670, 313
  * Spitler & Forbes (2009) Spitler, L. R., & Forbes, D. A. 2009, MNRAS, 392, L1
  * Swinbank et al. (2009) Swinbank, A. M., Webb, T. M., Richard, J., et al. 2009, MNRAS, 400, 1121
  * Swinbank et al. (2012) Swinbank, A. M., Smail, I., Sobral, D., et al. 2012, ApJ, 760, 130
  * van der Wel et al. (2014) van der Wel, A., Franx, M., van Dokkum, P. G., et al. 2014, ApJ, 788, 28
  * van Dokkum et al. (2015) van Dokkum, P. G., Abraham, R., Merritt, A., et al. 2015, ApJ, 798, L45
  * Vanzella et al. (2017) Vanzella, E., Calura, F., Meneghetti, M., et al. 2017a, MNRAS, 467, 4304
  * Vanzella et al. (2017) Vanzella, E., Castellano, M., Meneghetti, M., et al. 2017b, ApJ, 842, 47
  * Vanzella et al. (2019) Vanzella, E., Calura, F., Meneghetti, M., et al. 2019, MNRAS, 483, 3618
  * Vanzella et al. (2020) Vanzella, E., Caminha, G. B., Calura, F., et al. 2020, MNRAS, 491, 1093
  * Vanzi et al. (2008) Vanzi, L., Cresci, G., Telles, E., & Melnick, J. 2008, A&A, 486, 393
  * Webb & Leigh (2015) Webb, J. J., & Leigh, N. W. C. 2015, MNRAS, 453, 3278
  * Wisnioski et al. (2012) Wisnioski, E., Glazebrook, K., Blake, C., et al. 2012, MNRAS, 422, 3339
  * Wuyts et al. (2014) Wuyts, E., Rigby, J. R., Gladders, M. D., et al. 2014, ApJ, 781, 61
  * Zick et al. (2018) Zick, T. O., Weisz, D. R., & Boylan-Kolchin, M. 2018, MNRAS, 477, 480
  * Zick et al. (2020) Zick, T. O., Weisz, D. R., Ribeiro, B., et al. 2020, MNRAS, 493, 5653
  * Zitrin et al. (2011) Zitrin, A., Broadhurst, T., Coe, D., et al. 2011, MNRAS, 413, 1753