# The Origin of the Observed
Ly$\alpha$ EW Distribution of Dwarf Galaxies at $z\sim 2$

Christopher Snapp-Kolas,${}^{1,{\href https://orcid.org/0000-0002-9593-0053}}$
Brian Siana,${}^{1,{\href https://orcid.org/0000-0002-4935-9511}}$ Timothy
Gburek${}^{1,{\href https://orcid.org/0000-0002-7732-9205}}$ Anahita
Alavi${}^{2,{\href https://orcid.org/0000-0002-8630-6435}}$ Najmeh
Emami${}^{3,{\href https://orcid.org/0000-0003-2047-1689}}$ Johan
Richard${}^{4,{\href https://orcid.org/0000-0001-5492-1049}}$ Daniel P. Stark5
Claudia Scarlata${}^{6,{\href https://orcid.org/0000-0002-9136-8876}}$
1Department of Physics & Astronomy, University of California, Riverside, CA
92521, US
2IPAC, California Institute of Technology, 1200 E. California Boulevard,
Pasadena, CA 91125, USA
3Minnesota Institute for Astrophysics, University of Minnesota, Minneapolis,
MN, 55455, USA
4Univ Lyon, Univ Lyon1, Ens de Lyon, CNRS, Centre de Recherche Astrophysique
de Lyon UMR5574, F-69230, Saint-Genis-Laval, France
5Steward Observatory, University of Arizona, 933 N Cherry Ave, Tucson, AZ
85721, USA
6Minnesota Institute for Astrophysics, University of Minnesota, Minneapolis,
MN, 55455, USA E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present a rest-UV selected sample of 32 lensed galaxies at $z\sim 2$
observed with joint Keck/LRIS rest-UV and Keck/MOSFIRE rest-optical spectra
behind the clusters Abell 1689, MACS J0717, and MACS J1149. The sample pushes
towards the faintest UV luminosities observed ($-19\leq{\rm M_{\rm
UV}}\leq-17$) at this redshift. The fraction of dwarf galaxies identified as
Ly$\alpha$ emitters ($\rm EW\geq 20\ \rm\AA$) is ${\rm X_{\rm
LAE}}=25^{+15}_{-10}\%$. We use the Balmer lines and UV continuum to estimate
the intrinsic EW allowing us to distinguish the effects of the ionizing
spectra and escape fraction on the observed EW distribution. Fainter galaxies
(M${}_{\rm UV}>-19$) show larger intrinsic EWs and escape fractions than
brighter galaxies. Only galaxies with intrinsic EWs greater than $40\ \rm\AA$
have escape fractions larger than 0.05. We find an anti-correlation between
the escape fraction and $\rm A_{V}$ as well as UV spectral slope. The
volumetric escape fraction of our sample is $f_{\rm esc}^{\rm
Ly\alpha}=4.59^{+2.0}_{-1.4}\%$ in agreement with measurements found elsewhere
in the literature. About half of the total integrated Ly$\alpha$ luminosity
density comes from galaxies with ${\rm EW}_{\rm obs}>20$ Å.

###### keywords:

galaxies: evolution – galaxies: high-redshift

††pubyear: 2022††pagerange: The Origin of the Observed Ly$\alpha$ EW
Distribution of Dwarf Galaxies at $z\sim 2$–The Origin of the Observed
Ly$\alpha$ EW Distribution of Dwarf Galaxies at $z\sim 2$

##  1 Introduction

More than five decades ago, Ly$\alpha$ was proposed as a tracer of early stage
star forming galaxies (Partridge & Peebles, 1967). Since then Ly$\alpha$ has
been used to study star-forming galaxies locally and at the highest redshifts
observed (i.e. Shapley et al., 2003; Jung et al., 2020). It has been used to
study reionization (Bolton & Haehnelt, 2013; Mesinger et al., 2015; Mason et
al., 2018; Matthee et al., 2021), the interstellar medium (ISM) (Du et al.,
2021), the circumgalactic medium (Matsuda et al., 2012; Hayes et al., 2014),
and has confirmed the redshifts of the most distant star-forming galaxies
(Caruana et al., 2014; Hoag et al., 2019; Endsley et al., 2020; Jung et al.,
2020).

Ly$\alpha$ is a resonant line and has a large cross section. Therefore, it
will be scattered even by low column density gas. This makes it a great tracer
of gas distributions, but can result in difficulties in interpreting
observations. Furthermore, Ly$\alpha$ can be easily absorbed by dust in the
interstellar medium (ISM) reducing the strength of the line. Ly$\alpha$
observations are even more difficult when we try to observe galaxies during or
before the epoch of reionization where there are large amounts of neutral
hydrogen in the intergalactic medium (IGM) which can scatter whatever
Ly$\alpha$ manages to escape. Each of these effects can drastically hamper the
observability of Ly$\alpha$ and have strong effects on the distribution of
observed Ly$\alpha$ equivalent widths (EWs) (Stark et al., 2010, 2011). In
fact, at high redshift the attenuation of Ly$\alpha$ can be a useful probe of
the neutral fraction of hydrogen in the intergalactic medium. But in order to
measure the attenuation, one must first know the intrinsic distribution of
EWLyα to compare with the observed distribution. The evolution of the EWLyα
distribution has been used to study the end of reionization and is well
understood from z$=3-6$ and for "bright" galaxies with absolute UV magnitudes
M${}_{\rm UV}<-19$ (Stark et al., 2010, 2011; Pentericci et al., 2011; Curtis-
Lake et al., 2012; Ono et al., 2012; Schenker et al., 2012; Cassata et al.,
2015; De Barros et al., 2017; Haro et al., 2018; Caruana et al., 2018; Fuller
et al., 2020; De La Vieuville et al., 2020; Kusakabe et al., 2020; Zhang et
al., 2021). At $z=2$ there has been some study of the brighter star-forming
galaxies ($\rm M_{\rm UV}<-19$) (Reddy & Steidel, 2009; Cassata et al., 2015;
Hathi et al., 2016; Du et al., 2021; Zhang et al., 2021), but little work has
been done to study the fainter population ($\rm M_{\rm UV}>-19$). However,
according to the luminosity functions of Alavi et al. (2016), Konno et al.
(2016), and Bouwens et al. (2022) the faint galaxy population is larger at
$z=2$ and becomes more numerous with redshift. Therefore, if we want to
understand reionization we need more information on the faint galaxy
population, and $z=2$ galaxies can help to constrain the EWLyα distribution at
higher redshift where observation of Ly$\alpha$ is more difficult.

Ly$\alpha$ studies will either select galaxies via their rest-UV continuum
luminosity density (via broadband imaging), or their Ly$\alpha$ line emission
(via narrowband imaging, Berry et al., 2012; Konno et al., 2016; Sobral et
al., 2017; Sobral & Matthee, 2019; Hashimoto et al., 2017). Narrow-band
selected samples will be biased towards Ly$\alpha$ emitters (LAEs) as it
requires that the line be significantly detected in emission. Alternatively,
narrow-band studies of Ly$\alpha$ often select on other emission lines in the
optical (such as [OIII]5007) (i.e. Ciardullo et al., 2014; Trainor et al.,
2015; Weiss et al., 2021). Erb et al. (2016) show that [OIII]5007-selected
galaxies are biased towards LAEs by comparing the fraction of LAEs (XLAE) in
their [OIII]5007 selected sample with a sample of UV-selected galaxies. 80%
are found to be LAEs in their [OIII]5007 sample while only 9% are found to be
LAEs in their UV selected sample. This is further confirmed in Trainor et al.
(2019); Weiss et al. (2021), indicating that samples selected on nebular
emission lines are biased toward Ly$\alpha$ emission. A sample chosen
irrespective of the strength of nebular emission lines would be useful for
understanding the full distribution of Ly$\alpha$ EWs. With an unbiased
sample, it is possible to disentangle the effects of the stellar population
(e.g. age and metallicity) and that of the gas and dust on the observed EW
distribution.

Broad-band surveys, which select galaxies independent of emission line
strength, are able to probe down to ${\rm R}\sim 30$, but the followup
spectroscopy necessary for accurate escape fraction and EW analyses limit the
observations to $\rm R<25.5$ (i.e Reddy et al., 2008) in order to obtain
sufficient S/N in the continuum.

The Ly$\alpha$ EW distribution is dependent on the escape fraction of
Ly$\alpha$ photons. Past studies have measured the Ly$\alpha$ escape fraction
by comparing the Ly$\alpha$ luminosity density with the Ha or UV continuum
luminosity densities, obtained by integrating luminosity functions (Hayes et
al., 2010; Ciardullo et al., 2014; Konno et al., 2016). However, comparing
luminosity densities does not allow for analysis of individual galaxy escape
fractions or correlations with other properties of the population. Others make
use of narrowband imaging of Ly$\alpha$ and one of the Balmer lines (Trainor
et al., 2015) to make direct measurements of the Ly$\alpha$ escape fraction.
Narrow-band imaging is biased towards high Ly$\alpha$ EW galaxies, and
therefore does not represent the general population of star-forming galaxies
as evidenced by the rather high average Ly$\alpha$ escape fractions of 30%
(Kornei et al., 2010; Blanc et al., 2011; Wardlow et al., 2014; Trainor et
al., 2015) which differs greatly from other measures with more complete
samples (i.e.$\;5.3\%$, Hayes et al., 2010). Some try to model the escape
fraction as a function of Ly$\alpha$ EW (i.e. Sobral & Matthee, 2019), but
this method must assume a star formation history and intrinsic ionizing
continuum which are uncertain for low-mass galaxies. A direct measurement from
a UV-selected sample with spectroscopic followup avoids these uncertainties.

What is needed is a measure of Ly$\alpha$ emission properties of a UV-selected
sample of faint galaxies. We address this by selecting galaxies with
photometric redshifts determined from HST photometry described in Alavi et al.
(2014) and Alavi et al. (2016). We did not select on emission line flux and
therefore the sample is unbiased towards Ly$\alpha$ emitters. Furthermore, by
using gravitational lensing, we can study much fainter galaxies at $z=2$. The
goals of this paper are:

  * •

Measure the Ly$\alpha$ EW distribution and determine its dependence on the
properties of the ionizing sources and the Ly$\alpha$ escape fraction.

  * •

Calculate the volumetric escape fraction of dwarf galaxies

  * •

Identify trends in the EW distribution with absolute UV magnitude and redshift

The remainder of the paper is organized as follows. In $\S$2, we discuss the
observations, data reduction, and sample selection. In $\S$3, we discuss fits
to continua and emission lines, SED fitting, magnification, and dust
estimates. In $\S$4, we present our measured values of the EW distribution and
the Ly$\alpha$ escape fraction. In $\S$5, we discuss the implications of our
measurements. In $\S$6, we summarize the work. We adopt a $\Lambda$CDM
cosmology with $\Omega_{m}$ = 0.3, $\Omega_{\Lambda}=0.7$, and $h=0.7$
throughout the paper and all magnitudes are in the AB system.

##  2 Observations, Data Reduction, and Sample Selection

###  2.1 Observations & Data Reduction

Our sample consists of dwarf galaxies behind lensing clusters that have deep
HST data and are accessible to the Keck Observatory (Abell 1689, MACS J1149,
MACS J0717). Photometric redshifts were determined from HST photometry, as
described in Alavi et al. (2014, 2016). Our selection criteria were
photometric redshifts of $1.5<z<3.5$ and visual magnitudes brighter than
m${}_{\rm F625W}<26.3$. Slit allocation preference was given to galaxies with
high magnification and photometric redshifts where the rest-optical lines are
accessible in the YJHK atmospheric bands. In order to study physical
properties (i.e. metallicity, $f_{\rm esc}$, ionization parameter, etc.) of
these dwarf galaxies we obtained Keck/MOSFIRE spectra as described in Gburek
et al. (2019). We also obtained Keck/LRIS optical spectra, for which the
details are listed in Table 1. We observe eleven masks in three clusters with
LRIS. Exposure times varied greatly (from 4500s-18000s) depending on the
conditions and priority of objects in the mask, and seeing was typically $\sim
1^{\prime\prime}$. We decreased read noise by binning by 2 in the spectral
direction of the CCD.

Table 1: List of data acquired with Keck/LRIS. In order from left to right we have the mask name used, the number of spectra obtained in each mask, the exposure time in seconds, the modified julian date (MJD), and the seeing in arcseconds Mask | Spectra | Exposure Time | MJD | Seeing
---|---|---|---|---
(Name) | (Number) | (s) |  | (arcsec)
A1689$\\_$1 | 15 | 18000 | 55325 | 0.9
A1689$\\_$3 | 16 | 12286 | 55981 | 0.9
Macsj0717 | 11 | 12600 | 55981 | 1.1
Macsj1149$\\_$1 | 16 | 5400 | 57042 | 1.1
Macsj0717${\\_}2$ | 8 | 9000 | 57042 | 0.7
Macsj0717$\\_$1 | 8 | 9000 | 57042 | 0.9
A1689$\\_$4 | 10 | 9000 | 57042 | 1.4
Macsj1149$\\_$1_2 | 13 | 5400 | 57043 | 0.8
A1689$\\_z1\\_$1 | 12 | 12000 | 57043 | 0.8
Macsj1149$\\_$2 | 5 | 4500 | 57398 | 1.0
Macsj0717$\\_$3 | 8 | 6720 | 57398 | 0.7
Macsj1149$\\_$3 | 11 | 4860 | 57870 | 1.5
A1689$\\_$6 | 10 | 9600 | 57870 | 1.0

The data were reduced using a modified version of the PypeIt v1.x reduction
pipeline (Prochaska et al., 2020b). This pipeline performs bias subtraction,
flat fielding, cosmic ray rejection, wavelength calibration, sky subtraction,
and extracts the 1D spectra. The 1D spectra are extracted using an optimal
extraction b-spline fitting to the object profile along the spatial direction.
The 1D spectra are then flux calibrated using the spectrum of a standard star.
The 1D spectra from each frame are then combined using a weighted mean
algorithm in PypeIt. We perform a final absolute flux normalization by scaling
the spectra of compact bright continuum sources to the Hubble photometry. We
take the median (per mask) of these and scale the remaining spectra according
to this median. Most of our galaxies are faint and therefore may not have
their continua well-detected in the 1D spectra, so the use of a median
correction from bright continuum sources is necessary for the absolute
calibration.

We correct for slit losses by convolving the Hubble images with a Guassian
such that point sources in the final images would have a Guassian FWHM equal
to the seeing. We then measure the total flux and the flux within the slit. We
use the F435W and F475W bands of Hubble for the Abell 1689 and MACS J0717/MACS
J1149 masks respectively. We apply this correction to Ly$\alpha$, to obtain an
estimate of the total flux, but refrain from applying the correction when
displaying spectra.

When we have spectra of the same galaxy in different masks, we combine them
using an inverse variance weighted mean. Some of our galaxies are multiply
imaged, but combining multiple images involves demagnifying the spectra before
combining and results in a spectrum of arbitrary magnification. Furthermore,
the uncertainties in the magnification of high magnification galaxies would
result in large uncertainties in the flux with little gain in S/N. Therefore,
we remove highly magnified images for the multiply imaged galaxies.

The LRIS slit masks use box slits to align the masks on bright stars in the
field. Occasionally, the spectra of these alignment stars will contaminate
nearby slits. As a result some of the slits near the alignment stars are
rejected due to poor sky subtraction. After accounting for slits affected by
nearby box slits, our final count of 1D spectra is 127. The clusters are about
3’x3’ which takes up about one half of the LRIS detector. Because of this we
have about half the number of spectra typically expected from an LRIS mask. We
estimate the spectral resolution of our spectra given the slit width of our
observations is $\sim$1" with a plate scale of 0.135"/pix, however we have
binned along the wavelength direction such that the effective plate scale is
0.27"/pix. The wavelength spacing is $\sim 2.18\;\rm\AA\;\rm pix^{-1}$. So the
resolution of our observations is $\rm R\sim 500$ or $\Delta v\sim 600\;\rm
km\;\rm s^{-1}$. We show some example 1D spectra in figure 1.

Figure 1: Three example spectra from our sample of galaxies. The red line is
the fit for each spectrum. The fitting method is described in section 3. Top:
An example of an “emission" spectrum where we have Ly$\alpha$ in emission
only. Middle: An example of an “absorption" spectrum. Bottom: An example of a
“combination" spectrum where we see absorption and emission around Ly$\alpha$

###  2.2 Sub-Sample Selection

The remainder of the paper focuses on an analysis of XLAE and the escape
fraction of Ly$\alpha$ photons ($f_{\rm esc}$). In order to measure XLAE in an
unbiased manner we need to select a sub-sample of galaxies for which we can
identify galaxies with no Ly$\alpha$ emission. That is, a spectroscopic
redshift can be measured independent of detection of the Ly$\alpha$ line. We
call this sub-sample the XLAE sample. In order to measure $f_{\rm esc}$ we
must be able to measure H$\alpha$. This is a more stringent requirement which
eliminates a greater number of galaxies from the sample and would result in
greater uncertainty in XLAE were we to impose this condition on all of our
data. Therefore, we choose to create an additional sub-sample which we call
the H$\alpha$ sample.

####  2.2.1 XLAE sample

We choose a sub-sample of galaxies from our sample of 89 in order to avoid
biases in our Ly$\alpha$ equivalent width distribution. We account for biases
from magnification, slit-losses, and possible blending of sources. We mimic
the sub-sample selection methods of Emami et al. (2020), but with values
derived from our sample as follows:

  * •

We require secure redshifts for all the galaxies in this sample so that
galaxies with no Ly$\alpha$ or Ly$\alpha$ in absorption can be identified.
Therefore, only galaxies with confirmed redshifts from our MOSFIRE data are
kept. These redshifts are primarily determined by H$\alpha$ and
[OIII]$\lambda\lambda 5007$.

  * •

We remove galaxies which do not have spectroscopic coverage of Ly$\alpha$.
Atmospheric absorption at $\lambda<3200$ Å limits our analysis to galaxies at
$z\gtrsim 1.6$.

  * •

We remove galaxies for which there is possible confusion/blending with nearby
objects in the slit. These objects could have emission lines from one galaxy
and continuum from another, affecting the equivalent width measurements.

  * •

We remove galaxies with large magnification, as they could suffer from
differential magnification across the galaxy. This could introduce a bias into
our sample which is selected on UV luminosity density. We therefore remove
galaxies which have magnification $\mu>30$ for Abell 1689 and $\mu>15$ for
MACS J0717 and MACS J1149.

  * •

Finally, we remove galaxies with large slit loss correction in either LRIS or
MOSFIRE. If a large fraction of the galaxy flux is outside of a slit, then it
is not clear that a slitloss-corrected flux will reflect the true spectrum.
Most of the galaxies have slit losses $<2.2$ and we thus set this as our upper
limit. The slit loss distributions are shown in figure 2.

After this cleaning of the sample we obtain a XLAE sample of 32 galaxies with
a mean redshift $\langle z\rangle=2.27$.

Figure 2: The distribution of slit loss corrections to our H$\alpha$ and
Ly$\alpha$ spectra. The dashed line at 2.2 denotes the upper limit we set for
including galaxies in our sample.

####  2.2.2 H$\alpha$ sample

In order to get a measurement of the Ly$\alpha$ escape fraction, H$\alpha$
must be observable. Because of this we create a second sub-sample of galaxies
from the XLAE sub-sample for which H$\alpha$ could have been observed, which
limits the redshift range to $z\lesssim 2.6$. Additionally, The rest-UV HST
data are very deep and the KECK H$\alpha$ data are shallower which could bias
us against galaxies that are faint in H$\alpha$. To check whether we are
excluding galaxies that are too faint in H$\alpha$ we observe that our bright
MUV sample lies almost entirely above log(LHα/LUV)$\sim 13.4$, as shown in
figure 3, and therefore set this as our completeness limit for the sample. We
take the median H$\alpha=4.3\times 10^{-18}\;\rm erg\;s^{-1}\;cm^{-2}$ line
flux error multiplied by 3 as our 3$\sigma$ limit for detection of H$\alpha$.
We then check our galaxies magnification and absolute UV magnitude and ask
whether the galaxy would have been observable at log(LHα/LUV)$\sim 13.4$. If
the galaxy would not have been observable at $3\sigma$ at our completeness
limit it was then removed. No galaxy fell below this criterion and therefore
none were removed as shown in figure 3. We split our sample at M${}_{\rm
UV}=-19$ and measure the mean and error on the mean of the bright and faint
samples to be log(LHα/LUV)$=13.8\pm 0.003$ and log(LHα/LUV)$=13.5\pm 0.002$
respectively. While all but one of our galaxies have log(LHα/LUV)$>13.4$, 11
of the 18 galaxies fainter than M${}_{\rm UV}=-19$ could have been observed
below this line, given their magnifications. This suggests that there are
indeed very few galaxies with log(LHα/LUV)$<13.4$ and that our measured
increase of log(LHα/LUV) for faint galaxies is real, and not due to a bias
against galaxies with low log(LHα/LUV). Our final H$\alpha$ sample after all
of these considerations is 23 galaxies.

Figure 3: Plotted is the $\rm log(L_{H\alpha}/L_{UV})$ vs. absolute UV
magnitude. Points are color coded according to each galaxy’s magnification.
The black dashed line at $\rm log(L_{H\alpha}/L_{UV})=13.4$ denotes our
completeness limit for the sample. The angled colored space demonstrates the
3$\sigma$ H$\alpha$ sensitivity limit of our detector color coded according to
magnification

##  3 Methods

###  3.1 H$\alpha$ fits

The fits to the H$\alpha$ line were performed in a manner similar to Gburek et
al. (2019) on the 1D extracted MOSFIRE spectra. We briefly review the
procedure here. The spectra were fit using an MCMC sampler algorithm _emcee_
(Foreman-Mackey et al., 2013). Each of the J,H, and K bands were fit
separately and the emission lines were parameterized with a Gaussian and a
linear continuum fit. The redshift was taken to be the weighted average of the
values obtained between bands with emission lines present. The flux of the
H$\alpha$ line was taken from the Gaussian parameterization.

###  3.2 Ly$\alpha$ fits

We identified four types of spectra with respect to the Ly$\alpha$ line in our
dataset. Adopting the naming conventions of Kornei et al. (2010) we observe
Ly$\alpha$ in ’emission’, ’absorption’, ’combination’, and ’noise.’ We measure
the EWs of these spectra using the conventions of Du et al. (2018) with some
variation. For each type of galaxy we set the bounds of numerical integration
as follows:

  * •

For emission line galaxies we set the bounds to be where the emission line met
the continuum

  * •

For combination and absorption line galaxies we set an upper limit to the
width at 45 Å rest frame. The long-wavelength bound was set at the location
where the absorption profile met the continuum fit. The short-wavelength bound
was set at the blue end of the absorption wing or redder if this resulted in a
width greater than 45 Å rest-frame.

  * •

For noise spectra we have no information on the Ly$\alpha$ profile. Therefore,
we use boundaries derived by Kornei et al. (2010) at 1199.9 Å and 1228.8 Å
rest-frame for the short- and long-wavelength bounds respectively. Where the
redshift is determined from the rest-optical emission lines.

In addition, we performed a parameterized fit of the Ly$\alpha$ profile (for
emission and combination spectra) and continuum of each spectrum for our
galaxy sample. The fitting procedure is performed using a Markov chain Monte
Carlo code from pymc3. The wavelength range used for fitting varied based on
redshift as the short-wavelength bound was set to just blueward of the
Ly$\alpha$ emission line. The long-wavelength bound was set to be at $5450$ Å
in the observed frame, as the transmission decreases at longer wavelengths due
to the dichroic at 5600$\rm\AA$. Typical emission and absorption lines
(SiII$\lambda\lambda 1260$ Å, OI+SiII$\lambda\lambda 1303$ Å,
CII$\lambda\lambda 1334$ Å, SiIV$\lambda\lambda 1393$ Å, CIV$\lambda\lambda
1549$ Å, HeII$\lambda\lambda 1640$ Å, OIII$\lambda\lambda 1666$ AA,
CIII$\lambda\lambda 1909$ Å, and OIV$\lambda\lambda 1343$ Å) were masked when
fitting each spectrum. We set the width for these masks to be $800{\;\rm
km\;s^{-1}}$ except for the CIV$\lambda\lambda 1549$ Å line which we set to
have an asymmetric width of $1400{\;\rm km\;s^{-1}}$ with the blue side of the
line extending to $-1000{\;\rm km\;s^{-1}}$ because of the P-Cygni profile
that CIV$\lambda\lambda 1549$ Å tends to take. This process was repeated for
all galaxies observed. The ‘emission’ spectra are modelled simultaneously with
a power law continuum and a Gaussian emission line to handle correlations
between parameters,

$f_{\lambda}=A_{0}e^{-\frac{(\lambda-\lambda_{0})^{2}}{2\sigma^{2}}}+B_{0}(\frac{\lambda}{\lambda_{c}})^{\beta}$
(1)

For the ’combination’ spectra we add in a first order approximation to the
Ly$\alpha$ absorption profile,

$f_{\lambda}=\left\\{\begin{array}[]{ll}A_{0}e^{-\frac{(\lambda-\lambda_{0})^{2}}{2\sigma^{2}}}+a\lambda+b,&\lambda<\lambda_{s}\\\
B_{0}(\frac{\lambda}{\lambda_{c}})^{\beta},&\lambda>\lambda_{s}\end{array}\right.$
(2)

Where $\lambda_{s}$ is the wavelength where the linear approximation to the
Voigt profile meets the UV continuum. The remaining galaxies are parameterized
with a power law continuum redward of Ly$\alpha$ (identified either from
damped Ly$\alpha$ absorption or from the redshift estimate obtained with
Keck/MOSFIRE),

$f_{\lambda}=A_{0}(\frac{\lambda}{\lambda_{0}})^{\beta}$ (3)

We use an upper limit of FWHM = 4.9 Å rest-frame to obtain upper limits on the
flux from the ‘absorption’ and ‘noise’ spectra. The value of 4.9 Å is derived
from the distribution of 49 FWHMs from the sample for which we have Ly$\alpha$
in emission. 4.9 Å is greater than all but two of 49 (4%) of the FWHMs
measured and therefore serves as an upper limit on the FWHM. We determine the
Ly$\alpha$ EW and flux by numerical integration around Ly$\alpha$ using the
methods of Du et al. (2021) and then divide by the continuum level
extrapolated to rest-frame 1216 Å to determine the EW.

###  3.3 Magnification

Thanks to lensing, the signal from faint galaxies is magnified and we are able
to extend spectroscopic studies to fainter magnitudes at $z\sim 2$ in just a
few hours of Keck/LRIS and Keck/MOSFIRE time (see table LABEL:table1). Our
galaxies are observed behind three lensing clusters. In the XLAE and H$\alpha$
sub-samples the magnifications span $1.25\leq\mu\leq 29.12$. The median
magnifications are $\langle\mu\rangle_{\rm median}=7.11$ and
$\langle\mu\rangle_{\rm median}=7.45$ respectively. It is important to
properly model the lensing in order to accurately measure the UV luminosity of
the galaxies. Alavi et al. (2016) details the lens models used to measure the
magnifications of the galaxies behind MACS J0717, MACS J1149, and Abell 1689.
The models considered are constrained by the location and redshift of known
multiply-imaged sources. We use the models produced by the Clusters As
Telescopes (CATS)
collaboration111https://archive.stsci.edu/prepds/frontier/lensmodels/. As
Alavi et al. (2016) states, we use the models of Limousin et al. (2007),
Limousin, M. et al. (2016) and Jauzac et al. (2016) for Abell1689, MACS J0717,
and MACS J1149 respectively. These models are derived using LENSTOOL
222https://projets.lam.fr/projects/lenstool/wiki (Jullo et al., 2007).

###  3.4 SED fits

We fit spectral energy distributions (SEDs) to our Hubble photometry in the
near-UV, optical, and near-IR. Before SED-fitting we subtract off
contributions from nebular emission lines using our slit-loss corrected
spectra. We add an additional 3% flux error to all of our bands to account for
systematic errors in our photometry (Alavi et al., 2016). Using the code
FAST333https://w.astro.berkeley.edu/~mariska/FAST.html (Kriek et al., 2009),
we fit Bruzual & Charlot (2003) stellar population synthesis models to the
emission-line subtracted photometry under the following assumptions:

  * •

Constant star formation histories (SFHs)

  * •

A Chabrier (2003) initial mass function (IMF)

  * •

Stellar metallicities of either 0.2 $Z_{\odot}$ or 0.4 $Z_{\odot}$

  * •

A Gordon et al. (SMC 2003) dust attenuation curve

Galaxy redshifts are set by the fit curves to the spectroscopy.

###  3.5 Dust Correction

In our analysis of the H$\alpha$ sample we need to correct the H$\alpha$ line
and the UV continuum at 1700 Å for dust attenuation. We use the SMC curve of
Gordon et al. (2003) and AV estimates from our SED fitting to estimate the
attenuation of the UV continuum. The correction for H$\alpha$ is estimated by
direct measurement of the Balmer decrement (assuming Case-B recombination).
However, many of our galaxies have Balmer lines that are either too faint or
obscured by sky emission and we are unable to directly measure the Balmer
decrement of each galaxy. We observe that the Balmer decrement is well
determined for three of our five brightest galaxies , while our fainter
galaxies are less certain. Therefore, We split our H$\alpha$ sample into
bright (M${}_{\rm UV}<-19$) and faint (M${}_{\rm UV}>-19$) sub-samples for
which we estimate AV. For the faint sub-sample we normalize the spectra by
H$\alpha$ and perform a median stack of the galaxies. We use a Cardelli et al.
(1989) curve and measure $\langle{\rm A_{V}}\rangle=0.54$ and apply this to
all of our dwarf galaxies. Most of our reported results will be about the
dwarf galaxies for which we use this average attenuation value. We recognize
that there will be significant scatter around this value. Nonetheless, the
correction is small, a factor of 1.5, and does not affect the average trends
reported in this paper. For our more massive galaxies we measure AV for three
of the galaxies (${\rm A_{V}=1.14,\ 0.88,\ and\ 0.80}$). We then use the mean
($\langle{\rm A_{V}}\rangle=0.94$) on the remaining two galaxies for which we
did not have H${}_{\beta}\;S/N>5$ and therefore could not reliably calculate
the dust correction.

##  4 Results

As explained above, our larger $\rm X_{LAE}$ sample is used to measure the EW
distribution and the smaller Ha sub-sample is used to measure the escape
fraction. We compare our two samples in order to see if the two are similar
and therefore both representative of dwarf galaxies at this redshift. We find
that the median EWs are consistent with one another as shown in figure 4. The
difference in mean redshift is also small ($\langle z_{\rm
Ly\alpha}\rangle=2.28$ and $\langle z_{\rm H\alpha}\rangle=2.23$). This
suggests that the trends of the H$\alpha$ sample are likely representative of
the $\rm X_{\rm LAE}$ sample as well.

Figure 4: The green (orange) histogram is a normalized distribution of rest-
frame EWs of our $\rm X_{LAE}$ (H$\alpha$) sample. The colored dashed vertical
lines show the median EW of 5.8 (5.8) for the $\rm X_{LAE}$ (H$\alpha$)
sample. The H$\alpha$ median (orange dashed line) is under the green dashed
line. The Ly$\alpha$ distributions of both samples are consistent.

###  4.1 Ly$\alpha$ EW distribution

We compare our EW distribution with that of the higher luminosity sample of Du
et al. (2021) as shown in Figure 5. We find that our sample is skewed towards
larger Ly$\alpha$ EWs. The median EW for the Du et al. (2021) sample is $-6.0$
Å and the median for our sample is $5.8$ Å.

Figure 5: The green histogram is a normalized distribution of rest-frame EWs
of our $\rm X_{LAE}$ sample. The gold histogram is a normalized distribution
of rest-frame EWs of the brighter sample from Du et al. (2021). The vertical
dashed lines of corresponding color show the median Ly$\alpha$ EWs of each
sample. the median EWs are $\langle EW\rangle_{\rm median}=5.8\AA$ and
$\langle EW\rangle_{\rm median}=-6.0\AA$ respectively.

The cutoff for defining a LAE in the literature varies significantly depending
on the study (Stark et al., 2010, 2011; Caruana et al., 2018; Kusakabe et al.,
2020, etc.). However, the most common cuts are $\rm EW>20$ Å, $EW>25$ Å, and
$EW>55$ Å, so we will use these cuts as well. For these three cutoffs we
measure XLAE = $25^{+15}_{-10}\%$ (6/24), $25^{+15}_{-10}\%$ (6/24), and
$17^{+13}_{-8}\%$ (4/24), respectively. We assume a Poisson distribution to
estimate uncertainties on $\rm X_{LAE}$. XLAE measurements of brighter
galaxies are $\sim 11\%$ for $EW>20$ Å at $z\sim 2$ (12% from Reddy et al.
(2008), 11.1% from Hathi et al. (2016), and 10.7% Du et al. (2021)). For
brighter galaxies with $EW>25$ Å and $EW>55$ Å at $z\sim 2$ X${}_{\rm
LAE}=10\%$ and $4\%$ respectively (Cassata et al., 2015). These data suggest
that galaxies fainter than M${}_{\rm UV}=-19$ have greater numbers of LAEs in
their population. We explore this in more detail in section 5.3.

###  4.2 Escape Fraction

In our H$\alpha$ sample for which we are able to measure the escape fraction
of Ly$\alpha$ photons we assume a Case-B scenario for which the gas is taken
to be optically thick for the Lyman series. The ratio of the flux of
Ly$\alpha$ to the flux of H$\alpha$ under this assumption can vary depending
on the temperature and electron density of the galaxy. We choose to use a
ratio of 8.7 which is common in the literature (i.e. Matthee et al., 2016). We
then measure the escape fractions of individual galaxies as

$f_{\rm esc}=\frac{F_{\rm Ly\alpha}}{8.7\ F_{\rm H\alpha,cor}}$ (4)

where $\rm F_{Ly\alpha}$ is the Ly$\alpha$ flux and $\rm F_{H\alpha,cor}$ is
the dust corrected H$\alpha$ flux. The mean of the individual $f_{\rm esc}$
measurements is $4.3^{+1.6}_{-1.1}\%$. We estimate the uncertainty in the mean
via a bootstrap method. We perturb each measurement of the sample according to
a Gaussian distribution with standard deviation set by the uncertainty of the
measurement and then resample by randomly selecting with replacement from our
perturbed sample and then calculating the mean of the escape fractions. This
process is performed 100000 times. We then take the 16th and 84th percentiles
of the 100,000 iterations to be the uncertainty on the mean escape fraction.

However, this value may not be representative of the actual number of photons
escaping into the intergalactic medium. The intrinsic luminosity of galaxies
varies greatly and may correlate with Ly$\alpha$ escape fraction. Thus, the
mean escape fraction may be different from the net output from all galaxies,
which we refer to as the volumetric escape fraction. To calculate the
volumetric escape fraction, we first need to determine the galaxies’ intrinsic
luminosity and, thus, de-magnify each of our galaxies. We then replace the
fluxes in 4 with the sum of the respective luminosities. We measure a
volumetric escape fraction of $4.59^{+2.0}_{-1.4}\%$. We estimate the
uncertainty in a similar manner to the individual escape fraction estimate. We
perturb each measurement of the sample according to a Gaussian distribution
with standard deviation set by the uncertainty of the measurements, resample,
and then calculate the volumetric escape fraction. We take the 16th and 84th
percentiles of the 100,000 iterations to be the uncertainty on the volumetric
escape fraction. This method includes contributions from uncertainties in the
dust correction. This value agrees well with what is found in the literature
($5.3\%\pm 3.8\%$ in Hayes et al. (2010), $4.4^{+2.1}_{-1.2}\%$ in Ciardullo
et al. (2014), $5.1\%\pm 0.2\%$ in Sobral et al. (2017), and
$5.8^{+0.7}_{-0.5}\%$ in Weiss et al. (2021)).

By the use of gravitational lensing we were able to directly measure a UV
selected sample of dwarf galaxies at $z=2$. We are able to probe fainter
H$\alpha$ and Ly$\alpha$ luminosities and have a greater sample of joint
detections of H$\alpha$ and Ly$\alpha$ emitters than what was observed in
Hayes et al. (2010). Furthermore, our sample is not biased towards emitters as
in Ciardullo et al. (2014), Sobral et al. (2017), and Weiss et al. (2021).
Nevertheless, the volumetric escape fraction in each study is consistent with
our measurement, and all within 4%-6%.

##  5 Discussion

Here we seek to understand the origins of the observed EW distribution. Is the
spread in observed EW primarily due to varying escape fractions, or are
variations in stellar populations (in particular, starburst age), creating a
large scatter in intrinsic EWs as well.

###  5.1 Intrinsic Ly$\alpha$ EW

To investigate influences from the stellar populations we make use of the
intrinsic Ly$\alpha$ EW

$EW_{\rm int}=\frac{8.7\dot{}L_{\rm H\alpha,cor}}{L_{\rm\lambda,1216,cor}}$
(5)

where LHα,cor is the dust-corrected and de-magnified H$\alpha$ luminosity, and
Lλ,1216,cor is the dust corrected and de-magnified UV luminosity density
extrapolated to 1216 Å. Figure 6 shows the intrinsic EW as a function of
absolute UV magnitude.

Figure 6: The intrinsic equivalent width as a function of absolute UV
magnitude (MUV). Points are color coded according to the observed equivalent
width (see color bar on the right) with blue and purple points representing
galaxies observed as LAEs. The square gold points are average values of higher
luminosity galaxies and are derived from the $\xi_{\rm ion}$ measurements of
Shivaei et al. (2018). Galaxies brighter than -19 in absolute UV magnitude are
plotted with an x, while fainter galaxies are plotted with filled circles. We
show the mean intrinsic EW of Shivaei et al. (2018) and uncertainty in the
mean with a gold shaded region, and we show the mean intrinsic EW and
uncertainty in the mean of our dwarf galaxy sample in black. Five galaxies did
not have sufficient photometric coverage to perform SED fitting and are
removed from any intrinsic EW analysis.

We compare our measured $\rm EW_{int}$ values to those of more massive
galaxies at the same redshift. We convert the ionizing photon production
efficiency ($\xi_{\rm ion}$) of Shivaei et al. (2018) to $\rm EW_{int}$ by
converting $\xi_{\rm ion}$ to $\rm L_{Ly\alpha}/L_{UV}$. In Shivaei et al.
(2018) they calculate $\xi_{\rm ion}$ from weighted mean stacks of their
galaxies in bins of absolute UV magnitude. They are shown as gold squares in
figure 6. Shivaei et al. (2018) show a mean log$(\xi_{\rm ion})=25.36\pm 0.06$
for the SMC curve. This corresponds to an intrinsic EW of $23\rm\AA\pm 3$ Å.
Emami et al. (2020) have $\xi_{\rm ion}=25.22\pm 0.10$ which corresponds to an
intrinsic EW of $22\pm 5$ Å for a similar sample. For our faint galaxies our
mean intrinsic EW is $67\pm 8$ Å. Therefore, we see a significant increase in
the mean intrinsic EW for fainter galaxies. The faintest galaxies in our
sample may be skewing the sample toward higher $\rm EW_{int}$. We also show in
figure 6 the observed EWs via the colorbar where blue and purple points
indicate LAEs. From the plot we observe large intrinsic EWs ($\rm
EW_{Ly_{\alpha}}^{int}>40$ Å) facilitate observation of LAEs, but are not a
guarantee that a galaxy will be observed as a LAE. This implies that the
ionizing output of the stellar population in galaxies has significant bearing
on whether a galaxy is observed to be a LAE. However, it is clear that the
stellar population alone cannot account for the observability of LAEs.

###  5.2 Escape Fraction $f_{\rm esc}$

The other factor driving the observed distribution in Ly$\alpha$ EWs is
whether the photons actually escape from the galaxy. With our MOSFIRE spectra
we are able to measure this value directly. For our absorption galaxies in the
H$\alpha$ sample we set $f_{\rm esc}=0$. We show the escape fraction as a
function of absolute UV magnitude in figure 7.

Figure 7: The Ly$\alpha$ escape fraction as a function of absolute UV
magnitude (MUV). The color coding and markers are the same as in figure 6. The
mean escape fraction of the dwarf galaxies ($4.3^{+1.6}_{-1.1}\%$) and the
more massive galaxies ($0.5^{+0.3}_{-0.2}\%$) are displayed with shaded black
regions denoting the propagated uncertainty in the individual galaxies.
However, the bright galaxy sample has only 5 galaxies and is therefore
uncertain.

We see large scatter in the escape fraction at faint absolute UV magnitudes.
It is also the case that no escape fraction less than $\sim 10\%$ is observed
as a Ly$\alpha$ emitter. Therefore, it is clear that the escape of ionizing
photons plays a crucial role in the observed EW distribution and XLAE.

To determine if the escape fraction and EWint are correlated, we plot them in
figure 8.

Figure 8: The Ly$\alpha$ escape fraction as a function of intrinsic equivalent
width. The color coding and markers are the same as in figure 6. Galaxies with
no Ly$\alpha$ in emission are set to $f_{\rm esc}=0$. We observe that only
galaxies with intrinsic equivalent widths greater than 40 Å have escape
fractions greater than 0.05.

We see that 4/9 of the galaxies with high intrinsic EWs ($\geq 40$ Å) have
large escape fractions ($f_{\rm esc}>0.05$). No galaxy (0/8) with $\rm
EW_{int}<40$ Å has an escape fraction larger than 0.05. We can set two
conditions for LAEs; LAEs have escape fractions greater than 5% and intrinsic
EWs greater than $40$ Å. We caution that our sample size is small, so this
result could be made more robust by increasing the sample size. In order to
understand this a little deeper we can look at possible drivers of the escape
fraction.

Ly$\alpha$ can be heavily attenuated by dust. However, in our sample we have
shown that our galaxies are typically not very dusty. Nevertheless we check to
see if there is any relation between the dust attenuation estimated from SED
fits and the escape fraction. Figure 9 shows the escape fraction as a function
of ${\rm A_{V}}$. We observe an anti-correlation between $f_{\rm esc}$ and
${\rm A_{V}}$. However, nearly all of our galaxies have $\rm A_{V}\leq 0.2$.
With little variance in dust content we aren’t able to test this robustly.
Nevertheless, we don’t observe escape fractions greater than 0.05 for $\rm
A_{V}\geq 0.2$.

Figure 9: The Ly$\alpha$ escape fraction as a function of dust attenuation.
The color coding and markers are the same as in figure 6. Black diamonds show
the mean escape fraction in three bins of $\rm A_{V}$. The horizontal error
bars denote the sizes of the bins and the vertical error bars denote the
uncertainty in the mean. AV is here derived from SED fitting of Hubble
photometry and is in discreet steps.

The slope of the UV continuum can also be used as an indicator of dust and so
we look at possible correlations with UV slope as well. This is shown in
figure 10. Here we see a mild dependence in that the scatter of escape
fractions increases for $\beta<-0.5$. Together, these plots suggest that there
is a slight anti-correlation between dust and $f_{\rm esc}$ for faint
galaxies. This is consistent with figure 9 which shows an anti-correlation
between $f_{\rm esc}$ and AV.

Figure 10: The Ly$\alpha$ escape fraction as a function of UV spectral slope
($\beta$). The color coding and markers are the same as in figure 6. The black
dashed vertical line is set as a limit for LAEs. All LAEs have slopes bluer
than -0.5. There is large variance in beta for similar $f_{\rm esc}$, perhaps
suggesting bursty star-formation. The black diamonds are mean values of the
escape fraction in bins of $\Delta\beta=1.5$. The error bars are calculated by
propagating the statistical uncertainty of each datum. One object was excluded
due to highly uncertain $\beta$ from low S/N continuum.

###  5.3 Trends in XLAE

The decrement in XLAE from the expected XLAE at high redshift is often used to
infer the IGM neutral fraction (Stark et al., 2010, 2011, i.e.). However, in
order to investigate the number of faint galaxies that contribute to
reionization it is helpful to establish trends in XLAE with $\rm M_{UV}$ at
lower redshifts where Ly$\alpha$ is observable. To this end we compare with
the VUDS sample (Le Fèvre et al., 2015; Hathi et al., 2016; Cassata et al.,
2015) at higher UV luminosities. Figure 11 shows the VUDS data set in bins of
MUV along with our data set. We find no trend in the VUDS data for brighter
MUV, so we fit a constant line to the VUDS data set and find X${}_{\rm
LAE}=0.106\pm 0.011$. This is $1.4\sigma$ from our result of X${}_{\rm
LAE}=0.25^{+0.15}_{-0.10}$. More data between the VUDS faintest MUV and our
sample is necessary to determine any trend in XLAE with MUV.

Figure 11: The Ly$\alpha$ emitter fraction, XLAE, as a function of MUV. Our
data set is represented by a blue diamond and the VUDS dataset by blue
circles. We show a constant fit line, along with uncertainties in shaded blue,
to the VUDS data at X${}_{\rm LAE}=0.106\pm 0.011$ as there is no trend in the
brighter MUV sample.

We also investigate trends in redshift using the VUDS data set as shown in
figure 12. We fit each redshift bin with a constant value since we again see
no trend with MUV at any redshift. There is a clear trend towards larger XLAE
with redshift. The $X_{\rm LAE}$ values of low luminosity galaxies at $z\sim
2$ in our sample are consistent with more massive galaxies at $z\geq 3$.

Figure 12: The Ly$\alpha$ emitter fraction, XLAE, as a function of absolute UV
magnitude and redshift. The lines and shaded regions of corresponding color to
the data points are the constant fit curves to the data at each redshift.
There are no trends in MUV at any redshift in the VUDS data.

###  5.4 Observed Ly$\alpha$ EW

We investigate which galaxies are emitting most of the Ly$\alpha$ luminosity.
Figure 13 shows that about 50% of the integrated Ly$\alpha$ luminosity density
comes from galaxies with $\rm EW_{Ly\alpha}>20$ Å. This suggests that LAEs are
contributing a large fraction of the Ly$\alpha$ photons. This cumulative
distribution can also be calculated from narrow-band surveys for Ly$\alpha$.
However, those surveys can not determine the amount of total emissions from
low EW galaxies. Here we shos that $\sim 10$% of the Ly$\alpha$ luminosity
density EW${}_{\rm rest}<5$ÅḢowever, because of the size of our sample, this
result is quite uncertain.

Figure 13: The integrated Ly$\alpha$ luminosity density as a function of EW.
The uncertainty is given by the shaded green regions. We find that about 50%
of the integrated luminosity density comes from galaxies with $\rm
EW_{Ly\alpha}>20$ Å

##  6 Summary

In this paper we have investigated the distribution of Ly$\alpha$ EWs for
dwarf galaxies at $z\sim 2$. We selected (via UV continuum) magnified galaxies
behind lensing clusters, allowing us to obtain rest-UV spectroscopy for some
of the faintest galaxies (M${}_{\rm UV}<-17$, $\langle{\rm
M}^{\star}\rangle_{\rm median}=10^{8.4}\ {\rm M}_{\odot}$ ) observed at $z\sim
2$. We reduce our sample to avoid issues with differential magnification,
large slit loss uncertainties, incompleteness due to faint $H\alpha$, and the
observability of Ly$\alpha$ and H$\alpha$ within the bands of Keck/LRIS and
Keck/MOSFIRE. We then analyze the EW distribution of the sample and draw the
following conclusions:

  * •

The observed EW distribution of low UV luminosity galaxies is skewed towards
larger EWs than higher luminosity galaxies such as those in the KBSS sample of
Du et al. (2021). Our sample shows a median EW of 5.8 Å whereas the sample of
Du et al. (2021) shows a median EW of -6.0 Å.

  * •

The fraction of galaxies that are Ly$\alpha$ emitters XLAE is
$25^{+15}_{-10}\%,\;25^{+15}_{-10}\%,\;{\rm and}\;17^{+13}_{-8}\%$ for $\rm
EW>20$ Å, $\rm EW>25$ Å, and $\rm EW>55$ Å respectively. These values are
greater than XLAE in higher luminosity samples such as Reddy et al. (2008);
Hathi et al. (2016); Cassata et al. (2015); Du et al. (2021).

  * •

We investigate possible trends in XLAE with MUV and redshift using the VUDS
data set (Le Fèvre et al., 2015; Hathi et al., 2016; Cassata et al., 2015). We
find no trend with MUV between $-22\leq{\rm M_{UV}}\leq-19.5$, but find our
XLAE at M${}_{\rm UV}\sim-18$ to be $\sim 15\%$ greater at 1.4$\sigma$
significance. There is a trend towards larger XLAE with redshift in the VUDS
data. Further investigation of dwarf galaxies at higher redshift could show
increased XLAE relative to higher mass galaxies as well.

  * •

We find that the total integrated Ly$\alpha$ luminosity is about 50% for
galaxies with $\rm EW_{Ly\alpha}>20$ Å, suggesting that LAEs contribute a
large fraction of the Ly$\alpha$ photons. However, because of small numbers
this value is uncertain.

  * •

We estimate the intrinsic Ly$\alpha$ EW of galaxies, and show an increase in
the mean $\rm EW_{int}$ for faint galaxies ($\rm\langle EW\rangle=67\pm 8\AA$)
compared to the brighter sample of Shivaei et al. (2018, $\rm\langle
EW\rangle=23\pm 3\AA$) and the intermediate luminosity sample of Emami et al.
(2020, $\rm\langle EW\rangle=22\pm 5\AA$). This suggests that younger ages
and/or lower metallicities of the stellar populations of dwarf galaxies are
increasing the intrinsic EW and contributing to the larger XLAE.

  * •

We investigate the escape fraction of Ly$\alpha$ photons to further understand
what is driving the EW distribution. We observe a marginal increase in the
mean escape fraction with absolute UV magnitude. We also observe that only
galaxies with EW${}_{\rm int}>40\AA$ have $f_{\rm esc}>0.05$. This suggests
that only galaxies with higher EWint will be classified as LAEs, not just
because of their higher EWint, but also because only those galaxies have large
escape fractions.

  * •

We find an anti-correlation between $f_{\rm esc}$ and both $\rm A_{V}$ and
$\beta$. This implies that low dust-content facilitates the escape of
Ly$\alpha$ photons.

  * •

We observe a global volumetric escape fraction of $f_{\rm
esc}=4.59^{+2.0}_{-1.4}\%$ in our sample, in good agreement with values found
for other faint UV luminosity samples in the literature (i.e. Hayes et al.,
2010; Weiss et al., 2021, etc.).

We were able to disentangle to some extent two primary drivers of the
Ly$\alpha$ EW distribution, namely the ionizing sources (via the intrinsic
EW), and the ISM gas/dust content (via the escape fraction and its
dependencies). This sample can serve as a baseline with which to compare
higher redshift and higher mass samples. Larger sample sizes of dwarf galaxies
would serve to solidify some of the results of this work.

## Acknowledgments

This research made use of PypeIt,444https://pypeit.readthedocs.io/en/latest/ a
Python package for semi-automated reduction of astronomical slit-based
spectroscopy (Prochaska et al., 2020c, a).
Some of the data presented herein were obtained at the W. M. Keck Observatory,
which is operated as a scientific partnership among the California Institute
of Technology, the University of California and the National Aeronautics and
Space Administration. The Observatory was made possible by the generous
financial support of the W. M. Keck Foundation.
Based on observations made with the NASA/ESA Hubble Space Telescope, obtained
from the Data Archive at the Space Telescope Science Institute, which is
operated by the Association of Universities for Research in Astronomy, Inc.,
under NASA contract NAS5-26555. These observations are associated with
programs #9289, #11710, #11802, #12201, #12931, #13389, #14209.

## Data Availability

This paper is based on public data from the Hubble Space Telescope as well as
from programs 12201, 12931, 13389, 14209. Spectroscopic data from our survey
with the Keck Observatory. These data are available upon request from
Christopher Snapp-Kolas or Dr. Brian Siana.

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In this fashion, we may assume $G$ is connected, and further abelian by
hypothesis. In particular, we may assume $G$ is a trivial extension of a
multiplicative type group-scheme by an abelian unipotent one. By Section 4.3.1
and Theorem 4.14, we may further assume $G$ is of multiplicative type, so
isomorphic to a finite direct sum $\bigoplus_{i}\bm{\mu}_{p^{e_{i}}}$. Using
Section 4.3.1 once again, we may further suppose $G\cong\bm{\mu}_{p^{e}}$. Now
the result follows directly from Section 4.4 and Section 4.4. ∎

By iterating Theorem 4.26 until we exhaust the $F$-signature of $R$, we obtain
the main result of this work.

######  Theorem 4.27 (The existence of a maximal cover).

Work in Section 4. There exists a chain of finite extensions of strongly
$F$-regular $\mathcal{k}$-rational germs

$(R,\mathfrak{m},\mathcal{k},K)\subsetneq(R_{1},\mathfrak{m}_{1},\mathcal{k},K_{1})\subsetneq\cdots\subsetneq(R_{t},\mathfrak{m}_{t},\mathcal{k},K_{t})=(R^{\star},\mathfrak{m}^{\star},\mathcal{k},K^{\star}),$

where each intermediate extension
$(R_{i},\mathfrak{m}_{i},\mathcal{k},K_{i})\subsetneq(R_{i+1},\mathfrak{m}_{i+1},\mathcal{k},K_{i+1})$
is a $G_{i}$-torsor away from $IR_{i}$, and:

  1. (a)

$G_{i}$ is a linearly reductive group-scheme for all $i$,

  2. (b)

$[K^{\star}:K]$ is at most $1/s(R)$,

  3. (c)

any finite $G$-torsor over $(R^{\star},\mathfrak{m}^{\star})$ away from
$IR^{\star}$, so that $G^{\circ}$ is either trigonalizable or
nilpotent,404040Where $G^{\circ}$ denotes the connected component of $G$ at
the identity. is a torsor everywhere, i.e. it is the restriction of a
$G$-torsor over $\operatorname{{Spec}}R^{\star}$.

###### Proof.

By iterating Theorem 4.26 until $s(R)$ is exhausted, and using Section 4.1, we
obtain the weaker version of this result where in part (c) we assume that
$G^{\circ}$ is abelian. To see part (b), just notice that the transformation
rule yields

$1\geq
s\bigl{(}R^{\star}\bigr{)}=[K^{\star}:K_{t-1}]s(R_{t-1})=[K^{\star}:K_{t-1}][K_{t-1}:K_{t-2}]s(R_{t-2})\\\
=\cdots=[K^{\star}:K_{t-1}][K_{t-1}:K_{t-2}]\cdots[K_{1}:K]s(R)=[K^{\star}:K]s(R).$

It only remains to prove part (c) in its full generality, i.e. where we let
$G^{\circ}$ to be either trigonalizable and nilpotent. As in the proof of
Theorem 4.26, we may assume $G$ is connected, and so trigonalizable or
nilpotent. In this manner, it suffices to prove
$\varrho_{X^{\star}}(G)\colon\check{H}^{1}(X^{\star}_{\textnormal{ft}},G)\xrightarrow{\
\ }\check{H}^{1}(U^{\star}_{\textnormal{ft}},G)$ is surjective for all
connected trigonalizable and all connected nilpotent group-schemes
$G/\mathcal{k}$.

If $G$ is trigonalizable, it has a nontrivial normal subgroup $G^{\prime}$
which is isomorphic to either $\bm{\alpha}_{p}$ or
$(\mathbb{Z}/p\mathbb{Z})^{\oplus m}$ for some $m$; see [Mil15, Corollary
17.25]. If $G^{\prime}=G$ we are done, otherwise, we may proceed by induction
on the order of $G$ as $G^{\prime\prime}\coloneqq G/G^{\prime}$ is
trigonalizable.

If $G$ is nilpotent, by definition, it has a nontrivial central subgroup
$G^{\prime}$ (so normal and abelian), whose quotient is nilpotent. If
$G=G^{\prime}$ we are done, otherwise we may proceed by induction on the
order.

In either case, the proofs go very similar to the one for Theorem 4.14]. The
only difference is explaining why the twisted form of $G^{\prime}$ by a
$G$-torsor is trivial. In the proof of Theorem 4.14], we had this because
$G^{\prime}$ was taken central. Hence, the same argument works _mutatis
mutandis_ for the nilpotent case. The trigonalizable case requires a different
explanation though. The following claim does the job, using both $Y$ equal to
$X^{\star}$ and $U^{\star}$.

######  Claim 4.28.

If $Y$ a scheme with torsion-free Picard group and trivial étale fundamental
group, then the map $\check{H}^{1}(Y_{\textnormal{ft}},G)\xrightarrow{\ \
}\check{H}^{1}(Y_{\textnormal{ft}},\operatorname{Aut}G^{\prime})$ induced by
$G\xrightarrow{\ \ }\operatorname{Aut}G^{\prime}$ is trivial if either
$G^{\prime}=\bm{\alpha}_{p}$ or if $\operatorname{Aut}G^{\prime}$ is finite
étale, e.g. $G^{\prime}=(\mathbb{Z}/p\mathbb{Z})^{\oplus m}$.

###### Proof of claim.

Notice that
$\check{H}^{1}(Y_{\textnormal{ft}},\operatorname{Aut}G^{\prime})=0$ if
$\operatorname{Aut}G^{\prime}$ is finite étale, by the assumption on the étale
fundamental group of $Y$, this case is then trivial. On the other hand,
observe that $\operatorname{Aut}\bm{\alpha}_{p}=\mathbb{G}_{\textnormal{m}}$
[Mil17, 15.33], therefore
$\check{H}^{1}(Y_{\textnormal{ft}},\operatorname{Aut}\bm{\alpha}_{p})=\operatorname{Pic}Y$
[Mil80, III, 4.9]. Since $G$ is finite over $\mathcal{k}$, the map
$\check{H}^{1}(Y_{\textnormal{ft}},G)\xrightarrow{\ \ }\operatorname{Pic}Y$
sends $G$-torsors to torsion line bundles, which are trivial by assumption.
This proves the claim. ∎

This proves the theorem. ∎

###  4.6. Relation with Esnault and Viehweg’s Local Nori fundamental group-
scheme

It is natural to ask what the relationship is between our maximal cover
$R\subset R^{\star}$; see Theorem 4.27, and Esnault and Viehweg’s construction
of the _Local Nori fundamental group-scheme_ in [EV10]. To simplify our
discussion, let us consider abelian group-schemes only. In this case, the
property our cover has is that its abelian local Nori fundamental group-scheme
$\pi_{1,\textnormal{loc}}^{\textnormal{N,ab}}(U^{\star},X^{\star},x^{\star})$
is trivial. Nevertheless, it is not clear to the author whether or not this
implies that $\pi_{1,\textnormal{loc}}^{\textnormal{N,ab}}(U,X,x)$ is finite.

We are deeply thankful to Christian Liedtke who pointed out to us an example
suggesting this should not be always the case. He kindly shared with us the
following example of a surface $\textnormal{D}_{4}$ singularity in
characteristic $2$ that is a $\mathbb{Z}/2\mathbb{Z}$-quotient of
$\hat{\mathbb{A}}_{\mathcal{k}}^{2}$ but admitting a nontrivial
$\bm{\mu}_{2}$-torsor. The singularity is $\mathcal{k}\llbracket
x,y,x\rrbracket/(z^{2}-xyz-x^{2}y-xy^{2})$, the Weil divisor corresponding to
the prime ideal $(x,z)$ has index $2$. Therefore, its local abelian Nori
fundamental group-scheme cannot be $\mathbb{Z}/2\mathbb{Z}$ since it needs to
take into account $\bm{\mu}_{2}$ as well. Nonetheless, this singularity,
though $F$-pure, is not strongly $F$-regular. We invite the reader to look at
[ST14, Example 7.12] for a closer look into this particular singularity, cf.
[Art75].

###  4.7. Beyond the trigonalizable and nilpotent cases

It is unclear to the author how to go beyond the trigonalizable and nilpotent
cases, which are important instances of the solvable case. The general
solvable case remains open. On the other hand, the author is very grateful to
Axel Stäbler who made him aware of the classification of the (connected)
simple finite rank-1 group-schemes via the classification of simple restricted
Lie algebras in positive characteristic by Block–Wilson–Premet–Strade [BW88],
[SW91], [PS08], we recommend [Viv10] for a very nice, brief account. Then,
letting
$\varrho^{1}_{X}(G)\colon\check{H}^{1}(X_{\textnormal{ft}},G)\xrightarrow{\ \
}\check{H}^{1}(U_{\textnormal{ft}},G)$ denote the natural restriction map, the
following questions of interest are in order:

######  Question 4.29.

For which connected simple finite rank-1 group-schemes $G$ is
$\varrho^{1}_{X}(G)$ surjective?

######  Question 4.30.

If $G$ is a connected finite rank-1 group-scheme so that $\varrho^{1}_{X}(G)$
is not surjective, does Section 4.2 have affirmative answer?

######  Question 4.31.

Given a connected simple finite rank-1 group-scheme $G$, for which type of
($F$-)singularity $X$, if any, is $\varrho^{1}_{X}(G)$ naturally surjective?

##  5\. Applications to Picard groups

With the results developed in [CST16], we are only able to deduce that the
prime-to-$p$ torsion of the Picard group of $U$ is bounded by $1/s(R)$. Using
the additional analysis we have made in this paper for general
$\bm{\mu}_{n}$-torsors over $U$, we are able to bound all torsion.

######  Corollary 5.1.

The torsion of the Picard group $\operatorname{Pic}U$ is bounded by $1/s(R)$,
i.e. if $\mathcal{L}\in\operatorname{Pic}U$ has index $n$, then $n\leq
1/s(R)$. In particular, if $s(R)>1/2$ then $\operatorname{Pic}U$ is torsion-
free.

###### Proof.

Let $\mathcal{L}\in\operatorname{Pic}U$ with index $n$, say
$\varphi\colon\mathcal{O}_{U}\xrightarrow{\ \ }\mathcal{L}^{n}$ is an
isomorphism. Then, by Section 4.4, we have $n\cdot
s(R)=s\bigl{(}C(\mathcal{L},\varphi)\bigr{)}\leq 1$. ∎

In the same spirit, we also get a generalization of [CST16, Corollary 3.7].

######  Corollary 5.2.

Let $\mathcal{A}$ be an ample line bundle on a globally $F$-regular projective
variety $Y$ over $\mathcal{k}$, let $A\coloneqq\bigoplus_{i\geq
0}H^{0}(Y,\mathcal{A}^{i})$ be the associated section ring of $Y$. If
$\mathcal{A}=\mathcal{L}^{n}$ for another line bundle $\mathcal{L}$, then
$n\leq 1/s(A_{O})$, where $O\in\operatorname{{Spec}}A\eqqcolon C(Y)$ is the
origin of the affine cone C(Y).

###### Proof.

First of all, remember $Y$ is globally $F$-regular if and only if $A$ is
strongly $F$-regular, by the work [SS10]. In particular $s(A_{O})>0$. Let
$B\coloneqq\bigoplus_{i\geq 0}H^{0}(Y,\mathcal{L}^{i})$. Then $A$ is the
$n$-th Veronese subring of $B$. By taking (strict) Henselizations at the
origin we get an inclusion $A_{O}^{\textnormal{sh}}\subset
B_{O}^{\textnormal{sh}}$, which is cyclic of Veronese-type of index $n$.
Notice that $A_{O}^{\textnormal{sh}}$ is a strongly $F$-regular
$\mathcal{k}$-rational germ of $F$-signature $s(A_{O})$. Therefore, $1\geq
s(B_{O}^{\textnormal{sh}})=n\cdot s(A)$. ∎

######  Corollary 5.3.

With the same setup as in Section 5, the torsion of $\operatorname{Cl}Y$; the
divisor class group of $Y$, is bounded by $1/s(A_{O})$.

###### Proof.

We simply apply [Har77, Chapter II, Exercise 6.3]. Thus we have a SES

$0\xrightarrow{\ \ }\mathbb{Z}\overset{u}{\xrightarrow{\ \
}}\operatorname{Cl}Y\overset{v}{\xrightarrow{\ \
}}\operatorname{Cl}\bigl{(}C(Y)\bigr{)}\xrightarrow{\ \ }0.$

On the other hand,
$\operatorname{Cl}\bigl{(}C(Y)\bigr{)}\cong\operatorname{Cl}\bigl{(}\operatorname{{Spec}}A_{O}\bigr{)}\cong\operatorname{Pic}U$,
where $U$ is the regular locus of $\operatorname{{Spec}}A_{O}$. Therefore, by
Section 5, the torsion of this group is bounded by $1/s(A_{O})$.

These two facts together imply the torsion of $\operatorname{Cl}Y$ is bounded
by $1/s(A_{O})$. Indeed, let $D\in\operatorname{Cl}Y$ of order $n$. Then
$nv(D)=v(nD)=0$, so we have the index of $v(D)$, say $m\leq 1/s(A_{O})$,
divides $n$. In particular, $mD$ belongs to the kernel of $v$, so there is
$l\in\mathbb{Z}$ such that $u(l)=mD$, but then $(n/m)\cdot u(l)=(n/m)mD=nD=0$,
i.e. $u((n/m)l)=0$, which implies $l=0$. Hence, $mD=0$, so $n\mid m$, whence
$n=m$ and $n\leq s(A_{O})$ as desired. ∎

######  Remark 5.4.

Since $Y$ in Section 5 is normal, we have
$\operatorname{Pic}Y\subset\operatorname{Cl}Y$. Therefore, this result also
bounds the torsion of the Picard group of $Y$.

Recall that globally $F$-regular varieties are an analog of log-Fano varieties
in characteristic zero, see [SS10] for full details on this analogy. It is in
this regard that Section 5 should be compared to [Xu14, Proposition 1].
Concretely, if $D$ is an order $n$ Weil divisor414141Meaning $nD\sim 0$ and
$n$ is the smallest positive integer with this property. on $(Y,\Delta)$, a
log-Fano pair, then the corresponding Veronese-type cyclic cover
$Y^{\prime}\xrightarrow{\ \ }Y$ defines a finite surjective morphism of degree
$n$. Therefore, as a direct application of [Xu14, Proposition 1] and its
proof, we have

$n\leq C/\textnormal{vol}(K_{Y}+\Delta),$

where $C=C(r,\dim Y)$ is a constant depending only on $r$ the index of
$K_{Y}+\Delta$ (as $\mathbb{Q}$-Cartier divisor) and the dimension of $Y$.
This constant is shown to exists in the fundamental paper [HMX14, Corollary
1.8]. Then, it is natural to ask:

######  Question 5.5.

What is the relation between the numbers $1/s(A_{O})$ and
$C/\textnormal{vol}(K_{Y}+\Delta)$?

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