# EDGE: The direct link between mass growth history and the extended stellar
haloes of the faintest dwarf galaxies

Alex Goater1, Justin I. Read1, Noelia E. D. Noël1, Matthew D. A. Orkney2,
Stacy Y. Kim1, Martin P. Rey3, Eric P. Andersson4, Oscar Agertz5, Andrew
Pontzen6, Roberta Vieliute7, Dhairya Kataria8 and Kiah Jeneway9
1Department of Physics, University of Surrey, Guildford, GU2 7XH, UK
2ICCUB, Universitat de Barcelona, Martí i Franquès 1, 08028 Barcelona, Spain
3Sub-department of Astrophysics, University of Oxford, DWB, Keble Road,
Oxford, OX1 3RH, UK
4American Museum of Natural History, 200 Central Park West, New York, NY
10024, USA
5Lund Observatory, Division of Astrophysics, Department of Physics, Lund
University, Box 43, SE-221 00 Lund, Sweden
6Department of Physics and Astronomy, University College London, London, WC1E
6BT, UK
7School of Physics and Astronomy, University of Southampton, Southampton, SO17
1BJ, UK
8School of Physics Engineering and Computer Science, University of
Hertfordshire, College Lane, Hatfield, AL10 9AB, UK
9School of Physics and Astronomy, University of Kent, Canterbury, CT2 7NH, UK

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Ultra-faint dwarf galaxies (UFDs) are commonly found in close proximity to the
Milky Way and other massive spiral galaxies. As such, their projected stellar
ellipticity and extended light distributions are often thought to owe to tidal
forces. In this paper, we study the projected stellar ellipticities and faint
stellar outskirts of tidally isolated ultra-faints drawn from the ‘Engineering
Dwarfs at Galaxy Formation’s Edge’ (EDGE) cosmological simulation suite.
Despite their tidal isolation, our simulated dwarfs exhibit a wide range of
projected ellipticities ($0.03<\varepsilon<0.85$), with many possessing
anisotropic extended stellar haloes that mimic tidal tails, but owe instead to
late-time accretion of lower mass companions. Furthermore, we find a strong
causal relationship between ellipticity and formation time of an UFD, which is
robust to a wide variation in the feedback model. We show that the
distribution of projected ellipticities in our suite of simulated EDGE dwarfs
matches well with that of 21 Local Group dwarf galaxies. Given the ellipticity
in EDGE arises from an ex-situ accretion origin, the agreement in shape
indicates the ellipticities of some observed dwarfs may also originate from a
similar non-tidal scenario. The orbital parameters of these observed dwarfs
further support that they are not currently tidally disrupting. If the
baryonic content in these galaxies is still tidally intact, then the same may
be true for their dark matter content, making these galaxies in our Local
Group pristine laboratories for testing dark matter and galaxy formation
models.

###### keywords:

galaxies: dwarf; galaxies: formation; galaxies: stellar content; galaxies:
structure

††pubyear: 2022††pagerange: EDGE: The direct link between mass growth history
and the extended stellar haloes of the faintest dwarf galaxies–EDGE: The
direct link between mass growth history and the extended stellar haloes of the
faintest dwarf galaxies

##  1 Introduction

The Local Group (hereafter LG) of galaxies offers an excellent laboratory to
constrain the $\Lambda$CDM paradigm. The satellite systems orbiting the Milky
Way (hereafter MW) allow us to investigate the processes and feedback effects
governing galaxy formation and evolution in exquisite detail. In particular,
the LG is host to the smallest galaxies known to date, the ultra-faint dwarfs
(hereafter UFDs). With the faintest containing a few thousand stars and the
brightest having a luminosity of $\sim 10^{5}$ L⊙, UFDs represent the extreme
lower limit of the galaxy luminosity function (Simon, 2019), carrying key
evidence that can shed light on fundamental galactic processes (Battaglia et
al., 2013; Simon, 2019; Agertz et al., 2020; Sales et al., 2022). These
systems are the oldest, most chemically primitive (Bromm & Yoshida, 2011;
Kirby et al., 2013; Frebel et al., 2014; Chiti et al., 2021) and most dark
matter dominated (Brown et al., 2014; Bullock & Boylan-Kolchin, 2017;
Zoutendijk et al., 2021; Battaglia & Nipoti, 2022; Collins & Read, 2022)
systems in the Universe and, hence, make excellent laboratories to constrain
the nature of the mysterious dark matter.

The first major milestone in the search for UFDs was brought about by the
advent of digital surveys, such as the Sloan Digital Sky Survey (SDSS), where
searches for these faint systems in our LG were completed up to a surface
brightness limit of 25.5 mag arcsec-2 (Whiting et al., 2007; Koposov et al.,
2008; Willman et al., 2005; Belokurov et al., 2007, 2008, 2009, 2010;
McConnachie, 2012; Belokurov, 2013). Thanks to deep imaging and spectroscopy
with modern telescopes such as the Dark Energy Camera (DECam) (Flaugher et
al., 2015), the past decade has seen an explosion in the number of faint dwarf
galaxies discovered, with 68 now known around the MW, 9 around M31 and 15
around galaxies beyond the LG. These newly discovered UFDs in our LG, with
solar luminosities $\lesssim 10^{5}L_{\odot}$, have vastly improved our
understanding of these systems (Bechtol et al., 2015; Koposov et al., 2015;
Koposov et al., 2018; Drlica-Wagner et al., 2015, 2016; Martin et al., 2015;
Kim et al., 2015a, b; Kim & Jerjen, 2015; Laevens et al., 2015; Torrealba et
al., 2016, 2018; Homma et al., 2016, 2018; Simon, 2019; Drlica-Wagner et al.,
2021; Cerny et al., 2021; Collins et al., 2022; Sand et al., 2022).
While UFDs offer the promise of unique constraints on galaxy formation models
(Collins & Read, 2022) and the nature of dark matter (Simon, 2019), this is
made more challenging by the potential impact of tidal forces from nearby host
galaxies like the MW and M31 (Read et al., 2006; Łokas et al., 2013; Collins
et al., 2017; Mazzarini et al., 2020). These larger systems can severely
disrupt the environment of nearby smaller galaxies by stripping their dark
matter content and then their stellar content, thus deforming the structure of
the latter. Evidence for tides has been claimed for many nearby UFDs based on
their extended light profiles (Li et al., 2018; Mutlu-Pakdil et al., 2019;
Pozo et al., 2022), apparent tidal features (Muñoz et al., 2010; Muñoz et al.,
2012), velocity gradients in their outermost stars (Sand, 2017) and/or
constraints on their orbits (Küpper et al., 2017). However, the latest
constraints on UFD orbits suggest that a number are tidally isolated at
present (Simon, 2018; Fritz et al., 2018; McConnachie & Venn, 2020; Pace et
al., 2022). A notable example is Tucana II, a LG UFD located at a distance of
$\sim 58$ kpc away from the MW (Pace et al., 2022), which exhibits multiple
features characteristic of tidal isolation (Chiti et al., 2021). Chiti et al.
(2021, 2022) have recently reported member stars anisotropically distributed
around Tucana II, up to nine half-light radii from its galactic centre. These
member stars reach out to and even extend past our calculated tidal radius
estimates of Tucana II, $r_{\textnormal{t}}\approx 0.76$ kpc. This indicates
that there are plenty of stars not of tidal origin in the region
$r_{\textnormal{1/2}}<R<r_{\textnormal{t}}$, where $r_{\textnormal{1/2}}$ is
the half-light radius.
Similar extended structures have also been discovered around Boötes I (Filion
& Wyse, 2021; Longeard et al., 2022; Waller et al., 2023), Ursa Major I
(Waller et al., 2023), Coma Berenices (Waller et al., 2023), Ursa Minor
(Sestito et al., 2023a), Fornax (Yang et al., 2022), Hercules (Longeard et
al., 2023), and Sculptor (Sestito et al., 2023b). One possible explanation for
the presence of such anisotropic stars is dwarf-dwarf tidal interactions prior
to infall (Genina et al., 2022). However, it is interesting to ask whether
such extended light profiles and apparent ‘tidal distortions’ can occur via
other means for tidally isolated systems. For example, Tarumi et al. (2021)
argue that galaxy mergers could explain the extended stellar halo around
Tucana II.
Deason et al. (2022) discerned how different galaxy models affect the
contribution of accreted stars to dwarf galaxy haloes, finding that minor
mergers hold the strongest clues for dwarf galaxy models.

Before the aforementioned discoveries, the existence of stellar haloes at
these low mass scales remained inconclusive since they are thought to relate
to early mergers that are less common as one goes down the mass scale of
galaxies (Deason et al., 2022). The mere existence of stellar haloes around
dwarf galaxies helps to constrain the nature of galaxy formation and dark
matter on the lowest mass scales.
In this paper, we study the ellipticity and extended light around tidally
isolated UFDs drawn from the ‘Engineering Dwarfs at Galaxy Formation’s Edge
(EDGE) project (Agertz et al., 2020). These tidally isolated galaxies are
found to possess anisotropic and extended stellar outskirts, resembling the
structure of tidal tails. The existence of these stellar haloes in the EDGE
UFDs prompts us to look into their morphological origins, given that we
intrinsically rule out the possibility of tidal isolation. Furthermore, we
provide a comparison between the morphologies of the full suite of the EDGE
simulations and an observed sample of UFDs, we use a Maximum Likelihood
technique that was developed to calculate the observed structural parameters
of dwarf galaxies (Martin et al., 2008, 2016). If the projected ellipticities
for the entire EDGE simulation suite reasonably match observed dwarfs, then
given that the EDGE UFDs are designed to be isolated from other massive
systems, it is possible that tidal features in local UFDs are not necessarily
due to tides and instead originate from a non-tidal scenario.
This paper is organised as follows. In Section 2, we discuss the setup of the
EDGE simulations, the prerequisites we place on the selection of observed data
samples, as well as the methods we use to derive the EDGE structural
parameters in the fashion of an observational astronomer. In Section 3, we
describe the results obtained pertaining to the projected ellipticities of the
EDGE simulations; the relationship between ellipticity and formation time,
their comparison to observations, as well as the extended stellar light of the
faintest galaxies. We then examine these results and look towards their
implications, discussing the origin of stellar ellipticity in the EDGE UFDs.
Finally, we draw conclusions in Section 4.

##  2 Method

###  2.1 EDGE Simulations

The suite of simulations examined in this work belongs to the EDGE project.
These simulations were analysed using the tangos database package (Pontzen &
Tremmel, 2018) and the pynbody analysis package (Pontzen et al., 2013). A more
comprehensive review of the simulations, and their underlying sub-grid
physics, is found in Agertz et al. (2020).

The EDGE project is designed to study isolated UFDs with halo mass
$10^{9}<M/M_{\odot}<5\times 10^{9}$, in a simulated 50 Mpc void region. The
simulations are initialised to assume cosmological parameters
$\Omega_{\textnormal{m}}=0.309$, $\Omega_{\mathrm{\Lambda}}=0.691$,
$\Omega_{\mathrm{b}}=0.045$ and $H_{0}=67.77$ $\mathrm{km}$ $\mathrm{s^{-1}}$
$\mathrm{Mpc^{-1}}$, taken from the PLANCK satellite 2013 data release (Planck
Collaboration et al., 2014). The volume is initially simulated at a $512^{3}$
resolution, from $z=99$ to $z=0$. The largest void volume in this region is
then selected and resimulated to a resolution of $2048^{3}$, with the
inclusion of an appropriate small scale power to the grid. Within this
resimulated region, the HOP halo Finder (Eisenstein & Hut, 1998) is
implemented to find dark matter haloes at $z=0$. Once a suitably isolated
candidate has been confirmed, the halo is resimulated via the implementation
of a zoom-in simulation technique (Katz & White, 1993; Oñorbe et al., 2014),
up to redshift, $z=0$.

We approach a maximum spatial resolution of $\sim$3 pc in the hydrodynamic
grid. This high spatial resolution allows for the accurate injection of energy
from a supernova, thus reducing the need for mechanisms required to prevent
over-cooling of the supernovae-heated gas (see e.g Agertz et al., 2013; Agertz
et al., 2020; Kimm et al., 2015; Wheeler et al., 2019).

The adaptive mesh refinement hydrodynamics code, ramses (Teyssier, 2002), is
used to model the evolution of both baryonic matter and dark matter. The
baryonic physics model makes use of a Schmidt law (Schmidt, 1959) to describe
star formation in cells of gas that satisfy the required temperature and
density (see Agertz et al., 2020). Initially, each stellar particle represents
$300M_{\odot}$ and can be thought of as a mono-age stellar population
described by a Chabrier initial mass function (Chabrier, 2003).

The epoch of reionisation is modelled as a time-dependent uniform UV
background at $z=8.5$ (Haardt & Madau, 1996). The reader may refer to Rey et
al. (2020) for details on the specific implementation of this model.

###  2.2 Selection of observed candidates

To provide a clear comparison of the shape distribution of observed dwarfs to
the shape distribution of the EDGE simulation suite, we collate a refined list
of dwarf galaxies belonging to the MW, presented in Table 1. We place two
stringent constraints when creating this sample of galaxies. The first
constraint is related to the mass of the galaxy, where we only include
observed dwarfs that have stellar masses within approximately one order of
magnitude of the EDGE dwarfs. For the second constraint, we adopt a list of
observed galaxies that are thought to be tidally undisturbed (Kim et al., in
preparation), since the EDGE galaxies were specifically selected due to their
isolation. This second requirement is possible to meet thanks to the latest
orbits taken from Gaia EDR3 (Gaia Collaboration et al., 2021), which Pace et
al. (2022) then combine with accurate photometry to determine the systemic
proper motions, thus providing insight into their respective tidal
interactions. It should be noted that we use the pericentre values including
the influence of the LMC, to provide us with the most authentic orbital
scenarios.

The metric for isolation of an UFD is categorised as a ratio between its tidal
and half-light radius. However, the time variation of the tidal radius and its
dependency on the mass distribution of both systems involved are usually
poorly understood; hence the tidal radius remains ambiguously defined (Simon,
2019). A solution for this uncertainty is to approximate the tidal radius as
the position where the total force (from the satellite and host) matches the
centrifugal acceleration needed to stay on the same orbit as the satellite.
The tidal radius is given as follows (von Hoerner, 1957; Binney & Tremaine,
2008; Simon, 2019):

$r_{\textnormal{t}}=\left(\frac{m_{\mathrm{dwarf}}}{3M_{\textnormal{MW}}}\right)^{\frac{1}{3}}d,$
(1)

where $r_{\textnormal{t}}$ is the tidal radius, $m_{\textnormal{dwarf}}$ is
the dwarf galaxy mass, $M_{\textnormal{MW}}$ is the MW mass enclosed within
the dwarf orbital radius, and $d$ is the distance between the dwarf and the
galactic centre of the host system. It should be noted that Equation 1 is only
an approximation, as it assumes a point-mass approximation for the dwarf and
the MW, purely radial motion between the dwarf and the MW, and stars within
the dwarf moving on purely radial orbits (Read et al., 2006).

Following Simon (2019), we define dwarfs to be “tidally isolated” if they have
$\frac{r_{\textnormal{t}}}{r_{\textnormal{1/2}}}$ > 3\. In practice, even
these dwarfs are likely stripped to some degree. Shipp et al. (2023) recently
showed using the FIRE simulations that many "intact" looking satellites have
tidal tails, with $\sim 66\%$ of a total 64 stream progenitors being mistaken
for intact satellites. However, it will be challenging to detect any stripping
from their extended light distributions. The only exception to this is dwarfs
that interact with other dwarfs before infall to the MW. These can be on
apparently benign orbits, where the galaxy shows no present-day sign of prior
tidal interactions, despite having experienced significant stripping in the
past (Genina et al., 2022). At present, no method has been proposed for
distinguishing such dwarfs from genuinely tidally isolated systems. As such,
we must accept the possibility that some of our samples will be contaminated
by such tidally affected systems. Nevertheless, in a $\Lambda$CDM cosmology,
these are expected to be quite rare, with Genina et al. (2022) stating that
they find 9 out of 212 simulated luminous dwarfs as analogues of this
scenario.

Dwarf Galaxy | Ellipticity, $\varepsilon$ | Pericentres (kpc) | Half-light radius, $r_{1/2}$ (kpc) | Tidal radius, $r_{\textnormal{t}}$ (kpc) | Referencesa
---|---|---|---|---|---
Sculptor | 0.32 ± 0.03 | 44.9 ${}^{+4.3}_{-3.9}$ | 0.279 ± 0.016 | 1.050 | (1, 12)
Leo I | 0.21 ± 0.03 | 47.5 ${}^{+30.9}_{-24.0}$ | 0.270 ${}^{+0.017}_{-0.016}$ | 1.082 | (1, 12)
Leo II | 0.13 ± 0.05 | 61.4 ${}^{+62.3}_{-34.7}$ | 0.171 ± 0.010 | 0.970 | (1, 12)
Ursa Minor | 0.56 ± 0.05 | 55.7 ${}^{+8.4}_{-7.0}$ | 0.405 ± 0.021 | 1.421 | (1, 12)
Sextans | 0.35 ± 0.05 | 82.2 ${}^{+3.8}_{-4.3}$ | 0.456 ± 0.015 | 1.750 | (1, 12)
Carina | 0.33 ± 0.05 | 77.9 ${}^{+24.1}_{-17.9}$ | 0.311 ± 0.015 | 1.312 | (1, 12)
Canes Venatici I | 0.39 ± 0.03 | 84.5 ${}^{+53.6}_{-37.2}$ | 0.437 ± 0.018 | 1.717 | (2, 12)
Canes Venatici II | 0.52 ${}^{+0.10}_{-0.11}$ | 47.4 ${}^{+46.8}_{-29.7}$ | 0.071 ± 0.011 | 0.437 | (2, 12)
Draco | 0.31 ± 0.02 | 58.0 ${}^{+11.4}_{-9.5}$ | 0.231 ± 0.017 | 1.180 | (2, 12)
Ursa Major I | 0.80 ± 0.04 | 49.9 ${}^{+46.2}_{-15.6}$ | 0.295 ± 0.028 | 0.962 | (2, 12)
Leo T | 0.29 ${}^{+0.12}_{-0.14}$ | > 250 | 0.118 ± 0.011 | - | (2)
Leo IV | 0.49 ± 0.11 | 66.8 ${}^{+60.7}_{-44.1}$ | 0.114 ± 0.013 | 0.527 | (3, 12)
Leo V | 0.50 ± 0.15 | 165.8 ${}^{+5.8}_{-49.2}$ | 0.049 ± 0.016 | 0.636 | (3, 12)
Leo P | 0.52 | - | - | - | (4)
Reticulum II | 0.60 ± 0.10 | 37.0 ${}^{+2.9}_{-5.3}$ | 0.051 ± 0.003 | 0.262 | (5, 12)
Pisces II | 0.40 ± 0.10 | 130.5 ${}^{+70.1}_{-72.3}$ | 0.060 ± 0.010 | 0.991 | (6, 12)
Eridanus II | 0.48 ± 0.04 | 114.4 ${}^{+80.9}_{-67.6}$ | 0.246 ± 0.017 | 1.683 | (7, 12)
Boötes I | 0.68 ± 0.15 | 37.9 ${}^{+7.5}_{-6.8}$ | 0.191 ± 0.008 | 0.517 | (8, 12)
Hydrus I | 0.21 ${}^{+0.15}_{-0.07}$ | 45.8 ${}^{+16.1}_{-6.0}$ | 0.053 ± 0.004 | 0.270 | (9, 12)
Pegasus III | 0.46 ${}^{+0.18}_{-0.27}$ | 141.0 ${}^{+87.8}_{-79.3}$ | 0.078 ${}^{+0.031}_{-0.025}$ | 1.151 | (10, 12)
Hercules | 0.67 ± 0.03 | 67.4 ${}^{+15.5}_{-16.1}$ | 0.216 ± 0.020 | 0.877 | (11, 12)
Table 1:  Observed ellipticities, pericentres, half-light radii and calculated
tidal radii for 21 dwarf galaxies in our LG. aReferences: (1) Irwin &
Hatzidimitriou (1995), (2) Martin et al. (2008), (3) de Jong et al. (2010),
(4) McQuinn et al. (2015), (5) Mutlu-Pakdil et al. (2018), (6) Belokurov et
al. (2010), (7) Crnojević et al. (2016), (8) Longeard et al. (2021), (9)
McConnachie et al. (2006), (10) Kim et al. (2015b), (11) Sand et al. (2009),
(12) Pace et al. (2022).

###  2.3 Structural parameters

To derive the structural parameters of the EDGE simulations, we employ a
Maximum Likelihood technique similar to the one utilised by observational
astronomers to uncover the structural parameters for dwarf galaxies in the
SDSS (Martin et al., 2008) and PandAS (Martin et al., 2016) surveys111PandAS
is an astronomical survey focused on the content and structure of M31 and M33
(Ibata et al., 2014).. In this paper, the method is used in such a way as to
treat the simulations as if they were two-dimensional observations projected
onto the sky, so that the resemblance between the EDGE simulations and
observational data can be accurately analysed.

Equation 2 calculates the probability each star particle contributes to the
structural parameters, given its respective position. As the stellar particles
in EDGE are representative of a stellar mass on the order of
$10^{2}M_{\odot}$, we can determine how much each stellar particle should
contribute to the calculation of the structural parameters by weighting via
the stellar mass.

$l_{i}=\frac{1.68^{2}N_{*}}{2\pi
r_{\textnormal{1/2}}^{2}(1-\varepsilon)}\exp\left(\frac{-1.68r_{i}}{r_{\textnormal{1/2}}}\right)\frac{m_{i}}{M_{*}},$
(2)

where the relation between the half-light radius and the exponential scale
radius of the profile is $r_{\textnormal{1/2}}\approx 1.68r_{\textnormal{e}}$,
$N_{*}$ is the number of stars in the sample, $M_{*}$ is the total stellar
mass, $m_{i}$ is individual stellar mass, $\varepsilon$ is the ellipticity
defined as $\varepsilon=1-b/a$, with $b/a$ as the minor-to-major-axis ratio of
the system, $\theta$ is the position angle of the major axis, defined as East
of North, $r_{\textnormal{1/2}}$ is the half-light radius of its assumed
exponential radial profile, and $r_{i}$ is the elliptical radius. Here,
$r_{i}$, is related to the spatial positions $x_{i}$ and $y_{i}$ as follows,

$r_{i}=\left(\left(\frac{1}{1-\varepsilon}(x_{i}\cos\theta-
y_{i}\sin\theta)\right)^{2}+\left(x_{i}\sin\theta+y_{i}\cos\theta\right)^{2}\right)^{\frac{1}{2}}$
(3)

A more comprehensive mathematical approach may be found in Martin et al.
(2008), where a similar exponential model is used to describe a low stellar
density.

The total log-likelihood is calculated by taking the summation of all the
logged individual probabilities,

$\log\mathcal{L}=\sum_{i}\log l_{i}$ (4)

We determine the most likely shape parameters
$(\varepsilon,\theta,r_{\textnormal{1/2}})$ for each EDGE UFD with the emcee
code (Foreman-Mackey et al., 2013). This is a Markov Chain Monte Carlo (MCMC)
method, and it is found to definitively converge upon the structural
parameters of the galaxy when 50 walkers are run over a total of 600 steps.
Similar to Martin et al. (2016), we place flat priors for the three parameters
such that $0\leqslant\varepsilon<1$, $\theta$ is in an interval of 180
degrees, and $r_{\textnormal{1/2}}>0$.

###  2.4 Cutting on surface brightness

Since one of our main aims is to compare simulations to observations, we
attempt to replicate the same methods and techniques that observational
astronomers use. Therefore, during the MCMC calculation we apply a surface
brightness cut to the EDGE simulations. This creates the effect that our mock
observations of simulations are limited by surface brightness, just as
observations are through the use of telescopes.

Inspired by the literature, we place two different surface brightness cuts on
the simulated UFDs. The first cut is at 25.5 mag arcsec-2 since this was the
surface brightness limit of the SDSS telescope (Whiting et al., 2007; Koposov
et al., 2008) used to observe several MW dwarf galaxies in Table 1. We place
the next surface brightness cut at 30 mag arcsec-2 to highlight what we should
be able to predict with the most modern detection instruments. The latter cut
is a more optimistic approach inspired by the contemporary advances of
observational astrophysics in recent years, i.e. the Dark Energy Survey (DES)
(Abbott et al., 2021), the DECam Local Volume Exploration survey (DELVE)
(Drlica-Wagner et al., 2021), and within the next few years, the Vera C. Rubin
Observatory (Kahn, 2018).

###  2.5 Gaussian KDE fitting method

The ellipticity we calculate with our Maximum Likelihood technique is the
projected ellipticity of the UFD and not the true ellipticity. Accordingly, we
orient each EDGE galaxy around 100 random viewing angles to create a
probability distribution function (hereafter PDF) of projected ellipticities.

All our PDFs are created with a kernel density estimator (KDE), used to
convolve our data with a Gaussian kernel. A certain degree of smoothing is
employed to produce the PDFs of projected ellipticity for both EDGE and
observations. We use Silverman’s Rule to define the degree of this smoothing
so that the PDF provides a match to the underlying data. With each PDF, we
include a rug plot displaying the data points from which the distributions are
constructed.

##  3 Results

The GenetIC code (Stopyra et al., 2021) is utilised within the EDGE
simulations to ‘genetically modify’ the initial conditions for an EDGE UFD
($t_{\mathrm{form}}=2.4$ Gyrs) to form three unique variations at earlier
($t_{\mathrm{form}}=2.8$ Gyrs) and later times ($t_{\mathrm{form}}=$ 3.1 Gyrs,
3.6 Gyrs) (Rey et al., 2019). We define formation time as the time when the
galaxy has assembled 50% of its final mass at $z=0$.

Rey et al. (2019) studied the mass accretion histories for these UFDs, and
revealed that the later-forming variations assemble their stellar mass from
late-time dry mergers. Such an assembly history leads to extremely low surface
brightness and an increase in the half-light radius (i.e. an increase in the
size of the galaxy). Conversely, the mass accretion history of the earlier-
forming galaxy primarily consisted of stars that had formed in-situ, leading
to a higher surface brightness and a decrease in the half-light radius.

Figure 1: PDFs of projected ellipticity for a fiducial EDGE UFD (original -
dashed) and three variations of this simulated UFD, genetically modified to
have formation times; earlier (dotted), later (blue), and latest (green).
These four distributions of projected ellipticities are created at a surface
brightness cut of 30 mag arcsec-2. The coloured lines in the underlying rug
plot represent the individual projected ellipticities in the respective
distribution. Figure 2: PDFs of projected ellipticity for a fiducial EDGE UFD
created in an environment with limited feedback (original - purple dashed) and
three variations of this simulated UFD, genetically modified to have formation
times; earlier (pink dotted), later (blue), and latest (green), also created
with the ‘weak feedback’ model. Once again, these four distributions of
projected ellipticities are created at a surface brightness cut of 30 mag
arcsec-2.The coloured lines in the underlying rug plot represent the
individual projected ellipticities in the respective distribution.

###  3.1 EDGE projected ellipticity correlates with formation time

Figure 1 displays the PDFs of projected ellipticity for the same fiducial UFD
and three variations of this fiducial UFD at earlier and later times. The
projected ellipticities here are taken at a surface brightness cut of 30 mag
arcsec-2.

A systematic shift from a lower projected ellipticity to a higher projected
ellipticity is seen from the peak of the distributions. This shift in
projected ellipticity correlates with the formation time of the UFDs, i.e.
earlier assembly times have lower ellipticities, and later assembly times have
greater ellipticities. Our findings represent the first evidence in support of
a causal relationship between the time of main halo formation in an UFD and
the galactic stellar ellipticity of an UFD.

To provide complete clarity, the ellipticity of the stellar content is
directly related to the distribution of the stellar structure (Rey et al.,
2019). The distribution of the stellar content in UFDs will depend on whether
star-forming gas is available. If it is, the majority of the UFD forms via in-
situ star formation, and this leads to a rounder, more compact shape. However,
if the gas escapes the UFD, no more star formation will occur, and the UFD
will form via ex-situ mergers, leading to a fainter, more elliptical shape.
Therefore, the ellipticity is determined by the method of formation of the
stellar component, which is decided by the availability of gas in the galaxy.
However, this availability of gas is decided by the formation time of the main
halo and whether it is large enough at reionisation to retain its gas. And so,
the formation time of the main halo dictates the availability of gas, which in
turn decides whether the stars originate in-situ or ex-situ, i.e. where the
ellipticity arises from.

Furthermore, we must now consider that this relationship is affected by how
much stellar mass forms in-situ in the centre of the UFD, and this becomes
very sensitive to the choice of feedback model. A model with less feedback
will produce a lot more in-situ stars, as less feedback implies an increase in
star-forming gas. There will also be an increase in the late-time accreted
component of the stellar content, however, it becomes uncertain how our
relationship will now follow with ellipticity.

Figure 2 displays the PDFs of projected ellipticity at a surface brightness
cut of 30 mag arcsec-2 for the four variations of the UFD in Figure 1, now
implemented with a ‘weak-feedback’ model. This model artificially limits the
efficiency of the supernovae wind driving by placing numerical limits on the
maximum supernovae gas temperatures and velocities (see Agertz et al., 2013;
Agertz et al., 2020).

Once again, we see a clear systematic shift from a lower projected ellipticity
to a higher projected ellipticity from the peaks of the distributions. This
shift in projected ellipticity correlates with the formation time of the UFDs
going from lower ellipticities at earlier assembly times to higher
ellipticities at later assembly times. Therefore, even if our feedback model
does not provide an absolute description of reality, under the regime of this
‘weak feedback’ model, the relative ordering of ellipticity with mass growth
histories stands. Our relationship is robust to the feedback physics we use,
and the stars at large radii that make up these extended tails are found
within two different feedback models, making them a strong prediction from our
simulations.

We observe that the PDFs shown in Figure 1 have a great deal of overlap,
calculating a 12% chance to mistake the projected ellipticity of the earlier-
forming UFD with the latest-forming UFD. Meanwhile, there is only a 2% chance
to mistake the projected ellipticity of the earlier-forming UFD with the
latest-forming UFD, from the PDFs shown in Figure 2.

Figure 3: PDFs of projected ellipticity from the sample of observed dwarf
galaxies in Table 1 (black/grey) and the EDGE simulation suite (blue/red). The
median, 1$\sigma$, and 2$\sigma$ confidence intervals on the observed
distribution are calculated from the uncertainties of the ellipticities in
Table 1. There are two PDFs for the EDGE simulations, the blue dashed line
represents the distribution of projected ellipticities of the EDGE simulations
when observed up to a surface brightness limit of 25.5 mag arcsec-2, and the
red dotted line represents the distribution of projected ellipticities of the
EDGE simulations when observed up to a surface brightness limit of 30 mag
arcsec-2. The 1$\sigma$ and 2$\sigma$ confidence intervals on the EDGE data
result from a bootstrapping technique used to determine the variation in the
EDGE ellipticity distributions, which are randomly sampled from the weighting
by formation time. In the legend, the acronym ‘SB’ denotes surface brightness.
The coloured lines in the underlying rug plot represent the individual
projected ellipticities in the respective distribution.

###  3.2 Projected ellipticity in EDGE vs observed dwarf galaxies

The PDFs of the projected ellipticities for the EDGE simulation suite are
shown in Figure 3. Here, we use the full suite of the EDGE simulations to make
our comparison with observations. To see a tabulated summary of the 10 EDGE
UFDs, please refer to Rey et al. (2022). We note that only five of these ten
UFDs are unique realisations; we construct the additional five UFDs from
genetic modifications applied to the original five.
The EDGE PDFs are given at surface brightness cuts of 25.5 mag arcsec-2 (blue
dashed) and 30 mag arcsec-2 (red dotted). Also shown in Figure 3 is the PDF of
the projected ellipticities created from the observed ellipticity samples
(black) in Table 1, along with the respective confidence intervals at 68%
(1$\sigma$) and 95% (2$\sigma$) variance, calculated from the uncertainties of
the observed samples.
From the newly discovered relationship between an UFDs formation time and
ellipticity, we know that our more elliptical galaxies are later forming and
the less elliptical galaxies are earlier forming. Studying the populous of
haloes in the EDGE volume, it becomes clear that the haloes forming earlier
and later are significantly rarer. Therefore, when choosing to weight our EDGE
PDFs of projected ellipticity, we do so via their formation times opposed to
stellar mass. We assume that formation time is the dominant variable beyond
mass when it comes to the ellipticity of these galaxies. To weight the EDGE
PDFs by formation time, we use our probability density function of haloes in
the EDGE volume to weight the contribution of each EDGE UFD to the shape
distribution. We note that we only include the formation times of systems with
similarly sized halo masses to those of our EDGE samples. This weighting
ensures that the EDGE PDFs are not biased by formation time. The UFDs with a
more common formation time will contribute more to the EDGE PDFs, and the UFDs
with a rarer formation time will contribute less to the EDGE PDFs, thus
creating a more accurate comparison to the PDF from our sample of observed
dwarfs.
The comparison between EDGE PDFs presented in Figure 3 displays the peak of
the EDGE distribution shifting towards higher ellipticities at a fainter
surface brightness cut of 30 mag arcsec-2. As these galaxies are viewed out to
a fainter surface brightness limit, the number of stars increases in our
‘observations’ of the EDGE UFDs. Therefore, the increase in projected
ellipticity confirms that these galaxies have an elongated stellar
distribution along one axis.
It is important to highlight the inhomogeneity of surface brightness limits
the galaxies in our sample were observed at. A number of ellipticity
measurements were taken at a surface brightness limit $\sim$ 25.5 mag
arcsec-2, but for some galaxies observed more recently, the surface brightness
limits extend up to $\sim$ 30 mag arcsec-2, thanks to improved detection
instruments. Following the above-mentioned relation in EDGE, if some galaxies
were observed out to a fainter surface brightness limit, they would sit at
higher ellipticities on the observed PDF, and this could explain the existence
of the second smaller peak in the observed distribution.
The projected ellipticities of EDGE galaxies viewed at a more luminous surface
brightness limit of 25.5 mag arcsec-2 provide good agreement with a
significant number of dwarf galaxies from the observed sample, with one of the
peaks of the blue EDGE distribution laying well within the overlap of the
larger observed peak. We also see a good agreement between the smaller peak of
observed galaxies and the fainter distribution of EDGE galaxies at 30 mag
arcsec-2. These agreements point towards a relation between the shape of the
EDGE UFDs and the sample of observed dwarfs up to a surface brightness limit
of 30 mag arcsec-2. As our EDGE UFDs are tidally isolated, this suggests a
number of the galaxies in our sample of observed dwarfs have ellipticities
that are perhaps unduly attributed to tides. This possible tidal isolation is
further reinforced by the tidal radii we calculate from the large pericentres
of the observed sample.

####  3.2.1 A cautionary note on tidal effects in dwarf galaxies

Following our discussion in Section 2.2, even though we make an effort to try
to reduce the number of tidally affected galaxies in Table 1, there is still
the possibility that tides may have some influence on observations, and even
these dwarfs in our sample are likely stripped to some degree. For example,
Shipp et al. (2023) report from their simulations that more than 50% of
satellites have tidal tails at distances 50-200 kpc from their hosts.
Consequently, this may be the reason why we see the PDF of observed galaxies
skew to higher projected ellipticities in Figure 3, where we notice from the
underlying rug plot that there is a handful of galaxies lying beyond the
second peak in the observed distribution.

Alternatively, we also have to account that the present-day orbit does not
always provide the complete history of a galaxy. Given the density of the
environment that these dwarfs live in, it is plausible they experienced
previous interactions with close-by dwarf galaxies such as the LMC or even
with each other, which may yet be another reason explaining the high
ellipticity population in our observed sample. Using the example of Fornax,
Genina et al. (2022) show that systems appearing to be tidally isolated today
can have had significant galaxy-galaxy interactions in the past. Analogues of
this scenario appearing in our observed sample would shift the distribution of
observed galaxies to a more elliptical peak. However, as previously stated in
Section 2.2, this scenario is expected to be quite rare, with Genina et al.
(2022) finding a <5% chance of a Fornax analogue from their sample of 212
simulated dwarfs.

Figure 4: Spatial distributions of the stellar content in four EDGE UFDs, with
underlying surface brightness maps created via a kernel smoothing scheme. The
projection axes of the galaxies are oriented in such a way as to maximise the
ellipticity of the stellar distribution. Top left panel shows the spatial
distribution of the original fiducial simulation. Top right panel shows the
spatial distribution of the original UFD, genetically modified to form at an
earlier time. Bottom left panel shows the spatial distribution of the original
UFD, genetically modified to form at a later time. Bottom right panel shows
the spatial distribution of the original UFD, genetically modified to form at
the latest time of the four. Notice that in each case, the extended stellar
haloes are highly anisotropic. These look like tidal tails but owe instead to
the late-time accretion of lower-mass companions. Star particles that stem
from this accretion origin are coloured cyan. Meanwhile, star particles
coloured grey represent stellar content that formed in-situ. The radial
distance of the white dashed circle represents ten projected half-light radii.

###  3.3 Origin of extended stellar content in EDGE

Figure 4 represents the present-day spatial distributions for the same UFD in
Figure 1, along with its three variations with altered formation times, and
their underlying surface brightness maps. The surface brightness maps were
created with a kernel smoothing scheme using the py-sphviewer package Benitez-
Llambay (2015).
As these are projected spatial distributions, they are susceptible to the two-
dimensional effect of appearing seemingly less elliptical than their true
galactic shapes imply. To counter this, we ensured that the stellar contents
of all the galaxies were observed from their most elliptical viewing angle.
The first key result that stands out when observing the stellar content in
Figure 4 is the elongation that the systems possess along one axis. This
elongation coincides with the stars coloured cyan, which represent stars that
come from a late-time dry accretion origin. The stars coloured grey, on the
other hand, are stars that form from an in-situ origin and we see these are
much more central in the stellar distribution.
While the UFDs in the bottom two panels of Figure 4 are the most elliptical in
terms of the overall shape of the galactic centre, the extended stellar
distributions of the top two panels are still very clearly discerned upon
inspection.
The centres of the two UFDs residing in the top panels of Figure 4 are not as
elliptical, however, these galaxies still possess this extended distribution
feature, with detritus of member stars located up to $\sim$ 78 half-light
radii from the centre of the original galaxy and $\sim$ 56 half-light radii
from the centre of the earlier-forming galaxy. To provide a visual reference
for these large half-light radii, we include a white dashed circle in our
spatial distributions, for which the radial distance from the centre
represents ten half-light radii. From the surface brightness maps, we see that
these outer stars at these distances will become observable at a surface
brightness $\gtrsim$ 34 mag arcsec-2. We note how great the radial distance to
ten half-light radii is for the later-forming galaxies, implying how faint
they are compared to the earlier-forming UFDs.
The original UFD in the upper left panel of Figure 4 has an apparent elongated
distribution of stars and possesses a much larger spread of stars than either
of the later-forming galaxies. This original galaxy has an ex-situ mass
fraction of 2.96%.

The earlier-forming UFD in the upper right panel is much rounder at the
centre, which is supported by the low projected ellipticity distribution shown
in Figure 1. The earlier-forming galaxy has an ex-situ mass fraction of 1.05%,
making it the UFD with the lowest amount of ex-situ content compared to the
other three.

The later-forming UFD in the bottom left panel has less of an extended
distribution than the original and early-forming UFDs, however, its galactic
centre is more elliptical. The later-forming galaxy has an ex-situ mass
fraction of 7.31%.

Finally, the latest-forming UFD in the bottom right panel is strongly
elongated, with the most elliptical centre out of the four UFDs. The latest-
forming UFD has the highest content of ex-situ stars with a mass fraction of
95.8%, given that practically all stellar content originates from ex-situ dry
accretions. Also, the overall stellar mass for this galaxy is an order of
magnitude smaller than the other three UFDs. Justified from the peaks of the
PDFs in Figure 1, we can definitively say that the latest-forming UFD has the
most elliptical galactic centre. These PDFs consist of ellipticities taken at
a surface brightness cut of 30 mag arcsec-2, meaning they exclude the majority
of the extended stellar outskirts.

We trace back the origin of the stars in the EDGE UFDs and find that those
coloured cyan in Figure 4 accrete onto the systems through late-time dry
mergers and make up the extended part of the stellar distribution. The stellar
ellipticities in the EDGE UFDs thus originate from these late-time accretion
events of smaller haloes that had their gas quenched by reionisation before
they could merge onto the main halo. The bottom two panels of Figure 4,
showing the later-forming galaxies, exemplify this as they assemble primarily
through late-time dry accretion and have extended distributions of stars along
one axis around the galactic centres, giving them systematically larger
ellipticities.

On the other hand, the earlier-forming UFDs form their stellar content in-situ
and have less elliptical galactic centres, with systematically lower
ellipticities. However, in the spatial distributions of the two earlier-
forming galaxies, there is still a distribution of stellar material coloured
cyan, extending even further than the elongations depicted in the later-
forming galaxies. Therefore, we show that even though these galaxies formed
earlier, they still have elongated stellar content originating from the late-
time accretion events of smaller haloes. Such features clarify that the
extended starlight in EDGE is a natal characteristic of these ultra-faint
systems, and these cyan coloured stars from late-time dry accretion events are
the origin of this stellar ellipticity. For the first time, we show that the
accretion mechanism giving rise to the extended shapes emerges naturally in a
fully cosmological context, where several dark matter sub-haloes undergo
hierarchical minor mergers following $\Lambda$CDM cosmology. And the existence
of these stellar haloes around the smallest galaxies in the Universe
constrains $\Lambda$CDM at the smallest scales.

##  4 Conclusions

Within our local volume of the Universe, we find a multitude of the faintest,
most dark matter dominated galaxies. Given that these UFDs are scattered
around more massive systems, it is logical to assume that these systems are
extremely prone to tidal effects. Thus, the large projected ellipticities of
the stellar distribution and the stellar detritus of these systems can easily
be attributed to tidal deformation.

Utilising the EDGE simulations, we have shown that despite their tidal
isolation, our simulated dwarfs exhibit anisotropic extended stellar outskirts
that masquerade as tidal tails but are instead natal, owing to the origin of a
late-time dry accretion assembly. Furthermore, we revealed that UFDs with
later formation times have more elliptical stellar distributions, thus
establishing a novel connection between the shape of an UFD and its respective
formation time. This newly discovered relationship was robust to a wide
variation in the feedback model, making it a strong prediction from our
simulations. The above-mentioned results extend the conclusion found in Rey et
al. (2019), in which the authors discovered that UFDs with later formation
times have an extremely low surface brightness and a much larger stellar size.
These discoveries within the EDGE fully cosmological context are vital in
assessing the extent to which the smallest galaxies in the Universe have
mechanisms and features indistinguishable from more massive galaxies, such as
the MW, with respective fossil records detailing their history.

We studied the projected stellar ellipticities of 10 isolated UFDs in the EDGE
cosmological simulation suite by implementing a well-known observational
method to calculate structural parameters. The Maximum Likelihood technique,
developed and employed by observational astronomers (Martin et al., 2008,
2016), allowed us to make direct comparisons concerning the projected
ellipticity of our simulations, contrasted to a refined sample of observed MW
dwarf galaxies in our LG.
We sampled projected ellipticities from around 100 random viewing angles for
each EDGE UFD to acquire a representative distribution of orientations.
Observing the EDGE UFDs out to fainter surface brightness, we noticed an
increase in projected ellipticity. Therefore, if UFDs have extended stellar
distributions in reality, we should expect a similar increase in the projected
ellipticity of known UFDs when these galaxies are further uncovered with
deeper and better-resolved spectra of their surrounding stars.
The PDFs of projected ellipticity for EDGE and observations displayed a good
agreement, given that peaks of their distributions lay comfortably within the
confidence intervals. This agreement implies our simulated EDGE UFDs resemble
the shapes of a number of faint dwarfs belonging to the MW. As the EDGE UFDs
are designed to be isolated from more massive systems, it is possible that
tidal features in these observed dwarfs are not necessarily due to tides but
instead originate from a non-tidal scenario. This theorised isolation of our
select sample of MW dwarf galaxies is reinforced by their newfound orbital
parameters in Pace et al. (2022).
If a significant number of the nearby UFDs in our LG are tidally intact, as
our results may suggest, their baryonic and dark matter contents could remain
uninfluenced from more massive systems in their surrounding environments.
Therefore, many LG UFDs would serve as excellent natural laboratories for
probing dark matter and galaxy formation physics.

## Acknowledgements

This work was performed using the DiRAC Data Intensive service at Leicester,
operated by the University of Leicester IT Services, which forms part of the
STFC DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded by BEIS
capital funding via STFC capital grants ST/K000373/1 and ST/R002363/1 and STFC
DiRAC Operations grant ST/R001014/1. DiRAC is part of the National
e-Infrastructure. OA and EA acknowledge financial support from the Knut and
Alice Wallenberg Foundation and the Swedish Research Council (grant
2019-04659). EA also acknowledges support from the US NSF (grant AST18-1546).

## Data Availability

Data available upon reasonable request.

## References

  * Abbott et al. (2021) Abbott T. M. C., et al., 2021, ApJS, 255, 20
  * Agertz et al. (2013) Agertz O., Kravtsov A. V., Leitner S. N., Gnedin N. Y., 2013, ApJ, 770, 25
  * Agertz et al. (2020) Agertz O., et al., 2020, MNRAS, 491, 1656
  * Battaglia & Nipoti (2022) Battaglia G., Nipoti C., 2022, Nature Astronomy, 6, 659
  * Battaglia et al. (2013) Battaglia G., Helmi A., Breddels M., 2013, New Astron. Rev., 57, 52
  * Bechtol et al. (2015) Bechtol K., et al., 2015, ApJ, 807, 50
  * Belokurov (2013) Belokurov V., 2013, New Astron. Rev., 57, 100
  * Belokurov et al. (2007) Belokurov V., et al., 2007, ApJ, 654, 897
  * Belokurov et al. (2008) Belokurov V., et al., 2008, ApJ, 686, L83
  * Belokurov et al. (2009) Belokurov V., et al., 2009, MNRAS, 397, 1748
  * Belokurov et al. (2010) Belokurov V., et al., 2010, ApJ, 712, L103
  * Benitez-Llambay (2015) Benitez-Llambay A., 2015, py-sphviewer: Py-SPHViewer v1.0.0, doi:10.5281/zenodo.21703, http://dx.doi.org/10.5281/zenodo.21703
  * Binney & Tremaine (2008) Binney J., Tremaine S., 2008, Galactic Dynamics: Second Edition
  * Bromm & Yoshida (2011) Bromm V., Yoshida N., 2011, ARA&A, 49, 373
  * Brown et al. (2014) Brown T. M., et al., 2014, ApJ, 796, 91
  * Bullock & Boylan-Kolchin (2017) Bullock J. S., Boylan-Kolchin M., 2017, ARA&A, 55, 343
  * Cerny et al. (2021) Cerny W., et al., 2021, ApJ, 920, L44
  * Chabrier (2003) Chabrier G., 2003, PASP, 115, 763
  * Chiti et al. (2021) Chiti A., et al., 2021, Nature Astronomy, 5, 392
  * Chiti et al. (2022) Chiti A., et al., 2022, arXiv e-prints, p. arXiv:2205.01740
  * Collins & Read (2022) Collins M. L. M., Read J. I., 2022, Nature Astronomy,
  * Collins et al. (2017) Collins M. L. M., Tollerud E. J., Sand D. J., Bonaca A., Willman B., Strader J., 2017, MNRAS, 467, 573
  * Collins et al. (2022) Collins M. L. M., Charles E. J. E., Martínez-Delgado D., Monelli M., Karim N., Donatiello G., Tollerud E. J., Boschin W., 2022, arXiv e-prints, p. arXiv:2204.09068
  * Crnojević et al. (2016) Crnojević D., Sand D. J., Zaritsky D., Spekkens K., Willman B., Hargis J. R., 2016, ApJ, 824, L14
  * Deason et al. (2022) Deason A. J., Bose S., Fattahi A., Amorisco N. C., Hellwing W., Frenk C. S., 2022, MNRAS, 511, 4044
  * Drlica-Wagner et al. (2015) Drlica-Wagner A., et al., 2015, ApJ, 813, 109
  * Drlica-Wagner et al. (2016) Drlica-Wagner A., et al., 2016, ApJ, 833, L5
  * Drlica-Wagner et al. (2021) Drlica-Wagner A., et al., 2021, ApJS, 256, 2
  * Eisenstein & Hut (1998) Eisenstein D. J., Hut P., 1998, ApJ, 498, 137
  * Filion & Wyse (2021) Filion C., Wyse R. F. G., 2021, ApJ, 923, 218
  * Flaugher et al. (2015) Flaugher B., et al., 2015, AJ, 150, 150
  * Foreman-Mackey et al. (2013) Foreman-Mackey D., Hogg D. W., Lang D., Goodman J., 2013, PASP, 125, 306
  * Frebel et al. (2014) Frebel A., Simon J. D., Kirby E. N., 2014, ApJ, 786, 74
  * Fritz et al. (2018) Fritz T. K., Battaglia G., Pawlowski M. S., Kallivayalil N., van der Marel R., Sohn S. T., Brook C., Besla G., 2018, A&A, 619, A103
  * Gaia Collaboration et al. (2021) Gaia Collaboration et al., 2021, A&A, 649, A1
  * Genina et al. (2022) Genina A., Read J. I., Fattahi A., Frenk C. S., 2022, MNRAS, 510, 2186
  * Haardt & Madau (1996) Haardt F., Madau P., 1996, ApJ, 461, 20
  * Homma et al. (2016) Homma D., et al., 2016, ApJ, 832, 21
  * Homma et al. (2018) Homma D., et al., 2018, PASJ, 70, S18
  * Ibata et al. (2014) Ibata R. A., et al., 2014, ApJ, 780, 128
  * Irwin & Hatzidimitriou (1995) Irwin M., Hatzidimitriou D., 1995, MNRAS, 277, 1354
  * Kahn (2018) Kahn S., 2018, in 42nd COSPAR Scientific Assembly. pp E1.16–5–18
  * Katz & White (1993) Katz N., White S. D. M., 1993, ApJ, 412, 455
  * Kim & Jerjen (2015) Kim D., Jerjen H., 2015, ApJ, 808, L39
  * Kim et al. (2015a) Kim D., Jerjen H., Milone A. P., Mackey D., Da Costa G. S., 2015a, ApJ, 803, 63
  * Kim et al. (2015b) Kim D., Jerjen H., Mackey D., Da Costa G. S., Milone A. P., 2015b, ApJ, 804, L44
  * Kimm et al. (2015) Kimm T., Cen R., Devriendt J., Dubois Y., Slyz A., 2015, MNRAS, 451, 2900
  * Kirby et al. (2013) Kirby E. N., Cohen J. G., Guhathakurta P., Cheng L., Bullock J. S., Gallazzi A., 2013, ApJ, 779, 102
  * Koposov et al. (2008) Koposov S., et al., 2008, ApJ, 686, 279
  * Koposov et al. (2015) Koposov S. E., Belokurov V., Torrealba G., Evans N. W., 2015, ApJ, 805, 130
  * Koposov et al. (2018) Koposov S. E., et al., 2018, MNRAS, 479, 5343
  * Küpper et al. (2017) Küpper A. H. W., Johnston K. V., Mieske S., Collins M. L. M., Tollerud E. J., 2017, ApJ, 834, 112
  * Laevens et al. (2015) Laevens B. P. M., et al., 2015, ApJ, 813, 44
  * Li et al. (2018) Li T. S., et al., 2018, ApJ, 866, 22
  * Łokas et al. (2013) Łokas E. L., Gajda G., Kazantzidis S., 2013, MNRAS, 433, 878
  * Longeard et al. (2021) Longeard N., et al., 2021, arXiv e-prints, p. arXiv:2107.10849
  * Longeard et al. (2022) Longeard N., et al., 2022, MNRAS, 516, 2348
  * Longeard et al. (2023) Longeard N., et al., 2023, arXiv e-prints, p. arXiv:2304.13046
  * Martin et al. (2008) Martin N. F., de Jong J. T. A., Rix H.-W., 2008, ApJ, 684, 1075
  * Martin et al. (2015) Martin N. F., et al., 2015, ApJ, 804, L5
  * Martin et al. (2016) Martin N. F., et al., 2016, ApJ, 833, 167
  * Mazzarini et al. (2020) Mazzarini M., Just A., Macciò A. V., Moetazedian R., 2020, A&A, 636, A106
  * McConnachie (2012) McConnachie A. W., 2012, AJ, 144, 4
  * McConnachie & Venn (2020) McConnachie A. W., Venn K. A., 2020, Research Notes of the American Astronomical Society, 4, 229
  * McConnachie et al. (2006) McConnachie A. W., Arimoto N., Irwin M., Tolstoy E., 2006, MNRAS, 373, 715
  * McQuinn et al. (2015) McQuinn K. B. W., et al., 2015, ApJ, 812, 158
  * Muñoz et al. (2010) Muñoz R. R., Geha M., Willman B., 2010, AJ, 140, 138
  * Muñoz et al. (2012) Muñoz R. R., Padmanabhan N., Geha M., 2012, ApJ, 745, 127
  * Mutlu-Pakdil et al. (2018) Mutlu-Pakdil B., et al., 2018, ApJ, 863, 25
  * Mutlu-Pakdil et al. (2019) Mutlu-Pakdil B., et al., 2019, ApJ, 885, 53
  * Oñorbe et al. (2014) Oñorbe J., Garrison-Kimmel S., Maller A. H., Bullock J. S., Rocha M., Hahn O., 2014, MNRAS, 437, 1894
  * Pace et al. (2022) Pace A. B., Erkal D., Li T. S., 2022, arXiv e-prints, p. arXiv:2205.05699
  * Planck Collaboration et al. (2014) Planck Collaboration et al., 2014, A&A, 571, A16
  * Pontzen & Tremmel (2018) Pontzen A., Tremmel M., 2018, ApJS, 237, 23
  * Pontzen et al. (2013) Pontzen A., Roškar R., Stinson G., Woods R., 2013, pynbody: N-Body/SPH analysis for python, Astrophysics Source Code Library, record ascl:1305.002 (ascl:1305.002)
  * Pozo et al. (2022) Pozo A., Broadhurst T., Emami R., Smoot G., 2022, MNRAS, 515, 2624
  * Read et al. (2006) Read J. I., Wilkinson M. I., Evans N. W., Gilmore G., Kleyna J. T., 2006, MNRAS, 366, 429
  * Rey et al. (2019) Rey M. P., Pontzen A., Agertz O., Orkney M. D. A., Read J. I., Saintonge A., Pedersen C., 2019, ApJ, 886, L3
  * Rey et al. (2020) Rey M. P., Pontzen A., Agertz O., Orkney M. D. A., Read J. I., Rosdahl J., 2020, MNRAS, 497, 1508
  * Rey et al. (2022) Rey M. P., Pontzen A., Agertz O., Orkney M. D. A., Read J. I., Saintonge A., Kim S. Y., Das P., 2022, MNRAS, 511, 5672
  * Sales et al. (2022) Sales L. V., Wetzel A., Fattahi A., 2022, Nature Astronomy,
  * Sand (2017) Sand D., 2017, The Origin of Ultra-Faint Galaxies, HST Proposal id.15182. Cycle 25
  * Sand et al. (2009) Sand D. J., Olszewski E. W., Willman B., Zaritsky D., Seth A., Harris J., Piatek S., Saha A., 2009, ApJ, 704, 898
  * Sand et al. (2022) Sand D. J., et al., 2022, arXiv e-prints, p. arXiv:2205.09129
  * Schmidt (1959) Schmidt M., 1959, ApJ, 129, 243
  * Sestito et al. (2023a) Sestito F., et al., 2023a, arXiv e-prints, p. arXiv:2301.13214
  * Sestito et al. (2023b) Sestito F., Roediger J., Navarro J. F., Jensen J., Venn K. A., Smith S. E. T., Hayes C., McConnachie A. W., 2023b, MNRAS, 523, 123
  * Shipp et al. (2023) Shipp N., et al., 2023, ApJ, 949, 44
  * Simon (2018) Simon J. D., 2018, ApJ, 863, 89
  * Simon (2019) Simon J. D., 2019, ARA&A, 57, 375
  * Stopyra et al. (2021) Stopyra S., Pontzen A., Peiris H., Roth N., Rey M. P., 2021, ApJS, 252, 28
  * Tarumi et al. (2021) Tarumi Y., Yoshida N., Frebel A., 2021, ApJ, 914, L10
  * Teyssier (2002) Teyssier R., 2002, A&A, 385, 337
  * Torrealba et al. (2016) Torrealba G., et al., 2016, MNRAS, 463, 712
  * Torrealba et al. (2018) Torrealba G., et al., 2018, MNRAS, 475, 5085
  * Waller et al. (2023) Waller F., et al., 2023, MNRAS, 519, 1349
  * Wheeler et al. (2019) Wheeler C., et al., 2019, MNRAS, 490, 4447
  * Whiting et al. (2007) Whiting A. B., Hau G. K. T., Irwin M., Verdugo M., 2007, AJ, 133, 715
  * Willman et al. (2005) Willman B., et al., 2005, ApJ, 626, L85
  * Yang et al. (2022) Yang Y., Hammer F., Jiao Y., Pawlowski M. S., 2022, MNRAS, 512, 4171
  * Zoutendijk et al. (2021) Zoutendijk S. L., et al., 2021, arXiv e-prints, p. arXiv:2112.09374
  * de Jong et al. (2010) de Jong J. T. A., Martin N. F., Rix H.-W., Smith K. W., Jin S., Macciò A. V., 2010, ApJ, 710, 1664
  * von Hoerner (1957) von Hoerner S., 1957, ApJ, 125, 451


(multiplied by 100). We have also performed this test using the covariance
matrix of one of the mocks and found similar results. Given a covariance
matrix $C$ and correlation function $\xi$, we generate samples from the
multivariate normal distribution with mean $\xi$ and covariance $C$. These
samples are given by:

$\tilde{\xi}=\xi+A\vec{y},$ (A.1)

where the matrix $A$ is given by the Cholesky decomposition $C=AA^{T}$,
$\vec{y}$ is a vector of N independent standard normal variates, and N is the
size of C. These MC simulations of the correlation function are then fit using
the same model from Section 3.4.

We generated two populations of MC simulations for our test. The first was
created starting with the vector $\xi$ given by the stacked correlation
functions from 100 Ly$\alpha$CoLoRe mocks, $\xi_{\mathrm{stack}}$. The second
was created starting from the best-fit model to the stack of 100
Ly$\alpha$CoLoRe mocks, $\xi(\vec{\theta}_{\mathrm{best}})$, where
$\vec{\theta}_{\mathrm{best}}$ represents the best-fit parameters. The two
populations are then fit with the same model, and their resulting best-fit
$\chi^{2}$ distributions are shown in Figure 17.

We find that the population based on the best-fit model matches the expected
$\chi^{2}$ distribution with $9540-16=9524$ degrees of freedom very well. On
the other hand, the population created from the stacked correlation functions
is shifted to larger $\chi^{2}$ values and matches the observed $\chi^{2}$
distribution of our mocks. The covariance matrix used for these tests is
correct by construction (i.e. the noise was generated with this covariance),
and therefore, deviations from the expected $\chi^{2}$ distribution are caused
by the inability of the model to fit the input $\xi$. This shows that the
shift in the $\chi^{2}$ distribution measured from mocks is caused by our
model failing to fit the measured correlation functions across the entire
range of scales considered here ($10<r<180$ $h^{-1}\text{Mpc}$). We discuss
this further in Section 4.2.

Figure 17: Histograms of the best-fit $\chi^{2}$ statistic. We compare the
distribution measured from mocks (blue), with two distributions obtained from
Monte Carlo (MC) simulations of the correlation function. The first set of MC
simulations was generated based on the stack of measured Ly$\alpha$CoLoRe
correlations in mocks (red), while the second was generated based on the best-
fit model to the stack of 100 Ly$\alpha$CoLoRe mocks (black). The distribution
of MC simulations based on the model is consistent with the expected
$\chi^{2}$ distribution with $9540-16=9524$ degrees of freedom, while the
other two distributions are shifted to larger $\chi^{2}$ values. This
indicates that the shift in mock best-fit $\chi^{2}$ values is caused by the
model failing to accurately fit the mock correlation functions.

##  Appendix B Tests of the fiducial cosmology

In the DESI DR1 BAO analysis, a fiducial cosmology based on the Planck 2018
results [41] is used to transform redshifts and angles to co-moving
separations and as a template for the model. Here, we wish to validate that
our constraints on $D_{\rm H}/r_{\rm d}$ and $D_{\rm M}/r_{\rm d}$ are
independent of our choice of fiducial cosmology. To do this, we use a set of
20 Ly$\alpha$CoLoRe DESI Y5 mocks, created using the method described in
Section 2, but mirroring the footprint and exposure time (4000s) of the
expected DESI Y5 survey. For simplicity, we use mocks that do not contain any
of the contaminants discussed above (e.g., metal absorption, HCDs, etc.), but
that do go through the same continuum fitting procedure described in section
3.1. We only use the Ly$\alpha$(A)$\times$Ly$\alpha$(A) and
Ly$\alpha$(A)$\times$QSO correlations for this test.

To perform this test we use five cosmologies: Planck 2018 results [Column 5 of
Table 2 in 41], and four alternative cosmologies. Note that these are all
different from the true cosmology used to create the Ly$\alpha$CoLoRe mocks
used here (see Section 2.1). We highlight this by showing the truth (as
crosses) in our results in Figures 18 and 19. In the first two of the four
alternative cosmologies, we fix the sound horizon $r_{\rm d}$ and physical
matter density $\Omega_{\rm m}h^{2}$ and vary $\Omega_{\rm m}$ and $h$ to the
values given in Table 3. By fixing the sound horizon and physical matter
density, which are both very well constrained from CMB anisotropies, we ensure
the shape of our template power spectrum does not change. Thus we test the
assumption that our BAO constraints are independent of the cosmology we use to
make our coordinate transformation.

In the second set of alternative cosmologies, we instead fix $\Omega_{\rm m}$,
while varying the values of $\Omega_{b}h^{2}$, $\Omega_{c}h^{2}$ and $h$. This
changes $r_{\rm d}$ and $\Omega_{\rm m}h^{2}$, and therefore the shape of the
template matter power spectrum, without affecting the coordinate
transformation. The relevant values for this set are also given in Table 3. We
choose values of $h=0.6472$ and $0.70$, which produces an $\pm$8% change in
$\Omega_{\rm m}h^{2}$ \- roughly 10 times the error on the measurement from
CMB anisotropies in Planck (2018). We note that in practice we do not expect
such extreme differences between our template and the truth.

Parameter | Planck18 | Fix $r_{\rm d}$ | Fix $\Omega_{\rm m}$
---|---|---|---
$\Omega_{\rm m}$ | 0.315 | 0.26 | 0.37 | 0.315 | 0.315
$h$ | 0.6736 | 0.7415 | 0.6216 | 0.6472 | 0.7
$\Omega_{\rm m}h^{2}$ | 0.14297 | 0.1429 | 0.1429 | 0.1319 | 0.1545
$r_{\rm d}\;[\rm Mpc]$ | 147.08 | 147.08 | 147.08 | 150.09 | 144.16
$D_{\rm H}(z_{\rm eff}=2.3)[\rm Mpc]$ | 1289.3 | 1278.0 | 1297.5 | 1342.0 | 1240.5
$D_{\rm M}(z_{\rm eff}=2.3)[\rm Mpc]$ | 5712.8 | 5463.7 | 5918.8 | 5946.0 | 5496.8
$D_{\rm H}(z_{\rm eff}=2.3)/r_{\rm d}$ | 8.77 | 8.69 | 8.82 | 8.94 | 8.61
$D_{\rm M}(z_{\rm eff}=2.3)/r_{\rm d}$ | 38.84 | 37.14 | 40.24 | 39.62 | 38.13
Table 3: Main parameters of the cosmologies we used to test the BAO dependence
on the choice of fiducial cosmology. Planck18 is the cosmology used in the BAO
analysis on data, the two "Fix $r_{d}$" cosmologies change the coordinate
transformation without changing the model template, and the "Fix $\Omega_{\rm
m}$" cosmologies do the opposite. The first three rows contain primary
parameters that we vary, and the last 5 are derived from these.

Result | Planck18 | Fix $r_{\rm d}$ | Fix $\Omega_{\rm m}$
---|---|---|---
|  | $\Omega_{\rm m}=0.26$ | $\Omega_{\rm m}=0.37$ | $h=0.6472$ | $h=0.7$
$\alpha_{\parallel}$ | 0.9997$\pm$0.0017 | 1.0103$\pm$0.0018 | 0.9928$\pm$0.0017 | 0.9783$\pm$0.0016 | 1.0199$\pm$0.0018
$\alpha_{\perp}$ | 1.0011$\pm$0.0021 | 1.0445$\pm$0.0021 | 0.9680$\pm$0.0020 | 0.9821$\pm$0.0020 | 1.0191$\pm$0.0021
$D_{\rm H}/r_{\rm d}$ | 8.763$\pm$0.015 | 8.779$\pm$0.016 | 8.757$\pm$0.015 | 8.747$\pm$0.015 | 8.776$\pm$0.015
$D_{\rm M}/r_{\rm d}$ | 38.883$\pm$0.080 | 38.799$\pm$0.079 | 38.953$\pm$0.082 | 38.906$\pm$0.080 | 38.859$\pm$0.080
$A_{\rm BAO}$ | 0.976$\pm$0.011 | 1.078$\pm$0.012 | 0.885$\pm$0.011 | 0.8037$\pm$0.0095 | 1.171$\pm$0.014

Table 4: Measured scale parameters for each fiducial cosmology, along with the
corresponding $D_{\rm H}/r_{\rm d}$ and $D_{\rm M}/r_{\rm d}$ values, which
are consistent independent of the fiducial cosmology used. We also show the
measured BAO amplitude for each case.

The BAO scale parameters ($\alpha_{\parallel}$,$\alpha_{\perp}$) are defined
in Equation 3.25, as the ratio of the measured BAO peak position to that of
the template used in the analysis. Therefore, we expect them to shift when
fitting our mock datasets with different cosmologies. The results are
summarized in Table 4, and the recovered scale parameters from each of our
fits are shown on the left-hand panel of Figures 18 and 19. The crosses in
each case correspond to the predicted scale parameter locations, calculated
from the ratio $[D_{\rm H}/r_{\rm d}]_{\rm true}/[D_{\rm H}/r_{\rm d}]_{\rm
X}$, where X stands for the alternative cosmology in question. These plots
show the large shifts introduced by changing the fiducial cosmology, and how
well we can recover the expected values. The expected values (crosses) are
within the $68\%$ contour in all cases.

Figure 18: (left) Scale parameters obtained from measurements with the three
fiducial cosmologies with fixed $r_{\rm d}$ and different $\Omega_{m}$ and $h$
values. We also include crosses to mark the expected positions of the scale
parameters, based on the ratio of their template BAO to that of the template
used to create the mocks (Planck 2015). (right) Measured BAO distances
obtained by multiplying the scale parameters with the template BAO position.
This shows we are able to recover the true BAO position independent of the
cosmology used to compute comoving coordinates. Figure 19: Similar to Figure
18, but using three fiducial cosmologies with fixed coordinate transformation
(i.e. $\Omega_{\rm m}$), and different $r_{d}$ values. This shows that we are
able to recover the true BAO position independent of the cosmology used to
create the template.

In the right panel of Figures 18 and 19, we show the recovered BAO distances,
obtained by multiplying the measured scale parameter (contour in the left-hand
plot) by the template BAO position (table 3). Also highlighted on this plot is
the DESI DR1 $\sigma/3$, the limit given to variations around the baseline
analysis in DESI2024-Ly$\alpha$. These figures show that we are able to
recover the correct $D_{\rm H}/r_{\rm d}$ and $D_{\rm M}/r_{\rm d}$ values to
within $1/10$ of the DESI DR1 uncertainty. This also corresponds to the level
at which we can trust the current generation of mocks given the results in
Figure 7. Therefore, we conclude that at the level of DESI DR1, we do not
detect a significant systematic offset due to the input fiducial cosmology.

In Table 4, we also show the measured BAO amplitude $A_{\rm BAO}$, which we
treat as a free parameter for this test. This parameter modifies Equation 3.24
such that:

$\xi(r_{||},r_{\bot})=\hat{\xi}_{\mathrm{s}}(r_{||},r_{\bot})+A_{\rm
BAO}\hat{\xi}_{\mathrm{p}}(\alpha_{||}r_{||},\alpha_{\bot}r_{\bot}),$ (B.1)

and therefore it fits for differences between the template and the measured
amplitude of the BAO peak. Note that this is fixed to $A_{\rm BAO}=1$ for the
rest of this work, and also in DESI2024-Ly$\alpha$. As shown in Table 4, we
find that the recovered value for this parameter changes depending on the
fiducial cosmology. For the cases where we vary $\Omega_{m}h^{2}$, the ratio
$\Omega_{c}h^{2}$/$\Omega_{b}h^{2}$ also varies, and therefore the BAO
amplitude as well. For the cases where we change $\Omega_{m}$, it is not as
straightforward. It could be explained by the fact that f$\sigma_{8}$ changes,
which affects the value of $b_{Ly\alpha}$ and leads to different BAO
amplitudes, as $A_{\rm BAO}$ is correlated with $b_{Ly\alpha}$. However, as we
do not use this parameter to constrain cosmology, we defer an in-depth study
of its reliance on the fiducial cosmology to future work.

##  Appendix C Author Affiliations

1Center for Cosmology and AstroParticle Physics, The Ohio State University,
191 West Woodruff Avenue, Columbus, OH 43210, USA

2Departamento de Física, Universidad de Guanajuato - DCI, C.P. 37150, Leon,
Guanajuato, México

3Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of
Science and Technology, Campus UAB, 08193 Bellaterra Barcelona, Spain

4Department of Astronomy, The Ohio State University, 4055 McPherson
Laboratory, 140 W 18th Avenue, Columbus, OH 43210, USA

5Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720,
USA

6Consejo Nacional de Ciencia y Tecnología, Av. Insurgentes Sur 1582. Colonia
Crédito Constructor, Del. Benito Juárez C.P. 03940, México D.F. México

7IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France

8Physics Dept., Boston University, 590 Commonwealth Avenue, Boston, MA 02215,
USA

9Department of Physics and Astronomy, University of California, Irvine, 92697,
USA

10Department of Physics & Astronomy, University College London, Gower Street,
London, WC1E 6BT, UK

11Instituto de Física, Universidad Nacional Autónoma de México, Cd. de México
C.P. 04510, México

12Kavli Institute for Particle Astrophysics and Cosmology, Stanford
University, Menlo Park, CA 94305, USA

13SLAC National Accelerator Laboratory, Menlo Park, CA 94305, USA

14University of California, Berkeley, 110 Sproul Hall #5800 Berkeley, CA
94720, USA

15Departamento de Física, Universidad de los Andes, Cra. 1 No. 18A-10,
Edificio Ip, CP 111711, Bogotá, Colombia

16Observatorio Astronómico, Universidad de los Andes, Cra. 1 No. 18A-10,
Edificio H, CP 111711 Bogotá, Colombia

17Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain

18Institute of Cosmology and Gravitation, University of Portsmouth, Dennis
Sciama Building, Portsmouth, PO1 3FX, UK

19Institute of Space Sciences, ICE-CSIC, Campus UAB, Carrer de Can Magrans
s/n, 08913 Bellaterra, Barcelona, Spain

20Fermi National Accelerator Laboratory, PO Box 500, Batavia, IL 60510, USA

21Department of Physics, The Ohio State University, 191 West Woodruff Avenue,
Columbus, OH 43210, USA

22School of Mathematics and Physics, University of Queensland, 4072, Australia

23Sorbonne Université, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de
Hautes Energies (LPNHE), FR-75005 Paris, France

24Departament de Física, Serra Húnter, Universitat Autònoma de Barcelona,
08193 Bellaterra (Barcelona), Spain

25NSF NOIRLab, 950 N. Cherry Ave., Tucson, AZ 85719, USA

26Institució Catalana de Recerca i Estudis Avançats, Passeig de Lluís
Companys, 23, 08010 Barcelona, Spain

27Department of Physics and Astronomy, Siena College, 515 Loudon Road,
Loudonville, NY 12211, USA

28Department of Physics & Astronomy, University of Wyoming, 1000 E.
University, Dept. 3905, Laramie, WY 82071, USA

29Instituto Avanzado de Cosmología A. C., San Marcos 11 - Atenas 202.
Magdalena Contreras, 10720. Ciudad de México, México

30Department of Physics and Astronomy, University of Waterloo, 200 University
Ave W, Waterloo, ON N2L 3G1, Canada

31Perimeter Institute for Theoretical Physics, 31 Caroline St. North,
Waterloo, ON N2L 2Y5, Canada

32Waterloo Centre for Astrophysics, University of Waterloo, 200 University Ave
W, Waterloo, ON N2L 3G1, Canada

33Space Sciences Laboratory, University of California, Berkeley, 7 Gauss Way,
Berkeley, CA 94720, USA

34Instituto de Astrofísica de Andalucía (CSIC), Glorieta de la Astronomía,
s/n, E-18008 Granada, Spain

35Departament de Física, EEBE, Universitat Politècnica de Catalunya, c/Eduard
Maristany 10, 08930 Barcelona, Spain

36Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France

37Université Clermont-Auvergne, CNRS, LPCA, 63000 Clermont-Ferrand, France

38Department of Physics, Kansas State University, 116 Cardwell Hall,
Manhattan, KS 66506, USA

39Department of Physics and Astronomy, Sejong University, Seoul, 143-747,
Korea

40CIEMAT, Avenida Complutense 40, E-28040 Madrid, Spain

41Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA

42University of Michigan, Ann Arbor, MI 48109, USA

43Department of Physics & Astronomy, Ohio University, Athens, OH 45701, USA

44Excellence Cluster ORIGINS, Boltzmannstrasse 2, D-85748 Garching, Germany

45University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität,
Scheinerstr. 1, 81677 München, Germany

46National Astronomical Observatories, Chinese Academy of Sciences, A20 Datun
Rd., Chaoyang District, Beijing, 100012, P.R. China