# Galaxies lacking dark matter produced by close encounters in a cosmological
simulation

Jorge Moreno∗,1,2,3 Shany Danieli4,5 James S. Bullock2 Robert Feldmann6 Philip
F. Hopkins3 Onur Çatmabacak6 Alexander Gurvich7 Alexandres Lazar2 Courtney
Klein2 Cameron B. Hummels3 Zachary Hafen2 Francisco J. Mercado2 Sijie Yu2
Fangzhou Jiang3,8 Coral Wheeler9 Andrew Wetzel10 Daniel Anglés-Alcázar11,12
Michael Boylan-Kolchin13 Eliot Quataert4 Claude-André Faucher-Giguère7 & Dušan
Kereš15

###### Abstract

The standard cold dark matter plus cosmological constant model predicts that
galaxies form within dark-matter haloes, and that low-mass galaxies are more
dark-matter dominated than massive ones. The unexpected discovery of two low-
mass galaxies lacking dark matter immediately provoked concerns about the
standard cosmology and ignited explorations of alternatives, including self-
interacting dark matter and modified gravity. Apprehension grew after several
cosmological simulations using the conventional model failed to form adequate
numerical analogues with comparable internal characteristics (stellar masses,
sizes, velocity dispersions and morphologies). Here we show that the standard
paradigm naturally produces galaxies lacking dark matter with internal
characteristics in agreement with observations. Using a state-of-the-art
cosmological simulation and a meticulous galaxy-identification technique, we
find that extreme close encounters with massive neighbours can be responsible
for this. We predict that $\sim$30% of massive central galaxies (with at least
$10^{11}$ solar masses in stars) harbour at least one dark- matter-deficient
satellite (with $10^{8}–10^{9}$ solar masses in stars). This distinctive class
of galaxies provides an additional layer in our understanding of the role of
interactions in shaping galactic properties. Future observations surveying
galaxies in the aforementioned regime will provide a crucial test of this
scenario.

Department of Physics and Astronomy, Pomona College, Claremont, CA 91711, USA

Department of Physics and Astronomy, 4129 Reines Hall, University of
California, Irvine, CA 92697, USA

TAPIR, California Institute of Technology, Pasadena, CA 91125, USA

Department of Astrophysical Sciences, 4 Ivy Lane, Princeton University,
Princeton, NJ 08544, USA

Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA

Institute for Computational Science, University of Zurich, Winterthurerstrasse
190, Zurich CH-8057, Switzerland

Department of Physics & Astronomy and CIERA, Northwestern University, 1800
Sherman Ave, Evanston, IL 60201, USA

Carnegie Observatories, 813 Santa Barbara Street, Pasadena, CA 91101, USA

Department of Physics and Astronomy, California State Polytechnic University,
Pomona, Pomona, CA 91768, USA

Department of Physics & Astronomy, University of California, Davis, One
Shields Ave, Davis, CA 95616

Department of Physics, University of Connecticut, 196 Auditorium Road, U-3046,
Storrs, CT 06269-3046, USA

Center for Computational Astrophysics, Flatiron Institute, 162 5th Ave, New
York, NY 10010, USA

Department of Astronomy, The University of Texas at Austin, 2515 Speedway,
Stop C1400, Austin, TX 78712-1205, USA

Department of Astrophysical Sciences, Princeton University, Princeton, NJ
08544, USA

Department of Physics, Center for Astrophysics and Space Sciences, University
of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA

## Abstract

## Editor’s Summary

A cosmological simulation shows that low-mass galaxies can form with far less
dark matter than expected, with results matching some observed
characteristics. Roughly one-third of massive central galaxies may host at
least one such dark-matter- deficient satellite.

Figure 1: Galaxies lacking dark matter. Top left: mock stellar Hubble Space
Telescope $u/g/r$ composite images of the environments around our seven dark-
matter deficient galaxies at present time ($z=0$), with a cut-off of 28.9 mag
arcsec-2 (corresponding to $10^{5}L_{\odot}$ kpc-2). Bottom left: dark matter
mass surface density maps (down to $10^{5}M_{\odot}$ kpc-2). The satellites
are clearly visible in the stellar images but not in the dark matter maps.
Right: magnified mock stellar images (top) and masses inside the subhalo
radius versus time ($t$, bottom) of Bird (top) and Long Hair (bottom). The red
circles indicate the radii containing 50 and 80% of the stellar mass, and the
blue circle refers to the effective radius in the g-band. The grey vertical
lines mark pericentric passages, red vertical lines mark the closest
pericentric passage.

## Main

For over half a century, cold dark matter has been a key ingredient in our
understanding of galaxy formation[1, 2]. In the low-mass regime, the standard
cold dark matter paradigm[3] predicts that galaxies should be more dark-matter
dominated[4]. This is supported by observations in the nearby Universe[5]. For
this reason, the detection of DF2 and DF4, two low-mass galaxies devoid of
dark matter, was not anticipated[6, 7]. This intriguing discovery immediately
sparked several searches of numerical analogues in cosmological simulations[9,
10, 11, 8, 12, 13, 14, 15, 16, 17, 18], with limited success. Indeed, although
idealised numerical experiments[19, 20, 21, 22, 23] had hinted at the
possibility of the existence of such analogues within the standard model – no
cosmological simulation had successfully generated numerical galaxies whose
internal properties (stellar masses, velocity dispersions and morphologies)
simultaneously matched those of their observed counterparts. Unsurprisingly,
the absence of dark-matter deficient low-mass galaxies in some of the earlier
cosmological simulations also raised doubts on the validity of the standard
paradigm itself[8, 22] – although, these studies did not conduct an exhaustive
comparison to previous works (see Supplementary Information). In this paper we
demonstrate that a novel cosmological hydrodynamical simulation (which
presupposes this paradigm) naturally creates adequate numerical versions of
the observed dark matter-deficient galaxies. This simulation utilises the
‘Feedback In Realistic Environments’ (FIRE-2) physics model[24], which
successfully reproduces an array of galaxy properties[25] – and is state-of-
the-art in its ability to resolve the internal structure of individual
galaxies within a large cosmological environment (see Methods Section for
details). By directly comparing with observations, we confirm that one of our
simulated galaxies resembles DF2 and DF4 in arresting ways. This finding
alleviates the aforementioned concerns – and mitigates the exigency for
alternative explanations invoking new (non-standard) physics, such as self-
interacting dark matter[22] and modified gravity[29, 30, 8, 26, 27, 28].

We identify seven galaxies lacking dark matter within a volume of $10^{4}$
Mpc3. Fig. 1 shows present-time images of the environments around these
simulated dark-matter deficient galaxies (white circles), which we name in
honour of the seven Cherokee clans: Bird, Blue, Deer, Long Hair, Paint, Wild
Potato and Wolf. The top-left panels show mock Hubble Space Telescope
composite $u/g/r$ stellar images (down to 28.9 mag arcsec-2, corresponding to
$10^{5}L_{\odot}$ kpc-2) and the bottom-left panels display corresponding dark
matter mass surface density maps (down to $10^{5}M_{\odot}$ kpc-2). The clear
presence of these galaxies in the stellar images and not in the dark matter
maps is remarkable. To illustrate, the right-hand panels display two special
cases: Bird and Long Hair, the members with the most and least active
interaction histories in our set of seven.

At the present time (redshift $z=0$), Bird, Blue and Deer inhabit the same
galaxy group (virial mass $M_{\rm vir}=1.8\times 10^{13}M_{\odot}$) and are at
80 kpc, 66 kpc and 174 kpc away from its center, respectively. Long Hair also
lives in a group ($M_{\rm vir}=1.4\times 10^{13}M_{\odot}$) and has the
largest halo-centric distance (484 kpc) amongst the seven. Paint, Wild Potato
and Wolf exist in haloes slightly more massive than that of the Milky Way
($M_{\rm vir}=3.6,\,2.5,$ and $\,3.3\times 10^{12}M_{\odot}$) and have the
following halo-centric distances: 72 kpc, 57 kpc and 36 kpc. The two observed
galaxies lacking dark matter, DF2[6] and DF4[7], are at close projected
separations (80 kpc and 165 kpc, which set a lower limit to the 3D
separations) from NGC 1052[31, 32] (stellar mass $M_{\star}=1.05\times
10^{11}M_{\odot}$[33]). Following Ref.[34], we estimate that NGC 1052 has
$M_{\rm vir}=6.2^{+25.8}_{-4.2}\times 10^{12}M_{\odot}$ at 90% confidence,
which is consistent with the virial masses of our hosts. Although halo-centric
separations vary with time, their present-day values can provide information
about their current membership status as satellites (see below).

Figure 2: Fraction of mass in dark matter in simulated and observed galaxies.
Dark matter mass versus stellar mass, colour-coded by fraction of total mass
in dark matter ($M_{\rm dm}/M_{\rm total}$). Masses are determined within
$r^{\star}_{\rm 50}$, the radius containing 50% of the stellar mass. The solid
circles and triangles denote central and satellites galaxies (simulations).
Symbol dimensions scale inversely with $M_{\rm dm}/M_{\rm total}$. The light-
gray band represents the median and standard deviation (simulated centrals
only). The light-yellow region denotes the dark-matter deficient regime. The
yellow hexagons highlight our sample of simulated galaxies lacking dark
matter. The red arrows with error bars (1 s.d.) represent observations[6, 7]
conducted with the Dragonfly Telephoto Array.

Fig. 2 shows dark matter mass ($M_{\rm dm}$) versus stellar mass
($M_{\star}$). Hereafter we quote quantities within $r^{\star}_{\rm 50}$, the
radius containing 50% of the stellar mass, to compare directly with
observations[35] (dark-red arrows with error bars). We note, however, that our
seven galaxies are also dark-matter deficient out to larger galactocentric
radii (Supplementary Table 2 and Supplementary Fig. 9). The diagonal lines
represent constant dark-matter-to-stellar mass ratios. We select our sample of
seven dark-matter deficient galaxies (yellow hexagons) by requiring $M_{\rm
dm}<M_{\star}$ (light-yellow region below the $1:1$ line). Wolf’s stellar mass
is consistent with both DF2 and DF4. We do not include galaxies with
$M_{\star}<M_{\rm dm}<M_{\rm baryon}$ in this sample. These are likely
simulated analogues of the recently-discovered baryon-rich galaxies[36, 37,
38], which we plan to study separately. (We also discard one object – the
triangle in close proximity to Deer – because it is participating in an on-
going gas-rich merger, unlike DF2 and DF4.)

Figure 3: Comparison of internal properties between simulated and observed
galaxies. Quantities are measured within $r^{\star}_{\rm 50}$. The triangles
denote simulated satellites with $M_{\star}=10^{8-9}M_{\odot}$, their
dimensions scale with $M_{\rm total}$ and are colour-coded by $M_{\rm
dm}/M_{\rm total}$. The red stars with error bars (1 s.d.) represent Dragonfly
observations[7, 35]. The diagonal dashed lines show various fixed dynamical
masses according to an analytic mass estimator[39]: $M_{\rm
dyn}(<r^{\star}_{\rm 50})\propto(\sigma^{\rm 1D}_{\star})^{2}\,r^{\star}_{\rm
50}$. Figure 4: Conditions for creating a galaxy lacking dark matter. The
minimum halo-centric distance ever achieved (in units of the hosts’ virial
radius) versus the satellite-central stellar mass ratio. The triangles denote
simulated satellites with $M_{\star}=10^{8-9}M_{\odot}$, colour coded by
$M_{\rm dm}/M_{\rm total}$, with dimensions scaled by $M_{\star}$
(simulations). Quantities are within $r^{\star}_{\rm 50}$ for satellites and
$r^{\star}_{\rm 80}$ for central companions. The yellow rectangle, percentages
within parenthesis and values between downward arrows highlight the region of
parameter space our seven galaxies occupy (yellow hexagons) and the numbers in
parentheses indicate number of pericentric passages. The vertical red lines
(and bands) denote the stellar mass ratios (and uncertainty) between DF2/DF4
and NGC 1052 (observations).

Fig. 3 shows the 1D line-of-sight velocity dispersion ($\sigma^{\rm
1D}_{\star}$) versus the galactic size ($r^{\star}_{\rm 50}$). For
observations of DF2 and DF4, we use deprojected measurements[39] by Ref.[35]
and Ref.[7], respectively. For comparison, we also include simulated galaxies
with $M_{\star}=10^{8-9}M_{\odot}$ (within $r^{\star}_{\rm 50}$), in line with
our sample of seven. We note that although this stellar-mass regime contains
DF2 and DF4, it is “too generous”. Namely, only galaxies with $M_{\star}$
(within $r^{\star}_{\rm 50}$) close to $10^{8}M_{\odot}$ may qualify as
realistic numerical analogues of DF2 and DF4 (the rest should thus be regarded
as slightly more massive predictions.) To guide the eye, the dashed diagonal
lines track fixed dynamical masses within $r^{\star}_{\rm 50}$ according to an
analytic mass estimator[39]. Additionally, to quantify the morphology of our
seven galaxies, we performed Sérsic[40] profile fittings in the $g$-band. We
measure effective radii ($R^{\rm 2D}_{\rm e}$), which are in line with our
$r^{\star}_{\rm 50}$ values after deprojection (see the Supplementary Figs. 2
and 3, plus Supplementary Table 1). With this procedure, we also obtain Sérsic
indices ($n_{\rm S\acute{e}rsic}$), which range from $0.60-0.77$, in excellent
agreement with observations ($n_{\rm S\acute{e}rsic}=0.60$ and $0.79$ for DF2
and DF4, respectively). Our results thus far suggest that Wolf is the only
dark-matter deficient galaxy in the cosmological-simulation literature that
simultaneously matches observations on the following fronts: stellar mass
(within $r^{\star}_{\rm 50}$), 1D line-of-sight velocity dispersion, size and
morphology (Wild Potato has $\sigma^{\rm 1D}_{\star}=7.0$ km sec-1 along one
of the three orthogonal directions we report, but is has marginally higher
stellar mass than DF2 and DF4 – see Supplementary Fig. 4 and Supplementary
Table 1).

Previous cosmological simulations have not been able to match the
aforementioned internal properties of the two observed dark-matter deficient
galaxies. The most promising object in Ref.[17] has $\sim$66% more mass in
dark matter relative to stars – whilst the stellar mass and 1D velocity
dispersion in the simulations reported by Ref.[10] exceed observational
measurements by more than 1-$\sigma$. See Supplementary Fig. 9 for a direct
comparison against those two works. Furthermore, no other cosmological
simulation has been able to produce low-mass galaxies with dark-matter mass
fractions as low as ours: three members of our sample have dark-matter mass
fractions below 10% and one has only 1% of its mass in dark matter – rendering
them almost dark-matter free galaxies. One possible explanation is that our
simulation has the distinct advantage of being able to model the small-scale
($\sim 20$ pc) interstellar medium and the high densities ($>300$ cm-3) of
star-forming gas of individual galaxies within a cosmological region ($\sim
20$ Mpc on the side) large enough to contain several massive groups of
galaxies. See the Supplementary Fig. 9 and associated text for a detailed
comparison with results by other groups.

Below we show that close encounters are responsible for the creation of dark-
matter deficient galaxies in our simulation. Observationally, it is
challenging to infer if a galaxy has experienced close interactions in its
history. For instance, Bird exhibits ‘S-shaped’ low surface brightness tails,
whilst Long Hair appears undisturbed (see magnified examples in the right-hand
side of Fig. 1 – and the Supplementary Fig. 5 for a more detailed analysis of
Long Hair). However, our simulation reveals that close encounters with a
massive neighbour are responsible for dark-matter deficiency in all of our
seven galaxies. They all started out more massive, gas rich and with fairly
typical stellar-to-dark-matter mass ratios for galaxies of their mass – but
subsequently experienced multiple close interactions with their host galaxies
(see the mass-evolution insets in Fig. 1 and Supplementary Fig. 7). With the
exception of Deer, these galaxies lost all of their gas during this process,
in agreement with DF2[41]. These satellites also lost between $97.9-99.99\%$
of their dark matter mass, whilst losing only $45-97\%$ of their stellar mass
(within the subhalo radius) – see Supplementary Table 2 and Supplementary Fig.
8 for details). We suspect that stars are more resilient, as they move on more
circular orbits in contrast to the eccentric orbits of the dark matter
particles. Differences in elongation and dynamical times of the orbits
followed by dark matter particles relative to those followed by stars thus
leave them more susceptible to tidal stripping[42, 43, 44, 45].

To quantify the role of interactions, we evaluate how close satellite galaxies
come relative to their hosts and their stellar mass ratios relative to these
massive central companions. Fig. 4 shows $d_{\rm min}/R^{\rm host}_{\rm vir}$,
the minimum halo-centric distance ever attained by each satellite (in units of
the final virial radius of the host) versus $M_{\star}/M^{\rm
central}_{\star}$, the present-day satellite-central stellar-mass ratio. For
satellites, we continue to employ $r^{\star}_{\rm 50}$, whilst for centrals,
we switch to $r^{\star}_{\rm 80}$, which better captures their spatial
extent[46]. Our simulation suggests that, to become a dark-matter deficient
galaxy, a satellite must pierce within $\sim$5% (below horizontal black dashed
line) of the host virial radius (within $\sim 6-30$ kpc, depending on the
host). In other words, the satellite must transit through the host’s stellar
body. We also predict that their satellite-central stellar-mass ratio must be
$\sim 0.1\%$. The light-yellow region indicates these two conditions. The
stellar mass ratios of DF2 and DF4 relative to NGC 1052 are consistent with
our simulation; however, no orbital information is available to determine
their minimum halo-centric distance.

Whether or not the galaxies in our simulation were formed in the same way as
DF2 and DF4 is still unclear. Our results support a scenario where DF2 and DF4
became dark-matter deficient due to past close encounters with NGC 1052, the
only massive galaxy in their neighbourhood (note that our simulation allows
hosts to have multiple dark-matter deficient satellites). Their projected (80
kpc and 165 kpc) and line-of-sight separations ($2.1\pm 0.5$ Mpc)[31] – as
well as our estimates of the virial diameter of the NGC 1052 halo – suggest
that DF2 and DF4 could be satellites of the NGC 1052 group. Concretely, using
Ref.[34], we estimate this galaxy group to have a virial diameter of $0.7-1.7$
Mpc at 90% confidence, allowing for the possibility that NGC 1052 could
encompass both objects. This possibility is amplified if either DF2 or DF4 are
a ‘backsplash’[47] galaxy – i.e., a galaxy on an extremely radial orbit that
carried it beyond the virial volume of NGC 1052 after a close encounter (we
note that our simulated sample includes two recent backsplash galaxies: Long
Hair and Deer). Future investment in characterizing accurate 3D separations
between DF2 and DF4 relative to NGC 1052 will provide an important test of
this interaction-based scenario.

Although we have successfully created a galaxy (Wolf) in a cosmological
simulation that matches DF2 and DF4 remarkably (in terms of $M_{\star}$,
$r^{\star}_{\rm 50}$, $\sigma^{\rm 1D}_{\star}$ and $n_{\rm S\acute{e}rsic}$),
a few outstanding issues remain. In addition to being dark-matter deficient,
DF2 and DF4 are intriguing because they (1) inhabit the same group and (2) are
extremely similar to each other. Namely, they resemble each other in stellar-
mass, size, (extremely-low) velocity dispersion, Sérsic index, colour, and the
presence of anomalous globular cluster populations[48]. Our cosmological
simulation has insufficient resolution to explore the globular-cluster
problem, but zoom-in simulations with our model can potentially address
this[49, 50]. However, we are able to measure stellar-population properties in
our seven dark-matter deficient galaxies (see Supplementary Table 3). In
particular, Blue and Bird have stellar ages of 10 and 8.7 Gyr, in line with
DF2[6] ($8.9\pm 1.5$ Gyr). Indeed, just like DF2 and DF4, these two simulated
galaxies also inhabit the same group (Fig. 1). However, although they have
some of the lowest metallicities in our set of seven
(${\rm\langle[Fe/H]\rangle}=-0.61$ and $-0.34$), they are not as metal-poor as
DF2[51] (${\rm\langle[Fe/H]\rangle}=-1.35\pm 0.12$). One possibility is that
DF2 and DF4 had infall stellar-masses similar to Blue and Bird, but sat
initially on the lower envelop of their mass-metallicity relation. Confirming
the existence of such objects would require a larger simulation box or
multiple distinct realsations of this simulation (with the same physics, but
different initial conditions).

In summary, our results demonstrate that galaxies resembling DF2 and DF4 can
arise naturally within the standard cold-dark-matter based cosmological
paradigm. We note that we did not expect this to occur a priori (i.e., our
simulation was not originally designed for this purpose). Although there
certainly is still room for new physics beyond the standard paradigm,
discriminating between alternative models will now rely on predicted
differences between the expected properties of dark-matter deficient galaxies,
rather than their mere existence[8].

To efficiently uncover more galaxies devoid of dark matter, observers should
focus on satellites with $M_{\star}=10^{8-9}M_{\odot}$, near massive
companions ($M_{\star}\geq 10^{11}M_{\odot}$). Concretely, we predict 30% of
such massive galaxies to have at least one dark-matter deficient satellite.
Interestingly, we also predict that only 30% of haloes with virial masses
$\geq 2\times 10^{12}$ (the lower bound for NGC 1052) harbour such peculiar
objects. This mass range is slightly above the virial masses of the Milky Way
and Andromeda – which might explain why no analogueues of DF2 and DF4 have
been detected in the Local Group. However, we cannot exclude a situation where
a central-satellite pair outside this mass regime could have a sufficiently
unorthodox orbital configuration to produce the same effect. Verifying how
strict this regime is will require a larger volume (or more realisations).
More generally, to truly constrain the varied processes and underlying physics
governing these galaxies, ambitious observational campaigns aimed to enumerate
and characterise large samples of galaxies with unusual dark-matter properties
– will be desirable in the years ahead.

###  0.1 Our cosmological simulation.

We ran a high-resolution cosmological simulation of galaxy formation with box
size of $21$ comoving Mpc and the following number of baryonic and dark matter
particles: $N_{\rm b}=1,024^{3}$ and $N_{\rm dm}=1,024^{3}$. Initially the
baryonic particle masses are $m_{\rm baryon}=6.3\times 10^{4}\,M_{\odot}$ for
gas and star particles, whilst the dark matter particles have $m_{\rm
dm}=3.3\times 10^{5}\,M_{\odot}$. The force resolution is set at a fixed
$h_{\rm star}=12$ pc (physical) for star particles and 80 pc for dark matter
particles. For gas, the force resolution is set equal to the adaptive
smoothing length down to a minimum of 1.5 pc, which occurs only in the densest
regions of galaxies. We identify galaxies (and merger history trees) with the
AMIGA Halo Finder (AHF[1]), which uses an iterative unbinding procedure to
identify gravitationally-bound objects[53]. This method assumes the virial-
mass definition of Ref.[54] by construction. We used yt[55] to interface with
the particle data. The cosmological parameters are set to $\Omega_{\rm
m}=0.3089$, $\Omega_{\Lambda}=0.6911$, $\Omega_{\rm b}=0.0486$,
$\sigma_{8}=0.8159$ and $h=0.6774$.

We employ the ‘Feedback In Realistic Environments’ (FIRE-2) model[24] for
baryonic physics, which includes radiative cooling, star formation in dense
self-gravitating gas and accounts for stellar feedback in the form of
supernovae, stellar mass-loss and radiation interacting with the surrounding
gas (as described therein). We follow eleven separately-tracked species: H,
He, C, N, O, Ne, Mg, Si, S, Ca, and F. This model has been extensively
validated in a number of publications analyzing properties of galaxies across
a range in stellar masses and numerical resolutions[25]. In our run, star
formation occurs only in gas denser than $n=300$ cm-3. With this condition,
the inter-particle separation in star-forming gas is $\sim 20$ pc (or less)
and we are able to resolve giant-molecular-cloud complexes within the
interstellar medium of individual galaxies.

###  0.2 Our simulated galaxy samples.

In this work we focus on galaxies with at least 100 stellar particles at
redshift $z=0$. The actual number of particles per object is typically much
larger owing to an additional contribution of gas and dark matter particles
for each object, either at the present time or at some point in its history.
To avoid discarding galaxies lacking dark matter, our application of AHF
utilises all types of particles, not just dark matter. This comes at a price:
the halo finder falsely identifies large numbers of gas clumps within galaxies
as individual ‘galactic units’. To address this, first we discard objects with
baryon-to-total mass ratios greater than 0.5. This may naturally discard the
very objects we seek (and keep false clumps below this threshold). To
ameliorate this, we create surface density maps of every halo with
substructures and visually recover objects that might have been discarded
erroneously by our 50% cut. The images shown in Fig. 1 belong to this set. We
focus on manually deleting clumps embedded inside massive galaxies and on
recovering galaxies that are either clearly disjoint from their neighbours –
or in tidal tails, to avoid rejecting tidally-formed candidates[56]. We
highlight that every galaxy lacking dark matter studied in this paper is
recovered during this step. Lastly, because we are interested in the low-mass
regime, we constrain our sample to have stellar masses under $10^{9}M_{\odot}$
(within $r^{\star}_{\rm 80}$, the radius containing 80% of the stellar mass).
This produces our final parent sample, which contains 1,218 resolved galaxies:
886 centrals and 332 satellites. See Supplementary Fig. 1 for a visual
description of these two sets.

Our dark-matter deficient set consists of seven galaxies with $M_{\rm
dm}<M_{\star}$ that are not participating in a major merger (we discard an
eighth galaxy meeting the former condition, but not the latter). We adopt this
extra condition because the two observed Dragonfly galaxies are not currently
merging with a companion of similar mass. We also exclude 19 low-mass galaxies
(15 centrals and 4 satellites) with $M_{\rm dm}<M_{\rm baryon}$ (but $M_{\rm
dm}>M_{\star}$) – which are possibly numerical analogues of the recently-
observed baryon-rich galaxies[36, 37, 38]. The final set of seven are
satellites with $M_{\star}=10^{8-9}M_{\odot}$; they orbit five different
massive central galaxies with $M_{\star}>10^{11}M_{\odot}$. In total, our
simulation contains fifteen massive centrals that host 47 satellite galaxies
in the target stellar mass range ($10^{8-9}M_{\odot}$). Of these satellites,
$\sim 15\%$ belong to our set of seven, which orbit $\sim 30\%$ of the
available massive centrals.

The absence of dark-matter deficient satellite galaxies at stellar masses
below $10^{8}M_{\odot}$ could arise from the fact that, at lower masses,
galaxies are born as centrals with very high dark matter mass fractions, such
that environmental effects cannot act to remove dark matter without completely
destroying the galaxy. The absence of dark-matter deficient galaxies at lower
stellar masses is likely not driven by lack of resolution because our
simulation produces a large population of galaxies with at least 1,000 stellar
particles that are less massive than Wolf (which has 3,280 stellar particles
at redshift zero and had a factor of $\sim$35 higher before becoming a
satellite). For a discussion about potential numerical effects on tidal
disruption, see Supplementary Fig. 8 and associated text.

###  0.3 Galactic sizes.

We calculate galactic sizes as follows. For each object, we record the
distance to the nearest major companion (with stellar mass of at least a tenth
of that of our object of interest). If no such companion exists, we record the
virial radius (for centrals) or subhalo radius (for satellites). This step is
necessary because AHF considers substructures as part of their host. Once this
region is identified, we calculate the stellar center-of-mass. Next we
redefine the radius of this region by subtracting the distance from the origin
to this center-of-mass. Within this region, we then calculate cumulative mass
profiles (using 200 identical linear radial bins) and extract the radius
containing 80% of the stellar mass within this region by linearly
interpolating between the two radii below and above this threshold. The sizes
of our seven dark-matter deficient galaxies are listed in Supplementary Table
1 and displayed in Supplementary Figs. 2 and 3.

###  0.4 1D line-of-sight velocity dispersions.

We also calculate 1D line-of-sight velocity dispersions ($\sigma^{\rm
1D}_{\star}$) within $r^{\star}_{\rm 50}$. We perform this calculation along
three random directions (in the frame of each galaxy), corresponding to the
$x$, $y$ and $z$ directions in the frame of the box. We create histograms of
line-of-sight velocity minus the average line-of-sight velocity within the
$r^{\star}_{\rm 50}$. These histograms have bin-widths of $2$ km sec-1 and
extend from $-70$ to $70$ km sec-1. We then fit a Gaussian curve and record
its $\sigma$-value. In the main text we report the simple average of these
three numbers. The $\sigma^{\rm 1D}_{\star}$-values for our seven dark-matter
deficient galaxies are listed Supplementary Table 1 and displayed (along the
aforementioned three directions) in Supplementary Fig. 4

## Peer review

Peer review information. Nature Astronomy thanks the anonymous reviewers for
their contribution to the peer review of this work.

###  0.5 Data availability.

The datasets generated during and/or analyzed during the current study are
available from the corresponding author on reasonable request, contingent on
approval by the FIRE Collaboration on a case-by-case basis.

###  0.6 Code availability.

Below we list the URLs of the codes we use.

  * •

GIZMO: https://bitbucket.org/phopkins/gizmo-public/src/master/.

  * •

yt: https://yt-project.org/.

  * •

Amiga Halo Finder: http://popia.ft.uam.es/AHF/Download.html.

  * •

FIRE Studio: https://github.com/agurvich/FIRE_studio.

  * •

WebPlotDigitizer: https://apps.automeris.io/wpd/.

The Python scripts used to create the figures in the current study are
available from the corresponding author on reasonable request, contingent on
approval by the FIRE Collaboration on a case-by-case basis.

Sabbatical leave support from Pomona College and the Harry and Grace Steele
Foundation (J.M.). NASA Hubble Fellowship grant number HST-HF2-51454.001-A
awarded by the Space Telescope Science Institute, which is operated by the
Association of Universities for Research in Astronomy, Incorporated, under
NASA contract number NAS5-26555 (S.D.). NSF grant number AST- 1910346 (J.S.B.,
F.J.M. and S.Y.). Swiss National Science Foundation grant numbers
PP00P2_157591, PP00P2_194814 and 200021_188552; Swiss National Supercomputing
(CSCS) project IDs s697 and s698 (R.F.). NSF Research Grant numbers 1911233
and 20009234, NSF CAREER grant number 1455342, NASA grant numbers
80NSSC18K0562 and HST-AR-15800.001-A (P.F.H.). NASA grant number
80NSSSC20K1469 (A.L.). Gary McCue postdoctoral fellowship through the Center
for Cosmology at UC Irvine (Z.H.). National Science Foundation Graduate
Research Fellowship under grant number DGE-1839285 (C.K.). NSF CAREER grant
number 2045928; NASA ATP grant numbers 80NSSC18K1097 and 80NSSC20K0513; HST
grant numbers AR-15809 and GO-15902 from STScI; Scialog Award from the
Heising-Simons Foundation; Hellman Fellowship (A.W.). NSF grant number
AST-2009687 and by the Flatiron Institute, which is supported by the Simons
Foundation (D.A.-A). NSF CAREER award number AST-1752913, NSF grant number
AST-1910346, NASA grant numbers NNX17AG29G and HST-AR-15006, HST-AR- 15809,
HST-GO-15658, HST-GO-15901, HST-GO-15902, HST-AR-16159 and HST-GO-16226 from
STScI (M.B.-K.). Simons Investigator Award from the Simons Foundation; NSF
grant number AST-1715070 (E.Q.). NSF through grant numbers AST- 1715216 and
CAREER award AST-1652522; by NASA through grant number 17-ATP17-0067; by STScI
through grant number HST-AR-16124.001-A; and by the Research Corporation for
Science Advancement through a Cottrell Scholar Award and a Scialog Award
(C.-A.F.-G.). NSF grant number AST-1715101 (D.K.). Numerical calculations were
run on the Caltech compute cluster “Wheeler,” allocations FTA-HopkinsAST20016
supported by the NSF and TACC and NASA HEC SMD-16-7592. We acknowledge PRACE
for awarding us access to MareNostrum at the Barcelona Supercomputing Center
(BSC), Spain. This research was partly carried out via the Frontera computing
project at the Texas Advanced Computing Center. Frontera is made possible by
NSF award number OAC-1818253. We acknowledge access to Piz Daint at the Swiss
National Supercomputing Centre, Switzerland under the University of Zurich’s
share with the project ID uzh18. Additional computing support was provided by
S3IT resources at the University of Zurich. We thank Y. Jing for patiently
answering every question we had regarding their 2019 paper, and for sharing
the data we requested. We also thank P. van Dokkum, J. Hudgings, D. Whitaker,
D. Tanenbaum and R. Gaines for comments on an earlier draft, and C. Hayward
for data-transfer support. J.M. thanks J.S.B., P.F.H., P. Torrey and L.
Hernquist for sabbatical-leave hospitality. J.M. (an astronomer of Indigenous
ancestry, non-Cherokee) thanks D. Ingram (a Cherokee physicist) for sharing
knowledge about the Cherokee Nation and cultural appropriation. This work was
conducted on Tongva-Gabrielino land.

J.M. conducted the analysis and designed the manuscript, with substantial
input from S.D. and J.S.B. S.D. collected observational data from the
literature and verified the theory–observations comparisons. R.F. ran the
cosmological simulation used in this work. O.Ç. generated the halo catalogues
and merger trees. A.G. provided support for the creation of mock images. A.L.
and C.B.H. provided support with yt. C.K. calculated g-band effective radii,
Sérsic indices and surface brightness contours. C.K. and F.J.M. calculated
stellar ages, metallicities and colours; and S.D. conducted the observational
comparison. J.S.B, R.F., O.Ç. and F.J.M. worked on the discussion of numerical
stripping, with substantial input from F.J. Z.H. contributed to the literature
comparison. All authors contributed to the final creation of the manuscript
and figures.

The authors declare that they have no competing financial interests.

Correspondence and requests for materials should be addressed to
J.M. (email: jorge.moreno@pomona.edu).

Published online version available at
https://www.nature.com/articles/s41550-021-01598-4.

Supplementary material available at
https://doi.org/10.1038/s41550-021-01598-4.

DOI number: 10.1038/s41550-021-01598-4.

Scheduled for publication in Nature Astronomy on 14 February 2022 at 16:00
(London time), 14 February 2022 at 11:00 (US Eastern Time).

## References

  * [1] Zwicky, E. Die Rotverschiebung von extragalaktischen Nebeln. Helvetica Physica Acta 6, 110-127 (1933).
  * [2] Rubin, B. C. & Ford, W. K. Jr. Rotation of the Andromeda Nebula from a Spectroscopic Survey of Emission Regions. Astrophys. J. 159, 379 (1970).
  * [3] White, S. D. M. & Rees, M. J. Core condensation in heavy : a two-stage theory for galaxy formation and clustering. Mon. Not. R. Astron. Soc. 183, 341-358 (1978).
  * [4] Behroozi, P. S., Wechsler, R. H. & Conroy, C. The Average Star Formation Histories of Galaxies in Dark Matter Halos from z=0-8. Astrophys. J. 770, 57 (2013).
  * [5] Tollerud, E. et al. From Galaxy Clusters to Ultra-Faint Dwarf Spheroidals: A Fundamental Curve Connecting Dispersion-supported Galaxies to Their Dark Matter halos. Astrophys. J. 726, 2, 23 (2011).
  * [6] van Dokkum, P. et al. A galaxy lacking dark matter. Nature 555, 629-632 (2018).
  * [7] van Dokkum, P. et al. A Second Galaxy Missing Dark Matter in the NGC 1052 Group. Astrophys. J. Lett. 874, 1 (2019).
  * [8] Haslbauer, M., Banik, I., Kroupa, P. & Grishunin, K. The ultra-diffuse dwarf galaxies NGC 1052-DF2 and 1052-DF4 are in conflict with standard cosmology. Mon. Not. R. Astron. Soc. 489, 2634-2651 (2019).
  * [9] Yu, H., Ratra, B. & Wang, F.-J. Dark matter deficient galaxies in the Illustris flat-$\Lambda$ CDM model structure formation simulation. Preprint at https://arxiv.org/abs/1809.05938 (2018).
  * [10] Jing, Y., et al. Dark-matter-deficient galaxies in hydrodynamical simulations. Mon. Not. R. Astron. Soc. 488, 3298-3307 (2009).
  * [11] Haslbauer, M., et al. Galaxies lacking dark matter in the Illustris simulation. Astron. Astrophys. 626, A47 (2019).
  * [12] Carleton, T. et al. The Formation of Ultra Diffuse Galaxies in Cored Dark Matter Haloes Through Tidal Stripping and Heating. Mon. Not. R. Astron. Soc. 485, 382-395 (2019).
  * [13] Sales, L. V., et al. The formation of ultradiffuse galaxies in clusters. Mon. Not. R. Astron. Soc. 494, 2, 1848-1858 (2020).
  * [14] Saulder, C., et al. Isolated dark-matter-deprived galaxies in hydrodynamical simulations: real objects or artifacts? Mon. Not. R. Astron. Soc. 491, 1278-1286 (2020).
  * [15] Shin, E., et al. Dark Matter Deficient Galaxies Produced via High-velocity Galaxy Collisions in High-resolution Numerical Simulations. Astrophys. J. 899, 25 (2020)
  * [16] Applebaum, E., et al. Ultrafaint Dwarfs in a Milky Way Context: Introducing the Mint Condition DC Justice League Simulations. Astrophys. J. 909, 2, 96 (2021).
  * [17] Jackson, R. A., et al. Dark matter-deficient dwarf galaxies form via tidal stripping of dark matter in interactions with massive companions. Mon. Not. R. Astron. Soc. 502, 2, 1785-1796 (2021).
  * [18] Wright, A. C. The formation of isolated ultradiffuse galaxies in ROMULUS25. Mon. Not. R. Astron. Soc. 502, 4, 5370-5389 (2021).
  * [19] Ogiya, G. Tidal stripping as a possible origin of the ultra diffuse galaxy lacking dark matter. Mon. Not. R. Astron. Soc. 480, L106-L110 (2018).
  * [20] Huo, R. Refracting into ultra-diffuse galaxy NGC 1052-DF2 by passing near the centre of NGC 1052. Mon. Not. R. Astron. Soc. 495, L144-L148 (2020).
  * [21] Nusser, A. A Scenario for Ultradiffuse Satellite Galaxies with Low Velocity Dispersions: The Case of [KKS 2000]04. Astrophys. J. 893, 66, 11 (2020).
  * [22] Yang, D., Yu, H.-B. & An, H. Self-Interacting Dark Matter and the Origin of Ultradiffuse Galaxies NGC1052-DF2 and -DF4. Phys. Rev. Lett. 125, 11 (2020).
  * [23] Macciò, A. V., et al. Creating a galaxy lacking dark matter in a dark matter-dominated universe. Mon. Not. R. Astron. Soc. 501, 1, 693-700 (2021).
  * [24] Hopkins, P. F. et al. FIRE-2 simulations: physics versus numerics in galaxy formation. Mon. Not. R. Astron. Soc. 480, 1, 800-863 (2018).
  * [25] Faucher-Giguère, C.-A. Recent progress in simulating galaxy formation from the largest to the smallest scales. Nat. Astron. 2, 368-373 (2018).
  * [26] Kroupa, P. et al. Does the galaxy NGC1052-DF2 falsify Milgromian dynamics? Nature 561, E4 (2018).
  * [27] Moffat, J. W. & Toth, V. T. NGC 1052-DF2 and modified gravity (MOG) without dark matter. Mon. Not. R. Astron. Soc. 482, L1-L3 (2019).
  * [28] Khalifeh A. R. & Jimenez R. Dwarf galaxies without dark matter: constraints on modified gravity. Mon. Not. R. Astron. Soc. 501, 254-260 (2021).
  * [29] Famaey, B., McGaugh, S. & Milgrom M. MOND and the dynamics of NGC 1052-DF2. Mon. Not. R. Astron. Soc. 480, 473-476 (2018).
  * [30] Haghi, H. et al. A new formulation of the external field effect in MOND and numerical simulations of ultra-diffuse dwarf galaxies – application to NGC 1052-DF2 and NGC 1052-DF4. Mon. Not. R. Astron. Soc. 487, 2442-2454 (2019).
  * [31] Shen, Z. et al. A Tip of the Red Giant Branch Distance of 22.1 ± 1.2 Mpc to the Dark Matter Deficient Galaxy NGC 1052–DF2 from 40 Orbits of Hubble Space Telescope Imaging. Astrophys. J. Lett. 914, 1, L12, 9 (2021).
  * [32] Danieli, S. et al. A Tip of the Red Giant Branch Distance to the Dark Matter Deficient Galaxy NGC 1052-DF4 from Deep Hubble Space Telescope Data. Astrophys. J. Lett. 895, 1, L4, 8 (2020).
  * [33] Forbes, D. A. et al. The SLUGGS Survey: stellar masses and effective radii of early-type galaxies from Spitzer Space Telescope 3.6 $\mu$m imaging. Mon. Not. R. Astron. Soc. 464, 4611–4623 (2017).
  * [34] Zahid, H. J., Sohn, J. & Geller, M. J. Stellar Velocity Dispersion: Linking Quiescent Galaxies to Their Dark Matter Haloes. Astrophys. J. 859, 96, 10 (2018).
  * [35] Danieli, S. et al. Still Missing Dark Matter: KCWI High-resolution Stellar Kinematics of NGC1052-DF2. Astrophys. J. Lett. 874, 2, L12, 8 (2019).
  * [36] Mancera Piña, P. et al. Off the Baryonic Tully–Fisher Relation: A Population of Baryon-dominated Ultra-diffuse Galaxies. Astrophys. J. 883, L33 (2019).
  * [37] Guo, Q. et al. Further evidence for a population of dark-matter- deficient dwarf galaxies. Nat. Astron. 4, 246–251 (2020).
  * [38] Mancera Piña, P. et al. No need for dark matter: resolved kinematics of the ultra-diffuse galaxy AGC 114905. Preprint at https://arxiv.org/abs/2112.00017 (2021).
  * [39] Wolf, J. et al. Accurate masses for dispersion-supported galaxies. Mon. Not. R. Astron. Soc. 406, 1220–1237 (2010).
  * [40] Sérsic, J. L., Influence of the atmospheric and instrumental dispersion on the brightness distribution in a galaxy. Bol. Asoc. Argent. de Astron. 6, 41 (1963).
  * [41] Sardone, A. et al. Constraints on the HI Mass of NGC 1052-DF2. Astrophys. J. 871, 2, 31, 5 (2019).
  * [42] Peñarrubia, J. et al. The impact of dark matter cusps and cores on the satellite galaxy population around spiral galaxies. Mon. Not. R. Astron. Soc. 406, 1290–1305 (2010).
  * [43] Bryan, S. E. Influence of baryons on the orbital structure of dark matter haloes. Mon. Not. R. Astron. Soc. 422, 3, 1863-1979 (2013).
  * [44] Valluri, M. et al. Probing the shape and history of the Milky Way halo with orbital spectral analysis. Mon. Not. R. Astron. Soc. 419, 3, 1951-1969 (2013).
  * [45] Zhu, Q. et al. Baryonic impact on the dark matter orbital properties of Milky Way-sized haloes. Mon. Not. R. Astron. Soc. 466, 3876-3886 (2017).
  * [46] Mowla, L., van der Wel, A., van Dokkum, P. & Miller, T. B. A Mass-dependent Slope of the Galaxy Size–Mass Relation out to z$\sim$3: Further Evidence for a Direct Relation between Median Galaxy Size and Median Halo Mass. Astrophys. J. Lett. 872, L13 (2019).
  * [47] Wetzel, A. R. et al. Galaxy evolution near groups and clusters: ejected satellites and the spatial extent of environmental quenching. Mon. Not. R. Astron. Soc. 439, 2687–2700 (2014).
  * [48] Shen, Z. et al. A Complex Luminosity Function for the Anomalous Globular Clusters in NGC 1052-DF2 and NGC 1052-DF4. Astrophys. J. 909, 2, 179, 10 (2021).
  * [49] Kim, J. h. et al. Formation of globular cluster candidates in merging proto-galaxies at high redshift: a view from the FIRE cosmological simulations. Mon. Not. R. Astron. Soc. 474, 3, 4232-4244 (2018).
  * [50] Ma, X., et al. Self-consistent proto-globular cluster formation in cosmological simulations of high-redshift galaxies. Mon. Not. R. Astron. Soc. 493, 3, 4315-4332 (2020).
  * [51] Fensch, J. al. The ultra-diffuse galaxy NGC 1052-DF2 with MUSE II. The population of DF2: stars, clusters, and planetary nebulae. Astron. Astrophys. 625, A77 (2019).
  * [52] Knollmann, S. R. & Knebe, A. AHF: Amiga’s Halo Finder. Astrophys. J. Suppl. 182, 2, 608-624 (2009).
  * [53] Knebe, A. et al. Haloes gone MAD: The Halo-Finder Comparison Project. Mon. Not. R. Astron. Soc. 415, 3, 2293-2318 (2011).
  * [54] Bryan, G. L. & Norman, M. L. Statistical Properties of X-Ray Clusters: Analytic and Numerical Comparisons. Astrophys. J. 495, 1, 80 (1998).
  * [55] Turk, M. J. et al. yt: A Multi-code Analysis Toolkit for Astrophysical Simulation Data. Astrophys. J. Suppl. 192, 1, 9, 16 (2011).
  * [56] Elmegreen, B. G., Kaufman, M. & Thomasson, M. An Interaction Model for the Formation of Dwarf Galaxies and $10^{8}$ Msun Clouds in Spiral Disks. Astrophys J. 412, 90, (1993).

## Supplementary Information

Supplementary Figs. 1–9, Tables 1–4 and Discussion.

Supplementary Figure 5: Stellar masses of satellite and central galaxies. We
display the stellar mass of the central galaxy (within $r^{\star}_{\rm 80}$)
versus that of each satellite (within $r^{\star}_{\rm 50}$). We adopt the same
symbol and colour scheme as in Figure 4. The horizontal red line refers to NGC
1052’s stellar mass and the dark-red stars represent DF2 and DF4. The diagonal
light-gray dashed lines indicate constant central-to-satellite stellar mass
ratios. The brown diagonal lines (and bands) indicate the stellar-mass ratios
between DF2 or DF4 and NGC 1052 (and their uncertainties).

###  0.7 Central and satellite stellar masses.

Supplementary Fig. 5 shows the stellar mass of the central galaxies (hosting
galaxies in our low-mass sample) versus that of their satellites. We adopt
$r^{\star}_{\rm 80}$ for the former (to better capture the spatial extent of
the galaxy[46]) and $r^{\star}_{\rm 50}$ for the latter (for easy comparisons
with observations). The diagonal lines denote constant stellar-mass ratios.
Our seven dark-matter deficient galaxies (with stellar masses between
$10^{8-9}M_{\odot}$) accompany massive central galaxies in a narrow stellar-
mass regime between $(1.3-3.7)\times 10^{11}M_{\odot}$. The stellar mass
ratios of our seven galaxies are similar to that of DF2 and DF4 relative to
NGC 1052 (diagonal brown lines and bands).

Supplementary Figure 6: Images of the seven dark-matter deficient galaxies.
Mock Hubble Space Telescope $u/g/r$ composite stellar images (inner and outer
red circles denote $r^{\star}_{\rm 50}$ and $r^{\star}_{\rm 80}$, blue circle
represents $(4/3)R^{\rm 2D}_{\rm e}$ in the $g$-band; field-of-view equals the
subhalo radius).

Supplementary Figure 7: Cumulative mass profiles of the seven dark-matter
deficient galaxies. Dotted-teal, dot-dashed-orange and dashed-purple curves
denote gas, stars and dark matter. Solid black represents their sum. The dark
and light rectangles denote $r^{\star}_{\rm 50}$ and $r^{\star}_{\rm 80}$; the
blue vertical line denotes $(4/3)R^{\rm 2D}_{\rm e}$ ($g$-band).

Supplementary Figure 8: Velocity dispersions for the seven dark-matter deficient galaxies. Line-of-sight quantities within $r^{\star}_{\rm 50}$ (stellar component) along three random orthogonal directions ($x$, $y$ and $z$ directions in the frame of the box). | ${\rm Bird}$ | ${\rm Blue}$ | ${\rm Deer}$ | ${\rm LH}$ | ${\rm Paint}$ | ${\rm WP}$ | ${\rm Wolf}$ | ${\rm DF2}$ | ${\rm DF4}$
---|---|---|---|---|---|---|---|---|---
$M_{\star}\,(10^{8}M_{\odot})$ | ${\rm 3.8}$ | ${\rm 4.1}$ | ${\rm 5.3}$ | ${\rm 3.7}$ | ${\rm 2.1}$ | ${\rm 1.6}$ | ${\rm 1.0}$ | ${\rm 1.0}\pm{\rm 0.2}$ | ${\rm 0.75}\pm{\rm 0.4}$
$M_{\rm gas}\,(10^{8}M_{\odot})$ | ${\rm 0.0}$ | ${\rm 0.0}$ | ${\rm 0.5}$ | ${\rm 0.0}$ | ${\rm 0.0}$ | ${\rm 0.0}$ | ${\rm 0.0}$ | $-$ | $-$
$M_{\rm dm}\,(10^{8}M_{\odot})$ | ${\rm 1.0}$ | ${\rm 0.6}$ | ${\rm 3.1}$ | ${\rm 1.0}$ | ${\rm 0.02}$ | ${\rm 0.1}$ | ${\rm 0.03}$ | $-$ | $-$
$M_{\rm total}\,(10^{8}M_{\odot})$ | ${\rm 4.8}$ | ${\rm 0.6}$ | ${\rm 8.9}$ | ${\rm 4.7}$ | ${\rm 2.1}$ | ${\rm 1.7}$ | ${\rm 1.1}$ | $1.3\pm 0.8$ | $-$
$\sigma^{\rm 1D}_{\star}\,({\rm km}\,{\rm sec}^{-1})$ | ${\rm 22.3}$ | ${\rm 22.0}$ | ${\rm 11.7}$ | ${\rm 13.0}$ | ${\rm 10.2}$ | ${\rm 10.5}$ | ${\rm 8.6}$ | $8.5^{+2.3}_{-3.1}$ | $4.2^{+4.4}_{-2.2}$
$r^{\star}_{\rm 50}\,({\rm kpc})$ | ${\rm 1.5}$ | ${\rm 1.6}$ | ${\rm 10.7}$ | ${\rm 4.3}$ | ${\rm 3.8}$ | ${\rm 3.0}$ | ${\rm 2.2}$ | $-$ | $-$
$R^{\rm 2D}_{\rm e}\,({\rm kpc})$ | ${\rm 1.0}$ | ${\rm 1.2}$ | ${\rm 7.2}$ | ${\rm 2.4}$ | ${\rm 2.9}$ | ${\rm 1.6}$ | ${\rm 1.5}$ | $2.2\pm 0.1$ | $1.6\pm 0.1$
$n_{\rm S\acute{e}rsic}$ | ${\rm 0.77}$ | ${\rm 0.60}$ | ${\rm 0.70}$ | ${\rm 0.67}$ | ${\rm 0.58}$ | ${\rm 0.73}$ | ${\rm 0.60}$ | $0.6$ | $0.79$
$M^{\rm central}_{\star}\,(10^{11}M_{\odot})$ | ${\rm 3.1}$ | ${\rm 3.1}$ | ${\rm 3.1}$ | ${\rm 3.7}$ | ${\rm 2.0}$ | ${\rm 1.3}$ | ${\rm 1.8}$ | $1.0$ | $1.0$
$M^{\rm host}_{\rm vir}\,(10^{13}M_{\odot})$ | ${\rm 1.8}$ | ${\rm 1.8}$ | ${\rm 1.8}$ | ${\rm 1.4}$ | ${\rm 0.4}$ | ${\rm 0.3}$ | ${\rm 3.3}$ | $0.6^{+2.6}_{-0.4}$ | $0.6^{+2.6}_{-0.4}$
$d_{\rm host}\,({\rm kpc})$ | ${\rm 80.2}$ | ${\rm 66.3}$ | ${\rm 174.1}$ | ${\rm 483.8}$ | ${\rm 72.3}$ | ${\rm 56.9}$ | ${\rm 36.2}$ | $>80$ | $>165$
$d_{\rm min}\,({\rm kpc})$ | ${\rm 9.1}$ | ${\rm 10.7}$ | ${\rm 31.7}$ | ${\rm 7.4}$ | ${\rm 3.6}$ | ${\rm 10.4}$ | ${\rm 6.7}$ | $-$ | $-$
Supplementary Table 1: Characteristics of our dark-matter deficient galaxies.
‘LH’ and ‘WP’ denote Long Hair and Wild Potato. Measurements are taken within
$r^{\star}_{\rm 50}$, the radius containing 50% of the stellar mass, unless
stated otherwise. (i) Stellar mass; (ii) gas mass; (iii) dark matter mass;
(iv) total mass; (v) 1D line-of-sight velocity dispersion; (vi) galactic size;
(vii) effective 2D radius ($g$-band stellar light); (viii) Sérsic[40] index;
(ix) stellar mass of central within $r^{\star}_{\rm 80}$; (x) host’s virial
mass (90% confidence[34]); (xii) present-time halo-centric distance (projected
distance is used for observations); (xiii) minimum halo-centric distance.

###  0.8 Characteristics of our dark-matter deficient sample.

Supplementary Table 1 lists various properties for our seven dark-matter
deficient galaxies. Supplementary Fig. 6 shows mock Hubble space telescope
$u/g/r$ composite stellar images for this set. These images were created with
Fire Studio, an open source Python visual package. We employ the same surface
brightness limit as in Figure 1: 28.9 mag arcsec-2. The field of view
corresponds to the subhalo radius, defined as the truncation radius[1] where
the density profile experiences an upward “turn over.” The inner and outer red
circles denote $r^{\star}_{\rm 50}$ and $r^{\star}_{\rm 80}$, respectively.
The blue circles denote $R_{\rm e}$, the $g$-band effective radius (more on
how this is measured below). Supplementary Fig. 7 shows cumulative 3D-radial
mass profiles within the subhalo radius. The dark and light vertical
rectangles represent $r^{\star}_{\rm 50}$ and $r^{\star}_{\rm 80}$,
respectively. The vertical blue line represents $R_{\rm e}$. The dotted-teal,
dot-dashed-orange, dashed-purple and solid-black lines represent gas (when
available), stars, dark matter and the sum of these (the total mass).
Supplementary Fig. 8 shows histograms and Gaussian best-fitting lines for 1D
line-of-sight velocity dispersions with $r^{\star}_{\rm 50}$ along three
random orthogonal directions ($x$, $y$ and $z$ in the frame of the simulation
box). We report values using particles within a sphere with this radius. Using
a projected circular region yields very similar results.

With the exception of Long Hair, all of our simulated dark-matter deficient
galaxies exhibit low surface brightness tidal features (down to 28.9 mag
arcsec-2 – but see Supplementary Fig. 9 below for an analysis of this object
at deeper surface-brightness levels). Deer is the only member of our set with
a detectable (albeit subdominant) gaseous component. For Wolf, Wild Potato and
Paint, the stellar cumulative mass profiles dominate at all galactocentric
radii (the orange and black curves are almost indistinguishable). This is
particularly evident for Paint, the galaxy with the lowest dark-matter-to-
total mass fraction in our simulation (the purple curve is $\sim$2 dex below
the orange curve at all radii). Wolf, Wild Potato and Paint are the only
galaxies that achieve 1D line-of-sight velocity dispersions below 10 km sec-1
– which has been a challenge to produce in cosmological simulations[12, 13].
In addition to lacking low surface brightness features, Long Hair is also the
galaxy with the largest halo-centric distance today. Note that its line-of-
sight velocity dispersion histograms and curves along the three orthogonal
directions are almost indistinguishable, suggesting that this galaxy has had
enough time at distances far away from its host to become dynamically relaxed.

We recognize that it is observationally impossible to directly measure
stellar-mass-based 3D sizes. To emulate what is often done, we also measure
the effective radii ($R^{\rm 2D}_{\rm e}$) of the seven dark-matter deficient
galaxies by fitting a Sérsic[40] profile to the average surface brightness in
the $g$-band along each principal axis. In Supplementary Fig. 6, we report the
simple average of $R^{\rm 2D}_{\rm e}$ calculated along three (random)
orthogonal directions (blue circles, using a deprojection factor[39] of $4/3$)
– which agrees well with our 3D method. Supplementary Table 1 includes these
values, along with the associated Sérsic indices ($n_{\rm
S\acute{e}rsic}\simeq 0.60-0.77$), which are also in line with observations
($n_{\rm S\acute{e}rsic}=0.60$ and $0.79$ for DF2 and DF4, respectively).

Supplementary Figure 9:  Tidal features. Mock $g$-band image of Long Hair,
limited to 30.5 mag arcsec-2. The contours represent $g$-band surface-
brightness levels at 29, 28.5, 30 and 30.5 mag arcsec-2. Tidal features for
this object are evident at surface-brightness levels deeper than 30 mag
arcsec-2.

###  0.9 Tidal features.

Past interactions may leave a detectable imprint on galaxies. This has
instigated a debate on whether or not DF2 and DF4 do exhibit tidal features[2,
3, 4]. The majority of our dark-matter deficient simulated galaxies exhibit
tidal features down to 28.9 mag arcsec-2. One exception is Long Hair. This
object also has the largest halo-centric distance amongst our set of seven,
and is dynamically relaxed (see the upper-right panel of Supplementary Fig.
6). Supplementary Fig. 9 explores this object in more detail. The background
$g$-band image is now limited to 30.5 mag arcsec-2 and the contours correspond
to 29, 29.5, 30 and 30.5 mag arcsec-2 surface-brightness levels in the
$g$-band. We find that, at 30 and 30.5 mag arcsec-2, Long Hair does display
evident tidal features. We defer further studies on this issue to future work
because a careful analysis may require more realistic synthetic data
products[5].

###  0.10 The evolution of our dark-matter deficient sample.

Supplementary Fig. 10 shows halo-centric distance (in physical units) versus
cosmic time, between infall time (defined as the first time the galaxy became
a satellite) and today. Each panel displays one of our seven dark-matter
deficient galaxies. The dashed and dotted gray lines represent the growth of
the virial radius of the host and 5% thereof. The vertical thin-gray lines
indicate pericentric passages and the vertical thick-red line corresponds to
the minimum halo-centric distance. We confirmed the aformentioned number of
close passages by inspecting 3D representations of these orbits (not
included). Supplementary Fig. 11 show mass (within the subhalo radius) versus
time. The dotted-teal, dot-dashed-orange, dashed-purple and solid-black curves
represent mass in gas, stars, dark matter and the sum of these, respectively.
Before becoming satellites, our seven galaxies were more massive
($M_{\star}\sim 2-10\times 10^{9}M_{\odot}$), gas-rich ($M_{\rm gas}\sim
3-25\times 10^{9}M_{\odot}$) and dark-matter dominated ($M_{\rm dm}\sim
4-16\times 10^{10}M_{\odot}$). All of them lost over $99\%$ of their dark-
matter content between infall and the present time. See Supplementary Table 2
for numerical values.

Supplementary Figure 10: Halo-centric distance (in physical units) versus
cosmic time. Each panel corresponds to one of our seven dark-matter deficient
galaxies. The dashed and dotted gray curves represents the virial radius of
the host and 5% thereof. The time range extends from $t_{\rm infall}$, the
infall time when the galaxy became a satellite for the first time, to the
present time (13.8 Gyr). The vertical thin gray lines indicate pericentric
passages. The vertical thick red line highlights the time of the minimum halo-
centric distance.

Supplementary Figure 11: Mass evolution versus cosmic time. Each panel corresponds to one of our seven dark-matter deficient galaxies. The dotted-teal, dot-dashed-orange and dashed-purple curves indicate mass in gas, stars and dark matter. The solid black curve represents total mass. The time range extends from $t_{\rm infall}$, the infall time when the galaxy became a satellite for the first time, to the present time (13.8 Gyr). The vertical thin gray lines indicate pericentric passages. The vertical thick red line highlights the time of the minimum halo-centric distance. | ${\rm Bird}$ | ${\rm Blue}$ | ${\rm Deer}$ | ${\rm LH}$ | ${\rm Paint}$ | ${\rm WP}$ | ${\rm Wolf}$
---|---|---|---|---|---|---|---
$M^{z=0}_{\star}$ | ${\rm 7.0}$ | ${\rm 7.6}$ | ${\rm 10.0}$ | ${\rm 8.8}$ | ${\rm 3.9}$ | ${\rm 3.0}$ | ${\rm 2.0}$
$M^{z=z_{\rm infall}}_{\star}$ | ${\rm 39.4}$ | ${\rm 64.5}$ | ${\rm 19.5}$ | ${\rm 30.2}$ | ${\rm 98.5}$ | ${\rm 73.4}$ | ${\rm 68.8}$
$M^{z=0}_{\rm gas}$ | ${\rm 0.0}$ | ${\rm 0.0}$ | ${\rm 0.8}$ | ${\rm 0.0}$ | ${\rm 0.0}$ | ${\rm 0.0}$ | ${\rm 0.0}$
$M^{z=z_{\rm infall}}_{\rm gas}$ | ${\rm 26.7}$ | ${\rm 89.7}$ | ${\rm 91.8}$ | ${\rm 36.8}$ | ${\rm 247.2}$ | ${\rm 187.3}$ | ${\rm 176.5}$
$M^{z=0}_{\rm dm}$ | ${\rm 1.6}$ | ${\rm 1.2}$ | ${\rm 10.0}$ | ${\rm 6.9}$ | ${\rm 0.08}$ | ${\rm 0.3}$ | ${\rm 0.08}$
$M^{z=z_{\rm infall}}_{\rm dm}$ | ${\rm 353.2}$ | ${\rm 920.0}$ | ${\rm 480.8}$ | ${\rm 565.9}$ | ${\rm 1,622.6}$ | ${\rm 1,252.6}$ | ${\rm 1,252.9}$
$f^{\rm lost}_{\star}$ | ${\rm 82.36\%}$ | ${\rm 88.21\%}$ | ${\rm 45.40\%}$ | ${\rm 70.99\%}$ | ${\rm 96.08\%}$ | ${\rm 95.94\%}$ | ${\rm 97.17\%}$
$f^{\rm lost}_{\rm gas}$ | ${\rm 100\%}$ | ${\rm 100\%}$ | ${\rm 99.15\%}$ | ${\rm 100\%}$ | ${\rm 100\%}$ | ${\rm 100\%}$ | ${\rm 100\%}$
$f^{\rm lost}_{\rm dm}$ | ${\rm 99.56\%}$ | ${\rm 99.87\%}$ | ${\rm 97.85\%}$ | ${\rm 98.78\%}$ | ${\rm 99.99\%}$ | ${\rm 99.98\%}$ | ${\rm 99.64\%}$
$z_{\rm infall}$ | ${\rm 1.26}$ | ${\rm 1.23}$ | ${\rm 1.23}$ | ${\rm 0.76}$ | ${\rm 1.00}$ | ${\rm 0.56}$ | ${\rm 0.63}$
Supplementary Table 2: Mass evolution of our seven dark-matter deficient
galaxies. ‘LH’ and ‘WP’ denote for Long Hair and Wild Potato and $z_{\rm
infall}$ denotes the infall redshift. We display masses in units of
$10^{8}M_{\odot}$ to emphasize changes. Quantities are within $R_{\rm vir}$
(or $R_{\rm subhalo}$ after infall) unless stated otherwise. (i) Stellar mass
at $z=0$; (ii) stellar mass at infall; (iii) gas mass at $z=0$; (iv) gas mass
at infall; (v) dark matter mass at $z=0$; (vi) dark matter mass at infall;
(vii) fraction of stellar mass lost since infall; (viii) fraction of gas mass
lost since infall; (ix) fraction of dark matter mass lost since infall; (x)
infall redshift. Note: Strictly speaking, the final $z=0$ values are lower
limits, given that our galaxies might be impacted by artificial disruption.
However, the final $M^{z=0}_{\rm dm}/M^{z=0}_{\star}$ ratio is unlikely to be
affected by this effect (see below for a discussion).

###  0.11 Stellar populations.

Supplementary Table 3 compares properties of the stellar populations in our
seven dark-matter deficient galaxies against observations. For simulations, we
report quantities within $r^{\star}_{\rm 50}$. There are no measurements
available for the age or metallicity of DF4, but given its similarity in
colour to DF2, it is reasonable to expect that these two objects have similar
stellar populations. Bird and Blue have similar stellar ages to DF2. Our other
dark-matter deficient galaxies are younger (7.3 Gyr at most). In addition to
being the oldest, Bird and Blue also have below-average
$\rm\langle[Fe/H]\rangle$ values relative to the set of seven. However, we do
not achieve metallicities as low as DF2 (or $V-I$ colours as blue as this
object). Comparing to Local Group data[8], Bird, Blue and DF2 lie marginally
within the scatter of the local stellar mass-metallicity relation – with Blue
and Bird sitting on the upper envelop, and DF2 on the lower one. One
possibility is that DF2 and DF4 had lower than usual metallicities than Bird
and Blue at infall (within the scatter, which is also broader in the massive
regime, where the progenitors of these galaxies existed at infall).

| ${\rm Bird}$ | ${\rm Blue}$ | ${\rm Deer}$ | ${\rm LH}$ | ${\rm Paint}$ | ${\rm WP}$ | ${\rm Wolf}$ | ${\rm DF2}$ | ${\rm DF4}$
---|---|---|---|---|---|---|---|---|---
${\rm Age_{50}}$ | ${\rm 10.0}$ | ${\rm 8.7}$ | ${\rm 5.5}$ | ${\rm 7.3}$ | ${\rm 5.5}$ | ${\rm 4.3}$ | ${\rm 4.2}$ | $8.9\pm 1.5$ | $-$
$\langle[{\rm Fe/H}]\rangle$ | ${\rm-0.61}$ | ${\rm-0.34}$ | ${\rm-0.46}$ | ${\rm-0.35}$ | ${\rm-0.08}$ | ${\rm-0.08}$ | ${\rm-0.13}$ | $-1.35\pm 0.12$ | $-$
${\rm V-I}$ | ${\rm 0.70}$ | ${\rm 0.72}$ | ${\rm 0.60}$ | ${\rm 0.70}$ | ${\rm 0.75}$ | ${\rm 0.67}$ | ${\rm 0.67}$ | $0.37\pm 0.05$ | $0.32\pm 0.1$
Supplementary Table 3: The stellar populations of our seven dark-matter
deficient galaxies. ‘LH’ and ‘WP’ denote Long Hair and Wild Potato.
Measurements in simulations are taken within $r^{\star}_{\rm 50}$. (i) stellar
age (median for simulated galaxies in Gyr; see Ref.[51] for DF2’s stellar
light); (ii) stellar iron-to-hydrogen abundance, dex (see Ref.[6] for DF2, who
report this value only for the globular cluster population); (iii) V$-$I
colour (mag, see Ref.[6] for DF2 and Ref.[7] for DF4, who use $V_{\rm
606}-I_{\rm 814}$ from the Hubble Space Telescope Advanced Cameras for Surveys
in the AB system).

###  0.12 Resolution and tidal disruption.

Our dark-matter deficient galaxies are all tidal remnants of satellites which
experienced multiple pericentric passages. To resolve such tidal remnants
requires excellent numerical resolution. It has been shown that subhaloes with
insufficient number of particles simulated with non-optimal softening length
suffer from artificial tidal mass loss and even complete numerical
disruption[9]. Therefore, it is natural to be cautious about the robustness of
their dark-matter deficient status. Although it is quite possible that our
dark-matter deficient galaxies experienced some artificial tidal dark-matter
mass loss by $z=0$, most likely they were already in the dark-matter deficient
territory before artificial tidal mass loss became significant. Below we gauge
the severity of numerical noise and artificial tidal evolution in two ways.

First, we use the study of artificial disruption in Ref.[9] – especially the
results tabulated in their Figure 10, which report the timescales for
numerical disruption in terms of the initial particle number of the satellite
and the softening length. These authors find that, for subhaloes orbiting
within 10% of the virial radius of the host, if the initial particle number is
$\sim 3\times 10^{5}$ and the softening length is $\sim 0.03$ times the scale
radius of the subhalo at infall ($r_{\rm s,\,sub}$), then artificial tidal
stripping kicks in after $\sim 5$ Gyr of evolution (and disrupts the subhalo
completely after $\sim 10$ Gyr). To illustrate, we focus our attention on two
of our dark-matter deficient galaxies: Wolf (the one with the lowest final
stellar mass) and Paint (the one with the lowest final dark-matter mass
fraction). Both of these galaxies started with dark-matter masses of $\sim
1.6$ and $\sim 1.3\times 10^{11}M_{\odot}$ (corresponding to $\sim 5$ and
$\sim 4\times 10^{5}$ dark matter particles) at infall. They both have scale
radii of $\sim 2.5$ kpc, so our softening length of $80$ pc almost exactly
corresponds to $0.03r_{\rm s,\,sub}$. Additionally, their orbital pericentres
are approximately $0.1R^{\rm host}_{\rm vir}$. That is, the aforementioned
artificial disruption condition is applicable to these two galaxies. As our
Supplementary Fig. 11 shows, Paint and Wolf became dark-matter deficient
($M_{\star}/M_{\rm dm}>1$) after spending only $\sim 3$ Gyr in the host
potential, before artificial stripping has time to become significant. That
said, Paint and Wolf spent $\sim 6$ and $\sim 8$ Gyr in total in the host
potential – so during the latest couple of orbits, their associated dark-
matter subhaloes likely suffered from some artificial disruption and thus only
have a few particles left.

Supplementary Figure 12: Dark-matter versus stellar mass retention. Fraction
of retained dark-matter mass versus fraction of retained stellar mass, between
infall and today. The triangles represent simulated satellites with stellar
masses between $10^{8-9}M_{\odot}$ (as in Figures 3 and 4), their colours
represent infall redshift, and their symbol dimensions represent their dark-
matter mass at infall. Our seven dark-matter deficient galaxies (labeled)
experience higher $f^{\rm lost}_{\rm dm}/f^{\rm lost}_{\star}$ values than
other galaxies in our parent sample with comparable $M^{\rm infall}_{\rm dm}$
and $z_{\rm infall}$.

Secondly, we also address this issue with tidal evolutionary tracks[10, 11].
In particular, we estimate at what stage of their structural evolution they
start to deviate from their physical tidal evolutionary tracks. As Figure A1
of Ref.[11] shows, artificial deviation from the tidal track starts when the
maximum-circular-velocity radius $r_{\rm max}$ approaches 8 $\times$ the
softening length. In our case, this is $8\times 80$ pc $\sim 0.6$ kpc. Both
Paint and Wolf started with $r_{\rm max}\sim 5$ kpc. Therefore, according to
the tidal tracks in Figure A1 of Ref.[11], the dark-matter structural
evolution of these two galaxies can be trusted down to $r_{\rm max}/r^{\rm
infall}_{\rm max}\sim 0.1$ – or equivalently, when their bound mass is $1/100$
of the initial value. From our Supplementary Fig. 11, we can see that both
galaxies became dark-matter deficient ($M_{\star}/M_{\rm dm}>1$) before their
dark-matter mass dropped to 1% of that at infall.

We also note that the above studies adopt dark-matter-only configurations. It
would be interesting to see how those numerical experiments change in the
presence of an additional collision-less stellar population, plus a
hydrodynamic gaseous component and feedback physics.

Lastly, we underscore the fact that all of our dark-matter deficient galaxies
exhibit unusually stronger differential mass loss of dark-matter versus stars
relative to other satellites in our simulation. Supplementary Figure 12 shows
the fraction of retained dark-matter mass (defined as $M^{\rm today}_{\rm
dm}/M^{\rm infall}_{\rm dm}$) versus the fraction of retained stellar mass
(defined as $M^{\rm today}_{\star}/M^{\rm infall}_{\star}$). The diagonal
dashed lines depict ratios between these two quantities. The triangles
represent simulated satellites with $z_{\rm infall}>0$ (i.e., we exclude $z=0$
satellites that were not also satellites at the previous-to-last snapshot) and
$10^{8}<M_{\star}/M_{\odot}<10^{9}$ (as in Figures 3 and 4). The symbol
dimensions scale logarithmically with dark-matter mass at infall and the
colour scheme represents infall redshift. Clearly, our seven dark-matter
deficient galaxies all lose significantly more dark-matter mass than stellar-
mass when compared to other satellites with similar $M^{\rm infall}_{\rm dm}$
and $z_{\rm infall}$. That is, even if artificial disruption is significant,
it should affect both sets of galaxies similarly, and yet we still see a
stronger differential mass loss in our sample of seven. This suggest that the
environmental processes making responsible for transforming galaxies into
dark-matter deficient ones are qualitatively correctly captured.

###  0.13 Simulations by other groups.

Since the discovery of the DF2[6], several efforts to re-create analogues in
simulations have been published. What is novel about our work is that, for the
first time, we are able to generate a simulated low-mass galaxy (Wolf) that
resembles DF2 and DF4 (in terms of stellar mass, size, 1D line-of-sight
velocity dispersion and morphology) within a cosmological simulation. In other
words, it is not enough to merely create dark-matter deficient galaxies (which
is challenging on its own right); it is paramount for said galaxies to also
exhibit internal properties that match observations. We devote this Section to
a discussion of studies on dark-matter deficient galaxies using various
methods.

Works on this subject can be categorized into three classes: idealised (non-
cosmological) numerical experiments, zoom-in simulations and cosmological-
volume simulations. The first class[19, 20, 21, 15, 22, 12, 23] offers unique
insights at resolution levels often inaccessible to cosmological simulations.
In particular, some of these controlled experiments[21, 23] have been
instrumental at exposing how challenging it is to achieve velocity dispersions
as low as those measured in DF2 and DF4. The majority of these efforts (but
not all) also highlight the importance of satellite-host interactions, in line
with our conclusions. However, a major limitation of this approach is that it
relies on ad-hoc assumptions about (1) the initial structure of the
progenitors, (2) the properties of their neighboring companion (all of the
numerical experiments published thus far assume the presence of only one
companion), and (3) the orbital geometry of their interaction. Interestingly,
the idealised runs by Ref.[23] circumvent the first caveat by extracting the
progenitor structure from a cosmological box, but the other two caveats
remain. Nevertheless, the adoption of these assumptions does not reflect
realistic cosmological conditions and provides no information on the
likelihood or frequency that such configurations actually occur. Another key
limitation of such works (to-date) is that they operate under the presumption
that a single specific mechanism must be responsible for the creation of dark-
matter deficient low-mass galaxies. For example, unlike the majority of works
adhering to this approach, the idealised simulations by Ref.[15] assume a
priori that dark-matter deficient low-mass galaxies are created via gas-rich
major mergers (i.e., the companion is not a significantly more massive
central, but a galaxy of comparable mass – see also Ref.[12]). Please note
that we do not find any instances of dark-matter deficient low-mass galaxies
being created in this fashion. Overall, the above limitations in the
idealised-simulation method underscore the importance of conducting
investigations in a cosmological context.

The second class involves zoom-in simulations, where a sub-volume of a larger
cosmological box is re-simulated at higher resolution. By design, this method
generally only simulates a massive galaxy and a small number of galaxies in
its proximity, and is thus not well suited for finding objects with rare or
unusual properties. To illustrate, Ref.[16] use a suite of Milky-Way-like
zoom-in simulations and find one satellite with $M_{\star}>M_{\rm dm}$. Unlike
our sample of seven (or DF2 and DF4), this galaxy has a much lower stellar
mass ($\sim 10^{5}M_{\odot}$) and is orbiting a Milky-Way-mass host.
Unfortunately, this object is poorly resolved (its half-light radius is below
their adopted force resolution). If confirmed by these authors with future
simulations at higher resolution, this object will certainly become a
provocative prediction, because no dark-matter deficient galaxies orbiting
massive centrals at the Milky Way scale have ever been observed, and our
predictions reject hosts at this scale. On a different vein, zoom-in
simulations at the galaxy group scale (as suggested by our results and
observations) may also be of significant interest.

The third class is cosmological-volume simulations[9, 12, 10, 11, 13, 14, 15,
17]. Ref.[11] use an older version of a well-established cosmological
simulation to challenge the viability of the $\Lambda$-Cold-Dark-Matter
paradigm. These authors find zero dark-matter deficient galaxies that match
DF2 simultaneously in stellar mass and size. They do find a galaxy slightly
smaller than DF2 (by a factor of 0.8 in stellar-mass and size), but this
object is gas-rich (unlike DF2 and DF4) and its massive companion is over one
order-of-magnitude smaller in stellar mass than NGC 1052. Ref.[9] use the same
simulation and find a few galaxies lacking dark matter – but most are poorly
resolved, rendering their results inconclusive. These authors find one
resolved dark-matter deficient galaxy. However this object has $M_{\star}\sim
9\times 10^{10}M_{\odot}$, almost two orders of magnitude more massive than
DF2 and DF4, and it is the result of a numerical artifact[14]. Ref.[10] find
several dark-matter deficient galaxies in two cosmological simulations, but
these objects are in the $M_{\star}=10^{9-10}M_{\odot}$ regime, over an order-
of-magnitude above DF2 and DF4. Using a newer simulation, Ref.[13] instead
investigate the tidal evolution of low-mass galaxies in massive galaxy
clusters ($M_{\rm vir}\geq 10^{14}M_{\odot}$ – see also Ref.[12]). These
authors do not find any galaxies in the $M_{\star}=10^{8-9}M_{\odot}$ regime
with sufficiently low values of $\sigma^{\rm 1D}_{\star}$ to be consistent
with DF2 and DF4 (their Figure 5). It would be interesting if (1) this study
repeated at group scale found suitable candidates; and (2) if low-mass
galaxies lacking dark matter are observed in massive galaxy clusters. Ref.[15]
use the same simulation to test their high-velocity collision scenario,
finding only one such event, which results in a gas clump that never turns
into a galaxy.

The majority of the cosmological simulations we cite in this paper have
resolutions much lower than ours, which could be the culprit for the lack of
success in finding analogues of DF2 and DF4. Only two works[10, 17] employ
cosmological simulations with volumes and resolutions somewhat comparable to
ours. First we compare to Ref.[17], who employ a $10$-cMpc box (on the side)
with $m_{\rm DM}=10^{6}M_{\odot}$, $m_{\star}=10^{4}M_{\odot}$, maximum
spatial resolution of $34$ pc and star-forming gas density threshold of $10$
cm-3 (versus our $21$-cMpc box with $m_{\rm DM}=3.3\times 10^{5}M_{\odot}$,
$m_{\rm b}=6.3\times 10^{4}M_{\odot}$, maximum adaptive smoothing length
resolution of $1.5$ pc and star-forming gas density threshold of $300$ cm-3).
Below we demonstrate that none of the galaxies in Ref.[17] are consistent with
DF2 or DF4. We also note that their “DM def.” sample is “too generous”.
Namely, they require members of this subset to be satellites with $M_{\rm
dm}/M_{\star}<10$, rendering the majority of their objects dark-matter
dominated by construction (i.e., with $M_{\rm dm}/M_{\star}>1$). As a result,
all of their low-mass galaxies (with $M_{\star}<10^{9}M_{\odot}$) are also
dark-matter dominated. We note that their paper reports quantities out to
$r_{\rm subhalo}$, not within $r^{\star}_{\rm 50}$, as done here and in
observations.

Supplementary Figure 13:  Comparison with other cosmological simulations.
Stellar mass versus dark-matter-to-stars mass ratio. Quantities are measured
out to either the virial radius (centrals) or the subhalo radius (satellites).
The small circles (centrals) and triangles (satellites) represent our parent
sample (with $M_{\star}<10^{9}M_{\odot}$). The light-yellow region denotes the
dark-matter deficient regime in this stellar-mass range, which contains six of
our seven galaxies (large yellow circles), but none of the galaxies in
Ref.[17] (large light-blue hexagons). The datum shared with us by Y. Jing
(large orange pentagon, private communication) is dark-matter deficient, but
its stellar mass and velocity dispersion are too high compared with
observations (dark-red arrows).

Supplementary Fig. 13 reproduces Figure 1 in Ref.[17] with our data, theirs,
an object reported shared with us by Y. Jing (the lead author of Ref.[10],
private communication) and observations. We use WebPlotDigitizer, a free
online tool, to extract data from Ref.[17]. Quantities in this Supplementary
Figure are measured out to $R_{\rm vir}$ for centrals (circles), and $r_{\rm
subhalo}$ for satellites (triangles). The light-gray band represents the
$M_{\star}-M_{\rm halo}/M_{\star}$ relation for centrals. The light-yellow
region denotes the dark-matter deficient regime (with $M_{\rm
dm}/M_{\star}<1$). The dark-red arrows with error bars represent observations.
Here, the vertical location of these two arrows corresponds to
$M_{\star}=M_{\star}(r<r_{\rm subhalo})\equiv 2\times
M_{\star}(r<r^{\star}_{\rm 50})$. See Table 1 for the
$M_{\star}(r<r^{\star}_{\rm 50})$ values measured for DF2 and DF4. We note
that $M_{\rm dm}/M_{\star}$ has not been observed out to $r_{\rm subhalo}$ for
these two galaxies (i.e., the horizontal extent of the two dark-red arrows).
However, given that all of our seven galaxies are dark-matter deficient within
both $r^{\star}_{\rm 50}$ and $r_{\rm subhalo}$, it is reasonable to assume
that this is also the case for DF2 and DF4. The large yellow circles highlight
our seven dark-matter deficient galaxies, which span approximately two orders-
of-magnitude to the left of the $M_{\rm dm}/M_{\star}=1$ boundary (vertical
dashed black line). In contrast, all low-mass galaxies in the Ref.[17] sample
(large light-blue hexagons) are dark-matter dominated – i.e., they span about
one order-of-magnitude to the right of the $M_{\rm dm}/M_{\star}=1$ boundary.
In particular, their least dark-matter dominated galaxy in this Figure still
has $\sim 66\%$ more mass in dark-matter than stars.

Note that Supplementary Fig. 13 only displays galaxies in the
$M_{\star}=10^{8-9}M_{\odot}$ regime. Recall that our focus is on low-mass
galaxies with $M_{\star}<10^{9}M_{\odot}$ (within $r^{\star}_{\rm 50}$). This
is because observations[5] suggest that galaxies at higher stellar-mass
regimes tend to already have lower dark-matter contributions (within
$r^{\star}_{\rm 50}$), so discoveries of more-massive galaxies with low dark-
matter content are not that surprising. When we expand our galactocentric
extent out to $r_{\rm subhalo}$, Deer falls outside (above) the stellar-mass
range considered in this Supplementary Figure. This Supplementary Figure also
excludes three massive or intermediate-mass galaxies reported by Ref.[17]
(with $M_{\rm dm}/M_{\star}<1$). Concretely, these three objects have
approximately the following stellar masses: $1.2\times 10^{9}M_{\odot}$,
$1.6\times 10^{10}M_{\odot}$ and $1.1\times 10^{11}M_{\odot}$. The first of
the three lies right above our low-mass regime cut (at
$M_{\star}=10^{9}M_{\odot}$), but below theirs (at
$M_{\star}=10^{9.5}M_{\odot}$). Although the line separating low- and
intermediate-mass galaxies is arbitrary, this object is over one-order-of-
magnitude above observations – i.e., $12$-$\sigma$ above DF2 and $\sim
13$-$\sigma$ above DF4.

Nevertheless, this “mildly” dark-matter dominated population reported by
Ref.[17] (the large light-blue hexagons in Supplementary Fig. 13) is
interesting on its own right. This set is situated to the left of the light-
gray band. I.e., they are less dark-matter dominated than centrals at
comparable stellar masses. The authors show that this effect is caused by
tidal perturbations. We note that, unlike our set of seven, their objects
never transit within 5% of the host virial radius. Instead, their $d_{\rm
min}/R_{\rm vir}$ values are between 20% and 80% (compare the left panel of
their Figure 10 to our Figure 4). It is nonetheless striking to see that, even
though their satellite-central interactions have larger $d_{\rm min}/R_{\rm
vir}$ values than ours, they still contribute towards helping those galaxies
migrate to the left of the light-gray band. But more importantly, the absence
of low-mass galaxies in their sample that (1) cross to the left of the $M_{\rm
dm}/M_{\star}=1$ boundary and (2) plunge within 5% of their host virial radius
further supports our scenario. Namely, that satellites are required to pierce
through the innermost portions of their hosts to become dark-matter deficient
galaxies (as demonstrated in Figure 4).

We now discuss Ref.[10], who found an interesting object using a third
simulation (at higher resolution than the other simulations they study). This
simulation employs a $25$-cMpc box (on the side) with $m_{\rm DM}=1.21\times
10^{6}M_{\odot}$, $m_{\rm b}=2.26\times 10^{5}M_{\odot}$, softening length of
$350$ pc and star-forming gas density threshold of $10$ cm-3 (versus our
$21$-cMpc box with $m_{\rm DM}=3.3\times 10^{5}M_{\odot}$, $m_{\rm
b}=6.3\times 10^{4}M_{\odot}$, $h_{\star}=12$ pc, $h_{\rm DM}=80$ pc, maximum
adaptive smoothing length resolution of $1.5$ pc and star-forming gas density
threshold of $300$ cm-3). They report quantities within $2\times
r^{\star}_{\rm 50}$, which makes it difficult to compare directly with DF2 and
DF4 (which employ $r^{\star}_{\rm 50}$ instead). This object has
$M_{\star}(r<2r^{\star}_{\rm 50})=2.47\times 10^{8}M_{\odot}$, and $f_{\rm
dm}\equiv M_{\rm dm}(r<2r^{\star}_{\rm 50})/M_{\rm total}(r<2r^{\star}_{\rm
50})\sim 0.31$. We note that this object becomes dark-matter deficient through
the same mechanism as ours, by plunging within 5% of its host’s virial radius
(their Figure 6). The large orange hexagon in Supplementary Fig. 13 represents
this object, which has $M_{\star}(r<r_{\rm subhalo})=2.67\times
10^{8}M_{\odot}$ and $M_{\rm dm}/M_{\star}\sim 0.46$. Measurements of this
object out to $r_{\rm subhalo}$ are not published in Ref.[10]; rather these
were shared with us by Y. Jing, the lead-author of that work (via private
communication). Whilst this object has $M_{\rm dm}<M_{\star}$, its properties
do not match those of DF2 and DF4. Specifically, its stellar mass is $\sim
1.7$-$\sigma$ above DF2 and $\sim 1.5$-$\sigma$ above DF4. Furthermore, recall
that prior to our work, no other cosmological simulation has successfully
created galaxies with velocity dispersions as low as those of DF2 and DF4[12,
13]. This issue has also been explored by some idealised high-resolution
experiments. Ref.[23] find that to achieve this, very particular satellite-
central orbital configurations are required. Ref.[21], on the other hand,
invoke supernova feedback and ram-pressure stripping, aided by subsequent
tidal stripping, to achieve such low $\sigma^{\rm 1D}_{\star}$ values. The
velocity dispersion of this object (large orange pentagon in Supplementary
Fig. 13) is $\sigma_{\star}=22.4$ km sec-2. If we assume kinematic isotropy,
this would correspond to $\sigma^{\rm 1D}_{\star}=12.9$ km sec-2. In other
words, its 1D line-of-sight velocity dispersion is $\sim 1.9$-$\sigma$ above
DF2 and $\sim 2.0$-$\sigma$ above DF4. (Strictly speaking, such comparisons
require velocity dispersions within $r^{\star}_{\rm 50}$, but these
measurements have not been published.)

There are several possibilities that could explain why galaxies that resemble
the observed dark-matter deficient galaxies (such as Wolf) have not been
reported in past cosmological simulations. First, as mentioned in the main
text, our simulations are relatively unique in that they explicitly track
dense, molecular gas in low-mass galaxies within a volume that is large enough
to contain several massive group-size haloes. The simulation in Ref.[10]
probes a similar volume, but uses a lower resolution of 0.35 kpc – although it
is still higher than most. However, they also employ a cruder star-formation
treatment: a polytropic equation-of-state, with star-formation occurring at
gas-densities $>10$ cm-3. Ref.[17] also has similar formal spatial resolution
and volume to ours, and their star-formation prescription is more
sophisticated than that in Ref.[10] (it is similar to ours). However, their
gas component is also allowed to form stars at densities as low as $10$ cm-3
(as in Ref.[10]). What makes our simulation unique is that all of our star-
forming gas reaches densities $>300$ cm-3 – i.e., thirty times higher than
what is used in these comparably-sized simulations. Furthermore, our adaptive
smoothing length reaches $1.5$ pc (versus $34$ pc for Ref.[17] and $350$ pc
for Ref.[10]). This allows us to track the internal baryonic structure of
galaxies with higher precision. Another possibility is that some of these
simulations do in fact produce such extreme objects, but their automated
galaxy-recovery algorithms might pose an impediment to their query (recall
that we employ a visual-recovery technique, which may not be feasible for
larger runs). It would be interesting to carry out a visual search for dark-
matter deficient galaxies in the simulations discussed by Ref.[10] and
Ref.[17]. Indeed, additional identifications of more galaxies with internal
properties resembling DF2 and DF4 in other cosmological simulations will
certainly put the standard paradigm on firmer ground.

## References

  * [1] Knollmann, S. R. & Knebe, A. AHF: Amiga’s Halo Finder. Astrophys. J. Suppl. 82, 2, 608-624 (2009).
  * [2] Montes, M. et al. The Galaxy “Missing Dark Matter” NGC 1052-DF4 is Undergoing Tidal Disruption. Astrophys. J. 904, 114 (2020).
  * [3] Montes, M. et al. A disk and no signatures of tidal distortion in the galaxy ”lacking” dark matter NGC 1052-DF2. Astrophys. J. 919, 56 (2021).
  * [4] Keim, M. A. et al. Tidal Distortions in NGC1052-DF2 and NGC1052-DF4: Independent Evidence for a Lack of Dark Matter. Preprint at https://arxiv.org/abs/2109.09778 (2021).
  * [5] Bottrell, C. et al. Deep learning predictions of galaxy merger stage and the importance of observational realism. Mon. Not. R. Astron. Soc. 490 4, 5390 (2019).
  * [6] van Dokkum, P. et al. An Enigmatic Population of Luminous Globular Clusters in a Galaxy Lacking Dark Matter. Astrophys. J. 856 L30(2018).
  * [7] Cohen, Y. The Dragonfly Nearby Galaxies Survey. V. HST/ACS Observations of 23 Low Surface Brightness Objects in the Fields of NGC 1052, NGC 1084, M96, and NGC 4258. Astrophys. J. 868, 96 (2018).
  * [8] Kirby, E. N. et al. The universal stellar mass-stellar metallicity relation for dwarf galaxies. Astrophys. J. 779, 102, 21, (2013).
  * [9] van den Bosch, F. C. & Ogiya, G. Dark matter substructure in numerical simulations: a tale of discreteness nosation, runaway instabilities, and artificial disruption. Mon. Not. R. Astron. Soc. 475, 4066 (2018).
  * [10] Errani, R. & Peñarrubia, J. Can tides disrupt cold dark matter subhaloes? Mon. Not. R. Astron. Soc. 491, 4, 4591 (2020).
  * [11] Errani, R. & Navarro, J. F. The asymptotic tidal remnants of cold dark matter subhaloes. Mon. Not. R. Astron. Soc. 505, 1, 18 (2021).
  * [12] Lee, J., Shin, E-j & Kim J.-h. Dark Matter Deficient Galaxies and Their Member Star Clusters Form Simultaneously during High-velocity Galaxy Collisions in 1.25 pc Resolution Simulations. Astrophys. J. Lett. 917, L15, 8pp (2021).