perturbed. Therefore, to estimate the $\eta$ at which simulated dwarfs should
start appearing morphologically disturbed, we look at the evolution of their
aspect ratio (Equation 62). We need to bear in mind that even a uniform
external field can cause a MONDian dwarf to become deformed because the
potential of a point mass is not spherical once the EFE is considered (Banik &
Zhao, 2018a). N-body simulations of dwarfs experiencing the EFE but not tides
explicitly show that this process can yield axis ratios of $\approx 0.7$ (Wu
et al., 2017). This is very much in line with our lowest eccentricity orbit
with $R_{i}=150$ kpc, so the mild degree of flattening evident here is not
necessarily indicative of tidal effects. We find that models with $R_{i}=150$
kpc start to acquire significantly elongated morphologies throughout most of
their trajectories only when $\eta\ga 0.6$ (see column 3 in Fig. 13).
Therefore, we take $\eta_{\textrm{min, dist}}\approx 0.6$. This is slightly
higher than what our MCMC analysis requires ($\eta_{\textrm{min,
dist}}=0.24_{-0.19}^{+0.24}$). One possible explanation is that dwarfs with
higher $R_{i}$ start acquiring elongated morphologies at lower $\eta$.

To check if increasing the resolution would affect our results, we perform a
high-resolution rerun of one of our models for each $R_{i}$. This is shown
using the dotted line in each panel of Fig. 13. The only resolution-related
effect which we can observe is that the half-mass radius of a distant dwarf
expands less than at lower resolution. Because of this, we obtain slightly
lower pericentric $\eta$ values for the same orbit with higher resolution.
However, the evolution of the dwarf properties as a function of $\eta$ at
pericentre remains almost the same as for the low-resolution model. Therefore,
our conclusions should barely be affected by the resolution of the simulation.

##  8 Discussion

Observations of Fornax Cluster dwarf galaxies show that some of them present a
detectable level of disturbance in their morphology. Among the environmental
effects inside a galaxy cluster that could be causing this disturbance, we
found that gravitational tides from the cluster are the most likely cause
(Section 3). The condition for a dwarf galaxy in a galaxy cluster to be
tidally stable is approximately the same as the requirement that the dwarf’s
density exceed the average density of the cluster interior to the dwarf’s
orbit (Equation 7).121212The tidal stress $\Delta g_{c}/\Delta r$ is related
to the cluster mass profile $M_{c}\left(<R\right)$ by $\Delta g_{c}/\Delta
r=GM_{c}\left(2-\alpha\right)/R^{3}$, from which it follows that
$r_{\textrm{tid}}^{3}/M_{\textrm{dwarf}}\approx R^{3}/M_{c}$. Thus, a dwarf
with $r_{h}\approx r_{\textrm{tid}}$ has $M_{\textrm{dwarf}}/r_{h}^{3}\approx
M_{c}/R^{3}$. This should be the case for a $\Lambda$CDM dwarf in a cluster
because we expect the dwarf to be dominated by dark matter and to have formed
much earlier than the cluster, at which time the cosmic mean density was
higher. Therefore, in this paradigm, the dwarf galaxies in the Fornax Cluster
should be little affected by the tides it raises. This is indeed what our
calculations show (Fig. 2).

In MOND, the enhancement to the Newtonian gravity of an isolated dwarf is
similar to that provided by the dark matter halo in $\Lambda$CDM. However,
MONDian dwarfs in a galaxy cluster are also affected by the resulting EFE,
which weakens their self-gravity. As a result, they are more susceptible to
tides than dwarfs in $\Lambda$CDM, which has no EFE due to the strong
equivalence principle. Therefore, observations of Fornax dwarfs can be used to
compare which of the two models performs better.

To check if tides might be important in the Fornax Cluster, we plotted the
projected separation ($R_{\textrm{sky}}$) of each FDS dwarf against a measure
of its surface brightness (Fig. 3). This revealed a lack of low surface
brightness dwarfs in the central $\approx 200$ kpc even though such dwarfs are
evident further out, indicating that selection effects are not responsible for
the tentative tidal edge marked on this figure as a grey line. Just below
this, the proportion of apparently disturbed dwarfs is also much higher than
elsewhere in the cluster (see Fig. 4). We quantified this trend by plotting
the disturbed fraction as a function of the tidal susceptibility $\eta$ of
each dwarf (Equation 29), revealing a clear rising trend detected at
$2.9\sigma$ significance in $\Lambda$CDM and $3.5\sigma$ in MOND (Fig. 5).
These arguments suggest that the dwarf galaxy population in the FDS catalogue
has been significantly shaped by tides, as previously argued by Venhola et al.
(2022).

Figure 14: Joint inference on $\eta_{\textrm{min, dist}}$ and
$\eta_{\textrm{destr}}$ (Section 5.2.4). We show the $1\sigma$ (inner solid
line), $3\sigma$ (dashed line), and $5\sigma$ (outer dotted line) confidence
region for MOND (blue) and $\Lambda$CDM (orange). The thick orange line shows
the $\Lambda$CDM expectation that $\eta_{\textrm{destr}}\approx 1$. For MOND,
the corresponding expectation from our N-body simulations (Section 7) is that
$\eta_{\textrm{destr}}=1.70\pm 0.30$ (horizontal blue stripe). The grey shaded
region below the line of equality is not allowed by our choice of prior
because it is unphysical.

However, the overall distribution of $\eta$ only goes up to $\approx 0.5$ in
$\Lambda$CDM (Fig. 2). We expect a dwarf to be destroyed or severely disturbed
only if $\eta\approx 1$, as indicated by $\Lambda$CDM N-body simulations
(Peñarrubia et al., 2009; van den Bosch et al., 2018). We quantified this
discrepancy using our MCMC analysis, which shows that the tidal stability
limit of the Fornax dwarfs should be
$\eta_{\textrm{destr}}=0.25^{+0.07}_{-0.03}$ to match observations. Therefore,
$\Lambda$CDM dwarfs should be destroyed when the tidal force that they
experience is $\approx 0.25^{3}=1.56\times 10^{-2}$ times smaller than their
internal gravity (tidal force/internal gravity $\approx\eta^{3}$). Not only is
this unrealistic, but also such a low $\eta_{\textrm{destr}}$ is in $>5\sigma$
tension with the $\eta_{\textrm{destr}}$ value of 1 inferred from $\Lambda$CDM
N-body simulations (Fig. 14). The highest $\eta_{\textrm{destr}}$ value
achieved with our MCMC analysis for $\Lambda$CDM is only 0.60. This
corresponds to the $4.42\sigma$ upper limit because we ran $10^{5}$ MCMC
trials. Since the uncertainty on $\eta_{\textrm{destr}}$ towards higher values
from the mode is only 0.07, it is clear that $\eta_{\textrm{destr}}=1$ is
strongly excluded by the observations if the tidal susceptibilities are
calculated within the $\Lambda$CDM framework.

These calculations are based on Equation 11, which can be written in the
alternative form

$\displaystyle\frac{r_{\textrm{tid,
}\Lambda\textrm{CDM}}}{R}~{}=~{}\left(\frac{M_{\textrm{dwarf}}}{\beta
M_{c}\left(<R\right)}\right)^{1/3}\,,\quad\beta~{}=~{}2\left(2-\alpha\right)\,,$
(63)

where $\alpha=1.1$ (defined in Equation 8) is the logarithmic slope of the
Fornax Cluster mass profile $M_{c}\left(<R\right)$ based on hydrostatic
equilibrium of the gas around its central galaxy (Paolillo et al., 2002). This
implies $\beta=1.8$. Other workers use slightly different definitions for the
tidal radius, which affects the results somewhat because the calculated
$\eta\propto\beta^{1/3}$. For example, equation 6 of Wasserman et al. (2018)
gives $\beta=2-\alpha=0.9$ for radial orbits and $\beta=3-\alpha=1.9$ for
circular orbits. Allowing even a modest amount of eccentricity, it is clear
that $\beta$ in their tidal radius definition is smaller than our adopted 1.8,
so their formula generally gives even lower $\eta$ values, worsening the
problem for $\Lambda$CDM. Meanwhile, equation 3 of Peñarrubia et al. (2009)
gives $\beta=3$, though this is for circular orbits and lacks a rigorous
derivation (see section 3.1 of Peñarrubia et al., 2008). $\beta=2$ is more
appropriate to account for elongation in the potential along the radial
direction (Innanen et al., 1983; Zhao & Tian, 2006). However, even if we adopt
$\beta=3$, this would only raise our calculated $\eta$ values by a factor of
$\left(3/1.8\right)^{1/3}$, or equivalently imply that we can keep our
definition but should consider dwarfs to be destroyed at
$\eta_{\textrm{destr}}=\left(1.8/3\right)^{1/3}=0.84$. This is still well
above the value given by any of the $10^{5}$ trials in the MCMC analysis. A
more recent detailed derivation affirms that for circular orbits, the
appropriate value of $\beta=3-\alpha=1.9$ in the Fornax case (equation 5 of
van den Bosch et al., 2018), which is very similar to our adopted value of
1.8. Although this could be somewhat higher with a lower value for $\alpha$,
we can get $\beta=3$ only for circular orbits around a point mass
($\alpha=0$), which is not consistent with the Fornax Cluster having an
extended dark matter halo. Moreover, a dwarf on an elliptical orbit is exposed
to the pericentre value of $\eta$ for only a short time. We may intuitively
expect that a dwarf would be disrupted only if it experiences $\eta>1$ for a
significant duration, since otherwise there is not enough time for tidal
forces to disrupt the dwarf. This could explain why van den Bosch et al.
(2018) found that dwarfs are actually quite robust to tides, more so than in
many numerical simulations where apparent tidal destruction could be a
numerical artefact (see also Webb & Bovy, 2020). It could well be that the
appropriate $\eta_{\textrm{destr}}$ is slightly above 1, as in the MOND case.
Moreover, van den Bosch et al. (2018) found that galaxy-galaxy harassment is
much less damaging than the tidal shock from pericentre passage. While their
work addressed subhaloes in a MW-like halo and neglected hydrodynamics, it is
still very useful in showing that a subhalo can resist disruption even if the
energy it gains from harassment exceeds the binding energy, justifying our
neglect of the harassment scenario (Section 4.1).

In MOND, we obtained a tidal stability limit with the MCMC analysis of
$\eta_{\textrm{destr}}=1.88^{+0.85}_{-0.53}$, which is closer to the expected
value of $\approx 1$ based on analytic arguments (Equation 20). To check if
this limit is accurate, we performed several N-body simulations of a dwarf
orbiting a central potential similar to the Fornax Cluster (Section 7). These
simulations suggest that cluster tides would make Fornax dwarfs appear
disturbed when $\eta_{\textrm{min,dist}}\ga 0.6$ and destroy them at
$\eta_{\textrm{destr}}=1.70\pm 0.30$, which is in good agreement with our MCMC
results (see Fig. 14).

We considered several possible explanations for the discrepancy between the
low tidal susceptibility values of $\Lambda$CDM dwarfs and the fact that some
of the observed Fornax dwarfs appear disturbed. This could be due to the fact
that cluster tides are not the main effect responsible for the observed
morphological disturbances. However, there are several trends in the FDS that
suggest exactly this. These trends are as follows:

  1. 1.

There are fewer low surface brightness dwarfs towards the centre of the
cluster, where they are most susceptible to tides (Fig. 3). Since such dwarfs
are detectable further out, this feature cannot be ascribed to selection
effects. A related finding is that FDS dwarfs are typically larger towards the
cluster centre, which could be related to tidal heating (for a more detailed
discussion, see section 7.4 of Venhola et al., 2022); and

  2. 2.

The algorithm in charge of fitting the simulated Fornax system to the
observations clearly noticed a rising trend between $\eta$ and the probability
of disturbance ($P_{\textrm{dist}}$). This is shown by the fact that the
algorithm chose $P_{\textrm{dist, ceiling}}>P_{\textrm{dist, floor}}$ with
$\approx 3\sigma$ confidence in both $\Lambda$CDM and MOND (see Fig. 15), even
though we did not impose this condition a priori.

We have seen that these trends cannot be understood in $\Lambda$CDM as a
direct consequence of cluster tides given the very low $\eta$ values.
Moreover, the other major environmental effect that could be causing the
observed disturbance (galaxy-galaxy harassment) also presents very low $\eta$
values (see Section 4.1).

Figure 15: Joint inference on $P_{\textrm{dist, floor}}$ and $P_{\textrm{dist,
ceiling}}$ (Section 5.2.4). We show the $1\sigma$ (inner solid line),
$2\sigma$ (dashed line), and $3\sigma$ (outer dotted line) confidence region
for MOND (blue) and $\Lambda$CDM (orange). Physically, we expect to get values
above the solid red line of equality ($P_{\textrm{dist, ceiling}}\geq
P_{\textrm{dist, floor}}$), though this is not imposed as a prior. Even so,
this is favoured by the MCMC analysis, which gives a likelihood that
$P_{\textrm{dist, ceiling}}\leq P_{\textrm{dist, floor}}$ of only $3.14\times
10^{-3}$ ($2.95\sigma$) in MOND and $6.43\times 10^{-3}$ ($2.73\sigma$) in
$\Lambda$CDM. Both theories prefer a non-zero false positive rate of
$P_{\textrm{dist, floor}}\approx 0.1$, which is related to the similar
fraction of dwarfs classified as disturbed in the outskirts of the Fornax
Cluster where tides should be unimportant (Fig. 4).

Another possibility is that our results could be affected by some of the
assumptions or choices that we made during the analysis. To check if this is
the case, we repeat the procedures described in Section 5 but change some of
the assumed conditions and/or parameters in the following ways:

  1. 1.

Considering that the FDS dwarfs could have a lower dark matter fraction within
their optical radius: We consider the possibility that the dark matter
fraction of the FDS dwarfs is lower than assumed in our nominal case (this is
motivated in Section 8.1.1). Assuming that $\Lambda$CDM explains the
properties of isolated dwarfs, we use the velocity dispersions of nearby
isolated dwarfs to estimate their typical dark matter fraction, which returns
a somewhat lower value than assumed in our nominal analysis. Substituting this
fit (Equation 65) into our MCMC chain raises $\eta_{\textrm{destr}}$ slightly,
but it is still only $0.33^{+0.04}_{-0.05}$. We then consider a very
conservative scenario in which there is only $10\times$ as much dark matter as
stars within the optical extent of each dwarf, which requires altering
Equation 10 to $M_{\textrm{dwarf, }\Lambda\textrm{CDM}}=11~{}M_{\star}$. For
this very low dark matter fraction, we obtain that
$\eta_{\textrm{destr}}=0.54^{+0.19}_{-0.09}$, which reduces the tension
between observations and $\eta_{\textrm{destr}}=1$ (as expected from N-body
simulations) to $2.29\sigma$ (the triangle plot for this analysis is shown in
Fig. 21). While this is a significant improvement with respect to the
$>5\sigma$ tension in the nominal case, we see that even when considering one
of the most conservative assumptions for the amount of dark matter contained
within the optical radius of a dwarf, $\eta_{\textrm{destr}}\geq 1$ is still
excluded at 97.8% confidence. Moreover, we show in Section 8.1.1 that in a
recent high-resolution cosmological $\Lambda$CDM simulation, the dark matter
fraction within the stellar $r_{h}$ of a dwarf is far higher than this at the
relevant $M_{\star}$, and is actually quite close to our nominal assumption;

  2. 2.

Changing the lower limit to the distribution of dwarf densities in the test
mass simulation: To check if the adopted detection limit to the density of the
Fornax dwarfs significantly affects the results, we repeat the analysis using
a density threshold $\rho_{t}$ that is $5\sigma$ below the mean logarithmic
density. We also consider a density limit of $\rho_{\textrm{mean}}$ (grey line
in Fig. 6). For reference, the nominal $\rho_{t}$ in MOND is $2.88\sigma$
below the mean logarithmic density, while $\rho_{\textrm{mean}}$ is
$1.91\sigma$ below. The corresponding values in $\Lambda$CDM are $3.58\sigma$
and $2.56\sigma$, respectively (see Appendix E). Fig. 22 shows the triangle
plots comparing the results obtained using these two density limits with the
nominal one for $\Lambda$CDM and MOND. From these plots (described further in
Appendix F), we can see that choosing a lower $\rho_{t}$ worsens the tension
for $\Lambda$CDM while maintaining consistency in MOND. Using a higher
$\rho_{t}$ helps to increase the estimated values for $\eta_{\textrm{destr}}$
in $\Lambda$CDM. However, even if we use $\rho_{t}=\rho_{\textrm{mean}}$, the
inferred $\eta_{\textrm{destr}}$ is still significantly below the threshold of
$\approx 1$ required in N-body simulations, while the inference on
$\eta_{\textrm{min, dist}}$ hardly changes. Thus, choosing even higher
$\rho_{t}$ could perhaps help $\Lambda$CDM to reach a reasonable
$\eta_{\textrm{destr}}$. However, taking such high values for $\rho_{t}$ would
be in disagreement with observations as the whole point of $\rho_{t}$ is that
dwarfs are not detectable if they have a lower density, but dwarfs with a
lower density are clearly observed if we adopt such a high $\rho_{t}$;

  3. 3.

Changing the values of the deprojection parameters (see Appendix A): The
deprojection parameters in our nominal analysis were
$\textrm{offset}=0.4^{\circ}$ and $\textrm{nnuc}_{\textrm{floor}}=1.2^{\circ}$
based on fig. 6 of Venhola et al. (2019). We repeat our analysis using
deprojection values at the upper limit of the envelope in this figure:
$\textrm{offset}=0.5^{\circ}$ and
$\textrm{nnuc}_{\textrm{floor}}=1.5^{\circ}$. Fig. 23 shows the triangle plots
comparing the results for these two different deprojections in $\Lambda$CDM
and MOND. From these plots, we can see that these two deprojections give
almost the same results in either theory;

  4. 4.

Changing the ratio between present and pericentre distances (see Appendix B):
A related change we could make is to consider altering the assumed ratio of
0.29 between the average $R$ and the pericentre distance. This is valid for a
thermal eccentricity distribution with $\textrm{Slope}_{P_{e}}=2$, which is
expected theoretically but is the highest possible value (Equation 41). With a
lower $\textrm{Slope}_{P_{e}}$, the ratio would rise as orbits would typically
be more circular, reducing the calculated tidal susceptibility at pericentre.
This would worsen the problem for $\Lambda$CDM; and

  5. 5.

Increasing the resolution: In Section 5.1, we created a grid of $100\times
100$ cells for different values of the orbital eccentricity ($e$) and initial
distance to the cluster centre ($R_{i}$). We increase the resolution to
$200\times 200$ and repeat the analysis to check if this has any effect on the
results. The triangle plots showing the results in $\Lambda$CDM and MOND for
these two resolutions are shown in Fig. 24. From these plots, we can see that
the results are nearly identical for the high- and low-resolution cases.

From these tests, we infer that our results are not significantly affected by
modelling assumptions.

###  8.1 The dark matter content of dwarf galaxies in $\Lambda$CDM

Our conclusion that $\Lambda$CDM is inconsistent with the FDS dwarfs relies
heavily on their low values of $\eta$ in this paradigm, which in turn relies
on the assumption that they should be dominated by dark matter. We therefore
explore whether consistency could be gained by partially relaxing this
assumption in a manner consistent with other constraints.

To try and raise $\eta$ while continuing to use Newtonian gravity, we consider
the possibility that the FDS dwarfs are TDGs. Our results are presented in
Appendix D. We see that this scenario is also not viable because the
elliptical galaxies in the cluster must still contain substantial dark matter
haloes, leading to highly efficient disruption of dwarfs through galaxy-galaxy
harassment.

It thus seems clear that the FDS dwarfs should be primordial. In this case, we
may consider whether the dark matter density in their central regions could be
substantially less than assumed here, raising their tidal susceptibility
within the $\Lambda$CDM framework. The transformation of central cusps in the
dark matter density profile into cores is expected to be rather inefficient
for dwarfs with $M_{\star}\la 10^{7.2}\,M_{\odot}$ (Di Cintio et al., 2014;
Dutton et al., 2016; Tollet et al., 2016). Most FDS dwarfs have a lower
$M_{\star}$ (i.e., they lie below the red line in Fig. 7). This makes it
unlikely that baryonic feedback has substantially reduced the central dark
matter density of most FDS dwarfs, especially at the low mass and low surface
brightness end important to our argument about tidal stability. Adiabatic
contraction could actually raise the central dark matter density (Li et al.,
2022; Forouhar Moreno et al., 2022), as could tidal stripping of the dark
matter halo (Peñarrubia et al., 2008). The colours of the FDS dwarfs also
indicate that star formation stopped early, most likely due to ram pressure
stripping of the gas (Section 3). Thus, it would only be possible for strong
feedback to substantially reduce the baryonic potential depth once. This is
insufficient to substantially affect the central dark matter density even in
the extreme case that the entire gas disc is instantaneously removed (Gnedin &
Zhao, 2002). Multiple bursts of star formation would be required to
substantially affect the dominant dark matter halo (Pontzen & Governato,
2012), but it is very unlikely that this occurred in most FDS dwarfs.
Consequently, they should still have a significant amount of dark matter in
their central regions, as is the case with Galactic satellites whose star
formation ended early (Read et al., 2019). Moreover, the low surface
brightness nature of the FDS dwarfs considered here implies an atypically
large size at fixed $M_{\star}$, causing the baryonic portion of the dwarf to
enclose a larger amount of dark matter than for the more typical Illustris
galaxies considered by Díaz-García et al. (2016).

Another way in which FDS dwarfs could lose dark matter is through interactions
with a massive elliptical galaxy. This scenario has been shown to lead to a
dwarf like DF2 with an unusually low dark matter content (Shin et al., 2020).
However, such examples are rare in cosmological simulations (Haslbauer et al.,
2019a; Moreno et al., 2022). In addition, the possibility that most FDS dwarfs
lack dark matter altogether runs into severe difficulties based on simple
analytic arguments: Newtonian TDGs would be very fragile and easily disrupted
by interactions with massive cluster ellipticals, which must have substantial
dark matter haloes in a $\Lambda$CDM context (Appendix D). MOND seems to offer
the right level of tidal stability: neither too much such that all the dwarfs
are completely shielded from tides and the observed signs of tidal disturbance
remain unexplained, nor too little such that the dwarfs would have been
destroyed by now in the harsh cluster environment studied here. The FDS dwarfs
behave just as they ought to in MOND.

This conclusion is in agreement with the recent work of Keim et al. (2022),
which used the observed tidal disturbance of the dwarf galaxies NGC 1052-DF2
and NGC 1052-DF4 to argue that they must be ‘dark matter free’, since
otherwise their dark matter halo would have shielded them from tides. Phrased
in a less model-dependent way, these observations indicate much weaker self-
gravity than for a typical isolated dwarf, which is a clear prediction of MOND
due to the EFE (Famaey et al., 2018; Kroupa et al., 2018; Haghi et al.,
2019a). In the more isolated galaxy DF44, the self-gravity is stronger despite
a similar baryonic content (van Dokkum et al., 2019), but this too is in line
with MOND expectations (Bílek et al., 2019; Haghi et al., 2019b). Strong
evidence for the EFE has also been reported from the outer rotation curves of
spiral galaxies, which tend to be flat for isolated galaxies but have a
declining trend for galaxies in a more crowded environment (Haghi et al.,
2016; Chae et al., 2020, 2021).

Our results with the FDS are similar to those of Chilingarian et al. (2019)
and Freundlich et al. (2022), who also report signs of tidal disturbance in
some of the dwarf galaxies in the Coma cluster. Another case in point is the
recent study of the dwarf galaxy population in the Hydra I cluster, where the
proximity to the cluster centre seems to be affecting the morphology of the
dwarfs in a manner suggestive of tidal effects (e.g., larger half-mass radii
for dwarfs closer to the cluster centre; La Marca et al., 2022). Closer to
home, the MW satellites also show signs of tidal disturbance like elliptical
isophotes (McGaugh & Wolf, 2010). There is a good correlation between these
features and the value of $\eta$ in MOND, which moreover has a maximum value
very close to 1 (see their fig. 6). However, the maximum $\eta$ in
$\Lambda$CDM is $\la 0.2$, making it difficult to understand the observations
in this framework.

####  8.1.1 Revised dark matter fraction in $\Lambda$CDM dwarfs

Throughout our analysis, we followed the Díaz-García et al. (2016)
prescription that $4\%$ of the total dark matter halo of each dwarf lies
within its optical radius, with the total halo mass $M_{\textrm{halo}}$
following from $M_{\star}$ through the Moster et al. (2010) abundance matching
relation. The factor of $4\%$ was obtained by fitting to the dynamically
inferred dark matter masses $M_{\textrm{DM}}$ within the optical radii of S4G
galaxies, as shown in fig. 6 of Díaz-García et al. (2016). In this figure, we
can see that for low-mass galaxies ($M_{\star}\la 10^{9}M_{\odot}$), the
$M_{\textrm{DM}}/M_{\star}$ vs. $M_{\star}$ relation seems to flatten at
$M_{\textrm{DM}}\approx 10\,M_{\star}$. However, this is unclear because S4G
has very few well-observed galaxies with such a low mass.

We can use other surveys to extend the S4G results to even lower mass by using
measurements of the baryonic properties of dSph galaxies and their line of
sight velocity dispersion $\sigma_{\textrm{los}}$. The Newtonian dynamical
masses of galaxies from the other surveys are found using equation 2 in Wolf
et al. (2010):

$\displaystyle
M_{\textrm{dyn}}\left(<r_{h}\right)~{}=~{}\frac{3r_{h}\langle\sigma^{2}_{\textrm{los}}\rangle}{G}\,,$
(64)

where $M_{\textrm{dyn}}\left(<r_{h}\right)$ is the mass within the baryonic
$r_{h}$. Note that when using this to estimate $M_{\textrm{DM}}/M_{\star}$, we
account for the fact that only half the stellar mass is enclosed within
$r_{h}$.

Figure 16: The relation between stellar mass and Newtonian dynamical mass
(Equation 64). The semi-transparent coloured dots represent galaxies from four
different galaxy surveys as indicated in the legend (Falcón-Barroso et al.,
2011; McConnachie, 2012; Ryś et al., 2014; Toloba et al., 2014). The cyan dots
with error bars represent the logarithmic mean and dispersion of the
total/stellar mass ratio within the stellar $r_{h}$ of the dwarfs in each
$M_{\star}$ bin in the $\Lambda$CDM cosmological simulation Illustris TNG50
(Pillepich et al., 2018, 2019; Nelson et al., 2019a, b). These bins have a
width of 0.25 dex and cover the mass range
$\log_{10}\left(M_{\star}/M_{\odot}\right)=4.5-12$. The solid black line
represents the expected trend in $\Lambda$CDM with the nominal dark matter
fraction from abundance matching (Equation 10). The dashed black line
represents the fit to the dwarfs from the aforementioned galaxy surveys in the
mass range covered by the FDS dwarfs. The horizontal blue line at
$\log_{10}11$ shows our conservative assumption that
$M_{\textrm{DM}}=10\,M_{\star}$. In these three cases, the relations are only
plotted over the $M_{\star}$ range of the FDS dwarfs. Their median mass is
shown with the dashed green vertical line. The solid red vertical line
corresponds to the stellar mass below which core formation is inefficient in
$\Lambda$CDM (see the text).

To check the consistency between the assumed dark matter fraction and
observations of isolated dwarfs, we use Fig. 16 to plot
$M_{\textrm{dyn}}/M_{\star}$ of the galaxies in four different galaxy surveys
(semi-transparent coloured dots), assuming the Díaz-García et al. (2016)
result for the dark matter fraction as used in our nominal analysis (black
line; Equation 10), and assuming conservatively that
$M_{\textrm{DM}}=10\,M_{\star}$ (blue line). We can see that it is rather
unlikely that the FDS dwarfs generally have much less dark matter in their
baryonic region than we assumed, since the linear regression to the survey
data over the $M_{\star}$ range of the FDS dwarfs (dashed black line) is quite
close to our nominal dark matter fraction. We can also use the Illustris TNG50
cosmological simulation (Pillepich et al., 2018, 2019; Nelson et al., 2019a,
b) to check the dark matter fraction that we expect dwarfs to have in the
$\Lambda$CDM paradigm. We do this in Fig. 16, where we show the mean and
standard deviation of $M_{\textrm{DM}}/M_{\star}+1$ within the stellar $r_{h}$
in $M_{\star}$ bins of width 0.25 dex (cyan dots with error bars). The trend
followed by these simulated dwarfs is even steeper than that given by the
observed dwarfs, though both give a similar dark matter fraction at the low-
mass end crucial to our analysis (the median $M_{\star}$ of the FDS dwarfs is
shown by the vertical dashed green line at
$\log_{10}\left(M_{\star}/M_{\odot}\right)=6.96$). This further supports our
nominal choice for the dark matter fraction of FDS dwarfs. One reason for
their high expected dark matter fraction is that the vast majority of them
have too little stellar mass for efficient core formation, the threshold for
which is shown by the red vertical line at
$\log_{10}\left(M_{\star}/M_{\odot}\right)=7.2$ for the reasons discussed
above. All these arguments highlight that the $M_{\textrm{DM}}=10\,M_{\star}$
case is clearly very conservative given the steep relation followed by low-
mass galaxies that we expect from abundance matching arguments, Illustris
TNG50 results, and the velocity dispersions of nearby dwarfs.

To assess the sensitivity of our analysis in Section 5 to the assumed dark
matter fraction, we repeat it with the dark matter fraction given by the
linear fit (equations 18 and 19 of Banik & Zhao, 2018b) to the observed
isolated dwarfs in Fig. 16:

$\displaystyle\log_{10}\left(\frac{M_{\textrm{DM}}}{M_{\star}}+1\right)=-0.396+4.089~{}\log_{10}\left(\frac{M_{\star}}{M_{\odot}}\right)\,,$
(65)

where $M_{\textrm{DM}}/M_{\star}$ is the ratio of dark matter to stars within
the stellar $r_{h}$. The typical dwarf densities in this case are about 0.5
dex lower than with the nominal dark matter fraction. As a result, the
logarithmic mean is lower than in the nominal case by a similar amount: It is
now $\log_{10}\,\rho\left(M_{\odot}/\textrm{pc}^{3}\right)=-1.41$. In this
case, the density threshold $\rho_{t}=5.85\times
10^{-4}~{}M_{\odot}/\textrm{pc}^{3}$ is $2.44\sigma$ below the mean. To keep
our statistical analysis comparable to our nominal one, we use the same 6 bins
in $\eta_{\textrm{obs}}$ as before. In this way, we obtain that Equation 65
gives a slightly higher $\eta_{\textrm{destr}}=0.33^{+0.04}_{-0.05}$. The
maximum value achieved by the MCMC chain is only 0.59, which implies that the
$\Lambda$CDM model is still in $>5\sigma$ tension with the expected value of
1.

For completeness, we repeat our analysis with the very conservative assumption
that $M_{\textrm{DM}}=10\,M_{\star}$. In this case, the distribution of dwarf
densities is similar to that in MOND (Fig. 6) but scaled up $11\times$. Thus,
the logarithmic dispersion remains $\sigma=0.57$ dex and the density threshold
$\rho_{t}=4.66\times 10^{-4}~{}M_{\odot}/\textrm{pc}^{3}$ is still
$2.88\sigma$ below the mean $\log_{10}\,\rho$, which is now $-1.69$ in these
units. As expected, the $\rho_{t}$ value is $11\times$ higher than in the MOND
model $-$ and thus much less than in our nominal $\Lambda$CDM analysis. We
found that in this reduced density case,
$\eta_{\textrm{destr}}=0.54^{+0.19}_{-0.09}$ and the probability that
$\eta_{\textrm{destr}}\geq 1$ is $2.23\times 10^{-2}$ ($2.29\sigma$).

Appendix F shows the complete triangle plot with the distributions of the
model parameters and parameter pairs for the nominal $\Lambda$CDM analysis and
the two revised cases described above. There is little impact to the
inferences on parameters other than $\eta_{\textrm{destr}}$,
$\eta_{\textrm{min, dist}}$, and $\textrm{Slope}_{P_{e}}$.

Therefore, it is clear that assuming a lower dark matter fraction for the
$\Lambda$CDM dwarfs helps to alleviate the tension between observations and
N-body simulations only if this fraction is reduced significantly. However,
having a dark matter fraction of $M_{\textrm{DM}}/M_{\star}=10$ within the
optical radius is a very conservative assumption at odds with many other lines
of evidence, including cosmological simulations. Even with this assumption,
$\eta_{\textrm{destr}}\geq 1$ is still excluded by our MCMC analysis of the
FDS at 97.8% confidence.

##  9 Conclusions

We studied the tidal susceptibility of dwarf galaxies in the Fornax Cluster to
gravitational effects of the cluster environment in both $\Lambda$CDM and
MOND. In both theories, we found cluster tides to be the main effect. Thus,
cluster tides should be able to explain the observed morphological disturbance
of some Fornax dwarfs and the lack of low surface brightness dwarfs towards
the cluster centre (Fig. 3). By constructing a test mass simulation of the
Fornax system and performing a statistical analysis using the MCMC method, we
constrained the tidal susceptibility ($\eta\equiv r_{h}/r_{\textrm{tid}}$)
value at which a Fornax dwarf should get destroyed in order to match the
observations, which we call $\eta_{\textrm{destr}}$. We found that
$\eta_{\textrm{destr}}=0.25_{-0.03}^{+0.07}$ in $\Lambda$CDM and
$1.88_{-0.53}^{+0.85}$ in MOND.

The $\eta_{\textrm{destr}}$ value in $\Lambda$CDM falls significantly below
analytic expectations (Equation 11) and is in $>5\sigma$ tension with N-body
simulation results, which indicate that $\eta_{\textrm{destr}}\approx 1$
(Peñarrubia et al., 2009; van den Bosch et al., 2018). In other words, the
very low $\eta$ values of FDS dwarfs imply that they should be unaffected by
cluster tides, contradicting the observed signs of tidal disturbance. We also
found that the other major environmental influence of interactions with
individual massive galaxies in the cluster should not be a significant process
in $\Lambda$CDM (see also section 7.3.3 of Venhola et al., 2019). We discarded
the possibility that the above-mentioned discrepancy is due to the minimum
allowed density of the simulated sample of dwarfs being too low, the
deprojection parameters being different from our nominal ones, the resolution
of the test mass simulation not being high enough to get reliable results, and
the dwarfs having less dark matter than we assumed (Section 8). In particular,
the velocity dispersions of nearby isolated dwarfs suggest a slightly lower
dark matter fraction (dashed line in Fig. 16). Using this only slightly raises
$\eta_{\textrm{destr}}$ to $0.33^{+0.04}_{-0.05}$. Even if we conservatively
assume that the FDS dwarfs have only $10\times$ as much dark matter as stars
within their optical radius, we still get a $2.29\sigma$ tension with
expectations (Equation 11). Therefore, our results reliably show that the
$\Lambda$CDM paradigm is in serious tension with observations of perturbed
dwarf galaxies in the Fornax Cluster (observations which are strongly
suggestive of tidal effects, see also section 7.4 of Venhola et al., 2022).

An alternative model that assumes different properties for the dark matter
particles could perhaps reconcile the basics of the $\Lambda$CDM cosmology
with the observed morphological disturbances of some Fornax dwarfs. One of the
most popular alternatives is the ‘superfluid dark matter’ model (Berezhiani &
Khoury, 2015; Hossenfelder & Mistele, 2020). Like most hybrid models, it
attempts to reconcile the successes of MOND on galaxy scales with the
advantages of dark matter on larger scales, especially with regards to the CMB
anisotropies and galaxy cluster dynamics. However, this model also presents
its own problems, including orbital decay of stars in the Galactic disc from
Cherenkov radiation (Mistele, 2021) and that the LG satellite planes extend
beyond the estimated superfluid core radii of the MW and M31, making it
difficult to explain the high observed internal velocity dispersions of the
satellites in these planes (see section 5.6 of Roshan et al., 2021a). There
are also difficulties explaining the observed regularities in rotation curves
consistently with gravitational lensing results in a theory where baryons feel
extra non-gravitational forces that do not affect photons (Mistele, McGaugh &
Hossenfelder, 2022). Another possibility is that the dark matter particles are
fuzzy with a low mass and thus a long de Broglie wavelength, reducing their
density in the central region of a dwarf galaxy. However, ultralight bosons
(Hu et al., 2000; Hui et al., 2017) are in significant tension with
observations of the Lyman-$\alpha$ forest (Rogers & Peiris, 2021). More
generally, reducing the ability of dark matter to cluster on small scales
would make it difficult to form dwarf galaxies at high redshift and to explain
their high Newtonian dynamical $M/L$ ratios.

This brings us to the MOND case, in which the inferred $\eta_{\textrm{destr}}$
is much more consistent with analytic expectations (Equation 20). In order to
compare $\eta_{\textrm{destr}}$ with the results of N-body simulations as we
did for $\Lambda$CDM, we had to perform numerical MOND simulations ourselves
(though one pioneering study exists, see Brada & Milgrom, 2000). From our
simulations tailored to the properties of a typical dwarf galaxy in the Fornax
Cluster, we obtained that $\eta_{\textrm{destr}}=1.70\pm 0.30$, in excellent
agreement with the value required to fit the observational data according to
the MCMC method. We therefore conclude that MOND performs significantly better
than $\Lambda$CDM and is clearly the preferred model in all the tests that we
conducted throughout this work, even though it was not designed with the FDS
in mind. Nevertheless, MOND still needs an additional ingredient to explain
some of the observations on larger scales, especially the temperature and
pressure profiles in galaxy clusters and the CMB power spectrum (Famaey &
McGaugh, 2012). For this, several models have been proposed that complement
MOND. Some of the most promising ones are the relativistic MOND theory which
can fit the speed of gravitational waves and the CMB anisotropies but likely
cannot explain the dynamics of virialized galaxy clusters (Skordis & Złośnik,
2021); and the $\nu$HDM model that assumes MOND gravity and 11 eV sterile
neutrinos (Angus, 2009). These proposed particles would play the role of a
collisionless component that only aggregates at the scale of galaxy clusters,
helping to explain the Bullet Cluster (Angus et al., 2007) and other
virialized galaxy clusters (Angus et al., 2010), where the MOND corrections to
Newtonian gravity are generally small. MOND has also proved capable of
explaining several physical phenomena that $\Lambda$CDM has been failing to
describe, including the planes of satellite galaxies in the LG and beyond
(Pawlowski, 2021a, b), the weakly barred morphology of M33 (Sellwood et al.,
2019; Banik et al., 2020), and the pattern speeds of galaxy bars (Roshan et
al., 2021a, b). Using the $\nu$HDM extension, MOND can also explain the CMB
(Angus & Diaferio, 2011), the KBC void and Hubble tension (Haslbauer et al.,
2020), and the early formation of the interacting galaxy cluster El Gordo
(Katz et al., 2013; Asencio et al., 2021). Therefore, this later model is
capable of explaining both the CMB and the dynamics of galaxy clusters while
preserving the successes of MOND at galaxy scales (Banik & Zhao, 2022, and
references therein). In this study, we have shown that it should also be
capable of resolving the problem faced by $\Lambda$CDM with regards to the
observed signs of tidal disturbance in Fornax Cluster dwarf galaxies.

## Acknowledgements

EA is supported by a stipend from the Stellar Populations and Dynamics
Research Group at the University of Bonn. IB is supported by Science and
Technology Facilities Council grant ST/V000861/1, which also partially
supports HZ. IB acknowledges support from a “Pathways to Research” fellowship
from the University of Bonn. PK acknowledges support through the Deutscher
Akademischer Austauschdienst-Eastern European Exchange Programme. EA would
like to thank Prof. Xufen Wu for providing the initial conditions templates of
the dwarf galaxy used in the MOND N-body simulations. The authors are grateful
to Sara Eftekhari for providing the table of literature data shown in Fig. 16.
They are also grateful to the referee for comments which substantially
improved this paper.

## Data availability

The results presented can be reproduced by using the data available in the
Vizier catalogue131313https://vizier.u-strasbg.fr/viz-
bin/VizieR-3?-source=J/A%2bA/620/A165/dwarf&-out.max=50&-out.form=HTML%20Table&-out.add=_r&-out.add=_RAJ,_DEJ&-sort=_r&-oc.form=sexa
and following the methods described in this paper. For a user guide describing
how to install por and providing links from which it can be downloaded, we
refer the reader to Nagesh et al. (2021).

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##  Appendix A Deprojecting distances in the sky plane to 3D distances

In order to convert an observed 2D projected distance $R_{\textrm{sky}}$ into
a 3D distance $R$, we use a simplified version of the deprojection method
applied in Venhola et al. (2019). For convenience, we normalize distances to
$d_{\textrm{Fornax}}=20$ Mpc, the distance to the Fornax Cluster (Blakeslee et
al., 2009). Thus, we define

$\displaystyle\theta_{\textrm{2D}}~{}\equiv~{}\frac{R_{\text{sky}}}{d_{\textrm{Fornax}}}\,,\quad\theta_{\textrm{3D}}~{}\equiv~{}\frac{R}{d_{\textrm{Fornax}}}\,.$
(66)

Fig. 6 of Venhola et al. (2019) shows the relation between these quantities
for nucleated and non-nucleated dEs.141414Results are also shown for dwarf
irregulars, but we removed these from our sample. The relation for nucleated
dwarfs is almost parallel to the line of equality, but with an offset of
$\approx 0.4^{\circ}$. Therefore, we deproject a dwarf labelled as ‘nucleated’
using

$\displaystyle\theta_{\textrm{3D}}~{}=~{}\theta_{\textrm{2D}}+\textrm{offset}\,,$
(67)

with offset $=0.4^{\circ}$ in our nominal analysis.

In the case of non-nucleated dwarfs, $\theta_{\textrm{3D}}$ has a constant
floor value of $\approx 1.2^{\circ}$ until it joins the relation between
$\theta_{\textrm{3D}}$ and $\theta_{\textrm{2D}}$ followed by nucleated dwarfs
at $\theta_{\textrm{2D}}>\textrm{nnuc}_{\textrm{floor}}-\textrm{offset}$.
Therefore, for the non-nucleated dwarfs, we apply the following deprojection:

$\displaystyle\theta_{\textrm{3D}}=\begin{cases}\textrm{nnuc}_{\textrm{floor}},&\textrm{if}\
\theta_{\textrm{2D}}\leq\textrm{nnuc}_{\textrm{floor}}-\textrm{offset}\,,\\\
\theta_{\textrm{2D}}+\textrm{offset},&\textrm{if}\
\theta_{\textrm{2D}}\geq\textrm{nnuc}_{\textrm{floor}}-\textrm{offset}\,,\end{cases}$
(68)

where $\textrm{nnuc}_{\textrm{floor}}=1.2^{\circ}$ in our nominal analysis. As
with the nucleated dwarfs, we use offset $=0.4^{\circ}$.

##  Appendix B Obtaining $R_{\textrm{per}}$ from a 3D distance

Assuming a thermal eccentricity distribution (Jeans, 1919; Ambartsumian, 1937;
Kroupa, 2008), we have that the probability distribution of eccentricities is
$P_{e}=2e$. If the orbits are approximately Keplerian, the pericentre distance
$R_{\textrm{per}}=a\left(1-e\right)$, where $a$ is the semi-major axis and $e$
is the eccentricity. The time-average distance can be calculated as $\langle
R\rangle=a\left(1+e^{2}/2\right)$ (section 3 of Méndez & Rivera-Valentín,
2017). To obtain the relation between $\langle R\rangle$ and
$R_{\textrm{per}}$, we integrate over the whole eccentricity distribution:

$\displaystyle\frac{R_{\textrm{per}}}{\langle
R\rangle}=\int\left.\frac{R_{\textrm{per}}}{\langle
R\rangle}\right|_{e}P_{e}\;de=\int_{0}^{1}\left(\frac{1-e}{1+\frac{e^{2}}{2}}\right)2e\;de=0.29.$
(69)

We assume that the 3D distance of a dwarf inferred from its observed projected
distance (Appendix A) is about the same as its time-average distance. We
therefore obtain that for the FDS dwarfs, $R_{\textrm{per}}=0.29\,R$.

##  Appendix C Do two experiments have the same proportion of successes?

In Section 4.3, we encountered the problem that one experiment gives
$S_{\textrm{obs, 1}}$ ‘successes’ out of $T_{1}$ trials while another
experiment gives $S_{\textrm{obs, 2}}$ successes out of $T_{2}$ trials, with a
success defined as a dwarf galaxy that appears disturbed. The problem is to
test the null hypothesis that the proportion of successes ($x$) is the same in
both experiments assuming that $T_{1}$ and $T_{2}$ are set in advance
independently of the actual number of successes. We consider this problem in
two stages as follows:

  1. 1.

Keeping $x$ fixed, we evaluate the likelihood $P_{x}$ of obtaining data as bad
as or worse than the observed combination ($S_{\textrm{obs, 1}}$,
$S_{\textrm{obs, 2}}$) for the null hypothesis; and

  2. 2.

We then obtain a weighted mean value for $P_{x}$ by considering all plausible
$x$, each time weighting by the likelihood that the observed ($S_{\textrm{obs,
1}}$, $S_{\textrm{obs, 2}}$) arises with that $x$.

If we know $x$, we can use binomial statistics (Equation 48) to find the
likelihood of obtaining any combination ($S_{1}$, $S_{2}$). We obtain $P_{x}$
by adding the probabilities of all ($S_{1}$, $S_{2}$) combinations which are
as likely as or less likely than the observed combination ($S_{\textrm{obs,
1}}$, $S_{\textrm{obs, 2}}$). This follows the usual principle that if the
data seems unlikely given the null hypothesis, we should consider all the ways
in which it could look as bad or even worse.

If the null hypothesis were true, the probability distribution of its
parameter $x$ can be found more accurately by combining the two experiments to
obtain a single experiment with $\left(S_{\textrm{obs, 1}}+S_{\textrm{obs,
2}}\right)$ successes out of $\left(T_{1}+T_{2}\right)$ trials. We use
Equation 32 to calculate the mean $x_{0}$ and uncertainty $\sigma_{x}$ of the
resulting posterior inference on $x$ assuming a uniform prior. We then
consider all values of $x$ within the range $x_{0}\pm 5\sigma_{x}$ provided
this does not go outside the mathematically allowed range ($0-1$). Within the
considered range of $x$, we weight each $P_{x}$ determination by the binomial
likelihood $P_{\textrm{obs}}\left(x\right)$ of obtaining the observed
combination ($S_{\textrm{obs, 1}}$, $S_{\textrm{obs, 2}}$), so
$P_{\textrm{obs}}\left(x\right)$ is a product of the binomial likelihood from
each of the experiments. The idea is that each $P_{x}$ should be weighted by
how plausible the corresponding $x$ is given the data in the context of the
null hypothesis. This leads to our estimated $P$-value:

$\displaystyle P~{}=~{}\frac{\int
P_{x}P_{\textrm{obs}}\left(x\right)\,dx}{\int
P_{\textrm{obs}}\left(x\right)\,dx}\,.$ (70)

Since it is possible that no value of $x$ matches the observations very well
because the null hypothesis is wrong, $P_{\textrm{obs}}\left(x\right)$ might
not integrate to 1.

In the particular case of Section 4.3, calculating the significance $P$ in
this way only tells us how plausible it is that $f_{d}$ is the same in the low
$\eta$ and high $\eta$ subsamples, which is the null hypothesis. Our
alternative hypothesis specifies that $f_{d}$ should be higher in the high
$\eta$ subsample on physical grounds, not merely that $f_{d}$ should have some
sort of correlation with $\eta$. Since the inferred $f_{d}$ indeed rises with
$\eta$, we should bear in mind that the low likelihood of the null hypothesis
is caused by a deviation in just the sense expected on physical grounds under
the alternative hypothesis where tides are relevant. On the other hand, we
tried all possible choices of $\eta_{t}$ to maximize the significance of the
signal, leading to a look-elsewhere effect.

##  Appendix D Tidal susceptibility of Newtonian TDGs

As discussed in Section 8, our results indicate a higher level of tidal
susceptibility than is expected in $\Lambda$CDM. This could be a sign that the
Fornax dwarfs lack dark matter altogether, which is possible in this framework
if the FDS dwarfs are mostly TDGs. These are expected to be rather rare in
$\Lambda$CDM, so the scenario is not very plausible (Haslbauer et al., 2019b).
We nonetheless consider it for completeness.

If the dwarfs are of tidal origin, they would be free of dark matter (Barnes &
Hernquist, 1992; Wetzstein et al., 2007). However, the massive cluster
galaxies would still be surrounded by a dark matter halo. In this scenario,
the mass ratio between the dwarfs and the massive galaxies would be rather
extreme, suggesting a serious problem with the stability of the dwarfs.

To quantify this, we obtain the tidal radius of a dwarf by applying Equation
11 considering only its baryonic mass. Similarly, we can obtain the disruption
time-scale by applying Equation 13 and accounting for the fact that the terms
referring to the dwarf (those labelled with a subindex ‘dwarf’) should be
purely baryonic while the terms referring to the large galaxies (labelled with
a subindex ‘p’) should still account for the dark matter contribution to the
mass and half-mass radius. We can then substitute in these results to obtain
the susceptibility to cluster tides (Equation 29) and galaxy-galaxy harassment
(Equation 31). The results are shown in Fig. 17.

Figure 17: The distribution of tidal susceptibility values of Fornax Cluster
dwarfs in a Newtonian TDG scenario to cluster tides (top) and harassment
(bottom), with a bin width of 0.05 and 0.5, respectively.

As expected, the dwarfs are now much more susceptible to cluster tides (higher
$\eta_{\textrm{rtid}}$ than in Fig. 2). The distribution of
$\eta_{\textrm{rtid}}$ becomes very similar to MOND, suggesting that maybe the
Newtonian TDG scenario is plausible. However, the tidal susceptibility to
harassment ($\eta_{\textrm{har}}$) is very large in this scenario and greatly
exceeds 1 for the vast majority of the dwarfs. The high $\eta_{\textrm{har}}$
values arise because the dwarfs are completely unprotected: They do not have a
boost to their self-gravity either from MOND or from a dark matter halo. Given
their low surface brightness, this leads to very weak self-gravity. However,
in a $\Lambda$CDM universe, the large galaxies must still have dark matter
haloes. As a result, purely baryonic dwarfs governed by Newtonian gravity
should have already been destroyed by encounters with the massive cluster
galaxies. Therefore, we can consider that the TDG scenario in $\Lambda$CDM is
extremely unlikely. Note that in MOND, our analysis is not sensitive to
whether the dwarfs are TDGs or formed primordially $-$ they are purely
baryonic in either case.

##  Appendix E Distribution of dwarf densities in $\Lambda$CDM

Our MCMC analysis relies on an assumed distribution for the dwarf densities,
which are crucial to their tidal stability. We therefore need to repeat the
steps discussed in Section 5.2.3 for the case of $\Lambda$CDM. For this model,
we show the mass-luminosity relation (Fig. 18), the surface density-volume
density relation (Fig. 19), and the histogram of volume densities of the
dwarfs in the FDS catalogue (Fig. 20). The main difference is that the mass of
the dwarfs is higher since it includes the contribution of the dark matter
component within the optical radius (Equation 10). This raises their surface
and volume density. We found that $M/L_{r^{\prime}}=74.92\pm
52.38~{}M_{\odot}/L_{\odot,r^{\prime}}$, indicating a rather high dispersion.
Moreover, we can no longer approximate that the slope of the relation is 1 on
logarithmic axes, indicating non-linearity.

Figure 18: Similar to Fig. 7, but for $\Lambda$CDM instead of MOND and showing
only the linear regression. Figure 19: Similar to Fig. 8, but for $\Lambda$CDM
instead of MOND and showing only the linear regression. Figure 20: The
distribution of each dwarf galaxy’s mean density within its half-mass radius,
accounting for both baryonic and dark matter. The orange vertical line at
$-0.99$ shows the sample mean, while the magenta lines offset by $\pm 0.54$
dex show the standard deviation around it. Other lines have a similar meaning
to Fig. 6 and have been obtained similarly to the MOND case to allow a fair
comparison (see the text).

Due to these difficulties, we found that it would be unsuitable to repeat the
steps described in Section 5.2.3. To enable a fair comparison with MOND, we
nonetheless used as similar a procedure as possible. For this, we fixed the
logarithmic offset between the density of the least dense dwarf in our sample
($\rho_{\textrm{min,FDS}}$) and the adopted density threshold of the survey
($\rho_{t}$). As a result, the minimum observational limit (black line in Fig.
20) is 0.09 dex below $\rho_{\textrm{min,FDS}}$, the mean observational limit
(grey line in this figure) is 0.46 dex above $\rho_{\textrm{min,FDS}}$, and
the maximum observational limit (dashed black line in this figure) is 0.79 dex
above $\rho_{\textrm{min,FDS}}$. As in the MOND case, we choose the minimum
observational limit (black line in Fig. 20) as our nominal density limit for
the distribution since it is the only one of these three choices that implies
$\rho_{t}<\rho_{\textrm{min,FDS}}$, which is required of a realistic detection
threshold. Assuming instead the mean observational limit would make us lose 2
observed dwarf galaxies from the low-density tail of the distribution. Note
also that these dwarfs have a clear tidal morphology because we removed any
dwarfs where this is unclear (Section 2.1).

To summarize, our nominal $\rho_{t}$ in $\Lambda$CDM is $3.58\sigma$ below the
mean logarithmic density, while $\rho_{\textrm{mean}}$ is $2.56\sigma$ below.

##  Appendix F Triangle plots with alternative modelling choices

In this appendix, we rerun our MCMC analysis with different modelling
assumptions and show their impact using triangle plots similar to Fig. 10.
Instead of showing $\Lambda$CDM and MOND results on the same graph as done
there, our approach will be that each graph shows results for different
modelling assumptions but within the context of the same theory. We will use
different panels for the different theories. As before, we show only the
$1\sigma$ contour for each pair of parameters, though the full probability
distribution is shown when considering the posterior on one parameter
marginalized over all others. The results presented here are discussed in more
detail in Section 8.

In Fig. 21, we check how decreasing the dark matter fraction within the
optical radius of the FDS dwarfs affects the results. In particular, we
consider the revised dark matter fraction given in Equation 65 based on the
observed velocity dispersions of nearby dwarfs (Section 8.1.1). As discussed
there, we also consider the very conservative case
$M_{\textrm{DM}}=10\,M_{\star}$. The main impact is on the parameters
$\eta_{\textrm{destr}}$ and $\eta_{\textrm{min, dist}}$. The inference on the
slope of the eccentricity distribution is rather different for the case
$M_{\textrm{DM}}=10\,M_{\star}$, but otherwise the posteriors are not much
different to the nominal $\Lambda$CDM case in both revised analyses shown
here.

Figure 21: Triangle plot showing the inferred parameter values in the
$\Lambda$CDM model constrained by our statistical analysis assuming the
nominal dark matter fraction (Equation 10; blue), our fit to the empirically
determined dark matter fractions of nearby isolated dwarfs (Equation 65;
orange), and a very conservative scenario in which the mass of dark matter
within the optical radius of each dwarf is only $10\times$ that of the baryons
(green).

Figure 22: Triangle plot similar to Fig. 10, but this time showing
$\Lambda$CDM (left) and MOND (right) in separate panels. Each panel shows the
difference in the final results when using three different lower limits to the
density distribution of the simulated Fornax dwarfs. The results for the
$5\sigma$ lower limit are shown in green, those with the nominal density limit
are shown in blue, and results for the mean observational limit
($\rho_{\textrm{mean}}$) are shown in orange.

In Fig. 22, we compare the parameter inferences resulting from the MCMC
analysis assuming three different lower limits ($\rho_{t}$ values) to the
density distribution of the dwarfs:

  1. 1.

The lowest considered $\rho_{t}$ is set at $5\sigma$ below the mean
logarithmic density;

  2. 2.

The second-lowest considered $\rho_{t}$ is the nominal value used in the main
analysis; and

  3. 3.

The highest considered $\rho_{t}$ is the mean observational limit
($\rho_{\textrm{mean}}$), which we obtained in Section 5.2.3 and Appendix E
for MOND and $\Lambda$CDM, respectively.

In Fig. 23, we compare $\Lambda$CDM and MOND while assuming two different
values for the deprojection parameters ‘offset’ and ‘non-nucleated floor’ (see
Appendix A). In addition to the nominal values used in the main analysis, we
also consider a higher set of values corresponding to the highest plausible 3D
distance given the sky-projected distance (see fig. 6 of Venhola et al.,
2019). This entails setting $\textrm{nnuc}_{\textrm{floor}}=1.5^{\circ}$ and
offset $=0.5^{\circ}$ instead of the nominal
$\textrm{nnuc}_{\textrm{floor}}=1.2^{\circ}$ and offset $=0.4^{\circ}$.

Figure 23: Similar to Fig. 22, but showing the difference in the final results
for $\Lambda$CDM (left) and MOND (right) when two different sets of values for
the parameters ‘offset’ and ‘non-nucleated floor’ are used to deproject
distances (Appendix A). Results for the nominal deprojection (offset
$=0.4^{\circ}$, non-nucleated floor $=1.2^{\circ}$) are shown in blue, while
results with the revised deprojection parameters (offset $=0.5^{\circ}$, non-
nucleated floor $=1.5^{\circ}$) are shown in orange.

In Fig. 24, we check if increasing the resolution of the orbital elements in
the test mass simulation affects the results for $\Lambda$CDM and MOND. The
nominal resolution used is a grid of size $100\times 100$ for the eccentricity
$e$ and initial distance from the cluster centre ($R_{i}$), which also
corresponds to the semi-major axis. In the higher resolution case, this is
raised to $200\times 200$.

Figure 24: Similar to Fig. 22, but showing the difference in the final results
for $\Lambda$CDM (left) and MOND (right) with two different resolutions in
orbital elements. The nominal resolution case (blue) uses a grid of $100\times
100$ bins to generate orbits with different eccentricities and initial
positions/semi-major axes. The high-resolution case (orange) uses a grid of
$200\times 200$ bins.