# The effects of self-interacting dark matter
on the stripping of galaxies that fall into clusters

Ellen L. Sirks, Kyle A. Oman, Andrew Robertson, Richard Massey & Carlos Frenk
Institute for Computational Cosmology, Department of Physics, Durham
University, South Road, Durham DH1 3LE, UK E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We use the Cluster-EAGLE (c-eagle) hydrodynamical simulations to investigate
the effects of self-interacting dark matter (SIDM) on galaxies as they fall
into clusters. We find that SIDM galaxies follow similar orbits to their Cold
Dark Matter (CDM) counterparts, but end up with ${\sim}$25 per cent less mass
by the present day. One in three SIDM galaxies are entirely disrupted,
compared to one in five CDM galaxies. However, the excess stripping will be
harder to observe than suggested by previous DM-only simulations because the
most stripped galaxies form cores and also lose stars: the most discriminating
objects become unobservable. The best test will be to measure the stellar-to-
halo mass relation (SHMR) for galaxies with stellar mass
$10^{10-11}\,\mathrm{M}_{\odot}$. This is 8 times higher in a cluster than in
the field for a CDM universe, but 13 times higher for an SIDM universe. Given
intrinsic scatter in the SHMR, these models could be distinguished with noise-
free galaxy-galaxy strong lensing of ${\sim}32$ cluster galaxies.

###### keywords:

cosmology: theory – dark matter – galaxies: clusters: general – galaxies:
haloes

††pubyear: 2020††pagerange: The effects of self-interacting dark matter on the
stripping of galaxies that fall into clusters–References

##  1 Introduction

As galaxies fall into clusters, they are transformed, morphologically and
spectroscopically. Their gas content, hitting the intra-cluster gas, is
shocked. Turbulence causes a sudden, final burst of star formation — before
ram pressure and gravitational tides strip it away, quenching star formation
thereafter (e.g. McCarthy et al., 2008; van den Bosch et al., 2008; Oman et
al., 2021). The galaxies’ dark matter (DM) is also eventually stripped by
tidal gravity and gradually becomes incorporated into the (now slightly
larger) cluster. This is the key mechanism for the growth of structure in the
Universe; yet, the timescale for dark matter stripping and virialisation
remains poorly understood.

In the standard $\Lambda$CDM model of cosmology, dark matter particles
interact with each other only through gravity. The model successfully explains
all observables at large scales, such as the galaxy clustering signal (for a
review see Frenk & White, 2012) and the the cosmic microwave background
anisotropy (e.g. Planck Collaboration et al., 2016). However, there is no a
priori reason why DM particles should not interact with each other (Spergel &
Steinhardt, 2000; Burkert, 2000), and weak self-interactions are a natural
consequence of some particle physics theories for the origin of DM (for a
review see, e.g., Tulin & Yu, 2018). With a mean free path ranging from 1 kpc
to 1 Mpc, DM self-interactions would preserve the large scale success of
$\Lambda$CDM, and could resolve tensions between the results of DM-only
simulations and observations of dwarf and low-mass galaxies (for a review see
Bullock & Boylan-Kolchin, 2017).

Massive galaxy clusters are a promising environment to search for DM-DM
interactions, because the interaction rate would be proportional to the local
DM density and to the local velocity dispersion of DM particles (for a review
see Massey et al., 2010). Observations have placed several limits on the
strength of the SIDM cross-section per unit mass ($\sigma/m$) at the typical
velocities encountered in clusters, including $\sigma/m\lesssim
1\,\mathrm{cm}^{2}\,\mathrm{g}^{-1}$ (Peter et al., 2013, from cluster halo
shapes), $\sigma/m<1\,\mathrm{cm}^{2}\,\mathrm{g}^{-1}$ (Rocha et al., 2013,
from cluster core sizes), and
$\sigma/m<0.47\,\mathrm{cm}^{2}\,\mathrm{g}^{-1}$ (Harvey et al. 2015, from
DM-galaxy offsets in merging clusters, but see also Wittman et al. 2018).
Merging clusters are sufficiently rare that interpretation of them tends to be
limited by uncertainty in their orientation with respect to the line-of-sight
(Clowe et al., 2006; Bradač et al., 2008; Dawson et al., 2012). However, the
promising prospects revealed by Robertson et al. (2017a)’s detailed study of
high-velocity DM collisions motivates a search for more ubiquitous examples of
objects falling into clusters.

Whenever a galaxy falls into an SIDM cluster, interactions between its DM
particles and those in the cluster could scatter DM out of the galaxy. This
‘evaporation’ acts in addition to tidal stripping, and accelerates overall
mass loss. The orbits of infalling galaxies might also be changed. Galaxies
spiral toward the centre of a cluster due to dynamical friction, which has
strength proportional to the galaxy’s mass (Binney & Tremaine, 2008, chapter
8). If galaxies lose additional mass, they might sink less far or more slowly
into the cluster. On the other hand, drag due to the DM self-interactions
(which may be positive or negative Robertson et al., 2017a) could increase the
rate of decay; or inhibit the formation of trailing density wakes in the first
place (Di Cintio et al., 2017).

The aims of this paper are to study the differences in DM mass loss and
orbital dynamics of cluster galaxies, using hydrodynamical simulations with
CDM and SIDM physics — and to investigate whether the differences would be
observable. The only previous study of such effects used DM-only simulations
(Bhattacharyya et al., 2021).

This paper is organised as follows: in Section 2, we present the simulation
suite used in this work; in Section 3 we study the effects of self-
interactions by matching galaxies between our CDM and SIDM simulations; and in
Section 4 we investigate the effects on observables using the population of
galaxies at ${z=0}$. Finally, we discuss our results and present our
conclusions in Section 5.

##  2 Data

###  2.1 The EAGLE and Cluster-EAGLE simulations

We use the 50 Mpc EAGLE cosmological simulation (Schaye et al., 2015) and the
Cluster-EAGLE (c-eagle) zoom cosmological simulations of smaller volumes
centred on $\gtrsim 10^{14}M_{\odot}$ galaxy clusters (Bahé et al., 2017).
Both were run with a modified version of the GADGET-3 code that includes
radiative cooling, star formation, chemical evolution, and stellar and AGN
feedback (with the ‘AGNdT9’ feedback model Schaye et al. 2015; Crain et al.
2015). The DM particle mass is $9.7\times 10^{6}\,\mathrm{M}_{\odot}$, the
initial gas particle mass is $1.8\times 10^{6}\,\mathrm{M}_{\odot}$, and the
gravitational softening length was set to 2.66 comoving kpc before $z=2.8$,
and then kept fixed at 0.7 physical kpc at $z<2.8$. The simulations assume
cosmological parameters from Planck Collaboration et al. (2014).

The eagle volume and two of the c-eagle clusters, CE-05 and CE-12, have been
re-simulated from identical initial conditions in a $\Lambda$SIDM universe
(see Table 1 and Robertson et al. 2018 for more details). These two particular
c-eagle clusters are ‘relaxed’, based on their gas properties at $z=0.1$
(Barnes et al., 2017). Since CE-12 is slightly more massive, and has more
member galaxies, we shall quote the higher signal-to-noise statistics from
that cluster whenever we study the differences between CDM and SIDM at
${z=0}$. However, no data are available for that cluster at higher redshift,
so we shall use CE-05 whenever we trace the evolution of DM through time. Note
that the central galaxy in CE-05 happened to form early, and the central
density cusp has been retained in both CDM and SIDM. The central galaxy of
CE-12 formed later, and SIDM interactions created a $\sim 100$ kpc constant
density core by ${z=0}$.

Our implementation of SIDM assumes an isotropic, velocity-independent
interaction cross section, $\sigma/m=1\,\mathrm{cm}^{2}\,\mathrm{g}^{-1}$.
This is around the upper limit of values compatible with current measurements,
and therefore maximises the observable consequences. During each simulation
timestep, $\Delta t$, DM particles scatter elastically off neighbours within
radius $h_{\mathrm{SI}}=2.66\,\mathrm{kpc}$ (comoving) with probability

$P_{\mathrm{scat}}=\frac{(\sigma/m)~{}m_{\mathrm{DM}}\,v\,\Delta
t}{\frac{4\pi}{3}h_{\mathrm{SI}}^{3}}~{},$ (1)

where $v$ is the particles’ relative velocity and $m_{\mathrm{DM}}$ the DM
particle mass (for more details see Robertson et al., 2017b). We log the time
and particle IDs of all DM scattering events. This enables us to distinguish
between: DM particles that have not scattered; those that have scattered with
other DM particles from their own (sub)halo; and those that have scattered
with DM particles from elsewhere in the cluster.

###  2.2 Finding and tracking individual galaxies

We detect groups of particles in the simulations using a Friends-of-Friends
(FoF, Davis et al., 1985) algorithm with linking length 0.2, and identify
individual subhaloes (in all 30 simulation snapshots from $z=14$ to ${z=0}$)
using the subfind (Springel et al., 2001; Dolag et al., 2009) algorithm. For
subfind to identify a galaxy it must have at least 20 particles. We track
subhaloes between snapshots, and construct their merger trees using the
D-Trees algorithm (Jiang et al., 2014). This identifies each subhalo’s
$N_{\mathrm{link}}$ most bound particles of any species, with
$N_{\mathrm{link}}=\mathrm{min}(100,\mathrm{max}(0.1N_{\mathrm{gal}},10))$,
where $N_{\mathrm{gal}}$ is the total number of particles in the subhalo in
each snapshot. The descendant of a subhalo is the object that contains most of
these $N_{\mathrm{link}}$ particles in the next snapshot. A subhalo can have
multiple progenitors in the previous snapshot, but we define the main
progenitor as that for which the mass summed across all earlier snapshots is
the largest. The main branch of a subhalo is comprised of its main progenitors
and descendants. We use the main branches of subhaloes to trace their
properties through time.

We identify as ‘field galaxies’ all subfind central halos (rank 0 in a given
FoF group) in eagle that contain at least one star particle. We identify as
‘cluster member galaxies’ all subfind subhaloes in c-eagle that contain at
least one star particle and are within radius $2R_{200}$. We define their time
of infall as the first snapshot after they enter that radius for the first
time.

The mass of every galaxy is defined as the total mass, $M_{\mathrm{tot}}$, of
all particles gravitationally bound to it (i.e. the mass $M_{\textsc{sub}}$
assigned to the subhalo by the subfind algorithm). Its stellar mass,
$M_{\star}$, is defined as the total mass of stars within twice its half light
radius. Its location is defined by the location of its constituent particle
with the lowest gravitational potential energy.

Table 1: Properties of the CDM and SIDM versions of the two c-eagle clusters at redshift ${z=0}$. The mass $M_{200}$ is that enclosed within the sphere of physical radius $R_{200}$ whose mean density is 200 times the critical density of the Universe. Cluster member galaxies are the $N_{\mathrm{gal}}$ subhalos in the FoF group of the cluster that are within 2$R_{200}$ of the cluster centre and contain one or more star particles. Simulation | DM Type | $M_{200}/\mathrm{M}_{\odot}$ | $R_{200}/\mathrm{Mpc}$ | $N_{\mathrm{tot}}$
---|---|---|---|---
CE-05 | CDM | $1.38\times 10^{14}$ | 1.09 | 1442
SIDM | $1.36\times 10^{14}$ | 1.09 | 1183
CE-12 | CDM | $3.96\times 10^{14}$ | 1.55 | 3893
SIDM | $3.91\times 10^{14}$ | 1.54 | 2938
Figure 1: Cluster-centric radial velocity as a function of distance from the
cluster centre for the DM in a CDM satellite of CE-05 and its SIDM
counterpart, 2 Gyr (top) and 10.5 Gyr (bottom) after infall. Particles moving
outwards from the centre of potential of the cluster have positive radial
velocity. Plotted here is the DM that was in the satellite at infall. Left
column: the phase space properties of the DM in the CDM galaxy. Middle column:
the properties of the DM in the SIDM galaxy that has not scattered with the
cluster halo DM in the time since infall. Right column: same as middle column,
but for the SIDM that has scattered with the cluster halo since infall. The
location of the galaxy itself is indicated by a green cross on each panel.

###  2.3 The stellar-to-halo mass relation

Below we will compare the stellar-to-halo mass relations (SHMR) of galaxies in
our SIDM and CDM models. We fit the SHMR to a population of simulated galaxies
using the form of the Moster et al. (2013) relation derived from abundance
matching,

$M_{\star}(M_{\mathrm{tot}})=2NM_{\mathrm{tot}}\left[\left(\frac{M_{\mathrm{tot}}}{M_{1}}\right)^{-\beta}+\left(\frac{M_{\mathrm{tot}}}{M_{1}}\right)^{\gamma}\right]^{-1}.$
(2)

By numerically inverting equation (2), we also fit
$M_{\mathrm{tot}}(M_{\star})$, which can be measured observationally.

We use the Markov Chain Monte Carlo (MCMC) sampler emcee (Foreman-Mackey et
al., 2013) to obtain the best-fit values and posterior PDFs of the free
parameters, $M_{1}$, $N$, $\beta$, $\gamma$, as well as the free parameter,
$\sigma_{\mathrm{M}}$, the scatter in stellar mass (or in total mass for the
inverse fit), which we assume to be constant. The latter enters the fit
through the log likelihood function,

$\mathrm{log}\mathcal{L}=-\frac{1}{2}\sum^{N}_{i=1}\left(\frac{\mathrm{log}M_{i}-\mathrm{log}M_{i}^{\mathrm{mod}}}{\sigma_{\mathrm{M}}}\right)^{2}-\frac{N}{2}\mathrm{log}\,\left(2\pi\sigma_{\mathrm{M}}^{2}\right),$
(3)

where the summation is over the total number of galaxies, $N$; $M_{i}$ is the
stellar/total mass of galaxy $i$, and $M_{i}^{\mathrm{mod}}$ is the modelled
stellar/total mass of galaxy $i$, for a given set of parameters. When fitting
the SHMR, we truncate fits at the mass where each galaxy includes at least 10
star particles.

###  2.4 Matching galaxies between simulations

We match galaxies between the CDM and SIDM simulations, so their evolution can
be individually compared. In the snapshot after each galaxy crosses
$2R_{200}$, we identify its counterpart in the other simulation as that which
contains the highest fraction, $f_{\mathrm{match}}$, of shared particle IDs

$f_{\mathrm{match}}=\frac{N_{\mathrm{shared}}^{2}}{N_{\mathrm{CDM,tot}}N_{\mathrm{SIDM,tot}}},$
(4)

where $N_{\rm{shared}}$ is the number of DM particles the CDM galaxy and a
possible matching SIDM galaxy have in common, $N_{\rm{CDM,TOT}}$ the total
number of DM particles in the CDM galaxy, and $N_{\rm{SIDM,TOT}}$ the total
number of DM particles in the SIDM galaxy. To complete an association, we
require a bijective match: i.e. the CDM galaxy points to an SIDM galaxy that
points back to it. The paired CDM and SIDM galaxies inevitably have slightly
different infall masses and infall times. When we bin by these, we use the CDM
values. This is an arbitrary choice, but none of our results change
qualitatively when using either SIDM or common bins (with logarithmic bins of
1 dex in mass, only 10 per cent of galaxies are binned differently).

When analysing matched galaxies, we ignore any cluster galaxies that were
unmatched to cluster galaxies, and any field galaxies that were unmatched to a
central galaxy. In cluster CE-12, 96 out of 889 CDM cluster galaxies were
matched to the central galaxy of the SIDM cluster. For the field galaxies, 383
out of 9126 CDM galaxies were matched to a satellite galaxy in the SIDM
simulation.

Figure 2: The cumulative frequency of galaxies with a fraction of DM lost
smaller than $f$. Plotted in blue are the distributions for the CDM galaxies
of CE-05, and in red their SIDM counterparts. The left panel shows the results
at ${z=1}$, the right panel the results at ${z=0}$. The different line styles
represent different bins of mass at infall, as shown in the legend. A galaxy
that has been completely disrupted, i.e. merged with another galaxy or with
the main cluster halo is assigned $f=1$. The fraction of disrupted galaxies in
each infall mass bin is given by 1 minus the cumulative frequency at $f=1$, as
the cumulative frequency is plotted for fractions smaller than $f$.

##  3 Evolution of DM since infall

In this section we examine the effect of self-interactions on the DM mass of
galaxies after they fall into the clusters, by directly matching galaxies
between the CDM and SIDM simulations.

###  3.1 The behaviour of one example galaxy

To build intuition, we first track the DM halo of one galaxy in detail. We
identify a typical galaxy that fell into the cluster CE-05 at $z=1.99$ with
mass $2.7\times 10^{11}\mathrm{M}_{\odot}$, and track the 6D phase space
coordinates (cluster-centric radius and velocity) of all its DM particles to 2
Gyr ($z=1.15$) and 10.5 Gyr (${z=0}$) after infall. The result is illustrated
in Fig. 1 which shows that self-interactions increase the mass loss of the
SIDM galaxy compared to its CDM counterpart, but the orbit is unaffected.

####  3.1.1 Dark matter loss

The velocity dispersion of DM within the galaxy is reflected in the ‘Fingers
of God’ extending towards positive and negative radial velocities. Tidally
stripped DM extends both forwards and backwards along the galaxy’s orbit: by 2
Gyr, some particles have already passed through pericentre and are now moving
back out. On a phase-space diagram, tidally stripped material moves along the
same path as the galaxy it has been removed from, both in the case of CDM and
SIDM. However, the evaporated material should occupy a region distinct from
the galaxy and tidally stripped material.

We separate the SIDM into particles that have scattered with DM particles in
the cluster, and particles that have not (Fig. 1). Note that some scattering
events result in very low exchange of momentum or merely swap particle
trajectories, so the scattered particles include some that have barely been
perturbed. However, we find many DM particles that do not follow the tidally
stripped material and therefore must be evaporated DM. After 2 Gyr the CDM
galaxy has lost roughly 54 per cent of its DM mass since infall, whereas the
SIDM galaxy has lost approximately 76 per cent of its DM mass. By ${z=0}$,
these fractions have increased to 91 per cent and 99 per cent. Evaporation has
increased the mass loss in the SIDM galaxy with respect to its CDM
counterpart.

We find a much greater SIDM mass loss from galaxies in clusters, than Dooley
et al. (2016, fig. 9) found for dwarf galaxies in the Milky Way (with the same
SIDM cross-section, only a few per cent more than CDM, 10 Gyr after
accretion). This striking difference is probably due to the much greater DM
density and scattering rate in a cluster, but occurs despite the deeper
potential wells.

####  3.1.2 Orbital evolution

After 2 Gyr, the CDM galaxy has moved to a 3D cluster-centric radius of
${\sim}0.4$ physical Mpc (pMpc), with a mean radial velocity centred on about
$-500\,\mathrm{km}\,\mathrm{s}^{-1}$, i.e. the galaxy is moving towards the
centre of potential of the cluster (green cross on the top row of Fig. 1). Its
SIDM counterpart is within ${\sim}0.1$ pMpc and has a similar mean radial
velocity. By ${z=0}$ the CDM galaxy has moved to a radius of ${\sim}0.2$ pMpc,
with a mean velocity of about $+500\,\mathrm{km}\,\mathrm{s}^{-1}$. Its SIDM
counterpart has a mean velocity of about $-100\,\mathrm{km}\,\mathrm{s}^{-1}$,
but is located at about the same radius (green cross on the bottom row of Fig.
1). Indeed, we find that there is virtually no difference between the
evolution of the 3D cluster-centric radius over time of the CDM and SIDM
galaxy (not shown). Self-interactions increase the mass loss of the galaxy,
but do not have a significant effect on its orbit.

###  3.2 The behaviour of a population of galaxies

The galaxy used to produce Fig. 1 is just one example of the many member
galaxies of cluster CE-05. In this section, we investigate the effect of self-
interactions on the evolution of DM particles for a large sample of infalling
galaxies.

####  3.2.1 Dark matter loss

In Fig. 2 we plot the cumulative distribution at redshifts111To be precise,
there is no simulation snapshot at exactly ${z=1}$. The snapshot used here is
actually at ${z=1.02}$. ${z=1}$ and ${z=0}$ of the fraction of DM lost from
all CDM and SIDM galaxies that were within their cluster in one or more of the
30 simulation snapshots of CE-05. We separate the galaxies into logarithmic
bins of 1 dex in infall mass from $10^{9}$ to $10^{12}\mathrm{M}_{\odot}$.
When a galaxy merges with the cluster central galaxy or into the main branch
of some other galaxy, we consider it to have been completely disrupted and we
set the fraction of DM lost to 1.

At both redshifts, we find that for a given fraction of DM lost, $f$, a
greater fraction of the SIDM galaxies have lost a greater portion of their DM
than $f$ compared with the CDM galaxies, reflecting the increased mass loss
due to self-interactions. The biggest difference is between the low infall
mass CDM and SIDM galaxies (dotted lines on Fig. 2). By ${z=0}$, the mass loss
and the difference between the SIDM and CDM galaxies have increased relative
to ${z=1}$. We find that a larger fraction of SIDM than CDM cluster galaxies
have been disrupted across all mass bins and at both redshifts; see Table 2.
This is in line with our expectations, as increased mass loss from self-
interactions should lead to more disrupted cluster galaxies.

Table 2: Fraction of disrupted cluster member galaxies of the CDM and SIDM version of CE-05, at ${z=1}$ and ${z=0}$ and separated into bins of 1 dex in mass at infall. | $z=1$ | $z=0$
---|---|---
Mass range | $N_{\mathrm{disrupted}}/N_{\mathrm{tot}}$ | $N_{\mathrm{disrupted}}/N_{\mathrm{tot}}$
$M_{\odot}$ | CDM | SIDM | CDM | SIDM
$10^{9}-10^{10}$ | 0.15 | 0.24 | 0.23 | 0.34
$10^{10}-10^{11}$ | 0.11 | 0.13 | 0.11 | 0.18
$10^{11}-10^{12}$ | 0.1 | 0.17 | 0.14 | 0.19
$10^{9}-10^{12}$ | 0.15 | 0.22 | 0.2 | 0.31

The high mass galaxies (solid lines) have lost a greater fraction of their DM
than the galaxies in the other infall mass bins (the solid lines have a
different shape than the dotted and dashed lines). This is most likely a
consequence of the high mass galaxies having sunk further into the cluster,
where stripping becomes more efficient. The timescale for dynamical friction
scales with the inverse of the velocity dispersion of the galaxy cubed
(section 8.1.1 in Binney & Tremaine 2008), i.e. the time scale decreases as
the (infall) mass of the cluster galaxy increases.

For subfind to identify a galaxy it needs to have at least 20 particles. As a
consequence a $10^{8}\mathrm{M}_{\odot}$ galaxy can only lose approximately 90
per cent of its mass before it is already considered disrupted, compared to
approximately 99.9 per cent for a $10^{11}\mathrm{M}_{\odot}$ galaxy. As a
result, relatively fewer high mass galaxies disrupt compared to low mass
galaxies, even though the high mass galaxies tend to lose a larger fraction of
their DM overall.

The cluster galaxy used to produce Fig. 1 has an infall mass of $2.7\times
10^{11}\,\mathrm{M}_{\odot}$, placing it in the high mass bin of Fig. 2. By
${z=0}$, the CDM and SIDM version of this galaxy have lost approximately 91
per cent and 99 per cent of their DM mass at infall, corresponding to
cumulative frequencies of approximately 0.7 and 0.8 respectively. While both
have lost more of their DM than most galaxies of their (high) mass, the loss
is not remarkable.

####  3.2.2 Orbital evolution

We found that the CDM galaxy and its SIDM counterpart used to produce Fig. 1
followed nearly the same orbit. To determine whether galaxy orbits in general
are unaffected by self-interactions, we now consider the median evolution
since time of infall for a sample of galaxies orbiting in the cluster CE-05,
in Fig. 3. We use a sample of 396 matched cluster member galaxies (see Section
2.4) from CE-05 that have $M_{\star}\gtrsim 10^{7}\,\mathrm{M}_{\odot}$ at
${z=0}$. Depending on their infall redshift, the galaxies have spent a
different amount of time in the cluster, so a different number of galaxies
contribute to each point of Fig. 3.

SIDM galaxies start losing more mass than their CDM counterparts about 2 Gyr
after infall (bottom left panel of Fig. 3). By 9 Gyr after infall, CDM
galaxies have lost $\sim 75$ per cent of their mass, while SIDM galaxies have
lost $\sim 80$ per cent. However, we find no difference between the typical
orbits of CDM and SIDM galaxies that survive to $z=0$ (top right panel of Fig.
3; we shall later see very slight differences in the distribution of galaxies
that do not survive).

Results are indistinguishable (but noisier) for galaxies with
$M_{\star}\gtrsim 10^{10}\,\mathrm{M}_{\odot}$. Results are also very similar
in CE-12, where CDM galaxies have lost 80 per cent of their mass after 9 Gyr,
and SIDM galaxies have lost 90 per cent.

Figure 3: The median evolution since infall of cluster member galaxies with
$M_{\star}\gtrsim 10^{7}\,\mathrm{M}_{\odot}$ at ${z=0}$, in the CDM (solid
blue) and SIDM (dashed red) versions of cluster CE-05. Top left: cluster-
centric distance in units of $R_{200}$ versus galaxy mass in units of galaxy
mass at infall. The labels indicate the time passed since infall, and the
corresponding points on both tracks are encircled. Top right: cluster centric
distance in units of $R_{200}$ as a function of time since infall. Bottom
right: time since infall as a function of galaxy mass in units of the galaxy
mass at infall. Note that a different number of galaxies contribute to the
median at every point on the plot. Figure 4: Stellar-to-halo mass relation for
galaxy pairs with $>$$10$ star particles ($M_{\star}\gtrsim 5\times
10^{7}\,\mathrm{M}_{\odot}$), matched between CDM and SIDM simulations. Top
panels: number-density contours of the stellar mass versus total mass in
cluster CE-12 (left) and the field (right). Both are smoothed with the same
circular Gaussian kernel of width $\sigma=0.35$ dex: the increased scatter
inside a cluster is real. A version for all (including unmatched) galaxies
looks qualitatively similar. Bottom panels: the difference in total mass
(left) and stellar-to-halo mass (right) between the SIDM and CDM galaxy
populations. Pink points show matched galaxy pairs in cluster CE-12, with the
running median overlaid; grey points show pairs in the field. The effect of
SIDM is greatest for more massive galaxies.

##  4 Observable differences between cluster galaxies in CDM and SIDM

We saw in Section 3 that a galaxy made of SIDM has a higher rate of DM loss
than an identical galaxy made of CDM. However, observations of the real
Universe do not have the luxury of matched comparisons to a control sample or
null test. In this section we investigate whether the increased rate of mass
loss has observable effects on the population of galaxies in a cluster at
${z=0}$.

###  4.1 Stellar-to-Halo Mass Relation

At the mass scale of individual galaxies, the SHMR of field galaxies is
indistinguishable between CDM and SIDM simulations (Figs. 4 and 5). This is
expected because efficient gas cooling and star formation ensure that a
baryon-dominated core retains a deep gravitational well (Robertson et al.,
2019). Once a galaxy falls into a cluster, tidal forces preferentially remove
DM, which is more diffuse than stars.

We first investigate the SHMR for matched pairs of galaxies with more than 10
star particles at $z=0$ (Fig. 4). On average, SIDM cluster galaxies ended up
with ${\sim}$0.12 dex ($25$ per cent) lower masses than their CDM
counterparts. This effect increases to ${\sim}$0.2 dex ($35$ per cent) for the
most massive cluster member galaxies. We then fit the Moster et al. (2013)
relation, as described in Section 2.3. We fit all galaxies, not just those
matched between simulations (as would be done with observational data).
Because this adds some almost-stripped galaxies, this raises the normalisation
of the SHMR at low masses by a factor ${\sim}1.5$ for both CDM and SIDM, and
moves the location of the turnover within its (considerable) statistical
uncertainty. The best fits are shown in Fig. 5, and the best fitting
parameters are listed in Table 3.

Figure 5: Fits to the SHMRs using equation 2. Left: the SHMR as a function of
stellar mass. Fits to the cluster galaxies in CE-12 are shown as solid lines,
and to field galaxies fits as dashed lines. Blue and red lines represent the
CDM and SIDM versions of a given simulation respectively. Shaded regions
represent the 68 per cent confidence regions, obtained from the
$16^{\mathrm{th}}$ and $84^{\mathrm{th}}$ percentiles of the MCMC chain.
Right: similar to the left panel, but now for the SHMR as a function of halo
mass. The fits to the galaxies in CE-05 are similar but noisier, because that
cluster has fewer member galaxies.

The SHMRs for CDM and SIDM cluster galaxies are only distinguishable at the
high mass end, when binning by stellar mass. Fortunately, it is possible to
measure this observationally. We find that cluster galaxies within 2$R_{200}$
with stellar mass $10^{10-11}\,\mathrm{M}_{\odot}$ have
$M_{\star}/M_{\rm{tot}}$ 8 times higher than field galaxies in a CDM universe,
but 13 times higher in an SIDM universe. For cluster galaxies within
$R_{200}$, we find that these numbers increase to 10 and 20 (the best fitting
parameters are included in Table 3, but the fits are not shown on Fig.5).
There is considerable scatter in the SMHR at these masses, and it would
require noise-free measurements of ${\sim}32$ cluster galaxies to distinguish
between these values at 3$\sigma$, assuming that the SHMR for field galaxies
is well known. It would be more challenging to measure other quantities like
the slope of the SHMR at low masses, or the position of the turnover, because
these vary by less than five per cent with different DM.

Table 3: The best fit parameters of the SHMR 2 for field galaxies and for
cluster galaxies within $2R_{200}$ and $R_{200}$ of CE-12. The 68 per cent
confidence intervals are the difference between the $16^{\mathrm{th}}$ and
$84^{\mathrm{th}}$ percentiles of the marginalized 1D posteriors. Field
galaxies
---
| Fit to $M_{\mathrm{tot}}$($M_{\star}$) | Fit to $M_{\star}$($M_{\mathrm{tot}}$)
| CDM | SIDM | CDM | SIDM
$\mathrm{log_{10}}M_{1}$ | $12.09^{+0.06}_{-0.05}$ | $12.11^{+0.06}_{-0.05}$ | $12.22^{+0.05}_{-0.05}$ | $12.2^{+0.05}_{-0.05}$
$N$ | $0.022^{+0.001}_{-0.001}$ | $0.024^{+0.001}_{-0.001}$ | $0.021^{+0.002}_{-0.001}$ | $0.023^{+0.002}_{-0.001}$
$\beta$ | $0.81^{+0.02}_{-0.02}$ | $0.81^{+0.02}_{-0.02}$ | $0.84^{+0.02}_{-0.02}$ | $0.86^{+0.02}_{-0.02}$
$\gamma$ | $0.46^{+0.04}_{-0.04}$ | $0.48^{+0.04}_{-0.04}$ | $0.6^{+0.07}_{-0.07}$ | $0.57^{+0.07}_{-0.07}$
$\sigma_{\mathrm{M}}$ | $0.215^{+0.002}_{-0.002}$ | $0.214^{+0.002}_{-0.002}$ |  $0.278^{+0.003}_{-0.003}$ | $0.281^{+0.003}_{-0.003}$
Cluster galaxies ($R<2R_{200}$)
| Fit to $M_{\mathrm{tot}}$($M_{\star}$) | Fit to $M_{\star}$($M_{\mathrm{tot}}$)
| CDM | SIDM | CDM | SIDM
$\mathrm{log_{10}}M_{1}$ | $10.55^{+0.29}_{-0.24}$ | $10.84^{+0.47}_{-0.35}$ | $11.23^{+0.2}_{-0.24}$ | $11.02^{+0.25}_{-0.23}$
$N$ | $0.1^{+0.02}_{-0.03}$ | $0.26^{+0.37}_{-0.11}$ | $0.11^{+0.02}_{-0.02}$ | $0.26^{+0.29}_{-0.03}$
$\beta$ | $1.27^{+0.13}_{-0.17}$ | $1.27^{+0.13}_{-0.22}$ | $1.23^{+0.15}_{-0.18}$ | $1.27^{+0.15}_{-0.18}$
$\gamma$ | $0.0^{+0.19}_{-0.19}$ | $0.08^{+0.42}_{-0.43}$ | $0.66^{+0.34}_{-0.39}$ | $0.08^{+0.37}_{-0.4}$
$\sigma_{\mathrm{M}}$ | $0.36^{+0.01}_{-0.01}$ | $0.41^{+0.01}_{-0.01}$ | $0.62^{+0.02}_{-0.02}$ | $0.41^{+0.03}_{-0.03}$
Cluster galaxies ($R<R_{200}$)
| Fit to $M_{\mathrm{tot}}$($M_{\star}$) | Fit to $M_{\star}$($M_{\mathrm{tot}}$)
| CDM | SIDM | CDM | SIDM
$\mathrm{log_{10}}M_{1}$ | $10.53^{+0.29}_{-0.25}$ | $10.85^{+0.3}_{-0.2}$ | $11.25^{+0.29}_{-0.25}$ | $11.01^{+0.43}_{-0.43}$
$N$ | $0.14^{+0.04}_{-0.04}$ | $0.5^{+0.34}_{-0.14}$ | $0.16^{+0.04}_{-0.05}$ | $0.5^{+0.92}_{-0.09}$
$\beta$ | $1.28^{+0.14}_{-0.18}$ | $1.37^{+0.12}_{-0.17}$ | $1.19^{+0.2}_{-0.24}$ | $1.37^{+0.35}_{-0.55}$
$\gamma$ | $0.06^{+0.21}_{-0.2}$ | $0.41^{+0.38}_{-0.26}$ | $0.6^{+0.44}_{-0.51}$ | $0.41^{+0.53}_{-0.63}$
$\sigma_{\mathrm{M}}$ | $0.34^{+0.01}_{-0.01}$ | $0.39^{+0.01}_{-0.01}$ | $0.62^{+0.03}_{-0.03}$ | $0.39^{+0.04}_{-0.04}$

###  4.2 The stripping factor

Another measure used to express the mass lost from cluster galaxies is the
‘stripping factor’ (Niemiec et al., 2019)

$\tau_{\mathrm{strip}}(M_{\mathrm{\star}})\equiv
1-\frac{\widetilde{M}_{\mathrm{tot,cluster}}(M_{\mathrm{\star}})}{\widetilde{M}_{\mathrm{tot,field}}(M_{\mathrm{\star}})},$
(5)

where $\widetilde{M}_{\mathrm{tot,cluster}}(M_{\star})$ and
$\widetilde{M}_{\mathrm{tot,field}}(M_{\star})$ are the median total mass of
cluster and field galaxies in a bin of stellar mass $M_{\star}$. This
definition is motivated by a model in which a galaxy’s star formation is
quenched as it enters a cluster. Since no new stars are formed, field galaxies
of a given stellar mass act as the progenitors of cluster galaxies with the
same stellar mass.

We split our sample of cluster (CE-12) and field galaxies into logarithmic
bins of 1 dex in stellar mass ranging from $10^{6}$ to
$10^{11}\,\mathrm{M}_{\odot}$, and calculate the stripping factor in each bin;
the result is shown in Fig. 6. The errors on the stripping factors are
calculated using bootstrapping.

Figure 6: Top: $\tau_{\mathrm{strip}}$ (equation 5) as a function of mean
stellar mass in five stellar bins. The results for the galaxies in the CDM and
SIDM versions of CE-12 are plotted in blue and red respectively. The
horizontal solid blue and dashed red line are the mean $\tau_{\mathrm{strip}}$
of the CDM and SIDM galaxies respectively. The mean stripping factor has a
value of $0.86\pm 0.03$ and $0.87\pm 0.04$ for the CDM and SIDM satellites
respectively. The results for cluster CE-05 are $0.83\pm 0.04$ and $0.85\pm
0.04$. Bottom: histogram of number of galaxies in the same five stellar bins
as plotted in the top panel. Again blue represents CDM and red SIDM.

The difference between CDM and SIDM is not significant in this measure,
although the largest hint of a difference again appears to be in galaxies with
high stellar mass. The mean stripping factor of galaxies inside $2R_{200}$ at
$z=0$ is $0.86\pm 0.03$ and $0.87\pm 0.04$ for the CDM and SIDM version of
cluster CE-12 respectively (blue solid and red dashed horizontal lines in Fig.
6), and there is little scatter about this value in the different stellar mass
bins. For massive galaxies with $10^{10}M_{\sun}<M_{\star}<10^{11}M_{\sun}$,
the mean stripping factor for SIDM is $\mathcal{O}(10^{-2}$) higher than for
CDM, but this is much smaller than statistical uncertainty. More stripping
occurs in the inner parts of the cluster, and the stripping factors rises to
$0.88\pm 0.03$ and $0.90\pm 0.05$ for galaxies inside $R_{200}$. Again there
is little hope for observational discrimination.

Stripping factors are reduced in the lower mass cluster CE-05, to $0.83\pm
0.04$ and $0.85\pm 0.04$ for the CDM and SIDM versions of galaxies within
2$R_{200}$ with again little scatter about these values. A more massive
cluster seems to increase slightly both the stripping of mass and the effect
of self-interactions.

###  4.3 The number and radial distribution of cluster galaxies

There are ${\sim}20$ per cent fewer member galaxies in the SIDM version of a
given cluster at ${z=0}$ (Table 1). Most of the discrepancy is in the central
$\sim 100$ kpc, which is also where the most disruption takes place of SIDM
galaxies whose CDM counterparts survive (Figure 7). This is consistent with
our earlier findings that SIDM barely changes the orbits of galaxies, but
makes them more susceptible to disruption (Section 3.2). Cluster outskirts
contain similar numbers of galaxies, with the populations continually
replenished by objects infalling from the field.

It would be difficult to distinguish between CDM and SIDM using cluster
richness, given the intrinsic scatter in the mass-richness relation (Simet et
al., 2017; Murata et al., 2019; Hilton et al., 2021). It is probably also
difficult to distinguish between CDM and SIDM using the radial distribution of
cluster galaxies. We find that 33 per cent and 36 per cent of galaxies reside
inside $0.5R_{200}$ in the CDM version of clusters CE-05 and CE-12, compared
to 30 per cent and 26 per cent in the SIDM versions. More simulations are
needed to determine the population mean and intrinsic scatter, but the
difference is likely to be washed out by projection effects (of outlying
members in front of/behind the cluster core, and field galaxies onto cluster
outskirts).

Figure 7: Top: The radial distribution of galaxies that survive until $z=0$,
in CDM and SIDM versions of cluster CE-05. The only useful difference is the
slight reduction of SIDM galaxies inside the cluster core. Bottom: The last
known location of galaxies that did not survive until $z=0$. Cumulative number
of galaxies inside a given radius, in the simulation snapshot immediately
before they were disrupted.

##  5 Discussion & Conclusions

We studied the effects of self-interactions on the mass stripping of galaxies
as they fall into galaxy clusters by comparing cosmological simulations with
and without DM self-interactions. When a galaxy falls into a cluster, DM
interactions accelerate the rate of mass stripping. Over 33 per cent of
galaxies in an SIDM cluster can be entirely disrupted by the present time,
compared to 20 per cent in a CDM cluster. Unfortunately, the disrupted
galaxies (which are the most different between CDM and SIDM) are no longer
observable. The orbits of surviving galaxies are essentially unchanged, and
disrupted galaxies are continually replaced by new ones falling into the
cluster. When comparing matched galaxies between the CDM and SIDM versions of
a given cluster (Section 3), we find significant differences in mass loss.
However, when we only look at the population of galaxies remaining in the
cluster at ${z=0}$ (Section 4), we find considerably smaller differences. SIDM
galaxies are more susceptible to disruption, so there is a large group of
disrupted SIDM galaxies which does not contribute to the signal at ${z=0}$.

Potentially observable ways to discriminate between CDM and SIDM include the
(high mass normalisation of the) stellar-to-halo mass relation of galaxies in
clusters, compared to galaxies in the field, or the stripping factor, both of
which describe the mass of the DM in a galaxy of fixed stellar mass. We found
a 25 per cent increase in the ratio of stellar-to-total mass of SIDM galaxies
with stellar mass $M_{\star}>5\times 10^{7}\,\mathrm{M}_{\odot}$. The absolute
normalisation of the relation is likely to be needed to discriminate SIDM from
CDM, but this depends to some extent on the subgrid physics of the
simulations. However, as in the field the relation is nearly indistinguishable
for a CDM and SIDM universe, one could use the difference between the field
and cluster relations at a given stellar mass to try and discriminate between
the two models. From the left panel of Fig. 5, we find that, at approximately
the stellar mass of the Milky Way, $10^{10.5}\,\mathrm{M}_{\odot}$, the ratio
$M_{\star}/M_{\mathrm{tot}}$ is 8 and 13 times higher in the cluster compared
to the field for the CDM and SIDM versions of CE-12 respectively.

Previous, DM-only simulations (Bhattacharyya et al., 2021) predicted larger
differences between SIDM and CDM, probably because of the way stars were
assigned to galaxies after the simulation using a semi-analytic model. In DM-
only SIDM simulations subhaloes form cores more easily than when baryons are
included, making them more easily disrupted. In contrast, our simulations co-
evolved a population of baryons and SIDM. In the full hydrodynamical
simulation a large number of cluster galaxies fail to form cores or have their
cores re-contracted by baryons, and so they are more durable.

In the future, it would be informative to simulate more SIDM clusters. While
the c-eagle suite comprises 30 simulated CDM clusters, only two have been re-
run with SIDM. It is also important to note that a cross-section of
$\sigma/m=1$ $\mathrm{cm}^{2}\mathrm{g}^{-1}$ has arguably already been ruled
out at the $\mathcal{O}(1000\,\mathrm{km}\,\mathrm{s}^{-1}$) collision
velocities between particles typical in clusters. Performing the same tests
with simulations for a lower cross section would presumably produce smaller
differences and would require even higher signal-to-noise observations. Future
surveys, such as the Euclid (Laureijs et al., 2011), Rubin Legacy Survey of
Space and Time (LSST; LSST Science Collaboration et al., 2009), SuperBIT
(Romualdez et al., 2018), and JWST Cosmos-Webb survey (C. Casey & J.
Kartaltepe, pers. comm. 2021) will provide data with higher signal-to-noise
than ever before, potentially making such tests possible.

## Acknowledgements

We thank Anna Niemiec for discussing the physics of infall, and Yannick Bahé
and David Barnes for making the c-eagle data available.

ES and RM are supported by the Royal Society. AR is supported by the European
Research Council’s Horizon2020 project ‘EWC’ (award AMD-776247-6). KAO and CSF
acknowledge support by the European Research Council (ERC) through Advanced
Investigator grant to CSF, DMIDAS (GA 786910). This work used the DiRAC@Durham
facility managed by the Institute for Computational Cosmology on behalf of the
STFC DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded by BEIS
capital funding via STFC capital grants ST/K00042X/1, ST/P002293/1,
ST/R002371/1 and ST/S002502/1, Durham University and STFC operations grant
ST/R000832/1. DiRAC is part of the National e-Infrastructure.

## Data Availability

The CDM simulations were introduced by Bahé et al. (2017); the SIDM
simulations by Robertson et al. (2018). Data from those works are available
from the original papers.

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