# Coherent Self-Interactions of Dark Matter in the Bullet Cluster

Zachary Bogorad  0000-0001-9913-6474<EMAIL_ADDRESS>Stanford Institute
for Theoretical Physics, Department of Physics, Stanford University, Stanford,
CA 94305, USA  Kavli Institute for Particle Astrophysics & Cosmology,
Department of Physics, Stanford University, Stanford, CA 94305, USA  Peter W.
Graham  0000-0002-1600-1601<EMAIL_ADDRESS>Stanford Institute for
Theoretical Physics, Department of Physics, Stanford University, Stanford, CA
94305, USA  Kavli Institute for Particle Astrophysics & Cosmology, Department
of Physics, Stanford University, Stanford, CA 94305, USA  Harikrishnan Ramani
0000-0003-3804-8598<EMAIL_ADDRESS>Stanford Institute for Theoretical
Physics, Department of Physics, Stanford University, Stanford, CA 94305, USA

###### Abstract

Many models of dark matter include self-interactions beyond gravity. A variety
of astrophysical observations have previously been used to place limits on the
strength of such self-interactions. However, previous works have generally
focused either on short-range interactions resulting in individual dark matter
particles scattering from one another, or on effectively infinite-range
interactions which sum over entire dark matter halos. In this work, we focus
on the intermediate regime: forces with range much larger than dark matter
particles’ inter-particle spacing, but still shorter than the length scales of
known halos. We show that gradients in the dark matter density of such halos
would still lead to observable effects. We focus primarily on effects in the
Bullet Cluster, where finite-range forces would lead either to a modification
of the collision velocity of the cluster or to a separation of the dark matter
and the galaxies of each cluster after the collision. We also consider
constraints from the binding of ultrafaint dwarf galaxy halos, and from
gravitational lensing of the Abell 370 cluster. Taken together, these
observations allow us to set the strongest constraints on dark matter self-
interactions over at least five orders of magnitude in range, surpassing
existing limits by many orders of magnitude throughout.

###### Contents

  1. I Introduction
  2. II Self-Interaction Constraints from the Bullet Cluster
    1. II.1 Separation of the Dark Matter and the Standard Model
    2. II.2 Modification of the Gas Collision Velocity
    3. II.3 Halo Density Profiles
    4. II.4 Substructure
    5. II.5 The Collision Location
    6. II.6 Dynamical Friction
    7. II.7 Tidal Distortion
  3. III Other Constraints
    1. III.1 Halo Binding
    2. III.2 Gravitational Lensing
  4. IV Conclusion

##  I Introduction

Although the presence of dark matter in the universe has been firmly
established by a variety of experiments, its nature and properties remain
essentially unknown. In particular, while dark matter’s presence is inferred
from its gravitational effects, it remains undetermined whether dark matter
particles have other interactions with the Standard Model or with each other.

The presence of possible self-interactions of dark matter is interesting for a
variety of reasons. First, in the absence of large interactions between a dark
sector and the Standard Model, self-interactions might become the first
discernible feature of a dark sector. Second, certain short-scale properties
of galaxies appear inconsistent with the results of simulations in the absence
of dark matter self-interactions, including the core-cusp problem (see, for
example, Refs. [1, 2, 3]) and the missing satellites problem (Refs. [4, 5,
6]); see also Ref. [7] for a review of various related discrepancies. While
some recent work has suggested that these individual discrepancies can be
explained without the need for dark matter self-interactions, it remains
unclear whether existing astrophysical observations can be fully explained by
dark matter that interacts only through gravity; see, for example, Refs. [8,
9, 10, 11].

Self-interactions are also motivated by the existence of dark energy. In
particular, models of dark energy based on quintessence can lead to new long-
ranged interactions between dark matter particles [12, 13].

Self-interactions of dark matter have been probed via a variety of
astrophysical observations, leading to a range of constraints on self-
interaction couplings as a function of the force’s range (and, in some cases,
on other properties of the dark matter). However, most such constraints treat
these interactions in one of two limits: a short-range limit in which dark
matter particles interact with one another only individually; and a long-range
limit in which the force is summed over the entirety of the considered dark
matter halo. Intermediate ranges, in which collective interactions are
significant but the force range remains smaller than the halo size, have
rarely been considered. A simplistic interpolation between the short- and
long-range regimes would assume that forces which are exponentially suppressed
beyond their range would lead to exponentially suppressed effects in halos
larger than that range.

They key point of this work is that this is not the case: forces with ranges
that are large compared to the interparticle spacing but smaller than the size
of the containing dark matter halo can still have large collective effects,
due to the presence of density gradients within such halos. This will allow us
to set substantially strengthened constraints on dark matter self-interactions
with ranges below approximately $10~{}{\rm kpc}$, surpassing existing
constraints on self-interactions by many orders of magnitude in strength over
many orders of magnitude in range.

The rest of this work is organized as follows: In Section II, we present a
constraint on long-ranged attractive forces based on observations of the
Bullet Cluster. (Repulsive forces are dealt with in Section. III). As we will
show, the strongest constraint in the attractive, long-ranged regime arises
not from the scattering of dark matter particles between the two clusters, but
rather from a combination of the observed collision velocity of the clusters’
gases and the coincidence of the location of the dark matter and the galaxies
after the collision. We emphasize, however, that the behavior we obtain in
this section for the Bullet Cluster is in fact fairly general, and can be used
to extend a variety of existing constraints on long-ranged dark matter self-
interactions to finite ranges. In Section III we will consider two such
constraints: the self-binding of ultrafaint dwarf galaxy’s halos to constrain
repulsive interactions, and the agreement of mass measurements of the Abell
370 cluster using gravitational lensing and using its galaxies’ velocity
dispersion, applicable to both attractive and repulsive forces. Notably, the
latter will also allow us to illustrate how the existence of substructure
within halos can affect our conclusions, potentially precluding some bounds
from being extended to shorter ranges. Finally, in Section IV, we summarize
our results.

##  II Self-Interaction Constraints from the Bullet Cluster

The Bullet Cluster (1E 0657–56) is a pair of galaxy clusters whose mass and
velocity distributions indicate that they recently collided [14, 15, 16]. In
particular, while the mass distributions and luminous components of both
clusters are now well-separated [17, 18, 19], the intracluster gas components
of both clusters have now merged into a largely continuous distribution
between the clusters’ mass distributions, with prominent shock waves arising
from their supersonic collision [20, 21]. The fact that this did not occur for
the bulk of the clusters’ mass indicates the presence of a large mass of
sufficiently non-collisional matter in both clusters—dark matter—and sets
limits on how much this dark matter can interact with itself in order to pass
through the other cluster largely unaffected. Estimating this effect in the
presence of a hard scattering interaction between dark matter particles leads
to a largely model-agnostic constraint on the momentum transfer cross-section
$\sigma_{T}$ for dark matter particles of mass $m$ to scatter with one another
(see, for example, Refs. [22, 23, 24, 25]):

$\displaystyle\frac{\sigma_{T}}{m}\lesssim 1~{}\frac{{\rm cm}^{2}}{\rm g},$
(1)

up to order-one factors that differ between sources.

The momentum transfer cross-section $\sigma_{T}$ is straightforward to
evaluate for short-range interactions, where only collisions between
individual particles matter. In this work we are interested in the opposite
regime, where the new force’s range is large and collective effects start
becoming important. For concreteness, let us focus on Yukawa forces, such that
the potential $V$ between two particles of masses $m_{1,2}$ separated by a
distance $r$ is given by

$\displaystyle V$ $\displaystyle=-\alpha G\frac{m_{1}m_{2}}{r}e^{-r/\lambda},$
(2)

with $\alpha$ the new force’s strength relative to gravity, $\lambda$ its
range, and $G$ Newton’s constant. For force ranges much smaller than the
inter-particle spacing, the impact parameters that contribute to scattering
are also much smaller than the inter-particle spacing, and hence collective
effects are not important. However, when the force range $\lambda$ exceeds the
inter-particle spacing, each particle scatters coherently off a whole mass
distribution which greatly enhances the interaction.

An extremely conservative bound on these interactions can be still be set by
restricting ourselves to individual scatters with impact parameter much less
than the inter-particle spacing, i.e by setting the maximum impact parameter
$b_{\rm max}$ to obey $b_{\rm max}\ll n^{-1/3}$, where $n$ is the dark matter
number density. If we also require that $b_{\rm max}\ll\lambda$, we can
approximate the Yukawa interaction as a Coulomb interaction. In this limit,
$\sigma_{T}$ can be evaluated to be [26]

$\displaystyle\sigma_{T}$ $\displaystyle=\frac{8\pi\kappa^{2}}{m_{\rm
DM}^{2}v_{\rm rel}^{4}}\log\left(\frac{2\kappa^{2}+b_{\rm max}^{2}m_{\rm
DM}^{2}v_{\rm rel}^{4}}{2\kappa^{2}+b_{\rm min}^{2}m_{\rm DM}^{2}v_{\rm
rel}^{4}}\right)$ (3) $\displaystyle=\frac{16\pi\alpha^{2}G^{2}m_{\rm
DM}^{2}}{v_{\rm rel}^{4}}\ln\Lambda.$

Here $m_{\rm DM}$ is the mass of individual dark matter particles, $v_{\rm
rel}$ is the relative velocity of the collisions, $b_{\rm min}$ is the minimum
impact parameter set by the typical distance of closest approach of a
particle, $\kappa=\alpha Gm_{\rm DM}^{2}$ and $\log\Lambda$ is the Coulomb
logarithm, with

$\displaystyle\Lambda^{2}=\frac{2\kappa^{2}+b_{\rm max}^{2}m_{\rm
DM}^{2}v_{\rm rel}^{4}}{2\kappa^{2}+b_{\rm min}^{2}m_{\rm DM}^{2}v_{\rm
rel}^{4}}.$ (4)

The bound resulting from requiring that this momentum-transfer cross-section
be less than $1~{}{\rm cm}^{2}/{\rm g}$ is plotted for two example dark matter
masses in Figures 1 and 2, using

$\displaystyle\pi b_{\rm min}^{2}(2r_{A})\frac{\rho_{A}}{m_{\rm DM}}=1$
$\displaystyle b_{\rm
max}=\min\left(\lambda,~{}0.1\left(\frac{\rho_{A}}{m_{\rm
DM}}\right)^{-1/3}\right)$ (5)

to estimate the Coulomb logarithm, with $\rho_{A}$ and $r_{A}$ the central
density and the scale radius of the larger cluster in the Bullet Cluster
(discussed in more detail below Eq. (6)). Here, $b_{\rm min}$ is the smallest
distance at which a given dark matter particle expects to encounter one other
particle during its passage, and $b_{\rm max}$ is set by either the force’s
range or the inter-particle spacing (including a conservative safety factor of
$0.1$); whichever is smaller.

However, it is immediately clear that the above constraint was extremely
restrictive and inclusion of coherent effects should lead to much stronger
limits on the coupling $\alpha$.

One such effect is the change in the relative velocity of the clusters at the
time of collision due to extra acceleration felt by dark matter. This can lead
to one of two observable effects, depending on whether the dark matter and the
Standard Model components of each cluster separate or remain bound together
until the collision. Some of these effects have been considered before in the
context of forces with range much larger than a Mpc.

Our focus, however, will be on forces with ranges shorter than the length
scales of the clusters in the Bullet Cluster, such that their behavior differs
significantly from that of infinite-range forces of the same strength. This
regime has not, to our knowledge, been evaluated previously. We will
specifically focus on Yukawa forces (Eq. (2)), although most of our
conclusions can be straightforwardly extended to other potentials. Crucially,
despite the forces we consider falling off exponentially at distances beyond
their range, we show below that the resulting bounds on interaction strength
are suppressed only as the square of the ratio of cluster length scale to
force range. Note that, in this section, we will focus our discussion on
attractive forces: repulsive interactions of dark matter can be constrained
more strongly at all ranges by the straightforward arguments of Section III.1.

Measurements of dark matter halo profiles via gravitational lensing are
difficult, so the precise shapes and other properties of the two clusters
within the Bullet Cluster remain somewhat uncertain (see, for example, Refs.
[22, 27, 28, 29]). In this work, we will assume the same profiles used in Ref.
[28], as our primary purpose is to point out the finite-range behavior of an
effect it discusses, but we note that different profile assumptions may change
our results by order-one factors; see Sec. II.3. In particular, we will assume
that both clusters—the larger (hereafter “A”) cluster, and the smaller
“bullet” (hereafter “B”) cluster—possess an NFW density profile given by

$\displaystyle\rho(r)=\frac{M_{200}}{4\pi(r^{*})^{3}\zeta(c)}\frac{1}{(r/r^{*})(1+r/r^{*})^{2}}$
(6)

where $M_{200}$ is the virial mass of the cluster,
$\zeta(c)=\log(1+c)-c/(1+c)$, $r^{*}=r_{200}/c$, is the scale radius,
$r_{200}$ is the virial radius, and $c$ is the concentration parameter. We use
$M_{200,A}=1.5\times 10^{15}~{}M_{\odot}$, with radius
$r_{200,A}=2136~{}\text{kpc}$ and concentration $c_{200,A}=1.94$, and
$M_{200,B}=1.5\times 10^{14}~{}M_{\odot}$ with $r_{200,B}=995~{}\text{kpc}$
and $c_{200,B}=7.12$.

Having stated these assumptions, we next turn to computing the relative
velocity changes of the Bullet Cluster components, and the observational
consequences of these changes.

###  II.1 Separation of the Dark Matter and the Standard Model

We first consider the possibility of the dark matter separating from the
Standard Model components (both gas and stars) due to the presence of an extra
force felt by the dark matter alone. Note that this is distinct from what is
observed: the gas alone separates and the stars in galaxies and dark matter
remain together through the collision. In particular, the present-day
positions of both the dark matter and the stars of the two clusters are
observable via gravitational lensing and optical measurements. The separation
between the centroids of these two components for the smaller of the two
clusters is observed to be $25\pm 29~{}{\rm kpc}$ [24, 18]—significantly
smaller than the size of either cluster. This immediately precludes situations
in which the dark matter and the Standard Model gas within it become separated
during approach. We convert this into a limit on the dark matter coupling
$\alpha$ next.

Let us begin by considering a test dark matter particle of mass $m$ in cluster
B falling into cluster A. We use the dark matter profile described above for
the A cluster. For an order of magnitude estimate, we can treat the dark
matter-Standard Model separation as occurring around the scale radius of the A
cluster. The acceleration due to gravity alone on the dark matter in the B
cluster and the Standard Model in the B cluster are identical because of the
equivalence principle and hence will not be considered for evaluating the
separation. The net Yukawa potential obtained by integrating over the new
force’s Yukawa potential, Eq. (2), over the A cluster profile gives:

$\displaystyle\frac{V_{\rm new}(\mathbf{r_{A}^{*}})}{m}$
$\displaystyle=-\alpha G\int{\rm
d}^{3}\mathbf{r}\frac{\rho_{A}(\mathbf{r})}{|\mathbf{r}-\mathbf{r_{A}^{*}}|}e^{-|\mathbf{r}-\mathbf{r_{A}^{*}}|/\lambda}$
(7)

where $\mathbf{r_{A}^{*}}$ is any point at the scale radius of the A cluster.
The additional acceleration seen by the dark matter in cluster B while at
radii $r^{\prime}\sim r_{A}^{*}$ is, from Eq. (7), of order

$\displaystyle\begin{split}a_{\rm DM}&=\frac{{\rm d}}{{\rm
d}r^{\prime}}\left(-\alpha G\int{\rm
d}^{3}\mathbf{r}\frac{\rho_{A}(\mathbf{r})}{|\mathbf{r}-\mathbf{r^{\prime}}|}e^{-|\mathbf{r}-\mathbf{r^{\prime}}|/\lambda}\right)\\\
&\sim\frac{{\rm d}}{{\rm d}r^{\prime}}\left(-\alpha
G\frac{\lambda^{3}}{\lambda}\rho_{A}(r^{\prime})\right)\\\ &\sim-\alpha
G\lambda^{2}\frac{\rho_{A}(r_{A}^{*})}{r_{A}^{*}}\end{split}$ (8)

where we have taken advantage of the fact that the density of A does not vary
significantly over the order-$\lambda^{3}$ volume that contributes to the
potential at $\mathbf{r^{\prime}}$, but rather over the length scale
$r_{A}^{*}\gg\lambda$. For larger ranges ($\lambda>r_{A}^{*}$), the new force
behaves simply like gravity: $a_{\rm DM}\sim\alpha
G\rho_{A}(r_{A}^{*})r^{*}_{A}$; see Eq. (10) below.

Note that, as advertised, the acceleration scales as a power law in $\lambda$,
as seen in the last line of Eq. (8), despite the first line containing the
Yukawa exponent. This scaling can also be obtained from the following simple
scaling argument: For any power-law profile with length scale
$r_{A}^{*}\gg\lambda$, there are masses of order
$\rho_{A}(r_{A}^{*})\lambda^{3}$ within a distance of order $\lambda$ both
inward ($M_{\rm in}$) and outward ($M_{\rm out}$) of the chosen point, but the
inward direction is more dense by a fraction of order $\lambda/r_{A}^{*}$,
such that $M_{\rm in}-M_{\rm
out}\sim\rho_{A}(r_{A}^{*})\lambda^{3}(\lambda/r_{A}^{*})$. Thus, roughly, the
two sides provide acceleration

$\displaystyle\begin{split}a_{\rm DM}&\sim\frac{\alpha GM_{\rm
out}}{\lambda^{2}}-\frac{\alpha GM_{\rm in}}{\lambda^{2}}\\\
&\sim-\frac{\alpha
G}{\lambda^{2}}\left(\rho_{A}(r_{A}^{*})\lambda^{3}\right)\frac{\lambda}{r_{A}^{*}}\\\
&\sim-\alpha G\frac{\rho_{A}(r_{A}^{*})\lambda^{2}}{r_{A}^{*}}.\end{split}$
(9)

This reproduces the power-law behavior of Eq. (8).

The next question is whether the stars keep up with the dark matter being
accelerated. This happens purely gravitationally. There is a maximum
acceleration of dark matter that the stars can keep up with. Conservatively,
we can estimate this acceleration by evaluating the maximum acceleration of
stars as a function of separation from the center of the dark matter cluster.
It is easy to check that this is maximal around the scale radius of B, giving

$\displaystyle\begin{split}a_{g}&\sim-G\frac{\rho_{B}(r^{*}_{B})(r^{*}_{B})^{3}}{(r^{*}_{B})^{2}}\sim-G\rho_{A}(r^{*}_{A})r^{*}_{A}\left[\frac{\rho_{B}(r^{*}_{B})r^{*}_{B}}{\rho_{A}(r^{*}_{A})r^{*}_{A}}\right]\\\
&\sim-G\rho_{A}(r^{*}_{A})r^{*}_{A}\end{split}$ (10)

where we have taken advantage of the bracketed ratio being coincidentally
approximately equal to one. If $a_{\rm DM}$ in Eq. (8) exceeds $a_{g}$ in
(10), then the dark matter matter separates. The net separation between stars
and dark matter $\Delta x$, over the entire bullet cluster event duration
$t_{\rm bc}\approx r_{A}^{*}/(v_{A}+v_{B})$, is then given by

$\displaystyle\begin{split}\Delta x&\sim\left(a_{\rm DM}-a_{g}\right)t_{\rm
bc}^{2}\\\ &\sim a_{\rm DM}\left(1-\frac{a_{g}}{a_{\rm
DM}}\right)\left(\frac{r_{A}^{*}}{v_{A}+v_{B}}\right)^{2}\\\ &\sim
r_{A}^{*}\left(1-\frac{a_{g}}{a_{\rm DM}}\right)\end{split}$ (11)

In the last line we have approximated the velocities via $v^{2}/r_{A}^{*}\sim
a_{\rm DM}$. Thus, even if $a_{\rm DM}$ only exceeds $a_{g}$ by
$\mathcal{O}(1)$, the separations are $\mathcal{O}\left(r_{A}^{*}\right)$, in
severe tension with observation. Then, requiring that $a_{\rm DM}$ not exceed
$a_{g}$ in Eqs. (8) and (10), we rule out the couplings

$\displaystyle\alpha\gtrsim 1+\left(\frac{\lambda}{r_{A}^{*}}\right)^{2}$ (12)

where the $1$ in front is required for the correct asymptote at range larger
than the clusters; see the discussion below Eq. (8).

###  II.2 Modification of the Gas Collision Velocity

If the dark matter and gas components of the clusters remain together as the
clusters approach one another, the relative velocity of the dark matter is
(indirectly) observable thanks to the visible shock fronts in the colliding
gas, whose shapes are a function of the Mach number of that collision.
Historically, a disagreement between this observed velocity and the collision
velocity expected from gravitational infall between clusters of the observed
masses has been cited as evidence for long-ranged interactions of dark matter
[27], although improved observations of the clusters combined with detailed
simulations of the collision dynamics have brought these values into
reasonable agreement assuming gravitational interactions alone [28]. As Ref.
[28] notes, this agreement can thus be reinterpreted as a bound on new long-
ranged interactions, which could modify the collision velocity. This is
precisely the bound that we will be evaluating in this section. For the first
time, we also extend this bound to force ranges much shorter than the typical
length scales of the Bullet Cluster.

To evaluate this bound, let us now consider the case when $a_{g}$ exceeds
$a_{\rm DM}$, such that the dark matter and Standard Model arrive together. We
begin by considering the fall of a single particle from rest at infinity into
a cluster, in the limit of negligible friction due to scattering with other
individual particles, such that the velocity of the particle at any point can
be evaluated simply from the potential relative to infinity seen by the
particle at that point. Requiring that a new force not change the collision
velocity of that particle with the cluster by more than order unity is thus
equivalent to requiring that the potential at the collision point not be
changed by more than order unity, i.e.

$\displaystyle\frac{\left|V_{\rm
new}(\mathbf{r_{A}^{*}})\right|}{m}\lesssim\frac{\left|V_{\rm
g}(\mathbf{r_{A}^{*}})\right|}{m}=G\int{\rm
d}^{3}\mathbf{r}\frac{\rho_{A}(\mathbf{r})}{|\mathbf{r}-\mathbf{r_{A}^{*}}|}.$
(13)

where $V_{\rm new}(\mathbf{r_{A}^{*}})$ is given in Eq. (7). Before we proceed
further, let us obtain an order-of-magnitude estimate of Eq. (7) in order to
understand its scaling behavior. A simple approximation for the potential per
unit mass at radius $r_{A}^{*}$ can obtained from arguments similar to Eq. (9)
to obtain

$\displaystyle\frac{V_{\rm new}^{*}}{m}\sim a^{*}r_{A}^{*}\sim\alpha
G\rho_{A}(r_{A}^{*})\left(1+\left(\frac{\lambda}{r_{A}^{*}}\right)^{2}\right),$
(14)

Requiring this to be greater than $V_{g}\sim
G\rho_{A}(r_{A}^{*})\left(r_{A}^{*}\right)^{2}$ gives

$\displaystyle\alpha\gtrsim 1+\left(\frac{\lambda}{r_{A}^{*}}\right)^{2}$ (15)

which is very similar to Eq. (12).

While the order-of-magnitude estimates match for the two observations,
numerical evaluation shows that the constraint obtained from the non-
observation of the separation of dark matter and the stellar population is
stronger. This occurs primarily from the fact that even a small mismatch
between $a_{\rm DM}$ and $a_{g}$ results in a separation of
$\mathcal{O}\left(r_{A}^{*}\right)$, whereas this separation is constrained to
be a tiny fraction of $r_{A}^{*}$. In the spirit of staying conservative—e.g.
with respect to additional interactions between dark matter and the Standard
Model or to effects of substructure which might suppress the observation in
Section II.1—we use the weaker bound obtained from numerically evaluating the
bound in Eq. (13). This conservative constraint is plotted in Fig. 1.

There are several complications to this analytical description of the
collision in the presence of a new force. We discuss them over the next
several subsections.

Figure 1: Our constraints on attractive self-interactions of dark matter
beyond gravity from coherent interactions in the Bullet Cluster (see Section
II) and from mass measurements of the Abell 370 cluster (see Section III.2),
and, for comparison, previous constraints from the tidal streams of the
Sagittarius dwarf galaxy [30, 31, 32] and from incoherent scattering in the
Bullet Cluster [22, 23, 24, 25] at two assumed dark matter particle masses
(the largest mass at which promordial black holes can make up all of dark
matter, $\sim 10^{-10}~{}M_{\odot}$ [33], as well as $\sim
10^{-20}~{}M_{\odot}$ to illustrate the mass dependence). As a rough estimate
of the uncertainty on our bound, the collision velocity constraint is shown
assuming both NFW and King profiles, as well as for an NFW profile but
conservatively assuming that the gas collides at two times the scale radius
rather than at the scale radius of the A cluster. The shortest ranges at which
we plot the lensing and tidal tails constraints are approximate; see the
discussion in Section III. Note also that the upper edge of the parameter
space excluded by the Sagittarius Dwarf is presently unknown; Ref. [32] has
shown only that $\alpha=1$ is consistent with observation.

###  II.3 Halo Density Profiles

We default to NFW profiles for both clusters in this work due to their
widespread use, but marginally better lensing fits are, in fact, obtained
using King profiles [27]. This leads to only small changes in the result of
numerically evaluating the bound in Eq. (13), as seen in Fig. 1. The minimal
dependence on profile choice is explained by the scaling arguments that lead
to Eq. (9): any power-law profile with length scale $r_{A}^{*}\gg\lambda$ will
lead to accelerations and potentials which scale as
$\left(\lambda/r_{A}^{*}\right)^{2}$ for small $\lambda$.

###  II.4 Substructure

A related caveat to our estimate above is that it assumes that all dark matter
is smoothly distributed in the shape of this profile. In fact, this is
certainly not the case: the presence of galaxies within the two clusters
indicates that at least some fraction of the dark matter in each cluster is
located in dense galactic halos, and might therefore not interact with the
dark matter in galaxy halos of the other cluster unless the two galactic halos
happen to pass within a distance of order $\lambda$ of one another. So long as
at least an order-one fraction of the dark matter in the A cluster remains
outside its galaxies, however—as is expected from N-body simulations of
structure formation (see, for example, Ref. [34] for a review)—this should not
affect our results by more than order one.

More generally, our results hold only when the dark matter in the two clusters
is not fully sequestered in clumps separated by length scales longer than the
force range $\lambda$. Such large separations could result in dark matter
clumps of cluster B passing through cluster A without ever passing within
$\lambda$ of another clump, thus never being affected by the new force. While
this scenario appears unlikely and is not supported by structure formation
simulations [34], it may be possible to construct models in which our
constraint is evaded in this way. We emphasize, however, that while the
presence of a new finite-range attractive force affecting dark matter would
certainly increase the clustering of dark matter at shorter length scales, our
bound would not be modified significantly unless the smoothly-distributed
fraction of dark matter became much less than unity, and remained that way in
spite of subsequent tidal stripping.

###  II.5 The Collision Location

It is also worth commenting on our assumption that the relevant point at which
to evaluate the velocity of the B cluster is at the scale radius of the A
cluster. This is, of course, a simplification: hydrodynamic simulations of the
Bullet Cluster collision show significant dynamics even at larger separation
radii [28]. Nonetheless, one generically expects that the majority of the
collision dynamics will occur at radii not much larger than the scale radius.
The exact collision radius chosen should therefore not affect our conclusions
by more than order-one factors. To illustrate this, Fig. 1 also shows a
conservative version of our bound that instead assumes a collision at two
times the scale radius of the A cluster.

###  II.6 Dynamical Friction

Self-interactions of dark matter also lead to a competing effect on the
cluster infall velocity: the tendency of scattering to slow the two clusters’
relative velocities through dynamical friction, even for purely attractive
forces. Dynamical friction is the process through which an object moving
through a field of particles with which it interacts gradually transfers
kinetic energy to that background field, being slowed in the process [35, 36,
37, 38].

Dynamical friction tends to slow the passage of cluster B through cluster A,
counteracting the additional infall velocity acquired due to the attractive
force as discussed above. The magnitude of this effect can be estimated from
Chandrashekhar’s dynamical friction formula [35, 38],

$\displaystyle\frac{{\rm d}\mathbf{v}_{B}}{{\rm d}t}$
$\displaystyle=-\frac{16\pi^{2}}{3}\alpha^{2}G^{2}M_{B}\rho_{A}\frac{\mathbf{v}_{B}}{v_{B}^{3}}\ln\Lambda,$
(16)

where $M_{B}$ and $\mathbf{v}_{B}$ are the mass and velocity of the B cluster,
respectively, $\rho_{A}$ is the mass density of the A cluster (which we will
approximate as a constant, since only particles in cluster A within a range
$\lambda$ of cluster B contribute, and we are interested in the case where
both $\lambda$ and the scale radius of B are significantly smaller than the
scale radius of A), and $\Lambda$ is the ratio between the range of the new
force and a short-distance cutoff: either the scale radius of the B cluster or
the distance at which a cluster A particle would be deflected by $90^{\circ}$
by its interaction with the B cluster; whichever is larger. (There is no
dependence on the velocity distribution function of the A cluster, since the B
cluster is moving at A’s escape velocity while particles in A must have
smaller velocities.)

Note that $\mathbf{v}_{B}$ is modified, relative to its gravity-only value, in
the presence of a new force. In particular, once the potential at the
collision point is dominated by the new force rather than by gravity,
$v_{B}\propto\alpha^{1/2}$. This change in velocity also affects the time
taken by the infall, and thus the total amount of dynamical friction. The end
result of this is that the fractional velocity loss from dynamical friction
scales with $\alpha$ as

$\displaystyle\begin{split}\frac{1}{v_{B}}\frac{{\rm d}v_{B}}{{\rm
d}r}&=\frac{1}{v_{B}^{2}}\frac{{\rm d}v_{B}}{{\rm d}t}\\\
&=\frac{1}{v_{B}^{2}}\left|-\frac{16\pi^{2}}{3}\alpha^{2}G^{2}M_{B}\rho_{A}\frac{\mathbf{v}_{B}}{v_{B}^{3}}\ln\Lambda\right|\\\
&\propto\frac{\alpha^{2}}{v_{B}^{4}}\propto\alpha^{0},\end{split}$ (17)

i.e. it is independent of the strength of the new force! It is thus sufficient
to confirm that dynamical friction’s effects are subdominant at the smallest
values of $\alpha$ (for each $\lambda$) for which collision velocity could
lead to a constraint on new self-interactions to ensure that it is
insignificant at all larger values of $\alpha$ as well.

Integrating Eq. (16) numerically, we find that dynamical friction would only
lead to a velocity loss of order $4\%$ for cluster B passing entirely through
cluster A, or less than $1\%$ before reaching cluster A’s scale radius.
Dynamical friction therefore does not significantly change the constraint on
new attractive forces obtained above.

This is also why we do not present a constraint on new forces arising from
long-range scattering of dark matter particles in the Bullet Cluster:
dynamical friction is precisely the effect of (soft) scattering in this long-
range, classical regime, and one might hope to obtain a bound from the fact
that it could not have been strong enough to slow the dark matter by order
unity. The small effect of dynamical friction compared to the direct effect of
the additional attractive potential, however, means that the net effect of the
new force is an acceleration, not a deceleration, and thus such a bound would
not be physical. While both effects would tend to slow the dark matter in the
case of a repulsive potential, the smallness of dynamical friction would still
not significantly affect the collision velocity bound obtained above, and—as
we will see in Section III.1—this is not competitive with bounds on repulsive
forces obtained from ultrafaint dwarf galaxies.

###  II.7 Tidal Distortion

The discussion so far has ignored any internal dynamics of the two clusters
within the Bullet Cluster. However, deformation of the dark matter profiles of
the two clusters could lead to a change in the acceleration of the clusters’
gas contents, modifying the resulting collision velocity.

As we argued in Section II.1, the source of net new forces on cluster B is the
gradient of the dark matter density of cluster A, leading to a parametric
suppression of forces of range $\lambda<r^{*}_{A}$ by a factor of
$\lambda/r^{*}_{A}$ from the fractional change in density within a distance
$\lambda$ of the particle being accelerated. The variation of this force over
the size of cluster B is then suppressed by an additional factor of
$r^{*}_{B}/r^{*}_{A}\sim 0.1$. Since our bound corresponds to the new force
imparting an additional velocity to the dark matter in cluster B of order the
escape velocity from $r^{*}_{A}$ of cluster A, the largest possible result of
this would be a spread in velocities imparted to opposite sides of cluster B
of order $0.1$ times the escape velocity of cluster A, which is in turn
smaller than the escape velocity of cluster B by a factor of approximately $4$
due to the higher density of cluster B (see Eq. (14)). Cluster B’s dark matter
profile is thus disrupted by much less than order unity at our bound.
Moreover, since the quantities discussed above have identical scaling with
$\alpha$ above our bound (when the escape velocity is dominated by the new
force rather than by gravity), this remains the case even for larger $\alpha$,
ensuring that tidal deformation is never significant.

The dark matter in cluster A is similarly accelerated, and then also
decelerated, as it passes through cluster B, leading (in the absence of a
restoring force) to a displacement of order $\Delta r\sim\Delta
v(r_{B}/v_{B})\sim r_{B}$ at our bound. This would not be a large effect even
if it affected all of cluster A (since $r_{B}\sim 0.1~{}r_{A}$), and
additionally it affects only the fraction $(r_{B}/r_{A})^{2}\sim 0.01$ of
cluster A that passes through cluster B, so the change in the gravitational
potential seen by the gas in cluster A is negligible. There is thus no
significant change in the A cluster gas’s velocity compared to what would have
occurred in the presence of gravity alone.

##  III Other Constraints

The Bullet Cluster is far from the only constraint on long-range self-
interactions of dark matter. We now turn to some of the other constraints on
this parameter space, emphasizing the same key idea presented in Section II:
that the gradient of the dark matter density in galaxy and cluster halos
causes bounds on new dark matter self-interactions to persist to ranges
shorter than the structure length scale, with only a power-law (rather than an
exponential) suppression. We will focus on two specific constraints: halos
remaining bound even in the presence of new repulsive forces, and the
agreement between the cluster masses obtained through gravitational lensing
and through measurements of galactic velocities. The same principles should
apply to various other constraints on this parameter space [39], including the
mass-to-luminosity ratio of galaxies in the Local Group, tidal stripping of
galaxies within clusters, and the modification of rotation curves resulting
from galaxy formation. Conversely, see Refs. [40, 41] for discussion of a
bound at longer ranges based on the cosmological effects of a background field
of the force mediator, whose shorter-range behavior has not yet been
determined. Similarly, Refs. [30, 31] showed that forces with range
$\lambda\gtrsim 10~{}{\rm kpc}$ and strength $\alpha\gtrsim 0.2$ would result
in asymmetries in the tidal tails of the Sagittarius (Sgr) Dwarf Spheroidal
Galaxy inconsistent with observations—although Ref. [32] demonstrates that
this effect disappears once $\alpha\sim 1$, and an exact upper bound to this
constraint has yet to be determined—but it is unclear how this constraint
extends to shorter force ranges, due to the large effects of such a new force
on the dark matter profile in the vicinity of the Sgr dwarf; see Section
III.2. Other constraints on long-range self-interactions of dark matter are
discussed in Refs. [42, 13, 43].

Figure 2: Our constraints on repulsive self-interactions of dark matter from
self-binding of ultrafaint dwarf galaxies (see Section III.1) and from mass
measurements of the Abell 370 cluster (see Section III.2), and, for
comparison, previous constraints from the tidal streams of the Sagittarius
dwarf galaxy [30, 31, 32] and from incoherent scattering in the Bullet Cluster
[22, 23, 24, 25] at two assumed dark matter particle masses (the largest mass
at which promordial black holes can make up all of dark matter, $\sim
10^{-10}~{}M_{\odot}$ [33], as well as $\sim 10^{-20}~{}M_{\odot}$ to
illustrate the mass dependence). The shortest ranges at which we plot the
lensing and tidal tails constraints are approximate; see the discussion in
Section III. Note also that the upper edge of the parameter space excluded by
the Sagittarius Dwarf is presently unknown; Ref. [32] has shown only that
$\alpha=1$ is consistent with observation.

###  III.1 Halo Binding

A very straightforward bound on repulsive forces arises simply from the
existence of dark matter halos, which would unbind in the presence of a
sufficiently strong repulsive self-interaction [39]. The smallest observed
dark matter halos are the ultrafaint dwarf galaxies’ halos with scale radii of
approximately $1~{}{\rm kpc}$ [44, 45]. Repulsive forces with ranges longer
than this length scale must therefore be weaker than gravity.

The corresponding bound on shorter-range forces is, as in the Bullet Cluster
case, suppressed by a factor of $(1~{}{\rm kpc}/\lambda)^{2}$, as we can see
from requiring that the total acceleration at the scale radius is still
pointing radially inward:

$\displaystyle a_{\rm DM}(r^{*})+a_{\rm g}(r^{*})<0.$ (18)

These are given by the $a_{\rm DM}$ and $a_{g}$ in Eqs. (8) and (10) but now
for a single halo with some density $\rho$ and size $r^{*}\sim{\rm kpc}$. The
resulting constraint is then approximately

$\displaystyle\alpha\lesssim 1+\left(\frac{\rm kpc}{\lambda}\right)^{2},$ (19)

where the constant value of $1$ accounts for the large-$\lambda$ behavior
discussed below Eq. (8) and we have omitted order-one numerical factors since
these depend on the essentially unknown halo profiles of ultrafaint dwarf
galaxies. We plot this constraint in Fig. 2.

A closely related constraint on attractive forces arises from the need for
orbiting dark matter to be able to stay in orbit, rather than to collapse into
a black hole (i.e. for the necessary orbital velocity of the dark matter to be
$<1$). It is straightforward to see that this leads to bounds significantly
weaker than those calculated in Section II, however.

###  III.2 Gravitational Lensing

We next consider an example of a constraint on dark matter self-interactions
that does not obey the power-law behavior discussed thus far at sufficiently
small length scales due to qualitatively new dynamics in the short-range,
strong-coupling regime. In the very long-range regime, constraints on dark
matter self-interactions have been derived from the agreement of cluster mass
measurements using the velocities of galaxies (which would be affected by a
new force) and using gravitational lensing of light (which would not) [39]. We
now consider the short-range behavior of this constraint.

For force ranges $\lambda$ much shorter than the scale radius $r^{*}\sim
1.2~{}{\rm Mpc}$, the additional acceleration of dark matter at a radius
$r\sim 220\,{\rm kpc}$ with $\lambda\ll r\ll r^{*}$ can naively be obtained
similarly to Eq. (8), but now accounting for the faster variation of density
near the center of NFW halos:

$\displaystyle|a_{\rm DM}(r)|\sim\alpha
G\frac{\lambda^{3}}{\lambda^{2}}\left(\lambda\frac{{\rm d}}{{\rm d}r}\rho_{\rm
NFW}(r)\right)\sim\alpha G\frac{\rho_{\rm NFW}(r)\lambda^{2}}{r}.$ (20)

Here we have dropped various order-one factors since this is intended only as
a rough estimate of the result. Requiring that this change in acceleration be
no greater than the gravity-only prediction, i.e. that the velocity
distribution (and thus the mass estimate) is not modified by more than order
one, i.e. that

$\displaystyle|a_{\rm DM}(r)|<|a_{g}(r)|\sim G\rho(r)r,$ (21)

then leads to a constraint on $\alpha$; this is plotted in Figs. 1 and 2.

However, once $\lambda\lesssim 100$ kpc, the typical size of individual galaxy
halos, a limit cannot be derived as explained above because we can no longer
ignore the back-reaction of the galaxy halo on to the background dark matter
density: the galaxy halo is a significant over-dense region compared to the
ambient background density and hence dominates the total dark matter mass
contained within $\lambda$ of a typical galaxy halo. There is therefore
significant back-reaction on the cluster profile due to the presence of the
galaxy. While this does not preclude the possibility of there existing limits,
a detailed simulation would be required to evaluate this back-reaction and
estimate the galaxy velocity dispersion in presence of a new short-ranged
force stronger than gravity. For this reason, we truncate the limit in Figs. 1
and 2 at $100$ kpc.

It is important to point out that this back-reaction effect is unique to this
case: both the Bullet Cluster interactions considered in Sec. II and the halo
binding discussed in Sec. III.1 were based on the smooth components of those
objects, which lack significant over-densities.

##  IV Conclusion

Many previous works have presented constraints on interactions between dark
matter particles, but they have generally focused on one of two regimes:
short-range interactions which result in scattering between individual
particles, and long-range (or infinite-range) interactions which effectively
modify the strength of gravity within dark sectors. In this work, we have
considered the intermediate regime, in which dark matter particles carry a net
charge under a force with range between the dark matter’s interparticle
spacing and the length scale of their containing halos.

If dark matter halos had sharp cutoffs—that is, if they had approximately
constant density within some region and negligible density outside of it—the
net force resulting from such a new interaction would be exponentially
suppressed at distances from the halo greater than the interaction’s range.
Previous works have therefore occasionally assumed that meaningful constraints
cannot be obtained from astrophysical observations for such short-range
forces.

In this work, we have shown that this is not the case: the gradual variation
in the dark matter density of observed (and simulated) dark matter halos leads
only to a power-law suppression in the effects of a finite-range interaction,
as a dark matter particle within such a halo sees a net force resulting from
the difference in the dark matter density between the regions within $\lambda$
of it that are closer and farther from the halo center. Since halo profiles
are believed to vary with radius as power laws, this density contrast is also
suppressed only as a power law, and thus finite-range interactions can still
have significant collective effects at ranges shorter than the halo length
scale.

We then used this behavior to extend various constraints on (effectively)
infinite-range interactions to finite ranges; the resulting constraints are
plotted in Figs. 1 and 2. For illustrative purposes, we focused on the effects
of such a finite-range attractive force on the Bullet Cluster, where it
results in an acceleration of the smaller cluster in that collision in a way
that would result in either a collision velocity of the clusters’ gas
inconsistent with observations, or with a separation between the dark matter
and galaxies within each cluster. For repulsive forces, we noted that a strong
constraint can be obtained simply from the fact that ultrafaint dwarf
galaxies’ halos remain bound. The resulting bounds are the strongest existing
bounds on net-charged dark matter self-interactions over at least five orders
of magnitude in force range even for dark matter at the heaviest masses
allowed for primordial black hole dark matter, and over far greater ranges for
typical particle-mass dark matter.

As we have argued, this finite-range behavior is quite generic, with caveats
arising primarily from the unknown substructure of dark matter halos. The
results of such substructure do not change the two constraints mentioned above
by more than order one unless almost all dark matter is concentrated in
widely-separated substructures. Other constraints, however, can be much more
sensitive to substructure effects: for example, as we showed, constraints from
the agreement of galaxy cluster mass measurements between velocity-based and
gravitational-lensing based approaches become ineffective once the majority of
the dark matter mass within $\lambda$ of a galaxy is bound to that galaxy
already, as the galaxy begins to have a large impact on the distribution of
dark matter in its vicinity. Even in this case, however, it may be possible to
set constraints on dark matter self-interactions via N-body simulations of
galactic dynamics; we leave this to future work.

More generally, there are many ways to both refine the constraints we present
and extend others to finite ranges. The focus of this work has been the
parametric behavior of various present-day constraints at finite ranges, so we
have computed those constraints only up to order-one factors. It may be
possible to strengthen them somewhat, however, by performing N-body
simulations of the bullet cluster collision, galaxy cluster dynamics, or
similar. Moreover, there are many other constraints which have been computed
at infinite ranges but which we have not considered here. In particular, the
existence of a new finite-range dark matter self-interaction stronger than
gravity would likely significantly modify structure formation; we leave the
study of such cosmological effects to future work.

###### Acknowledgements.

The authors acknowledge support by NSF Grants PHY-2310429 and PHY-2014215,
Simons Investigator Award No. 824870, DOE HEP QuantISED award #100495, the
Gordon and Betty Moore Foundation Grant GBMF7946, and the U.S. Department of
Energy (DOE), Office of Science, National Quantum Information Science Research
Centers, Superconducting Quantum Materials and Systems Center (SQMS) under
contract No. DEAC02-07CH11359. ZB is supported by the National Science
Foundation Graduate Research Fellowship under Grant No. DGE-1656518 and by the
Dr. HaiPing and Jianmei Jin Fellowship from the Stanford Graduate Fellowship
Program.

## References

  * de Blok [2009] W. J. G. de Blok, Advances in Astronomy 2010, 789293 (2009).
  * Flores and Primack [1994] R. A. Flores and J. R. Primack, The Astrophysical Journal 427, L1 (1994), arXiv:astro-ph/9402004 [astro-ph] .
  * Moore [1994] B. Moore, Nature 370, 629 (1994).
  * Kauffmann  _et al._ [1993] G. Kauffmann, S. D. M. White, and B. Guiderdoni, onthly Notices of the Royal Astronomical Society 264, 201 (1993).
  * Klypin  _et al._ [1999] A. Klypin, A. V. Kravtsov, O. Valenzuela, and F. Prada, The Astrophysical Journal 522, 82 (1999).
  * Moore  _et al._ [1999] B. Moore, S. Ghigna, F. Governato, G. Lake, T. Quinn, J. Stadel, and P. Tozzi, The Astrophysical Journal 524, L19 (1999).
  * Perivolaropoulos and Skara [2022] L. Perivolaropoulos and F. Skara, New Astronomy Reviews 95, 101659 (2022).
  * Kaplinghat  _et al._ [2020] M. Kaplinghat, T. Ren, and H.-B. Yu, JCAP 06, 027, arXiv:1911.00544 [astro-ph.GA] .
  * Kendall and Easther [2020] E. Kendall and R. Easther, Publ. Astron. Soc. Austral. 37, e009 (2020), arXiv:1908.02508 [astro-ph.CO] .
  * Banerjee  _et al._ [2022] A. Banerjee _et al._ , in  _Snowmass 2021_ (2022) arXiv:2203.07049 [astro-ph.CO] .
  * Del Popolo and Le Delliou [2021] A. Del Popolo and M. Le Delliou, Galaxies 9, 123 (2021), arXiv:2209.14151 [astro-ph.CO] .
  * Farrar and Peebles [2004] G. R. Farrar and P. J. E. Peebles, The Astrophysical Journal 604, 1 (2004).
  * Amendola and Tocchini-Valentini [2002] L. Amendola and D. Tocchini-Valentini, Phys. Rev. D 66, 043528 (2002).
  * Tucker  _et al._ [1995] W. H. Tucker, H. Tananbaum, and R. A. Remillard, Astrophys. J. 444, 532 (1995).
  * Tucker  _et al._ [1998] W. Tucker, P. Blanco, S. Rappoport, L. David, D. Fabricant, E. E. Falco, W. Forman, A. Dressler, and M. Ramella, The Astrophysical Journal 496, L5 (1998).
  * Barrena  _et al._ [2002] R. Barrena, A. Biviano, M. Ramella, E. E. Falco, and S. Seitz, Astronomy and Astrophysics 386, 816 (2002), arXiv:astro-ph/0202323 [astro-ph] .
  * Clowe  _et al._ [2004a] D. Clowe, A. Gonzalez, and M. Markevitch, The Astrophysical Journal 604, 596 (2004a).
  * Clowe  _et al._ [2006] D. Clowe, M. Bradač, A. H. Gonzalez, M. Markevitch, S. W. Randall, C. Jones, and D. Zaritsky, The Astrophysical Journal 648, L109 (2006), arXiv:astro-ph/0608407 [astro-ph] .
  * Bradač  _et al._ [2006] M. Bradač, D. Clowe, A. H. Gonzalez, P. Marshall, W. Forman, C. Jones, M. Markevitch, S. Randall, T. Schrabback, and D. Zaritsky, The Astrophysical Journal 652, 937 (2006).
  * Markevitch  _et al._ [2002] M. Markevitch, A. H. Gonzalez, L. David, A. Vikhlinin, S. Murray, W. Forman, C. Jones, and W. Tucker, The Astrophysical Journal 567, L27 (2002), arXiv:astro-ph/0110468 [astro-ph] .
  * Govoni  _et al._ [2004] F. Govoni, M. Markevitch, A. Vikhlinin, L. VanSpeybroeck, L. Feretti, and G. Giovannini, Astrophys. J. 605, 695 (2004), arXiv:astro-ph/0401421 .
  * Markevitch  _et al._ [2004] M. Markevitch, A. H. Gonzalez, D. Clowe, A. Vikhlinin, W. Forman, C. Jones, S. Murray, and W. Tucker, The Astrophysical Journal 606, 819 (2004).
  * Knapen  _et al._ [2017] S. Knapen, T. Lin, and K. M. Zurek, Phys. Rev. D 96, 115021 (2017).
  * Randall  _et al._ [2008] S. W. Randall, M. Markevitch, D. Clowe, A. H. Gonzalez, and M. Bradač, The Astrophysical Journal 679, 1173 (2008).
  * Harvey  _et al._ [2015] D. Harvey, R. Massey, T. Kitching, A. Taylor, and E. Tittley, Science 347, 1462 (2015).
  * [26] P. W. Graham and H. Ramani, Forthcoming .
  * Farrar and Rosen [2007] G. R. Farrar and R. A. Rosen, Phys. Rev. Lett. 98, 171302 (2007).
  * Springel and Farrar [2007] V. Springel and G. R. Farrar, Monthly Notices of the Royal Astronomical Society 380, 911 (2007), https://academic.oup.com/mnras/article-pdf/380/3/911/2794819/mnras0380-0911.pdf .
  * Clowe  _et al._ [2004b] D. Clowe, G. De Lucia, and L. King, Monthly Notices of the Royal Astronomical Society 350, 1038 (2004b), https://academic.oup.com/mnras/article-pdf/350/3/1038/2924905/350-3-1038.pdf .
  * Kesden and Kamionkowski [2006a] M. Kesden and M. Kamionkowski, Phys. Rev. Lett. 97, 131303 (2006a).
  * Kesden and Kamionkowski [2006b] M. Kesden and M. Kamionkowski, Phys. Rev. D 74, 083007 (2006b).
  * Keselman  _et al._ [2009] J. A. Keselman, A. Nusser, and P. J. E. Peebles, Phys. Rev. D 80, 063517 (2009).
  * Bird  _et al._ [2023] S. Bird, A. Albert, W. Dawson, Y. Ali-Haïmoud, A. Coogan, A. Drlica-Wagner, Q. Feng, D. Inman, K. Inomata, E. Kovetz, A. Kusenko, B. V. Lehmann, J. B. Muñoz, R. Singh, V. Takhistov, and Y.-D. Tsai, Physics of the Dark Universe 41, 101231 (2023).
  * Vogelsberger  _et al._ [2020] M. Vogelsberger, F. Marinacci, P. Torrey, and E. Puchwein, Nature Rev. Phys. 2, 42 (2020), arXiv:1909.07976 [astro-ph.GA] .
  * Chandrasekhar [1943a] S. Chandrasekhar, Astrophys. J. 97, 255 (1943a).
  * Chandrasekhar [1943b] S. Chandrasekhar, Astrophys. J. 97, 263 (1943b).
  * Chandrasekhar [1943c] S. Chandrasekhar, Astrophys. J. 98, 54 (1943c).
  * Binney and Tremaine [2008] J. Binney and S. Tremaine,  _Galactic Dynamics_ (Princeton University Press, Princeton, 2008).
  * Gradwohl and Frieman [1992] B.-A. Gradwohl and J. A. Frieman, Astrophys. J. 398, 407 (1992).
  * Archidiacono  _et al._ [2022] M. Archidiacono, E. Castorina, D. Redigolo, and E. Salvioni, Journal of Cosmology and Astroparticle Physics 2022 (10), 074.
  * Bottaro  _et al._ [2023] S. Bottaro, E. Castorina, M. Costa, D. Redigolo, and E. Salvioni, Unveiling dark forces with the large scale structure of the universe (2023), arXiv:2309.11496 [astro-ph.CO] .
  * Mainini and Bonometto [2006] R. Mainini and S. Bonometto, Phys. Rev. D 74, 043504 (2006).
  * Frieman and Gradwohl [1991] J. A. Frieman and B.-A. Gradwohl, Phys. Rev. Lett. 67, 2926 (1991).
  * Stegmann  _et al._ [2020] J. Stegmann, P. R. Capelo, E. Bortolas, and L. Mayer, Monthly Notices of the Royal Astronomical Society 492, 5247 (2020), https://academic.oup.com/mnras/article-pdf/492/4/5247/32368866/staa170.pdf .
  * Chiti  _et al._ [2021] A. Chiti, A. Frebel, J. D. Simon, D. Erkal, L. J. Chang, L. Necib, A. P. Ji, H. Jerjen, D. Kim, and J. E. Norris, Nature Astronomy 5, 392 (2021).