# Astrophysical Plasma Instabilities induced by Long-Range Interacting Dark
Matter

A. Cruz  Matthew McQuinn

###### Abstract

If dark matter is millicharged or darkly charged, collective plasma processes
may dominate momentum exchange over direct particle collisions. In particular,
plasma streaming instabilities can couple the momentum of the dark matter to
counter-streaming baryons or other dark matter and result in the counter-
streaming fluids coming to rest with each other, just as happens for baryonic
collisionless shocks in astrophysical systems. While electrostatic plasma
instabilites (such as the two-stream) are highly suppressed by Landau damping
when dark matter is millicharged, in the cosmological situations of interest,
electromagnetic instabilities such as the Weibel can couple the momenta,
assuming that the linear instability saturates in the manner typically found
for baryonic plasmas. We find that the streaming of dark matter in the pre-
Recombination universe is affected more strongly by direct collisions than
collective processes, validating previous constraints. However, when
considering unmagnetized instabilities the properties of the Bullet Cluster
merger and other merging cluster systems (which show dark matter streaming
through itself) are likely to be substantially altered if
$[q_{\chi}/m_{\chi}]\gtrsim 10^{-4}$, where $[q_{\chi}/m_{\chi}]$ is the
charge-to-mass ratio of the dark matter relative to that of the proton. When a
magnetic field is added consistent with cluster observations, the Weibel and
Firehose instabilities result in sufficiently fast growth to reach saturation
for $[q_{\chi}/m_{\chi}]\gtrsim 10^{-12}-10^{-11}$. The Weibel growth rates
are even faster in the case of a dark-$U(1)$ charge (because “hot” electrons
do not damp the instability), potentially ruling out
$[q_{\chi}/m_{\chi}]\gtrsim 10^{-14}$ in the Bullet Cluster system, in
agreement with [1]. The strongest previous limits on millicharged dark matter
(mDM) arise from considering the spin-down of galactic disks [2]. We show that
plasma instabilities or tangled background magnetic fields could lead to
diffusive propagation of the dark matter, weakening these spin-down limits.
Thus, plasma instabilities may place some of the most stringent constraints
over much of the millicharged, and our results corroborate previous extremely
stringent potential constraints on the dark-charged parameter space.

##  1 Introduction

Multiple, orthogonal astrophysical observational probes provide evidence for
the dark matter: galaxy rotation curves [3], the cosmic microwave background
(CMB) [4], gravitational lensing [5], and the matter power spectrum [6]. This
collection of observations indicates that dark matter (DM) interacts
gravitationally, but leaves room for some other interactions. In well-
motivated DM models such as weakly interacting massive particles (WIMPS), DM
also interacts with the Standard Model (SM) through mediators at the weak
scale. The QCD axion, on the other hand, has a weak effective electromagnetic
interaction but no electric charge. If instead DM has a small “millicharge”
and has long-range electromagnetic (EM) interactions with the SM, there can be
significant effects on astrophysical and cosmological scales. One such model
being the dark photon, which gives rise to kinetic mixing and naturally
produces DM with a small electromagnetic charge [7]. This DM model is known in
the literature as millicharged DM (mDM) and allows for long-range EM
interactions between DM and the SM. In this model, the particle DM has mass
$m_{\chi}$ and a small electromagnetic charge $q_{\chi}$ as well as its
antiparticle. While the nomenclature used in the literature is to refer to
this DM model as “millicharged", the considered range of $|q_{\chi}|$ spans
many orders of magnitude.

One benchmark millicharge occurs in freeze-in models [8, 9], where over a
large range of dark matter masses a coupling with $q_{\chi}/e\sim 10^{-11}$ is
able to produce the relic abundance of DM via decays from SM particles. Such
minuscule charge-to-mass ratios can occur in string theoretic constructions
[10]. Freeze-in models have received a lot of interest as a way of producing
light DM for which there are many new experimental search techniques, with
$m_{\chi}\sim 10-1000$ keV [11]. Freeze-in models with dark matter masses
$m_{\chi}\lesssim 0.1$ MeV are ruled out by stellar cooling constraints [12].
CMB and large-scale structure observations have the potential to rule out
somewhat larger masses, as direct collisions transfer momentum and damp
density fluctuations on small scales [12]. Perhaps most excitingly, all
freeze-in parameter space up to $100$ GeV may be ruled out by spin-down of the
Milky Way galaxy [2]. We will revisit this spin-down constraint in light of
our calculations.

Another possibility is that the dark matter is charged in a $U(1)$-gauged dark
sector that does not interact with SM particles [e.g. 13].111A weak mixing of
this sector with the SM hypercharge is often the case for how millicharge is
created. In this case, whether we are in this limit or the millicharged limit
depends on whether dark matter-baryon or dark matter-dark matter interactions
are stronger. This possibility avoids the anomaly conditions and tight
observational constraints that arise from SM hypercharge. In a purely darkly
charged model in thermal equilibrium with an associated massless dark photon,
constraints arise from limits on the number of degrees of freedom at Big Bang
Nucleosynthesis and at Recombination [14]. Models with sectors that mirror the
SM complexity provide additional constraints [15, 16, 17]. Some constraints
still apply if the dark photon that mediates the interaction is ultralight but
not massless. Limits on a charged dark sector (including sectors also with SM
millicharge) are reviewed in [13].

Many of the cosmological constraints on millicharged and dark charged
particles have largely focused on direct collisions. For the baryonic plasmas
we are most familiar with, collisions are rarely the most effective mechanism
for transferring momentum. Rather, there are collective processes that couple
the momenta of counter-streaming plasmas. This results in collisionless
shocks, where the width of the shock is much narrower than the collisional
mean free path. These shocks occur ubiquitously in astrophysics [18]. Just
like with the baryons, long range collective forces between DM particles or –
in the case of mDM – with the baryonic matter can couple the momentum of
counter-streaming flows. This collective behavior could dominate DM momentum
coupling that assume only collisions, and could improve CMB and the Bullet
Cluster constraints on such coupling, such as [19].

We consider plasma streaming instabilities that would act to couple the
momentum of the DM to either the gas or to the DM itself. Streaming
instabilities have been previously considered for mDM in [20, 21, 22, 23, 24].
When DM is millicharged, electrostatic instabilities, or instabilities with
any significant longitudinal component, are highly Landau damped owing to the
streaming velocity (and hence the phase velocity of the excited Langmuir
waves) being not much larger than the particle velocity dispersions for the
cosmological situations of interest [24]. Thus, electromagnetic instabilities
are the most promising when dark matter is millicharged. On the other hand,
when DM has a dark-$U(1)$ charge [1] has shown that the growth of the
electrostatic two-stream instability grows at the dark matter plasma
frequency. On the other hand, electrostatic instabilities, like the two-
stream, generally saturate by flattening the bump on the tail of the
distribution. Such saturation may occur before substantial momentum exchange
[e.g. 25]. Thus, we also focus on the electromagnetic instability in this
case. We consider primarily the Weibel instability [26, 27]. Indeed, the
Weibel instability is thought to mediate collisionless shocks in astrophysics
[28], saturating in a manner where the penetrating plasmas come to rest.
Although this case has been considered in the literature [29], we expand upon
previous analysis and verify the cold approximation for Weibel instability
growth rate for the Bullet Cluster system. We also consider the Firehose
instability, which requires an initial background magnetic field.

In this paper we examine the long-range collective electromagnetic
interactions of mDM and dark-$U(1)$ charged DM in galaxies, clusters of
galaxies, and in the CMB. Our paper is outlined as follows:

  1. 1.

In Section 2, we provide a brief introduction of plasma instabilities. We then
discuss the time required to reach instability saturation, and the time-scales
associated with two particle collisions, which alter the collective plasma
effects and destroy the growth of instabilities.

  2. 2.

In Section 3 we discuss electrostatic plasma beam instabilities and show that
they are likely significantly damped if DM is millicharged.

  3. 3.

This brings us to Section 4, where we discuss electromagnetic plasma
instabilities. In particular, the relevant instabilities are the Weibel
instability (§4.2) and the Firehose instability (§4.5). In the former case we
consider three astrophysical cases: mDM or $U(1)$ interactions during cluster
mergers, streaming DM in the Recombination-era plasma, and we briefly discuss
mDM falling onto the Milky Way. We calculate instability growth rates and
estimate when these growth rates are fast enough to grow within the age of the
system and to not be suppressed by two particle collisions.

  4. 4.

In Section 5, we revisit previous dark charged constraints in light of our
findings. In particular, we revisit the strong mDM constraint from the spin-
down of galactic disks of [2] in the limit where DM has a mean free path to
scatter off magnetic inhomogeneities that is much smaller than the size of the
halo, as may happen if the aforementioned plasma instabilities are able to
reach saturation.

  5. 5.

In Section 6 we discuss our mDM and dark-$U(1)$ parameter space constraints on
the $q_{\chi}$-$m_{\chi}$ plane. Additionally, we review and discuss previous
constraints set in the literature in light of our findings.

  6. 6.

Finally, in Section 7 we discuss our constraints, illustrating the new mDM and
darkly charged parameter space that is excluded by our study.

##  2 Plasma Instabilities

Charged particles participate in long-range collective electromagnetic
interactions that enhance their momentum exchange rate relative to particle-
particle interactions. Streaming instabilities, which arise when a plasma of
charged particles streams through a plasma at rest, are one of the most
studied manifestations of these collective forces. For electrons and protons,
the collective interactions dominate over single particle interactions and
lead to momentum redistribution between the inter-penetrating plasmas on
timescales much shorter than the mean collision time. Single particle
collisions between the mDM and baryons goes as the square of the charge-to-
mass ratio, whereas instability growth rates can be linear or even the $2/3$
power in the mDM charge-to-mass ratio relative to protons, which we denote as
$[q_{\chi}/m_{\chi}]$. This suggests that instabilities may be the dominant
mode for momentum exchange in the interesting parameter space of
$[q_{\chi}/m_{\chi}]\ll 1$.

However, a plasma instability will only grow if its growth rate is faster than
the rate at which two-particle collisions dissipate the waves. For mDM, the
collisions that dissipate waves the fastest are electron-electron collisions.
Electron-electron collisions damp waves excited by the instability with a rate
of [30]

$\Gamma_{\rm
coll}^{ee}=0.3{\rm~{}s^{-1}}\left(\frac{n_{e}}{10^{3}{\rm\;cm}^{-3}}\frac{\Lambda}{30}\right)\left(\frac{T}{5000{\rm\;K}}\right)^{-3/2}.$
(2.1)

If this rate is faster than the linear instability growth rate, the
instability will be quenched. As the electron-electron collision time is much
shorter than the baryon-dark matter collision time for relevant mDM parameter
space, we never need to consider whether the growth rate of plasma
instabilities is faster than the collisional coupling rate of the dark plasma
to the baryonic plasma. For DM charged under a dark-$U(1)$, the analogous dark
matter–dark matter collision time is too long to damp the instability for the
small dark charges of interest.

Another timescale of relevance is the age of the system. The instability has
to happen within 1% of the age of the system to reach saturation, as $100$
e-foldings is likely needed.222To derive the rough ${\cal N}\sim 100$
e-foldings criteria, a conservative assumption is that each electromagnetic
mode starts with a (minuscule) thermal amplitude of $k_{b}T$, where $T$ is
some characteristic temperature. The Weibel instability grows by ${\cal N}$
e-foldings until the magnetic field reaches equipartition [31] so that $k_{\rm
max}^{3}k_{b}T\exp[{\cal N}]\sim B_{\rm eq}^{2}$, where $k_{\rm max}$ is the
fasting growing mode, and $B_{\rm eq}^{2}\sim nk_{b}T$, where $n$ is the
particle number density. This reduces to the condition that ${\cal
N}=\ln(n/k_{\rm max}^{3})\approx 80-0.5\ln(n/10^{-5}{\rm
cm}^{-3})-1.5\ln([q_{\chi}/m_{\chi}]/10^{-3})$, where we have used our later
result that $k_{\rm max}\sim w_{p\chi}/c$ (§ 4). In the case of a successful
Weibel instability, saturation corresponds to growing magnetic fields to the
point at which they can deflect streaming particles on the scale of the
instability.

plasma frequency |  $\omega_{\mathrm{p}j}$ = $\sqrt{4\pi n_{j}q_{j}^{2}/m_{j}}$ | Alfvén speed | $v_{A}\equiv\Sigma_{j}B_{0}/\sqrt{4\pi\rho_{j}}$
---|---|---|---
Larmor frequency | $\Omega_{j}\equiv q_{j}B_{0}/m_{j}c$ | Larmor radius | r${}_{\mathrm{L},j}\equiv m_{j}cv_{j}/q_{j}B_{0}$
thermal speed | $\sigma_{T,j}\equiv\sqrt{2kT_{j}/m_{j}}$ | Debye screening length | $\lambda_{\mathrm{D},j}\equiv\sqrt{kT_{j}/4\pi n_{j}q_{j}^{2}}$
Table 1:  Definitions of commonly used plasma parameters in this work, in CGS-
gaussian units.  Figure 1: An illustration of the Weibel instability driven by
the mDM beam in a perpendicular shock and in the absence of a background
magnetic field. A magnetic fluctuation $\delta$B is assumed to oscillate in
$x$ and $t$ and to point perpendicular to the direction of the mDM beam, which
is in the $\hat{\bf y}$ direction. The magnetic fluctuation is thus in the
$x$-$z$ plane. A background plasma, composed of either electrons/protons or
positive/negative charged DM is not shown, but the presence of the background
plasma is critical for instabilities.

##  3 Electrostatic Plasma Instabilities

We briefly discuss electrostatic plasma beam instabilities here and why they
are likely significantly damped in the mDM case (although not necessarily in
the dark-$U(1)$ case). Streaming can excite (electrostatic) Langmuir and ion
acoustic waves that have phase velocities near the streaming velocity. The
cosmological situations of interest have particle velocity dispersions that
are comparable to the phase velocity of the waves. Thus, when DM is
millicharged, the excited waves are Landau damped by the ample supply of
charged particles traveling with the waves. For example, in the two-stream
instability (the most-famous-instability-of-all!), the growth rate $\gamma$
for a warm dark matter beam with velocity dispersion $\sigma_{T\chi}$ and warm
background plasma with $\sigma_{Tj}$ (which could be electrons, protons or
dark matter – often really all three333In the case of all three, their damping
effects should be summed in the following equation.) satisfies the
proportionality [25]

$\gamma\propto\overbrace{(V_{b\chi}-\omega_{r}/k)\exp\left[-\frac{({\omega_{r}}/{k}-V_{b\chi})^{2}}{2\sigma_{T\chi}^{2}}\right]}^{\rm
growth}-\overbrace{\left(\frac{\omega_{{\rm
p}j}^{2}\sigma_{T\chi}^{3}}{\omega_{{\rm
p}\chi}^{2}\sigma_{Tj}^{3}}\frac{\omega_{r}}{k}\right)\exp{\left[-\frac{\omega_{r}^{2}}{2k^{2}\sigma_{Tj}^{2}}\right]}}^{\text{Landau
damping}}$ (3.1)

where $\gamma$ is the imaginary component of the solution that leads to
instability and $\omega_{r}$ is the real frequency of the instability which
needs to satisfy $V_{b\chi}>\omega_{r}/k$ for there to be any instability, but
the instability weakens considerably once $V_{b\chi}\gg\omega_{r}/k$ owing to
the exponential factor in the first term on the right-hand side. However, if
$\omega_{r}/k\sim V_{b\chi}$ the dark matter needs to be very cold with
$\sigma_{T\chi}^{3}/\sigma_{Tj}^{3}\ll\omega_{{\rm p}\chi}^{2}/\omega_{{\rm
p}j}^{2}\approx[q_{\chi}/m_{\chi}]^{2}/[q_{j}/m_{j}]^{2}$ for the second term
on the right-hand side, due to Landau damping, not to dominate and suppress
instability, in the interesting limit for mDM damped by electrons,
$[q_{\chi}/m_{\chi}]/[q_{e}/m_{e}]$ is very small. Note that velocity
dispersion $\sigma_{Tj}$, the plasma frequency $\omega_{{\rm p}j}$ for a given
species $j$ and other relevant plasma parameters used in this work are defined
in Table 1. Equation 3.1 is for growth in the warm limit, and does not capture
the contribution from the principal value of the integral that is in the
plasma dispersion function. However, owing to $\sigma_{T\chi}\sim\sigma_{Tj}$
in the cosmological situations we consider where streaming instabilities can
occur444The early universe case being the exception but suffers from a huge
collision rate that damps these instabilities., we expect that even when this
component is included, the electrostatic instabilities for mDM are highly
damped. However, in the case where DM has a dark-$U(1)$ charge,
$[q_{\chi}/m_{\chi}]^{2}/[q_{j}/m_{j}]^{2}=1$ since $j=\chi$ and so
electrostatic instabilities can survive Landau damping [1].

The conclusion that electrostatic instabilities are significantly suppressed
in the mDM case echo [22], who considered electrostatic instabilities for the
case of mDM being excited by supernovae blast waves. The authors found that
these were significantly damped, with only the electromagnetic instabilities
showing growth. Similar damping results for instabilities in which the wave
vector is at an oblique angle to the beam, as the electric field along the
beam is again Landau damped [25, 32], the primary mathematical difference
being the parallel component of wave vector appears in the Landau damping term
in an analogous growth rate equation to that of Equation 3.1.

The situation is different for electromagnetic instabilities for which the
electromagnetic fields are perpendicular to the wave vector, as
electromagnetic waves can either have phase velocities that are much larger
than the particle velocities or instead because the electromagnetic streaming
instability can be purely imaginary, as is the case of the Weibel instability,
which we especially focus on.

##  4 Electromagnetic Plasma Instabilities

In this section, we turn to the possibility that charged DM (millicharged or
dark-$U(1)$) excites transverse EM waves in both the absence and presence of a
background magnetic field. As in [22], we consider only the case where wave
propagation is along ${\bf B_{\rm 0}}$ (i.e. ${\bf k}\parallel{{\bf B_{\rm
0}}}$) and we restrict our analysis to two cases: the Weibel instability
(where ${\bf V}_{b\chi}\perp{\bf B_{\rm 0}}$) and the Firehose instability
(where ${\bf V}_{b\chi}\parallel{\bf B_{\rm 0}}$). In both limiting cases, we
follow standard practices and search for purely imaginary solutions to the
linear dispersion relation as expressed in [32]. In either case, we search for
analytic solutions when possible; when this search is stifled, we instead
search for numerical solutions using the method detailed in the section below.

###  4.1 Numerical Method

Since the linear dispersion is a complex function, we look to find purely
imaginary frequency solutions where the square modulus of the linear
dispersion, $D(k,\omega)$, is zero, for a given wavenumber. It is far from
trivial to find roots or to know whether the root an algorithm returns
represents the fastest growing mode. Thus, finding solutions is a fraught
exercises that depends somewhat on the algorithm. Thus, we provide some
details on the algorithm we use to find roots. We note that, except in the
Bullet cluster case with a magnetic field, we use an analytic analysis to
motivate the starting point of the numerical root finder.

We minimize the square modulus of the dispersion $|D|^{2}$ using the Nelder-
Mead algorithm, a simplex search algorithm for multidimensional unconstrained
optimization of a given nonlinear function
$f:\mathbb{R}^{n}\rightarrow\mathbb{R}$ [33] (in our case
$|D|^{2}:\mathbb{R}^{2}\rightarrow\mathbb{R}$). A simplex $S$ in
$\mathbb{R}^{n}$ is defined as the smallest subset of $\mathbb{R}^{n}$ that
contains each whole line segment joining any two points of $n+1$ vertices
$x_{0},...,x_{n}\in\mathbb{R}^{n}$; i.e. a simplex is a triangle in
$\mathbb{R}^{2}$ and a tetrahedron in $\mathbb{R}^{3}$. The Nedler-Mead
algorithm constructs an initial working simplex $S$ around an initial input
point $x_{0}\in\mathbb{R}^{n}$. In our application of the algorithm, the
simplex is initialized around an initial frequency, $\omega_{0}$, chosen to be
an analytic solution to the linear dispersion at hand (except for the
magnetized Bullet cluster case where we do not have such a solution; see below
for discussion of these solutions). At each iteration of the algorithm, the
vertices of the simplex are evaluated and transformed via reflection,
contraction, or expansion towards the vertex which minimizes the non-linear
function $|D|^{2}$ most, and then a new simplex is constructed. This process
is repeated until the simplex $S$ is sufficiently small, the function values
are close for each vertex in the simplex, or the number of iterations exceeds
a maximum value, which ever comes first. We tried other numerical methods such
as gradient-decent and Newton’s method and discovered that the Nelder-Mead
algorithm produced the most stable results.

###  4.2 Weibel Instability

The growth rate of the linear Weibel instability can be calculated from the
electromagnetic dispersion relation. This relation for the case where the
particle distribution functions can each be modeled as Maxwellian with
velocity dispersion $\sigma_{T,j}$ is [see for example reference 32]:

$\begin{split}0=D^{\pm}(k,\omega)=c^{2}k^{2}-\omega^{2}-\sum_{b=i^{+},e^{-}}\omega_{pb}^{2}\big{(}\frac{\omega}{k\sigma_{T,b}}\big{)}Z(\xi_{b}^{\pm})\\\
-\sum_{s=\chi^{+},\chi^{-}}\omega_{ps}^{2}\bigg{[}\bigg{(}\frac{\omega}{k\sigma_{T,s}}\bigg{)}Z(\xi_{s}^{\pm})+f_{\chi}\bigg{(}\frac{V_{b\chi}}{\sigma_{T,s}}\bigg{)}^{2}(1+\xi_{s}^{\pm}Z(\xi_{s}^{\pm}))\bigg{]},\end{split}$
(4.1)

where $Z(\xi)$ is the plasma dispersion function (see Appendix A),
$\xi_{b}^{\pm}=\frac{\omega\pm\Omega_{b}}{k\sigma_{T,b}}$,
$\xi_{s}^{\pm}=\frac{\omega\pm\Omega_{s}}{k\sigma_{T,\chi}}$ and $f_{\chi}$ is
the fraction of dark matter streaming. Sometimes we will write a solution
using the notation $\omega=\omega_{r}+i\gamma$. Note that we have chosen to
account for the half of the dark matter that has like charge in the number
density that appears in $\omega_{p\chi}$. Even though the dark matter is not
thermal, in dark matter halos it is often reasonably described by a
Maxwellian, and in the early universe the solution does not depend on
$\sigma_{T,\chi}$. Even though we start by considering the unmagnetized limit,
Equation 4.1 allows for a homogeneous beam-aligned magnetic field $B_{0}$; we
will consider the case that it is nonzero in § 4.4. When the background
magnetic field is zero then of course the cyclotron frequencies $\Omega_{j}=0$
for all species $j$.

###  4.3 Unmagnetized Weibel Instability

We first consider the unmagnetized electromagnetic Weibel instability. The
streaming motion of dark matter plasma relative to other particles can drive
this instability [26, 27]. Figure 1 illustrates the instability. We consider a
scenario where a fraction $f_{\chi}$ of the dark matter participates in a
neutral mDM beam with equal parts $+q_{\chi}$ and $-q_{\chi}$ and with beam
velocity $V_{b\chi}$ that is streaming toward the other $1-f_{\chi}$ of the
dark matter plus the baryons. We assume the dark matter to baryon ratio is the
cosmological ratio of $\Omega_{d}/\Omega_{b}$ $\approx 0.2$. Figure 1
illustrates that a small fluctuation in the magnetic field $\delta B$
perpendicular to the dark matter “beam” results in the plus and minus charges
to separate into filaments (indeed, Weibel is often called the filamentation
instability). This separation in turn enhances the amplitude of the magnetic
field fluctuation, causing the filamentation and magnetic field fluctuations
to grow. The classical Weibel solution is purely growing, i.e. Re$[\omega]=0$,
such that it is not suppressed by Landau damping. This instability is found to
saturate once the Larmor radius of the dark matter is comparable to the length
scale of the instability [31], resulting in the dark matter no longer being
able to stream relative to the background plasma. Such momentum coupling
cannot happen for dark matter streaming relative to the baryons in several
cosmological scenarios (such as the pre-Recombination plasma or in the Bullet
Cluster), potentially allowing us to put limits on the dark matter’s charge.
In order to find solutions to Equation 4.1, we consider the limiting behavior
of $Z(\xi_{j})$ (detailed in Appendix A) for different particle species $j$,
where $\Omega_{j}=0$ and thus
$\xi_{j}^{\pm}=\xi_{j}=\frac{\omega}{k\sigma_{T,j}}$. Note that $\xi_{j}$
indicates whether the instability is cold/warm, meaning it grows faster/slower
than the thermal motion of particles across the instability scale.

##### Warm limit, where $\xi_{j}\ll 1$ for all participating species $j$:

Let us first treat the unmagnetized Weibel instability in the warm limit where
the $\xi_{j}\ll 1$ such that $Z(\xi_{j})=i\sqrt{\pi}+{\cal O}(\xi_{j})$.
(Expansions of $Z(\xi_{j})$ are given in Appendix A.) Only keeping the
dominant $j$ particle and the dark matter steaming term Equation 4.1 reduces
to

$0=c^{2}k^{2}-\omega^{2}-i\sqrt{\pi}\omega_{\mathrm{p}j}^{2}\bigg{(}\frac{\omega}{k\sigma_{T,j}}\bigg{)}-f_{\chi}\omega_{\mathrm{p}\chi}^{2}\bigg{(}\frac{V_{b\chi}}{\sigma_{T,\chi}}\bigg{)}^{2},$
(4.2)

where the subscript $j$ of the participating species are summed over. In most
applications, $j=e^{-}$, but $j=\chi^{+}+\chi^{-}$ for a dark charge, with
$n_{j}=n_{\chi}=2n_{\chi^{\pm}}$ such that
$\omega_{\mathrm{p}j}^{2}=\omega_{\mathrm{p}\chi}^{2}=2\omega_{\mathrm{p}\chi^{\pm}}^{2}$.

Equation 4.2 has the solutions

$\omega=-\frac{i\sqrt{\pi}}{2}\bigg{(}\frac{\omega_{\mathrm{p}j}^{2}}{k\sigma_{T,j}}\bigg{)}\pm\frac{1}{2}\sqrt{-\pi\left(\frac{\omega_{\mathrm{p}j}^{2}}{k\sigma_{T,j}}\right)^{2}-4\left(\omega_{\mathrm{p}\chi}^{2}f_{\chi}\bigg{(}\frac{V_{b\chi}}{\sigma_{T,\chi}}\bigg{)}^{2}-c^{2}k^{2}\right)}.$
(4.3)

Keeping the Weibel mode (${\rm Re}[\omega]\approx 0$, $\rm{Im}[\omega]>0$) in
the limit that the second term under the radical is small yields

$\gamma^{\rm
W}=\frac{1}{\sqrt{\pi}}\left(\frac{k\sigma_{T,j}}{\omega_{\mathrm{p}j}^{2}}\right)\omega_{\mathrm{p}\chi}^{2}f_{\chi}\bigg{(}\frac{V_{b\chi}}{\sigma_{T,\chi}}\bigg{)}^{2},$
(4.4)

where $\gamma^{\rm W}={\rm Im}[\omega]$ and the superscript “W” stands for
“warm” and we have dropped the $c^{2}k^{2}$ term, valid when $ck\lesssim
ck_{\rm
max}\equiv\omega_{\mathrm{p}\chi}(f_{\chi})^{1/2}({V_{b\chi}}/{\sigma_{T,\chi}})$.
Numerically we find that instability goes away when not in this limit.
Evaluating at $k=k_{\rm max}$ leads to an estimate for the maximum growth rate
assuming $j=e^{-}$ of

$\displaystyle\gamma_{\rm max}^{\rm W}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\pi}}\left(\frac{\omega_{\mathrm{p}\chi}^{3}f_{\chi}^{3/2}\sigma_{T,j}}{c\omega_{\mathrm{p}j}^{2}}\right)\bigg{(}\frac{V_{b\chi}}{\sigma_{T,\chi}}\bigg{)}^{3},$
$\displaystyle\approx$ $\displaystyle 8.3\times
10^{-14}s^{-1}\left(\frac{[q_{\chi}/m_{\chi}]}{10^{-4}}\right)^{3}\left(\frac{n_{e}}{10^{-3}{\rm
cm}^{-3}}\right)^{1/2}\left(\frac{f_{\chi}}{0.5}\right)^{3/2}\bigg{(}\frac{V_{b\chi}}{\sigma_{T,\chi}}\bigg{)}^{3},$

where we have evaluated at values most applicable for our mDM Bullet Cluster
case (and the fiducial parameters are given in Table 2), and we remind the
reader that $[q_{\chi}/m_{\chi}]$ is the dark charge-to-mass ratio with
respect to the proton’s.555After the fact, one can verify that the
approximations that led to our expression for $\gamma_{\rm max}^{\rm W}$ hold.
For example, our approximation $Z(\xi)\approx i\sqrt{\pi}$ only holds for the
branch where ${\rm Im}[\xi]<1/{\rm Re}[\xi]$ (see Appendix A). While the
growth we found is purely imaginary solution so that ${\rm Re}[\xi]=0$ clearly
holds, keeping the next order in the expansion for $Z(\xi)$ results in ${\rm
Re}[\xi]/{\rm Im}[\xi]\sim\xi_{j}$, where for our application with $j=e$. Thus
since ${\rm Re}[\xi]\ll{\rm Im}[\xi]\ll 1$ for our solution, we are safely in
the desired limit. The growth in this warm limit scales with the dark matter
charge-to-mass to the cubic power, unlike the limits that follow where the
growth scales linearly.

##### Warm-cold limit, where electrons have $\xi_{e}\ll 1$ and the dark matter
$\xi_{\chi}\gg 1$:

There is a second limit that we find is applicable in the pre-Recombination
plasma – when the dark matter is cold and other species are warm. In order to
be cold in the unmagnetized case $\omega>k\sigma_{T,\chi}$, which using our
expressions for $k_{\rm max}$ and $\gamma_{\rm max}$ reduces to the condition
at the wavenumber of maximum growth that

$[q_{\chi}/m_{\chi}]\gtrsim(\sigma_{T,\chi}/V_{b\chi})^{2}(\sigma_{T,\chi}/\sigma_{T,j})f_{S}^{-1/2}.$
(4.6)

This inequality is satisfied in the early universe (before dark matter halo
formation) as the dark matter velocity dispersion, $\sigma_{T,\chi}$ is
essentially zero. In this case, Equation 4.1 reduces to

$c^{2}k^{2}-\omega^{2}-i\sqrt{\pi}\omega_{\mathrm{p}j}^{2}\bigg{(}\frac{\omega}{k\sigma_{T,j}}\bigg{)}+f_{\chi}\omega_{\mathrm{p}\chi}^{2}\left(\frac{V_{b\chi}k}{\omega}\right)^{2}=0,$
(4.7)

and the solution to the quartic equation in the limit that $c^{2}k^{2}$ and
$\omega^{2}$ terms are smaller than the two final terms is

$\gamma^{\rm
WC}\approx\left(\frac{f_{\chi}}{\sqrt{\pi}}\frac{\omega_{p\chi}^{2}}{\omega_{pj}^{2}}(k\sigma_{T,j})(kV_{b\chi})^{2}\right)^{1/3},$
(4.8)

where the solution is valid for $ck<ck_{\rm
max}^{\prime}\equiv\omega_{pj}(\sqrt{\pi}f_{\chi}(\omega_{p\chi}/\omega_{pj})^{2}V_{b\chi}^{2}/\sigma_{T,j}^{2})^{1/6}$.
Plugging in $k_{\rm max}^{\ \rm WC}$ into Equation 4.8 yields

$\displaystyle\gamma^{\rm WC}_{\rm max}$ $\displaystyle\approx$
$\displaystyle\omega_{pj}\pi^{-1/6}\left(f_{\chi}\right)^{1/2}\left(\frac{\omega_{p\chi}}{\omega_{pj}}\right)\left(\frac{V_{b\chi}}{c}\right),$
(4.9) $\displaystyle=$ $\displaystyle 1.2\times 10^{-1}{\rm
s}^{-1}~{}f_{\chi}^{1/2}\left(\frac{[q_{\chi}/m_{\chi}]}{10^{-2}}\right)\left(\frac{n_{e}}{10^{3}{~{}\rm
cm}^{-3}}\right)^{1/2}\left(\frac{V_{b\chi}}{40~{}{\rm km~{}s^{-1}}}\right),$
(4.10)

where we have evaluated at values most applicable for our CMB (see Table 2)
scenario described below (§ 4.3.1), the scenario that we find this warm-cold
limit applies.

##### Cold limit, where $\xi_{j}\gg 1$ for all participating species $j$:

Finally, let us consider the case where all species are cold, the limit which
yields the shortest instability times. In this limit
$Z(\xi_{j})\approx-1/\xi_{j}$ \+ ${\cal O}(1/\xi_{j})$ and the dispersion
relation given by Equation 4.1 reduces to:

$0=c^{2}k^{2}-\omega^{2}+\omega_{\mathrm{p}p}^{2}+\omega_{\mathrm{p}e}^{2}+\omega_{\mathrm{p}\chi}^{2}+f_{\chi}\frac{\omega_{\mathrm{p}\chi}^{2}}{2}\left(\frac{V_{b\chi}k}{\omega}\right)^{2},$
(4.11)

which has solutions

$\omega=\frac{\pm
1}{\sqrt{2}}\left((\omega_{\mathrm{p}p}^{2}+\omega_{\mathrm{p}e}^{2}+\omega_{\mathrm{p}\chi}^{2}+k^{2}c^{2})\pm\left[(\omega_{\mathrm{p}p}^{2}+\omega_{\mathrm{p}e}^{2}+\omega_{\mathrm{p}\chi}^{2}+k^{2}c^{2})^{2}+2f_{\chi}\omega_{\mathrm{p}\chi}^{2}V_{b\chi}^{2}k^{2}\right]^{1/2}\right)^{1/2}.$

The positive, purely imaginary root is the classical Weibel solution, which
has magnitude [26]:

$\gamma^{\rm
C}\approx\frac{f_{\chi}^{1/2}\omega_{\mathrm{p}\chi}V_{b\chi}k}{\sqrt{2}\left(\omega_{\mathrm{p}j}^{2}+c^{2}k^{2}\right)^{1/2}},$
(4.12)

which has a maximum of

$\gamma_{\rm max}^{\rm
C}\approx\omega_{\mathrm{p}\chi}f_{\chi}^{1/2}\frac{V_{b\chi}}{c}=2.8\times
10^{-14}\rm{s}^{-1}\left(\frac{\rho_{DM}}{0.1GeVcm^{-3}}\right)^{1/2}\bigg{(}\frac{[q_{\chi}/m_{\chi}]}{10^{-3}}\frac{V_{b\chi}}{4000~{}{\rm
km~{}s^{-1}}}\bigg{)},$ (4.13)

where the solution is valid for $|\xi_{j}|\gg 1$, which corresponds to wave
numbers much greater than

$\frac{\gamma^{\rm
C}(k=k_{\mathrm{max}})}{k_{\mathrm{max}}\sigma_{T,j}}=1~{}~{}\longrightarrow~{}~{}ck_{\mathrm{max}}\approx\Bigg{(}\bigg{(}\frac{f_{\chi}^{1/2}\omega_{\mathrm{p}\chi}V_{b\chi}}{\sqrt{2}\sigma_{T,j}}\bigg{)}^{2}-\omega_{\mathrm{p},j}^{2}\Bigg{)}^{1/2}.$
(4.14)

When $\omega_{\mathrm{p},j}^{2}$ is the dark matter plasma frequency, a real
$k_{\rm max}$ exists such that our solution holds. We find that this limit
applies for the dark-$U(1)$ Bullet Cluster case (§ 4.3.2).

We now detail how these growth rates apply to three astrophysical systems of
interest.

####  4.3.1 mDM streaming in the Early Universe

Dark matter is moving relative to baryons at the time of Recombination with an
RMS velocity difference of $40\;$km s-1 [34]. This Recombination-era dark
matter cannot exchange significant momentum with the baryons or else it would
disturb the percent level agreement between calculations in the $\Lambda$CDM
model and the CMB. Indeed, this observation has been used to constrain the DM
millicharge under the approximation of collisional momentum exchange [35, 36,
37, even if only $\sim 1\%$ of the DM is millicharged].

We find that plasma streaming instabilities are unlikely to improve on the CMB
constraints from direct collisions. The “CMB” row in Table 2 shows the
parameters used for our fiducial Recombination Era; the density of
$10^{3}$cm-3 corresponds to a redshift of $1700$. Figure 2 shows the growth
rates for these parameters, evaluating the Weibel growth rate in the warm-cold
limit (Equation 4.8) for $[q_{\chi}/m_{\chi}]=10^{-2}$ – a charge-to-mass
ratio easily ruled out by other constraints.666Unlike in the other physical
situations we consider, for this CMB case we do not show a direct numerical
evaluation of the full dispersion relation, Equation 4.1, however, we note
that we were able to find numerical solutions near $k\approx k_{\rm max}$,
verifying our analytic solution in this regime.  The bottom panel in Figure 2
shows that the warm-cold limit is applicable as $\xi_{e}\ll 1$ and
$\xi_{\chi}\gg 1$. Unfortunately, two-particle electron collisions damp the
waves excited by the Weibel instability before the instability can grow,
except for large charge-to-mass ratios that are ruled out with other methods.
For our choice of $[q_{\chi}/m_{\chi}]=10^{-2}$, the electron-electron
collision rate is $\Gamma_{\rm coll}^{ee}\sim 1\;$s-1 (shown by the dotted
horizontal line). The growth rate must be in excess of this damping rate in
order for the instability to grow, but Figure 2 shows that is not the case for
$[q_{\chi}/m_{\chi}]=10^{-2}$: The growth rate is a factor of few below the
collision rate at $k_{\rm max}$, the maximum wavenumber where our calculation
for the warm-cold growth rate applies (see Equation 4.9 and the preceding
text). We find that growth can happen faster than collisions at $k\sim k_{\rm
max}$ only for $[q_{\chi}/m_{\chi}]\gtrsim 0.1$. However, these large
$[q_{\chi}/m_{\chi}]$ that would be ruled out by our instability analysis are
already easily ruled out by studies that considered just collisions [35, 36,
37].

Figure 2: Millicharge DM instability growth and damping rates in pre-Recombination universe/CMB scenario calculated for $[q_{\chi}/m_{\chi}]=10^{-2}$ such that $\omega_{\mathrm{p}\chi}=7.9\times 10^{2}$ Hz, with all other relevant fiducial parameters listed in Table 2. Top panel: The black solid line show the warm-cold Weibel growth rate as a function of wavenumber(Equation 4.8). This analytic expression applies until approximately $k_{\rm max}$, indicated with the vertical dashed black line. The electron-electron collision rate $\Gamma_{\rm coll}^{ee}\sim 1$s-1 is shown with the dotted black line. At, $[q_{\chi}/m_{\chi}]=10^{-2}$ the collision rate is always larger than the growth rate, damping the waves excited by the instability before the instability has time to grow. Instabilities will only occur at somewhat larger values of $[q_{\chi}/m_{\chi}]$ than shown, values easily ruled out by other mDM bounds. Thus, accounting for collective effects does not strengthen mDM constraints from CMB observations. Bottom panel: Demonstration that the warm-cold limit used in the top panel applies. This limit holds when $|\xi_{\chi^{\pm}}|\gg 1$ and $|\xi_{e}|\ll 1$, which are both satisfied. system |  $V_{b,\chi}$ $[{\rm km~{}s^{-1}}]$ |  $\sigma_{T,\chi}$ $[{\rm km~{}s^{-1}}]$ |  $\sigma_{T,e}$ $[{\rm km~{}s^{-1}}]$ |  $n_{e}$ $[{\rm cm}^{-3}]$ | $f_{\chi}$
---|---|---|---|---|---
CMB | $4\times 10^{1}$ | ${\rm negligible}$ | $4.6\times 10^{2}$ | $1000$ | $1$
MW | $4\times 10^{2}$ | ${\rm variable}$ | $8.6\times 10^{3}$ | $10^{-4}$ | $0.5$
BC | $4\times 10^{3}$ | $(1-3)\times 10^{3}$ | $4.3\times 10^{4}$ | $2\times 10^{-4}$ | $0.5$
Table 2:  Fiducial parameters for the three astrophysical scenarios in which
dark matter streaming can occur relative to baryons or other dark matter: The
pre-Recombination universe as observed by the Cosmic Microwave Background
(CMB; § 4.3.1), the Milky-Way (MW; § 4.3.3), and the Bullet Cluster (BC; §
4.3.2). Here, $V_{b,\chi}$ is the relative velocity between the baryons and
the DM, $\sigma_{T,\chi}$ is the thermal velocity of the DM, $\sigma_{T,e}$ is
the thermal velocity of the electrons, and $n_{e}$ is the electron number
density of the systems. We discuss how our results depend on these choices.

####  4.3.2 Bullet Cluster

The Bullet Cluster (1E 0657-56) has famously been used to place some of the
strongest constraints on interacting dark matter models. It consists of two
galaxy clusters that are colliding, where the dark matter of the smaller
cluster has passed through the larger (as indicated by lensing observations),
whereas its gas has not passed through and is visibly shocking in the interior
(as indicated by X-ray emission). These clusters have velocity dispersions of
$\approx 2000$ km s-1 and are streaming through one another with velocity of
$\approx 4000$ km s-1 [38]. The $\approx 2000$ km s-1 is from adding in
quadrature and dividing by $\sqrt{2}$ the internal velocity dispersions of the
two clusters, which we estimate to be $1500~{}$km s-1 and $2200$ km s-1
calculated from estimates of their temperatures from X-ray observations [39]
and the virial relation for ionized gas
$\sigma_{T,\chi}^{2}=2k_{b}T/(0.59\,m_{p})$. Given the crudeness of
approximating the two-cluster system as having a single temperature in our
calculations, we consider the instability for the range $1000-3000\,$km s-1.
If the Weibel instability can operate, it would have coupled the dark matter
to the gas (or the dark matter to the dark matter in the case of a dark-$U(1)$
force) and would not be allowed to stream through itself, as is observed.
Thus, we can use this system to place constraints on mDM. We use the “BC”
parameters in Table 2 to evaluate the potential for instability.

mDM-induced instability: First, we consider the SM millicharge case, where the
system parameters are in the limit of the Weibel instability for which both
the SM particles and the dark matter are warm ($\xi_{j}\ll 1$). Using Equation
4.4 and Equation 2.1, we can solve for when the electron collision time is
longer than the growth rate of the instability, indicating the instability can
occur. We also solve the full Weibel dispersion (Equation 4.1) using our
numerical method detailed in Section 4.1, which agrees well with our derived
analytic solution given by Equation 4.4. This is illustrated in Figure 3. We
find that $[q_{\chi}/m_{\chi}]\gtrsim 10^{-4}$ is excluded if the plasma is
unmagnetized. However, galaxy clusters are known to have micro-Gauss large-
scale magnetic fields [40]. In § 4.4 we show that this allows for modes that
grow even faster, resulting in even stronger limits.

Figure 3:  Millicharge DM instability growth and damping rates for the Bullet
cluster, calculated for $[q_{\chi}/m_{\chi}]=10^{-4}$ such that
$\omega_{\mathrm{p}\chi}=3.5\times 10^{-3}$ Hz, and $\sigma_{T,\chi}=2000$ km
s-1, with all other relevant fiducial parameters listed in Table 2. Top panel:
The black solid line shows the warm-warm Weibel growth rate as a function of
wavenumber (Equation 4.4), the black points are the numerical solutions to the
full dispersion given by Equation 4.1. This analytic expression applies until
approximately $k_{\rm max}$, where we define $\gamma^{\rm W}_{\rm max}$
(Equation 4.3), indicated with the vertical dashed black line. The electron-
electron collision rate $\Gamma_{\rm coll}^{ee}$ is shown with the dotted
black line. At $[q_{\chi}/m_{\chi}]=10^{-4}$ the collision rate is just
slightly larger than the growth rate. Instabilities will only occur at
somewhat larger values of $[q_{\chi}/m_{\chi}]$ than shown, resulting in our
Bullet Cluster constrain $[q_{\chi}/m_{\chi}]\gtrsim 10^{-4}$. Bottom panel:
Demonstration that the warm-warm limit used in the top panel applies. The
$|\xi_{j}|=1$ horizontal line is shown in black for reference. Indeed, the
electrons and dark matter are warm with $|\xi_{j}|\sim 10^{-9}$ for the
fiducial parameters (see gray curve).

Dark-$U(1)$-induced instability: Next, we consider the case where there is a
dark-$U(1)$ charge with a massless dark photon, but no SM millicharge. This
case turns out to be much more constraining, as the fast motions of electrons
don’t damp the instability as in the SM millicharge case. In particular, we
find as long as $[q_{\chi}/m_{\chi}]\gtrsim 10^{-14}$, that there are always
modes that are able to grow and would reach a nonlinear amplitude in the age
of the system. This is shown in Figure 4, which considers a charge-to-mass of
$[q_{\chi}/m_{\chi}]=10^{-13}$, near our claimed constraint. The horizontal
dotted line is $10^{3}H_{0}$, the rate required to grow $100$ $e$-foldings in
the cluster crossing time, taken to be $0.1(H_{0})^{-1}$ or approximately a
billion years. (The size of the system divided by the infall velocity yields a
similar timescale.) The light blue, medium blue, and black solid lines are our
analytic cold-limit estimates for $\sigma_{T,\chi}=1000$, 2000, and 3000 km
s-1, respectively.

The light blue, medium blue, and black points correspond to the respective
velocities and are numerical solutions to the full dispersion relation. We
find these solutions using the method detailed in Section 4.1, with the
initial simplex for the numerical method started around
$\omega_{0}=\gamma^{\rm C}$, our derived analytic solution given in Equation
4.12. In our fiducial case where $\sigma_{T,\chi}$ = 2000 km s-1, we are at
the borderline of the cold limit with $|\xi_{\chi}|\sim 1$. For smaller values
of $\sigma_{T,\chi}$, we are able to reproduce the cold limit, finding
numerical solutions that are within $\sim 20\%$ of $\gamma^{\rm C}$ until
$k\approx k_{\rm max}$ (given by Equation 4.14), above which the cold limit,
$|\xi_{j}|\gg 1$, no longer holds. This is illustrated by the
$\sigma_{T,\chi}$ = 1000 km s-1 points in Figure 4, which fall on the
corresponding theory curve below $k_{\rm max}$. In the fiducial case where
$\sigma_{T,\chi}$ = 2000 km s-1, we are no longer in the cold limit and
$|\xi_{\chi}|\sim 1$ (see bottom panel of Figure 4), but we still find
numerical solutions which are within a factor of two of $\gamma^{\rm C}$. As
we push to larger values of $\sigma_{T,\chi}>2000$ km s-1, where
$|\xi_{\chi}|<1$, our numerical solution slowly moves away from $\gamma^{\rm
C}$. This shows that the growth rate is not very sensitive to the assumed cold
limit and, hence, our parameter choices. This is in contrast to the two-stream
(longitudinal) instability, which shuts off for
$\sigma_{T,\chi}/V_{b,\chi}>1/\sqrt{3}\approx 0.6$ [25]. The Bullet cluster
system is likely above this threshold.

####  4.3.3 the Milky Way system

We briefly comment on the likelihood of instability in the Milky Way halo and
other galactic-mass dark matter halos. In falling dark matter is likely more
unstable to Weibel than in the Bullet cluster system just considered because
the velocity dispersion of the satellites or unvirialzed DM falling onto the
Milky Way can be considerably smaller relative to the in fall velocity, making
the DM effective colder and, hence, more unstable. Thus, for the unmagnetized
case, we expect instability happens in the Milky Way in the SM mDM case for
$[q_{\chi}/m_{\chi}]>10^{-4}$ and for the dark-$U(1)$ charge for
$[q_{\chi}/m_{\chi}]>10^{-13}$. The constraints on the former case are even
much stronger when we include magnetic fields, as discussed in § 4.4, and
Milky Way-like dark matter halos may harbor $\sim 1\mu$G magnetic fields [41].

If instabilities occur in the Milky Way or other galactic mass halos, we
expect the dark matter to behave more like a gas. Behaving like a gas may
reduce the triaxiality of the dark matter halo, and it will also affect the
strong constraint on mDM from disk spin-down. Constraints from these
considerations are discussed in § 5.

Figure 4: The Bullet Cluster system Weibel instability growth rates for dark
matter charged under a dark-$U(1)$ with a massless dark photon. Calculations
use $[q_{\chi}/m_{\chi}]=10^{-13}$ and $\omega_{\mathrm{p}\chi}=3.5\times
10^{-12}$ Hz, considering our fiducial and more extreme velocity dispersion,
$\sigma_{T,\chi}$, for the dark matter in the merging systems; the extreme
values are chosen to show that our numerical solution slowly runs away from
our analytic solution with increasing velocity dispersion. All other
parameters used in the calculation are listed in Table 2. Top panel: The light
blue, medium blue, and black points are the numerical solutions of the full
Weibel dispersion relation Equation 4.1 with the $\sigma_{T,\chi}$ = 1000,
2000 and 3000 km s-1, respectively. The light blue, medium blue, and black
solid lines show the analytic solution given by Equation 4.12 in the cold
limit for the three respective $\sigma_{T,\chi}$. Our analytic approximation
overpredicts the growth rate by $\approx 20\%$ for $\sigma_{T,\chi}=1000$ km
s-1 up to $k\approx k_{\rm max}$ (Equation 4.14; the dashed vertical lines are
colored by the corresponding $\sigma_{T,\chi}$), where the cold approximation
used to derive our analytic solution no longer holds. In our fiducial case
where $\sigma_{T,\chi}$ = 2000 km s-1, we are no longer in the cold limit as
$|\xi_{\chi}|\sim 1$ but we continue to find solutions that are within a
factor of two of $\gamma^{\rm C}$. When $\sigma_{T,\chi}$ = 3000 km s-1 (and
for $\sigma_{T,\chi}$ > 2000 km s-1), $|\xi_{\chi}|<1$ and we find solutions
that slowly depart from $\gamma^{\rm C}$. Bottom panel: Evaluation of
$\xi_{\chi}$ at $\gamma^{\rm C}$ shown in light blue, medium blue, and black
solid lines corresponding to $\sigma_{T,\chi}$ = 1000 km s-1 2000, and 3000 km
s-1, respectively. The light blue, medium blue, and black points correspond to
$\xi_{\chi}$ evaluated at the numerical solution $\gamma^{\rm C}$ at a given
$k$. This demonstrates the values of $\sigma_{T,\chi}$ that are in the cold
limit and those which are not.

###  4.4 Weibel Instability with non-zero background magnetic field

We now turn to the case where there is a non-zero background magnetic field,
$B_{0}$. The presence of the magnetic field introduces magnetic degrees of
freedom and timescales (set by the Larmor frequencies of charged species, see
Table 1). Beyond this, magnetic fields make plasmas look colder perpendicular
to $B_{0}$, as in their presence, charged particles become confined; this
should help Weibel instabilities to grow on shorter time scales at a given
charge-to-mass ratio. Indeed, [22] showed that mDM-baryon streaming
instabilities in supernovae shocks can be hugely amplified in the presence of
magnetic fields. We note that while we consider the presence of magnetic
fields in the mDM case, we do not consider them in the dark-$U(1)$ case, as
there is no strong motivation for large scale magnetic fields there.

The magnetized Weibel instability has a homogenous field that is perpendicular
to the direction of propagation. The presence of a perpendicular field is a
bit counterintuitive for the problem at hand, as such a large-scale field
would not allow charged particles to stream. However, when the two systems are
streaming through each other (the most applicable situation being two clusters
falling into each other), sometimes the field will be parallel and sometimes
perpendicular. In the perpendicular case, without instability it is likely the
streaming will exert force on the perpendicular field lines that aligns them
with the direction of streaming, as there is so much energy in the streaming
dark matter. The presence of the Weibel instability would not allow this
realignment of the large-scale field, as it excites small scale fields that
couple the fluids and likely magnetic turbulence. The other case, where the
field is aligned with the streaming motion, is discussed in the § 4.5, where
we find similar growth rates to here.

With this setup, let us now consider the mathematics in various limits. In the
cold limit for all species $\xi_{j}\gg 1$ (and ignoring helium and metals), we
can expand the Weibel plasma dispersion relation, Equation 4.1:

$\begin{split}0=c^{2}k^{2}-\omega^{2}+\omega_{\mathrm{p}p}^{2}\left(\frac{\omega}{\omega\pm\Omega_{p}}\right)+\omega_{\mathrm{p}e}^{2}\left(\frac{\omega}{\omega\pm\Omega_{e}}\right)\\\
+\omega_{\mathrm{p}\chi}^{2}\left(\frac{\omega^{2}}{\omega^{2}-\Omega_{\chi}^{2}}\right)+f_{\chi}\omega_{\mathrm{p}\chi}^{2}V_{b\chi}^{2}k^{2}\frac{\omega^{2}+\Omega_{\chi}^{2}}{(\omega^{2}-\Omega_{\chi}^{2})^{2}},\end{split}$
(4.15)

The cold or marginally cold limit when $\xi\gtrsim 1$ is the only interesting
limit as the effect of magnetic fields is to make the plasma appear colder and
enhance instability. Let us additionally take the limit where
$\omega\ll\Omega_{b}$, as we anticipate Weibel-like solutions with small
growth rates, such that the sum over the baryonic species term reduces to:

$\begin{split}\sum_{b=i^{+},e^{-}}\omega_{\mathrm{p}b}^{2}\bigg{(}\frac{\omega}{\omega\pm\Omega_{b}}\bigg{)}\approx\sum_{b=i^{+},e^{-}}\frac{\omega_{\mathrm{p},b}^{2}\omega}{\Omega_{b}}\bigg{(}1-\omega/\Omega_{b}\bigg{)},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\\\
=\omega\bigg{[}\frac{\omega_{\mathrm{p},e}^{2}}{\Omega_{e}}\bigg{(}1-\frac{\omega}{\Omega_{e}}\bigg{)}+\frac{\omega_{\mathrm{p},i}^{2}}{\Omega_{i}}\bigg{(}1-\frac{\omega}{\Omega_{i}}\bigg{)}\bigg{]}=-\omega^{2}c^{2}\bigg{(}\frac{\sqrt{4\pi(\rho_{e}+\rho_{i})}}{B_{0}}\bigg{)}^{2}=-\frac{\omega^{2}c^{2}}{v_{A}^{2}},\end{split}$
(4.16)

where we have used that $q_{e^{-}}=-q_{i^{+}}$ and
${\omega_{\mathrm{p},j}^{2}}/{\Omega_{j}}=4\pi n_{j}q_{j}c/{B_{0}}$, where
$B_{0}$ is the homogeneous background magnetic field oriented along the
direction of the perturbation and orthogonally to the streaming motion.
Equation 4.15 becomes:

$\begin{split}0=c^{2}k^{2}-\omega^{2}\bigg{(}1+\frac{c^{2}}{v_{A}^{2}}\bigg{)}+\omega_{\mathrm{p}\chi}^{2}\left(\frac{\omega^{2}}{\omega^{2}-\Omega_{\chi}^{2}}\right)+f_{\chi}\omega_{\mathrm{p}\chi}^{2}V_{b\chi}^{2}k^{2}\frac{\omega^{2}+\Omega_{\chi}^{2}}{(\omega^{2}-\Omega_{\chi}^{2})^{2}}.\end{split}$
(4.17)

We now consider two limits of this equation. First, in the very low frequency
limit where $\omega\ll{}\Omega_{\chi}$, the previous equation becomes:

$\begin{split}0=c^{2}k^{2}-\omega^{2}\bigg{(}1+\frac{c^{2}}{v_{A}^{2}}\bigg{)}-\omega_{\mathrm{p}\chi}^{2}\left(\frac{\omega}{\Omega_{\chi}}\right)^{2}+\frac{f_{\chi}\omega_{\mathrm{p}\chi}^{2}V_{b\chi}^{2}k^{2}}{\Omega_{\chi}^{2}}\bigg{(}1+3\frac{\omega^{2}}{\Omega_{\chi}^{2}}\bigg{)},\end{split}$
(4.18)

which has solution

$\begin{split}\omega^{2}=\bigg{(}c^{2}+\frac{f_{\chi}\omega_{\mathrm{p}\chi}^{2}V_{b\chi}^{2}}{\Omega_{\chi}^{2}}\bigg{)}k^{2}/\bigg{(}1+\frac{c^{2}}{v_{A}^{2}}+\frac{\omega_{\mathrm{p}\chi}^{2}}{\Omega_{\chi}^{2}}-\frac{3f_{\chi}\omega_{\mathrm{p}\chi}^{2}V_{b\chi}^{2}k^{2}}{\Omega_{\chi}^{4}}\bigg{)}.\end{split}$
(4.19)

We can derive a criterion on the maximum wavenumber where our solution holds
by requiring $\xi_{j}=1$:

$\begin{split}k_{\rm
max}^{2}=\bigg{[}\bigg{(}1+\frac{c^{2}}{v_{A}^{2}}+\frac{\omega_{\mathrm{p}\chi}^{2}}{\Omega_{\chi}^{2}}\bigg{)}-\bigg{(}\frac{c^{2}}{\sigma_{T,j}^{2}}+\frac{f_{\chi}\omega_{\mathrm{p}\chi}^{2}V_{b\chi}^{2}}{\Omega_{\chi}^{2}\sigma_{T,j}^{2}}\bigg{)}\bigg{]}/\bigg{(}\frac{3f_{\chi}\omega_{\mathrm{p}\chi}^{2}V_{b\chi}^{2}}{\Omega_{\chi}^{4}}\bigg{)}.\end{split}$
(4.20)

Unfortunately, there are no modes which yield imaginary solutions to Equation
4.19 and simultaneously obey Equation 4.20. We conclude that this very low
frequency limit does not yield an interesting solution.

Next, we consider the opposite higher frequency limit where
$\Omega_{\chi}\ll\omega$. In this limit Equation 4.17 becomes

$\begin{split}0=c^{2}k^{2}-\omega^{2}\bigg{(}1+\frac{c^{2}}{v_{A}^{2}}\bigg{)}-\omega_{\mathrm{p}\chi}^{2}+\frac{f_{\chi}\omega_{\mathrm{p}\chi}^{2}V_{b\chi}^{2}k^{2}}{\omega^{2}},\end{split}$
(4.21)

which has solutions:

$\begin{split}\omega^{2}=\Bigg{(}(c^{2}k^{2}-\omega_{\mathrm{p},\chi}^{2})\pm\sqrt{(c^{2}k^{2}-\omega_{\mathrm{p},\chi}^{2})^{2}+4f_{\chi}\omega_{\mathrm{p},\chi}^{2}V_{\mathrm{b},\chi}^{2}k^{2}\bigg{(}1+\frac{c^{2}}{v_{A}^{2}}\bigg{)}}\Bigg{)}/\bigg{(}2(1+\frac{c^{2}}{v_{A}^{2}})\bigg{)},\end{split}$
(4.22)

which again yields the ordinary cold Weibel growth rate given by Equation 4.12
in the limit that the second term under the square root is small compared to
the $c^{2}k^{2}-\omega_{{\rm p},\chi}^{2}$ term and for the purely imaginary
root. Thus, this limit also does not yield an interesting solution.

Since we were unable to find interesting solutions in the previous limits, we
consider the marginally cold case where $\omega\sim i\Omega_{\chi}$.
Unfortunately, in this case a similar analytic approach is not possible, and
instead we are forced to search blindly for numerical solutions.

Of the situations we consider, the presence of a large-scale magnetic field is
most motivated for the Bullet Cluster, as one does not expect the
Recombination era gas to be strongly magnetized. We use $B_{0}=1\mu$G, as
observations show that cluster magnetic fields are at the $\mu$G level [40].
We initialize our numerical method as detailed in Section 4.1 around the
order-of-magnitude guess $\omega_{0}=i\Omega_{\chi}$. This guess is motivated
by the mDM-baryon streaming instability growth rates found in [22] in the case
of supernovae shocks. However, fortuitously, this guess does find interesting
solutions: We find solutions with growth rates exceeding the electron-electron
collision rate down to $[q_{\chi}/m_{\chi}]=10^{-11}$ (right upper panel of
Figure 5). However, when $[q_{\chi}/m_{\chi}]\leq 10^{-12}$ (upper left panel
of Figure 5), the electron-electron collision rate exceeds the growth rate of
the instability such that collisions will damp the instability. Thus, we find
$[q_{\chi}/m_{\chi}]\gtrsim 10^{-12}(B_{0}/\mu)$G is excluded if the plasma is
magnetized, seven orders of magnitude lower than when the mDM-plasma is
unmagnetized as detailed in Section 4.3.2.

Figure 5: Numerical evaluation of Weibel dispersion for the Bullet Cluster
system for the parameters in Table 2, at $\sigma_{T,\chi}=2000$ km s-1, and
assuming a background magnetic field with $B_{0}=1\mu$G as motivated by
cluster observations. The left panel assumes $[q_{\chi}/m_{\chi}]=10^{-12}$
such that $\omega_{\mathrm{p}\chi}=3.5\times 10^{-11}$Hz and
$\Omega_{\chi}=9.6\times 10^{-15}$Hz, and the right panel
$[q_{\chi}/m_{\chi}]=10^{-11}$ such that $\omega_{\mathrm{p}\chi}=3.5\times
10^{-10}$ Hz and $\Omega_{\chi}=9.6\times 10^{-14}$ Hz. Top panels: The black,
solid line corresponds to our initial guess frequency,
$\omega_{0}=i\Omega_{\chi}$, when solving the magnetized Bullet Cluster
dispersion case numerically. The numerical solutions are relatively stable
when wavenumber is between $0.1-10[\omega_{{\rm p},\chi}/$c] and peak with
maximum growth rates of $\gamma_{\rm max}\sim i\Omega_{\chi}$. When
$[q_{\chi}/m_{\chi}]=10^{-12}$, $\Gamma^{ee}_{\rm coll}$ is larger than
$\gamma_{\rm max}$, and the Weibel instability is unable to grow. However,
$\gamma_{\rm max}>\Gamma^{ee}_{\rm coll}$ for $[q_{\chi}/m_{\chi}]=10^{-11}$,
which sets our lower bound on the charge-to-mass ratio of $\chi$ for the
magnetized BC. This lower bound is approximately seven orders of magnitude
lower than the unmagnetized Bullet Cluster bound we set in § 4.3.2. Bottom
panels: Evaluation of $\xi_{j}$ at $\gamma_{\rm num}$ for all participating
species $j$. Since we find $|\xi_{j}|\lesssim 1$, the analytic approach taken
elsewhere would not work to find these solutions.

###  4.5 Firehose Instability

In the case where the mDM propagation direction is parallel to the background
magnetic field, ${\bf B}_{0}$ (i.e. ${\bf V}_{\rm{b},\chi}\parallel{\bf
B}_{0}$), we expect the enhancement of EM perturbations to give way to a beam-
fire hose instability. Physically, the firehose instability is induced by a
back reaction against the centrifugal force as charged baryons and dark matter
move along curving magnetic lines of force. The result of the instability is
likely plasma turbulence, with the beam coupled to the background plasma.

A small perturbative field ${\bf\delta B}$ redirects charged particles with a
Lorentz force and produces a drift current in the direction of
$\nabla\times{\bf B}$ [42]. In this case, the dispersion relation is given by

$0=D^{\pm}(k,\omega)=c^{2}k^{2}-\omega^{2}-\sum_{b=i^{+},e^{-}}\omega_{\mathrm{p}b}^{2}\bigg{(}\frac{\omega}{k\sigma_{T,b}}\bigg{)}Z(\xi_{b})-\sum_{s=\chi^{+},\chi^{-}}\omega_{\mathrm{p},s}^{2}\bigg{(}\frac{\omega-
kV_{b,\chi}}{k\sigma_{T,\chi}}\bigg{)}Z(\xi_{s})$ (4.23)

where $\xi_{b}=(\omega\pm\Omega_{b})/k\sigma_{T,b}$ and $\xi_{s}=(\omega-
kV_{b,\chi}\pm\Omega_{s})/k\sigma_{T,\chi}$ and $Z(\xi_{j})$ is the plasma
dispersion function defined in Appendix A.

We again consider the Bullet Cluster with a background magnetic field of
${B}_{0}=1\mu$G. In this case we considered only numerical solutions with
$\gamma\approx i\Omega_{\chi}$ as in [22]. Indeed, solving Equation 4.23
numerically, we find numerical solutions with $\gamma_{\rm max}\approx
i\Omega_{\chi}$ over orders of magnitude in $[q_{\chi}/m_{\chi}]$ down to
$[q_{\chi}/m_{\chi}]\approx 10^{-11}$. Below this, we find electron-electron
collisions, given by Equation 2.1 damp waves excited by the instability.

##  5 Effects of plasma instabilities on previous constraints

Some of the strongest constraints on mDM arise from the interaction with
magnetic fields in galaxies, possibly spinning down the galactic disk [2], and
magnetic fields in clusters, potentially resulting in different halo profiles
[43]. The plasma instabilities we have considered excite strong small-scale
magnetic fields, which may alter these constraints. We briefly comment on
these constraints in light of our results.

###  5.1 Halo Profile Constraints

Ambient magnetic fields could change the radial density profiles of galaxy
clusters for the case of mDM. Indeed, [43] claims the strongest constraints on
mDM by requiring the mDM gyro radius for observed cluster ambient magnetic
fields of $\sim 1\,\mu G$ exceed the $\sim 1$ Mpc size of a galaxy cluster.

Dark matter-baryon plasma instabilities, excited as dark matter falls onto the
galaxy, would increase the dark matter baryon coupling and perhaps strengthen
any deviations from the collisionless-limit NFW halo profiles. For example, if
the instability was induced, the ambient small-scale magnetic field strength
could increase causing the gyro radius, $r_{L\chi}$, to decrease further.
Then, to maintain the inequality $r_{L\chi}\gtrsim 1$ Mpc from [43], the
derived bound on $[q_{\chi}/m_{\chi}]$ would decrease. On the other hand, it
is unclear to us that even the hydrodynamic limit (small gyro radius) is ruled
out, as hydrodynamic-only simulations produce similar profile halos to NFW
[e.g. 44]. So, we avoid deriving constraints from halo radial profiles.

Another interesting constraint owes to the observed asphericity of galaxy
clusters. Simulations show that galaxy clusters should be more spherical if
the dark matter is interacting, which may be in conflict with observations
[45]. Even in the hydrodynamic limit of many interactions, the timescale for a
sound wave to cross the virialized system and restore isotropy is still the
dynamical time, the same timescale for a collisionless system. Thus, this
constraint relies on the ${\cal O}(1)$ difference in the exact anisotropy-
damping timescale for an interacting and truly collisionless system. Whether
constraints apply to saturating plasma instabilities, which for example may
not be exactly the hydrodynamic limit, is unclear. Additionally, for mDM,
anisotropy could be further enhanced by astrophysical feedback, which will
have more of an effect on the dark matter since the dark matter is more
coupled to the gas. This enhanced coupling could perhaps even play a role in
the coring of dwarf galaxy halos and the cusp-core problem [e.g 46].

###  5.2 Disk spin-down constraints

It was argued in [2] that as mDM passes through the interstellar medium (ISM)
disk of spiral galaxies, the mDM will be deflected by embedded ordered
magnetic fields. In particular, it is argued that as mDM passes through the
disk of a spiral galaxy, it will be deflected by the magnetic fields and thus
angular momentum will be exchanged between the rotating ISM and the slowly
rotating mDM halo. This exchange would then cause the entire ISM disk to
spiral inwards, in contrast with observations, allowing for constraints to be
placed on the charge-to-mass ratio of mDM. Stebbins [2] considered the case of
ballistic trajectories into the Milky Way disk, deriving the tight bound

$\frac{q_{X}}{e}\lesssim
10^{-12}\left(\frac{m_{X}}{m_{p}}\right)\left(\frac{v}{300{\rm~{}km~{}s^{-1}}}\right)\left(\frac{\mu
G}{B}\right)\left(\frac{\rm kpc}{\ell_{B}}\right),$ (5.1)

where $\ell_{B}$ is coherence length of galactic magnetic field. This bound is
roughly set by the dark matter having a Larmor radius comparable to the width
of the Galactic disk for a $\sim\mu$G Galactic magnetic field.

If plasma instabilities couple the dark matter to the gas within halos, the
natural saturation of such instabilities is likely with the magnetic field
coming into kinetic energy equipartition with the gas, which in the Milky Way
halo results in $B\sim\sqrt{6\pi nkT_{\rm vir}}\sim 1\mu G$. (Equipartition
magnetic fields may also be generated by astrophysical processes.) The dark
matter then diffuses through the resulting magnetic inhomogeneities, rather
than traveling ballistically through the Galactic disk as assumed in [2]. The
Weibel instabilities excites small scale magnetic inhomogeneities, below the
Larmor radius of the dark matter. The later nonlinear stages of the
instability likely lead to turbulence and a very inhomogeneous magnetic field,
and we assume there is some component that points in different directions on
the scale of the dark matter Larmor radius. In this circumstance, a simple
picture for how the dark matter travels takes that the dark matter scatters in
a different direction each time it traverses its gyro-radius leading to the
estimate $D_{\chi}\sim v_{\chi}r_{L\chi}$ for the diffusion coefficient, where
$r_{L\chi}=m_{\chi}cv_{\chi}/[q_{\chi}B]$ is the DM gyro-radius. Plugging in
values yields

$D=v_{\chi}r_{L\chi}=1.4\times 10^{-2}{\rm~{}kpc}^{2}\;{\rm
Gyr}^{-1}\left(\frac{v_{X}}{200{\rm~{}km~{}s^{-1}}}\right)^{2}\left(\frac{[q_{\chi}/m_{\chi}]}{10^{-8}}\right)^{-1}\left(\frac{B}{1~{}\mu
G}\right)^{-1}.$ (5.2)

The mechanism of [2] uses that the ISM baryons in the galactic disk transfers
its angular momentum to the halo. In their case, they considered ballistic
trajectories for which dark matter travels from far out in the halo into the
disk. Instead, especially if instabilities like those considered in this paper
occur (although a tangled magnetic field from astrophysical processes can do
the same), the dark matter diffuses through the halo, exchanging momentum
between adjacent regions and bringing them into solid body rotation. For
substantial spin-down of the disk, we need $\sim 10\%$ of the dark matter to
come into solid body rotation with the disk, since $\sim 10$ percent of the
mass in our halo is in the disk. Very roughly, this requires the inner $R\sim
30~{}$kpc to come into solid body rotation (i.e. assuming a $r^{-2}$ profile
that extends out to $300$kpc), which occurs over a timescale of

$\tau_{\rm solidbody}\sim\frac{R^{2}}{D}=70\;{\rm
Gyr}\;\left(\frac{R}{30~{}{\rm
kpc}}\right)^{2}\left(\frac{v_{\chi}}{200{\rm~{}km~{}s^{-1}}}\right)^{-2}\left(\frac{[q_{\chi}/m_{\chi}]}{10^{-11}}\right)\left(\frac{B}{1~{}\mu
G}\right)$ (5.3)

Equating this with the $\sim 10~{}$Gyr age of the Milky Way and evaluating at
our fiducial values of $R\sim 30~{}{\rm kpc}$ and $v_{\chi}\sim 200$ km s-1,
millicharges with $[q_{\chi}/m_{\chi}]\gtrsim 10^{-12}$ may not be ruled out
by disk spin-down. We conclude that much of the parameter space ruled out by
[2] could be affected by halo magnetic fields leading to the dark matter
taking more diffusive trajectories and, hence, slowing spin-down.

##  6 Discussion

Figure 6: mDM bounds in the $q_{\chi}$–$m_{\chi}$ plane. Above the white and
black solid lines, the mDM charge-to-mass ratio $[q_{\chi}/m_{\chi}]$ is
sufficient to drive Weibel instabilities in the Bullet Cluster when the
magnetic field is 0 and 1 $\mu$G, respectively. Since 1 $\mu$G is inline with
observations, we consider the latter much more stringent constraint to be our
most realistic and, hence, the pink shaded region indicates our exclusion
region. Above the upper dashed, gray line, Stebbins [2] argues
$[q_{\chi}/m_{\chi}]$ is robustly ruled out as the mDM would spin-down the
Milky Way ISM. In Section 5.2, we conclude that when
$[q_{\chi}/m_{\chi}]\gtrsim 10^{-12}$, mDM could be affected by halo magnetic
fields, leading to mDM taking diffusive, rather than ballistic trajectories as
assumed in [2], pushing their constraints below the black solid line to the
gray dashed line. The solid teal line represents the parameter space for
freeze-in production of mDM, where annihilation of SM particles gives rise to
the abundance after inflation and before matter-radiation equality [9, 12].
The gray shaded region shows previous constraints on the existence of mDM by
the SLAC mDM experiment [47], stellar cooling [48], and supernova 1987a [49].
Additional constraints are shown from CMB decoupling [50, 51], and electron
and nuclear recoil direct detection in Xenon10/100 [52, 53] and Xenon1Tc,
respectively. Figure 7: Bounds in the $q_{\chi}$–$m_{\chi}$ plane on dark
matter that is $U(1)$ charged with a massless dark photon. In the pink shaded
region above the solid black electromagnetic line, the dark-$U(1)$ DM charge-
to-mass ratio is sufficient to drive Weibel instabilities in the Bullet
cluster system as detailed in section 4.2. This instability was considered
previously in [1] in the cold limit and we showed that it still grows
sufficiently fast in the warm limit. Also shown is the electrostatic
instability growth rate [1], which can grow faster by a factor of
c/$V_{\rm{b},\chi}$, but we argue is a less robust constraint in the Bullet
cluster. The gray shaded Coulomb region represents the parameter space where 2
$\rightarrow$ 2 Coulomb collisions would have a significant impact in DM halos
as detailed in [1]. The solid gray shaded region is disfavored by kinetic
decoupling constraints detailed in [19]. While our solutions considered a
massless dark photon, we note that these results should hold for a massive
photon whose Compton wavelength is larger than the length scale of the
instability, $\sim c/\omega_{p\chi}$.

Our main findings are summarized in Figures 6 and 7 along with constraints
derived in previous works. Starting with mDM constraints in Figures 6, the
solid teal line in Figure 6 shows the required mDM charges and masses for
freeze-in production of all the dark matter. Freeze-in is a scenario where the
DM is created from the byproducts of annihilation of SM particles after
inflation and before matter-radiation equality [9, 12]. The dark gray regions
in Figure 6 represent collected bounds on mDM production. Bounds from a
dedicated beam dump preformed at SLAC are shown for charges
$q_{\chi}>10^{-5}q_{e}$ [47]. Stellar energy losses due to the emission of mDM
pairs by plasmon decay would have reduced the neutrino pulse of SN1987A, owing
to constraints derived in [49]. Low mass constraints from anomalous cooling
from the emission of mDM from white dwarves, red giants, and horizontal branch
stars are shown and labeled as stellar cooling [48]. Additional constraints
are shown from CMB decoupling [50, 51], and electron and nuclear recoil direct
detection in Xenon10/100 [52, 53] and Xenon1Tc, respectively. Above the gray
dashed line, Stebbins [2] argues $[q_{\chi}/m_{\chi}]$ is robustly constrained
by ballistic mDM, ISM spin-down. However, in Section 5.2, we conclude that
when $[q_{\chi}/m_{\chi}]\gtrsim 10^{-12}$, mDM could be affected by halo
magnetic fields, leading to mDM taking diffusive, rather than ballistic,
trajectories as assumed in [2], pushing the constraints from Milky Way spin-
down to the band between the dashed gray line and the solid black line. Above
the white solid line, we show our weaker unmagnetized-Weibel constraint
derived earlier in this work (§ 4.3.2) in the absence of a magnetic field. Our
more realistic, magnetized exclusion region from electromagnetic streaming
instabilities derived are indicated by the pink shaded region above black
solid lines, where the mDM charge-to-mass ratio $[q_{\chi}/m_{\chi}]$ is
sufficient to drive Weibel instabilities in the Bullet Cluster when the
magnetic field is 1 $\mu$G, consistent with observations (§ 4.4).

Now turning to dark-$U(1)$ constraints detailed in Figure 7. The gray shaded
regions show constraints from $2\rightarrow 2$ Coulomb collisions that would
significantly impact DM halos [1]. The shaded region labeled “decoupling” show
constraints from [19] derived by considering kinetic decoupling. Specifically,
the authors obtain a temperature of kinetic decoupling, $T_{\rm kin}$, by
equating the Hubble rate to the Compton scattering rate for the dark plasma.
They then require that decoupling happens above $T_{\rm kin}>640$ eV, so that
DM-dark radiation coupling only significantly influences CMB multipoles above
$l>2500$. The pink shaded region above the solid black Electromagnetic Bullet
Cluster line represents our derived constraint on the dark-$U(1)$ DM charge-
to-mass ratio that is sufficient to drive unmagnetized Weibel instabilities in
the Bullet cluster system. These constraints corroborate the earlier bounds of
[1, 29], and further serve to confirm these results by showing that the growth
rate of the instability in the cold limit (Equation 4.12) is a reasonable
approximation for the numerical parameters consistent with the Bullet Cluster
(e.g. values listed in Table 2). This is even though the Bullet cluster
systems parameters is only at the borderline of the cold limit. We further
showed that even for higher velocity dispersions than seem reasonable for this
system, the growth rates do not change so significantly, indicating the
robustness of the linear Weibel instability to the assumed parameters.

However, the strongest constraints on dark-$U(1)$ DM come if electrostatic
instabilities are able to couple the counter-streaming momenta. Electrostatic
instabilities grow faster than the electromagnetic instabilities by a factor
of $c/V_{b,\chi}$, or $\sim 1000$ in the Bullet cluster system. An estimate
for the parameter space ruled out by electrostatic instabilities assuming
$100$ efoldings charge-to-mass ratios is given by the black line shown in
Figure 7 [1]. However, electrostatic instabilities, like the two-stream,
generally saturate by flattening the bump on the tail of the distribution.
Such saturation may occur before substantial momentum exchange [e.g. 25]. In
addition, such instabilities will not grow at all if
$\sigma_{T,\chi}/V_{b,\chi}>0.57$ [25], where the Bullet cluster system is
near equality. Thus, the electromagnetic Weibel instability considered here
may be more robust.

Finally, we briefly discuss the general case when the dark matter has both a
dark and millicharge. The dispersion relations are worked out for this
possibility in Appendix B. We find that for the purely dark Weibel dispersion
relation to apply, as we had considered in § 4.3.2, the ratio of the dark
matter’s dark charge to its SM charge, $Q$, must satisfy $Q\gg 1$. Thus, our
dark Bullet Cluster constraint is only valid if the dark-$U(1)$ charge is much
greater than the SM millicharge. On the other hand, our mDM constraints are
only valid when the dark charge is either zero or very small, such that $Q\ll
1$. Zero is most likely case since mDM models where the millicharge is created
by a slight mixing with a dark charge should have a significant $Q$. See
Appendix B for more discussion.

##  7 Conclusions

We derived new and corroborated previous bounds on mDM and dark-$U(1)$ DM with
an extremely light dark photon by considering plasma streaming instabilities.
These instabilities act to couple the momenta of counter-streaming plasmas,
likely resulting in the dark matter becoming unable to penetrate either
baryonic plasma or another dark matter plasma. Such coupling would conflict
with cosmological situations where the dark matter is observed to stream
through the baryons or other dark matter. Landau damping likely highly damps
electrostatic instabilities when the DM is millicharged, owing to the fact
that the streaming velocity dispersion of particles is comparable to their
streaming motion for the cosmological situations where such counter-streaming
may occur. When DM instead has a dark-$U(1)$ charge, electrostatic
instabilities are not prohibited by Landau damping and grow faster than
electromagnetic instabilities [1], although electromagnetic instabilities may
be more likely to couple the counter-streaming plasmas’ momenta. Thus, we
focused on electromagnetic instabilities.

We considered three physical situations where dark matter is streaming
relative to itself or other dark matter.

  1. 1.

The first is the cosmic microwave background, where the baryons and dark
matter are moving relative to each other at Recombination. However, we found
that electron two-particle collisions suppress streaming instabilities in this
situation. Thus, the streaming of dark matter in the pre-Recombination
universe is affected more strongly by direct two-particle collisions rather
than collective processes, validating previous constraints.

  2. 2.

The second situation was the Bullet Cluster, where the dark matter in one
cluster is streaming through the dark matter and baryons in the other. First,
considering the instructive case of the Weibel instability in an unmagnetized
plasma with mDM, the properties of the Bullet Cluster merger and other merging
cluster systems would result in fast enough growth of the linear instability
to likely couple the momenta if $[q_{\chi}/m_{\chi}]\gtrsim 10^{-4}$. For
smaller $[q_{\chi}/m_{\chi}]$, electron collisions damp the waves excited by
the instability. When a homogeneous magnetic field is considered with
amplitude consistent with cluster observations, both the Weibel and Firehose
instabilities have much larger growth rates of approximately the cyclotron
frequency owing to the new magnetic waves that can be excited. This more
realistic magnetized case results in our final constraint being
$[q_{\chi}/m_{\chi}]\gtrsim 10^{-12}$.

  3. 3.

Finally, we considered a dark-$U(1)$ charge and the mutual coupling of the
counter-streaming dark matter. We find that the Weibel instability is likely
fast enough to grow to saturation in the age of the Bullet cluster system for
$[q_{\chi}/m_{\chi}]\gtrsim 10^{-14}$, in agreement with [1]. Our work
extended the study of Lasenby [1] and Ackerman [29] by verifying that the cold
approximation for electromagnetic Weibel growth still roughly holds despite
the parameters of this system being only at the borderline of this limit. We
found that as we pushed to increasingly larger values of $\sigma_{T,\chi}$
above $2000$ km s-1 where $|\xi_{\chi}|<1$ such that the cold limit does not
hold, our numerical solution only slowly moved away from the cold growth rate,
which indicates the robustness of the linear Weibel instability. While our
solutions considered a massless dark photon, our results should hold for a
massive photon whose Compton wavelength is larger than the length scale of the
instability, $\sim c/\omega_{p\chi}$.

The final physical situation we considered is the Milky Way and its
surrounding halo. We showed that the previous constraints from Milky Way ISM
spin-down of [2] are weakened in a scenario where plasma instabilities, or
even just tangled background magnetic fields, couple mDM to the gas within a
dark matter halo. If such coupling happens, the dark matters propagation is
likely to instead be diffusive, instead of traveling on ballistic
trajectories, making spin-down timescales longer than the age of the Milky Way
unless $[q_{\chi}/m_{\chi}]\gtrsim 10^{-12}$. We further discussed whether
other observations of dark matter halos may rule out such diffusive
propagation.

We conclude with one caveat to our analysis. We assumed that if the linear
growth rate of a considered instability is below the collision rate that damps
its excited waves, then the instability grows until it saturates with a
significant enough magnetic field to deflect particles. For the Weibel
instability, many studies have found that it saturates when it grows to large
enough amplitudes to deflect the particles participating on the small scale of
the excited modes, and such a saturation mechanism makes physical sense in the
context of this purely growing instability. However, since our study considers
scenarios far from the baryonic case that has been studied in detail, it is
possible that nonlinear parasitic processes are able to damp these slowly
growing instabilities before they reach large enough magnetic fields for
deflection. More work is required to address this possibility.

##  8 Acknowledgments

AC was partially supported by the National Science Foundation Graduate
Research Fellowship under Grant No. DGE-1762114. The authors would like to
thank Robert Lasenby for comments on an early draft of this work.

##  Appendix A Plasma dispersion function and expansions

For Maxwellian distributions functions (and “drifting” Maxwellians), it is
useful to define the plasma dispersion function,

$Z(\xi_{j})\equiv\frac{1}{\sqrt{\pi}}\int_{-\infty}^{\infty}\frac{e^{-x^{2}}}{x-\xi_{j}}$
(A.1)

The plasma dispersion function is the Hilbert transform of a Gaussian. It is
useful to express this function in its limiting cases:

$Z(\xi_{j})=i\sqrt{\pi}\mathrm{exp}(-\xi_{j}^{2})-2\xi_{j}+\frac{4}{3}\xi_{j}^{3}-\frac{8}{15}\xi_{j}^{5}+\dots,\hskip
14.22636pt\mathrm{\hskip 14.22636ptfor\hskip 5.69046pt}|\xi_{j}|<1$ (A.2)

and

$Z(\xi_{j})=i\sigma\sqrt{\pi}\mathrm{exp}(-\xi_{j}^{2})-\frac{1}{\xi_{j}}-\frac{1}{2\xi_{j}^{3}}-\frac{3}{4\xi_{j}^{5}}+\dots,\hskip
14.22636pt\mathrm{\hskip 14.22636ptfor\hskip 5.69046pt}|\xi_{j}|>1$ (A.3)

where $\xi_{j}=x+iy$ and

$\sigma=\left\\{\begin{array}[]{ccc}0&&y>1/|x|;\\\ 1&&|y|<1/|x|;\\\
2&&y<-1/|x|.\end{array}\right.$

These expansions are used in evaluating the limiting growth rates.

##  Appendix B Dark matter with dark and millicharge

When the dark matter has both a dark and millicharge, the dispersion relation
for the right and left circular electromagnetic modes is even more complicated
but can be calculated from the determinant of the dielectric tensor to find
modes that satisfy $D_{ij}E_{j}=0$, where $\vec{E}_{\rm
all}=(E^{+},E^{-},{\cal E}^{+},{\cal E}^{-})$ where ${\cal E}$ is the dark
electric field in the left and right circular polarization basis and

$\mathbf{D}=\begin{pmatrix}\tilde{\omega}^{2}-k^{2}(S^{+}_{\rm
SM}+S^{+}_{\chi})&0&-k^{2}QS^{+}_{\chi}&0\\\
0&\tilde{\omega}^{2}-k^{2}(S^{-}_{\rm
SM}+S^{-}_{\chi})&0&-k^{2}QS^{-}_{\chi}\\\
-k^{2}QS^{+}_{\chi}&0&\tilde{\omega}^{2}-k^{2}Q^{2}S^{+}_{\chi}&0\\\
0&-k^{2}QS^{-}_{\chi}&0&\tilde{\omega}^{2}-k^{2}Q^{2}S^{-}_{\chi}\end{pmatrix}$
(B.1)

where $\tilde{\omega}^{2}=c^{2}k^{2}-\omega^{2}$. To satisfy our equation
$D_{ij}E_{j}=0$ for some $E_{j}$ requires $0=\det(\mathbf{D})$, which reduces
to the dispersion relation

$0=\left[c^{2}k^{2}-\omega^{2}-k^{2}(S^{\pm}_{\rm
SM}+S^{\pm}_{\chi})\right]\left[c^{2}k^{2}-\omega^{2}-k^{2}Q^{2}S^{\pm}_{\chi}\right]-k^{4}Q^{2}(S^{\pm}_{\chi})^{2},$
(B.2)

where the $S^{\pm}$ can be identified from our dispersion relations for the
Weibel and Firehose cases (Equations 4.1 and 4.23), as these previous
dispersion relations are derived from the determinant of the upper $2\times 2$
quadrant of $\mathbf{D}$, and the $Q$ is the ratio of the dark matter’s dark
charge to its SM charge. Note that the dark electric field terms have the same
polarization for the $\xi$ as the particle paths are still shaped by the
background SM magnetic field. (If the dark matter has no SM charge, then
$\Omega_{\chi}=0$ and this appendix is not relevant.) Equation (B.2) has the
familiar form of the dispersion relation we used for the millicharged case,
times that for the dark-$U(1)$ case, but then minus a cross term
$k^{2}Q^{2}S^{-}_{\chi}$. If we can neglect this cross term, then we can
safely consider the limit of millicharged or dark instabilities, as done in
the main paper. This of course can be done for $Q=0$ but also the limit of
$Q=\infty$.777To take the $Q=\infty$ limit, note $Q^{n}S^{\pm}_{\chi}$ goes to
zero for $n<2$ and is finite for $n=2$ such that Equation B.2 reduces to
multiplying two independent dispersion relations, one for the dark matter and
the other for the baryons.

For the Weibel $S^{\pm}$, the dispersion relation becomes

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\Bigg{(}c^{2}k^{2}-\omega^{2}-\sum_{b=i^{+},e^{-}}\omega_{pb}^{2}\big{(}\frac{\omega}{k\sigma_{T,b}}\big{)}Z(\xi_{b}^{\pm})-\sum_{s=\chi^{+},\chi^{-}}(\omega^{\rm
SM}_{ps})^{2}\bigg{[}\bigg{(}\frac{\omega}{k\sigma_{T,s}}\bigg{)}Z(\xi_{s}^{\pm})$
$\displaystyle+$ $\displaystyle
f_{\chi}\bigg{(}\frac{V_{b\chi}}{\sigma_{T,s}}\bigg{)}^{2}(1+\xi_{s}^{\pm}Z(\xi_{s}^{\pm}))\bigg{]}\Bigg{)}\Bigg{(}c^{2}k^{2}-\omega^{2}$
$\displaystyle-$ $\displaystyle\sum_{s=\chi^{+},\chi^{-}}(\omega^{\rm
dark}_{ps})^{2}\bigg{[}\bigg{(}\frac{\omega}{k\sigma_{T,s}}\bigg{)}Z(\xi_{s}^{\pm})+f_{\chi}\bigg{(}\frac{V_{b\chi}}{\sigma_{T,s}}\bigg{)}^{2}(1+\xi_{s}^{\pm}Z(\xi_{s}^{\pm}))\bigg{]}\Bigg{)}$
$\displaystyle-$ $\displaystyle\left(\sum_{s=\chi^{+},\chi^{-}}\omega^{\rm
dark}_{ps}\omega^{\rm
SM}_{ps}\bigg{[}\bigg{(}\frac{\omega}{k\sigma_{T,s}}\bigg{)}Z(\xi_{s}^{\pm})+f_{\chi}\bigg{(}\frac{V_{b\chi}}{\sigma_{T,s}}\bigg{)}^{2}(1+\xi_{s}^{\pm}Z(\xi_{s}^{\pm}))\bigg{]}\right)^{2},$

where we now distinguish between the dark and SM plasma frequencies for the
dark matter with superscripts ‘SM’ and ‘dark’. This has the form of the
dispersion relationship we solved for the millicharged case times the
dispersion relation we assumed for the dark case, minus a cross term.

For a purely dark instability to hold we require $\gamma_{\rm
max}^{2}\gg[k^{2}S_{\chi}]_{\rm max}$ where ‘max’ indicates that we are
checking at the fastest growing mode where $k\sim\omega_{p\chi}/c$ for our
unmagnetized Weibel, such that the zero is approximated by the solution to the
dark case. This reduces to the intuitive condition $Q\gg 1$ for the
unmagnetized cold Weibel. Thus, only when the dark charge is much greater than
the SM millicharge is our dark Bullet Cluster constraint valid. Similarly,
$\gamma_{\rm max}^{2}\gg Q^{2}[k^{2}S_{\chi}]_{\rm max}$ for the SM
electromagnetic instability. This condition can be used to put limits on what
dark charges are allowed to not damp our mDM instability, the most relevant
case being the magnetized Weibel. While we do not investigate this condition
in detail, our mDM constraints are only valid with the dark charge is either
zero or very small such that $Q\ll 1$. Since zero is most likely since mDM
models where the millicharge is created by a slight mixing with a dark charge
generate a significant $Q$, we consider our mDM constraints only valid for
dark matter with pure SM charge.

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