Astrophysical Plasma Instabilities induced by Long-Range Interacting Dark Matter

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.

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]

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.

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]

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.

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.

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.

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.

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.