#  The impact of electric currents on Majorana dark matter at freeze out

Lukas Karoly  David C. Latimer  Department of Physics, University of Puget
Sound, Tacoma, WA 98416-1031

###### Abstract

Thermal relics with masses in the GeV to TeV range remain possible candidates
for the Universe’s dark matter (DM). These neutral particles are often assumed
to have vanishing electric and magnetic dipole moments so that they do not
interact with single real photons, but the anapole moment can still be
nonzero, permitting interactions with single virtual photons. This anapole
moment allows for p-wave annihilation of DM into standard model particles, and
the DM interacts with external electric currents via the anapole moment.
Moving beyond their static electromagnetic properties, these particles
generically have non-zero polarizabilities which mediate interactions with two
real photons; in particular, spin-dependent polarizibilities admit s-wave
annihilation of the DM into two photons. When the Universe cools from a
temperature on the order of the DM mass to freeze out, the DM is in thermal
equilibrium with the background plasma of particles, but the comoving DM
density decreases due to annihilation. If a collection of initially
unpolarized DM particles were subjected to an electric current, then the DM
medium would become partially polarized, according to the Boltzmann
distribution, with a slight excess of anapole moments aligned with the
current, relative to those anti-aligned. For this region of partially
polarized DM particles, the s-wave annihilation mode becomes partially
suppressed because it requires a state of vanishing angular momentum. As a
consequence, the decreased DM annihilation rate in this region will result in
an excess of DM density, relative to an unpolarized region, as DM drops out of
thermal equilibrium. We explored this relative change of DM density for DM
that is subjected to an electric current through freeze out.

##  I Introduction

A concordance of observations point to a universe whose matter content is
overwhelmingly comprised of some new type of particles, outside the standard
model R. L. Workman, et al. (2022). The constraints on this new matter are
few: it must be non-relativistic, stable, and relatively weakly interacting. A
recent history of dark matter, including a discussion of various dark matter
(DM) models, can be found in Ref. Bertone and Hooper (2018). Early on, one
class of DM models, weakly interacting massive particles (WIMPs), was
theoretically well motivated, in part because of its potential tie in with
supersymmetry Jungman et al. (1996). Additional motivation for WIMPs stemmed
from a coincidence often called the “WIMP miracle.” If DM were a thermal
relic, then weak-scale masses and annihilation cross sections for the DM
candidate naturally result in the correct relic density of DM observed today.
This coincidence is compelling, but we note that a much wider class of
WIMPless models can satisfy the same relic density constraint, e.g. Ref. Feng
and Kumar (2008).

Despite a multimodal approach to DM detection, no definitive DM signal yet
exists, though some observations show tantalizing hints. Perhaps because of
theoretical prejudice, many direct DM detection experiments focus upon the
WIMP parameter space, broadly construed to include DM masses between the GeV
scale up to a few TeV. Aside from a few exceptions Bernabei et al. (2008,
2010); Aalseth et al. (2011, 2013); Agnese et al. (2013), decades of direct
detection experiments have found no evidence for DM, and as a result, strict
limits on the DM-nucleus interaction cross section Akerib et al. (2017);
Aprile et al. (2018); Meng et al. (2021) have ruled out many WIMP DM models.
In addition to direct detection experiments, particle colliders also place
stringent constraints on DM models. In particular, the LHC has produced no
particles outside the standard model, making tenuous the notion that DM is
comprised by supersymmetry’s neutralino Peskin (2015). One other tack to
assess the presence of dark matter is via indirect detection experiments in
which telescopes search for high energy cosmic rays or photons. If signals
cannot be attributable to standard astrophysical sources, then they may be due
to DM annihilation. In some instances, this results on constraints on the DM
annihilation cross section, as with the observations of dwarf spheroidal
galaxies from the Fermi Large Area Telescope (Fermi-LAT) for DM masses below
100 GeV Albert et al. (2017), while Fermi-LAT observations of the Galactic
Center hint at the possibility of a DM annihilation Hooper and Goodenough
(2011). Additionally, observations of antiprotons in the AMS-02 detector
Aguilar et al. (2016) could also signal DM annihilation for a DM mass around
80 GeV Cuoco et al. (2017).

Because of the severe constraints imposed by direct detection experiments and
particle colliders, axion dark matter models are, perhaps, eclipsing WIMP
models in terms of their favorability Marsh (2016), but the WIMP paradigm is
not entirely dead because there is still viable parameter space remaining
Roszkowski et al. (2018); Arcadi et al. (2018); Blanco et al. (2019). With a
narrowing parameter space, present-day modelers are opting to explore the WIMP
paradigm with either simplified models or from the perspective of effective
field theory (EFT) in which the modelers remain agnostic to a particular UV
completion of the theory De Simone and Jacques (2016).

In an EFT analysis of DM, one couples DM directly to standard model (SM)
particles at low enegies via, often, dimensionful effective couplings that
depend on a high-energy scale $\Lambda$. As long as interaction energies are
well below $\Lambda$, the effective interactions faithfully capture the
relevant physics. For neutral dark matter, the leading order electromagnetic
interactions in an EFT occur through their static electromagnetic properties.
Electric and magnetic dipole moments proceed through mass dimension-5
operators, and several DM modelers have explored the possibility that DM
predominantly interacts through such moments Raby and West (1987, 1988);
Pospelov and ter Veldhuis (2000); Sigurdson et al. (2004, 2006); Massó et al.
(2009); Heo (2010); Fitzpatrick and Zurek (2010); Barger et al. (2011, 2012);
Fortin and Tait (2012); Weiner and Yavin (2013); Del Nobile et al. (2014);
Kopp et al. (2014); Gresham and Zurek (2014); Matsumoto et al. (2014); Antipin
et al. (2015). If the DM candidate is a Majorana fermion, then both the
electric and magnetic dipole moments must vanish identically, and its sole
static electromagnetic property is its anapole moment Kayser (1984); Nieves
(1982). Anapole interactions of DM have been studied in Refs. Pospelov and ter
Veldhuis (2000); Fitzpatrick and Zurek (2010); Ho and Scherrer (2013); Del
Nobile et al. (2014); Kopp et al. (2014); Gresham and Zurek (2014); Gao et al.
(2014); Matsumoto et al. (2014); Sandick et al. (2016); Gelmini (2017); Alves
et al. (2018); Blanco et al. (2019). Because the anapole interaction arises
from a dimension-6 operator, the anapole moment is suppressed by $\Lambda^{2}$
which further suppresses these leading order interactions for Majorana fermion
DM. Additionally, one can choose model parameters so that annihilation
precedes primarily through p-wave modes Ho and Scherrer (2013), and because
its annihilation is velocity suppressed, it can more easily evade indirect
detection constraints Blanco et al. (2019).

Moving beyond the static electromagnetic properties of fermions, higher order
electromagnetic interactions with Majorana fermions are mediated by operators
that are dimension-7 and beyond Pospelov and ter Veldhuis (2000); Weiner and
Yavin (2013); De Simone and Jacques (2016). In particular, a fermion can
interact with two real photons via its polarizabilities. There are six two-
photon interactions, two spin independent and four spin dependent, that are
separately invariant under charge conjugation ($\mathcal{C}$), parity
($\mathcal{P}$), and time reversal ($\mathcal{T}$) Ragusa (1993), and there an
additional ten more polarizability terms that are either $\mathcal{C}$-odd and
$\mathcal{T}$-odd or $\mathcal{C}$-odd and $\mathcal{P}$-odd Gorchtein (2008).
The self-conjugate nature of the Majorana fermion does limit its interaction
with two real photons somewhat, requiring the four $\mathcal{C}$-odd and
$\mathcal{P}$-odd polarizabilites to vanish, but that still leaves a dozen
modes unconstrained Latimer (2016).

In this paper, we will focus upon Majorana fermion DM with a non-zero anapole
moment and non-zero polarizabilities, but we will restrict our considerations
to the six polarizabilities that separately preserve $\mathcal{C}$,
$\mathcal{P}$, and $\mathcal{T}$ symmetries. Of these six, the two spin-
independent polarizabilities arise from a dimension-7 interaction,
$\mathcal{L}_{\text{SI pol}}\sim\frac{1}{\Lambda^{3}}\chi^{\dagger}\chi
F^{\mu\nu}F_{\mu\nu}$, that gives rise to the low-energy interaction
Hamiltonian, expressed in terms of the electric and magnetic fields,
$H_{\text{SI pol}}\sim\alpha_{E}E^{2}+\beta_{M}B^{2}$. The spin-independent
electric and magnetic polarizabilities, $\alpha_{E}$ and $\beta_{M}$, are
order $\mathcal{O}\left(\frac{1}{\Lambda^{3}}\right)$. The four spin-dependent
polarizabilities arise from interactions that depend upon derivatives of the
electric and magnetic fields. As a consequence, the four spin-dependent
polarizabilities, $\gamma_{j}$, are nominally
$\mathcal{O}\left(\frac{1}{\Lambda^{4}}\right)$, though their precise mass
dependence depends on the particular UV completion of the theory Latimer
(2016). It is these spin-dependent polarizabilities that allow for s-wave
annihilation of two Majorana fermions into two photons. Nominally, this s-wave
mode is suppressed relative to p-wave annihilation, but depending upon the
particular UV completion, s-wave annihilation can be comparable to the p-wave
mode Latimer (2017).

The coupling between an anapole moment and a real photon vanishes, so the
leading order electromagnetic interaction for a Majorana fermion is via the
exchange of a virtual photon with a charged particle. Thus, at low energies,
Majorana fermions do not couple to electric or magnetic fields; they only
couple to electric currents. If a spin-$\frac{1}{2}$ Majorana fermion is
immersed in a persistent electric current, there is a difference in the two
spin states of the fermion, with the lower energy state corresponding to the
one in which the anapole moment is aligned with the current. In the presence
of a background current, a collection of Majorana fermions can be undergo a
some level of polarization, at least in principle.

This polarization can only be achieved if there are mechanisms that allow the
Majorana fermion to change spin states. For particles that are not in thermal
equilibrium, such as DM after freeze out, only irreversible processes can
allow the higher-energy anti-aligned anapole moments to flip spin to the lower
energy aligned state. Spontaneous two-photon emission or single-photon
emission via virtual Compton scattering are two such irreversible mechanisms;
however, because the photon coupling occurs through the polarizabilities, the
rates of these irreversible processes are extremely small Walter et al.
(2022). However, before freeze out when the DM is in equilibrium with the
thermal bath, reversible spin-flip processes can lead to a partially polarized
DM medium in the presence of a background current, assuming the spin-flip
interactions happen at a sufficient rate. The Boltzmann distribution would
guarantee a slight excess of lower-energy states with spins aligned with the
current.

Our focus in this paper will be on DM in the early Universe, specifically when
the DM is in thermal equilibrium with the Universe from a temperature around
the DM mass, $T\sim m_{\chi}$, until freeze out. As the Universe cools and
expands, whenever its temperature is around the DM mass, most SM particles no
longer have sufficient energy to produce DM via annihilation. As a
consequence, the comoving DM density decreases as it continues to annihilate
into SM particles. This decrease in comoving DM density continues until DM
drops out of thermal equilibrium with the rest of the Universe at freeze out.
At this point, because DM annihilations become so rare, the comoving DM
density becomes constant, yielding the relic density present today. This relic
density is largely determined by the DM annihilation cross section; a larger
cross section results in a smaller relic density, and vice versa.

In this paper, we explore the consequence that a persistent local background
current can have on DM in this time before freeze out. If the DM polarization
induced by the current is sufficiently large, then the s-wave mode of DM
annihilation can be suppressed because the s-wave mode requires the
annihilating particles to have opposite spin states. Overall, the DM
annihilation rate will be somewhat smaller in the presence of the current than
otherwise, which would result in a local relic density that is higher than
regions without a background current.

##  II EFT interactions

Anapole interactions arise from an effective Lagrangrian term
$\mathcal{L}_{\text{ana}}=\frac{1}{2}\frac{g}{\Lambda^{2}}\bar{\chi}^{\dagger}\gamma^{\mu}\gamma^{5}\partial^{\nu}F_{\mu\nu}$
Zeldovich (1957); Kayser (1984); Nieves (1982). The anapole moment is the
dimension-2 coefficient $f_{a}=\frac{g}{\Lambda^{2}}$ that results from a UV
complete theory in which the neutral fermion effectively couples to the photon
field through, at least, a one-loop process. In the UV-complete Lagrangian,
there must be a parity-violating Zeldovich (1957) trilinear term that couples
the Majorana fermion to a charged fermion and (vector or scalar) boson. In
terms of the mass scale $\Lambda$ for the anapole moment, it is set by the
dominant mass of the charged particle to which the Majorana fermion couples.
Examples of how this coefficient depends on the underlying UV physics can be
found in Refs. Giunti and Studenikin (2015); Cabral-Rosetti et al. (2016);
Latimer (2016).

At tree-level, Majorana fermions scatter charged particles via the exchange of
a virtual photon. At low energies, the resulting interaction Hamiltonian is
$H_{\text{ana}}=-f_{a}\bm{\sigma}\cdot\mathbf{J}$, where $\bm{\sigma}$ are the
Pauli spin matrices and $\mathbf{J}$ is the current density associated with
the charged particle Zeldovich (1957); Kayser (1984); Nieves (1982). Given
this, a background current establishes an energy difference
$\mathcal{E}=2f_{a}J$ between the Majorana fermion states aligned and
antialigned with the current.

Similarly, at tree level, Majorana fermion annihilation into a charged SM
particle-antiparticle pair proceeds through the coupling of the anapole moment
to a virtual photon. This p-wave cross section has been computed previously in
Refs. Ho and Scherrer (2013); Gao et al. (2014), and we quote the results
here. The thermally averaged cross section is

$\langle\sigma_{p}|v|\rangle=16\alpha
N_{\text{eff}}f_{a}^{2}m_{\chi}^{2}\left(\frac{T}{m_{\chi}}\right).$ (1)

The factor $N_{\text{eff}}$ accounts for all the kinematically available final
states, weighted by the square of the particles’ charges. For $m_{\chi}<80$
GeV, all final states are fermionic. Annihilation into an electron-positron
pair contributes $1$ to $N_{\text{eff}}$; annihilation into a quark-antiquark
pair, whose charges are $\pm qe$, contributes $3q^{2}$ to $N_{\text{eff}}$,
where the factor of 3 accounts for color degrees of freedom. For $m_{\chi}>80$
GeV, we must include the possibility that the Majorana fermions can annihilate
into $W$ bosons. We can accommodate this in $N_{\text{eff}}$ with the term
$\frac{3}{4}m_{\chi}^{2}/m_{W}^{2}$ if we employ the approximation, as in Ref.
Gao et al. (2014), that $m_{W}\gg m_{\chi}$.

Moving beyond the anapole moment, we consider two photon interactions with a
Majorana fermion. Spin-independent interactions arise from the dimension-7
effective Lagrangian discussed above. We are interested in the spin-dependent
two-photon interactions because s-wave annihilation proceeds through these
channels. The spin-dependent terms that couple the Majorana fermion to two
photons involve derivatives of the electromagnetic field, and they have been
characterized in Ref. Prange (1958) and elsewhere. The four coefficients of
these dimension-8 terms, the spin-dependent polarizabilities $\gamma_{j}$,
carry mass dimension $[M]^{-4}$. Ostensibly, these polarizabilities would seem
to scale as $\Lambda^{-4}$, but in considering an explicit simplified UV
complete theory, the reality is somewhat more nuanced. As with the anapole
moment, the effective two-photon coupling to a Majorana fermion arises from a
direct coupling of this fermion to charged particles. Let’s suppose these
charged particles have mass $\Lambda$ and $m$ with $\Lambda>m$. From a
simplified UV-completion Latimer (2017), one finds that the polarizabilities
can scale as $\sim\frac{1}{\Lambda^{4}}$ and $\sim\frac{1}{\Lambda^{2}m^{2}}$.

These two mass scales in the spin-depenent polarizability coeffecients also
make an appearance in the s-wave annihilation cross section. Dimensional
analysis suggests
$\langle\sigma_{s}|v|\rangle\sim\gamma^{2}m_{\chi}^{6}\sim\tilde{g}^{2}\frac{m_{\chi}^{6}}{\Lambda^{8}}$
where $\gamma$ is some linear combination of the polarizabilities and
$\tilde{g}$ is a dimensionless coefficient. From a UV complete theory, we note
that there are scenarios in which the s-wave annihilation cross section is not
as suppressed as one might naively assume:
$\langle\sigma_{s}|v|\rangle\sim\tilde{g}^{2}\frac{m_{\chi}^{6}}{\Lambda^{4}m^{4}}$
Latimer (2017). We will adopt the EFT s-wave annihilation rate to be

$\langle\sigma_{s}|v|\rangle=\tilde{g}^{2}\frac{m_{\chi}^{2}}{\Lambda^{4}\mu^{4}},$
(2)

where $\mu:=\frac{m}{m_{\chi}}$ with $1<\mu<\frac{\Lambda}{m_{\chi}}$.

We would like to compare the relative s- and p-wave annihilation rates in
order to determine their impact upon the relic density for a thermal WIMP. In
the anapole interaction, the dimensionless coupling, $g$, must be small enough
in order for perturbative calculations to be viable; we will set $g=1$. For
the polarizabilities, the corresponding dimensionless coupling, $\tilde{g}$,
is relatable to $g$ in a UV-complete theory. The polarizabilities arise from,
at least, a four-vertex Feynman diagram while the anapole moment comes from a
three-vertex diagram. Given this, $\tilde{g}$ should involve an extra factor
of $e$ relative to $g$ which would suppress $\tilde{g}$ by a factor
$\mathcal{O}(10^{-1})$ relative to $g$. But, at the same time, $\tilde{g}$ may
incorporate corrections that are logarithmic in the relevant mass scale that
could be $\mathcal{O}(10)$ Latimer (2016). Given this, we will also take
$\tilde{g}=1$, admitting that a particular UV-complete theory might deviate
from this value by a factor of 10.

With these couplings fixed, we find that s-wave annihilation into photons is
larger than or comparable to p-wave annihilation into charged particles
whenever

$\mu^{4}\lesssim\frac{1}{16}\frac{1}{\alpha
N_{\text{eff}}}\frac{m_{\chi}}{T}.$ (3)

To see what size $\mu$ makes the two annihilation channels comparable, we
consider DM masses, $m_{\chi}$, between 5 GeV and 80 GeV because, in this mass
range, $N_{\text{eff}}$ is fixed at 6.67. As a figure of merit, thermal WIMPs
typically fall out of thermal equilibrium in the early Universe for a
temperature around $T_{f}\sim\frac{m_{\chi}}{20}$. Given this, we see that
s-wave annihilation can exceed the p-wave process for $m\leq 2.3m_{\chi}$. The
upshot is that the s-wave annihilation mode is subdominant unless $m$ is
$\mathcal{O}(m_{\chi})$.

##  III  Rate of reversible spin-flip processes

Before freeze out, DM is in thermal equilibrium with the rest of the universe,
and if the DM medium were within a background external current, we would
expect a slight polarization of the medium by virtue of the Boltzmann
distribution. But, if a DM medium is initially unpolarized and then subjected
to a current, there must be a sufficient rate of spin-flip interactions to
assist in establishing polarization. In particular, the spin-flip rate must be
much larger than the Universe’s expansion rate, $\Gamma_{\text{spin flip}}\gg
H$. In the radiation-dominated era, we have

$H=1.66g_{\star}^{\frac{1}{2}}\frac{T^{2}}{M_{\text{Pl}}},$ (4)

where $g_{*}$ represents the relativistic degrees of freedom at temperature
$T$ and $M_{\text{Pl}}$ is the Planck mass Kolb and Turner (1990).

The anapole moment is spin dependent, so interaction, via a single virtual
photon, with a charged SM particle in the relativistic plasma can change a
Majorana fermion’s spin orientation. We first focus upon the interaction
between the Majorana fermion $\chi$ and one species of relativistic fermion
$\psi$ with charge $qe$. We assume the background current is
$\mathbf{J}=J\hat{\mathbf{z}}$, and we average over the initial spin of the
charged fermion and sum over its final spin states. Below, we compute the
amplitude for the process:
$\chi(p,\downarrow)+\psi(k)\to\chi(p^{\prime},\uparrow)+\psi(k^{\prime})$. To
do so, we make several simplifying assumptions. In particular, we neglect the
mass of the charged fermion because it is relativistic. Also, the Majorana
fermion is non-relativistic which implies
$|\mathbf{p}|,|\mathbf{p}^{\prime}|\ll m_{\chi}$. Implementing these
approximations, the leading order contribution to the squared amplitude for
this process is

$|\mathcal{M}_{\psi}|^{2}\approx
4q^{2}e^{2}f_{a}^{2}m_{\chi}^{2}(E_{\psi}E_{\psi}^{\prime}-k_{z}k_{z}^{\prime}).$
(5)

Integrating over the phase space for the final states, the total DM spin-flip
cross section is

$\sigma_{\psi\text{ flip}}=q^{2}\alpha f_{a}^{2}E_{\psi}^{2},$ (6)

where $\alpha$ is the fine structure constant.

The rate at which the charged SM fermion can flip the spin of Majorana fermion
depends on the total interaction cross section as well as the flux of incident
charged particles, $\Gamma_{\psi\text{ flip}}=n_{\psi}|v|\sigma_{\psi\text{
flip}}$. At temperature $T$, the fermion number density in the radiation-
dominated early Universe is given by

$\displaystyle n_{\text{fermion}}=$
$\displaystyle\frac{3}{4}\frac{\zeta(3)}{\pi^{2}}g_{\text{dof}}T^{3},$ (7)

where $\zeta$ is the Riemann zeta function and $g_{\text{dof}}$ represents the
degrees of freedom for the particular species Kolb and Turner (1990). In the
plasma, these fermions are incident upon the Majorana fermion from all
directions with a thermal distribution of momenta, so we average over all
possible $\mathbf{k}$ in the distribution. The thermally averaged cross
section becomes

$\langle\sigma_{\psi\text{
flip}}\,|v|\rangle=\frac{15\zeta(5)}{\zeta(3)}q^{2}\alpha f_{a}^{2}T^{2}.$ (8)

With this averaged cross section, we now have the thermally averaged spin-flip
rate, $\langle\Gamma_{\psi\text{ flip}}\rangle$, due to interaction with a
single species of fermion in the early Universe. At a given temperature, there
could be a host of different relativistic fermion species. We incorporate this
through an incoherent sum of spin-flip rates induced by each individual
relativistic species appropriately weighted by its charge, $q$.

For temperatures above 80 GeV, the $W$ boson is relativistic, so it can
appreciably contribute to the spin-flip rate of the Majorana fermion. As with
the charged fermion, the interaction between the $W$ boson and Majorana
fermion involves the exchange of a virtual photon,
$\chi(p,\downarrow)+W(k)\to\chi(p^{\prime},\uparrow)+W(k^{\prime})$. When
computing the cross section for this proces, we make several simplifying
assumptions. As above, we keep only leading order terms for the fermion spin
line, taking $|\mathbf{p}|,|\mathbf{p}^{\prime}|\ll m_{\chi}$, and we assume
the $W$ boson to be highly relativistic with
$|\mathbf{k}|,|\mathbf{k}^{\prime}|\gg m_{W}$. For these relativistic bosons,
the longitudinal polarization state will dominate with
$\varepsilon_{L}^{\mu}\to\frac{k^{\mu}}{m_{W}}$. Implementing these
assumptions, the leading contribution to the squared amplitude is

$|\mathcal{M}_{W}|^{2}\approx
4e^{2}f_{a}^{2}\frac{m_{\chi}^{2}}{m_{W}^{4}}(k\cdot
k^{\prime})^{2}|\mathbf{S}\cdot(\mathbf{k}+\mathbf{k}^{\prime})|^{2},$ (9)

where we define the three-vector $\mathbf{S}:=(1,-i,0)$ Integrating over the
phase space of the final states, we find the total cross section to be

$\sigma_{W\text{ flip}}\,|v|=\frac{1}{6}\alpha
f_{a}^{2}\frac{E_{W}^{4}}{m_{W}^{2}}(5-\cos 2\theta),$ (10)

where $\theta$ is the polar angle of the initial boson’s momentum,
$\mathbf{k}$. When relativistic, the boson number density in the radiation-
dominated early Universe is given by Kolb and Turner (1990)

$\displaystyle n_{\text{boson}}=$
$\displaystyle\frac{\zeta(3)}{\pi^{2}}g_{\text{dof}}T^{3}.$ (11)

Averaging over the thermal distribution of relativistic $W$ bosons we find

$\langle\sigma_{W\text{ flip}}\,|v|\rangle=300\alpha
f_{a}^{2}\frac{\zeta(7)}{\zeta(3)}\frac{T^{4}}{m_{W}^{2}}.$ (12)

For temperatures beyond $m_{W}$, we add to the thermally averaged fermion
spin-flip rate the contributions from the interactions with the $W$ boson,
$\langle\Gamma_{W\text{ flip}}\rangle=n_{\text{boson}}\langle\sigma_{W\text{
flip}}\,|v|\rangle$.

Figure 1: (a) In the top panel, the curve plots the anapole moment at which
the spin-flip rate of a non-relativistic Majorana fermion equals that of the
Hubble parameter for a given temperature in the radiation-dominated era of the
early Universe. (b) Setting $f_{a}=\frac{g}{\Lambda^{2}}$ with $g=1$, the
curve in the lower panel shows the energy scale $\Lambda$ at which the spin-
flip rate of a non-relativistic Majorana fermion equals that of the Hubble
parameter for a given temperature in the radiation-dominated era of the early
Universe.

We now consider the rate at which all relativistic particles can flip the spin
of a non-relativistic Majorana fermion in the early Universe. In comparing the
spin-flip rate to the Universe’s expansion rate, we find
$\frac{\langle\Gamma_{\text{ flip}}\rangle}{H}\approx\alpha
f_{a}^{2}M_{\text{Pl}}T^{3}$ whenever only charged fermions are relativistic,
$T<m_{W}$. Beyond that temperature, we have $\frac{\langle\Gamma_{\text{
flip}}\rangle}{H}\approx\alpha f_{a}^{2}\frac{M_{\text{Pl}}}{m_{W}^{2}}T^{5}$,
accurate to within a factor of a few. Using the full expression for the spin-
flip rate, we plot in Fig. 1(a) the anapole moment at which the spin-flip rate
equals the Hubble parameter as a function of temperature. For anapole moments
above this curve, spin-flip interactions are sufficiently frequent to allow a
collection of Majorana fermions to partially polarize in a background current
via thermalization.

Treating the anapole moment as an effective interaction,
$f_{a}=\frac{g}{\Lambda^{2}}$, we can constrain the energy scale for UV
completion. Setting $g=1$, we plot in Fig. 1(b) the energy $\Lambda$ for which
the Majorana fermion’s spin-flip rate in the early Universe is equal to the
Hubble expansion rate. From the figure, we may determine, for a given
temperature, the upper limit on $\Lambda$ for which spin-flip interactions are
sufficient to achieve Majorana fermion polarization. This energy scale can
also be used to inform our knowledge of additional, higher-order, effective
electromagnetic interactions with the Majorana fermion. In particular, it sets
the scale for the DM’s polarizabilities.

Considering a fixed $\Lambda$, then Fig. 1(b) shows the lowest temperature at
which the spin-flip and expansion rates are equal because
$\frac{\langle\Gamma_{\text{ flip}}\rangle}{H}\sim T^{3}$ (or
$\frac{\langle\Gamma_{\text{ flip}}\rangle}{H}\sim T^{5}$ for higher
temperatures, $T>m_{W}$). In what follows, we would like this spin-flip rate
to be sufficiently large for temperatures through freeze out, so that
polarization can be achieved up until the point at which dark matter decouples
from the background thermal bath. If we are to interpret the temperature in
Fig. 1(b) as the freeze-out temperature for a given dark matter candidate,
then the constraints on $\Lambda$ will be more stringent. We estimate these
more stringent constraints below.

The freeze-out temperature, $T_{f}$, for a DM candidate is determined
primarily by the thermally averaged annihilation cross section
$\langle\sigma_{\text{ann}}|v|\rangle$. The cross section can be expanded in a
power series for velocity because DM is non-relativistic when it decouples
from the thermal background. Here we assume one velocity mode dominates the
annihilation cross section and parametrize the cross section in terms of the
background temperature by virtue of $v\sim T^{\frac{1}{2}}$, viz.,
$\langle\sigma_{\text{ann}}|v|\rangle=\sigma_{0}x^{-n}$ where
$x=\frac{m_{\chi}}{T}$. If s-wave annihilation dominates, then $n=0$; for
p-wave, $n=1$; and so on.

To precisely determine freeze out and the relic DM density, one must solve the
Boltzmann equation, as discussed below in Sec. IV. However, estimates,
accurate to a few percent, do exist. In particular, the freeze-out temperature
and relic number density $Y=n/s$ (relative to the entropy density $s$) are
given by

$\displaystyle x_{f}$
$\displaystyle\approx\log[(n+1)a\lambda]-\left(n+\tfrac{1}{2}\right)\log[\log[(n+1)a\lambda]]$
(13) $\displaystyle Y_{\infty}$
$\displaystyle\approx\frac{(n+1)}{\lambda}x_{f}^{n+1}$ (14)

where $a=0.289\,g_{*}^{-1}$ and
$\lambda=0.264\,g_{*}^{1/2}M_{\text{Pl}}m_{\chi}\sigma_{0}$ Scherrer and
Turner (1986); Kolb and Turner (1990).

For the models under consideration herein, we assume the p-wave contribution
to the cross section to dominate, so we set $n=1$ and use the cross section in
Eq. (1). Given a particular DM mass $m_{\chi}$, we can determine what value of
energy scale $\Lambda$ yields a given freeze-out temperature $T_{f}>m_{\chi}$.
Figure 2 contains these results for a range of DM masses from 1 GeV to 1 TeV.
We superpose on this plot the curve from Fig. 1(b) which shows the upper limit
for $\Lambda$ at which the spin-flip rate is sufficient to achieve
thermalization. Not surprisingly, for the models under consideration, the
spin-flip rate is sufficiently large through the freeze-out temperature.
Additionally, we consider the values of $\Lambda$ and $m_{\chi}$ that
reproduce the relic DM mass density present today
$\rho_{\text{DM}}=\Omega_{\text{DM}}\rho_{\text{crit}}$, where
$\Omega_{\text{DM}}$ is the DM fraction of the energy budget and
$\rho_{\text{crit}}$ is the critical energy density R. L. Workman, et al.
(2022). If we assume our DM candidate is to reproduce the relic DM density,
then the energy scale $\Lambda$ is sufficiently small so that the DM candidate
interacts through freeze out to thermalize in a background current.

Figure 2: For a given DM mass $m_{\chi}$, the black curves show the value of
$\Lambda$ that yields a given freeze-out temperature, $T_{f}$. For the dotted
curve, the mass is 1 GeV; for the dashed curve, the mass is 10 GeV; for the
dot-dashed curve, the mass is 100 GeV; for the solid (black) curve, the mass
is 1 TeV. The value of $\Lambda$ that reproduces the relic DM density for a
given mass is denoted by $\bigstar$. The solid gray curve represents the spin-
flip constraints on $\Lambda$ for a given temperature, reproduced from Fig.
1(b)

##  IV DM density in a current

We must use the Boltzmann equation to precisely determine the evolution of the
DM density from the time it becomes non-relativistic through freeze out. As
above, we express as $Y=n/s$ the DM number density relative to the entropy
density, $s$. Given this, the first moment of the Boltzmann equation can be
written as

$\frac{\mathrm{d}}{\mathrm{d}x}Y=-\left(0.602g_{*}^{-\frac{1}{2}}\frac{M_{\text{Pl}}}{m_{\chi}^{2}}\right)\langle\sigma_{\text{ann}}|v|\rangle
sx(Y^{2}-Y_{\text{eq}}^{2}).$ (15)

The term $Y_{\text{eq}}(x)$ tracks the equilibrium number density which we
take as $Y_{\text{eq}}=a\,x^{\frac{3}{2}}e^{-x}$ in the non-relativistic
regime ($x\gg 3$), where again $a=0.289g_{*}^{-1}$. In what follows, we would
like to determine the impact of a locally polarized region of DM on the local
relic DM density, but the the Boltzmann equation is derived under the
assumptions of homogeneity and isometry. The presence of a local current
clearly violates that, but because the inhomogeneities we introduce are so
small, we will treat the current as a local perturbative term in the Boltzmann
equation.

Though the p-wave annihilation cross section typically dominates, Eq. (1), we
must also consider the s-wave mode, Eq. (2). For what follows, it will be
useful to factor the cross section as

$\displaystyle\langle\sigma_{\text{ann}}|v|\rangle=$
$\displaystyle\sigma_{0}x^{-1}(1+bx),$ (16)

where $\sigma_{0}=\langle\sigma_{\text{p}}|v|\rangle x$ and
$b=\langle\sigma_{\text{s}}|v|\rangle/\sigma_{0}$. For the DM interactions
considered herein, we find from Eq. (1) that $\sigma_{0}=16\alpha
m_{\chi}^{2}N_{\text{eff}}/\Lambda^{4}$. Then, from Eq. (2), we compute
$b=\tilde{g}^{2}/(16\alpha N_{\text{eff}}\mu^{4})$. If the s-wave process is
to be a subdominant correction through freeze out, then we must require
$bx_{f}\ll 1$. This constrains the parameter $\mu$: $\mu\gg[x_{f}/(16\alpha
N_{\text{eff}})]^{\frac{1}{4}}\sim 2.6$. Substituting the factored
annihilation cross section in Eq. (15), we have

$\frac{\mathrm{d}}{\mathrm{d}x}Y=-\lambda(1+bx)x^{-3}(Y^{2}-Y_{\text{eq}}^{2}),$
(17)

where again
$\lambda=\left(0.264\,g_{*}^{\frac{1}{2}}M_{\text{Pl}}m_{\chi}\right)\sigma_{0}$.

Figure 3:  Bounds on $\mu$ derived from observational limits on the s-wave DM
annihilation mode. The solid (black) curve uses data from Ref. Ackermann et
al. (2015); the dashed (red) curve uses data from Ref. D’Amico et al. (2018);
and the dotted (blue) curve uses data from Ref. Slatyer (2016).

Aside from our desire to keep the s-wave annihilation mode sub-dominant, there
are observational constraints on the annihilation of DM into two photons that
come from a variety of sources. In search of mono-energetic gamma rays from
annihilating DM in the galactic halo, Fermi-LAT places the most stringent
constraints on the annihilation into two photons; we use the most stringent
constraints from the R3 region of interest in Ref. Ackermann et al. (2015).
Additionally, precise measurements of the cosmic microwave background (CMB)
anisotropies from the Planck satellite Ade et al. (2016) severely constrain
the energy injection from DM annihilation at the time of recombination. An
analysis of the CMB has resulted in stringent constraints on the s-wave DM
annihilation cross section considered herein Slatyer (2016). Finally, an
absorption feature has been observed in the 21-cm spectrum at high redshift;
this constrains DM s-wave annihilation because energy injection from
annihilation would wash out the feature D’Amico et al. (2018). We use these
constraints on the s-wave cross section to place lower bounds on the parameter
$\mu$ in Eq. (2). To do so, we assume that, for a given DM mass $m_{\chi}$,
the energy-scale $\Lambda$ is set by reproducing the DM relic density through
p-wave annihilation only. Then the limits from Fermi-LAT, the CMB, and 21-cm
data bound $\mu$ as show in Fig. 3

In the presence of a local current density, we need to modify the s-wave
contribution to the cross section because it requires a spin-zero initial
state and a current can partially polarize the DM medium before freeze out. In
particular, suppose a current $J$ exists in a region. In thermal equilibrium,
the number density of DM particles with spins aligned with the current,
$n_{\uparrow}$, will exceed those anti-aligned, $n_{\downarrow}$, by an amount
$n_{\uparrow}/n_{\downarrow}=\exp[\mathcal{E}/T]\approx 1+\mathcal{E}/T$ where
$\mathcal{E}=2f_{a}J$ is the energy difference between the aligned and anti-
aligned states. The fractional relative difference in the two spin states is
$\epsilon:=(n_{\uparrow}-n_{\downarrow})/n\approx\mathcal{E}/(2T)$. Then, in
the Boltzmann equation, Eq. (17), the s-wave annihilation in the presence of a
current is suppressed by a factor of $(1-\epsilon)$

$\frac{\mathrm{d}}{\mathrm{d}x}Y=-\lambda[1+b(1-\epsilon)x]x^{-3}(Y^{2}-Y_{\text{eq}}^{2}),$
(18)

where $\epsilon=f_{a}J/T$.

Before exploring the impact of a current in detail, it is worth considering
the most extreme possibility in which s-wave annihilation is prohibited by
virtue of complete polarization of the DM medium; that is, we compare the
$\epsilon=1$ scenario (full polarization) with the $\epsilon=0$ scenario (no
polarization). We will do this, first, using the estimates of $x_{f}$ and
$Y_{\infty}$ from Eqs. (13) and (14) extended to include both s- and p-wave
annihilation Kolb and Turner (1990)

$\displaystyle x_{f}$
$\displaystyle\approx\log[2a\lambda]-\tfrac{3}{2}\log[\log[2a\lambda]]+\log[1+b\log[2a\lambda]]$
(19) $\displaystyle Y_{\infty}$
$\displaystyle\approx\frac{2}{\lambda}x_{f}^{2}\frac{1}{(1+2bx_{f})}$ (20)

If the p-wave process dominates freeze out, then $x_{f}\sim\log[2a\lambda]$,
and if we continue with our previous approximation $bx_{f}\ll 1$, then the
s-wave channel modifies the freeze-out temperature by
$x_{f}\stackrel{{\scriptstyle\sim}}{{\mapsto}}x_{f}+b\log[2a\lambda]$. Upon
including the s-wave annihilation mode, the relic DM density should decrease
by $Y_{\infty}\stackrel{{\scriptstyle\sim}}{{\mapsto}}Y_{\infty}(1-2bx_{f})$.
It is most useful to cast these changes in terms of the fractional change in
the relic density; full polarization of the DM medium (which turns off the
s-wave mode) relative to the no-current scenario results in a fractional
change of

$\frac{Y_{\infty}^{\epsilon=1}-Y_{\infty}^{\epsilon=0}}{Y_{\infty}}\approx
2bx_{f}.$ (21)

Using some exemplar parameters, we would like to numerically integrate the
Boltzmann equation, Eq. (18), to confirm the accuracy of the estimates in Eqs.
(19 \- 21). Because Eq. (18) is an extremely stiff differential equation, it
is easier to set $W=\log Y$ and instead integrate the equation Steigman et al.
(2012)

$\frac{\mathrm{d}}{\mathrm{d}x}W=\lambda[1+b(1-\epsilon)x]x^{-3}\left(e^{(2W_{\text{eq}}-W)}-e^{W}\right).$
(22)

For parameters, we set $m_{\chi}=100$ GeV. Annihilation dominated by the
p-wave process reproduces the observed DM relic density for $\Lambda=579$ GeV
with a freeze-out temperature around $x_{f}\sim 23.1$. If we use the most
stringent constraints on $\mu$ derived from the Fermi LAT data Ackermann et
al. (2015), then $\mu=11.2$. From the estimate in Eq. (21), we expect a
fractional increase in the relic DM density of 0.4% for a fully polarized DM
medium. Numerical calculations produce the same order of magnitude change.
Computing $Y(x=1000)$ for both scenarios, we find that the p-wave only (full
polarization) relic density is a factor of 0.1% larger than when both p- and
s-wave annihilations (no current scenario) are considered. (We note that the
estimate of $Y_{\infty}$ from Eq. (20) relative to the numerical computation
of $Y(x=1000)$ differs by 3.5%; however, the relative fractional change in the
estimates using Eqs. (14, 20) for the p-wave only and p- and s-wave
computations for $Y_{\infty}$ yield the correct order of magnitude result.)

We now discuss the impact of a local current upon the local relic DM density.
In order to do so, we must supply some details about the form of the current.
We treat the current classically, assuming that it can be represented as the
net drift of the relativistic charged species in the plasma,
$J=en_{q}v_{\text{drift}}$. The factor $n_{q}$ is the sum over the number
density of all charged relativistic species, Eqs. (7) and (11), in the plasma
at temperature $T$ weighted by their absolute charge. We suppose that the
current exists from $x=1$ ($T=m_{\chi}$) through freeze out, $x\sim 20-25$,
and we assume that the drift velocity suffers a redshift due to expansion so
that $v_{\text{drift}}(x)=\frac{1}{x}v_{\text{drift}}(1)$. With these
assumptions, the current scales like $J\sim x^{-4}$, and overall, the
perturbing term in Eq. (18) scales with $x$ as $\epsilon=f_{a}J/T\sim x^{-3}$.

Before we integrate Eq. (18) with the classical current, we will develop some
approximations that allow us to estimate the relative change in the local
relic density. Following the arguments in Ref. Kolb and Turner (1990) that
yield Eqs. (19) and (20), we can achieve the order of magnitude estimates

$\displaystyle x_{f}^{\epsilon}\approx$ $\displaystyle
x_{f}^{\epsilon=0}-b\epsilon\log[2a\lambda]$ (23) $\displaystyle
Y^{\epsilon}(x_{f})\approx$ $\displaystyle
Y^{\epsilon=0}(x_{f})\left[1+\frac{1}{2}b\epsilon x_{f}\right]$ (24)

where $\epsilon$ is evaluated at $x_{f}^{\epsilon=0}$. The fractional change
in the local relic density in the presence of the assumed classical current is

$\frac{Y^{\epsilon}(x_{f})-Y^{\epsilon=0}(x_{f})}{Y(x_{f})}\approx\frac{1}{2}b\epsilon
x_{f}.$ (25)

We would like to compare this crude estimate with a more robust solution of
Eq. (18). The current introduces a small perturbation in the DM density
$\delta Y=Y^{\epsilon}-Y^{\epsilon=0}$, and it is sufficient to linearize Eq.
(18) with respect $\delta Y$, neglecting small quantities

$\frac{\mathrm{d}}{\mathrm{d}x}\delta Y=-2\lambda[1+bx]x^{-3}Y\delta
Y+\epsilon\lambda bx^{-2}(Y^{2}-Y_{\text{eq}}^{2}),$ (26)

where $Y=Y^{\epsilon=0}$ is the no-current DM density. This first order non-
homogenous ordinary differential equation can be solved with an integrating
factor. Defining the functions

$\displaystyle p(x)=$ $\displaystyle 2\lambda[1+bx]x^{-3}Y$ (27)
$\displaystyle q(x)=$ $\displaystyle\epsilon\lambda
bx^{-2}(Y^{2}-Y_{\text{eq}}^{2}),$ (28)

then the solution to Eq. (26) is

$\delta Y(x)=\int_{1}^{x}P(x)^{-1}P(s)q(s)\mathrm{d}s$ (29)

where

$P(x)=\exp\left[\int^{x}_{1}p(s)\mathrm{d}s\right].$ (30)

Examining Eq. (29), we see that the approximation for $\delta Y$ is explicitly
linear in $v_{0}$ and manifestly positive because $q(x)$ is positive.

To perform the integral in Eq. (29), we first consider the factor

$P(x)^{-1}P(s)=\exp\left[-\int_{s}^{x}p(t)\mathrm{d}t\right].$ (31)

For the parameters under consideration $p(x)$ is extremely large, ranging from
$\sim 10^{18}$ around $x=1$ to $\sim 10^{8}$ around $x=x_{f}$ for the DM mass
of 100 GeV (and associated parameters considered above). Because $p(x)$ is so
large, the factor $P(x)^{-1}P(s)$ vanishes, for all practical purposes, except
at $s=x$, where $P(x)^{-1}P(s=x)=1$. Because of this feature, only the value
$q(x)$ is of any consequence in the integrand

$\delta Y(x)\approx q(x)\int_{1}^{x}P(x)^{-1}P(s)\mathrm{d}s.$ (32)

The remaining integral in Eq. (32) can be accurately estimated by a Taylor
expansion of the argument of the exponential in Eq. (31) about $s=x$

$P(x)^{-1}P(s)\approx\exp\left[(s-x)p(x)\right].$ (33)

We can then estimate the integral

$\int_{1}^{x}P(x)^{-1}P(s)\mathrm{d}s\approx\frac{1}{p(x)}$ (34)

for $x\gg 1$. This yields an estimate for $\delta Y$ of

$\delta Y(x)=\frac{q(x)}{p(x)}$ (35)

for $x\gg 1$. Using the definitions of $p$ and $q$ in Eqs. (27, 28), we find
at freeze out

$\delta Y(x_{f})=\frac{\epsilon
bx_{f}[Y^{2}(x_{f})-Y^{2}_{\text{eq}}(x_{f})]}{2[1+bx_{f}]Y(x_{f})},$ (36)

where $\epsilon$ is evaluated at $x_{f}$. For small $b$, we can approximate
this as

$\frac{\delta Y(x_{f})}{Y(x_{f})}=\frac{1}{2}\epsilon
bx_{f}\left[1-\frac{Y^{2}_{\text{eq}}(x_{f})}{Y^{2}(x_{f})}\right],$ (37)

In Eq. (37), if we further neglect $Y_{\text{eq}}^{2}$ relative to $Y^{2}$ at
freeze out, then we recover our crude estimate from Eq. (25): $\dfrac{\delta
Y(x_{f})}{Y(x_{f})}\approx\frac{1}{2}\epsilon bx_{f}$. Executing the
calculation for $\delta Y/Y$ in Eq. (37) for DM mass $m_{\chi}=100$ GeV (and
associated parameters considered above), we find that the result is about
$0.58$ times the crude estimate, $\frac{1}{2}\epsilon bx_{f}$.

Figure 4:  The local relative change in DM density at freeze out due to the
presence of a classical current with $v_{\text{drift}}(1)=1$. For a given
$m_{\chi}$, the parameters are chosen to reproduce the observed relic DM
density and satisfy constraints on s-wave annihilation derived from Fermi-LAT
data (the solid (black) curve) Ackermann et al. (2015), 21-cm spectral data
(the dashed (red) curve) D’Amico et al. (2018), and CMB data (the dotted
(blue) curve) Slatyer (2016).

With Eq. (37), we are now able to determine how a local electric current can
impact the local DM density through freeze out. As noted previously, we model
the current classically as a steady current from $x=1$ through freeze out,
modulo a decreasing plasma density and red shifted drift velocity, whose
initial value is $v_{\text{drift}}(1)$. For our calculations, we take
$v_{\text{drift}}(1)=1$. Because $\delta Y/Y$ is manifestly linear in
$v_{\text{drift}}(1)$, one can easily scale our results to accommodate more
realistic values for the drift velocity. With this assumption, we present our
results in Fig. 4. For a given DM mass, we determine the mass scale $\Lambda$
by fixing the relic DM density to the observed value, assuming p-wave
annihilations determine this. Then, we set the mass ratio $\mu$ to satisfy the
constraints on s-wave annihilation derived from Fermi-LAT data Ackermann et
al. (2015), CMB data from the Planck satellite Ade et al. (2016); Slatyer
(2016), and observations of the 21-cm line D’Amico et al. (2018).

##  V Discussion and conclusion

Herein, we considered the impact that a local current might have on the
evolution of the DM density around DM freeze out. The anapole moment of a
Majorana fermion DM candidate tends to align with external currents provided
there are sufficient spin-flip interactions to thermalize the DM. We find for
the model parameters under consideration DM can thermalize, so that the DM
states, described by a Boltzmann distribution, can lead to a partially
polarized DM medium. This partial polarization is of consequence for the
available DM annihilation channels.

Rather generically, a Majorana fermion DM candidate can interact with real
photons through higher order processes. In particular, DM can annihilate into
two real photons in an s-wave process by virtue of its spin-dependent
polarizabilities. In a partially polarized DM medium, this s-wave annihilation
channel is somewhat suppressed because it requires the initial DM states to
have opposite spins. As a consequence, the overall DM annihilation rate is
smaller than it would be if no current were present, and as the Universe cools
and expands, the lower annihilation rate will result in a slightly higher
relic DM density than in a no-current region.

Referring to Fig. 4, we find that, for an initial charge carrier drift speed
approaching the speed of light, the relative over density of DM in the current
bearing region ranges from $\sim 10^{-13}$ to $\sim 10^{-6}$ depending on the
DM mass and s-wave constraint used. Generally, the constraints on $\mu$
decrease with DM mass, and because $\delta Y\sim\mu^{-4}$, the size of $\delta
Y$ increases substantially with $m_{\chi}$. A density variation of order
$10^{-6}$ is extremely large at this time in the early Universe, but this is
based upon an unrealistic drift speed. Our results scale linearly with initial
drift speed, so it is trivial to determine the impact upon more realistic
current magnitudes.

In terms of the assumptions in our calculations, we assume a current that
persists from temperature $T=m_{\chi}$ to freeze out
$T\sim\frac{1}{20}m_{\chi}$. The density change $\delta Y$, however, is most
impacted by the existence of the current around freeze out, so the assumed
longevity of the current in our calculations could be relaxed without
significantly impacting the results. Additionally, we modeled our current as a
simple classical current density; more complex models could also fit into our
existing work. Finally, throughout we assumed that the p-wave annihilation
mode determines the relic density in our calculations. This is an excellent
approximation when using the constraints on $\mu$ derived from the
observations in Ref. Ackermann et al. (2015), but for DM masses greater than a
few hundred GeV, the approximation is strained if using the constraints on
$\mu$ derived from Refs. Slatyer (2016); D’Amico et al. (2018).

The premise of our entire work requires the presence of substantial electric
currents in the early Universe around DM freeze out, but the actual existence
of such currents is beyond the scope of this work. We do find, in the
literature, arguments supporting the existence of electric currents in the
early Universe specifically during the inflationary period Turner and Widrow
(1988), around the electroweak phase transition Sigl et al. (1997); Baym et
al. (1996), and around the QCD phase transition Quashnock et al. (1989); Cheng
and Olinto (1994); Sigl et al. (1997); de Souza and Opher (2008). These
currents should be short-lived, compared to the Hubble time Siegel and Fry
(2006). For our purposes, long-lived currents are not crucial to the results,
rather the strength of the current and the time at which it occurs are the
more important factors.

##  VI ACKNOWLEDGMENTS

DCL thanks the Kavli Institute for Theoretical Physics for its hospitality
during the completion of this work. This research was supported in part by the
National Science Foundation under Grant No. NSF PHY-1748958.

## References

  * R. L. Workman, et al. (2022) R. L. Workman, et al. (Particle Data Group), PTEP 2022, 083C01 (2022).
  * Bertone and Hooper (2018) G. Bertone and D. Hooper, Rev. Mod. Phys. 90, 045002 (2018).
  * Jungman et al. (1996) G. Jungman, M. Kamionkowski, and K. Griest, Phys. Rep. 267, 195 (1996).
  * Feng and Kumar (2008) J. L. Feng and J. Kumar, Phys. Rev. Lett. 101, 231301 (2008).
  * Bernabei et al. (2008) R. Bernabei et al. (DAMA), Eur. Phys. J. C 56, 333 (2008).
  * Bernabei et al. (2010) R. Bernabei et al. (DAMA, LIBRA), Eur. Phys. J. C 67, 39 (2010).
  * Aalseth et al. (2011) C. E. Aalseth, P. S. Barbeau, N. S. Bowden, B. Cabrera-Palmer, J. Colaresi, J. I. Collar, S. Dazeley, P. de Lurgio, J. E. Fast, N. Fields, et al. (CoGeNT Collaboration), Phys. Rev. Lett. 106, 131301 (2011).
  * Aalseth et al. (2013) C. E. Aalseth, P. S. Barbeau, J. Colaresi, J. I. Collar, J. Diaz Leon, J. E. Fast, N. E. Fields, T. W. Hossbach, A. Knecht, M. S. Kos, et al. (CoGeNT Collaboration), Phys. Rev. D 88, 012002 (2013).
  * Agnese et al. (2013) R. Agnese, Z. Ahmed, A. J. Anderson, S. Arrenberg, D. Balakishiyeva, R. Basu Thakur, D. A. Bauer, J. Billard, A. Borgland, D. Brandt, et al. (CDMS Collaboration), Phys. Rev. Lett. 111, 251301 (2013).
  * Akerib et al. (2017) D. S. Akerib, S. Alsum, H. M. Araújo, X. Bai, A. J. Bailey, J. Balajthy, P. Beltrame, E. P. Bernard, A. Bernstein, T. P. Biesiadzinski, et al. (LUX Collaboration), Phys. Rev. Lett. 118, 251302 (2017).
  * Aprile et al. (2018) E. Aprile, J. Aalbers, F. Agostini, M. Alfonsi, L. Althueser, F. D. Amaro, M. Anthony, F. Arneodo, L. Baudis, B. Bauermeister, et al. (XENON Collaboration 7), Phys. Rev. Lett. 121, 111302 (2018).
  * Meng et al. (2021) Y. Meng, Z. Wang, Y. Tao, A. Abdukerim, Z. Bo, W. Chen, X. Chen, Y. Chen, C. Cheng, Y. Cheng, et al. (PandaX-4T Collaboration), Phys. Rev. Lett. 127, 261802 (2021).
  * Peskin (2015) M. E. Peskin, Proc. Natl. Acad. Sci. U.S.A. 112, 12256 (2015).
  * Albert et al. (2017) A. Albert, B. Anderson, K. Bechtol, A. Drlica-Wagner, M. Meyer, M. Sánchez-Conde, L. Strigari, M. Wood, T. M. C. Abbott, F. B. Abdalla, et al., Astrophys. J. 834, 110 (2017).
  * Hooper and Goodenough (2011) D. Hooper and L. Goodenough, Phys. Lett. B 697, 412 (2011).
  * Aguilar et al. (2016) M. Aguilar, L. Ali Cavasonza, B. Alpat, G. Ambrosi, L. Arruda, N. Attig, S. Aupetit, P. Azzarello, A. Bachlechner, F. Barao, et al. (AMS Collaboration), Phys. Rev. Lett. 117, 091103 (2016).
  * Cuoco et al. (2017) A. Cuoco, M. Krämer, and M. Korsmeier, Phys. Rev. Lett. 118, 191102 (2017).
  * Marsh (2016) D. J. Marsh, Phys. Rep. 643, 1 (2016).
  * Roszkowski et al. (2018) L. Roszkowski, E. M. Sessolo, and S. Trojanowski, Rept. Prog. Phys. 81, 066201 (2018).
  * Arcadi et al. (2018) G. Arcadi, M. Dutra, P. Ghosh, M. Lindner, Y. Mambrini, M. Pierre, S. Profumo, and F. S. Queiroz, Eur. Phys. J. C 78, 203 (2018).
  * Blanco et al. (2019) C. Blanco, M. Escudero, D. Hooper, and S. J. Witte, Journal of Cosmology and Astroparticle Physics 2019, 024 (2019).
  * De Simone and Jacques (2016) A. De Simone and T. Jacques, Eur. Phys. J. C76, 367 (2016).
  * Raby and West (1987) S. Raby and G. B. West, Phys. Lett. B 194, 557 (1987).
  * Raby and West (1988) S. Raby and G. West, Phys. Lett. B 200, 547 (1988).
  * Pospelov and ter Veldhuis (2000) M. Pospelov and T. ter Veldhuis, Phys. Lett. B480, 181 (2000).
  * Sigurdson et al. (2004) K. Sigurdson, M. Doran, A. Kurylov, R. R. Caldwell, and M. Kamionkowski, Phys. Rev. D 70, 083501 (2004).
  * Sigurdson et al. (2006) K. Sigurdson, M. Doran, A. Kurylov, R. R. Caldwell, and M. Kamionkowski, Phys. Rev. D 73, 089903 (2006).
  * Massó et al. (2009) E. Massó, S. Mohanty, and S. Rao, Phys. Rev. D 80, 036009 (2009).
  * Heo (2010) J. H. Heo, Phys. Lett. B 693, 255 (2010).
  * Fitzpatrick and Zurek (2010) A. L. Fitzpatrick and K. M. Zurek, Phys. Rev. D 82, 075004 (2010).
  * Barger et al. (2011) V. Barger, W.-Y. Keung, and D. Marfatia, Phys. Lett. B 696, 74 (2011).
  * Barger et al. (2012) V. Barger, W.-Y. Keung, D. Marfatia, and P.-Y. Tseng, Phys. Lett. B 717, 219 (2012).
  * Fortin and Tait (2012) J.-F. m. c. Fortin and T. M. P. Tait, Phys. Rev. D 85, 063506 (2012).
  * Weiner and Yavin (2013) N. Weiner and I. Yavin, Phys. Rev. D 87, 023523 (2013).
  * Del Nobile et al. (2014) E. Del Nobile, G. B. Gelmini, P. Gondolo, and J.-H. Huh, JCAP 1406, 002 (2014).
  * Kopp et al. (2014) J. Kopp, L. Michaels, and J. Smirnov, JCAP 04, 022 (2014).
  * Gresham and Zurek (2014) M. I. Gresham and K. M. Zurek, Phys. Rev. D 89, 016017 (2014).
  * Matsumoto et al. (2014) S. Matsumoto, S. Mukhopadhyay, and Y.-L. S. Tsai, JHEP 10, 155 (2014).
  * Antipin et al. (2015) O. Antipin, M. Redi, A. Strumia, and E. Vigiani, JHEP 07, 039 (2015).
  * Kayser (1984) B. Kayser, Phys. Rev. D30, 1023 (1984).
  * Nieves (1982) J. F. Nieves, Phys. Rev. D26, 3152 (1982).
  * Ho and Scherrer (2013) C. M. Ho and R. J. Scherrer, Phys. Lett. B 722, 341 (2013).
  * Gao et al. (2014) Y. Gao, C. M. Ho, and R. J. Scherrer, Phys. Rev. D 89, 045006 (2014).
  * Sandick et al. (2016) P. Sandick, K. Sinha, and F. Teng, JHEP 10, 018 (2016).
  * Gelmini (2017) G. B. Gelmini, Rep. Prog. Phys. 80, 082201 (2017).
  * Alves et al. (2018) A. Alves, A. C. O. Santos, and K. Sinha, Phys. Rev. D 97, 055023 (2018).
  * Ragusa (1993) S. Ragusa, Phys. Rev. D47, 3757 (1993).
  * Gorchtein (2008) M. Gorchtein, Phys. Rev. C77, 065501 (2008).
  * Latimer (2016) D. C. Latimer, Phys. Rev. D94, 093010 (2016).
  * Latimer (2017) D. C. Latimer, Phys. Rev. D 95, 095023 (2017).
  * Walter et al. (2022) K. Walter, K. Hall, and D. C. Latimer, Phys. Rev. D 106, 096021 (2022).
  * Zeldovich (1957) Ya. B. Zeldovich, Sov. Phys. JETP 6, 1184 (1957), [Zh. Eksp. Teor. Fiz. 33,1531 (1957)].
  * Giunti and Studenikin (2015) C. Giunti and A. Studenikin, Rev. Mod. Phys. 87, 531 (2015).
  * Cabral-Rosetti et al. (2016) L. G. Cabral-Rosetti, M. Mondragón, and E. Reyes-Pérez, Nucl. Phys. B907, 1 (2016).
  * Prange (1958) R. E. Prange, Phys. Rev. 110, 240 (1958).
  * Kolb and Turner (1990) E. Kolb and M. Turner, _The Early Universe_ (Addison-Wesley, Redwood City, CA, 1990).
  * Scherrer and Turner (1986) R. J. Scherrer and M. S. Turner, Phys. Rev. D33, 1585 (1986), [Erratum: Phys. Rev. D34, 3263 (1986)].
  * Ackermann et al. (2015) M. Ackermann, M. Ajello, A. Albert, B. Anderson, W. B. Atwood, L. Baldini, G. Barbiellini, D. Bastieri, R. Bellazzini, E. Bissaldi, et al., Phys. Rev. D 91, 122002 (2015).
  * D’Amico et al. (2018) G. D’Amico, P. Panci, and A. Strumia, Phys. Rev. Lett. 121, 011103 (2018).
  * Slatyer (2016) T. R. Slatyer, Phys. Rev. D 93, 023527 (2016).
  * Ade et al. (2016) P. A. R. Ade et al. (Planck), Astron. Astrophys. 594, A13 (2016).
  * Steigman et al. (2012) G. Steigman, B. Dasgupta, and J. F. Beacom, Phys. Rev. D 86, 023506 (2012).
  * Turner and Widrow (1988) M. S. Turner and L. M. Widrow, Phys. Rev. D 37, 2743 (1988).
  * Sigl et al. (1997) G. Sigl, A. V. Olinto, and K. Jedamzik, Phys. Rev. D 55, 4582 (1997).
  * Baym et al. (1996) G. Baym, D. Bödeker, and L. McLerran, Phys. Rev. D 53, 662 (1996).
  * Quashnock et al. (1989) J. M. Quashnock, A. Loeb, and D. N. Spergel, Astrophys. J. Lett. 344, L49 (1989).
  * Cheng and Olinto (1994) B. Cheng and A. V. Olinto, Phys. Rev. D 50, 2421 (1994).
  * de Souza and Opher (2008) R. S. de Souza and R. Opher, Phys. Rev. D 77, 043529 (2008).
  * Siegel and Fry (2006) E. R. Siegel and J. N. Fry (2006), URL https://arxiv.org/abs/astro-ph/0609031.